Reflection Positivity and Phase Transitions
Daniel Ueltschi
Department of Mathematics, University of Warwick
“Positive Reflections”, in memory of Robert SchraderETH Zurich, 23 May 2016
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 1 / 16
Lattice spin systems
Condensed matter: Atoms form a regular latticeand electrons move in this periodic potential
Each electron carries a “spin”, i.e. an intrinsicmagnetic moment
Crystalline structure ofsodium chloride (table salt)
Source: Wikipedia
Heisenberg model: There is exactly one localised electron on eachatom, that is characterised by its spin
Interactions are nearest-neighbour only, and they are invariant underspin rotations
Important model for magnetism, and more generally for condensedmatter physics and quantum information
Further simplification: classical spin models
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 2 / 16
Classical spin models
Lattice Λ = {−L2 + 1, . . . , L2 }
d with periodic boundary conditions
State space: (SN−1)Λ
Hamiltonian: HΛ(σ) =∑
xy∈Λ(σx − σy)2
Special cases: N = 0: self-avoiding random walk; N = 1: Ising;N = 2: XY model; N = 3: classical Heisenberg
Ising model: Phase transitions in dimension 2 and higher; exactly twoextremal Gibbs states at low temperatures
For N > 2: model has continuous symmetry, that prevents ordering in2D (Mermin-Wagner); spontaneous magnetisation is expected in d > 3
N = 2: [Frohlich, Pfister ’83] prove that, whenever the free energy isdifferentiable in β, there is either a unique extremal Gibbs state, or allextremal states are labelled by the elements of the symmetry groupSO(2)
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 3 / 16
Results from RP & IRB
Spectacular results come from the method of reflection positivityand infrared bounds: the model has spontaneous magnetisation atlow enough temperatures!
Finite volume Gibbs state: 〈·〉 = 1Z(Λ)
∫(SN−1)Λ · e−βHΛ(σ) dσ
Theorem [Frohlich, Simon, Spencer ’76]
1
|Λ|2∑x,y∈Λ
〈σxσy〉 > 1− 3
β|Λ|∑
k∈Λ∗\{0}
1
ε(k)
Here, Λ∗ = 2πL {−
L2 + 1, . . . , L2 }
d and ε(k) = 2∑d
i=1(1− cos ki)
As a consequence,
lim infL→∞
⟨( 1
|Λ|∑x∈Λ
σx
)2⟩> 1− 3
(2π)dβ
∫[−π,π]d
dk
ε(k)
Hence there cannot be a single extremal state! (d > 3, β large)D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 4 / 16
Overview of the method of proof
Let κ(x) = 〈σ30σ
3x〉 the two-point fct and κ(k) its Fourier transform:
κ(k) =∑x∈Λ
e−ikx κ(x), k ∈ Λ∗
Notice that
κ(x) =1
|Λ|∑k∈Λ∗
eikx κ(k)
The goal is to show that1
|Λ|κ(0) > c > 0
For v ∈ RΛ, let
Z(v) =
∫(SN−1)Λ
exp{−β
∑xy∈Λ
(σx + vx − σy − vy)2}
dσ
Notice that Z(v = 0) = Z(Λ)D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 5 / 16
Overview of the method of proof
One proves a series of 4 lemmas:
Lemma 1: Reflection positivity
Z(v1, v2)2 6 Z(v1, Rv1)Z(Rv2, v2)
Pv1 v2
L2: RP ⇒ gaussian domination
Z(v) 6 Z(0)
L3: GD ⇒ infrared bound
κ(k) 61
βε(k)
Lemma 4: IRB ⇒ theorem1
|Λ|κ(0) >
1
3− 1
β|Λ|∑
k∈Λ∗\{0}
1
ε(k)
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 6 / 16
Reflection positivity
Lemma 1: Reflection positivity
Z(v1, v2)2 6 Z(v1, Rv1)Z(Rv2, v2)
Pv1 v2
Z(v1,v2) =
∫Λ1
dσ1
∫Λ2
dσ2 e−F (σ1,v1)−F (σ2,v2)
exp{−β
∑xy∈P
d∑i=1
(σi1x + vi1x − σi2y − vi2y)2}
=( ∏xy∈P
d∏i=1
∫ ∞−∞
dξixy√2π
e−12 (ξixy)2
)
·∫
Λ1
dσ1 e−F (σ1,v1) exp{
i√
2β∑xy∈P
d∑i=1
ξixy(σi1x − vi1x)
}∫Λ2
dσ2 . . .
Lemma follows from Cauchy-SchwarzD. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 7 / 16
From reflection positivity to gaussian domination
L2: RP ⇒ gaussian domination
Z(v) 6 Z(0)
Suppose (v1, v2) is maximiser. Then (v1, Rv1) is also maximiser by RP
Pv1 v2
(Periodic boundary conditions are not shown here)D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 8 / 16
From reflection positivity to gaussian domination
L2: RP ⇒ gaussian domination
Z(v) 6 Z(0)
Suppose (v1, v2) is maximiser. Then (v1, Rv1) is also maximiser by RP
Pv1 v2
(Periodic boundary conditions are not shown here)D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 8 / 16
From reflection positivity to gaussian domination
L2: RP ⇒ gaussian domination
Z(v) 6 Z(0)
Suppose (v1, v2) is maximiser. Then (v1, Rv1) is also maximiser by RP
(Periodic boundary conditions are not shown here)D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 8 / 16
From reflection positivity to gaussian domination
L2: RP ⇒ gaussian domination
Z(v) 6 Z(0)
Suppose (v1, v2) is maximiser. Then (v1, Rv1) is also maximiser by RP
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 8 / 16
From reflection positivity to gaussian domination
L2: RP ⇒ gaussian domination
Z(v) 6 Z(0)
Suppose (v1, v2) is maximiser. Then (v1, Rv1) is also maximiser by RP
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 8 / 16
From reflection positivity to gaussian domination
L2: RP ⇒ gaussian domination
Z(v) 6 Z(0)
Suppose (v1, v2) is maximiser. Then (v1, Rv1) is also maximiser by RP
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 8 / 16
From reflection positivity to gaussian domination
L2: RP ⇒ gaussian domination
Z(v) 6 Z(0)
Suppose (v1, v2) is maximiser. Then (v1, Rv1) is also maximiser by RP
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 8 / 16
From reflection positivity to gaussian domination
L2: RP ⇒ gaussian domination
Z(v) 6 Z(0)
Suppose (v1, v2) is maximiser. Then (v1, Rv1) is also maximiser by RP
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 8 / 16
From reflection positivity to gaussian domination
L2: RP ⇒ gaussian domination
Z(v) 6 Z(0)
Suppose (v1, v2) is maximiser. Then (v1, Rv1) is also maximiser by RP
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 8 / 16
From reflection positivity to gaussian domination
L2: RP ⇒ gaussian domination
Z(v) 6 Z(0)
Suppose (v1, v2) is maximiser. Then (v1, Rv1) is also maximiser by RP
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 8 / 16
From reflection positivity to gaussian domination
L2: RP ⇒ gaussian domination
Z(v) 6 Z(0)
Suppose (v1, v2) is maximiser. Then (v1, Rv1) is also maximiser by RP
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 8 / 16
From reflection positivity to gaussian domination
L2: RP ⇒ gaussian domination
Z(v) 6 Z(0)
Suppose (v1, v2) is maximiser. Then (v1, Rv1) is also maximiser by RP
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 8 / 16
From reflection positivity to gaussian domination
L2: RP ⇒ gaussian domination
Z(v) 6 Z(0)
Suppose (v1, v2) is maximiser. Then (v1, Rv1) is also maximiser by RP
There is a space-invariant maximiser!
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 8 / 16
From gaussian domination to the infrared bound
L3: GD ⇒ infrared bound
κ(k) 61
βε(k)
Z(v) =
∫(SN−1)Λ
dσ exp{−β∑
(σx−σy)2−2β(σ,∆v)−β∑
(vx−vy)2}
Z(v) 6 Z(0) ⇔ 〈 e2β(σ,∆v) 〉 6 e−β(v,∆v)
Take vx = cos kx, for fixed k ∈ Λ∗. Then −∆v = ε(k)vFor small field ηv, the inequality becomes
1 + η2β2ε(k)2κ(k)‖v‖2 +O(η4) 6 1 + η2βε(k)‖v‖2 +O(η4)
This implies the IRBD. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 9 / 16
From the infrared bound to spontaneous magnetisation
Lemma 4: IRB ⇒ theorem1
|Λ|κ(0) >
1
3− 1
β|Λ|∑
k∈Λ∗\{0}
1
ε(k)
We have 〈(σ30)2〉 = 1
3 ; by the inverse Fourier transform,
1
3= κ(0) =
1
|Λ|κ(0) +
1
|Λ|∑
k∈Λ∗\{0}
κ(k)
The lemma follows immediately
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 10 / 16
Quantum Heisenberg models
Lattice Λ = {−L2 + 1, . . . , L2 }
d with periodic b.c. as beforeHilbert space HΛ =
⊗x∈Λ C2S+1, S ∈ 1
2N
Spin operators S1, S2, S3 on C2 such that [S1, S2] = iS3, etc...Six = Si ⊗ IdΛ\{x}Hamiltonian:
H(u)Λ = −
∑{x,y}∈E
(S1xS
1y + uS2
xS2y + S3
xS3y
)u = 1: Heisenberg ferromagnet
u = −1: unitarily equivalent to Heisenberg antiferromagnet
u = 0: quantum XY model
Gibbs state 〈a〉 = Tr a e−βH(u)Λ
/Tr e−βH
(u)Λ
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 11 / 16
Difficulties in the quantum case
Let κ(x) = 〈S30S
3x〉. The infrared bound cannot hold!
Indeed, in the case S = 12 ,
〈~S0 · ~S0︸ ︷︷ ︸=3/4
〉 − 〈~S0 · ~Sx︸ ︷︷ ︸6 1/4
〉 > 12
However, assuming the infrared bound to hold,∣∣〈~S0 · ~S0〉 − 〈~S0 · ~Sx〉∣∣ =
1
|Λ|
∣∣∣∑k∈Λ∗
(1− eikx )κ(k)∣∣∣
61
|Λ|const
β
∑k∈Λ∗\{0}
|1− eikx |ε(k)
which goes to 0 as β →∞, contradiction
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 12 / 16
Extension to quantum systems
Theorem [Dyson, Lieb, Simon ’78] Assume that u ∈ [−1, 0]. Then
1
|Λ|2∑x,y∈Λ
〈S3xS
3y〉 > 1
3S(S + 1)− 1√2|Λ|
∑k∈Λ∗\{0}
√εu(k)
ε(k)
− 1
2β|Λ|∑
k∈Λ∗\{0}
1
ε(k)
where εu(k) =∑d
i=1
((1− u cos ki)〈S1
0S1ei〉+ (u− cos ki)〈S2
0S2ei〉)
Since εu(k) 6 S(S + 1), the lower bound is positive for S large enough
A better lower bound was proposed in [Kennedy, Lieb, Shastry ’88]
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 13 / 16
Changes in the quantum case
The proof follows [Frohlich, Simon, Spencer ’76] but with some changes
First, the infrared bound holds for the Duhamel two-point function:
(S30 , S
3x) =
1
Z(Λ)
∫ β
0TrS3
0 e−sH(u)Λ S3
x e−(β−s)H(u)Λ
Unlike 〈S30S
30〉, we do not have (S3
0 , S30) = const
Solution: Falk-Bruch inequality [Falk, Bruch ’69]. WithΦ(s) =
√s coth 1√
s,
2〈A∗A+AA∗〉〈[A∗, [H,A]]〉
6 Φ( 4(A,A)
〈[A∗, [H,A]]〉
)This allows to transfer the infrared bound for the Duhamel function, toa lower bound for the usual two-point function
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 14 / 16
Reflection positivity for quantum systems
LemmaHilbert space H = H1 ⊗H2, finite-dimensional, with H1 ' H2.Matrices A,B,Ci, Di in small space H1. Then∣∣∣TrH eA⊗1l+1l⊗B−
∑k(Ci⊗1l−1l⊗Di)2∣∣∣2 6 TrH eA⊗1l+1l⊗A−
∑k(Ci⊗1l−1l⊗Ci)2
· TrH eB⊗1l+1l⊗B−∑k(Di⊗1l−1l⊗Di)
2
Here, A is the complex conjugate of A
This gives the restriction u ∈ [−1, 0], which excludes the quantumHeisenberg ferromagnet!
More general approach in [Frohlich, Israel, Lieb, Simon ’78–80]
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 15 / 16
Conclusion
Reflection positivity allows to get infrared bound on spincorrelations, which allows to prove the occurrence of phasetransitions in models with continuous symmetries
Applies to classical and quantum models
Does not apply to irregular lattices, nor to quantum Heisenbergferromagnet
Gives interesting results for quantum systems in 2D [Neves, Perez
’86]
Further developments in statistical physics: chessboard estimates;matrix models (spin nematics); random loop models
Our knowledge of statistical physics would be much poorerwithout the methods of reflection positivity!
THANK YOU!
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 16 / 16
Conclusion
Reflection positivity allows to get infrared bound on spincorrelations, which allows to prove the occurrence of phasetransitions in models with continuous symmetries
Applies to classical and quantum models
Does not apply to irregular lattices, nor to quantum Heisenbergferromagnet
Gives interesting results for quantum systems in 2D [Neves, Perez
’86]
Further developments in statistical physics: chessboard estimates;matrix models (spin nematics); random loop models
Our knowledge of statistical physics would be much poorerwithout the methods of reflection positivity!
THANK YOU!
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 16 / 16
Conclusion
Reflection positivity allows to get infrared bound on spincorrelations, which allows to prove the occurrence of phasetransitions in models with continuous symmetries
Applies to classical and quantum models
Does not apply to irregular lattices, nor to quantum Heisenbergferromagnet
Gives interesting results for quantum systems in 2D [Neves, Perez
’86]
Further developments in statistical physics: chessboard estimates;matrix models (spin nematics); random loop models
Our knowledge of statistical physics would be much poorerwithout the methods of reflection positivity!
THANK YOU!
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 16 / 16
Conclusion
Reflection positivity allows to get infrared bound on spincorrelations, which allows to prove the occurrence of phasetransitions in models with continuous symmetries
Applies to classical and quantum models
Does not apply to irregular lattices, nor to quantum Heisenbergferromagnet
Gives interesting results for quantum systems in 2D [Neves, Perez
’86]
Further developments in statistical physics: chessboard estimates;matrix models (spin nematics); random loop models
Our knowledge of statistical physics would be much poorerwithout the methods of reflection positivity!
THANK YOU!
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 16 / 16
Conclusion
Reflection positivity allows to get infrared bound on spincorrelations, which allows to prove the occurrence of phasetransitions in models with continuous symmetries
Applies to classical and quantum models
Does not apply to irregular lattices, nor to quantum Heisenbergferromagnet
Gives interesting results for quantum systems in 2D [Neves, Perez
’86]
Further developments in statistical physics: chessboard estimates;matrix models (spin nematics); random loop models
Our knowledge of statistical physics would be much poorerwithout the methods of reflection positivity!
THANK YOU!
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 16 / 16
Conclusion
Reflection positivity allows to get infrared bound on spincorrelations, which allows to prove the occurrence of phasetransitions in models with continuous symmetries
Applies to classical and quantum models
Does not apply to irregular lattices, nor to quantum Heisenbergferromagnet
Gives interesting results for quantum systems in 2D [Neves, Perez
’86]
Further developments in statistical physics: chessboard estimates;matrix models (spin nematics); random loop models
Our knowledge of statistical physics would be much poorerwithout the methods of reflection positivity!
THANK YOU!
D. Ueltschi (Univ. Warwick) RP and Phase Transitions Zurich, 23.5.16 16 / 16