New Horizons in Black Hole Physics and Holography - Institute of ...

5 The planar Reissner-Nordström-AdS black hole 13 

5 The planar Reissner-Nordström-AdS The Maxwell potential black holeof the solution 13is 

New Horizons in Black Hole 

⇥ 

Physics 

The Maxwell potential of the solution is 

r 

⇥ 

A = µ 1 dt . (5.5) 

r 

r 

A = µ 1 

and 

dt . 

r + 

Holography 

+ 

(5.5) 

We have required the Maxwell potential to vanish on the horizon, A t (r + )= 

We have required the Maxwell potential 0. The tosimplest vanish onargument the horizon, for Athis t (r + condition )= is that otherwise the holonomy 

0. The simplest argument for this condition of the potential is that around otherwise the the Euclidean holonomy time circle would remain nonzero when 

of the potential around the Euclidean the time circle circle collapsed would remain at thenonzero horizon, when indicating a singular gauge connection. 

the circle collapsed at the horizon, The indicating planaraReissner-Nordström-AdS singular gauge connection. solution is characterized by two scales, 

The planar Reissner-Nordström-AdS the solution chemical is characterized potential µ =lim by two r scales, 0 A t and the horizon radius r + . From the 

the chemical potential µ =lim r 0 Adual t and field thetheory horizonperspective, radius r + . From it is the more physical to think in terms of the 

dual field theory perspective, it is temperature more physical than to think the horizon in terms radius of the 

temperature than the horizon radius 

Black Branes and Field 

Black 

Theory 

Branes and Field Theory 

at 

T 

Finite 

= 1 r 

3 

+µ 

4⇥r + 

Density 

2 2 ⇥ 

2 2 . 

at 

T 

Finite 

= 1 r 

3 

+µ 

(5.6) 

Density 

2 2 ⇥ 

4⇥r + 2 2 . (5.6) 

The black hole is illustrated in figure 4 below. This black hole, which can 


Saturday, November 10, 12 

Sunday, September 2, 2012 

Figure 4 The planar Reissner-Nordström-AdS Figure 4black Thehole. planar The Reissner-Nordström-AdS charge density 

is sourced entirely by flux emanating sityfrom is sourced the black entirely hole horizon. by flux emanating from the black hole 

black hole. The charge den- 

horizon. 

additionally carry a magnetic charge, additionally was the starting carry apoint magnetic for holographic charge, was the starting point for holographic 

Shamit Kachru 

approaches to finite density condensed 

Shamit Kachru 

Shamit Kachru 

approaches 

Stanford & 

matter 

SLAC 

[27, to finite 28]. density condensed 

Stanford & 

matter 

SLAC 

[27, 28]. 

Because the underlying UV theory Because is scaleStanford invariant, the underlying the and only UVdimension- 

less quantity that we can discuss isless thequantity ratio T/µ. that In we order canto discuss answerisour 

the ratio T/µ. In order to answer our 

SLAC 

theory is scale invariant, the only dimension- 

Based on: 

Based on: 

basic question aboutarXiv:1201.1905, the IR physics basic 1201.4861, at low question temperature, 1202.6635 aboutarXiv:1201.1905, the we must IR physics take the 1201.4861, at low temperature, 1202.6635 we must take the 

limit T/µ ⇥ 1 of the solution. Welimit thereby T/µ obtain ⇥ 1 of the the extremal solution. Reissner- We thereby obtain the extremal ReissnerarXiv 

: 1201.1905, 1201.4861, 1202.6635 & to appear 

Nordström-AdS black hole with Nordström-AdS black hole with 

f(r) =1 4 

r 

r + 

⇥ 3 

+3 

⇥ 

r 4 ⇥ 

r 3 

. f(r) =1 4 (5.7) +3 

r + r + 

r 

r + 

⇥ 4 

. (5.7)

I. Introduction 

An ab initio interesting question in theoretical physics is 

the classification of states of matter. Interesting new states 

are thought to arise in finite density QCD; in materials of 

modern interest in condensed matter physics; and 

probably other places as well. 

Saturday, November 10, 12

At first, the richness of phases of quantum matter is an 

An a priori different question is the classification of black 

embarassment for holography. 

hole solutions in general relativity (or its extensions with 

simple matter sectors). In asymptotically flat 4d Minkowski 

space, This impressive is because, results famously, were “black obtained holes by have the early no hair”: 1970s. 

Tuesday, October 2, 12 


Gauge/gravity duality invites us to connect the two 

questions. 

Thus, our line of attack will be as follows. In AdS/CFT, finite 

Finite temperature field theory is represented by the AdS/ 

temperature field theory is represented by the AdS 

Schwarzschild black brane: 

Thus, our line of attack will be as follows. In AdS/CFT, finite 


temperature field theory is represented by the AdS 


ds 2 = L2 

z 2 ( 

ds 2 = L2 

z 2 ( 

fdt2 + dx 2 + dz2 

f ) 

fdt2 z d + dx 2 + dz2 

f ) 

f 7=1 

T H E SC H WA RZSC HIL D SO LU T IO N A N D BL A C K H O L ES 

f 7=1 

T H E SC H WA RZSC HIL D SO LU T IO N A N D BL A C K H O L ES 

of the external 

zH 

d (asymptotically flat) universe, at least as seen from 

“Doping” by adding charge density, one is led to study 

see this 

of the 

(infinitely 

external (asymptotically 

long) history in 

flat) 

a 

universe, 

finite amount 

at leastof as 

their 

seen from 

prope 

th 

that gets see this to (infinitely them as long) they history approach in arfinite − is amount infinitely of their blueshifted. proper tim 

to believe (although I know of no proof) that any non-sphericall The 

relativity that there are that comes to believe into(although a Reissner-Nordstrøm I know of 

of 

no proof) 

such 

black that hole any 

branes: 

will non-spherically violently dist sy 

the Reissner-Nordstrøm go through the steps metric, explicitly, andbut is given merelyby 

quote 

described. 

the final that answer. comes It’sThe into hard 

solution a Reissner-Nordstrøm to say known 

whatas 

the black actual hole geometry will violently willdisturb 

holes that there are extremal avatars of such branes: 

look 

the Reissner-Nordstrøm metric, and is given by described. It’s hard to say what the actual geometry will look like 

good reason to believe that it must contain an infinite number of a 

ds 2 = −∆dt 2 + ∆ −1 dr 2 + r 2 dΩ 2 good , reason to believe that (7.110) it must contain an infinite number of asym 

ds 2 = −∆dt 2 + ∆ −1 dr connecting 2 + r 2 dΩ 2 , to each other via (7.110) various wormholes. 

connecting to each other via various wormholes. 

where 

Case Three — GM 2 = p 2 + q 2 

where 

∆ =1− 2GM 

Case Three — GM 2 = p 2 + q 2 

+ G(p2 + q 2 ) This 

. 

case is known as the ∆ =1− 2GM + G(p2 + q 2 ) This 

r r r 2 . 

case is known as the 

(7.111) 

extreme Reissner-Nordstrøm solution 

black r 2 black hole”). hole”). The The mass mass is is exactly balanced in some sense by by the the charg 

In this expression, In thisMexpression, once again M is once interpreted again interpreted as the mass the 

exact of mass exact theof solutions hole; the solutions hole; q qthe consisting the total total 

of of electric 

several extremal black holes holes which which rema r 

charge, and pcharge, is the total 

and p 

magnetic 

is the total 

charge. 

magneticIsolated charge. Isolated 

magnetic 

magnetic 

spect 

charges 

charges 

to each 

(monopoles) 

(monopoles) 

other for all 

have 

have 

time. 

never 

never 


spect to each other for all time. 


been observed in nature, but that doesn’t stop us from writing down the metric that they On 

On the one hand the 

the 

extremal 

extremal 

hole 

hole 

is a 

been observed in nature, but that doesn’t stop us from writing toy; down these the solutions metric arethat oftenthey 

examined in studies of the information 

Adding charge density, one is led instead to study a charged 

7 THE SCHWARZSCHILD SOLUTION AND BLACK HOLES 202 

instead black a charged 

Adding charge brane. It is black 

density, familiar brane. 

one is from Reissner-Nordstrom It 

led instead 

familiar 

to study 

from 

a charged 

flat black space 

7 THE SCHWARZSCHILD SOLUTION AND BLACKthat HOLES gets to them as they approach 202 r − infinitely blueshifted. 

go through the 

holes 

steps black explicitly, brane. 

that 

but 

there 

merelyIt quote is familiar 

are 

the 

extremal 

final answer. from The Reissner-Nordstrom 

avatars 

solution of 

known 

such 

as black 

branes: 


z d H 

z d

Figure 6 The zero temperature holographic superconductor. The electric 

flux is sourced entirely by the scalar field condensate. 

The no-hair 

finds 

theorems 

that theory (6.1) admits 

fail in 

Lifshitz 

AdS 

solutions 

space. 

with the dynamical 

Charged black 

critical exponent z given by solutions to 

brane 

8(V T 

horizons 

3) + 4(V 2 

can take a variety of forms: 

T 4V T + 12)z +(V T 2 +8V T 24)z 2 + VT 2 z 3 =0. (6.6) 

12 Horizons, holography and condensed matter 

Here we introduced 

given electric flux at the boundary, leads to gravitational physics that is 

V T = ⇥ 2 L 2 V (⇤ ⇥ )+m 2 ⇤ 2 ⇥ 

⇥ , VT = ⇥2 L 

V (⇤ ⇥ )+2m 2 ⇥ 

interesting in its own right. 

⇤ ⇥ . (6.7) 

e 

Figure 7 The electron star. The electric flux is sourced entirely by a fluid 

Thus of the fermions. dynamical The fluid critical is present exponent at all radii is determined for which the bylocal the chemical value of the 

potential andis its greater firstthan derivative the fermion at the mass. fixed point value of ⇤ ⇥ , which is in 

turn determined by the equations of motion. In order for the scaling (6.5) to 

have charge a is straightforward carried by the interpretation fermions ratherasthan a renormalisation lying behind a horizon. 

!" transformation, 

one 

Secondly, 

should have 

consider 

z>0. 

heating 5))5))5 

The null 

up the 

energy 

system. 

condition 

At leading #$%&'() 

in the 

order 

bulk 

in 

furthermore 

semiclassical 

01(#.&-# *(+,-./ 

implies z 

limit, 

> 1 

this 

[46]. 

means 

Even 

67)2&89)'&%:-./) placing 

if (6.6) 

a finite 

gives 

temperature 2134 

physical solutions 

black hole 

for z, 

horizon 

it is 


not 

*/+%9-#, 

guaranteed 

the interior of 

that 

the 

the 

spacetime. 

corresponding 

The fluid 

Lifshitz 

remains 

solution 

at zero 

is 

temperature, 

realised as the 

as the 

near 

fluid must be in thermal equilibrium with the Hawking radiation of 

horizon geometry. An instructive simple case to consider is m 2 the black 

> 0 and 

hole, the e ects of which are negligible in the semiclassical limit. At finite 

V = 0. One obtains in this case [46, 45] 

temperature, a fraction of the charge is carried by the black hole horizon, 

2 

which Figure subsequently 3 The basic pushes question the 

z = 

2 L 2 m 2 , in fermion finite ⇤2 density fluid 

⇥ = 1 a finite holography: distance 6z useaway the 

e 2 L 2 (1 + z)(2 + z) . gravitational 

Theequations star becomes to motion a band to of findfluid the with interior an IR inner geometry and angiven outerthe 

radius 

from (6.8) the 

horizon. 

boundary condition that there is an electric flux at infinity. 

[66, 67]. At nonzero temperature we can dial the ratio T/µ. We can expect 

The Lifshitz solutions are seen to exist so long as the scalar is not too heavy, 

that 

L 2 m 2 at su⌅ciently 

< 

2 . As L 2 high 

m 2 values of this ratio, the star will collapse to form a 

Tuesday, ⌅ 0, we see that z ⌅ 1 and an emergent relativistic 

black 

October 

hole. This 

2, 12 

will be analogous 

AdS 4 is obtained. As L 2 m 2 to 

⌅ 

2 the maximal mass of spherical neutron 

from below, z ⌅⇧and the extremal 

stars; in 

AdS 2 ⇥R 2 5global Therather planarthan Reissner-Nordström-AdS planar AdS, the mass scale black canhole 

be compared 

geometry is recovered. However, recall from (6.2) that AdS 2 ⇥R 2 is 

to the radius of the spatial boundary 

stable 

The minimal 

against 

framework 

⇤ condensing 

capable 

if 

2 of 

m 

describing 2 sphere. 

L 2 ⇤ 3 In that case there is a first 

2 

. 

the 

Extremal 

physics 

Reissner-Nordström 

of electric flux in 

is 

an 

likely 

asymptotically 

the ground 

AdS 

state 

geometry 

in this case. 

is Einstein-Maxwell 

It follows that 

theory 

the Lifshitz 

with a 

geometries 

negative 

cosmological constant [26]. The Lagrangian density can be written 

(6.8) realized as IR scaling regimes in this theory with a positive quadratic 

L = 1 

2⇥ 2 R + 6 ⇥ 

1 

L 2 4e 2 F µ⇥F µ⇥ . (5.1) 

Here ⇥ and e are respectively the Newtonian and Maxwell constants while 

L sets the cosmological constant lengthscale. 

There is a unique regular solution to the theory (5.1) with electric flux 

at infinity and that has rotations and spacetime translations as symmetries. 

represe 

The problem of classifying low-energy phases of doped 

matter can be mapped to the problem of classifying 


extremal black brane geometries. 

conduc 

Gubser; 

liquid 

dual ... to

While attempts at classification (of 2d cfts, string vacua, 

etc.) have a checkered history, we will put aside suspicion 

and pursue this line of thought: 

II. The basic horizon 

A. Near-horizon limit 

B. Interesting physics (non-Fermi liquids) 

III. More general homogeneous, isotropic geometries 

A. Lifshitz geometries (and linear resistivity) 

B. Hyperscaling violation (& entanglement) 

C. Instabilities 

IV. Homogeneous, anisotropic geometries 

A. Bianchi horizons 

B. More general 4-algebras 


The Fermi liquid fixed point is infr 

the Cooper channel instability). Th 

metals. 

At various times, we will take detours to explore the 

physical properties of the horizons we’ve discovered. 

We will often comment on their possible But a variety relationship of experiments to yield 

le of a system which enjoys such “shifted d” decidedly in “non-Fermi liquid” beha 

eal world, theories is a Fermi of liquid. Fermi Basic surfaces picture: and 

where 

non-Fermi 

Fermi liquid 

liquids. 

theory would 

rmi liquids A. J. Schofield 2 

simply using Fermi’s golden 

ll’s equations) and I have inho 

would like to see where 

coming from. 

w is as follows. I begin with 

id theory itself. This thevery 

good description of a 

s of Fermi particles (i.e. that 

ciple) where the interactions 

nimportant. The reason is 

eally describing are not the 

tron-like quasiparticles that 

gas of electrons. Despite its 

Figure 1: The ground state of the free Fermi gas in momentum 

space. All the states below the Fermi surface 

ate the subject of non-Fermi 

successful theory at describsome, 

like UPt 3 , where the 

are filled with both a spin-up and a spin-down electron. 

A particle-hole excitation is made by promoting 

iginal electrons are very imen 

to fail in other materials 

an electron from a state below the Fermi surface to an 

empty one above it. 

ptionsion to a general rule one but surface in k-space divides occupied Sunday, September 2, 2012 

esting materials 

It 

known. 

is 

As 

important to remember that this is a problem of 

ilure in the metallic state of 

2. Fermi-Liquid Theory: the electron quasiparticle 

rconductors. 

mples 

ccupied 

which, from a theo-stateste non-Fermi liquid metals. 

Only the orthogonal direction 

The need for a Fermi-liquid theory dates from the 

roperties 

classical 

first applications 

which can not be 

& 


quantum 

quantum mechanics to metallic 

gravity in its own right, however, and 

state. There were two key problems. Classically each 

t kly interacting a given electron-like point electron should contributethe 3k B /2 to theFermi specific heat surface. It is like a 

capacity of a metal—far more than is actually seen experimentally. 

In addition, as soon as it was realized 

tum critical point. When a 

should be understood on its own terms. 

ens rfaces at temperatures close worth to of 1+1 dimensional CFTs. 

that the electron had a magnetic moment, there was 

the puzzle of the magnetic susceptibility which did not 

show the expected Curie temperature dependence for 

iparticles scatter so strongly 

ave in the way that Fermiedict. 

ion–the Luttinger liquid. In 

s, electrons are unstable and 

ate particles (spinons and 


free moments: χ ∼ 1/T . 

These puzzles were unraveled at a stroke when 

Pauli (Pauli 1927, Sommerfeld 1928) (apparently 

reluctantly—see Hermann et al. 1979) adopted Fermi 

statistics for the electron and in particular enforced the 

??

mR = ∆ − 3 2 

II. The basic horizon 

In AdS/CFT, one represents a (global) U(1) symmetry of the 

field theory by a bulk Abelian gauge field. 

(1) 

where R is the AdS curvature radius. The spectral function 

of O at finite charge density can then be extracted 

by solving the Dirac equation for ψ in the charged AdS 

black hole geometry. Which pairs of (q, ∆) arise depends 

on the specific dual CFT. However, since we are working 

with a universal sector common to many gravity theories, 

we will take the liberty of considering an arbitrary pair 

of (q, ∆), scanning many possible CFTs. 

A. Black hole geometry 

The action for a vector field A M coupled to AdS 4 gravity 

can be written as 

S = 1 ∫ 

2κ 2 d 4 x √ −g 

[R − 6 ] 

R 2 − R2 

F MN F MN (2) 

where gF 2 is an effective dimensionless gauge coupling4 . 

The equations of motion following from (2) are solved by 

the geometry of a charged black hole [18, 19] 5 

ds 2 = r2 

R 2 (−fdt2 + dx 2 i )+ R2 dr 

planar analogue of the Reissner-Nordstrom 2 

r 2 (3) black hole. 

f 

g 2 F 

The zero-temperat 

√ 

3. At zero tempe 

becomes AdS 2 × R 

given by 

So the simplest theory where one can ask about field 

theory at finite density is: 

B 

To compute the 

the quadratic actio 

∫ 

S spinor = d d 

where 

¯ψ = ψ † Γ t , 

D 

and ω abM is the s 

action (10) depend 

This theory has simple, exact solutions which represent the 


4 It is defined so that for a typical supergravity Lagrangian it is a 

constant of order O(1) 

5 For a generic embedding of (2) into 4d N =2supergravity,this 

6 We will use M and 

space indices respe 

the boundary direc 

matrices refer to ta

gof yof dimension gravity can be written as 

ermi 

the 

r in the low-energy of corre- 

S mapped = to 

IR CFT. When operator is where (1) 

R CFT. When the operator is where g 2 gF 

2 is an effective dimensionless gauge coupling7 

IR ts give CFT), a nice understanding quasi-particle of the is unstable. low-energy andF 

is an effective dimensionless gauge coupling7 

S = R 1 ∫ 

is the curvature radius of AdS. The equations of 

arly avior theproportional around quasi-particle Fermi to its issurface. unstable. energyThe andscaling 

the and Rmotion is the 2κfollowing 2 d d+1 x √ [ 

] 

rface. toa x −g field A M in AdS. − −g + 

d(dR − 2 

R 2 g 

1) 

2 2 

F MN F 

tate. In for a field A MN MN M F to AdS 

he n The the scaling 

d+1 

of nderstanding quasi-particle of the is sur- 

a charged is where ang black 2 F is an gauge gauge coupling7 coupling7 

the radius of AdS. The equations of 

2κ 2 d d+1 ad U(1) 

x √ x gauge −g field R + A M in AdS. − surface. The scaling 

−g R + be −g as R + R 2 

R 2κ 

2 R 

2 g 

curvature from radius (1) areof solved RAdS. 2 − R2 

byThe g 2 F MN 

F 

F 2 2 

F MN F 

In contrast, there is no The action for a vector field A MN MN M coupled F to AdS d+1 

when dimension the dimension of the corre- is com- 

gravity can be written as 

F 

MN (1) (1) 

nsion Fthe equations geometry of of 

ortional idue re FT. of en the approaching is 

controlled vanishes to 

by 

its approaching the 

energy 

dimension 

and theof fermi the 

the suroperator 

corre- 

The black hole solution takes the form: (1) 

When of the corren 

the operator is 

operator is 

where motion a charged following black from hole (1) [5, 9] are solved by the geometry 

(1) 


shes 

the irrelevant, 

IR CFT. When is 

the the fermi quasi-particle 

the operator is 

2 is an gauge coupling7 

he hole [5, 9] 

article 

isscaling nal theFermi IR 

irrelevant, 

toCFT), toward its the quasi-particle 

the thequasi-particle 

Fermi 

The and surface is unstable. 

with 

is the 2κ and 

following 2 d d+1 x [ 

] (1) 

d(d − 1) 

where g 2 gF 

2 is an effective dimensionless −g R + R is the curvature 

from radius (1) ds 

oward 

particle 

the 

residue. 

Fermi 

When 

surface 

the operator 

with 

is 

≡ g MN dx M dx N = r2 areof solved R AdS. 2 − R2 

coupling7 

of the is unstable. low-energy andF 

is effective dimensionless 

The 

byThe gequations 2 Fgauge MN F MN coupling7 

S = R 1 ∫ 

is the curvature radius of AdS. The equations of 

ds 

esidue. 2 ≡ MN dx M dx N = r2 R 2 (−fdt2 + d⃗x 2 )+ R2 dr 2 of 

the equations geometry of of 

by Fermi 

linearly proportional to its energy and the motion (1) are solved by geometry 

R 2 (−fdt2 + d⃗x 2 )+ R2 r 2 of 

dr f 2 (2) 

its le approaching the 

to 

residue energy 

dimension 

its issurface. unstable. energyThe vanishes theofand scaling 

approaching fermi surelevant, 

corre- 


F 

Rmotion is the 2κ following 2 d d+1 x √ [ 

] 

The AdS/RN solution −g has R + a 

d(d 

metric: 

− 1) 

curvature from radius (1) takes areof solved Rthe AdS. 2 − R2 

form: byThe g 2 F MN F MN 

Fthe equations geometry 

the fermi surthe 

operator functionis irrelevant, then has the the quasi-particle 

form for a 

charged a charged black from black hole (1) [5, 9] are 9] solved by the geometry 

(1) 

of of 


IR spproaching the 

ctral energy 

dimension 

r 2 (2) 

hing 

CFT. When and theof the the fermi quasi-particle 

the fermi the 

the 

operator surlevant, 

corre- The black hole solution takes the form: 

motion is awhere charged following g 

sur- 

F 

liquid” introduced in the phenomeno- 

chargedwith 

black 2 is black anfrom effective hole (1) [5, dimensionless 9] are solvedgauge by the coupling7 geometry 

(1) 


f 

able, 

CFT. When hole [5, 9] 

rd ing the scaling quasi-particle toward the Fermi surface with 

ction normal thenstate hasof the high form T c cuprates for a 

residue. When the operator [8]. ds 

is 

2 ≡ g MN dx M dx N ( ) 

ntroduced emphasizingintwo theimportant phenomenol 

state is that of high TIR, c cuprates the theory [8]. has not 

r 2d−2 − M (−fdt2 + d⃗x 2 )+ R2 dr the the the Fermi fermi quasi-particle 

features of 

e spectral function then has the form for a f =1+ 

Q2 

first 

the operator is where g 

surface surthe 

is unstable. 

with 

and isF 

a charged black 2 is an effective dimensionless gauge 2 coupling7 

the curvature radius 

r 2 (2) 

with ds an “emblackening factor” and gauge f 

ue. field: 

ermi 

When 

surface operator 

with 

is 

2 g MN dx M dx N = r2 of AdS. The equations 

( ) 

zing scaling twosymmetry important butfeatures an SL(2,R) of con- f =1+ 

Q2 

at in IR, the theory has not 

r 2d−2 − M r d , A t = µ 1 − rd−2 0 

r d−2 . (3) 

ermi liquid” introduced in phenomenoy 

of thenormal operator state of high is 

2 MN dx M dx N = r2 R 2 (−fdt2 + d⃗x 2 )+ R2 dr 2 of 

ds with 

he quasi-particle is unstable. 

hole [5, 9] 

ortional thequasi-particle 

Fermi to itssurface energy and with the 

and R is the curvature radius 

motion following from (1) are solved by the geometry 

T 

r d , A t = µ 1 − rd−2 

c cuprates [8]. 

R 2 (−fdt2 + d⃗x 2 )+ R2 r 2 of 

dr 2 (2) 

shes approaching fermi sur- 

ds f 

e. charged black hole [5, 9] 

nis then 

When 

irrelevant, surface has 

the 

the 

operator with form for 

is 

2 ≡ g MN dx M dx N = r2 of AdS. The equations 

quasi-particle 

( 0 ) r 2 (2) 

duced in phenomeno-dte of high T c cuprates [8]. ds 

f 

n 

with 

oward 

r d−2 . (3) 

worth as 

the the 

(maybe the 

operator Fermi surface 

emphasizing even formVirasoro two is 

2 ≡ g MN dx M dx N = r2 R 2 (−fdt2 + d⃗x 2 )+ R2 dr 2 of 

tional to its energy and the motion following from (1) are solved by the geometry 

with 

important algebra). features Theof 

r 0 isf the =1+ horizon Q2 radius determined by the largest positive 

ymmetry al The behavior first is that 

but (including IR, SL(2,R) around the theory 

con- 

Fermi has not root of the redshift r 2d−2 − M r d , A t = µ 1 − rd−2 0 

r d−2 . (3) 

esidue. When the operator is 

2 ≡ g MN dx M dx N = 

R r22 

(−fdt2 + d⃗x 2 )+ R2 r 2 of 

dr 2 (2) 

es approaching the fermi surirrelevant, 

then has the the quasi-particle 

form for a 

a charged black hole [5, 9] 

f 

( ) 

n 

pears rgent two the 

scaling 

phenomenoeven 

important 

Virasoro frequency, symmetry features but 

algebra). 

notan inSL(2,R) the 


Thespatial 

conmetry 

T (maybe even Virasoro algebra). The 

gh r 0 

f is=1+ 

Q2 

the r 0 is horizon the horizon radius radius determined by bythe the largest positive 

the IR, c cuprates theory [8]. has not 

r 2d−2 − M R 2 (−fdt2 + d⃗x 2 )+ R2 dr 2 

r 2 (2) 

uced in phenomenoes 

of the high form 

with 

r 2 (2) 

f 

ard the Fermi surface with 

f 

ction then 

Thas c cuprates for a 

the form 

[8]. 

an ds “emblackening factor” and gauge field: 

idue. When the operator for is 

2 ≡ g MN dx M dx N = r2 

a 

( 

or (including around the Fermi root of the redshift f(r 0 )=0, → M r0 d largest + Q2 positive ) 

critical behavior (including around the Fermi root of the redshift factor 

r (4) 

Monday, July 11, 2011 

portant 

he lyetry appears 

but 

frequency, infeatures an 

theSL(2,R) not frequency, 


in thenot connzing 

Virasoro twofeatures important algebra). of features Theof 

r 0 isf the =1+ horizon Q2 radius determined by the largest 

spatial in thef spatial =1+ 

Q2 

0 

the theory has not 

r 2d−2 − M r d , A ( 

t = µ 1 − rd−2 ) 

0 

r d−2 . (3) 

ntroduced wo theimportant phenomenohhl 

state TIR, c cuprates of the high theory T 

in features phenomeno- 


with f =1+ 

Q2 

f(r 

f(r 0 )=0, r0 d + Q2 

ere may exist certain operator dimensions or 

0 )=0, r d , A t = µ 1 − rd−2 

c 

[8]. cuprates has not [8]. 

r 2d−2 − M R 2 (−fdt2 + d⃗x 2 )+ R2 dr 2 

r 2 (2) 

with an “emblackening factor” and gauge ffield: 

ion then has form for a 

( 0 ) 

→ M r0 d + Q2 r d−2 . (3) 

ortant 

r d−2 (4) 

positive 

at 0 

d−2 (4) 

ncluding IR, 0 

tnot anarise SL(2,R) inaround the theory 

a consistent con- 

gravity Fermi has not root of the redshift r 2d−2 − factor r t = µ 1 − rd−2 0 

f =1+ 

Q2 

r d−2 . (3) 

he theory has not theory with r 2d−2 − M r d , A t = µ 

( 

1 − rd−2 0 

r d−2 ) 

. (3) 

troduced phenomenostate 

try but high SL(2,R) T 

with 

r d , A t = µ 1 − rd−2 

c cuprates con- [8]. 

( 0 ) 

r d−2 . (3) 

ng Virasoro two important algebra). features Theof 

r 0 isf the =1+ horizon Q2 radius determined by the largest positive 

ymmetry 7 

exactly zero frequency G R (ω = 0, 

ist certain operator dimensions ⃗ It is defined so that for a typical supergravity Lagrangian it is a 

oro cluding equency, the IR, but 

is there algebra). 

notan inSL(2,R) may exist certain Thespatial 

conevenSL(2,R) 

around the theory 

an operator dimensions r or 

Virasoro algebra). con- 

Fermi has not 

The k) 0 is the roothorizon of the redshift r 

radius 2d−2 − M factor r d , A t = µ 1 − rd−2 0 

r 

is conysics, 

not by IR CFT. 

with 0 is the horizon radius determined by the 

r constant of order O(1). g F 

by the largest d−2 . (3) 

positive 

is related in boundary theory to 

ich do not arise in a consistent gravity theory 

tion. around the Fermi root of the redshift f(r normalization 0 )=0, of the two point function of J µ . 

in a consistent gravity theory with 

7 

ior at exactly zero frequency G R (ω = 0, ⃗ It is defined so → M = r d largest 

that for a typical supergravity Lagrangian 

0 + Q2 positive 

quency, metry but 

or o (including algebra). 

notan inSL(2,R) the 

around Thespatial 

conven 

Virasoro algebra). Theis the r 0 is horizon the horizon radius radius determined by bythe the largest 

the Fermi root of the redshift factor 

r d−2 (4) 

Monday, July 11, 

r 0 2011 

it is a 

positive 

k) is con- 

V physics, not by the IR CFT. 

7 constant of order O(1). g F is related in the boundary theory 0 

he to 

around , not frequency, the not spatial Fermi in spatial root of 

ro G R (ω = 0, ⃗ Ithe is defined redshift f(r 

the normalization so 0 )=0, 

that → M = r d largest 

for of the a typical two point supergravity function of J µ 

Lagrangian 0 + Q2 positive 

(including around . 

it is a 

k) is cony 

the IR CFT. 

thef(r f(r 

constant of order O(1). g F is related in the boundary theory to 

not 

normalization 0 )=0, 

of the two point function r0 d of + Q2 

ertaininoperator the spatial dimensions or 

0 )=0, → M = r0 d + Q2 r0 

d−2 (4) 

Monday, the Fermi 

July 11, 2011 root of the redshift factor 

e frequency, not in the spatial 

J r d−2 (4) 

µ . 

r0 

d−2 (4) 

0 

consistent gravity theory with 

f(r 

f(r 7 

equency G R (ω = 0, 

erator dimensions ⃗ It is defined 0 )=0, r d so that for a typical supergravity 0 + Q2 

rtain operator dimensions or 

0 )=0, → M = r0 d + Q2 

Lagrangian r d−2 (4) 

Friday, July 20, 2012 

it is a 

certain operator dimensions or 

r d−2 (4) 

0 

onsistent gravity theory with 

k) is con- constant of order O(1). g F is related in the boundary 0 theory to 

in a consistent gravity theory 

IR CFT. 

with 7 the normalization of the two point function of J µ . 

uency gravity G theory with 

7 

ro frequency G R (ω = 0, ⃗ It is defined so that for a typical supergravity Lagrangian it is a 

R (ω = 0, 

rator dimensions ⃗ It is defined so that for a typical supergravity Lagrangian it is a 

t certain operator dimensions or 

k) is con- constant of order O(1). g F is related in the boundary theory to 

a consistent gravity theory k) is with cony 

the IR CFT. 

7 constant of order O(1). g F is related in the boundary theory to 

R CFT. 

R(ω = 0, ⃗ It is defined the normalization 

the normalization so that forof of the a the typical two 

two point supergravity point function 

function of J µ of J 

. µ . it is a 

gravity theory with 

7 

frequency Gk) R (ω is= con- 0, ⃗ It is defined so that for a typical supergravity Lagrangian it is a 

k) is con-constanthe IR CFT. 

7 constant of order O(1). g F is related in the boundary theory to 

of order O(1). g F is related in the to 

. 

the of the two point of R(ω = 0, ⃗ It is defined the normalization so that for of the a typical two point supergravity function of J µ 

Lagrangian . 

it is a 

k) is con- constant of order O(1). g F is related in the boundary theory to 

the normalization of the two point function of J µ . 

S = 1 

d(d d(d −−1) 

1) 

with gauge field and “emblackening factor”: 

Chamblin, Emparan, 

Johnson, Myers 

The extremal limit arises when Q reaches a value where f 

develops a double zero at the outer horizon. This black 

brane represents the zero temperature ground state. 


4⇤r + 2 2 


4 

4 

4 4 

4 

./&0,$+0'1/23 

!"#$%&' 

(&)*+,- 

Figure 4 The planar Reissner-Nordström-AdS black hole. The charge den- 

At extremality, the near-horizon geometry simplifies 

sity is sourced entirely by flux emanating from the black hole horizon. 

to AdS 2 R 2 : 

additionally carry a magnetic charge, was the starting point for holographic 

approaches to finite density condensed matter [27, 28]. 

Because the underlying UV theory is scale invariant, the only dimensionless 

quantity that we can discuss is the ratio T/µ. In order to answer our 

basic question about the IR physics at low temperature, we must take the 

ds 2 = 

r 2 dt 2 + dr2 

r 2 + dx 2 + dy 2 

limit T/µ ⇥ 1 of the solution. We thereby obtain the extremal Reissner- 

Nordström-AdS black hole with 

f(r) =1 4 

r 

r + 

⇥ 3 

+3 

r 

r + 

⇥ 4 

. (5.7) 

It has some interesting (peculiar) features: 

The near-horizon extremal geometry, capturing the field theory IR, follows 

1. The spatial slices don’t contract at the horizon. This 

leads directly to an extensive ground state entropy. 


2. A natural notion at RG fixed points is dynamical scaling. 

We’re used to fixed points of the renormalization group 

with a scale invariance: 

x x, t t 

In a doped system, even if you have IR rotation invariance, 

you may well expect instead: 

x x, t z t, z ⇥= 1 

By a simple change of variables, you can see that the 

IR extremal geometry here realizes this with 

z = 


ead of the more familiar scale invariance which arises in the conformal group 

Naively, this too could indicate that such a fixed point is 

t → λt, x → λx . (1 

fine tuned. Large z is hard to realize in weakly coupled 

oy model which exhibits this scale invariance (and which is analogous in many wa 

QFT. E.g. to get z=2, one can consider: 

he free scalar field example of a standard conformal field theory) is the Lifshitz fi 

ory: 

∫ 

L = d 2 xdt ( (∂ t φ) 2 − κ(∇ 2 φ) 2) . (1 

s theory has a line of fixed points parametrized by κ [1] and arises at finite temperatu 

lticritical points in the phase diagrams of known materials [2,1]. It enjoys the anisotro 

One has tuned away the standard gradient term. Higher z 

le invariance (1.1) with z =2. 

would require tuning away more such operators. 

This fixed point and its interacting cousins have become a subject of renewed inter 

the context of strongly correlated electron systems. 

For instance, in the Rokhs 

But, this is just weakly coupled intuition, and could be 

elson dimer model [3], there is a zero-temperature quantum critical point which l 

misleading at strong coupling. 

he universality class of (1.3) [4] (for a nice general exposition of the importance 

ntum critical points, see [5]). Similar critical points also arise in more general latt 

dels of strongly correlated electrons [6,7,8]. The correlation functions in these mod 


1 

transition is T c a defect 

1 

dialed freely; while k F 

[6], where a defect is the lattice spacing for defect fermions, and can be 

3. These geometries 

a itinerant 

can give rise to holographic 

is another free parameter, where a itinerant is the lattice spacing 

for semi-holographic itinerant free fermions, which can also be adjusted independently. 

Thus we make a hierarchy a defect ⇥ a itinerant . 

non-Fermi liquids. 

S.S. Lee; Liu, 

McGreevy, Vegh; 

Leiden group 

Consider a field theory whose action takes the general 

Consider a quantum field theory whose action takes 

Easiest 

We now semi-holographically 

to see on the 

couple 

the “field 

this 

form: 

large 

schematic theory” 

N field theory 

form: side. 

to the free 

Consider: 

band fermion in 

the spirit of [8]: 7 S = S strong + 

⇥ 

+g 

J 

J,J 

dt 

⇥ 

dt 

⇤c † J (i J,J ⇧ t + µ J,J + t J,J )c J 

⌅ 

⇤ 

⌅ 

c † J OF J + (Hermitian conjugate) . (11) 

Here t J,J characterizes the band structure of the originally free fermion c sector, which now 

* There Here you is a strongly should think coupled of the sector c fields which as free we’ll lattice assume is 

mixes with the large N dimer model through the coupling constant g. 

A free fermion is coupled to a strongly coupled critical 

fermions, a large N with theory lattice that sites we can indexed describe by J, using and Ogravity. 

F as J 

The key insight of [8] is that large N factorization of the field theory (which would work 

theory by a coupling g. 

an even operator small ’t in Hooft the coupling) strongly can be used coupled to infer the field modifications theory, to theinteracting 

two-point 

with * functions There the 

of the 

free is a conducting free fermion (lattice) c fermions the 

arising fermion Jth 

fromlattice thewith coupling 

site. a g. Fermi The 

In 

g 

our 

= surface. 

0 Green’s 

setup, 

We assume the CFT has correlators governed by 

it could be a fermionic operator on the Jth D5 or anti-D5 

6 At very low frequency G( ) will still approach zero on general grounds, but it may do so with a di⇥erent 

* We deform these AdS2 two symmetries. theories by coupling them with 

scaling dimension 

in even more 

brane, 

complicated 

for 

ways. 

instance. 

7 

coupling 

For notational simplicity, 

constant 

we made the 

“g”; 

free c fermions 

c should 

live on the 

couple 

same lattice sites 

to 

as 

the 

the defect 

lowest 

fermions 

do. As just mentioned, however, we should really make the c fermions live on a much finer lattice to get 


1 1

∑ 

between the free fermion dt c † (with its Fermi liquid behaviour) 

J (iδ tum numbers to c, o 

J,J ′ ∂ t + µδ J,J ′ + t J,J ′)c J ′ 

dimension fermioni 

and In the perturbation 

J,J 

strongly interacting ′ theory sector. g, the leading For instance, effect on there are 

+ g ∑ ∫ 

ized sector. 

dt (c † 

fermion a set of propagator tree graphs comes that 

J OF J + Hermitian conjugate) . (12) Returning to the 

renormalize from the “mixing” the c propagator: diagrams: 

that the idea above 

J 

imation. By approp 

In (12), we are coupling a normal theory of a weakly coupled (under the Z 4 lattic 

Fermi surface (governing the excitations of the c fermion) to preserved by the br 

a) + + + ... 

the strongly coupled locally critical sector, through the coupling 

constant g mixing c with (in any natural theory) the low- 

10 

no lower ∆ operato 

in the previous subs 

to 

Normally, 

instead havewe zest ⇥dimension would 

N]. 

b) + While fermionic 

have 

in many operator 

+ to 

cases O 

include 

F these that has deformations the right quantum 

c fermions numbers 

interactions + ... maycorrelators leave 

coming 

the(14) essent 

off 

afte 

function 

At large 

for the 

N, this 

is 

geometric 

to couple to c. 

series is the exact answer. 

clude that we can w 

physics 

the 


strong 

the fermion Usingspectral O lines. 

function 

But in 

unchanged 

a large 

(see 

N 

[8] 

strong 

for a nicesector, discussion), 

such 

it is al 

1 ⇥large ⇤ N factorization, it is then easy to show that the 

G 

reasonable 

0 (k, ) ⇥ i dte 

Figure to 2: a) The geometric sum leading to the fermion correlator (2.5). The solid line 

to 

find 

instead 

other 

haveways z ⇥ N]. 

that i t ik·(x J x 

While 

the essential ) J 

⌃c J (t)c † J 

N 

in many cases 

insights (0)⌥ 1 

a marginal Fermi l 

g =0Green’s function of the c fermion g=0 ⌅ 

fermionic (12) operator i 

l.s. 

theseof deformations 

[10–13] v|k can k F 

may 

(k)| be 

leave 

reproduced 

the essent 


J,J 

above. This has ∆ 

more with robust k F (k) physics setting. the point of the The onfermion the AdS Fermi 2 spectral regions surface, function spanned closest unchanged toby the the argument D5- (see and [8] k, for and anti-D5-branes aNnice l.s. the 

tion, discussion), number 

this dimension in the it is to al 

ω − v|k − k 

down 

Then, 

holographic reasonable 

the 

dimer the resulting 

to find model two-point other of ways [6] 

“dressed” 

provide that the function 

F (k)| 

Backreaction U 

of lattice sites. Then we find that for finite coupling an essential alternative g, 

c 

after 

propagator 

insights of summing way Oof F 

[10–13] a: 

geometric obtain can 

can 

the be series reproduced 

be 

same of physi in 

is modified to 

action of the impur 

exhibiting more the robust strange setting. metallic The behavior AdS discussed in Refs. [6, 8]. The calculation uses only 

Here, tree-level 

written 

we explore mixing 

purely 

this diagrams, in a semi-holographic 

in terms 

2 regions spanned by the D5- and anti-D5-branes in the to 


setting 

the two-point 

following [8], and 

function 

we abstract each other. 

ofthe Thus ma 

the factorization property and so would be equally1 

true in weakly coupled large-N theories. 

down holographic G dimer model of [6] provide an alternative way to obtain the same physi 

features of the top-down 

O in 

g (k, ω) ∼ 

the 

Gmodel g (k, 

strongly 

) to ⌅ω −include v|k − kmore F 

1 

coupled 

(k)| −generic g 2 G(k, 

sector: 

possibilities. 

ω) , (14) single impurity inte 

Now Here, we we canexplore see whatthis happens in aifsemi-holographic thev|k AdS 2 ⇥k R F (k)| 2 strongly setting g 2 G(k, coupled following ) . gravity 

theory[8], is replaced and weby 

abstract (13) side exhibit 

branes each wrap the ma an 

We begin an AdS with 4 theory, where a large with ⇤N anfield operator theory, of dimension governed ⇥ inby thesome 2 + 1 dimensional action S strong CFT. , The withFaulkner, 

the followi 

In particular, features for G(k, of the)=c top-down 2 1 with model to ⇤ include 1, one finds morea dominant generic possibilities. low-frequency calbehavior 

quantum Polchinskicritical 

Thursday, January correlator 

∫ 

6, 2011 

features: 

characteristic 

V. On 

We of 

large 

begin a non-Fermi with 

N 

aG(ω) liquid 

FL/NFL 

⌅0|T large ⇤(⌅x, = Nwhich t)⇤ field dt † (0, ehas transitions 

theory, iωt 0)|0⇧ 〈O vanishing =(x J F (t)O governed 2 F tquasiparticle 2 † 

J ) (0)〉 by 

in . some 

our 

action residue (15) nontrivial field theo 

system 

S[with strong (2.6) , marginal with the followi 

Wednesday, May 18, 2011 

tree-level gravity so 

features: 

+ 

interactions are suppressed by powers of 1/N! 

Then, 

represents 

the c-fermion 

G 0 and the dashed 

self-energy 

line represents G 0 . b) 1The can 

geometric 

be written 

sum (3.2), where 


an 

terms 

⇥ 


represents the double-trace perturbation. G 0 (k, ω) ∼ 

(13) 

+ 

implies 

2 

Fermi liquid behavior precisely at = 1, when the naive 1 is modified to have log( ) 

G 0 ( ⌅ k, ⇥) =A(⇥)(k 2 ⇥ 2 ) 

3/2 , (2.7) 

behaviour]. For > 1, the residue does not vanish, but the theory is still novel in that 

vary the external parameters such as temperature [see ⌅ Fig.2 for the (1+1)-dimension 

1. There is a lattice of defect fermions which undergoes a dimerization transition as 


+ 

1. There is a lattice of defect fermions which undergoes a dimerization transition as w

vary the1. external There isparametersa lattice of defect such as fermions temperature which undergoes [see Fig.2 a dimerization for the (1+1)-dimensiona 

transition as we 

Now, If we let make us consider the strong the behaviour dynamical of two-point assumption functions that the 

the quasiparticle width does not agree with that of standard Fermi liquid theory. As we 

strongly In a 

for 

“locally 

natural 

critical” 

fermionic 

theory, 

operators 

i.e. 


one 

our setup. 

with infinite 

described above, coupled these results sector are trueexhibits in a regime where local k quantum 

easily generalize. dynamical 5 exponent, one 

F ⇧ ⇧ 1 

a 

has: 

defect 

, criticality, 

where the 4 

zero-temperature 

then 

Green’s 

the two-point 

functions used above 

function 

should be a 

is 

goodconstrained: 

approximation to the true 

vary the external parameters such as temperature [see Fig.2 for the (1+1)-dimensional 

case]. We will focus on the cases for which this parameter is temperature, but one can 

case]. We will focus on the cases for which this parameter is temperature, but one can 

easily generalize. 5 

9 

(finite- a) In the phase: The operator lives on an slice 

2. 

Thisbut Theretwo-point low-temperature) 

exist fermionic 

function answers. 

operators 

is 

Ofixed by the scaling 

of the bulk F AdSsymmetry 2 of 

J 

2. There exist fermionic operators geometry. OJ the LC theory to be G(ω) = c F The two-point 

correlation 

correlation 

functions 

functions 

in the undimerized ∆ ω 2∆−1 where ∆ is the di- 

in the are undimerized 

phase 

constrained phase 

are known 

are known 


to behave and 

gapless: 

advocated in [8]: 

gapless: 

mension of O F (and, importantly, G(ω) ∼ c ω log(ω) in as: the 

4 

degenerate case ∆ dte =1). 

This creates i t ⌅Odte J F (t)O i 

a t ⌅Onon-Fermi F F† 

J 

(0)⇧ J (t)O F = † 

J 

(0)⇧ i J,J = i G(⌅), 

liquid 

J,J G(⌅), 

if 

The correction term in the denominator of G g will dominateprecisely 

function 

o-point dte i t ⌃OJ F (t)OF † 

J 

the low-frequency 

is1, fixed one byhas the 

behavior a scaling marginal if 

symmetry ≤Fermi 1. Unitarity 

ofliquid allows with 

theory 

FIG. 1: Thursday, Predimerization January to6, 2011 be 

transition. (a)Disconnected configuration dominates at high temperature. 

3. 

G(ω) 

In the dimerized phase, is gapped and 

(b)Connected 

3. any Inconfiguration the ∆dimerized ≥ 1 = c 

dominates at low temperature. 

2 

and phase, this ∆ ω 

thescaling 2∆−1 where apple 1. ∆ is the diof 

O 

spectrumdimension is gapped andis a free parameter 

in F (and, importantly, G(ω) ∼ c ω log(ω) in the 

the general A. Disconnected approaches configuration 

of exhibiting power-law behavior, lim lim dte i t ⌅OJ F (t)O F † 

⇥0 

of [4, 7]. The marginal Fermi liq- 

obvious candidate behavior stable configuration of [2] of suchappears a pair ⇥0 

ate case ∆ =1). 

lim dte ⇥0 

G(k, 

i t ⌅O F Theuid is just twoin separated the J (t)O F † 

configurations 

of the sort consideredHere, in Sec.II BA with J,J ⇥ ⇥ = ¯⇥¯⇥ [see Fig.1(a)]. Its free energy is just 

case that the dimension of 

orrection ednesday, May 

twice original O F 18, 2011term in the denominator of G g will domilow-frequency 

is 

that of the single k D5-brane: 

Here, F . precisely 

A J,J dimerization. 

behavior 

1. Therefore, 

if ∆ ≤ 1. 

the 

Unitarity 

question 

allows 

is, are there natural 

circumstances in which 

⇤ 

⌅ 

F D5 + F D5 ¯ = 2 L4 sin 3 ⇥ ⇥ vol(S 4 ) 

≥ 1 r 

(2⇤) 5 g 

⇥3 + 

the theory S 

. (3.1) LC (A, B, Q, ˜Q) has a 

2 dimerization. and this scalingfermion dimension Greens s 

is a freefunction. 

parameter 

Note that leading it is independent 

Forfermionic example, 

of the separation 

for 

x. 

the operator literal D5of probe ∆ theory =1which in AdS 5 can S 5 , we couple can take 

eneral approaches of [4, 7]. The marginal Fermi liqvior 

example, of The [2] for theories appears literal weD5 have probeconstructed theory in AdS 

to c? 

For 

B. Connected 

inconfiguration 

the case that the dimension 5 

above S 5 , weof 

naturally can take come 

recisely 

inwith Kondodefect lattice models operators discussedof in [14] ∆ and =1, references as indicated therein. by our calculation 

in ofwhich the KKthespectrum theoryx =( 


1. 

behavior 

Therefore, 

conjectured 

the question is, 

by 

are 

Varma natumstances 

Another candidate solution with the given boundary condition is a reconnecting solution 

[see Fig.1(b)]: a reconnecting D5-brane starts at r = ⇤ with ⇥ x 

S on , 0, 0), dips into the 

2 

the probe M2 ′ LC (A, B, Q, ˜Q) et al in 1989. 

hasbranes. a It is in- 

Now, we are in a position to add one simple observation localized at onthe topJth of the lattice basic site, picture whose thermal 

Fermi surface. The main point is that the low frequency behavior of the Green’s function 

with G(⌅) with G(⌅) ⇥ ⌅ 2 ⇥ ⌅ 1 2 for 1 for ⌅ ⇤⌅ ⇤T. T. (8) 

(0)⌥ changes drastically in the dimerization transition. In the undimerized 

bulk, and then comes back to r = ⇤ now with ⇥ x =(+ x , 0, 0), e⇥ectively reversing its 

2 

localized at the Jth lattice site, whose therma 

in holographic models which undergo a dimerization transition as in 

The unitarity bound on the dimension is 1/2; for any value 

Sec. II, the phase transition also drives an interesting transition in the structure of the 

less than 1, one obtains a non-Fermi liquid, and if Delta is 

state, we will have non-Fermi liquid behavior just as in [8]. However, in the dimerized phase, 

Varma et al 

(1989) 

the spectrum in the dimer sector is gapped. This means that at low frequencies, instead 

)=A for some J (0)⇧ nonzero = iA J,J constant . A. Thus (9) 

This is evident because the simple pole (and hence the 

J (0)⇧ = iA J,J . (9 

in this phase, we have a conventional Fermi liquid whose Fermi surface is shifted from the 

is nonzero (generically if and) only if J = J ⇤ or J and J ⇤ are paired up via 

spectral weight of the quasiparticles) disappears from the 

is nonzero (generically if and) only if J = J ⇤ or J and J ⇤ are paired up via 

Therefore, in this semi-holographic setting, the dimerization transition of Sec. II becomes 

a transition between a conventional Fermi liquid phase (dimerized) and a non-Fermi liquid 

phase (undimerized). These transitions are somewhat 

OJ F = ⇤ † J ⇥ reminiscent of the phase transitions 

N =4(J)⇤ J (10) 

In the special “marginal” case, this gives a first derivation 

O F J = ⇤ † J ⇥ N =4(J)⇤ J (10 

Finally, we note that if one is purely interested in finding realizations of the non-Fermi 


5 For instance, one can consider driving such a transition by going to finite chemical potential for the large 

N gauge fields at T = 0, at the cost of introducing Reissner-Nordström black branes. At su ciently

Note that the Fermi surface physics here is a modeldependent 

1/N effect. 

III. More general homogeneous, isotropic horizons 

Gravity + Maxwell is the minimal content to study doped 

holographic matter. 

It is quite reasonable to expect additional light fields in the 

bulk! How does this effect the horizon? 

We’ll make some ad hoc choice of matter. The geometries 

we find emerge more universally, for many choices. 


hile 

ear-horizon 

the full black-brane 

behaviour 

solutionsof with 

extremal 

AdS asymptotics 

branedid not admit a sim 

alytical form that we could find, we are able to provide an analytical description 

Near-horizon eLet near-horizon us consider geometry solution adding for the extremal a neutral charged scalar dilaton “dilaton” branes 4 . field We use to a notati the 

d formalism similar to that in [14], in describing these near-horizon solutions. 

e the full black-brane system. solutions To begin withwith, AdS asymptotics we consider: 

For our bulk action, we take 

did not admit a si 

tical form that we could ⇤ find, we are able to provide an analytical descriptio 

ear-horizon geometry S = for d 4 xthe ⇤ extremal g R 2(⌅⇥) charged 2 edilaton 2 ⇥ F 2 branes 2 ⇥ . 4 . We use a nota (2 

formalism similar A. to Theories that in [14], with in describing no dilaton these potential 

M. Taylor; 

Goldstein, S.K., 

near-horizon solutions. 

Prakash, Trivedi 

3 or In the ourcontext bulk of action, holographic we take superconductors [10, 11], another class of black branes with vanish 

The maximally symmetric solution of this action has vanishing 

ropy at T = 0 was recently ⇤ found 

S = d 4 x ⇤ in [12, 13]. 

4 By the near-horizon gauge field, region constant we mean the region of spacetime 

g dilaton, R 2(⌅⇥) and 2 an e 2 AdS ⇥ “close 

F 2 metric to” 

2 ⇥ where the g 

. with tt compon 

the metric vanishes. 

curvature radius L related to the cosmological term via: 

the context of holographic superconductors [10, 11], another class of black branes with van 

The maximally symmetric solution of this action has vanishing 

py at T = 0 was recently found in [12, 13]. 

yermined the near-horizon by = 

gauge field, We region constant 3 restrict . we We mean consider 

= 

the dilaton, region metrics a 3 L 

. 

4 of and 2 spacetime of of 

an the 

AdS “close form: 

L 2 metric to” wherewith 

the g tt comp 

metric vanishes. 

aximally symmetric vacuum solution is then of course AdS space with Ad 

imally symmetric vacuum solution is then of course AdS space with Ad 

ined 

curvature 

by = 

We instead radius 3 . WeL consider 

related 

ato metric 

the cosmological 

of the form 

L term via: 

ds 2 = search 2 a(r) 2 for dt 2 solutions + a(r) 2 dr with 2 + b(r) a metric 2 (dx 2 of + dy the 2 ) form 

ximally symmetric ds 2 = a(r) vacuum 2 dt 2 + solution a(r) 2 is drthen of course AdS space with A 

mined 

= 3 2 + b(r) 2 (dx 2 + dy 2 ) 

gauge L 

4 

2 . 

ally symmetric by field of = thevacuum 3 form . We consider solutiona is metric then of the course formAdS space with A 

L 2 

uge and 

ed by fieldgauge of = the 3 field: form 

L 2 . We consider a metric of the form 

We instead 2 search 2for 2solutions 2 with 2 a metric 2 2 of the 2 form 


2 ⌅ Q

TheL maximally determinedsymmetric by = 

4 L 

L vacuum 2 . We consider solutiona is metric then of the course form 

L = 3 AdS space with AdS scale (2.8) 

L determined by = 

We instead 3 . We consider a metric of the form 

Lsearch for solutions with a metric of the form 

One can solve 

ds 2 = 2 a(r) 

for 2 dt 

the 2 + a(r) 

gauge 2 dr 2 field 

+ b(r) 2 using 

(dx 2 + dyGauss’ 2 ) 

law: 

(2.2) 

In the discussion below it will be convenient to set L = 1. When needed the dependence 

on L can beds determined 2 = a(r) 2 bydtdimensional 2 + a(r) 2 dr analysis. 

2 + b(r) 2 (dx 2 + dy 2 ) (2.2) 

and a gauge It is field most of the useful form to first find an approximate, near-horizon 

To find the near-horizon limit, we proceed with a scaling ansatz. Define the nearhorizon 

extremal variablesolution w form = r r to the equations. 

and a gauge and fieldgauge of the field: 

e 2 ⌅ F = 

Q Let us guess that near 

H where r H is the radius of the horizon. Let us make the 

dt ⇥ dr. (2.3) 

ansatz that the near-horizon w=0, scalingsthe b(r) 

e 2 ⌅ F = 

Q of the 2solution functionstakes a, b, ⇥ are the given form: by 

These are electrically charged dt ⇥ dr. (2.3) 

b(r) black 2 branes. 

That is, we are looking for a = electrically C charged black branes. 

The 

Monday, April 12, 2010 

2 w ⇤ ,b= C 1 w ⇥ , ⇥ = K log(w)+C 3 , (2.9) 

equations result of is motion solutions can be easily with inferred near-horizon from (107)-(110) geometry: 

of [14]. We find 

That is, we are looking for electrically charged black branes. 

that 

The equations where of the motion Cs are can be just easily integration inferred from constants. (107)-(110) of [14]. We find 

(a 2 b 2 ) ⇥⇥ = 4 b 2 (2.4) 

that 

5 

a ⇠ r , b ⇠ r , = K log(r) 

together take with the values the first order constraint, 

⇤ r (a 2 b 2 ⇤ r ⇥)= 

together with the first order a 2 b ⇥2 constraint, + 1 2 a2⇥ b 2⇥ = ⇥ ⇥2 a 2 b 2 


(a 2 2 ) ⇥⇥ = 4 b 2 (2.4) 

b ⇥⇥ 

b = (⇤ r⇥) 2 (2.5) 

A little bit of algebra now shows that one can exactly 

where one easily finds: 

solve the system of equations from the previous slide, 

b ⇥⇥ 

where C 1 ,C 2 ,C 3 are constants. 

if one 

A 

chooses 

little algebra 

exponents: 

then shows that an exact solution to 

the equations (2.4), (2.5), (2.6) and the constraint eq.(2.7) is obtained if the exponents 

b = (⇤ r⇥) 2 (2.5) 

⇤ r (a 2 b 2 ⇤ r ⇥)= e 2 ⌅ Q2 

b , (2.6) 

2 

e 2 ⌅ Q2 

b 2 , (2.6) 

2 

⇤ =1,K= 

1+( 2 

) 2 , ⇥ = ( 2 )2 

1+( 2 

) . (2.10) 

2 

e 2 ⌅ Q2 

b 2 b 2 . (2.7) 

By rescaling w, t, x and y appropriately the constant C 3 can be set to zero and C 1 to 

unity. One then gets that two remaining parameters C 1 and Q are determined in terms 

Even 

a 2 b ⇥2 + 1 a 2⇥ b 2⇥ = ⇥ ⇥2 a 2 b 2 e 2 ⌅ Q2 

b 2 Monday, April though 12, 2010 

of by 

we will be interested in the near-horizon limit, 

. 

it is important to keep 

(2.7) 

2

After a simple change of variables, the metric takes the 

form of a “Lifshitz” metric representing emergent 

dynamical scaling with dynamical exponent z: 

ds 2 = 

r 2z dt 2 + r 2 (dx 2 + dy 2 )+ dr2 

t ! z t, (x, y) ! (x, y), r ! r , z = 1 

r 2 

SK, Liu, 

Mulligan 

The near-horizon geometry glues into asymptotically AdS 

space with constant dilaton. 

Comparing to AdS/RN: 


As was mentioned above the solution eq.(2.40) arises as the near horizon limit of 

a slightly non-extremal black brane solution. The entropy density can be expressed 

1. as aThe function (tunable) of two dimensionful value of parameters z controls for scaling the non-extremal laws for solution, the 

temperature T and 

thermodynamic chemical potential 

observables: 

µ. Both of these are intensive variables 

with dimensions [Mass] 1 . It follows from eq.(2.43) and dimensional analysis that 

the entropy density of a slightly non-extremal black brane is given by 

s ⇥ T 2 µ 2 2 . (2.44) 

The entropy can be found from the classical action eq.(2.14) evaluated on-shell. 

This action can be calculated by scaling out the dependence on L -theradiusof 

2. The finite value of z relaxes the extensive ground state 

AdS space - and then working with dimensionless quantities. As a result the action 

and the entropy entropy will haveof a prefactor the Einstein/Maxwell which goes like, L 2 /Gtheory. 

N where G N is the four 

dimensional Newton’s constant. Putting all this together gives the entropy density 

of the slightly non-extremal black brane to be 

s = aCT 2 µ 2 2 , (2.45) 

3. Fermions in these geometries can also naturally 

deform to non-Fermi liquids, in a way that is 

reminiscent of field theory models: 

–12– 

Hartnoll, 

Polchinski, 

Silverstein, 

Tong 


E.g. consider the simplest model of a Fermi surface coupled 

to a critical boson: 

FIG. 1. The shaded region represents the occupied states inside a Fermi surface. Fluctuations 

the order parameter φ at wavevectors parallel to ⃗q couple most strongly to fermions near the Ferm 

surface points ± ⃗ k 0 .Thesefermionsaredenotedψ ± . 

FIG. 1. The shaded region represents the occupied states inside a Fermi surface. Fluctuations of 

the order parameter φ at wavevectors parallel to ⃗q couple most strongly to fermions near the Fermi 

surface points ± ⃗ k 0 .Thesefermionsaredenotedψ ± . 

Lagrangian becomes 

( 

L k0 = ψ † +σ ∂ τ − iv F ∂ x − 1 

Lagrangian becomes 

( 

L k0 = ψ † +σ ∂ τ − iv F ∂ x − 1 

2m ∂2 y 

2m ∂2 y 

) ( 

ψ +σ + ψ † −σ ∂ τ + iv F ∂ x − 1 

) 

ψ +σ + ψ † −σ 

( 

∂ τ + iv F ∂ x − 1 

2m ∂2 y 

) 

ψ −σ 

Here v F and m are the Fermi velocity and the band mass at k 0 , while d ±σ = d ±k0 σ,andwe 

have added a subscript k 0 to L emphasize that this is the Lagrangian for the patch near 

± ⃗ k 0 . 

We should emphasize here that all the fields in Eq. (2.4) are 2+1 dimensional quantum 

fields, with full dependence upon x, y, andτ i.e. the fields are φ(x, y, τ) andψ ±σ (x, y, τ). In 

2 

2m ∂2 y 

− d φψ +σψ † +σ − d −σ + 1 2 (∂ yφ) 2 + r 0 

2 φ2 (2.4) 

− d +σ φψ +σψ † +σ − d −σ φψ −σψ † −σ + 1 2 (∂ yφ) 2 + r 0 

2 φ2 (2. 

Here v F and m are the Fermi velocity and the band mass at k 0 , while d ±σ = d ±k0 σ,andw 

have added a subscript k 0 to L emphasize that this is the Lagrangian for the patch nea 


) 

ψ −σ

y 

The fermion correction to the boson propagator: 

FIG. 2: The one-loop boson self energy. 

FIG. 3: The one-loop fermion self energy. Here the boson propagator 

is a dressed propagator which include the one-loop self energy 

correction in Fig. 2. 

Note that η has dropped out of the final result. We are interested above on 

contribution to Π 0 , and this is insensitive to orders of integration: so unlike 

order, we have integrated over l x before l τ . We include the RPA polariza 

into the bosonic propagator to obtain 

3 

fermion self energy, the dressed boson propagator has been 

used because the boson self energy is of the order of 1. Inthe 

low energy limit, the leading terms of the quantum effective 

action are invariant under the scale transformation, 

D(q) = 1 N 

k 0 = b −1 k ′ 0 , 

( 

) −1 

|q τ | 

c b 

|q y | + q2 y 

e + r . 

2 

k x = b −2/3 k x, 

′ fermion self energy, the dressed boson propagator has b 

k y = b −1/3 k y, 

′ used8 

because the boson self energy is of the order of 1. In 

ψ a (b −1 k ′ 0 ,b−2/3 k ′ x ,b−1/3 k ′ y ) = b4/3 ψ ′ low 

a (k′ 0 ,k′ x ,k′ y ), energy limit, the leading terms of the quantum effec 

a(b −1 k 0,b ′ −2/3 k x,b ′ −1/3 k y) ′ = b 4/3 action are invariant under the scale transformation, 

a(k 0,k ′ x,k ′ y). ′ (3) 

k 

yields a theory with 

The singular 

z=3 

self energies render 

dynamical 

the terms, 

scaling. 

0 = b −1 k ′ 0 , 

k x = b −2/3 k x, 

′ 

ik 0 ψj ∗ (k)ψ j (k), 

FIG. 2: The one-loop[ boson self 

k 

2 

0 + kx 

2 ] energy. 

k y = b −1/3 k y, 

′ 

a ∗ (k)a(k) (4) 

ψ a (b −1 k ′ 0 ,b−2/3 k ′ x ,b−1/3 k ′ y ) = b4/3 ψ ′ a (k′ 0 ,k′ x ,k′ y ), 

irrelevant in the low energy limit. Usually, it is expected that a(b −1 k 0,b ′ −2/3 k x,b ′ −1/3 k y) ′ = b 4/3 a(k 0,k ′ x,k ′ y). 

′ 

one restores the same low energy quantum effective action if 

one drops the irrelevant terms from the beginning. However, The singular self energies render the terms, 

this is not true in this case. If one drops the irrelevant termsin 

the bare action, then the resulting theory becomes completely 

localized in time and one can not have a propagating mode. If 

there is no frequency dependence in the bare action, the fre- 

of the Fermi surface. The Fermi surface is on v x k x +v y ky 2 =0 

ik 0 ψj ∗ (k)ψ j (k), 

[ 

as is shown in Fig. 1. Thisisa‘chiralFermisurface’where 

k 

2 

0 + k 2 ] 

x a ∗ (k)a(k) 

the x-component of Fermi velocity is always positive. a is the quency dependent singular self energies can not be generated 

transverse component of an emergent U(1) gauge 

FIG. 

boson 

3: The 


one-loop either. fermion Therefore, selfto energy. restoreHere the full the boson low energy propagator 

is a dressedhas propagator keep which a minimal include information the one-loop that self the theory energy is not one com- 

restores the same low energy quantum effective actio 

dynamics, irrelevant one in the low energy limit. Usually, it is expected 

the Coulomb gauge ∇ · a =0.Weignorethetemporalcomponent 

of the gauge field which is screened to a short correction rangein Fig. pletely 2. localized in time. It turns out that the followingone action drops the irrelevant terms from the beginning. Howe 

interaction. The transverse gauge field is massless without a given by 

this is not true in this case. If one drops the irrelevant term 

fine tuning due to an emergent U(1) symmetry associated with 

the dynamical suppression of instantons 21,22 the bare action, then the resulting theory becomes comple 

. e is the 



the Fermi surface. The Fermi surface ψj ∗ (η∂ is τ on −viv x ∂ x − v y ∂y)ψ 2 x k x +v y ky 2 =0 j localized in time and one can not have a propagating mod 

L = ∑ between 

Feeding 

fermions and the critical boson. 

this back 

as is shown in Fig. 


1. 

to 

Thisisa‘chiralFermisurface’where 

the 

j 

fermionic 

In the one-loop order, singular self energies are generated 

from the diagrams Figs. 2 and 3,andthequantumeffective 

+ e 

there is no 

sector 

frequency dependence in the bare action, the 

the x-component of Fermi velocity √ is always ∑ 

aψ ∗positive. j ψ j + a(−∂ a is y 2 )a, the quency (5) dependent singular self energiesS.S. canLee; 

not be gener 

transverse component of an emergent 

action becomes 

N U(1) gauge boson in either. Therefore, to restore the full 

j 

Metlitski, 

low energy 

Sachdev; 

dynamics, 

Γ = ∑ ∫ [ 

dk i c yields 

the Coulomb 

a 

gauge 

non-Fermi 

∇ · a =0.Weignorethetemporalcomponent 

of the liquid. 

has to keep a minimal information that the theory is not c 

N sgn(k 0)|k 0 | 2/3 + ik 0 

is gauge the minimal field which local is theory screened which to restores a short the range one-looppletely quantumtransverse 

effective action gauge(2). fieldHere is massless η is a parameter withoutwhich a has given theby 

] fine tuning due 

localized in time. It turns out that the 

.... 

following ac 

interaction. The 

j 

+ v x k x + v y ky 

2 ψj ∗ (k)ψ dimension to an emergent −1/3 U(1) according symmetry to the associated scaling (3). with 

j(k) the dynamical suppression Since the time of instantons derivative 21,22 term . eis isirrelevant, the coupling η will flow to L = ∑ Saturday, November 10, 12 

ψj ∗ (η∂ τ − iv x ∂ x − v y ∂y)ψ 2 j

We can also use gauge/gravity duality to see 

The the set-up effect is the of a following. small z critical One inserts sector a on single fermions. ‘probe’ 

For large enough mass (i.e. small enough v 0 ), the flavor brane still shrinks to a point 

D-brane into the Lifshitz space-time with dynamical 

correspond to strings stretching from the flavor brane at v 

Insert a “probe” flavor-brane in a Lifshitz 

0 to the horizon at v 

geometry 

+ . 

exponent z. The probe changes the dual field 

ds 2 = 

correspond to strings stretching from the tip of the cigar down to the Poincaré horizon 

v = ⇥. At finite temperature, a black hole horizon arises at a finite radial position v + . 

outside the black hole horizon. Charge carriers in this 0

energy gap E gap to exciting charge carriers which is large compared to t 

. If the conductivity in a Lifshitz system is linear in the density J t of 

By a scaling argument, if the conductivity is linear in the 

density of charge carriers: 

then by the scaling given above we can conclude that the resistivity 

e 

⌅ T 2/z 

J t . 

lt is independent of the spatial dimension d. Here we are using the fa 

In This the high would density, give linear strongly resistivity interacting if z=2. regime, However, it is far in the from 

obvious real materials, that one charge-charge should have such self-interactions linearity. Nevertheless, are crucial; 

a non-trivial so why gravity is the calculation conductivity (using linear techniques in the density? developed 

g the energy gap should not lead to larger conductivity in order to excl 

gap-dependent contributions to (2.7). (Contributions to the conductivit 

with E gap are negligible in the limit of large E gap /T .) 

by Karch & co.) shows that such linearity does hold, giving 

In linear the HPST resistivity calculation, at z=2 even this comes in the regime out of basic of strongly facts about selfinteracting 

the charge large density carriers. expansion. Even 

the probe DBI action 

at large density, one gets linear resistivity at z=2. 

the chemical potential µ ⇥ T , linearity of the conductivity in the de 

te: this regime corresponds to very low density, where the conductivity 

ause it is simply the sum of the individual contributions of non-interacting 

Thursday, Saturday, January November 12, 10, 2012

II. 

EINSTEIN-MAXWELL-DILATON THEORY 

= f ⇥ (r) ⇥ (r) 

2f(r) 

+ h⇥ (r) 2 Z ⇥ ( (r)) 

4f(r) 

g ⇥ (r) ⇥ (r) 

2g(r) 

We begin with a discussion 

B. 

ofHyperscaling the basic characteristicsviolation 

of the EMD theory of compressible 

quantum states [14, 25] in d spatial dimensions. As reviewed in Ref. [37], the globally 

conserved U(1) charge of the compressible state ⇤ is holographically ⌅ realized by a U(1) gauge 

field A µ . In addition, the EMD ⇧ 2 ⇤Stheories EMD also include a scalar field (the ‘dilaton’) [66] which 

is dual to a relevant perturbation L d ⇤h(r) 

= d h ⇥ (r)Z( (r)) 

on the UVdrconformal r d⇧ f(r)g(r) field theory, and which allows access 

to a wider range of IR scaling reasonable behavior in compressible to postulate: 

states. 

So we consider the holographic Lagrangian 

So far, we considered vanishing dilaton potential. It is more 

defined on a (d + 2)-dimensional spacetime with Ricci scalar R, with Maxwell flux F µ⇥ 

associated with A µ , and a dilaton field with potential V ( ) and coupling Z( ). 

We use co-ordinates (t, r, x i ), where t is the time direction, r is the emergent holographic 

direction, and x i (i =1...d) are the flat spatial directions. We will examine solutions with 

metric 

⇤ 

⌅ 

ds 2 = L 2 f(r)dt behavior 2 + g(r)drto 2 + obey 

dx2 i 

, (2.2) 

r 2 

Rather We now discuss natural the structure choices of the for solutions the in functions the IR limit, r(motivated ⇤⌅. As we discussed by e.g. in 

only the temporal component of the gauge field non-zero 

A t = eL ⇥ 

1 

4 g(r)V ⇥ ( (r)) + ⇥⇥ (r) d 

Finally, the only non-zero Maxwell equation is Gauss’ Law, which yields 

h(r), (2.3) 

and a dilaton field (r) dependent only upon r. Under these conditions, we can work with 

action per unit spacetime volume of the boundary theory 

⇥ (r) 

r 

=0. 

=0. (2.7) 

The integration constant in (2.7) is set by the charge density on the boundary [36], and so 

we have 

L EMD = 1 ⇥ 

R 2(⌅ ) 2 V ( ) 

⇥ 

L d 1 h ⇥ (r)Z( (r)) Z( ) 

2⇥ 2 L 2 4e F µ⇥F µ⇥ (2.1) 

⇧e r d⇧ = Q. 

f(r)g(r) 2 (2.8) 

The dependence of the solutions on the charge density Q will be crucial to our purposes. 

Section 

gauged 

I, we are 

supergravity), 

interested in solutions which obey 

yield 

(1.1) 

scaling this limit. 

solutions, 

Extending theare: 

results 

of Ref. [25] to general d, we can deduce that such a solution will emerge from the equations 

of motion provided we choose the large 

with 


Charmousis, 

Z( ) = Z 0 exp ( ) 

, as ⇤⌅. 

Gouteraux, Kim, 

(2.9) 

Kiritsis, Meyer 

V ( ) = V 0 exp ( ⇥ ) 

, ⇥ > 0, and the exponents z and ⌅ are then given by 

⌅ = 

d 2 ⇥ 

(2.10)

The first parameter, in the gauge coupling function, 

transmuted into a dynamical critical exponent z. 

With two parameters in L, we find solutions that reflect 

two critical exponents: z and a “hyperscaling violation” 

.1 Metrics with scale covariance parameter ✓ : 

Metrics with scale covariance 

s we reviewed before, the gravity side is characterized by a metric of the form 

e reviewed before, the 

ds 2 gravity 

d+2 = r 2(d side )/d is characterized ⇥ 

r 2(z 1) dt 2 by 

+ dr 2 a metric 

+ dx 2 of the form 

i . 

⇥ 

ds 

his is the most general 2 d+2 = r 

metric that 2(d is )/d r 

spatially 2(z homogeneous 1) dt 2 + dr 2 + dx 

and 2 i . 

covariant under the 

ansformations 

is the most These general are metric conformal that is spatially to scale homogeneous invariant and metrics: covariant under th 

sformations x i ⌅ ⇥x i ,t⌅ ⇥ z t, r⌅ ⇥r , ds⌅ ⇥ /d ds . 

hus, z plays the role xof i ⌅the ⇥x dynamical i ,t⌅ ⇥exponent, z r⌅ ⇥r and , ds⌅ is the ⇥ /d hyperscaling ds . violation 

ent. 

s, z plays the role of the dynamical exponent, and is the hyperscaling violatio 

. The dual (d + 1)-dimensional field theory lives on a background spacetime identified 

surface of constant r in (2.1). The radial coordinate is related to the energy scale 

he dual The (d finite + 1)-dimensional temperature field theory deformations lives on a background are also spacetime known: identifi 

ual theory. For example, an object of fixed proper energy E pr and momentum ⌘p pr red 

rface of constant r in (2.1). The radial coordinate is related to the energy scale 

cording to 


l theory. For example, an object of fixed proper energy E and momentum ⌘p re

Starting now from the metric F (2.1) with hyperscaling violation, the black hole solution 

Starting now from the metric (2.1) with hyperscaling violation, the black hole solution 

becomes [17, 34] 

becomes [17, 34] 

ith 

⇥ 

⇥ d+z 2 /d /d 

r 

⇥ 

⇥ 

ds ds 2 f(r) 2 

d+2 R2 

=1 

. (5.3) 

d+2 = R2 r 

rr r 2(z 2(z 1) 1) f(r)dt 2 + dr2 

h 

f(r)dt 2 + dr2 

rr 2 2 r f(r) + dx2 i , (5.2) 

F f(r) + dx2 i , (5.2) 

tarting from a solution with f(r) = 1 and matter content general enough to allow for 

with 

rbitrary 

with 

(z, ), 14 one can show, using the results in ⇥ d+z the Appendix, that (5.3) still gives a 

rr 

olution. Concrete examples will be f(r) presented =1 in §6. As. usual, the relation between (5.3) the 

r h 

⇥ d+z 

. (5.3) 

r 

emperature and r h follows by expanding r h r = u h 2 , and demanding that near the horizon 

Starting from a general enough to allow for 

he metric Starting is ds from 2 ⇧ adusolution 2 + u 2 d⇤with 2 , where f(r) ⇤ = =(2⇥T 1 and)it. matter The content result isgeneral enough to allow for 

arbitrary arbitrary (z, (z,), 14 ), 14 one onecan canshow, using the resultsin inthe the Appendix, Appendix, that that (5.3) (5.3) still still gives gives a a 

solution. Concrete examples will be presented in §6. As usual, the relation between the 

solution. Concrete examples will be presented 

temperature and r h follows by expanding T = 1 |d + z in | §6. As usual, the relation between the 

r h r = u 2 , and demanding that near the horizon 

temperature and 

the metric is ds 2 r 

. (5.4) 

h follows 

⇧ du 2 by 

+ u 2 expanding 

d⇤ 2 , where 4⇥ r =(2⇥T h r z r = u 

)it. The 2 , and demanding that near the horizon 

h result is 

the metric is ds 2 ⇧ du 2 + u 2 d⇤ 2 , where ⇤ =(2⇥T )it. The result is 

These expressions imply that the thermal 

T = 1 |d entropy, + z | which is proportional to the area of 

. (5.4) 

he black hole, becomes 

This results in 

T 

an 

= 4⇥ 

entropy 1 |d + 

rh 

z z | 

. 

These expressions imply Sthat T ⌅ the (Mthermal Pl R) d Ventropy, T that scales like: 

(5.4) 

4⇥ r (d h 

z )/z 

which . is proportional to the area of (5.5) 

the These black expressions hole, becomes imply that the thermal entropy, r F which is proportional to the area of 

hus, the a positive black hole, specific becomes heat imposesS T 

the ⌅ (M condition 

Pl R) d V T (d )/z 

. (5.5) 

r F 

S T ⌅ 

Thus, a positive specific heat imposesd 

(M Pl R) d V T (d )/z 

. (5.5) 

the condition 

⇤ 0 . r F (5.6) 

z 

Thus, a positive specific heat imposes the d condition 

⇤ 0 . (5.6) 

e see that the branch {0 < z < 1, ⇤ zd + z} that was consistent with the NEC is 

hermodynamically 

We see that the 

unstable. Hyperscaling 

d 

branch 

On 

{0 < 

the 

z 

other 

< 1, 

hand, violation ⇤ d + 

⇤{z z} 

0 ⇥. that 

0, 

was 

⇤shifts d} 

consistent 

is still d! allowed 

with the 

by 

NEC 

(5.6). 

is 

(5.6) It 

z 

ould bethermodynamically interesting to study unstable. this case On the in other morehand, detail {zto ⇥decide 0, ⇤ d} whether is still allowed it is consistent by (5.6). It – the 

ntanglement Wewould see that be entropy interesting the branch analysis to study {0 of < §4 this z suggested case < 1, in more ⇤and detail instability + z} tothat decide for was whether all consistent >d. it consistent with the– the NEC is 

thermodynamically entanglement entropy unstable. analysisOn of §4 thesuggested other hand, instability {z ⇥ 0, for ⇤ all d} is>d. 

still allowed by (5.6). It 


An example of a system which enjoys such “shifted d” in 

the real world, is a Fermi liquid. Basic picture: 

Non-Fermi liquids A. J. Schofield 2 

can be obtained relatively simply using Fermi’s golden 

rule (together with Maxwell’s equations) and I have included 

these for readers who would like to see where 

some of the properties are coming from. 

The outline of this review is as follows. I begin with 

a description of Fermi-liquid theory itself. This theory 

tells us why one gets a very good description of a 

metal by treating it as a gas of Fermi particles (i.e. that 

obey Pauli’s exclusion principle) where the interactions 

are weak and relatively unimportant. The reason is 

that the particles one is really describing are not the 

original electrons but electron-like quasiparticles that 

emerge from the interacting gas of electrons. Despite its 

recent failures which motivate the subject of non-Fermi 

liquids, it is a remarkably successful theory at describing 

many metals including some, like UPt 3 , where the 

interactions between the original electrons are very important. 

However, it is seen to fail in other materials 

and these are not just exceptions to a general rule but 

are some of the most interesting materials known. As 

an example I discuss its failure in the metallic state of 

the high temperature superconductors. 

I then present four examples which, from a theoretical 

perspective, generate non-Fermi liquid metals. 

These all show physical properties which can not be 

understood in terms of weakly interacting electron-like 

objects: 

Figure 1: The ground state of the free Fermi gas in momentum 

space. All the states below the Fermi surface 

are filled with both a spin-up and a spin-down electron. 

A particle-hole excitation is made by promoting 

an electron from a state below the Fermi surface to an 

empty one above it. 

A codimension one surface in k-space divides occupied 

2. Fermi-Liquid Theory: the electron quasiparticle 

from unoccupied states. Only the orthogonal direction 

The need for a Fermi-liquid theory dates from the 

first applications of quantum mechanics to the metallic 

state. There were two key problems. Classically each 

electron should contribute 3k B /2 to the specific heat 

capacity of a metal—far more than is actually seen experimentally. 

In addition, as soon as it was realized 

“scales” at a given point on the Fermi surface. It is like a 

surfaces worth of 1+1 dimensional CFTs. 

• Metals close to a quantum critical point. When a 

phase transition happens at temperatures close to 

the puzzle of the magnetic susceptibility which did not 

absolute zero, the quasiparticles scatter so strongly 

show the expected Curie temperature dependence for 

that they cease to behave in the way that Fermiliquid 

theory would predict. 

free moments: χ ∼ 1/T . 

These puzzles were unraveled at a stroke when 

• Metals in one dimension–the Luttinger liquid. In Pauli (Pauli 1927, Sommerfeld 1928) (apparently 

one dimensional metals, electrons are unstable and 

decay into two separate particles (spinons and 

holons) that carry the electron’s spin and charge 

reluctantly—see Hermann et al. 1979) adopted Fermi 

statistics for the electron and in particular enforced the 

exclusion principle which now carries his name: No two 

respectively. 

electrons can occupy the same quantum state. In the 


absence of interactions one finds the lowest energy state

For 

We start with a review of basic ideas and properties of entanglement entropy. 

Next we divide the total system into two subsystems A and B. 

✓ = d 1 

the thermodynamic functions of In the spin black chain 

example, we just artificially cut off the chain at some point and divide the lattice points 

brane match those of a theory with a Fermi surface. 

2.1 Definition of Entanglement Entropy 

Next we divide the total system into two subsystems A and B. 

into two groups. Notice that physically we do not do anything to the system and the 

cutting 

Consider 

procedure 

a quantum 

is an 

mechanical 

imaginary 

system 

process. 

with 

Accordingly 

many degrees 

the total 

of freedom 

Hilbert 

such 

space 

as 

can 

spin 

be 

chains. 

written 

More 

as 

generally, 

a direct product 

we can consider 

of two spaces 

arbitrary 

H tot 

lattice 

= H A 

models 

⊗ H B 

or 

corresponding 

QFTs including 

to 

CFTs. 

those of 

One other subsystems hallmark A and B. Theof observer fermions who is onlyis accessible an anomalous to the subsystem scaling A will feel asof 

One place where this shows up quantitatively is in the 

the entanglement entropy: entropy. 

We put the system at zero temperature and then the total quantum system is described 

by 

if 

the 

the 

pure 

total 

ground 

system 

state 

is described 

|Ψ〉. Weassumenodegeneracyofthegroundstate.Then,the 

by the reduced density matrix ρ A 

density matrix is that of the pure state ρ A =tr B ρ tot , (2.2) 

where the trace is taken only over the ρ tot 

Hilbert = |Ψ〉〈Ψ|. space H B . 

(2.1) 

Now we define the entanglement entropy of the subsystem A as the von Neumann 

The entropy von Neumann of the reduced entropy density of thematrix total system ρ A 

is clearly zero S tot = −tr ρ tot log ρ tot =0. 

S A = −tr A ρ A log ρ A . (2.3) 

This quantity provides us with a convenient 

4 

way to measure how closely entangled (or how 

“quantum”) a given wave function |Ψ〉 is. Notice also that in time-dependent backgrounds 

the density matrix ρ tot and ρ A are time dependent as dictated by the von Neumann 

equation. Thus we need to specify the time t = t 0 when we measure the entropy. In this 

paper, we always study static systems and we can neglect this issue. 

“quantum”) It is a given also possible wave function to define|Ψ〉 theis. entanglement Notice also entropy that in time-dependent S A (β) at finite temperature backgrounds 

T = β −1 . This can be done just by replacing (2.1) with the thermal one ρ thermal = e −βH , 

where H is the total Hamiltonian. When A is the total system, S A (β) is clearly the same 

as the thermal entropy. 

In the spin chain 

example, we just artificially cut off the chain at some point and divide the lattice points 

into two groups. Notice that physically we do not do anything to the system and the 

cutting procedure is an imaginary process. Accordingly the total Hilbert space can be 

written as a direct product of two spaces H tot = H A ⊗ H B corresponding to those of 

subsystems A and B. The observer who is only accessible to the subsystem A will feel as 

if the total system is described by the reduced density matrix ρ A 

where the trace is taken only over the Hilbert space H B . 

ρ A =tr B ρ tot , (2.2) 

Now we define the entanglement entropy of the subsystem A as the von Neumann 

entropy of the reduced density matrix ρ A 

S A = −tr A ρ A log ρ A . (2.3) 

This quantity provides us with a convenient way to measure how closely entangled (or how 

A 

the density matrix ρ tot and ρ A are time dependent as dictated by the von Neumann 

equation. Thus we need to specify the time t = t 0 when we measure the entropy. In this 


Tuesday, October 23, 12 

B

ce the entanglement between A and B occurs at t 

In general, one expects an “area law” for the entanglement 

result (2.8) was originally found from numerical com 

(with a UV-cutoff dependent coefficient). This represents 

ny later arguments (see e.g. recent works [27, 28, 29 

the entanglement of microscopic d.o.f. across the boundary. 

area law (2.8), however, does not always describe t 

One well-known violation of the area law occurs in 2D 

The Fermi liquid actually famously gives a logarithmic 

CFTs. There, the result is: 

violation of the “area law” which is universal. Eg in d=2: 

y in generic situations. As we will discuss in detai 

ent entropy of 1D quantum systems at criticality sca 

linear size l of A, S A ∼ c 3 

A 

------------------------ 

log l/a where c is the 

Holzhey, 

cen 

Wolf; Larsen, 

we can match the entanglement thermodynamic B entropy. behavior of a 

Cardy; 

Fermi 

Holzhey, 

Saturday, Tuesday, October November 23, 10, 12 12 

S L ⇠ k F L log(L)+ L ✏ + ··· 

6 

Gioev, KlichWilczek 

By setting 

This can roughly be understood from the fact that 1+1 

= d 1 

Huijse, Sachdev, 

Swingle 

dimensional CFTs have an anomalous log in their 

Larsen, Wilczek 

surface, in terms of T-scaling of entropy, heat capacity, etc.

he 

he 

n- 

er 

ot 

m 

an 

d, 

as 

]. 

ce 

er 

a 

en 

be 

ed 

er 

ly, 


w 

with 

papers [12, 13]. Define the entanglement entropy S A in 

a CFT on R 1,d (or R×S d ) for a subsystem A that has an 

arbitrary d − 1 dimensional boundary ∂A ∈ R d (or S d ). 

In this setup we 

entropy 

proposein the 

holography: 

following ‘area law’ 

There is a formula for computing the entanglement 


S A = Area of γ A 

4G (d+2) 

N 

Ryu, 

, (1.5) 

Takayanagi 

where γ A is the d dimensional static minimal surface in 

A 

AdS d+2 whose boundary is given by ∂A, and G (d+2) 

N 

is 

the d + extending 2 dimensional into Newton the bulk constant. of AdS Intuitively, space. this 

suggests that the minimal surface γ A plays the role of a 

holographic screen for an observer who is only accessible 

to the subsystem A. We show explicitly the relation (1.5) 

in the lowest dimensional case d = 1, where γ A is given 

by a geodesic line in AdS 3 . We also compute S A from the 

gravity side for general d and compare it with field theory 

results, which is successful at least qualitatively. From 

(1.5), it is readily seen that the basic properties of the 

entanglement entropy (i) S = S (B is the complement 

the minimal surface with appropriate boundary,

Calculating this for the 

✓ = d 1 

metric yields a matched 

anomalous entanglement entropy! 

Are the hyperscaling violating geometries 

theories with a Fermi surface at leading order? 

Ogawa, Takayanagi, 

Ugajin; Huijse, 

Sachdev, 

Swingle 

C. Instability of the dilatonic metrics 

In the theories we’ve discussed so far, the dilaton is running 

to extreme values close to the horizon: 

= K log(r) 


In electric (magnetic) black branes, this implies that there is 

weak (strong) coupling at the horizon: 

g = e 

⇥ magnetic: 

g $ 1 g 

Both are potentially problematic. 

Electric case: 

At weak coupling in string theory, higher-derivative 

terms are important: 

M string = gM Planck 

L = ⌅ g R + R 2 + ···⇥ 


S = d 4 x g R 2(⌥⇤) 2 e 2 ⇥ F 2 2 . (1.3) 

They are 

In these 

known 

theories, although 

to play 

the metric 

an important 

in the IR takes the 

role 

Lifshitz 


form 

asymptotically 

with z = 1+( 2 )2 

,the 

( 2 

) 

In these theories, although the metric in the IR takes the Lifshitz form with z = 1+( 2 )2 

,the 

2 

scalar dilaton is logarithmically running. Both electrically and magnetically ( charged black 

branes flat giveblack rise to such hole geometries: solutions in the former with thefew dilatoncharges. 

2 

) 2 

scalar dilaton is logarithmically running. Both electrically and magnetically charged 

runs towards weak 

Sen; black Dabholkar, 


branes give rise to such geometries: in the former Kallosh, Maloney 

at the horizon (in the sense that g ⇤ e ⇥ the dilaton runs towards weak coupling 

⇧ 0), while in the latter, the dilaton runs 

at the horizon (in the sense that g ⇤ e ⇥ ⇧ 0), while in the latter, the dilaton runs 

towards 

towards 

strong 

strong 

coupling. 

coupling. Related 

Related 

solutions 

solutions 

with 

with 

Lifshitz 

Lifshitz 

asymptotics 

asymptotics 

were 

were 

first 

first 

discussed 

discussed 



[13], [13], and and several other papers papersexploring exploringclosely closely related related solutions solutions have have subsequently subsequently appeared appeared 

[14, 15, 21, Magnetic 22]. case: 

[14, 15, 16, 17, 18, 19, 20, 21, 22]. 

It It was noted already in in [11, 12] 12] that thathe the running of the of the dilaton dilaton means means that that one cannot one cannot 

trust the Lifshitz form of the solutions to tothe the action action (1.3) (1.3) in the in the very very deep deep IR. InIR. theIncase 

case 

many similar cases such singularities are known to be harmless and physically admissible 

of of the magnetically charged branes, this this is because as as g grows, g grows, quantum quantum corrections corrections should should 

be expected to grow in importance – see §4.2 of [12]. For electrically charged branes, on the 

⇥ corrections (i.e., higher-derivative 

Now, For magnetic g flows to branes, strong in coupling some cases at the the horizon. “very near-horizon 

[10], it is an open question in this case what the correct interpretation of the singularities 

be expected Then, we 

can neglect 

fate” 

is. 1 to grow in importance – see §4.2 of [12]. For electrically charged branes, on the 

The 


results 

higher 

the 


emergent 

this note will not apply 

derivative 

Lifshitz 

to the solutions, 

terms, 

solution 

like those of 

but not 

(with 

[6], which 

g corrections 

running 

have exact 

other 

Lifshitz hand, 

scaling it would be 

symmetry. 

expected that thatin instring string theory theory 

⇥ corrections (i.e., higher-derivative 

terms) 

More would 

generally, become important. 

these dilaton) 

metrics also emerge 

is as 

naturally 

follows: 

in relativistic systems which are 

doped We 

We 

have 

have by finite already 

already charge seen 

seendensity. 

cases 

cases to 



string For string the 

instance, theory 

theory action. 

where asymptotically where 

⇥ corrections AdS extremal “resolve” black a horizon 

Harrison, SK, 

branes which 

Wang 


⇥ corrections “resolve” a horizon which 

is naively singular [23]. Here, we discuss an analogous phenomenon for black branes. Instead 

⇥ 

of corrections we will focus on the quantum corrections to the near-horizon geometry of 

the magnetically charged black branes in quantum-corrected versions of the theory (1.3). As 

a simplest toy model for these corrections in g, we will promote the gauge kinetic term in 

(1.3) 

e 2 ⇥ F 2 ⇧ f(⇤)F 2 (1.4) 

iswhose naively near-horizon singular [23]. geometry Here, we is of discuss the Lifshitz an analogous form were phenomenon found in [11, for 12] black by studying branes. the Instead 

⇥ 

ofsolutions corrections of the theory we will with focus action 

the quantum corrections to the near-horizon geometry of 

the magnetically charged black ⇤ branes 

S = d 4 x ⌃ in quantum-corrected versions 

g R 2(⌥⇤) 2 e 2 ⇥ F 2 2 ⇥ of the theory (1.3). As 

a simplest toy model for these corrections in g, we will promote . the gauge kinetic(1.3) 

term in 

(1.3) 

e 2 ⇥ F 2 ⇧ f(⇤)F 2 

In these theories, although the metric in the IR takes the Lifshitz form with z = 1+( (1.4) 

2 )2 

,the 

( 2 

) 

with 2 scalar the dilaton “gauge is coupling logarithmically function” running. f(⇤) Both taking electrically the formand magnetically charged black 

branes give rise to such geometries: in the former the dilaton runs towards weak coupling 

at the f(⇤) horizon =e 2 (in ⇥ + the ⇥ sense that g ⇤ e 1 + ⇥ 2 e 2 ⇥ + ⇥ 3 e 4 ⇥ + ⇧···= 0), while 1 in the latter, the dilaton runs 

towards strong coupling. Related solutions with Lifshitz g + ⇥ 2 1 + ⇥ 2 g 2 + ⇥ 3 g 4 + ··· (1.5) 

asymptotics were first discussed in 

[13], and several other papers exploring closely related solutions have subsequently appeared 

The new terms ⌅ ⇥ 

[14, 15, 16, 17, 18, 19, i in the gauge coupling function are meant to mock up the quantum 

20, 21, 22]. 

corrections which become important as the coupling constant grows near the horizon. We 

It was noted already in [11, 12] that the running of the dilaton means that one cannot 

with the “gauge coupling function” f(⇤) taking the form 

f(⇤) =e 2 ⇥ + ⇥ 1 + ⇥ 2 e 2 ⇥ + ⇥ 3 e 4 ⇥ + ···= 1 g 2 + ⇥ 1 + ⇥ 2 g 2 + ⇥ 3 g 4 + ··· (1.5) 

The new terms ⌅ ⇥ i in the gauge coupling function are meant to mock up the quantum

The corrections yield “attractor fixed points” for the 

dilaton in the near-horizon region: 

Ferrara, Kallosh, 

Strominger 

20 

15 

10 

5 

2 4 6 8 10 12 14 

Figure 1: Evolution of the dilaton from various initial conditions at infinity to a common fixed 

point at r =0. 

So the radial flows one finds: 


* Asymptote to 

AdS 4 

Harrison, 

SK, Wang 

* Exhibit Lifshitz-scaling with the expected “z” over many 

decades in energy 

* End up in AdS 2 R 2 (with much reduced entropy) 

0.75 

0.7 

a r⇥ 

10 4 b r⇥ 

0.65 

0.6 

1000 

0.55 

0.5 

100 

0.45 

10 8 10 10 10 12 10 14 r 

10 

10 8 10 10 10 12 10 14 r 

Figure 2: Here we see a log-log plot of the crossover from Lifshitz scaling to AdS 4 in the 

metric functions a(r) andb(r). The crossover occurs around r =10 11 . For r10 11 , the solution becomes 

AdS 4 . Left: a (r); right: b(r). The flow in a(r) just reflects the fact that the coe⌅cient of 

the linear term in a(r) ⇤ r is di erent in the Lifshitz and AdS 4 regions. The change in slope 

in the log-log plot for b(r) indicates the di erence between a solution with dynamical scaling 

(z = 5, for our choice of parameters) and the z = 1 characteristic of AdS 4 . 


The hyperscaling metrics can similarly be “IR completed” by 

AdS2 regions, in some cases. 

Bhattacharya, 

Cremonini, 

Sinkovics 

But as AdS2 itself is under suspicion of various instabilities, 

this leads to an Ouroborosian scenario... 

Some of the instabilities of AdS2 that have been discussed 

in the literature, break spatial translations. This 

provides a natural transition to our final topic... 

Donos, 

Gauntlett, 

Pantelidou 


IV. Homogeneous, anisotropic geometries 

All the phases we’ve discussed (and most discussed in the 

literature) have enjoyed normal translation symmetry. But 

in real systems, often the low T states break naive 

translations: 


As a starting point for a classification, we would like to 

classify the most general homogeneous, anisotropic 

extremal black brane geometries. 

Iizuka, SK, 

Kundu, Narayan, 

Sircar, Trivedi 

Homogeneous: For a theory in d spatial dimensions, there 

should be a d-dimensional group action which relates each 

point to its neighbors. 

That, is there should be d Killing vectors whose 

commutators give rise to a Lie algebra. Only the trivial 

algebra gives “normal” translations. 


sed Translations and The Bianchi Classification 

inning with a simple example illustrating the kind of generalised 

metry we would like to explore. Example: Suppose we are in three dimensional 

inning 

by coordinates 

with a simple 

x 1 ,x 2 ,x 

example 3 and suppose 

illustrating 

the system 

the kind 

of interest 

of generalised 

has the 

mmetry 

nal symmetries 

we would 

along 

like to 

the 

explore. 

x 2 ,x 3 directions. 

Suppose we 

It 

are 

also 


has 

three 

an 

dimensional 

additional 

nstead 

by coordinates 

of being 

x 

the 1 ,x 

usual 2 ,x 3 and 

translation 

suppose 


the 

the 

system 

x 1 direction 

of interest 

it is 

has 

now 

the 

a 

nal 

mpanied 

symmetries 

by a rotation 

along the 


x 

the 2 ,x 

x 3 2 directions. 

− x 3 plane. 

It 

The 

also 

translations 

has an additional 

along 

ion 

nstead 

are 


generated 

being the 

by 

usual 

the vectors 

translation 

fields, 

in the x 1 direction it is now a 

mpanied by a transverse rotation in theplane, x 2 − x 3 to plane. get a The symmetry. translations along 

ion are generatedξ 1 by= the ∂ 2 , vectors ξ 2 = ∂ 3 fields, , (2.1) 

Imagine in our three-dimensional space, we enjoy usual 

translation symmetries along two of the directions. But 

along the third, one must translate as well as rotating in the 

rd symmetry is generated ξ 1 = ∂ 2 , byξ 2 the = ∂vector 3 , field, 

(2.1) 

rd symmetry isξgenerated 3 = ∂ 1 + x 2 by ∂ 3 the − xvector 3 ∂ 2 . field, translations (2.2) 

ξ 3 = ∂ 1 + x 2 ∂ 3 − x 3 ∂ 2 . (2.2) 

–4– 

–4– 

vector fields which 

generate our generalised 

These generate a homogeneous space, in the sense that 

each point in an infinitesimal neighborhood can be 

transported to each other point. 


It is easy to see that the three transformations generated by these vectors transform 

that any requiring point in that the three this is dimensional true for the plane three to symmetry any other point generators in its immediate mentioned 

see 

bove, However, neighborhood. eq.(2.1), eq.(2.2), the Thiscommutation property, leads towhich the condition akin relations to that of φdefine usual is a constant translations, a Lie independent algebra is called 

fwhich the homogeneity. coordinates, is invariantly However, x 1 ,x 2 ,xthe 3 . distinct Now, symmetry suchfrom group a constant we the arealgebra configuration dealing with of here for translations: 

φis is clearly invariant different 

the from usual usual translation translations, alongsince x 1 , generated the commutators by ∂ 1 , of besides the generators, being invariant eq.(2.1,2.2) also 

nder 

nder take a rotation the formin the x 2 −x 3 plane (and the other rotations). Thus we see that with 

scalar order parameter [ξ a 1 situation , ξ 2 ]=0; where [ξ 1 , ξ 3 ]=ξ the 2 generalised ; [ξ 2 , ξ 3 ]=−ξ translations 1 , are unbroken (2.3) 

ight as well be thought of as one that preserves the usual three 

and do not vanish, as they would have for the usual translation 3 translations along 

. 

ith additional rotations. 

In this paper we will be interested in exploring such situations in more generality 

th In Now where additional fact, suppose these symmetries rotations. that generalised instead are different of being translations from a scalar the usual the order could translations parameter leave butinvariant specifying still preserve the a 

vector 

stem Now homogeneity, issuppose a 

order 

vectorallowing that V , 

parameter 

which instead anywe point denote of being (whose the by asystem V i scalar ∂ i . 

expectation 

Under ofthe interest the order transformation to parameter bevalue transformed “breaks” specifying generated to any 

y aother vector byfield a symmetry ξ i this transforms transformation as 4 . 

normal translations): 

a vector When field canξ i 

one this expect transforms such a symmetry as group to arise? Let us suppose that there 

is a scalar order parameter, φ(x), δV which = ɛ[ξspecifies i ,V]. the system of interest. Then (2.5) this 

order parameter must be invariant under the unbroken symmetries. The change of 

equiring this order thatparameter these commutators under an infinitesimal vanish for all transformation three ξ i ’s leads generated to theby conditions, the vector 

field ξ i is given by 

V 1 = constant, V 2 = V 0 

δφ 

cos(x 

= ɛξ 1 + 

i (φ) 

δ),V 3 = V 0 sin(x 1 + δ) 

(2.4) 

(2.6) 

d this would have to vanish for the transformation to be a symmetry. It is e 

see that requiring that this is true for the three symmetry generators mentio 

ove, eq.(2.1), eq.(2.2), leads to the condition that φ is a constant independ 

the coordinates, x 1 ,x 2 ,x 3 . Now, such a constant configuration for φ is invari 

der the usual translation along x 1 , generated by ∂ 1 , besides being invariant a 

der a rotation in the x 2 −x 3 plane (and the other rotations). Thus we see that w 

calar order parameter a situation where the generalised translations are unbro 

ght as well be thought of as one that preserves the usual three translations al 

stem is a vector V , which we denote by V i ∂ i . Under the transformation genera 

δV = ɛ[ξ i ,V]. (2 

quiring that these commutators vanish for all three ξ i ’s leads to the condition 

V 1 = constant, V 2 = V 0 cos(x 1 + δ),V 3 = V 0 sin(x 1 + δ) (2 

3 In fact this symmetry group is simply the group of symmetries of the two dimensional Euclidean 

and this would have to vanish for the transformation to be a symmetry. It is easy 

to see that requiring that this is true for the three symmetry generators mentioned 

3 In 4 Infact a connected this symmetry space group this follows is simply from thegroup requirement of symmetries that anyof point the two is transformed dimensionalto Euclid any 

above, eq.(2.1), eq.(2.2), leads to the condition that φ is a constant independent 

lane, its two translations and one rotation. 


Insulator 

Fig. 1. 

electron theory. 

k 

Or in a picture: 

Schematic of the phases of matter which can be described by extensions of the independent 

Fermi energy E F . The states with energy equal to E F define a (d 1)-dimensional 

‘Fermi surface’ in momentum space (in spatial dimension d), and the low energy 

excitations across the Fermi surface are responsible for the metallic conduction. 

When the Fermi energy lies in an energy gap, then the occupied states completely 

fill a set of bands, and there is an energy gap to all electronic excitations: this defines 

a band insulator, and the band-filling criterion requires that there be an even number 

of electrons per unit cell. Both the metal and the band insulator are states which are 

adiabatically connected to the states of free electrons illustrated in Fig. 1. Finally, 

in the Bardeen-Cooper-Schrie er theory, a superconductor is obtained when the 

Happily, for the application to phases in 3 spatial 

dimensions, all possible algebras of this sort have been 

classified. This is the Bianchi classification, also of use in 

theoretical cosmology. 


ostly we will work in 5 dimensions, this corresponds to setting d = 4 in eq.(1. 

addition, to keep the discussion simple, we assume that the usual translatio 

mmetry along The the basic timemathematical direction is preserved structure so that the is metric as follows: is time independ 

nd also assume that there are no off-diagonal components between t and the oth 

irections in the metric. This leads to 

* For each of the 9 inequivalent algebras, there are three 

“invariant one-forms” 

! i 

three isometries. 

left invariant under all 

ds 2 = dr 2 − a(r) 2 dt 2 + g ij dx i dx j (3 

ith the indices i, j taking values 1, 2, 3. For any fixed value of r, t, we get a th 

imensional subspace spanned by x i . We will take this subspace to be a homogeneo 

* A metric expressed in terms of these forms with constant 

ace, corresponding to one of the 9 types in the Bianchi classification. 

As discussed coefficients in [2], for each will of automatically the 9 cases therebe areinvariant. 

three invariant one form 

i ,i=1, 2, 3, which are invariant under all 3 isometries. A metric expressed in ter 

f these one-forms with x i independent coefficients will be automatically invaria 

nder the isometries. * The Forone-forms future reference satisfy we also the note relations: that these one-forms sati 

e relations 

dω i = 1 2 Ci jk ωj ∧ ω k (3 

here Cjk i are the structure constants of the group of isometries [2]. 

with C the structure constants of the algebra. 

We take the metric to have one additional isometry which corresponds to sc 

Saturday, variance. November 10, 12 An infinitesimal isometry of this type will shift the radial coordin

,i=1, 2, 3, which are invariant under all 3 isometries. A metric expressed in term 

these one-forms with x i independent coefficients will be automatically invaria 

der the Now, isometries. in holography For future reference we really we also have notean that extra these spatial one-forms satis 

e relations 

dimension. We also have time. Natural assumptions: 

dω i = 1 2 Ci jk ωj ∧ ω k (3. 

ere C r r + ,t e t⇥ t ; 

jk i are the structure constants of the group of isometries Maintain [2]. as 

t t + const . 

symmetries 

We take the metric to have one additional isometry which corresponds to sca 

variance. An infinitesimal isometry of this type will shift the radial coordina 

r → r + ɛ. In addition we will take it to rescale the time direction with weig 

, t →Then te −βtɛ the 

, andgeneral assume that 

“allowed” 

it acts on 

near-horizon 

the spatial coordinates 

metrics 

x i such 

take 

that th 

variant one forms, ω i transform with weights β 

the form: 

i under it, 5 ω i → e −βiɛ ω i . 

These properties fix the metric to be of the form 

ds 2 = R 2 [dr 2 − e 2β tr dt 2 + η ij e (β i+β j )r ω i ⊗ ω j ] (3. 

th η ij being a constant matrix which is independent of all coordinates. 

I.e. In given the previous the Bianchi section wetype, saw that there a situation are a where finite theset usual of translations constants we 

t preserved 

one 

butmust the generalised 

solve for 

translations 

to get the 

werescaling unbrokenmetric. 

requires a vector ( 

ssibly tensor) order parameter. A simple setting in a gravity setting which cou 

d to such a situation is to consider an Abelian gauge field coupled to gravity. W 


* Can find solutions in most Bianchi types in Einstein 

gravity + massive vector, by solving algebraic equations. 

* 7 of the types, as far as we can tell, are entirely new as 

symmetries of black-brane solutions. 

* Close analogues of type VII have been previously found as 

instabilities of other holographic phases. 

Domokos, Harvey; 

Nakamura, Ooguri, Park; 

Donos, Gauntlett 

* Gluing these solutions into AdS seems possible. We have 

done it for type VII (numerically). 


Generalization to 4-algebras: 

In fact, the Bianchi classification isn’t quite what we ordered 

for 5-geometries dual to 4d field theories. Even if we wish 

to study static geometries, we really have 4d spatial slices in 

the bulk. 

We should study real 4d Lie algebras with 3d subalgebras 

that give the symmetry preserved in the “field theory” 

spatial directions. 


These were classified ~60 years after Bianchi. 

There are 12 4-algebras and a myriad of possible 

embeddings of 3-algebras within. 

MacCallum; 

Patera, Winternitz 

In the following, we follow the notation of [26] in naming the algebras – we call them 

A 4,k with k =1, ··· , 12. We list only the non-vanishing structure constants (up to obvious 

permutation of indices). 

• A 4,1 : C24 1 =1,C34 2 =1 

e 1 = @ 1 e 2 = @ 2 e 3 = @ 3 e 4 = x 2 @ 1 + x 3 @ 2 + @ 4 

! 1 = dx 1 x 4 dx 2 + 1 2 x2 4dx 3 ! 2 = dx 2 x 4 dx 3 ! 3 = dx 3 ! 4 = dx 4 

e 1 = @ 1 e 2 = @ 2 

1 

2 3@ 1 e 3 = @ 3 + 1x 2 2@ 1 

e 4 =2x 1 @ 1 +(x 2 + x 3 )@ 2 + x 3 @ 3 + @ 4 

! 1 = e 2x4 (dx 1 + 1x 2 2dx 3 1x 2 3dx 2 ) ! 2 = e x4 (dx 2 x 4 dx 3 ) 

! 3 = e x4 dx 3 ! 4 = dx 4 

• A a 4,2: C14 1 = a, C24 2 =1,C34 2 =1,C34 3 =1 

e 1 = @ 1 e 2 = @ 2 e 3 = @ 3 e 4 = ax 1 @ 1 +(x 2 + x 3 )@ 2 + x 3 @ 3 

! 1 = e ax4 dx 1 ! 2 = e x4 (dx 2 x 4 dx 3 ) ! 3 = e ax4 dx 3 ! 4 = dx 4 

• A 4,8 : C23 1 =1,C24 2 =1,C34 3 = 1 

e 1 = @ 1 e 2 = @ 2 

1 

2 3@ 1 e 3 = @ 3 + 1x 2 2@ 1 

e 4 = x 2 @ 2 x 3 @ 3 + @ 4 

! 1 = dx 1 + 1x 2 2dx 3 1 2 dx3 dx 2 ! 2 = e x4 dx 2 ! 3 = e x4 dx 3 ! 4 = dx 4 

• A 4,3 : C 1 14 =1,C 2 34 =1 

e 1 = @ 1 e 2 = @ 2 e 3 = @ 3 e 4 = x 1 @ 1 + x 3 @ 2 + @ 4 

! 1 = e x4 dx 1 ! 2 = dx 2 x 4 dx 3 ! 3 = dx 3 ! 4 = dx 4 

• A 4,4 : C14 1 =1,C24 1 =1,C24 2 =1,C34 2 =1,C34 3 =1 

e 1 = @ 1 e 2 = @ 2 e 3 = @ 3 

e 4 =(x 1 + x 2 )@ 1 +(x 2 + x 3 )@ 2 + x 3 @ 3 + @ 4 

! 1 = e x4 (dx 1 x 4 dx 2 + 1 2 x2 4dx 3 ) ! 2 = e x4 (dx 2 x 4 dx 3 ) ! 3 = e x4 dx 3 ! 4 = dx 4 

• A a,b 

4,5: C14 1 =1, C24 2 = a, C34 3 = b 

e 1 = @ 1 e 2 = @ 2 e 3 = @ 3 e 4 = x 1 @ 1 + ax 2 @ 2 + bx 3 @ 3 + @ 4 

! 1 = e x4 dx 1 ! 2 = e ax4 dx 2 ! 3 = e bx4 dx 3 ! 4 = dx 4 

• A a,b 

4,6: C14 1 = a, C24 2 = b, C24 3 = 1 

e 1 = @ 1 e 2 = @ 2 e 3 = @ 3 

e 4 = ax 1 @ 1 +(bx 2 + x 3 )@ 2 +(bx 3 x 2 )@ 3 + @ 4 

! 1 = e ax4 dx 1 ! 2 = e bx4 [cos(x 4 )dx 2 sin(x 4 )dx 3 ] 

! 3 = e bx4 (cos(x 4 )dx 3 + sin(x 4 )dx 2 ) ! 4 = dx 4 

• A 4,7 : C 1 14 =2,C 2 24 =1,C 2 34 =1,C 3 34 =1,C 1 23 =1 

• A b 4,9: C23 1 =1,C14 1 =1+b, C24 2 =1,C34 3 = b 

e 1 = @ 1 e 2 = @ 2 

1 

2 3@ 1 e 3 = @ 3 + 1x 2 2@ 1 

e 4 =(1+b)x 1 @ 1 + x 2 @ 2 + bx 3 @ 3 + @ 4 

! 1 = e (b+1)x4 (dx 1 + 1x 2 2dx 3 1x 2 3dx 2 ) ! 2 = e x4 dx 2 ! 3 = e bx4 dx 3 ! 4 = dx 4 

• A 4,10 : C23 1 =1,C24 3 = 1, C34 2 =1 

e 1 = @ 1 e 2 = @ 2 

1 

2 3@ 1 e 3 = @ 3 + 1x 2 2@ 1 

e 4 = x 2 @ 3 + x 3 @ 2 + @ 4 

! 1 = dx 1 + 1x 2 2dx 3 1x 2 3dx 2 ! 2 = cos(x 4 )dx 2 sin(x 4 )dx 3 ! 3 = cos(x 4 )dx 3 + sin(x 4 )dx 2 

! 4 = dx 4 

• A a 4,11: C23 1 =1,C14 1 =2a, C24 2 = a, C24 3 = 1, C34 2 =1,C34 3 = a 

e 1 = @ 1 e 2 = @ 2 

1 

2 3@ 1 

e 3 = @ 3 + 1x 2 2@ 1 e 4 =2ax 1 @ 1 +(ax 2 + x 3 )@ 2 +(ax 3 x 2 )@ 3 + @ 4 

! 1 = e 2ax4 (dx 1 + 1x 2 2dx 3 1x 2 3dx 2 ) ! 2 = e ax4 (cos(x 4 )dx 2 sin(x 4 )dx 3 ) 

! 3 = e ax4 (cos(x 4 )dx 3 + sin(x 4 )dx 2 ) ! 4 = dx 4 

• A 4,12 : C13 1 =1,C23 2 =1,C14 2 = 

e 1 = @ 1 

1, C24 1 =1 

e 2 = @ 2 e 3 = @ 3 + x 1 @ 1 + x 2 @ 2 

e 4 = @ 4 + x 2 @ 1 x 1 @ 2 

! 1 = e x3 (cos(x 4 )dx 1 sin(x 4 )dx 2 ) ! 2 = e x3 (cos(x 4 )dx 2 + sin(x 4 )dx 1 ) 

! 3 = dx 3 ! 4 = dx 4 

2.2 Three-dimensional subalgebras 


4 

We would like the bulk geometry to reflect homogeneity of the spatial slices in the dual 

field theory. For this to happen, we wish to embed a three-dimensional real Lie algebra

What we can hope to accomplish in the near future: 

* Refine classification of homogeneous but possibly 

anisotropic solutions using 4-algebras. Show that new 

horizons can be supported by reasonable bulk matter and 

embedded into asymptotically AdS space. 

* Try to relate to symmetries of observed phases. 

* Construct/classify extremal inhomogeneous phases in 

Einstein gravity; hopefully some that are analytically 

tractable. 

c.f. Horowitz, 

Santos, Tong

New Horizons in Black Hole Physics and Holography - Institute of ...

Create successful ePaper yourself

Delete template?

Save as template?