Susceptibility of the Two-Dimensional Ising Model · 1.De nition of 2D Ising Model 2.Why is the nearest neighbor zero- eld 2D Ising model exactly solvable? (Lars Onsager and Bruria

Susceptibility of the Two-Dimensional Ising Model

Craig A. TracyUC Davis

September 2014

Outline

1. Definition of 2D Ising Model

2. Why is the nearest neighbor zero-field 2D Ising model exactlysolvable? (Lars Onsager and Bruria Kaufman, 1944–1949)

3. Spontaneous magnetization—some interesting history of mathematics

4. Toeplitz determinants and spin-spin correlation functions

5. Massive scaling limit and connection with Painleve III

6. The Ising susceptibility and the natural boundary conjecture

Only #6 reports on new developments—joint with Harold Widom

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Consider the square lattice Z2. A spin configuration σ is an assignment of±1 to each site (i , j) ∈ Z2; that is, σ ∈ {−1, 1}Z2

.

The energy of a configuration σ in box Λ is

EΛ(σ) = −J1

∑i ,j∈Λ

σi jσi j+1 − J2

∑i ,j∈Λ

σi jσi+1 j − h∑i ,j∈Λ

σi j

The Gibbs measure gives the probability of configuration σ in box Λ atinverse temperature β:

PΛ(σ) :=exp(−βE(σ))

ZΛ(β, h), ZΛ is called the partition function

Remarks:

I We assume Ji > 0 so the system is ferromagnetic, i.e. like spinsfavored.

I The coefficient h gives the coupling of the system to an externalmagnetic field.

I In practice we will take periodic boundary conditions which means thesystem is defined on a torus with m rows and n columns.

I This defines the 2D Ising model with nearest neighbor interactions onthe square lattice in a magnetic field.


.


EΛ(σ) = −J1

∑i ,j∈Λ

σi jσi j+1 − J2

∑i ,j∈Λ


σi j




Remarks:






.


EΛ(σ) = −J1

∑i ,j∈Λ

σi jσi j+1 − J2

∑i ,j∈Λ


σi j




Remarks:






.


EΛ(σ) = −J1

∑i ,j∈Λ

σi jσi j+1 − J2

∑i ,j∈Λ


σi j




Remarks:






.


EΛ(σ) = −J1

∑i ,j∈Λ

σi jσi j+1 − J2

∑i ,j∈Λ


σi j




Remarks:






.


EΛ(σ) = −J1

∑i ,j∈Λ

σi jσi j+1 − J2

∑i ,j∈Λ


σi j




Remarks:






.


EΛ(σ) = −J1

∑i ,j∈Λ

σi jσi j+1 − J2

∑i ,j∈Λ


σi j




Remarks:





Thermodynamic quantities and correlation functions

I Free energy per lattice site

−βf (β, h) = lim|Λ|→∞

1

|Λ|logZΛ(β, h)

I Magnetization and spontaneous magnetization

M(β, h) := −∂f∂h, M0(β) := lim

h→0+M(β, h)

I Susceptibility

χ(β, h) =∂M

∂h(β, h)

I Spin-spin correlation function

〈σ00σMN〉 = lim|Λ|→∞

EΛ (σ00σMN) = lim|Λ|→∞

∑σ σ00σMN e−βE(σ)∑

σ e−βE(σ)



−βf (β, h) = lim|Λ|→∞

1

|Λ|logZΛ(β, h)


M(β, h) := −∂f∂h, M0(β) := lim

h→0+M(β, h)

I Susceptibility

χ(β, h) =∂M

∂h(β, h)





σ e−βE(σ)



−βf (β, h) = lim|Λ|→∞

1

|Λ|logZΛ(β, h)


M(β, h) := −∂f∂h, M0(β) := lim

h→0+M(β, h)

I Susceptibility

χ(β, h) =∂M

∂h(β, h)





σ e−βE(σ)



−βf (β, h) = lim|Λ|→∞

1

|Λ|logZΛ(β, h)


M(β, h) := −∂f∂h, M0(β) := lim

h→0+M(β, h)

I Susceptibility

χ(β, h) =∂M

∂h(β, h)





σ e−βE(σ)

Why is the zero-field h = 0 2D Ising model exactly solvable?

Or stated differently, what’s the problem for h 6= 0?

Since we don’t know f (β, h) as a function of h, how do we findM0 and χ(β, 0)?

Second question a bit easier to answer:

M20 (β) = lim

N→∞〈σ00σNN〉

zero-field susceptibility: χ(β) =∑

M,N∈Z

[〈σ00σMN〉 −M2

0

]





M20 (β) = lim



M,N∈Z

[〈σ00σMN〉 −M2

0

]





M20 (β) = lim



M,N∈Z

[〈σ00σMN〉 −M2

0

]

Method of Transfer Matrices

On a torus of m rows and n columns can write

Zmn(β, h) = Tr(Vm) (?)

where V is a 2n × 2n matrix.

To see this let ~sα represent the configuration of row α,~sα = (s1, s2, . . . , sn), sj = ±1. Define the 2n × 2n matrix V1 byincorporating the Boltzmann factors in row α:

V1(~s, ~s ′) = δ~s,~s ′ ·n∏

α=1

eβJ1sαs′α+1

For Boltzmann factors on columns we introduce

V2(~s, ~s ′) =n∏

j=1

eβJ2sj s′j

and for the magnetic field

V3(~s, ~s ′) = δ~s,~s ′ ·n∏

j=1

eβhsj

Method of Transfer Matrices

On a torus of m rows and n columns can write


where V is a 2n × 2n matrix.

To see this let ~sα represent the configuration of row α,~sα = (s1, s2, . . . , sn), sj = ±1. Define the 2n × 2n matrix V1 byincorporating the Boltzmann factors in row α:

V1(~s, ~s ′) = δ~s,~s ′ ·n∏

α=1

eβJ1sαs′α+1

For Boltzmann factors on columns we introduce

V2(~s, ~s ′) =n∏

j=1

eβJ2sj s′j

and for the magnetic field

V3(~s, ~s ′) = δ~s,~s ′ ·n∏

j=1

eβhsj

DefiningV = V1V2V3

it is easy to check that


Thus the problem “reduces” to the spectral theory of V ; or more precisely,the largest eigenvalue of V in computing f (β, h) in the thermodynamiclimit.

For h 6= 0 this is an open problem.

For h = 0; namely, V3 = I , a diagonalization was first accomplished byLars Onsager (1944). His analysis was subsequently simplified by BruriaKaufman (1949).

It is Kaufman’s point of view which we now summarize.

DefiningV = V1V2V3

it is easy to check that


Thus the problem “reduces” to the spectral theory of V ; or more precisely,the largest eigenvalue of V in computing f (β, h) in the thermodynamiclimit.

For h 6= 0 this is an open problem.

For h = 0; namely, V3 = I , a diagonalization was first accomplished byLars Onsager (1944). His analysis was subsequently simplified by BruriaKaufman (1949).

It is Kaufman’s point of view which we now summarize.

Kaufman’s Analysis

Stated concisely here is what Kaufman realized:

The 2n × 2n matrices V1 and V2; and hence V = V1V2, are spinrepresentations of rotations in the orthogonal group O(2n).Furthermore, V is a spin representative of a product ofcommuting plane rotations. Thus the spectral analysis is reducedto that of a 2n × 2n orthogonal matrix. Indeed, due to thetranslational invariance of the interaction energy, the spectraltheory reduces to solving quadratic equations! 1

V is not a spin-representative of a rotation when h 6= 0.

1Actually, this statement is true for a certain direct sum decompositionV = V+ ⊕ V−. The statements apply to V±.

Combinatorial Approach to 2D Ising Model

Kasteleyn’s theory (1963) of dimers on planar lattices: Partition functionis expressible as a Pfaffian.2

Fisher (1966) building on work of Kac and Ward (1952) showed the 2DIsing model (as defined above) is equivalent to a dimer problem.

Figure: A six-site cluster that may be used to convert the Ising problem into adimer problem. See McCoy & Wu, The Two-Dimensional Ising Model for details.

2For modern treatment see work of Rick Kenyon.

The Spontaneous Magnetization: Some history3

Onsager, well-known for being cryptic, announced in a discussion sectionat a conference in Florence (1949)4 that he and Kaufman had recentlyobtained an exact formula for the spontaneous magnetization:

M0 = (1− k2)1/8, k := (sinh 2βJ1 sinh 2βJ2)−1 (??)

Onsager gave no details to how he and Kaufman obtained (??).

Figure: Spontaneous magnetization. k = 1 defines the critical temperature.

3See, P. Deift, A. Its and I. Krasovsky, Toeplitz Matrices and Toeplitz Determinantsunder the Impetus of the Ising Model: Some History and Some Recent Results.

4And on a blackboard at Cornell on 23 August 1948.

What was later revealed was that Onsager and Kaufman first showed that〈σ00σ0N〉 was expressible as a Toeplitz determinant. Unpublished, theyalso showed the diagonal correlation is a Toeplitz determinant and theexpression is a bit simpler:

〈σ00σNN〉 = det (ϕm−n)m,n=0,...,N−1

with

ϕm =1

2π

∫ 2π

0e− imθ ϕ(ei θ) dθ, ϕ(z) =

[1− k/z

1− kz

]1/2

Today we know the strong Szego limit theorem (plus some conditions onϕ)

limN→∞

det(ϕm−n)

µN= exp

( ∞∑k=1

k(logϕ)k(logϕ)−k

)and a simple application gives M2

0 .

But Szego had not yet proved his strong limit theorem (1952)!


〈σ00σNN〉 = det (ϕm−n)m,n=0,...,N−1

with

ϕm =1

2π

∫ 2π


[1− k/z

1− kz

]1/2


limN→∞

det(ϕm−n)

µN= exp

( ∞∑k=1



0 .



〈σ00σNN〉 = det (ϕm−n)m,n=0,...,N−1

with

ϕm =1

2π

∫ 2π


[1− k/z

1− kz

]1/2


limN→∞

det(ϕm−n)

µN= exp

( ∞∑k=1



0 .



〈σ00σNN〉 = det (ϕm−n)m,n=0,...,N−1

with

ϕm =1

2π

∫ 2π


[1− k/z

1− kz

]1/2


limN→∞

det(ϕm−n)

µN= exp

( ∞∑k=1



0 .


I Onsager had, in fact, derived (nonrigorously) the limit formula. Hecommunicated this result to Shizuo Kakutani who communicated theresult to Szego. (I verified this story when I met, many years later,Kakutani at Yale.)

I Before all this was revealed(!), C. N. Yang (1952) gave anindependent derivation which is a subtle perturbation argument. Toquote Yang from his Selected Works 1945–1980 :

I was thus led to a long calculation, the longest in my career. Fullof local, tactical tricks, the calculation proceeded by twists andturns. There were many obstructions. But always, after a fewdays, a new trick was somehow found that pointed to a new path.The trouble was that I soon felt I was in a maze and was not surewhether in fact, after so many turns, I was anywhere nearer thegoal than when I began. This kind of strategic overview was verydepressing, and several times I almost gave up. But each timesomething drew me back, usually a new tactical trick thatbrightened the scene, even though only locally.

Finally, after six months of work off and on, all the pieces

suddenly fitted together, producing miraculous cancellations, and I

was staring at the amazingly simple final result . . .


I Before all this was revealed(!), C. N. Yang (1952) gave anindependent derivation which is a subtle perturbation argument.

Toquote Yang from his Selected Works 1945–1980 :






I Before all this was revealed(!), C. N. Yang (1952) gave anindependent derivation which is a subtle perturbation argument. Toquote Yang from his Selected Works 1945–1980 :





Spin-spin correlation functions

I Though both the row and diagonal correlations are expressible asToeplitz determinants, there is no Toeplitz representation for thegeneral case 〈σ00σMN〉.

I If we use the Case-Geronimo-Borodin-Okounkov (CGBO) formulathat expresses a Toeplitz determinant as a Fredholm determinant(times a normalization constant), we arrive at new representations forthese correlations. It is this type of expression that generalizes. ForT < Tc it takes the form

〈σ00σMN〉 =M20 det(I − KMN)

This was first derived by Wu, McCoy, Barouch & CT (1976) and thenput on a rigorous footing by Palmer and CT (1981).5

5See, Planar Ising Correlations by John Palmer, Progress in Mathematical Physics,2007.

Spin-spin correlation functions

I Though both the row and diagonal correlations are expressible asToeplitz determinants, there is no Toeplitz representation for thegeneral case 〈σ00σMN〉.

I If we use the Case-Geronimo-Borodin-Okounkov (CGBO) formulathat expresses a Toeplitz determinant as a Fredholm determinant(times a normalization constant), we arrive at new representations forthese correlations. It is this type of expression that generalizes. ForT < Tc it takes the form

〈σ00σMN〉 =M20 det(I − KMN)

This was first derived by Wu, McCoy, Barouch & CT (1976) and thenput on a rigorous footing by Palmer and CT (1981).5

5See, Planar Ising Correlations by John Palmer, Progress in Mathematical Physics,2007.

Massive Scaling Functions & Universality

I Most interest lies for T , the temperature, close to the criticaltemperature Tc . It is in this limit we expect to see universality.

I Precisely, let ξ(T ) denote the correlation length, which is known todiverge at T → T±c , then the massive scaling limit6 from below Tc is

F−(r) := limscaling

〈σ00σMN〉M2

0

= det(I − K<)

where “limscaling” is

M,N →∞, ξ(T )→∞ such that r :=√

M2 + N2/ξ(T ) is fixed.

I Somewhat similar formula exists for F+(r).I Note that the scaling functions are rotationally invariant.I The scaling functions F± are expected to be universal for a large class

of 2D ferromagnetic systems. There is no proof (as far as I know),but it is generally accepted by physicists using nonrigorousrenormalization group arguments.

6We state this for the case J1 = J2.





〈σ00σMN〉M2

0

= det(I − K<)


M,N →∞, ξ(T )→∞ such that r :=√M2 + N2/ξ(T ) is fixed.








〈σ00σMN〉M2

0

= det(I − K<)



I Somewhat similar formula exists for F+(r).

I Note that the scaling functions are rotationally invariant.I The scaling functions F± are expected to be universal for a large class







〈σ00σMN〉M2

0

= det(I − K<)



I Somewhat similar formula exists for F+(r).I Note that the scaling functions are rotationally invariant.

I The scaling functions F± are expected to be universal for a large classof 2D ferromagnetic systems. There is no proof (as far as I know),but it is generally accepted by physicists using nonrigorousrenormalization group arguments.






〈σ00σMN〉M2

0

= det(I − K<)






Connection with integrable differential equations, Painleve III

There are alternative expressions for F± (WMTB, 1976; MTW, 1977):

F−(r) = coshψ(r)/2 exp

(1

4

∫ ∞r

[m2 sinh2 ψ(x)−

(dψ

dx

)2]x dx

}F+(r) = (tanhψ(r)/2) F−(r)

whered2ψ

dr2+

1

r

dψ

dr=

1

2sinh(2ψ), ψ(r) ∼ 2K0(r), r →∞.

I η(r) = e−ψ(r) is a Painleve III function.

I M. Sato, T. Miwa & M. Jimbo (1978–79) gave an isomonodromydeformation analysis interpretation for the appearance of Painleve IIIin the 2D Ising model. They also derived a total system of PDEs forthe n-point scaling functions. The asymptotic analysis of these PDEsis an open problem.




(1

4

∫ ∞r

[m2 sinh2 ψ(x)−

(dψ

dx

)2]x dx

}F+(r) = (tanhψ(r)/2) F−(r)

whered2ψ

dr2+

1

r

dψ

dr=

1

2sinh(2ψ), ψ(r) ∼ 2K0(r), r →∞.






(1

4

∫ ∞r

[m2 sinh2 ψ(x)−

(dψ

dx

)2]x dx

}F+(r) = (tanhψ(r)/2) F−(r)

whered2ψ

dr2+

1

r

dψ

dr=

1

2sinh(2ψ), ψ(r) ∼ 2K0(r), r →∞.






(1

4

∫ ∞r

[m2 sinh2 ψ(x)−

(dψ

dx

)2]x dx

}F+(r) = (tanhψ(r)/2) F−(r)

whered2ψ

dr2+

1

r

dψ

dr=

1

2sinh(2ψ), ψ(r) ∼ 2K0(r), r →∞.



The problem of the Ising susceptibility

The zero-field susceptibility χ(T ) is defined by

χ(T ) :=∂M(T ,H)

∂H

∣∣∣∣H=0+

.

To distinguish between T < Tc and T > Tc we write χ− and χ+,respectively.

Sinceβ−1χ(T ) =

∑M,N∈Z

[〈σ00σMN〉 −M2

0

],

and〈σ00σMN〉 =M2

0 det(I − KMN), T < Tc ,

we study ∑M,N∈Z

[det(I − KMN)− 1]

But first some history of the problem

The problem of the Ising susceptibility

The zero-field susceptibility χ(T ) is defined by

χ(T ) :=∂M(T ,H)

∂H

∣∣∣∣H=0+

.

To distinguish between T < Tc and T > Tc we write χ− and χ+,respectively.

Sinceβ−1χ(T ) =

∑M,N∈Z

[〈σ00σMN〉 −M2

0

],

and〈σ00σMN〉 =M2

0 det(I − KMN), T < Tc ,

we study ∑M,N∈Z

[det(I − KMN)− 1]

But first some history of the problem

•M. Fisher in 1959 initiated the analysis of the analytic structure of χnear the critical temperature Tc by relating it to the long-distanceasymptotics of the correlation function at Tc (a result known to Kaufmannand Onsager).

•WMTB (1973–76) derived the exact form factor expansion of χ andrelated the the coefficients C± in the asymptotic expansion

χ±(T ) = C± |1− T/Tc |−7/4 + O(|1− T/Tc |−3/4

), T → T±c

to integrals involving a Painleve III function.

•The analysis of χ as a function of the complex variable T was initiated byGuttmann and Enting (1996). By use of high-temperature expansionsthey conjectured that χ+(T ) has a natural boundary in the complexT -plane.

•B. Nickel (1999,2000) analyzed the n-dimensional integrals appearingin the form factor expansion and identified a class of complex singularities,now called Nickel singularities, that lie on a curve and which become evermore dense with increasing n. This provides very strong support for theexistence of a natural boundary for χ±—curve is |k | = 1.



χ±(T ) = C± |1− T/Tc |−7/4 + O(|1− T/Tc |−3/4

), T → T±c






χ±(T ) = C± |1− T/Tc |−7/4 + O(|1− T/Tc |−3/4

), T → T±c






χ±(T ) = C± |1− T/Tc |−7/4 + O(|1− T/Tc |−3/4

), T → T±c




•Orrick, Nickel, Guttmann & Perk (2001) and Chan,Guttmann, Nickel & Perk (2011) on the basis of high- andlow-temperature expansions (300+ terms!) give the following conjecturefor the critical point behavior:

Set τ = 12 (√k − 1

√k)

χ± := kBTχ± = C0,± |τ |−7/4F± + B

where B is of the form (“short-distance” terms)

B =∞∑q=0

b√qc∑p=0

b(p,q)(log |τ |)pτq

with numerical approximations to b(p,q), p ≤ 5, and

F±=k14

[1 +

τ2

2− τ4

12+

(647

15360− 7C6±

5

)τ6 −

(296813

11059200− 4973C6±

3600

)τ8

+

(23723921

1238630400− 100261C6±

115200− 793C10±

210

)τ10 + · · ·

]with high-precision decimal estimates for C6± and C10±.

•Orrick, Nickel, Guttmann & Perk (2001) and Chan,Guttmann, Nickel & Perk (2011) on the basis of high- andlow-temperature expansions (300+ terms!) give the following conjecturefor the critical point behavior: Set τ = 1

2 (√k − 1

√k)

χ± := kBTχ± = C0,± |τ |−7/4F± + B


B =∞∑q=0

b√qc∑p=0



F±=k14

[1 +

τ2

2− τ4

12+

(647

15360− 7C6±

5

)τ6 −

(296813

11059200− 4973C6±

3600

)τ8

+

(23723921

1238630400− 100261C6±

115200− 793C10±

210

)τ10 + · · ·



2 (√k − 1

√k)

χ± := kBTχ± = C0,± |τ |−7/4F± + B


B =∞∑q=0

b√qc∑p=0


with numerical approximations to b(p,q), p ≤ 5,

and

F±=k14

[1 +

τ2

2− τ4

12+

(647

15360− 7C6±

5

)τ6 −

(296813

11059200− 4973C6±

3600

)τ8

+

(23723921

1238630400− 100261C6±

115200− 793C10±

210

)τ10 + · · ·



2 (√k − 1

√k)

χ± := kBTχ± = C0,± |τ |−7/4F± + B


B =∞∑q=0

b√qc∑p=0



F±=k14

[1 +

τ2

2− τ4

12+

(647

15360− 7C6±

5

)τ6 −

(296813

11059200− 4973C6±

3600

)τ8

+

(23723921

1238630400− 100261C6±

115200− 793C10±

210

)τ10 + · · ·


Diagonal susceptibility: A simpler problem7

From now on we restrict to T < Tc :

βχd =∑N∈Z

[〈σ00σNN〉 −M2

0

]

〈σ00σNN〉 = det(TN(ϕ))GCBO

= M20 det(I − KN)

KN = HN(Λ)HN(Λ−1), Λ(ξ) = ϕ−(ξ)/ϕ+(ξ) =√

(1− kξ)(1− k/ξ).

Here ϕ = ϕ+ · ϕ− is the Wiener-Hopf factorization of ϕ and HN(ψ) is theHankel operator with entries (ψN+i+j+1)i ,j≥0.

Sum to be analyzed:

S :=∞∑

N=1

[det(I − KN)− 1] .

7First introduced by Assis, Boukraa, Hassani, van Hoeiji, Maillard &McCoy (2012).

Diagonal susceptibility: A simpler problem7

From now on we restrict to T < Tc :

βχd =∑N∈Z

[〈σ00σNN〉 −M2

0

]

〈σ00σNN〉 = det(TN(ϕ))GCBO

= M20 det(I − KN)

KN = HN(Λ)HN(Λ−1), Λ(ξ) = ϕ−(ξ)/ϕ+(ξ) =√

(1− kξ)(1− k/ξ).

Here ϕ = ϕ+ · ϕ− is the Wiener-Hopf factorization of ϕ and HN(ψ) is theHankel operator with entries (ψN+i+j+1)i ,j≥0.

Sum to be analyzed:

S :=∞∑

N=1

[det(I − KN)− 1] .

7First introduced by Assis, Boukraa, Hassani, van Hoeiji, Maillard &McCoy (2012).

Proposition (T–Widom): Let HN(du) and HN(dv) be two Hankelmatrices acting on `2(Z+) with i , j entries∫

xN+i+j du(x),

∫yN+i+j dv(y)

respectively, where u and v are measures supported inside the unit circle.Set KN = HN(du)HN(dv). Then

∞∑N=1

[det(I − KN)− 1] =∞∑n=1

(−1)n

(n!)2

∫· · ·∫ ∏

i xiyi1−

∏i xiyi

×

(det

(1

1− xiyj

))2∏i

du(xi )dv(yi )

Ideas in proof:

1. First use Fredholm expansion of det(I − KN).

2. Then use the Andreief identity:

∫···

∫det(φj (xk ))j,k=1,...,N ·det(ψj (xk ))j,k=1,...,N dx1···dxN=N! det(

∫φj (x)ψk (x) dx)

j,k=1,...,N

3. Symmetrization argument

Proposition (T–Widom): Let HN(du) and HN(dv) be two Hankelmatrices acting on `2(Z+) with i , j entries∫

xN+i+j du(x),

∫yN+i+j dv(y)

respectively, where u and v are measures supported inside the unit circle.Set KN = HN(du)HN(dv). Then

∞∑N=1

[det(I − KN)− 1] =∞∑n=1

(−1)n

(n!)2

∫· · ·∫ ∏

i xiyi1−

∏i xiyi

×

(det

(1

1− xiyj

))2∏i

du(xi )dv(yi )

Ideas in proof:

1. First use Fredholm expansion of det(I − KN).

2. Then use the Andreief identity:

∫···

∫det(φj (xk ))j,k=1,...,N ·det(ψj (xk ))j,k=1,...,N dx1···dxN=N! det(

∫φj (x)ψk (x) dx)

j,k=1,...,N

3. Symmetrization argument

Applying this to KN = HN(Λ)HN(Λ−1) followed by a deformation ofcontours gives S =

∑∞n=1 Sn with

Sn =1

(n!)2

κ2n

π2n

∫ 1

0· · ·∫ 1

0

∏i xiyi

1− κn∏

i xiyi

(det

(1

1− κxiyj

))2

×∏i

Λ1(xi )

Λ1(yi )

∏i

dxidyi

where

κ := k2 and Λ1(x) =

√(1− x)(1− κx)

x

Alternate representation of Sn (Cauchy determinant identity):

F1

(n!)2

κn(n+1)

π2n

∫ 1

0· · ·∫ 1

0

∏i xiyi

1− κn∏

i xiyi

∆(x)2∆(y)2∏i ,j(1− κxiyj)2

∏i

Λ1(xi )

Λ1(yi )dxidyi

Theorem (T–Widom): The unit circle |κ| = 1 is a natural boundary for S.

Proof proceeds by four lemmas.


∑∞n=1 Sn with

Sn =1

(n!)2

κ2n

π2n

∫ 1

0· · ·∫ 1

0

∏i xiyi

1− κn∏

i xiyi

(det

(1

1− κxiyj

))2

×∏i

Λ1(xi )

Λ1(yi )

∏i

dxidyi

where

κ := k2 and Λ1(x) =

√(1− x)(1− κx)

x


F1

(n!)2

κn(n+1)

π2n

∫ 1

0· · ·∫ 1

0

∏i xiyi

1− κn∏

i xiyi

∆(x)2∆(y)2∏i ,j(1− κxiyj)2

∏i

Λ1(xi )

Λ1(yi )dxidyi




∑∞n=1 Sn with

Sn =1

(n!)2

κ2n

π2n

∫ 1

0· · ·∫ 1

0

∏i xiyi

1− κn∏

i xiyi

(det

(1

1− κxiyj

))2

×∏i

Λ1(xi )

Λ1(yi )

∏i

dxidyi

where

κ := k2 and Λ1(x) =

√(1− x)(1− κx)

x


F1

(n!)2

κn(n+1)

π2n

∫ 1

0· · ·∫ 1

0

∏i xiyi

1− κn∏

i xiyi

∆(x)2∆(y)2∏i ,j(1− κxiyj)2

∏i

Λ1(xi )

Λ1(yi )dxidyi




∑∞n=1 Sn with

Sn =1

(n!)2

κ2n

π2n

∫ 1

0· · ·∫ 1

0

∏i xiyi

1− κn∏

i xiyi

(det

(1

1− κxiyj

))2

×∏i

Λ1(xi )

Λ1(yi )

∏i

dxidyi

where

κ := k2 and Λ1(x) =

√(1− x)(1− κx)

x


F1

(n!)2

κn(n+1)

π2n

∫ 1

0· · ·∫ 1

0

∏i xiyi

1− κn∏

i xiyi

∆(x)2∆(y)2∏i ,j(1− κxiyj)2

∏i

Λ1(xi )

Λ1(yi )dxidyi



Let ε 6= 1 be a nth root of unity and we wish to consider behavior of S asκ→ ε radially. Look at d`Sn/dκ`—main contribution will come from

FF∫ 1

0· · ·∫ 1

0

∏i xiyi

(1− κn∏

i xiyi )`+1

∆(x)2∆(y)2∏i ,j(1− κxiyj)2

∏i

Λ1(xi )

Λ1(yi )dxidyi

Lemma 1: The integral FF is bounded when ` < 2n2 − 1 and it is oforder log(1− |κ|)−1 when ` = 2n2 − 1.

Main idea: Various approximations bound FF by the integral∫ 2nδ

0r2n2−`−2 dr

which will give first part of lemma.

Lemma 2:

(d

dκ)2n2−1Sn ≈ log(1− κ)−1

In differentiating F the other terms are O(1).


FF∫ 1

0· · ·∫ 1

0

∏i xiyi

(1− κn∏

i xiyi )`+1

∆(x)2∆(y)2∏i ,j(1− κxiyj)2

∏i

Λ1(xi )

Λ1(yi )dxidyi



0r2n2−`−2 dr


Lemma 2:

(d

dκ)2n2−1Sn ≈ log(1− κ)−1



FF∫ 1

0· · ·∫ 1

0

∏i xiyi

(1− κn∏

i xiyi )`+1

∆(x)2∆(y)2∏i ,j(1− κxiyj)2

∏i

Λ1(xi )

Λ1(yi )dxidyi



0r2n2−`−2 dr


Lemma 2:

(d

dκ)2n2−1Sn ≈ log(1− κ)−1


Lemma 3: If εm 6= 1, then(d

dκ

)2n2−1

Sm = O(1).

All integrals are bounded as κ→ ε.

Lemma 4: (Main lemma)

∑m>n

(d

dκ

)2n2−1

Sm = O(1)

For κ sufficiently close to ε, all integrals we get by differentiating theintegrals for Sm are at most Ammm. (A can depend upon n but not on m.)The extra (m!)2 appearing in F gives a bounded sum.

Lemma 3: If εm 6= 1, then(d

dκ

)2n2−1

Sm = O(1).

All integrals are bounded as κ→ ε.

Lemma 4: (Main lemma)

∑m>n

(d

dκ

)2n2−1

Sm = O(1)

For κ sufficiently close to ε, all integrals we get by differentiating theintegrals for Sm are at most Ammm. (A can depend upon n but not on m.)The extra (m!)2 appearing in F gives a bounded sum.

What about χ?

For T < Tc :

β−1χ− =∑

M,N∈Z

[〈σ00σMN〉 −M2

0

]=M2

0

∑M,N∈Z

[det(I − KM,N)− 1]

= M20

∞∑n=1

χ(2n)

In the last equality we used the Fredholm expansion and product formulasfor the determinants appearing in the integrands to find

χ(n)(s) =1

n!

1

(2π i)2n

∫Cr· · ·∫Cr

(1 +∏

j x−1j )(1 +

∏j y−1j )

(1−∏

j xj)(1−∏

j yj)×

∏j<k

(xj − xk)(yj − yk)

(xjxk − 1)(yjyk − 1)

∏j

dxjdyjD(xj , yj ; s)

D(x , y ; s) = s + s−1 − (x + x−1)/2− (y + y−1)/2, s := 1/√k

What about χ?

For T < Tc :

β−1χ− =∑

M,N∈Z

[〈σ00σMN〉 −M2

0

]=M2

0

∑M,N∈Z

[det(I − KM,N)− 1]

= M20

∞∑n=1

χ(2n)

In the last equality we used the Fredholm expansion and product formulasfor the determinants appearing in the integrands to find

χ(n)(s) =1

n!

1

(2π i)2n

∫Cr· · ·∫Cr

(1 +∏

j x−1j )(1 +

∏j y−1j )

(1−∏

j xj)(1−∏

j yj)×

∏j<k

(xj − xk)(yj − yk)

(xjxk − 1)(yjyk − 1)

∏j

dxjdyjD(xj , yj ; s)

D(x , y ; s) = s + s−1 − (x + x−1)/2− (y + y−1)/2, s := 1/√k

Nickel Singularities

Standard estimates show that χ−(s) is holomorphic for |s| > 1 (|k| < 1).

Definition: A Nickel singularity of order n is a point s0 on the unit circlesuch that the real part of s0 is the average of the real parts of two nthroots of unity.

Note that D(x , y ; s) vanishes when

<(s) =<(x) + <(y)

2

Theorem (T–Widom) When n is even χ(n) extends to a C∞ function onthe unit circle except at the Nickel singularities of order n.

Some ideas in the proof

I A partition of unity allows one to localize.

I Each potential singular factor in the integrand of χ(n) is representedas an exponential integral over R+.

I The gradient of the exponent in the resulting integrand isapproximately a linear combination with positive coefficients of certainvectors—one from each factor.

I Unless s is a Nickel singularity the convex hull of these vectors doesnot contain 0—this allows a lower bound for the length of thegradient.

I Application of the divergence theorem gives the bound O(1)—samebound after differentiating with with respect to s any number of times.

I Follows that χ(n) extends to a C∞ function excluding the Nickelsingularities.




































Remarks:

I What we don’t have is a “Lemma 4” that says the sum of the otherterms don’t cancel the Nickel singularities.

I It appears that χ(n) satisfies a linear differential equation with onlyregular singular points. (This follows from work of Kashiwara.) Usingthis result we get that for n even χ(n) extends analytically across theunit circle except at the Nickel singularities.

Thank you for your attention!

Remarks:




Remarks:




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