asymptotic foel for the heisenberg model on boxes...review heisenberg model ferromagnetic ordering...

Review Heisenberg modelFerromagnetic ordering of energy levels

Linear Spin Wave Approx

Asymptotic FOEL for the Heisenberg model onboxes

Mathematical Many Body Theory, June 13–18

BCAM, Bilbao, Spain

Shannon Starr

University of Alabama at Birmingham

June 16, 2016

https://arxiv.org/abs/1509.00907

I Bruno Nachtergaele, U. C. Davis,

I Wolfgang Spitzer, FernUniversitat Hagen.

1 / 28

https://arxiv.org/abs/1509.00907



Outline

(1) Review Heisenberg model on graph G = (V ,E )

(2) Define the “ferromagnetic ordering of energy levels” condition(FOEL)

(3) Linear spin wave approximation for {1, . . . , L}d ⊆ Zd atenergies on the order of 1/L2

2 / 28



DefinitionHistoryUsual Ordering of Energy Levels

1. Define Heisenberg model

Graph: G = (V ,E ), |V | <∞,

E ⊆{{x , y} : x ∈ V , y ∈ V , x 6= y

}.

Let: V = {x1, . . . , xN} with |V | = N.

Hilbert space: HV = (C2)⊗N = C2 ⊗ · · · ⊗ C2.

S (1) =

[0 1/2

1/2 0

], S (2) =

[0 −i/2i/2 0

], S (3) =

[1/2 0

0 −1/2

]Operators S

(a)xk : HV → HV ,

S(a)xk = 1C2⊗· · ·⊗1C2⊗S (a)⊗1C2⊗· · ·⊗1C2 = 1

⊗(k−1)C2 ⊗S (a)⊗1⊗N−kC2 ,

for a ∈ {1, 2, 3}, k ∈ {1, . . . ,N}.3 / 28



DefinitionHistoryUsual Ordering of Energy Levels[

S(1)x , S

(2)y

]=

i

2δx ,y S

(3)x , and cyclic permutations.

Define S(a)V =

∑x∈V S

(a)x .

For each {x , y} ∈ E , define hx ,y : HV → HV ,

hx ,y =1

41− ~Sx · ~Sy =

1

41−

3∑a=1

S(a)x S

(a)y .

Then the total Hamiltonian is

HG =∑{x ,y}∈E

hx ,y .

This is a model such that[S

(a)V , HG

]= 0 for a = 1, 2, 3.

4 / 28




Brief review: history for Heisenberg model

I Frohlich, Simon and Spencer (Comm. Math. Phys., 1976)• classical Heisenberg ferromagnet• proved LRO for d ≥ 3 and sufficiently small T > 0• invented reflection positivity technique to prove• Gaussian domination for 2-point correlation functions.

I Dyson, Lieb, Simon (J. Stat. Phys., 1978) proved the quantumantiferromagnet is reflection positivity, not ferromagnet.See, also, Eugene Speer, (Lett. Math. Phys., 1985.)

I Physicists use the linear spin wave approximation.Mathematicians have worked to justify this:Correggi, Giuliani, Seiringer, (Comm. Math. Phys. 2015).

5 / 28




Usual ordering of energy levels: Lieb-Mattis theorem

Suppose G is bipartite: G = (V ,E ) with V = A t B and

E ⊆{{x , y} : x ∈ A , y ∈ B

},

and assume |A| = |B|, balanced.

Then Lieb and Mattis (J. Math. Phys., 1962) proved that, for theantiferromagnet −HG , the ground state is a spin singlet.

More generally, they allow antiferromagnetic couplings between thetwo parts and ferromagnetic couplings with single parts

HV (J) =∑{x ,y}∈E

Jx ,yhx ,y .

Jx ,y ≤ 0 if (x , y) ∈ (A× B) ∪ (B × A),Jx ,y ≥ 0 if (x , y) ∈ (A× A) ∪ (B × B).

6 / 28




For each s ∈ {0, 1, . . . ,N/2}, define the total-spin s subspace

HtotspV (s) = ker

(3∑

a=1

(S

(a)V

)2− s(s + 1)1

).

Then HGHtotspV (n) ⊆ Htotsp

V by SU(2) symmetry.

Lieb and Mattis also proved

min spec(HG � Htotsp

V (0))≤ . . . ≤ min spec

(HG � Htotsp

V (1

2N)).

More generally if |A| and |B| are not balanced, then they provedthe ground state has total spin 1

2

∣∣|A| − |B|∣∣ and

.min spec(HG � Htotsp

V

(∣∣|A| − |B|∣∣2

))≤ . . . ≤ min spec

(HG � Htotsp

V (N

2)).

7 / 28



Spectral gapDigression: CLR theoremThe FOEL property

Ferromagnetic ordering of energy levels

Denote E0(G , s) = min spec(HG � Htotsp

V (s))

.

For G = (V ,E ), N = |V | the ground state energy is E0(G , 12N).

For open chains

V = {1, . . . , L} ⊂ Z , E ={{k, k + 1} : 1 ≤ k ≤ L− 1

},

Koma and Nachtergaele (Lett. Math. Phys., 1997) proved that thespectral gap is E0([1, L], 1

2L− 1).

So

E0

([1, L],

1

2L

)≤ E0

([1, L],

1

2L− 1

)≤ min

s< 12L−1E0 ([1, L], s) .

8 / 28




Induction argument

Consider increasing chain from [1, L] to [1, L + 1].

The definition of each pair interaction

hx ,y =1

4−~Sx ·~Sy =

01/2 −1/2−1/2 1/2

0

x ,y

⊗⊗

z 6∈{x ,y}

(1C2)z ,

is positive semi-definite and has ground state energy 0.

So E0([1, L], 12L) = 0.

Koma and Nachtergaele calculatedE0([1, L], 1

2L− 1) = 1− cos(π/L) = 2 sin2(π2L

),

strictly decreasing.9 / 28




Suppose ψ ∈ Htotsp[1,L+1](

12 (L + 1)− 2) is an eigenvector of H[1,L+1].

Then

∃ψ1, ψ2 ∈ Htotsp[1,L]

(1

2L− 2

)+Htotsp

[1,L]

(1

2L− 1

),

such that ψ = ψ1 ⊗ |↑〉+ ψ2 ⊗ |↓〉,where |↑〉 and |↓〉 denote the standard basis of C2.

So, since hL,L+1 ≥psd 0, this means H[1,L] ≤psd H[1,L+1].

〈ψ , H[1,L+1]ψ〉 ≥ 〈ψ , H[1,L]ψ〉 = 〈ψ1 , H[1,L]ψ1〉+ 〈ψ2 , H[1,L]ψ2〉 .

But for k ∈ {1, 2},

〈ψk , H[1,L]ψk〉‖ψk‖2

≥ min

({E0

([1, L],

1

2L− 2

), E0

([1, L],

1

2L− 1

)})10 / 28




So this proves

E0

([1, L + 1],

1

2(L + 1)− 2

)≥ min

({E0

([1, L],

1

2L− 2

), E0

([1, L],

1

2L− 1

)})If, as an induction hypothesis, we assume

E0

([1, L],

1

2L− 1

)≤ E0

([1, L],

1

2L− 2

),

then this means

E0

([1, L + 1],

1

2(L + 1)− 2

)≥ E0

([1, L],

1

2L− 1

).

11 / 28




But Koma and Nachtergaele showed

E0

([1, L + 1],

1

2(L + 1)− 1

)≤ E0

([1, L],

1

2L− 1

),

i.e., sin2( π2(L+1) ) ≤ sin2( π2L).

So we deduce the induction step

E0

([1, L + 1],

1

2(L + 1)− 2

)≥ E0

([1, L + 1],

1

2(L + 1)− 1

).

12 / 28




Aldous’s conjec; Caputo, Liggett, Richthammer’s thm

Handjani and Jungreis rediscovered this argument(J. Theor. Prob., 1996).

The Heisenberg ferromagnet is unitarily equivalent to the SEP.

Aldous conjectured, for any graph G = (V ,E ), with N = |V |,

E0

(G ,

1

2N − 1

)≤ min

s< 12N−1E0(G , s) .

Caputo, Liggett and Richthammer proved this (J. A. M. S., 2010).

13 / 28




CLR’s network reduction

Caputo, Liggett and Richthammer consider weighted graphs.

Rates cxy ≡ coupling constants Jxy .

Network reduction G = (V , c) −→ Gz = (V \ {z} , c(z))

∀x , y ∈ V \ {z} , c(z)xy = cxy +

cxzcyz∑w∈V \{z} cwz

Example:1

1

1

1

x3

x1

x2

x4

−→11

1/2x3

x1

x2

14 / 28




The definition of property “FOEL-n”

We say a graph G = (V ,E ), with N = |V |, satisfies FOEL-n if

E0

(G ,

1

2N − n

)≤ min

s< 12N−nE0(G , s) .

The FOEL-n property is a property that G may or may not satisfy,for each n ∈ {0, . . . , b1

2Nc}.

By the same reasoning as Koma and Nachtergaele’s argument,

I if we can grow our graph one vertex at a timeG2n,G2n+1, . . . ,GN = G ,

I and if E0(G2n, 0) ≥ E0(G2n+1,12 ) ≥ · · · ≥ E0(G , 1

2N − n),

I then G satisfies FOEL-n.

15 / 28




Example [1, L] a la Koma and Nachtergaele

Using this argument, the graphs Koma and Nachtergaeleconsidered, [1, L] ⊆ Z, satisfy FOEL-n for each n ≤ b1

2Lc.

Note, by the CLR theorem, all graphs satisfy FOEL-1.(By Lieb-Mattis all graphs also satisfy FOEL-0.)

Koma and Nachtergaele actually considered quantum groupSUq(2) = Uq(sl2) symmetric XXZ model for q ∈ (0,∞).

This representation comes with dual canonical basis, thatTemperley and Lieb called “Hulthen bracket basis.” In this basisone has tree-like1(?) behavior of E0([1, L], 1

2L− n) for all n.

See Nachtergale, Spitzer, S (J. Stat. Phys., 2004).1Wielandt minimax for Perron-Frobenius roots

16 / 28




Counterexamples for n ≥ 2

Since FOEL-1 (as well as FOEL-0) is true for all graphs, is FOEL-2?

No, not for the hexagon.

Let CN be G = (V ,E ) for V = {1, . . . ,N} and

E ={{1, 2} , {2, 3} , . . . , {N − 1,N} , {1,N}

}.

Then C2n violates FOEL-(n − 1) for n > 2:true for n = 3,numerically verified by Lanczos iteration for n = 4, 5, 6, 7(Tran,Spitzer,S; J. Math. Phys.,2012),and n = 8 (Dhar and Shastry, Phys. Rev. Lett., 2000).

Bethe’s ansatz question (Sutherland, Phys. Rev. Lett., 1995).

17 / 28




Asymptotic FOEL-n

If you have a sequence of graphs G2n,G2n+1, . . . , and if

E0

(GN ,

1

2N − n

)≤ min

M∈{2n,...,N}E0

(GM ,

1

2M − n

),

then GN satisfies FOEL-n.

Lemma Suppose there are constants p > 0 and 0 = C0 < C1 < . . .such that the sequence of graphs above satisfy

E0

(GN ,

1

2N − n

)∼ CnN

−p , as N →∞,

for each n ∈ {0, 1, . . . }.Then, for each n ∈ {0, 1, . . . }, there is Nn such that GN satisfiesFOEL-n for each N ≥ Nn.

18 / 28



Energies of O(1/L2) on boxes Bd (L)Toth representationFilling in

LSW approximation for energies of order 1/L2

Consider a box Bd(L) = {1, . . . , L}d ⊂ Zd .

Then the energies of order 1/L2 in Htotsp(Bd(L), 12L

d − n) areasymptotically

π2

2L2

(‖~κ1‖2 + · · ·+ ‖~κn‖2

)ranging over ordered n-tuples (~κ1, . . . , ~κn) ∈ (Zd \ {0})n, modulothe action of Sn.

In particular, E0

(Bd(L),

1

2Ld − n

)∼ nπ2

2L2.

This means E0(GN ,12N − n) ∼ CnN

−p for Cn = nπ2/2, p = 2/d .

19 / 28




Disclaimer

If you prove the claim, you do not actually need the inductiveargument for FOEL.

Correggi, Giuliani and Seiringer (Comm. Math. Phys., 2015),“Validity of the Spin-Wave Approximation for the Free Energy ofthe Heisenberg Ferromagnet,” proved ∃c > 0 such that for all Land all n

E0

(Bd(L) ,

1

2Ld − n

)≥ cn

L2.

20 / 28




Graph Laplacian of the configuration graph

Given a graph G = (V ,E ), the graph Laplacian is the operator−∆G : `2(V )→ `2(V )

−∆G f (x) =1

2

∑y∈V

1E ({x , y})[f (x)− f (y)] ,

so that

〈f , −∆G f 〉 =1

2

∑{x ,y}∈E

‖f (x)− f (y)‖2 .

One may define a new graph Φn(G ) = (V n,Φ(n)(E )), whereΦ(n)(E ) is the set of all {(x1, . . . , xn), (y1, . . . , yn)},

∃k ∈ {1, . . . , n} , s.t. {xk , yk} ∈ E , and xj = yj for j 6= k.

21 / 28




The configuration graph

The n particle configuration graph isΘ(n)(G ) = (Θ(n)(V ),Θ(n)(E )), where

Θ(n)(V ) ={

(x1, . . . , xn) ∈ V n : j 6= k ⇒ xj 6= xk

},

and Θ(n)(E ) is the set of edges

{(x1, . . . , xn), (y1, . . . , yn)} ∈ Φ(n)(E ) ,

such that (x1, . . . , xn), (y1, . . . , yn) ∈ Θ(n)(V ).

22 / 28




Then, for any function f ∈ `2(Θ(n)(V )) which is symmetric

∀π ∈ Sn , f (xπ(1), . . . , xπ(n)) = f (x1, . . . , xn) ,

we can map T(n)V : `2

sym(Θ(n)(V ))→ HV ∩ ker(S(3)V − 1

2N + n) by

T(n)V f =

1√n!

∑(x1,...,xn)∈Θ(n)(V )

f (x1, . . . , xn)S−x1· · · S−xn

(⊗x∈V|↑〉x

),

where S±x = S(1)x ± iS

(2)x .

Then T(n)V is a unitary transformation, and

HGT(n)V f = T

(n)V

(−∆Θ(n)(G)f

).

23 / 28




Another tool from CGS, and “filling in”

If we know the spectrum of −∆G then it is trivial to determine thespectrum of −∆Φ(n)(G).

Given a f ∈ `2(Θ(n)(V )) with “low energy” we could extend byzero to `2(Φ(n)(V )).

Then we have to worry about∑x∈Θ(n)(V )

∑y∈Φ(n)(V )\Θ(n)(V )

1Φ(n)(E)({x, y})‖f (x)− f (y)‖2 .

For G = Bd(L), with d ≥ 3, Correggi, Giuliani and Seiringer haveanother powerful result: for any eigenfunction Ψ of HG witheigenvalue λ,

‖ρ‖∞ ≤ Cλd‖ρ‖1 ,

where ρ(x , y) = 〈Ψ , S−x S+x S−y S+

y Ψ〉 is the 2-particle density.24 / 28




If d ≤ 2, then we can “fill in.”

For any x ∈ Φ(n)(V ) \Θ(n)(V ), just define f (x) to be an averageof f (y) for nearby y ∈ Θ(n)(V ).

25 / 28




26 / 28




We raise the energy by a constant factor (depending on n).

But we just use the energy bound with the Chebyshev inequality torestrict the number of eigenmodes we must keep to have a goodapproximation to f in the eigenvector decomposition relative to−∆Φ(n)(Bd (L)).

We already have uniform bounds for eigenfunctions of−∆Φ(n)(Bd (L)) and their norms restricted to

Φ(n)(Bd(L)) \Θ(n)(Bd(L)) are small.

Then eigenfunctions f of −∆Θ(n)(Bd (L)) with energy C/L2 must beclose in norm to linear combinations of of eigenfunctions of−∆Φ(n)(Bd (L)).

So the min-max theorems prove that the restrictions of the spectrato [0,C/L2] are close.

27 / 28




Thanks for your attention!

28 / 28

asymptotic foel for the heisenberg model on boxes...review heisenberg model ferromagnetic ordering...

Documents