Spontaneous Symmetry Breaking And Goldstone Modes

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Spontaneous Symmetry BreakingImplications of SSBConclusionSpontaneous Symmetry Breaking andGoldstone ModesSören PetratMay 26, 2009Sören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionOutline1Spontaneous Symmetry BreakingSymmetries in PhysicsThe phenomenon of SSB2Implications of SSBGoldstone ModesGoldstone TheoremMermin-Wagner Theorem3ConclusionSören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionSymmetries in PhysicsThe phenomenon of SSBWhat is a symmetry in Physics?Invariance of a physical law under transformationsSymmetries can arise from physics . . .e.g. Galilei-invariance in Newtonian Mechanics. . . or from mathematicse.g. gauge-invariance in Electrodynamics: fAµ0 Aµ xµSymmetries form a groupSören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionSymmetries in PhysicsThe phenomenon of SSBDifferent kinds of symmetrieswe differentiate between:global symm.: acts simultaneously on all variableslocal symm.: acts independently on each variablefurthermore:continuous symm., e.g. rotations (SO(n))discrete symm., e.g. spin group (Z2 )Lie group: differentiable manifold that is also a grouprespecting the continuum properties of the manifoldSören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionSymmetries in PhysicsThe phenomenon of SSBThe concept of SSBObservation: our world is (mostly) not symmetric! General concept in modern physicsoriginal law is symmetricbut the solutions are not!i.e. the symmetry is broken (by some mechanism realizedin our world)Sören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionSymmetries in PhysicsThe phenomenon of SSBExamplesUnification of fundamental forces in particle physicsearly universe: only one forceuniverse cooled down separation to 4 fundamentalforcesorigin: superconductivity (Anderson 1958)today: applications in condensed matter physics(superconductivity, superfluidity, BEC) and QFT (particlephysics, Standard Model)Sören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionSymmetries in PhysicsThe phenomenon of SSBDefinition of SSBexample: Ising modelH0 1Xσi Jij σj2 ij(1)σi 1, i Zd , d: dimension, no external field hereH0 is Z2 -invariant (Z2 : global discrete symm.)if Jij J( Ri Rj ) lattice symm.: Zd (Bravais)we know: phase transitions for d 2order parameter: magnetization mSören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionSymmetries in PhysicsThe phenomenon of SSBDefinition of SSBH0 is Z2 -inv., but solution for m not: it changes sign!mean-field: m τ β , βmf 12 for T TcHelmholtz free energy a(m, T ) former symm. is broken!Sören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionSymmetries in PhysicsThe phenomenon of SSBDefinition of SSBThe general concept:Definition (SSB)G a global symm. group of HThen SSB occurs if in stable TD equilibrium state:m : M 6 0(2)M: observable not G-inv.m: order parameter (of the new phase)Sören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionSymmetries in PhysicsThe phenomenon of SSBHow can SSB occur?Problem: There schould not be any SSB!Why? Averages w.r.t. ρ and ρ ρ(H) m M Tr (Mρ(H)) 0(3)example: Ising model:m 1 X1 X Xσ i σ i · exp βH({σi }) 0N iN · ZN i σ 1i(4)Sören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionSymmetries in PhysicsThe phenomenon of SSBHow can SSB occur?Solution: take TD limit, but this isn’t enough!one needs: symm. breaking field h extra term: h · MDefinition (SSB (Bogolyubov))lim lim M N,h m 6 0h 0 N (5)Limits cannot be interchanged!Sören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionSymmetries in PhysicsThe phenomenon of SSBRemarksSSB long range order, e.g. in ferro-/antiferro-magnetsSSB phase transitionBut there are phase transitions without SSB, e.g.liquid-vapour: both fully rotational and translationalinvariant!“order parameter” ρl ρv Sören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionGoldstone ModesGoldstone TheoremMermin-Wagner TheoremGoldstone ModesSSB of continuous symm. new excitations:Goldstone Modes (cost little energy)example: Spin waves in Heisenberg modelXXH JSi Sj Jcos(θij ), Si R3 , Si 1ijij(6)ferromagnetic phase (SSB) Groundstate: all spins inone directionSören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionGoldstone ModesGoldstone TheoremMermin-Wagner TheoremGoldstone Modesenergy cost to rotate one spin: Eg J(1 cos(θ)),(infinitesimal small angle θ)energy cost to rotate all spins: Nothing!(due to cont. symm. O(3)) long-wavelength spin-wavesremark: cannot happen with discrete broken symm.!e.g. Ising model: Eg J (every excitation costs finiteenergy)Sören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionGoldstone ModesGoldstone TheoremMermin-Wagner TheoremGoldstone Theoremgeneral result proving existence of long-range orderconsider correlation functions Gijαβ , e.g. spin-spin corr. i : order parameterm hi : symm. breaking fieldGibbs free energy: G {h} ln(Z )Sören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionGoldstone ModesGoldstone TheoremMermin-Wagner TheoremProof of Goldstone TheoremP ivertex function: F {m} G {h} i hi m(Legendre transformation)take F G-invariant (in zero external field) F {m} hiα mαi2G 1χijcorr. function: Gijαβ h α hβ hi 0 βij(Fluctuation-Dissipation Theorem)Sören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionGoldstone ModesGoldstone TheoremMermin-Wagner TheoremProof of Goldstone Theoremgeneral property of Legendre transf.:2F 1 αβ m α m)ijβ (Gi 2F m22 ( hG2 ) 1jFourier transformation:P2F [G 1 ( q )]αβ i e i q(Ri Rj ) m α mβijnow: zero field case, uniform order parameter , i.e. q 0) and infinitesimal( mi const. mtransformation gβα id tβα (Lie group)tβα : non-trivial transformation ( ˆ transversal modes!)acting on mα gβα (mβ ) mα tβα mβ {z }δmαSören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionGoldstone ModesGoldstone TheoremMermin-Wagner TheoremProof of Goldstone TheoremP2F Fβvariation yields: 0 δ[ m[ m α mα] i m δmβ ]mβjiijX [G 1 ( q 0)]αβ tγβ mγ 0(7)βγ 0no SSB trivial, since m“longitudinal” transformation trivial, since tγβ mγ 0 6 0 detG 1 ( q 0) 0 detG( q 0) but if mSören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionGoldstone ModesGoldstone TheoremMermin-Wagner TheoremGoldstone TheoremTheorem (Goldstone)If Lie symm. group is spontaneously broken and orderparameter is uniform then the order parameter-orderparameter response function G mm develops a pole.long range orderappearence of Goldstone Modesgapless excitation spectrum (“zero mass”)Sören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionGoldstone ModesGoldstone TheoremMermin-Wagner TheoremGoldstone Theorem: RemarkIn the language of field theory:Rpartition function: ZR D{φ}exp[S{φ}]with action S{φ} d d xL[φ(x)],order parameter field φ(x)PLagrange density: L 21 i φi 2 P[Cn (φ)]2propagators: ij ( x , y ) δλiδ( xΩ{λ})δλj ( y )Ω ln(Z), λi : source term external field det( 1 (p 0)) 0Sören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionGoldstone ModesGoldstone TheoremMermin-Wagner TheoremGoldstone Theorem: Remarkremark: order parameter modulated with wave-vector QdetG 1 (Q) 0 (above: special case Q 0)e.g. liquid-solid phase transitionconsider ρ̃(q) order parameter ρ̃(G ) (G: reciprocallattice vector)Goldstone modes?Sören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionGoldstone ModesGoldstone TheoremMermin-Wagner TheoremGoldstone Theorem: Remarklimq 0 ω(q) 0: consequence of short range forces(nothing special, usual sound waves that appear in liquidsand solids)Goldstone Modes: “Umklapp” phonons at q GSören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionGoldstone ModesGoldstone TheoremMermin-Wagner TheoremGoldstone Theorem: ExampleExample: superconductorsField operators Ψ̂σ ( r )global U(1) gauge symmetry: Ψ̂σ ( r ) Ψ̂σ ( r )e iθorder parameter sc ( r ) Ψ̂ ( r )Ψ̂ ( r ) 6 0short range forces: collective density excitations withlimq 0 ω(q) 0long range forces (more realistic, e.g. Coulomb force):Goldstone Modes pushed to the plasma frequency:limq 0 ω(q) Ωpl ( Cooper Pairs!)caveat: for long range forces: ω(q 0) 6 limq 0 ω(q)spectrum has a gap minimum mass ( the same inHiggs mechanism [except that we have our particles!])Sören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionGoldstone ModesGoldstone TheoremMermin-Wagner TheoremMermin-Wagner TheoremSSB, phase transitions and the role of dimension .Theorem (Mermin-Wagner)If we have:SSB of a Lie symm. groupP 2short range forces ( i Ri Ji0 )poisson bracket structure (classical) or (anti-)commutatorstructure (quantum mechanics) [fulfilled for all standardHamiltonians]Then there is no phase transition (associated with a longrange order!) for dimension d 2 (for T 0).Sören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionGoldstone ModesGoldstone TheoremMermin-Wagner TheoremMermin-Wagner TheoremTheorem (Mermin-Wagner)There is no phase transition (associated with a long rangeorder!) for dimension d 2 (for T 0).Proof uses Bogolyubov’s inequalitye.g. Heisenberg model:Zddk2Tm22S P 2dk 2 S 2 i RBZ (2π) i Ji0 h m (8) diverges for d 2 (if there is SSB; for zero field)d 2: there is phase transition associated with“quasi-long range” order Kosterlitz-ThoulessSören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Spontaneous Symmetry BreakingImplications of SSBConclusionWhat can you take home?SSB is a general concept in modern physics applicable toa variety of fieldsSSB new excitations: Goldstone Modes/Bosons (e.g.Higgs mechanism)SSB remarkably general results about phase transitions(Mermin-Wagner)Thank you for your attentionSören PetratSpontaneous Symmetry Breaking and Goldstone Modes

Goldstone Theorem Mermin-Wagner Theorem Mermin-Wagner Theorem Theorem (Mermin-Wagner) There is no phase transition (associated with a long range order!) for dimension d 2 (for T 0). Proof uses Bogolyubov’s inequality e.g. Heisenberg model: S2 Z BZ ddk (2ˇ)d 2Tm2 j kj2S 2 P i j R ijjJ i

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