Gil Young Cho - Institute of Condensed Matter Theory at the U of I

UIUC, Feb 5 th (2013)
Superconductivities of
doped Weyl semimetals
Phys. Rev. B. 86, 214514 (2012) - Editors’ suggestion
Gil Young Cho
UC Berkeley
Jens H Bardarson
Yuan-Ming Lu
Joel E Moore

Plan.
• Part 1 : superconductivities of doped WSM
– Motivations, Weyl semimetals?
– BCS or FFLO superconductivity?
– Topological Defects in FFLO SC-ivity
UIUC, Feb 5 th (2013)
• Part 2 : Proximate phases of Z2 spin liquid on
Kagome lattice (GYC, YML and AV, in preparation)
• Z2 spin liquid in Kagome lattice
• Proximate phases of Z2 spin liquid
Ashvin
Vishwanath

Part 1.
I. Superconductivity
of doped Weyl semimetals
Hedgehogs in momentum space

Weyl semimetals = 3D graphene?
• graphene? Dirac fermion in 2D
Honeycomb lattice of graphene
H = v D (σ x k x + σ y k y ) − μ Here σ
a
acts on sublattice basis
(A-B sublattice)

Weyl semimetals = 3D graphene?
graphene: H = v D (σ x k x + σ y k y ) − μ
Weyl SM: H = v D (σ x k x + σ y k y + σ z k z ) − μ
~ two-component Weyl fermion in HEP
In general, WSM is described by
H = v a ⋅ q σ a
a=1..3
with the chirality c = sgn ( v 1 ⋅ v 2 × v 3 )
Material: iridates, layered top. Ins., TlBi Se 1−x S x 2 , HgCr2Se3

What is interesting in WSM?
Ref. Wan et. al. (2011), Hosur et. al. (2012), Burkov et.al. (2011), Yang et.al. (2011)
A. “robust-ness” of Weyl nodes:
- no mass term for a single Weyl node: e.g H = v σ ⋅ k
- “pair annihilation” is the only way to gap out!
B. “flat” fermi arc surface state
C. three-dimensional anomalous Hall effect: σ xy ~ Δθ e2
D. interesting transport phenomena
E. Need to break time-reversal or/and inversion symmetry
h
Review article: Ari Turner, and Ashvin Vishwanath, arxiv: 1301.0330 (2013)

Where can we find
Weyl semimetal phase?
Topological insulator
+ “large enough” Time-reversal symmetry breaking
Material: Fe/Cr doped TlBi Se 1−x S x 2
Ref. Burkov et.al. (2011), Gil Young Cho (2011)

“Top. Ins.”-based Weyl semimetals
• Minimal model for Topological Insulators
H = v D τ z σ ⋅ k + τ x M
Ref. Gil Young Cho (2011)
e.g. near the Γ-point in Bi 2 Se 3
τ a : orbital degrees of freedom
σ a : spin degrees of freedom
• We add a time-reversal breaking term coming
from magnetic impurities (e.g. Fe or Cr) in bulk
H zeeman = mτ 0 σ z

“Top. Ins.”-based Weyl semimetals
H = v D τ z σ ⋅ k + τ x M + mσ z
• Spectrum for m > M
E
K Z ∗ = ± m 2 − M 2
away from the high sym. point
Winding # is -1
Γ
Winding # is +1
Kz
Ref. Gil Young Cho (2011)

“Top. Ins.”-based Weyl semimetals
Insulator
Insulator
M
m
a. Top. Ins. – Triv. Ins.
transition ≈ a fourcomponent
Dirac
fermion.
b. A four-component
Dirac fermion can
be “splitted” into a
pair of twocomponent
Weyl
fermions
Material: Fe- or Cr- doped TlBi Se 1−x S x 2
Ref. Gil Young Cho (2011)

Superconductivities of
doped Weyl semimetal?
Fermi surfaces due to doping
+
Attractive interactions
between the electrons

Why superconductivities?
GY Cho, J Bardarson, YM Lu, and JE Moore, Phys. Rev. B. 86, 214514 (2012)
I. Strong spin-orbit coupling
e.g. A doped topological Insulator can
become a topological superconductor
II. Topological Winding # around Weyl nodes
III. Disconnected Fermi pockets
for small doping

GY Cho, J Bardarson, YM Lu, and JE Moore, Phys. Rev. B. 86, 214514 (2012)
In this talk:
Study possible superconducting
states emerging from a doped
Weyl semimetal

Our approach is…
Following the spirit of Fu and Berg (2010)
A. Consider a “minimal” model for a Weyl
semimetal & dope the model slightly
B. Add a “phenomenological” attractive
interaction to the model
C. Classify possible superconductivities
D. Compare energies of SC states

Minimal Lattice model for Weyl semimetals
Ref. Yang et.al (2011)
A. The minimal model has two bands touching each other at two Weyl points
E
It can be modeled with “spin-ful”
electrons c ↑,↓ with spin-orbit coupling in
a cubic lattice :
Γ
Kz
c i,↑
c j,↓
site i
site j
H = −iλ SOI
{ c i,α ∗ σ x αβ c i+x,β + c i,α
∗
σ y αβ c i+y,β } + h. c
+M c i,α ∗ (σ z )c i,β − t {c i,↑ ∗ c j,↑ − c i,↓ ∗ c j,↓ }
- Pauli matrices σ a acts on the spin index ↑, ↓ of the electron c ↑/↓

Minimal Lattice model for Weyl semimetals
Ref. Yang et.al (2011)
Minimal model on cubic lattice (broken time-reversal symmetry)
H = H 1 k x , k y , M + H 2 (k z , m)
H 1 k x , k y , M = σ x sin k x + σ y sin k y + Mσ z (2 − cos k x − cos k y )
H 2 k z , m
= mσ z (cos k z − cos Q)
Important properties of the Hamiltonian
I. Two Weyl nodes at k z = ±Q and k x = k y = 0
II. no spin rotational symmetry ( spin~〈σ a 〉 )
(~ strong spin-orbit coupling)
III. symmetry = lattice symmetry C 4h
(~ pairing should be classified by C 4h )

Doping the minimal model
A. There are two “fermi pockets” around the nodes with
“spin-momentum locking”
e.g. near the upper node
H + = σ x k x + σ y k y + σ z (k z −Q) − μ
Note: winding around the node is ±1
B. To study a superconducting instabilities, we
add a “simplest” on-site attractive interaction
δH = −V 0
r
n ↑ (r)n ↓ (r)
e.g. phonon-mediated attractive interaction

Possible BCS superconducting states
A. The interaction is completely “local” and thus
the electrons are paired in “singlet” channel
(due to Pauli’s exclusion principle).
Pairing interaction is local in real space
= pairing gap is constant in momentum space
B. So the mean-field state should be “singlet” BCS pairing
δH = Δ ψ ∗ u k ψ∗ d
−k + h. c.
note: the total momentum carried by the pairing is zero;
so it should be “inter”-nodal pairing

Possible BCS superconducting states
BCS state: “inter-nodal” + “singlet” (trivial rep of the lattice sym.)
Note: To be paired in singlet channel, spins need to be anti-parallel
The nodes are “Weyl” nodes !

Possible FFLO superconducting states
A. There is another competing superconducting state
for the doped Weyl semimetal ~ FFLO pairing
(FFLO state: “intra-nodal” + “singlet” )
δH = Δ ± ψ ∗ u ±Q + k ψ∗ d
±Q − k + h. c.
note: the pairing “Δ ± ” carry
crystal momenta ±2Q
note: this FFLO is fully gapped
B. Why this state can be better than BCS state?
- the effective “DOS” participating to the BCS
pairing is reduced because the spin states at k
and −k are not anti-parallel.
- However, FFLO state connects anti-parallel
spins

Mean-field Energy of superconducting states

So..
FFLO state wins against BCS state
in the doped Weyl semimetal

Physical properties of
FFLO superconductivity in WSM
??

FFLO superconducting states
A. We have two independent pairings “Δ ± ” for each node
δH = Δ ± ψ ∗ u ±Q + k ψ∗ d
±Q − k + h. c.
which can be compactly written as a wave in real space
Δ r
= Δ cos(2Q ⋅ r + δφ)
a. so there should be “density modulation” in FFLO state
b. center of momentum is fixed by the positions of Weyl nodes
(in the iridates, we have 24 Weyl nodes)
B. FFLO state has a half-quantum vortex and a usual full quantum vortex
half-quantum vortex : a unit winding in only one of the two pairings
full quantum vortex: a composite of two half-quantum vortices
Any “exotic” interesting bound state to these vortices?

Lesson From superconducting doped Top. Ins.
Any interesting bound mode to vortex?
Ref. Fu and Berg (2010)
Example. Topological SC from a doped Topological Insulator CuBiSe
E
i) Fermi surface is two-fold
degenerate, related by T-
symmetry
kx
ii)
Each fermi surface encloses a
non-trivial winding # +1 or -1
iii) Vortex realizes “two zero modes”
related by T-symmetry, or “a
helical Majorana mode”
γ ↑,k = γ ∗ ↑,−k
, E = +vk
γ ↓,k = γ ∗ ↓,−k
, E = −vk
Vortex

Similarty & Difference from doped Top. Ins.
Similarities
i) Fermi surface encloses a winding #
ii)
Fully gapped (robust topological feature)
Differences
i) No time-reversal symmetry
= no helical state; at best chiral or gapped
states at vortex
ii)
Order parameter space = S 1 × S 1 in FFLO,
instead of S 1 in usual SC
= more topological defect types in FFLO
= half-quantum vortex in FFLO
e.g. spinful p+ip superconductor; we have
a direction d ∈ S 1 and a phase e iθ ∈ S 1
see Ivanov (2000)

Half-quantum vortex in FFLO superconducting states
A. Half-quantum vortex corresponds to
arg(Δ + ) → arg(Δ + ) + 2π
arg Δ − → arg(Δ − )
i.e., only the phase of Δ + winds once
B. The Fermi surface around the node carries
a unit “topological” winding number
C. Hence there is ‘a zero mode’ or a chiral Majorana
mode in the half-quantum vortex.
(i.e., γ k = γ ∗ , E = +vk)
−k
A quick way to see a chiral zero mode in the vortex:
One node of the doped Weyl semimetal is “half” of
the “topological superconductor”

Possible Majorana mode at a full vortex
A. Bring two half-quantum vortices with opposite chiral Majorana modes
Half-quantum vortices
B. In the vortex, we have a “helical” Majorana mode which is not protected
C. The helical Majorana mode will be gapped
D. There are two possibilities for the gapped phase;
weak pairing and strong pairing phases
ref. Kitaev (2000)
E. In the weak pairing phase, we have a bound Majorana zero mode
at the end of the vortex
A dangling
Majorana
mode
Full quantum vortex

Conclusion
I. We found FFLO state is competing with BCS
state (FFLO is energetically better than BCS
in our model)
II. FFLO state
I. Density modulation with the momenta fixed by
position of Weyl nodes
II.
Chiral Majorana mode at half-quantum vortex
III. Possible Majorana zero mode at full vortex

UIUC, Feb 5 th (2013)
Part 2.
II. Proximate phases
of Z 2 spin liquid on Kagome lattice
(Gil Young Cho, Yuan-Ming Lu and Ashvin Vishwanath, in preparation)
Yuan-Ming
Lu
Ashvin
Vishwanath

Z 2 spin liquid on Kagome lattice
• Heisenberg interaction on Kagome lattice :
Materials: dMIT, Herbertsmithite
DMRG studies found a Z2 spin liquid!
A. Gapped
B. Topological Entanglement Entropy
Ref. Yan Huse and White (2011)
Jiang Wang and Balents (2012)

Which Spin liquid ?
Ref. Hastings (2000)
Ran, Hermele, Lee, and Wen (2007)
Hermele, Ran, Lee, and Wen (2008)
• Many different spin liquids from the fermionic rep. of spin-1/2
The (relatively) low energy state
is, “U(1) Dirac spin liquid“
(among the fermionic SL ansatz)
This state features:
A. Dirac spectrum for fermions
B. U(1) Gauge theory
Not consistent with DMRG result

Is there a Z2 spin liquid near the U(1)
Dirac spin liquid?
Ref. Lu, Ran, and Lee (2011)
Requirement 1: gauge theory should be broken from U(1) to Z2
Pairing of Dirac fermions!
Requirement 2: Invariant under the lattice symmetry operation
+ Invariant under the spin rotation operation
Requirement 3: Pairing should gap out the Dirac fermion
There is one and only one such
pairing satisfying the requirements!
s-wave pairing of Dirac fermion

What can we tell about this Z2 state?
(Gil Young Cho, Yuan-Ming Lu and Ashvin Vishwanath, in preparation)
Our claim is:
If this Z2 spin liquid is the spin liquid found in DMRG study,
it should have very specific proximate phases separated
by a continuous transition from the spin liquid.
A. Q=0 non-collinear magnetic ordered state
B. VBS phase with the very specific bond-bond correlation

Proximate phases
Special 5-tuplets of masses of Dirac fermions
= WZW term for 5-tuplet masses
= Unconventional second order transitions
Ref. Wiegmann and Abanov (2000), Senthil and Fisher (2006), Grover and Senthil
(2008), Ryu, Mudry, Hou, and Chamon (2009), Herbut (2010) etc.
Underlying physics of WZW term:
• topological defect in one phase carries the quantum numbers
related to the other phase
• Condensation of the defect = destroying one ordering
+ inducing the other order

Q=0 non-collinear magnetic order
and Z2 spin liquid
Monopole quantum #s of U(1) SL
Vison∼ spin-1/2
Q = 0 state

Q=0 non-collinear magnetic orders
n r =
V=

Result :
Sachdev (1992)
=
Continuous transition
?!

Nature of the VBS phase
Order parameters for VBS pattern N are at M-points in BZ
• At least two of M-points should participate
• All the lattice symmetries (except translational symmetries) are broken
• 12-site unit cell (“quadrupled” unit cell)
e.g. Diamond pattern found in DMRG has 12-site unit cell patterns
Two Dirac nodes
BZ
ky
M- points
kx

Ex. Bond ordering patterns at M-points
BZ

Ex. Bond ordering patterns at M-points
BZ

Conclusion
I. We have studied the proximate symmetry
broken phases of a particular Z2 spin liquid
II. Q=0 non-collinear magnetic ordered state
- This allows us to identify the fermionic Z2 spin liquid as
the bosonic Q1=Q2 spin liquid
III. VBS state
- superposition of VBS patterns at M-points
- 12-site unit cell with broken lattice symmetries
IV. More direct probe?

Thanks!
My previous research:
(1) Topological BF theory description of topological Insulator (2011)
(2) Quantum Phase transition and fractionalization in a topological insulator thin
film with Zeeman and excitonic masses (2011)
(3) Weyl semimetal in magnetically doped topological insulator (2011)
(4) Dyon condensation in topological Mott insulator (2012)
(5) Gapless edge state of BF field theory and Z2 spin liquids, (2012)
(6) Superconductivities of doped Weyl semimetals, (2012)
(7) Two dimensional symmetry protected phases with PSU(N) and time reversal
symmetry, (2012)
(8) Proximate phases of Z2 spin liquid on Kagome lattice, in preparetion
My collaborators:
C. Xu
(UCSB)
J. E. Moore
(UCB)
Y.B. Kim
(U Toronto)
Y.-M. Lu
(UCB)
A. Vishwanath
(UCB)
J.H Bardarson
(UCB)