validity of spin wave theory for the quantum heisenberg model · 2017. 6. 6. · validity of spin...
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![Page 1: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/1.jpg)
Validity of spin wave theory for the quantumHeisenberg model
Alessandro Giuliani
Based on joint work withM. Correggi and R. Seiringer
GGI, Arcetri, May 30, 2014
![Page 2: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/2.jpg)
Outline
1 Introduction: continuous symmetry breaking and spin waves
2 Main results: free energy at low temperatures
![Page 3: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/3.jpg)
Outline
1 Introduction: continuous symmetry breaking and spin waves
2 Main results: free energy at low temperatures
![Page 4: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/4.jpg)
Spontaneous symmetry breaking
General question: rigorous understanding of thephenomenon of spontaneous breaking of acontinuous symmetry.
Easier case: abelian continuous symmetry.Several rigorous results based on:
reflection positivity,
vortex loop representation
cluster and spin-wave expansions,
by Frohlich-Simon-Spencer, Dyson-Lieb-Simon, Bricmont-Fontaine-
-Lebowitz-Lieb-Spencer, Frohlich-Spencer, Kennedy-King, ...
![Page 5: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/5.jpg)
Spontaneous symmetry breaking
General question: rigorous understanding of thephenomenon of spontaneous breaking of acontinuous symmetry.
Easier case: abelian continuous symmetry.Several rigorous results based on:
reflection positivity,
vortex loop representation
cluster and spin-wave expansions,
by Frohlich-Simon-Spencer, Dyson-Lieb-Simon, Bricmont-Fontaine-
-Lebowitz-Lieb-Spencer, Frohlich-Spencer, Kennedy-King, ...
![Page 6: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/6.jpg)
Spontaneous symmetry breaking
General question: rigorous understanding of thephenomenon of spontaneous breaking of acontinuous symmetry.
Easier case: abelian continuous symmetry.Several rigorous results based on:
reflection positivity,
vortex loop representation
cluster and spin-wave expansions,
by Frohlich-Simon-Spencer, Dyson-Lieb-Simon, Bricmont-Fontaine-
-Lebowitz-Lieb-Spencer, Frohlich-Spencer, Kennedy-King, ...
![Page 7: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/7.jpg)
Spontaneous symmetry breaking
General question: rigorous understanding of thephenomenon of spontaneous breaking of acontinuous symmetry.
Easier case: abelian continuous symmetry.Several rigorous results based on:
reflection positivity,
vortex loop representation
cluster and spin-wave expansions,
by Frohlich-Simon-Spencer, Dyson-Lieb-Simon, Bricmont-Fontaine-
-Lebowitz-Lieb-Spencer, Frohlich-Spencer, Kennedy-King, ...
![Page 8: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/8.jpg)
Spontaneous symmetry breaking
General question: rigorous understanding of thephenomenon of spontaneous breaking of acontinuous symmetry.
Easier case: abelian continuous symmetry.Several rigorous results based on:
reflection positivity,
vortex loop representation
cluster and spin-wave expansions,
by Frohlich-Simon-Spencer, Dyson-Lieb-Simon, Bricmont-Fontaine-
-Lebowitz-Lieb-Spencer, Frohlich-Spencer, Kennedy-King, ...
![Page 9: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/9.jpg)
Spontaneous symmetry breaking
General question: rigorous understanding of thephenomenon of spontaneous breaking of acontinuous symmetry.
Easier case: abelian continuous symmetry.Several rigorous results based on:
reflection positivity,
vortex loop representation
cluster and spin-wave expansions,
by Frohlich-Simon-Spencer, Dyson-Lieb-Simon, Bricmont-Fontaine-
-Lebowitz-Lieb-Spencer, Frohlich-Spencer, Kennedy-King, ...
![Page 10: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/10.jpg)
Spontaneous symmetry breaking
General question: rigorous understanding of thephenomenon of spontaneous breaking of acontinuous symmetry.
Easier case: abelian continuous symmetry.Several rigorous results based on:
reflection positivity,
vortex loop representation
cluster and spin-wave expansions,
by Frohlich-Simon-Spencer, Dyson-Lieb-Simon, Bricmont-Fontaine-
-Lebowitz-Lieb-Spencer, Frohlich-Spencer, Kennedy-King, ...
![Page 11: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/11.jpg)
Spontaneous symmetry breaking
Harder case: non-abelian symmetry.Few rigorous results on:
classical Heisenberg (Frohlich-Simon-Spencer by RP)
quantum Heisenberg antiferromagnet (Dyson-Lieb-Simon by RP)
classical N-vector models (Balaban by RG)
Notably absent: quantum Heisenberg ferromagnet
![Page 12: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/12.jpg)
Spontaneous symmetry breaking
Harder case: non-abelian symmetry.Few rigorous results on:
classical Heisenberg (Frohlich-Simon-Spencer by RP)
quantum Heisenberg antiferromagnet (Dyson-Lieb-Simon by RP)
classical N-vector models (Balaban by RG)
Notably absent: quantum Heisenberg ferromagnet
![Page 13: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/13.jpg)
Spontaneous symmetry breaking
Harder case: non-abelian symmetry.Few rigorous results on:
classical Heisenberg (Frohlich-Simon-Spencer by RP)
quantum Heisenberg antiferromagnet (Dyson-Lieb-Simon by RP)
classical N-vector models (Balaban by RG)
Notably absent: quantum Heisenberg ferromagnet
![Page 14: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/14.jpg)
Spontaneous symmetry breaking
Harder case: non-abelian symmetry.Few rigorous results on:
classical Heisenberg (Frohlich-Simon-Spencer by RP)
quantum Heisenberg antiferromagnet (Dyson-Lieb-Simon by RP)
classical N-vector models (Balaban by RG)
Notably absent: quantum Heisenberg ferromagnet
![Page 15: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/15.jpg)
Spontaneous symmetry breaking
Harder case: non-abelian symmetry.Few rigorous results on:
classical Heisenberg (Frohlich-Simon-Spencer by RP)
quantum Heisenberg antiferromagnet (Dyson-Lieb-Simon by RP)
classical N-vector models (Balaban by RG)
Notably absent: quantum Heisenberg ferromagnet
![Page 16: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/16.jpg)
Spontaneous symmetry breaking
Harder case: non-abelian symmetry.Few rigorous results on:
classical Heisenberg (Frohlich-Simon-Spencer by RP)
quantum Heisenberg antiferromagnet (Dyson-Lieb-Simon by RP)
classical N-vector models (Balaban by RG)
Notably absent: quantum Heisenberg ferromagnet
![Page 17: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/17.jpg)
Quantum Heisenberg ferromagnet
The simplest quantum model for the spontaneoussymmetry breaking of a continuous symmetry:
HΛ :=∑〈x ,y〉⊂Λ
(S2 − ~Sx · ~Sy)
where:
Λ is a cubic subset of Z3 with (say) periodic b.c.
~Sx = (S1x ,S
2x ,S
3x ) and S i
x are the generators of a (2S + 1)-dimrepresentation of SU(2), with S = 1
2 , 1,32 , ...:
[S ix ,S
jy ] = iεijkS
kx δx,y
The energy is normalized s.t. inf spec(HΛ) = 0.
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Quantum Heisenberg ferromagnet
The simplest quantum model for the spontaneoussymmetry breaking of a continuous symmetry:
HΛ :=∑〈x ,y〉⊂Λ
(S2 − ~Sx · ~Sy)
where:
Λ is a cubic subset of Z3 with (say) periodic b.c.
~Sx = (S1x ,S
2x ,S
3x ) and S i
x are the generators of a (2S + 1)-dimrepresentation of SU(2), with S = 1
2 , 1,32 , ...:
[S ix ,S
jy ] = iεijkS
kx δx,y
The energy is normalized s.t. inf spec(HΛ) = 0.
![Page 19: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/19.jpg)
Quantum Heisenberg ferromagnet
The simplest quantum model for the spontaneoussymmetry breaking of a continuous symmetry:
HΛ :=∑〈x ,y〉⊂Λ
(S2 − ~Sx · ~Sy)
where:
Λ is a cubic subset of Z3 with (say) periodic b.c.
~Sx = (S1x ,S
2x ,S
3x ) and S i
x are the generators of a (2S + 1)-dimrepresentation of SU(2), with S = 1
2 , 1,32 , ...:
[S ix ,S
jy ] = iεijkS
kx δx,y
The energy is normalized s.t. inf spec(HΛ) = 0.
![Page 20: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/20.jpg)
Quantum Heisenberg ferromagnet
The simplest quantum model for the spontaneoussymmetry breaking of a continuous symmetry:
HΛ :=∑〈x ,y〉⊂Λ
(S2 − ~Sx · ~Sy)
where:
Λ is a cubic subset of Z3 with (say) periodic b.c.
~Sx = (S1x ,S
2x ,S
3x ) and S i
x are the generators of a (2S + 1)-dimrepresentation of SU(2), with S = 1
2 , 1,32 , ...:
[S ix ,S
jy ] = iεijkS
kx δx,y
The energy is normalized s.t. inf spec(HΛ) = 0.
![Page 21: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/21.jpg)
Ground states
One special ground state is
|Ω〉 := ⊗x∈Λ|S3x = −S〉
All the other ground states have the form
(S+T )n|Ω〉, n = 1, . . . , 2S |Λ|
where S+T =
∑x∈Λ S
+x and S+
x = S1x + iS2
x .
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Ground states
One special ground state is
|Ω〉 := ⊗x∈Λ|S3x = −S〉
All the other ground states have the form
(S+T )n|Ω〉, n = 1, . . . , 2S |Λ|
where S+T =
∑x∈Λ S
+x and S+
x = S1x + iS2
x .
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Excited states: spin waves
A special class of excited states (spin waves) isobtained by raising a spin in a coherent way:
|1k〉 :=1√
2S |Λ|
∑x∈Λ
e ikxS+x |Ω〉 ≡
1√2S
S+k |Ω〉
where k ∈ 2πL Z
3. They are such that
HΛ|1k〉 = Sε(k)|1k〉
where ε(k) = 2∑3
i=1(1− cos ki).
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Excited states: spin waves
A special class of excited states (spin waves) isobtained by raising a spin in a coherent way:
|1k〉 :=1√
2S |Λ|
∑x∈Λ
e ikxS+x |Ω〉 ≡
1√2S
S+k |Ω〉
where k ∈ 2πL Z
3. They are such that
HΛ|1k〉 = Sε(k)|1k〉
where ε(k) = 2∑3
i=1(1− cos ki).
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Excited states: spin waves
More excited states?
They can be looked for in the vicinity of
|nk〉 =∏k
(2S)−nk/2 (S+k )nk√nk!|Ω〉
If N =∑
k nk > 1, these are not eigenstates.
They are neither normalized nor orthogonal.
However, HΛ is almost diagonal on |nk〉 in thelow-energy (long-wavelengths) sector.
![Page 26: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/26.jpg)
Excited states: spin waves
More excited states?
They can be looked for in the vicinity of
|nk〉 =∏k
(2S)−nk/2 (S+k )nk√nk!|Ω〉
If N =∑
k nk > 1, these are not eigenstates.
They are neither normalized nor orthogonal.
However, HΛ is almost diagonal on |nk〉 in thelow-energy (long-wavelengths) sector.
![Page 27: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/27.jpg)
Excited states: spin waves
More excited states?
They can be looked for in the vicinity of
|nk〉 =∏k
(2S)−nk/2 (S+k )nk√nk!|Ω〉
If N =∑
k nk > 1, these are not eigenstates.
They are neither normalized nor orthogonal.
However, HΛ is almost diagonal on |nk〉 in thelow-energy (long-wavelengths) sector.
![Page 28: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/28.jpg)
Spin waves
Expectation:
low temperatures ⇒⇒ low density of spin waves ⇒⇒ negligible interactions among spin waves.
The linear theory obtained by neglecting spin waveinteractions is the spin wave approximation,in very good agreement with experiment.
![Page 29: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/29.jpg)
Spin waves
Expectation:
low temperatures ⇒⇒ low density of spin waves ⇒⇒ negligible interactions among spin waves.
The linear theory obtained by neglecting spin waveinteractions is the spin wave approximation,in very good agreement with experiment.
![Page 30: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/30.jpg)
Spin waves
In 3D, it predicts
f (β) ' 1
β
∫d3k
(2π)3log(1− e−βSε(k))
m(β) ' S −∫
d3k
(2π)3
1
eβSε(k) − 1
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Spin waves
In 3D, it predicts
f (β) 'β→∞
β−5/2S−3/2
∫d3k
(2π)3log(1− e−k
2
)
m(β) 'β→∞
S − β−3/2S−3/2
∫d3k
(2π)3
1
ek2 − 1
How do we derive these formulas?
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Spin waves
In 3D, it predicts
f (β) 'β→∞
β−5/2S−3/2
∫d3k
(2π)3log(1− e−k
2
)
m(β) 'β→∞
S − β−3/2S−3/2
∫d3k
(2π)3
1
ek2 − 1
How do we derive these formulas?
![Page 33: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/33.jpg)
Holstein-Primakoff representation
A convenient representation:
S+x =√
2S a+x
√1− a+
x ax2S
, S3x = a+
x ax − S ,
where [ax , a+y ] = δx ,y are bosonic operators.
Hard-core constraint: nx = a+x ax ≤ 2S .
![Page 34: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/34.jpg)
Holstein-Primakoff representation
A convenient representation:
S+x =√
2S a+x
√1− a+
x ax2S
, S3x = a+
x ax − S ,
where [ax , a+y ] = δx ,y are bosonic operators.
Hard-core constraint: nx = a+x ax ≤ 2S .
![Page 35: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/35.jpg)
Holstein-Primakoff representation
In the bosonic language
HΛ = S∑〈x ,y〉
(−a+
x
√1− nx
2S
√1− ny
2Say
−a+y
√1− ny
2S
√1− nx
2Sax + nx + ny −
1
Snxny
)
≡ S∑〈x ,y〉
(a+x − a+
y )(ax − ay)− K ≡ T − K
The spin wave approximation consists in neglectingK and the on-site hard-core constraint.
![Page 36: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/36.jpg)
Holstein-Primakoff representation
In the bosonic language
HΛ = S∑〈x ,y〉
(−a+
x
√1− nx
2S
√1− ny
2Say
−a+y
√1− ny
2S
√1− nx
2Sax + nx + ny −
1
Snxny
)
≡ S∑〈x ,y〉
(a+x − a+
y )(ax − ay)− K ≡ T − K
The spin wave approximation consists in neglectingK and the on-site hard-core constraint.
![Page 37: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/37.jpg)
Holstein-Primakoff representation
In the bosonic language
HΛ = S∑〈x ,y〉
(−a+
x
√1− nx
2S
√1− ny
2Say
−a+y
√1− ny
2S
√1− nx
2Sax + nx + ny −
1
Snxny
)
≡ S∑〈x ,y〉
(a+x − a+
y )(ax − ay)− K ≡ T − K
The spin wave approximation consists in neglectingK and the on-site hard-core constraint.
![Page 38: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/38.jpg)
Previous results
HΛ = S∑〈x ,y〉
(a+x − a+
y )(ax − ay)− K
For large S , the interaction K is of relative sizeO(1/S) as compared to the hopping term.
Easier case: S →∞ with βS constant (CG 2012)
Harder case: fixed S , say S = 1/2. So far, not evena sharp upper bound on the free energy was known.Rigorous upper bounds, off by a constant, weregiven by Conlon-Solovej and Toth in the early 90s.
![Page 39: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/39.jpg)
Previous results
HΛ = S∑〈x ,y〉
(a+x − a+
y )(ax − ay)− K
For large S , the interaction K is of relative sizeO(1/S) as compared to the hopping term.
Easier case: S →∞ with βS constant (CG 2012)
Harder case: fixed S , say S = 1/2. So far, not evena sharp upper bound on the free energy was known.Rigorous upper bounds, off by a constant, weregiven by Conlon-Solovej and Toth in the early 90s.
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Previous results
HΛ = S∑〈x ,y〉
(a+x − a+
y )(ax − ay)− K
For large S , the interaction K is of relative sizeO(1/S) as compared to the hopping term.
Easier case: S →∞ with βS constant (CG 2012)
Harder case: fixed S , say S = 1/2. So far, not evena sharp upper bound on the free energy was known.Rigorous upper bounds, off by a constant, weregiven by Conlon-Solovej and Toth in the early 90s.
![Page 41: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/41.jpg)
Previous results
HΛ = S∑〈x ,y〉
(a+x − a+
y )(ax − ay)− K
For large S , the interaction K is of relative sizeO(1/S) as compared to the hopping term.
Easier case: S →∞ with βS constant (CG 2012)
Harder case: fixed S , say S = 1/2. So far, not evena sharp upper bound on the free energy was known.Rigorous upper bounds, off by a constant, weregiven by Conlon-Solovej and Toth in the early 90s.
![Page 42: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/42.jpg)
Bosons and random walk
Side remark: the Hamiltonian can be rewritten as
HΛ = S∑〈x ,y〉
(a+x
√1− ny
2S− a+
y
√1− nx
2S
)·
·(ax
√1− ny
2S− ay
√1− nx
2S
)i.e., it describes a weighted hopping process ofbosons on the lattice. The hopping on an occupiedsite is discouraged (or not allowed).
The spin wave approximation corresponds to theuniform RW, without hard-core constraint.
![Page 43: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/43.jpg)
Bosons and random walk
Side remark: the Hamiltonian can be rewritten as
HΛ = S∑〈x ,y〉
(a+x
√1− ny
2S− a+
y
√1− nx
2S
)·
·(ax
√1− ny
2S− ay
√1− nx
2S
)i.e., it describes a weighted hopping process ofbosons on the lattice. The hopping on an occupiedsite is discouraged (or not allowed).
The spin wave approximation corresponds to theuniform RW, without hard-core constraint.
![Page 44: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/44.jpg)
Bosons and random walk
Side remark: the Hamiltonian can be rewritten as
HΛ = S∑〈x ,y〉
(a+x
√1− ny
2S− a+
y
√1− nx
2S
)·
·(ax
√1− ny
2S− ay
√1− nx
2S
)i.e., it describes a weighted hopping process ofbosons on the lattice. The hopping on an occupiedsite is discouraged (or not allowed).
The spin wave approximation corresponds to theuniform RW, without hard-core constraint.
![Page 45: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/45.jpg)
Outline
1 Introduction: continuous symmetry breaking and spin waves
2 Main results: free energy at low temperatures
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Main theorem
Theorem [Correggi-G-Seiringer 2013](free energy at low temperature).
For any S ≥ 1/2,
limβ→∞
f (S , β)β5/2S3/2 =
∫log(
1− e−k2) d3k
(2π)3.
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Remarks
The proof is based on upper and lower bounds.It comes with explicit estimates on theremainder.
Relative errors: • O((βS)−3/8) (upper bound)
• O((βS)−1/40+ε) (lower bound)
We do not really need S fixed. Our bounds areuniform in S , provided that βS →∞.
The case S →∞ with βS =const. is easier andit was solved by Correggi-G (JSP 2012).
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Remarks
The proof is based on upper and lower bounds.It comes with explicit estimates on theremainder.
Relative errors: • O((βS)−3/8) (upper bound)
• O((βS)−1/40+ε) (lower bound)
We do not really need S fixed. Our bounds areuniform in S , provided that βS →∞.
The case S →∞ with βS =const. is easier andit was solved by Correggi-G (JSP 2012).
![Page 49: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/49.jpg)
Remarks
The proof is based on upper and lower bounds.It comes with explicit estimates on theremainder.
Relative errors: • O((βS)−3/8) (upper bound)
• O((βS)−1/40+ε) (lower bound)
We do not really need S fixed. Our bounds areuniform in S , provided that βS →∞.
The case S →∞ with βS =const. is easier andit was solved by Correggi-G (JSP 2012).
![Page 50: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/50.jpg)
Remarks
An important consequence of our proof is aninstance of quasi long-range order:
〈S2 − ~Sx · ~Sy〉β ≤ 278 |x − y |2e(S , β) ,
where e(S , β) = ∂β(βf (S , β)) is the energy:
e(S , β) 'β→∞
32S−3/2β−5/2
∫dk
(2π)2log
1
1− e−k2
Therefore, order persists up to length scales ofthe order β5/4. Of course, one expects order topersist at infinite distances, but in absence of aproof this is the best result to date.
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Remarks
An important consequence of our proof is aninstance of quasi long-range order:
〈S2 − ~Sx · ~Sy〉β ≤ 278 |x − y |2e(S , β) ,
where e(S , β) = ∂β(βf (S , β)) is the energy:
e(S , β) 'β→∞
32S−3/2β−5/2
∫dk
(2π)2log
1
1− e−k2
Therefore, order persists up to length scales ofthe order β5/4. Of course, one expects order topersist at infinite distances, but in absence of aproof this is the best result to date.
![Page 52: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/52.jpg)
Remarks
An important consequence of our proof is aninstance of quasi long-range order:
〈S2 − ~Sx · ~Sy〉β ≤ 278 |x − y |2e(S , β) ,
where e(S , β) = ∂β(βf (S , β)) is the energy:
e(S , β) 'β→∞
32S−3/2β−5/2
∫dk
(2π)2log
1
1− e−k2
Therefore, order persists up to length scales ofthe order β5/4. Of course, one expects order topersist at infinite distances, but in absence of aproof this is the best result to date.
![Page 53: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/53.jpg)
Remarks
An important consequence of our proof is aninstance of quasi long-range order:
〈S2 − ~Sx · ~Sy〉β ≤ 278 |x − y |2e(S , β) ,
where e(S , β) = ∂β(βf (S , β)) is the energy:
e(S , β) 'β→∞
32S−3/2β−5/2
∫dk
(2π)2log
1
1− e−k2
Therefore, order persists up to length scales ofthe order β5/4. Of course, one expects order topersist at infinite distances, but in absence of aproof this is the best result to date.
![Page 54: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/54.jpg)
Ideas of the proof
The proof is based on upper and lower bounds.In both cases we localize the system in boxes ofside ` = β1/2+ε.
The upper bound is based on a trial densitymatrix that is the natural one, i.e., the Gibbsmeasure associated with the quadratic part ofthe Hamiltonian projected onto the subspacesatisfying the local hard-core constraint.
The lower bound is based on a preliminary roughbound, off by a log. This uses an estimate onthe excitation spectrum
HB ≥ (const.)`−2(Smax − ST )
![Page 55: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/55.jpg)
Ideas of the proof
The proof is based on upper and lower bounds.In both cases we localize the system in boxes ofside ` = β1/2+ε.
The upper bound is based on a trial densitymatrix that is the natural one, i.e., the Gibbsmeasure associated with the quadratic part ofthe Hamiltonian projected onto the subspacesatisfying the local hard-core constraint.
The lower bound is based on a preliminary roughbound, off by a log. This uses an estimate onthe excitation spectrum
HB ≥ (const.)`−2(Smax − ST )
![Page 56: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/56.jpg)
Ideas of the proof
The proof is based on upper and lower bounds.In both cases we localize the system in boxes ofside ` = β1/2+ε.
The upper bound is based on a trial densitymatrix that is the natural one, i.e., the Gibbsmeasure associated with the quadratic part ofthe Hamiltonian projected onto the subspacesatisfying the local hard-core constraint.
The lower bound is based on a preliminary roughbound, off by a log. This uses an estimate onthe excitation spectrum
HB ≥ (const.)`−2(Smax − ST )
![Page 57: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/57.jpg)
Summary
The preliminary rough bound is used to cutoffthe energies higher than `3β−5/2(log β)5/2. Inthe low energy sector we pass to the bosonicrepresentation.
In order to bound the interaction energy in thelow energy sector, we use a new functionalinequality, which allows us to reduce to a 2-bodyproblem. The latter is studied by random walktechniques on a weighted graph.
![Page 58: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/58.jpg)
Summary
The preliminary rough bound is used to cutoffthe energies higher than `3β−5/2(log β)5/2. Inthe low energy sector we pass to the bosonicrepresentation.
In order to bound the interaction energy in thelow energy sector, we use a new functionalinequality, which allows us to reduce to a 2-bodyproblem. The latter is studied by random walktechniques on a weighted graph.
![Page 59: Validity of spin wave theory for the quantum Heisenberg model · 2017. 6. 6. · Validity of spin wave theory for the quantum Heisenberg model Alessandro Giuliani Based on joint work](https://reader035.vdocuments.us/reader035/viewer/2022071101/5fda5d30549d9411380bc033/html5/thumbnails/59.jpg)
Thank you!