Condensed-Matter Physics: Inscrutably alluring to ... - Tarun Chitra

What is Condensed-Matter Physics?Condensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraCondensed-Matter Physics is the study of matter that is in thecondensed phase — Matter which is not gaseous.

What is Condensed-Matter Physics?Condensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraCondensed-Matter Physics is the study of matter that is in thecondensed phase — Matter which is not gaseous.One can define a gaseeous phase in many ways, with a simpleway being defined by the container that a gas is kept in. A gascan theoretically expand to an arbitrarily large volume.

QM, for the winCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraWith the advent of Quantum Mechanics in the 1920s, theoristsbegan to tackle the physics of solids with their newfangled toolsTheory of solids was ’finalized’ by physicists such as Felix Blochand Léon Brillouin. Enrico Fermi made important contributionto the theory of Fermionic gasesLiquid state, other phases of nature left out from this early-20thcentury boom in solid-state physics1978: APS changes the name of ’Solid-State Physics’ to’Condensed-Matter Physics’

Active Areas of Research in Condensed-Matter(C-M)Condensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun Chitra

Active Areas of Research in Condensed-Matter(C-M)Condensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraSuperfluidity and SupersoliditySuperconductivity

Active Areas of Research in Condensed-Matter(C-M)Condensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraSuperfluidity and SupersoliditySuperconductivityProtein PhysicsSupercooled Liquids

So. . . How does one formulate problems in C-M?Condensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraCondensed-Matter Physics involves the study of systems withlots of particles. Forget the three-body problem — consider the10 22 -body problem

So. . . How does one formulate problems in C-M?Condensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraCondensed-Matter Physics involves the study of systems withlots of particles. Forget the three-body problem — consider the10 22 -body problemSolid-state systems: Mathematically ’easy’ because solids have adefinite, static atom density that they can be approximated by alattice. Lattice discretizes phase space and real space, enforcesperiodicity conditions.

So. . . How does one formulate problems in C-M?Condensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraCondensed-Matter Physics involves the study of systems withlots of particles. Forget the three-body problem — consider the10 22 -body problemSolid-state systems: Mathematically ’easy’ because solids have adefinite, static atom density that they can be approximated by alattice. Lattice discretizes phase space and real space, enforcesperiodicity conditions.The liquid phase and various non-classical phases tend to bemuch more difficult to study – cannot force a lattice structureon a liquid and various non-classical phases need not behomogeneous.

All hope is not lost!Condensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraEven without a nice atomic-level description of all liquids, scientistshave been fairly adept at studying liquids macroscopic since the timeof the Bernoullis. Two key ingredients to studying liquids:

All hope is not lost!Condensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraEven without a nice atomic-level description of all liquids, scientistshave been fairly adept at studying liquids macroscopic since the timeof the Bernoullis. Two key ingredients to studying liquids:1 Approximate liquids as a ’limiting’ phase of solids and/or gases.Classical fluid dynamics: a liquid is a solid that ’deforms’ easilyEarly QM: Fermi liquids are ideal Fermi gases (e.g n-bodysystems with a non-interacting, SO(3)-invariant potentials)which are perturbed by an attractive interacting potential.

All hope is not lost!Condensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraEven without a nice atomic-level description of all liquids, scientistshave been fairly adept at studying liquids macroscopic since the timeof the Bernoullis. Two key ingredients to studying liquids:1 Approximate liquids as a ’limiting’ phase of solids and/or gases.Classical fluid dynamics: a liquid is a solid that ’deforms’ easilyEarly QM: Fermi liquids are ideal Fermi gases (e.g n-bodysystems with a non-interacting, SO(3)-invariant potentials)which are perturbed by an attractive interacting potential.2 Elucidate the symmetry-breaking that takes place in a liquid. Thetheory of solids is particularly nice because the system has quite a bitof symmetry; if we can figure out which symmetries of a solid stillapply to liquids, then we can ’naturally’ write down a Lagrangian fora liquid system

Basic Mathematical Framework: LatticesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansLet’s become a lot more formal with the definitions of the pastfew slides. 1Tarun Chitra1 There are far more general definitions of a lattice such as:’Any discrete subgroup Γ of a Lie Group G such that G/Γ has finite Haar measure’

Basic Mathematical Framework: LatticesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraLet’s become a lot more formal with the definitions of the pastfew slides. 1DefinitionAn lattice L in R n is a set of Z-translates of a basis {⃗e i } n i=1 . That is,L = {∑ ni=1 a ∣i⃗e i ai ∈ Z}. A cell is any Z n -translate of theparallelpiped spanned by {⃗e i } n i=1 (the unit cell)1 There are far more general definitions of a lattice such as:’Any discrete subgroup Γ of a Lie Group G such that G/Γ has finite Haar measure’

Basic Mathematical Framework: LatticesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraLet’s become a lot more formal with the definitions of the pastfew slides. 1DefinitionAn lattice L in R n is a set of Z-translates of a basis {⃗e i } n i=1 . That is,L = {∑ ni=1 a ∣i⃗e i ai ∈ Z}. A cell is any Z n -translate of theparallelpiped spanned by {⃗e i } n i=1 (the unit cell)In solid-state physics, we assume that virtually all systems canlocally be described by a lattice. By construction, a latticeadmits a free, transitive Z n -action which means that one needonly study physics in the unit cell Span(⃗e 1 , . . . ,⃗e n ) and then usethe Z n -action to translate the dynamics to any other cell. Thisis analogous to the setup for the NT algorithm that Anton relieson.1 There are far more general definitions of a lattice such as:’Any discrete subgroup Γ of a Lie Group G such that G/Γ has finite Haar measure’

Basic Mathematical Framework: SymmetriesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun Chitra

Basic Mathematical Framework: SymmetriesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansSo what is a symmetry? To answer that we need to recall a bitof classical mechanicsTarun Chitra

Basic Mathematical Framework: SymmetriesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraSo what is a symmetry? To answer that we need to recall a bitof classical mechanicsLet’s recall that informal statement of Hamilton’s Principle ofLeast Action: ’Particles are lazy, they want to take the shortestpossible path while expending the least energy’

Basic Mathematical Framework: SymmetriesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraSo what is a symmetry? To answer that we need to recall a bitof classical mechanicsLet’s recall that informal statement of Hamilton’s Principle ofLeast Action: ’Particles are lazy, they want to take the shortestpossible path while expending the least energy’How do we formalize this? One way is to use a Lagrangian.

Basic Mathematical Framework: SymmetriesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraSo what is a symmetry? To answer that we need to recall a bitof classical mechanicsLet’s recall that informal statement of Hamilton’s Principle ofLeast Action: ’Particles are lazy, they want to take the shortestpossible path while expending the least energy’How do we formalize this? One way is to use a Lagrangian.Physically, a Lagrangian L is fairly simple to define:L = ∑ {Kinetic Energy Terms} − ∑ {Potential Energy Terms}

Basic Mathematical Framework: SymmetriesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraSo what is a symmetry? To answer that we need to recall a bitof classical mechanicsLet’s recall that informal statement of Hamilton’s Principle ofLeast Action: ’Particles are lazy, they want to take the shortestpossible path while expending the least energy’How do we formalize this? One way is to use a Lagrangian.Physically, a Lagrangian L is fairly simple to define:L = ∑ {Kinetic Energy Terms} − ∑ {Potential Energy Terms}We need to define the arguments of a Lagrangian: Particle’spath x(t), velocity ẋ(t) and possibly an explicit dependence ontime t and further spatial and time derivatives

Basic Mathematical Framework: SymmetriesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraSo what is a symmetry? To answer that we need to recall a bitof classical mechanicsLet’s recall that informal statement of Hamilton’s Principle ofLeast Action: ’Particles are lazy, they want to take the shortestpossible path while expending the least energy’How do we formalize this? One way is to use a Lagrangian.Physically, a Lagrangian L is fairly simple to define:L = ∑ {Kinetic Energy Terms} − ∑ {Potential Energy Terms}We need to define the arguments of a Lagrangian: Particle’spath x(t), velocity ẋ(t) and possibly an explicit dependence ontime t and further spatial and time derivativesFormal definition: A Lagrangian is a C ∞ map between ann-manifold M and a target manifold T (usually R, C, H or C 2n orthe cotangent bundle of any symplectic manifold)

’Physicists can’t get enough action’ — TBKCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun Chitra

’Physicists can’t get enough action’ — TBKCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraSo now we need some way to define ’the shortest possible paththat costs the least energy’ in a way that is manifestlycompatible with Newton’s Laws

’Physicists can’t get enough action’ — TBKCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraSo now we need some way to define ’the shortest possible paththat costs the least energy’ in a way that is manifestlycompatible with Newton’s LawsIntroducing the action S. Mathematically, the action is definedas S = ∫ R L(x(t), ∂ αx(t), ∂ α ∂ α x(t), . . . , t)dt

’Physicists can’t get enough action’ — TBKCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraSo now we need some way to define ’the shortest possible paththat costs the least energy’ in a way that is manifestlycompatible with Newton’s LawsIntroducing the action S. Mathematically, the action is definedas S = ∫ R L(x(t), ∂ αx(t), ∂ α ∂ α x(t), . . . , t)dtHamilton and Lagrange showed (in different ways) that byminimizing the Lagrangian with respect to its arguments, onecould obtain a particle’s path x(t) and velocity ẋ(t) that isconsistent with what one would expect from Newtonianmechanics. This makes life a lot easier, because for a lot ofproblems it is much easier to ’guess’ kinetic and potentialenergies as opposed to computing all forces

Actions and SymmetryCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun Chitra

Actions and SymmetryCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansAction: S[x(t), ẋ(t), t].Tarun Chitra

Actions and SymmetryCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraAction: S[x(t), ẋ(t), t].Since x(t), ẋ(t) ∈ R 3 , any group that acts freely/transitively on R 3can act on the trajectories (pointwise). Representation of G:Λ : G → Aut(R 3 ). G generates a symmetry of S if:S[x(t), ẋ(t), t] = S[Λ(g)x(t), Λ(g)ẋ(t), t]∀g ∈ G

Actions and SymmetryCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraAction: S[x(t), ẋ(t), t].Since x(t), ẋ(t) ∈ R 3 , any group that acts freely/transitively on R 3can act on the trajectories (pointwise). Representation of G:Λ : G → Aut(R 3 ). G generates a symmetry of S if:S[x(t), ẋ(t), t] = S[Λ(g)x(t), Λ(g)ẋ(t), t]∀g ∈ GMinimizing the action stems from a somewhat empirical consideration— in many classical physical systems, the time expectations ofkinetic and potential energy were approximately equal. That is,E t [T (ẋ)] ≈ E t [V (x, ẋ)]

Noether’s Theorem (Physics Version)Condensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun Chitra

Noether’s Theorem (Physics Version)Condensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraWhat about conservation laws? How do we force Lagrangianmechanics (as vaguely defined) to have conservation laws

Noether’s Theorem (Physics Version)Condensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraWhat about conservation laws? How do we force Lagrangianmechanics (as vaguely defined) to have conservation lawsNoether’s Theorem (Emmy Noether, 1918): ’For everysymmetry of an action, there exists a conserved quantityassociated to the symmetry.’ In classical mechanics this quantityis the conserved momentum:P(x, ẋ) = ∑ i∂L∂ẋ iK i (x))where K i (q) is the linear ’component’ (think Taylor’s theorem)of the symmetry (e.g. Λx = x + ɛK(x))

ExampleCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraConsider (ẋ the Lagrangian for throwing a ball up in the air,L = m 2 2+ ẏ 2 + ż 2) − mgz. This is invariant under translations ofx, y, e.g. x ↦→ x + ɛ, y ↦→ y + ɛ ′ . These two symmetries generate twodifferent conserved momenta (one with K x = 1, K y = K z = 0 and theother with K y = 1:

ExampleCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraConsider (ẋ the Lagrangian for throwing a ball up in the air,L = m 2 2+ ẏ 2 + ż 2) − mgz. This is invariant under translations ofx, y, e.g. x ↦→ x + ɛ, y ↦→ y + ɛ ′ . These two symmetries generate twodifferent conserved momenta (one with K x = 1, K y = K z = 0 and theother with K y = 1:P 1 (x, y, z, ẋ, ẏ, ż) = ∂L∂ẋ K x + ∂L∂ẏ K y + ∂L∂ż K z = mẋ (1)P 2 (x, y, z, ẋ, ẏ, ż) = ∂L∂ẋ K x + ∂L∂ẏ K y + ∂L∂ż K z = mẏ (2)

Noether’s Theorem (Math Version)Condensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun Chitra

Noether’s Theorem (Math Version)Condensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraHamiltonian System (M, ω, H): Symplectic manifold (M, ω)with Hamiltonian H ∈ C ∞ (M). Every Hamiltonian H gives riseto a Hamiltonian Vector Field, X H = ω −1 (dH).

Noether’s Theorem (Math Version)Condensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraHamiltonian System (M, ω, H): Symplectic manifold (M, ω)with Hamiltonian H ∈ C ∞ (M). Every Hamiltonian H gives riseto a Hamiltonian Vector Field, X H = ω −1 (dH).Infinitesmal Symmetry: Smooth vector field V ∈ Γ(M, TM)with flow θ : D → M such that L XH ω = ω, θ ∗ H = H.

Noether’s Theorem (Math Version)Condensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraHamiltonian System (M, ω, H): Symplectic manifold (M, ω)with Hamiltonian H ∈ C ∞ (M). Every Hamiltonian H gives riseto a Hamiltonian Vector Field, X H = ω −1 (dH).Infinitesmal Symmetry: Smooth vector field V ∈ Γ(M, TM)with flow θ : D → M such that L XH ω = ω, θ ∗ H = H.Conserved Quantity: f ∈ C ∞ (M) such that f is constantalong any integral curve (read: trajectory) of X H . Also definedas f ∈ C ∞ (M) ∋ {H, f } = 0.

Noether’s Theorem (Math Version)Condensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraHamiltonian System (M, ω, H): Symplectic manifold (M, ω)with Hamiltonian H ∈ C ∞ (M). Every Hamiltonian H gives riseto a Hamiltonian Vector Field, X H = ω −1 (dH).Infinitesmal Symmetry: Smooth vector field V ∈ Γ(M, TM)with flow θ : D → M such that L XH ω = ω, θ ∗ H = H.Conserved Quantity: f ∈ C ∞ (M) such that f is constantalong any integral curve (read: trajectory) of X H . Also definedas f ∈ C ∞ (M) ∋ {H, f } = 0.Theorem(Noether) If f ∈ C ∞ (M) is a conserved quantity, then itsHamiltonian vector field X f is an infinitesmal symmetry. Conversely ifH 1 (M; R) = 0, then every infinitesmal symmetry is the Hamiltonianvector field of a conserved quantity.

Symmetries of LatticesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun Chitra

Symmetries of LatticesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraSet of isometries (distance-preserving) maps = physical intuitionthat rotation/translation of coordinate system shouldn’t affectthe physics. Distance-preservation is req’d to ensure that onedoesn’t change an energy functional, as most energy functionalsrequire a metric.

Symmetries of LatticesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraSet of isometries (distance-preserving) maps = physical intuitionthat rotation/translation of coordinate system shouldn’t affectthe physics. Distance-preservation is req’d to ensure that onedoesn’t change an energy functional, as most energy functionalsrequire a metric.What are the isometries of a cubic unit cell? Hint: It’s not adihedral group, but it decomposes similarly (has a subgroup ofreflections ∼ = Z 2 )

Symmetries of LatticesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraSet of isometries (distance-preserving) maps = physical intuitionthat rotation/translation of coordinate system shouldn’t affectthe physics. Distance-preservation is req’d to ensure that onedoesn’t change an energy functional, as most energy functionalsrequire a metric.What are the isometries of a cubic unit cell? Hint: It’s not adihedral group, but it decomposes similarly (has a subgroup ofreflections ∼ = Z 2 )Proper Rotations (24): Identity rotations about an axis fromcenter to face, rotation about an axis from the center of anedge, to an opposite edge, rotation about a body diagonal

Symmetries of LatticesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraSet of isometries (distance-preserving) maps = physical intuitionthat rotation/translation of coordinate system shouldn’t affectthe physics. Distance-preservation is req’d to ensure that onedoesn’t change an energy functional, as most energy functionalsrequire a metric.What are the isometries of a cubic unit cell? Hint: It’s not adihedral group, but it decomposes similarly (has a subgroup ofreflections ∼ = Z 2 )Proper Rotations (24): Identity rotations about an axis fromcenter to face, rotation about an axis from the center of anedge, to an opposite edge, rotation about a body diagonalRotation Plus a Reflection (24)

Symmetries of LatticesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraSet of isometries (distance-preserving) maps = physical intuitionthat rotation/translation of coordinate system shouldn’t affectthe physics. Distance-preservation is req’d to ensure that onedoesn’t change an energy functional, as most energy functionalsrequire a metric.What are the isometries of a cubic unit cell? Hint: It’s not adihedral group, but it decomposes similarly (has a subgroup ofreflections ∼ = Z 2 )Proper Rotations (24): Identity rotations about an axis fromcenter to face, rotation about an axis from the center of anedge, to an opposite edge, rotation about a body diagonalRotation Plus a Reflection (24)Math: Isometry Group of Cube: BC 3∼ = S4 ⋊ Z 2 is the 3rdCoxeter Group, with Coxeter-Dynkin Diagram, ❡ 4

Other types of latticesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansBut what if they’re not orthonormal? Bieberbach proved whatphysicists already knew: There are only a few types of crystalstructures:Tarun Chitra

Other types of latticesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansBut what if they’re not orthonormal? Bieberbach proved whatphysicists already knew: There are only a few types of crystalstructures:Tarun Chitra

Symmetry-breaking and LatticesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun Chitra

Symmetry-breaking and LatticesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraIntuitively a lattice is ’different’ from an arbitrary particlebecause it is ’rigid’. How can symmetries help us quantify this?

Symmetry-breaking and LatticesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraIntuitively a lattice is ’different’ from an arbitrary particlebecause it is ’rigid’. How can symmetries help us quantify this?First things first: What are the set of isometries of a single,thermally-isolated, classical particle that is sitting in a vacuumR n ?Rotations — Represented in n-dimensions by the Lie groupSO(n)Translations — represented by R n , treated as Abelian groupunder addition

Symmetry-breaking and LatticesCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraIntuitively a lattice is ’different’ from an arbitrary particlebecause it is ’rigid’. How can symmetries help us quantify this?First things first: What are the set of isometries of a single,thermally-isolated, classical particle that is sitting in a vacuumR n ?Rotations — Represented in n-dimensions by the Lie groupSO(n)Translations — represented by R n , treated as Abelian groupunder additionThese are differentiable symmetries, as the sets of rotations andtranslations turn out to be Lie groups and hence smoothmanifolds (so that differentiation can be defined).

The Poincaré GroupCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun Chitra

The Poincaré GroupCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraThis total set of symmetries of a free particle is known as thePoincaré Group P(R n ). However, note that P(R 3 ) ≁ = SO(3) × R3and in general, P(R n ) ≁ = SO(n) × R n for n > 1. To see whyconsider two rotations and translations(R, y), (˜R, ỹ), R, ˜R ∈ SO(3), y, ỹ ∈ R 3 . A rotation plustranslation acts on a vector x ∈ R 3 as (R, y) · x = Rx + y.Hence we have:(R, y) ∗ P (˜R, ỹ) · x = (R, y) · (˜Rx + ỹ)= R ˜Rx + Rỹ + y = (R ˜R, Rỹ + y) · x

The Poincaré GroupCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraThis total set of symmetries of a free particle is known as thePoincaré Group P(R n ). However, note that P(R 3 ) ≁ = SO(3) × R3and in general, P(R n ) ≁ = SO(n) × R n for n > 1. To see whyconsider two rotations and translations(R, y), (˜R, ỹ), R, ˜R ∈ SO(3), y, ỹ ∈ R 3 . A rotation plustranslation acts on a vector x ∈ R 3 as (R, y) · x = Rx + y.Hence we have:(R, y) ∗ P (˜R, ỹ) · x = (R, y) · (˜Rx + ỹ)= R ˜Rx + Rỹ + y = (R ˜R, Rỹ + y) · xHence we have P(R 3 ) ∼ = R 3 ⋊ ϕ SO(3) where ⋊ ϕ is thesemi-direct product with respect to ϕ : SO(3) → Aut(R 3 )defined as ϕ(R)(y) = Ry

Symmetry-BreakingCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun Chitra

Symmetry-BreakingCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraNow we can define symmetry-breaking: When dynamics force aphysical system to have a smaller group (subgroup) of the staticsymmetries.

Symmetry-BreakingCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraNow we can define symmetry-breaking: When dynamics force aphysical system to have a smaller group (subgroup) of the staticsymmetries.Physical system on Lattice: We can think of our system asembedded in R 3 , but with a much smaller symmetry group (e.g.BC 3 ⋊ ϕ Z n )

Symmetry-BreakingCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraNow we can define symmetry-breaking: When dynamics force aphysical system to have a smaller group (subgroup) of the staticsymmetries.Physical system on Lattice: We can think of our system asembedded in R 3 , but with a much smaller symmetry group (e.g.BC 3 ⋊ ϕ Z n )Practical Implementation:Start with a Lagrangian with a lot of symmetry L s and solve forthe trajectory/dynamicsPerturb L s with a non-symmetric Lagrangian, L p by trying tocompute the dynamics of L = L s + ɛL p, ɛ > 0, ɛ ≪ 1.Likely: You will be unable to solve the Euler-Lagrange equationsfor this Lagrangian. Using Taylor’s Theorem repeatedly (and Imean repeatedly) will make the new approximate solution for Ldepend heavily on the solution for L s

Example of Symmetry-Breaking: SuperfluidityCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun Chitra

Example of Symmetry-Breaking: SuperfluidityCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansPhysics: Liquid 3 He has no viscosity (or creep) at T < 3mK. Assuch, there exist at least three phases of matter of 3 He. This isa quantum phenomena, so we need to consider spin.Tarun Chitra

Example of Symmetry-Breaking: SuperfluidityCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraPhysics: Liquid 3 He has no viscosity (or creep) at T < 3mK. Assuch, there exist at least three phases of matter of 3 He. This isa quantum phenomena, so we need to consider spin.Symmetry: Local SO(3) × SO(3) × U(1) symmetry [Shouldremind you of the Standard Model, SU(3) × SU(2) × U(1)!!]

Example of Symmetry-Breaking: SuperfluidityCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraPhysics: Liquid 3 He has no viscosity (or creep) at T < 3mK. Assuch, there exist at least three phases of matter of 3 He. This isa quantum phenomena, so we need to consider spin.Symmetry: Local SO(3) × SO(3) × U(1) symmetry [Shouldremind you of the Standard Model, SU(3) × SU(2) × U(1)!!]Lagrangian density for Free Energy (Lagrangian of Superfluid3 He, Phys. Rev. B, 37(7), S. Theodorakis):L = k 1 [Ω ∗ µlΩ µl ] + ik 2 [Ω ∗ µlA µl − Ω µl A ∗ µl] + k 3 H µ ɛ µνσ A vi Ω ∗ σir+ k 3 H µ ɛ µνσ A ∗ νiΩ σi − F d − F mag + g S22

Example of Symmetry-Breaking: SuperfluidityCondensed-Matter Physics:Inscrutablyalluring toPhysicists,Chemists andevenMathematiciansTarun ChitraPhysics: Liquid 3 He has no viscosity (or creep) at T < 3mK. Assuch, there exist at least three phases of matter of 3 He. This isa quantum phenomena, so we need to consider spin.Symmetry: Local SO(3) × SO(3) × U(1) symmetry [Shouldremind you of the Standard Model, SU(3) × SU(2) × U(1)!!]Lagrangian density for Free Energy (Lagrangian of Superfluid3 He, Phys. Rev. B, 37(7), S. Theodorakis):rL = k 1 [Ω ∗ µlΩ µl ] + ik 2 [Ω ∗ µlA µl − Ω µl A ∗ µl] + k 3 H µ ɛ µνσ A vi Ω ∗ σi+ k 3 H µ ɛ µνσ A ∗ νiΩ σi − F d − F mag + g S22Symmetry-breaking: Contributions from A, Ω are invariant underSO(3) × SO(3) × U(1); F d , F mag are not. They contribute to anenergetically unfavorable liquid state at low T