The free energy of the quantum Heisenberg ferromagnet: validity of the spin wave approximation Alessandro Giuliani Based on joint work with M. Correggi and R. Seiringer Warwick, March 19, 2014
Outline 1 Introduction: continuous symmetry breaking and spin waves 2 Main results: free energy at low temperatures 3 Sketch of the proof: upper and lower bounds
Outline 1 Introduction: continuous symmetry breaking and spin waves 2 Main results: free energy at low temperatures 3 Sketch of the proof: upper and lower bounds
Spontaneous symmetry breaking General question: rigorous understanding of the phenomenon of spontaneous breaking of a continuous symmetry . Easier case: abelian continuous symmetry . Several rigorous results based on: reflection positivity, vortex loop representation cluster and spin-wave expansions, by Fr¨ ohlich-Simon-Spencer, Dyson-Lieb-Simon, Bricmont-Fontaine- -Lebowitz-Lieb-Spencer, Fr¨ ohlich-Spencer, Kennedy-King, ...
Spontaneous symmetry breaking General question: rigorous understanding of the phenomenon of spontaneous breaking of a continuous symmetry . Easier case: abelian continuous symmetry . Several rigorous results based on: reflection positivity, vortex loop representation cluster and spin-wave expansions, by Fr¨ ohlich-Simon-Spencer, Dyson-Lieb-Simon, Bricmont-Fontaine- -Lebowitz-Lieb-Spencer, Fr¨ ohlich-Spencer, Kennedy-King, ...
Spontaneous symmetry breaking General question: rigorous understanding of the phenomenon of spontaneous breaking of a continuous symmetry . Easier case: abelian continuous symmetry . Several rigorous results based on: reflection positivity, vortex loop representation cluster and spin-wave expansions, by Fr¨ ohlich-Simon-Spencer, Dyson-Lieb-Simon, Bricmont-Fontaine- -Lebowitz-Lieb-Spencer, Fr¨ ohlich-Spencer, Kennedy-King, ...
Spontaneous symmetry breaking General question: rigorous understanding of the phenomenon of spontaneous breaking of a continuous symmetry . Easier case: abelian continuous symmetry . Several rigorous results based on: reflection positivity, vortex loop representation cluster and spin-wave expansions, by Fr¨ ohlich-Simon-Spencer, Dyson-Lieb-Simon, Bricmont-Fontaine- -Lebowitz-Lieb-Spencer, Fr¨ ohlich-Spencer, Kennedy-King, ...
Spontaneous symmetry breaking General question: rigorous understanding of the phenomenon of spontaneous breaking of a continuous symmetry . Easier case: abelian continuous symmetry . Several rigorous results based on: reflection positivity, vortex loop representation cluster and spin-wave expansions, by Fr¨ ohlich-Simon-Spencer, Dyson-Lieb-Simon, Bricmont-Fontaine- -Lebowitz-Lieb-Spencer, Fr¨ ohlich-Spencer, Kennedy-King, ...
Spontaneous symmetry breaking General question: rigorous understanding of the phenomenon of spontaneous breaking of a continuous symmetry . Easier case: abelian continuous symmetry . Several rigorous results based on: reflection positivity, vortex loop representation cluster and spin-wave expansions, by Fr¨ ohlich-Simon-Spencer, Dyson-Lieb-Simon, Bricmont-Fontaine- -Lebowitz-Lieb-Spencer, Fr¨ ohlich-Spencer, Kennedy-King, ...
Spontaneous symmetry breaking General question: rigorous understanding of the phenomenon of spontaneous breaking of a continuous symmetry . Easier case: abelian continuous symmetry . Several rigorous results based on: reflection positivity, vortex loop representation cluster and spin-wave expansions, by Fr¨ ohlich-Simon-Spencer, Dyson-Lieb-Simon, Bricmont-Fontaine- -Lebowitz-Lieb-Spencer, Fr¨ ohlich-Spencer, Kennedy-King, ...
Spontaneous symmetry breaking Harder case: non-abelian symmetry . Few rigorous results on: classical Heisenberg (Fr¨ ohlich-Simon-Spencer by RP) quantum Heisenberg antiferromagnet (Dyson-Lieb-Simon by RP) classical N -vector models (Balaban by RG) Notably absent: quantum Heisenberg ferromagnet
Spontaneous symmetry breaking Harder case: non-abelian symmetry . Few rigorous results on: classical Heisenberg (Fr¨ ohlich-Simon-Spencer by RP) quantum Heisenberg antiferromagnet (Dyson-Lieb-Simon by RP) classical N -vector models (Balaban by RG) Notably absent: quantum Heisenberg ferromagnet
Spontaneous symmetry breaking Harder case: non-abelian symmetry . Few rigorous results on: classical Heisenberg (Fr¨ ohlich-Simon-Spencer by RP) quantum Heisenberg antiferromagnet (Dyson-Lieb-Simon by RP) classical N -vector models (Balaban by RG) Notably absent: quantum Heisenberg ferromagnet
Spontaneous symmetry breaking Harder case: non-abelian symmetry . Few rigorous results on: classical Heisenberg (Fr¨ ohlich-Simon-Spencer by RP) quantum Heisenberg antiferromagnet (Dyson-Lieb-Simon by RP) classical N -vector models (Balaban by RG) Notably absent: quantum Heisenberg ferromagnet
Spontaneous symmetry breaking Harder case: non-abelian symmetry . Few rigorous results on: classical Heisenberg (Fr¨ ohlich-Simon-Spencer by RP) quantum Heisenberg antiferromagnet (Dyson-Lieb-Simon by RP) classical N -vector models (Balaban by RG) Notably absent: quantum Heisenberg ferromagnet
Spontaneous symmetry breaking Harder case: non-abelian symmetry . Few rigorous results on: classical Heisenberg (Fr¨ ohlich-Simon-Spencer by RP) quantum Heisenberg antiferromagnet (Dyson-Lieb-Simon by RP) classical N -vector models (Balaban by RG) Notably absent: quantum Heisenberg ferromagnet
Quantum Heisenberg ferromagnet The simplest quantum model for the spontaneous symmetry breaking of a continuous symmetry: ( S 2 − � � S x · � H Λ := S y ) � x , y �⊂ Λ where: Λ is a cubic subset of Z 3 with (say) periodic b.c. � S x = ( S 1 x , S 2 x , S 3 x ) and S i x are the generators of a (2 S + 1)-dim representation of SU (2), with S = 1 2 , 1 , 3 2 , ... : [ S i x , S j y ] = i ǫ ijk S k x δ x , y The energy is normalized s.t. inf spec ( H Λ ) = 0.
Quantum Heisenberg ferromagnet The simplest quantum model for the spontaneous symmetry breaking of a continuous symmetry: ( S 2 − � � S x · � H Λ := S y ) � x , y �⊂ Λ where: Λ is a cubic subset of Z 3 with (say) periodic b.c. � S x = ( S 1 x , S 2 x , S 3 x ) and S i x are the generators of a (2 S + 1)-dim representation of SU (2), with S = 1 2 , 1 , 3 2 , ... : [ S i x , S j y ] = i ǫ ijk S k x δ x , y The energy is normalized s.t. inf spec ( H Λ ) = 0.
Quantum Heisenberg ferromagnet The simplest quantum model for the spontaneous symmetry breaking of a continuous symmetry: ( S 2 − � � S x · � H Λ := S y ) � x , y �⊂ Λ where: Λ is a cubic subset of Z 3 with (say) periodic b.c. � S x = ( S 1 x , S 2 x , S 3 x ) and S i x are the generators of a (2 S + 1)-dim representation of SU (2), with S = 1 2 , 1 , 3 2 , ... : [ S i x , S j y ] = i ǫ ijk S k x δ x , y The energy is normalized s.t. inf spec ( H Λ ) = 0.
Quantum Heisenberg ferromagnet The simplest quantum model for the spontaneous symmetry breaking of a continuous symmetry: ( S 2 − � � S x · � H Λ := S y ) � x , y �⊂ Λ where: Λ is a cubic subset of Z 3 with (say) periodic b.c. � S x = ( S 1 x , S 2 x , S 3 x ) and S i x are the generators of a (2 S + 1)-dim representation of SU (2), with S = 1 2 , 1 , 3 2 , ... : [ S i x , S j y ] = i ǫ ijk S k x δ x , y The energy is normalized s.t. inf spec ( H Λ ) = 0.
Ground states One special ground state is | Ω � := ⊗ x ∈ Λ | S 3 x = − S � All the other ground states have the form ( S + T ) n | Ω � , n = 1 , . . . , 2 S | Λ | where S + x ∈ Λ S + x and S + x = S 1 x + iS 2 T = � x .
Ground states One special ground state is | Ω � := ⊗ x ∈ Λ | S 3 x = − S � All the other ground states have the form ( S + T ) n | Ω � , n = 1 , . . . , 2 S | Λ | where S + x ∈ Λ S + x and S + x = S 1 x + iS 2 T = � x .
Excited states: spin waves A special class of excited states ( spin waves ) is obtained by raising a spin in a coherent way: 1 1 � ˆ e ikx S + S + √ | 1 k � := x | Ω � ≡ k | Ω � � 2 S | Λ | 2 S x ∈ Λ where k ∈ 2 π L Z 3 . They are such that H Λ | 1 k � = S ǫ ( k ) | 1 k � where ǫ ( k ) = 2 � 3 i =1 (1 − cos k i ).
Excited states: spin waves A special class of excited states ( spin waves ) is obtained by raising a spin in a coherent way: 1 1 � ˆ e ikx S + S + √ | 1 k � := x | Ω � ≡ k | Ω � � 2 S | Λ | 2 S x ∈ Λ where k ∈ 2 π L Z 3 . They are such that H Λ | 1 k � = S ǫ ( k ) | 1 k � where ǫ ( k ) = 2 � 3 i =1 (1 − cos k i ).
Excited states: spin waves More excited states? They can be looked for in the vicinity of (2 S ) − n k / 2 (ˆ S + k ) n + � √ n k ! | Ω � |{ n k }� = k If N = � k n k > 1, these are not eigenstates. They are neither normalized nor orthogonal. However, H Λ is almost diagonal on |{ n k }� in the low-energy (long-wavelengths) sector.
Excited states: spin waves More excited states? They can be looked for in the vicinity of (2 S ) − n k / 2 (ˆ S + k ) n + � √ n k ! | Ω � |{ n k }� = k If N = � k n k > 1, these are not eigenstates. They are neither normalized nor orthogonal. However, H Λ is almost diagonal on |{ n k }� in the low-energy (long-wavelengths) sector.
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