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The Mermin-Wagner TheoremAndreas WernerJune 24, 2010

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionConclusionThe Mermin-Wagner TheoremIn one and two dimensions, continuous symmetries cannot bespontaneously broken at finite temperature in systems withsufficiently short-range interactions.Andreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionContents1How symmetry breaking occurs in principle2Actors3Proof of the Mermin-Wagner TheoremThe Bogoliubov inequalityThe Mermin-Wagner Theorem4DiscussionAndreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionFor systems in statistical equilibrium the expectation value of anoperator A is given by hAi lim tr e βH AV If the Hamiltonian displays a continuous symmetry S it commuteswith the generators ΓiS of the corresponding symmetry group H, ΓiS 0If some operator is not invariant under the transformations of S, B, ΓiS C i 6 0the average of the commutator C i vanishes:Ci 0Andreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionFor systems in statistical equilibrium the expectation value of anoperator A is given by hAi lim tr e βH AV If the Hamiltonian displays a continuous symmetry S it commuteswith the generators ΓiS of the corresponding symmetry group H, ΓiS 0If some operator is not invariant under the transformations of S, B, ΓiS C i 6 0the average of the commutator C i vanishes:Ci 0Andreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionIt turns out that such averages may be unstable under aninfinitesimal perturbation of the HamiltonianHν H νH 0 µN̂one can define the quasi-average: hAiq lim lim tr e βHν Aν 0 V The quasi-average does not need to coincide with the normalaverage C i q lim tr e βHν H, ΓiS 6 0ν 0Andreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionAn Example:H XJij Si · Sjijit is invariant under rotations in spin-space[H, S] 0fromhαS ,Sβi 0and[S x , S y ] i S zwe find that the conventional average of the magnetizationvansihes.Adding a symmetry breaking fieldB0 B0 ezwe may study quasi-averages and find spontaneous symmetrybreaking.Andreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner 35)PierreHohenberg(1934)Andreas WernerSidneyColeman(1937-2007)The Mermin-Wagner TheoremNikolaiBogoliubov(1909-1992)

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner TheoremFor the proof of the Mermin-Wagner Theorem we will use theBogoliubov inequalityhi hi12[C , H] , C † [C , A] β A, A†2 Andreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner TheoremThe idea for proofing the Bogoliubov inequality is to define anappropriate scalar product and then exploit the Schwarz inequality:(A, B) EXDWm Wnn A† m hm B niE n Emn6 mwithWn e βEnTr (e βH )A scalar product has four defining axioms:1(A, B) (B, A) This is valid since DE DEn B † m hm A ni n A† m hm B niAndreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner Theorem2The linearity follows directly from the linearity of the matrixelement3It is also obvious that(A, A) 04From A 0 it naturally follows that (A, A) 0. The converseis not necessarily trueIn conclusion this shows that we have constructed a semidefinitescalar product.Andreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner TheoremTo exploit the Schwarz inequality, we calculate the terms occurringin it: (A, B) 2 (A, A) (B, B)We now choosehiB C †, HAndreas Werner The Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner TheoremFirst we calculate(A, B) XD XD X D†n A mEhim C †, H n6 mnWm WnEn EmEDEn A† m m C † n (Wm Wn )n,mDE XDEWm m C † A† m Wn n A† C † nmnC † A† A† C †Ehi†† C ,A Substituting B C † , H , we find(B, B) h†C , [H, C ] Andreas Werneri 0The Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner TheoremFor (A, A) we use the following approximation:Wm WnEn Em 1 e βEm e βEn e βEm e βEn Tr e βHEn Eme βEm e βEn βWm Wntanh(En Em ) En Em20 Since tanh x x for x 0, we find that0 Wm Wnβ (Wn Wm )En Em2Andreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner TheoremWe can now estimate the scalar product:(A, A) Eβ XDn A† m hm A ni (Wn Wm )2n6 mEβ XDn A† m hm A ni (Wn Wm )2 n,m DE DE βXWn n A† A n n AA† n2 nThis finally leads toβ(A, A) 2Andreas Wernerh†A, Ai The Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner TheoremPutting what we found in the Schwarz inequality, we find that weproofed the Bogoliubov inequality E D E1 D β A, A [C , H] , C 2Andreas Werner[C , A] The Mermin-Wagner Theorem2

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner TheoremWe now want to find out whether the isotropic Heisenberg modelgives a spontaneous magnetization. The starting point is theHamiltonianXXH Jij Si · Sj bSiz e iK·Rii,jiWe are interested in the magnetizationµB X iK·Ri zehSi iT ,B0Ms (T ) lim gJB0 0 iAndreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner TheoremFor the following analysis, we assume that the exchange integralsJij decrease sufficiently fast with increasing distance Ri Rj sothat the quantityQ 1 X Ri Rj 2 Jij Ni,jremains finite.Andreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner TheoremWe will now prove the Mermin-Wagner Theorem by using theBogoliubov inequality for the operatorsA S ( k K) A† S (k K)C S (k) C † S ( k)Where the spin operators in k-space are defined byXS α (k) Siα e ikRiiFrom this we find the commutation relations S (k1 ), S (k2 ) 2 S z (k1 k2 ) z S (k1 ), S (k2 ) S (k1 k2 )Andreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner TheoremWe now evaluate the three individual terms of the BogoliubovinequalityD ES (k), S ( k K) [C , A] 2 hS z (K)iX 2 e iKRi hSiz i i22 NgJ µBAndreas WernerM(T , B0 )The Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner TheoremiX hX D E†A, A S ( k K), S (k K) k k XXei(k K)(Ri Rj )DSi Sj i,jk 2NX 2NX(Six )2 (Siy )2iS2ii2 2 N 2 S(S 1)Andreas WernerThe Mermin-Wagner Theorem Sj Si E

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner TheoremNow we calculate the double commutatorhi†[C , H], C First we will evaluateX z z iKRmSm , H Jim 2Si Sm Siz Sm SmSi bSmeiUsing this, we evaluate the double commutatorh iX Sm, H , Sp 2 2Jip δmp Si Sp 2Siz Spz i z z 2 Jmp SmSp 2SmSp 2 2 bδmp Spz e iKRp2Andreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner TheoremThis leads to the following intermediate result for the expectationvalue we are looking forhhiiX [C , H], C † e ik(Rm Rp )Sm , H , Sp m,p 2 2 bXSpz e iKRpp 2 2X z z Sp 2SmSpJmp 1 e ik(Rm Rp ) Smm,pAndreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner TheoremTo find a simple upper bound we may add to the right-hand sidethe same expression with k replaced by k:hi†[C , H], C Xz2 4 bSp e iKRpp 4 2Xz zJmp (1 cos (k (Rm Rp ))) Sm Sp SmSpm,pAndreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner TheoremWe can simplify the right hand side using the triangle inequalityhi†[C , H], C 2Spz 4 bNX 4 2 Jmp (1 cos (k (Rm Rp ))) hSm Sp i z zSmSpm,p2 4 bN SpzX 4 2 Jmp 1 cos (k (Rm Rp )) 2 S(S 1) 2 S 2m,p2 4 bNSpz 8 2 S(S 1)X Jmp 1 cos (k (Rm Rp )) m,pAndreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner TheoremTherewith we have foundhi†[C , H], C 2 4 B0 M(T , B0 ) X Jmp k 2 Rm Rp 2 8 2 S(S 1)2m,p 4 2 B0 M(T , B0 ) 4Nk 2 4 QS(S 1)Andreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner TheoremSubstituting what we have found in the Bogoliubov inequality andsumming over all the wavevectors of the first Brillouin zone we get:βS(S 1) 1M2 X2222 B0 M k 2 NQS(S 1)N gj µBkWe are finally ready to prove the Mermin-Wagner Theorem. In thethermodynamic limit we find:S(S 1) m 2 v d Ωdβ(2π)d gj2 µ2BZAndreas Werner0k0k d 1 dk B0 M k 2 2 QS(S 1)The Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner TheoremAll that is left to do is to evaluate the integrals. This can be doneexactly; in one dimension we find: q 2 Q S(S 1)arctank20 B0 m m v1pS(S 1) 222β2πgj µBQ S(S 1) B0 m We are specifically interested in the behaviour of the magnetizationfor small fields B0 :1/3 m(T , B0 ) const.Andreas WernerB0,T 2/3as B0 0The Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner TheoremFor a two-dimensional lattice we find: q Q 2 S(S 1)k02 B0 m ln B0 m m2 v2S(S 1) 2Q 2 S(S 1)β2πgj2 µ2Bfrom which for small fields we get m(T , B0 ) const. T lnAndreas Werner const.0 B0 m B0 m 1/2The Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThe Bogoliubov inequalityThe Mermin-Wagner TheoremFrom the previous two expressions we conclude that there is nospontaneous magnetization in one and two dimensions:msp lim m(T , B0 ) 0 for T 6 0B0 0Thus, the Mermin-Wagner Theorem is proved.Andreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussion123The proof is valid only for T 0. For T 0 our inequalitiesmake no predictions.Via the factor e iKRi the proof also forbids long-range order inantiferromagnets.We cannot make any predictions for d 2, but Roepstroffstrengthened the proof to find an upper bound for themagnetization in d 3.Andreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussion4The theorem is valid for arbitrary spin S.5The theorem is valid only for the isotropic Heisenberg model.The proof is not valid even for a weak anisotropy. Thisexplains the existence of a number of two-dimensionalHeisenberg ferromagnets and antiferromagnets like K2 CuF4 .6The theorem is restricted only to the non-existence ofspontaneous magnetization. It does not necessarily excludeother types of phase transitions. For example the magneticsusceptibility may diverge.Andreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionThis is the end of my presentationThank you for your attention.Andreas WernerThe Mermin-Wagner Theorem

How symmetry breaking occurs in principleActorsProof of the Mermin-Wagner TheoremDiscussionReferencesColeman, S. There are no goldstone bosons in two dimensions. Comm. Math. Phys. 31 (1973), 259–264.Gelfert, A., and Nolting, W. The absence of finite-temperature phase transitions in low-dimensionalmany-body models: a survey and new results. Journal of Physics: Condensed Matter 13, 27 (2001), R505.Hohenberg, P. C. Existence of long-range order in one and two dimensions. Phys. Rev. 158, 2 (Jun1967), 383–386.Mermin, N. D., and Wagner, H. Absence of ferromagnetism or antiferromagnetism in one- ortwo-dimensional isotropic heisenberg models. Phys. Rev. Lett. 17, 22 (Nov 1966), 1133–1136.Nolting, W., and Ramakanth, A. Quantum Theory of Magnetism. Springer, 2009.Roepstorff, G. A stronger version of bogoliubov’s inequality and the heisenberg model. Comm. Math.Phys. 53, 2 (1977), 143–150.Andreas WernerThe Mermin-Wagner Theorem

Andreas Werner The Mermin-Wagner Theorem. How symmetry breaking occurs in principle Actors Proof of the Mermin-Wagner Theorem Discussion The Bogoliubov inequality The Mermin-Wagner Theorem 2 The linearity follows directly from the linearity of the matrix element 3 It is also obvious that (A;A) 0 4 From A 0 it naturally follows that (A;A) 0. The converse is not necessarily true In .

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