correlation functions of integrable …correlation functions of integrable models, combinatorics,...

CORRELATION FUNCTIONS OF INTEGRABLE

MODELS, COMBINATORICS, AND RANDOM

WALKS.

Cyril Malyshev(in collaboration with Nikolay M. Bogoliubov)

Steklov Mathematical Institute, RAS

St.-Petersbug, RUSSIA

RAQIS'16Recent Advances in Quantum Integrable Systems

Geneva, 22-26 August 2016

ABSTRACT

Certain quantum integrable models solvable by the Quantum InverseScattering Method demonstrate a relationship with combinatorics,partition theory, and random walks. Special thermal correlation functionsof the models in question are related with the generating functions forplane partitions and self-avoiding lattice paths. The correlation functionsof the integrable models admit interpretation in terms of nests of thelattice paths made by random walkers.

TABLE OF CONTENT

I. COMBINATORIAL PRELIMINARIES

II.1 FREE FERMION LIMIT OF XXZ HEISENBERG MODEL

II.2 CORRELATION FUNCTIONS AND VICIOUS WALKERS

II.3 RANDOM WALKS AS THE CORRELATION FUNCTIONS

III.1 WALKS ON SIMPLICIAL LATTICES

III.2 GENERALIZED PHASE MODEL AND ITS SOLUTION

III.3 TOTALLY ASYMMETRIC ZERO RANGE MODEL

IV. CONCLUDING REMARKS

**********************************

THE TALK IS BASED ON THE PAPERS

[7] N. Bogoliubov, C. Malyshev, The correlation functions of strongly

correlated bosons and random walks over simplicial lattices, PDMIpreprint 4/2016.[6] N. Bogoliubov, C. Malyshev, Integrable models and combinatorics,Russian Math. Surveys 70 (2015), 789.[5] N. M. Bogoliubov, C. L. Malyshev, Correlation functions of XX0Heisenberg chain, q-binomial determinants, and random walks, Nucl.Phys. B, 879 (2014), 268-291.[4] N. Bogoliubov, Calculation of correlation functions in totally

asymmetric exactly solvable models on a ring , Theoretical andMathematical Physics 175 (2013), 755.[3] N. M. Bogoliubov, C. Malyshev, The correlation functions of the XXZ

Heisenberg chain in the case of zero or innite anisotropy, and random

walks of vicious walkers, St.-Petersburg Math. J. 22 (2011), 359-377.[2] N. M. Bogoliubov, C. Malyshev, The correlation functions of the XX

Heisenberg magnet and random walks of vicious walkers, Theor. Math.Phys., 159 (2009), No. 2, 179-192.[1] N. M. Bogoliubov, T. Nassar, On the spectrum of the non-Hermitian

phase-dierence model , Phys. Lett. A, 234 (1997), 345-350.

I. COMBINATORIAL PRELIMINARIES: q-BINOMIAL

DETERMINANTS AND GENERATING FUNCTIONS OF PLANE

PARTITIONS

• • • Let us begin with the denitions of the generating functions ofboxed plane partitions and self-avoiding lattice walks.

An array (πi,j)i,j≥1 of non-negative integers that are non-increasingas functions of both i and j (i, j = 1, 2, . . . ) is called boxed planepartition π. Plane partition is represented by cubes arranged as stackswith coordinates (i, j) and with height equal to πi,j . The plane partitionis contained inside a box B(L,N,M) provided that i ≤ L, j ≤ N andπi,j ≤M for all cubes of the plane partition.

The generating function of plane partitions inside B(L,N, P ) is denedas formal series Zq(L,N, P ) ≡

∑π q

|π| (summation over all partitions

inside the box), and it takes the form:

Zq(L,N, P ) =

L∏j=1

N∏k=1

P∏i=1

1− qi+j+k−1

1− qi+j+k−2=

L∏j=1

N∏k=1

1− qP+j+k−1

1− qj+k−1.

The limit q → 1 leads to the MacMahon formula:

A(L,N, P ) =

L∏j=1

N∏k=1

P∏i=1

i+ j + k − 1

i+ j + k − 2=

L∏j=1

N∏k=1

P + j + k − 1

j + k − 1.

• • • To calculate the correlation functions, we need the determinantof a block-matrix (T)1≤j,k≤N given by entries:

Tkj =1− q(P+1)(j+k−1)

1− qj+k−1, 1 ≤ k ≤ L, 1 ≤ j ≤ N ,

Tkj = qj(N−k) , L+ 1 ≤ k ≤ N, 1 ≤ j ≤ N ,

where P and L ≤ N are arbitrary. The matrix (T)1≤j,k≤N consists oftwo blocks of the sizes L×N and (N − L)×N .

Several denitions are in order.

The q-binomial determinant

(ab

)q

is dened by

(ab

)q

≡(a1, a2, . . . aSb1, b2, . . . bS

)q

≡ det

([ajbi

])1≤i,j≤S

,

where a and b are ordered tuples: 0 ≤ a1 < a2 < · · · < aS and

0 ≤ b1 < b2 < · · · < bS . The entries

[ajbi

]are the q-binomial coecients:

[Nr

]≡ (1− qN )(1− qN−1) . . . (1− qN−r+1)

(1− q)(1− q2) . . . (1− qr), q ∈ R .

The q-binomial coecients are replaced at q → 1 by the binomial

coecients

(ajbi

). The q-binomial determinant is transformed to the

binomial determinant:(ab

)≡(a1, a2, . . . aSb1, b2, . . . bS

)= det

((ajbi

))1≤i,j≤S

.

The binomial determinant is positive at bi ≤ ai, ∀i.

Ðèñ.: S-Tuple (w1, w2, . . . , wS) of self-avoiding walks for S = 5.

Binomial determinant gives the number of self-avoiding walks acrosstwo-dimensional lattice. Each path wi from a tuple (w1, w2, . . . , wS)goes from Ai = (0, ai) to Bi = (bi, bi), 1 ≤ i ≤ S. Only steps to northand to east are allowed.

So, let us consider (T)1≤j,k≤N given by the entries:

Tkj =1− q(P+1)(j+k−1)

1− qj+k−1, 1 ≤ k ≤ L, 1 ≤ j ≤ N ,

Tkj = qj(N−k) , L+ 1 ≤ k ≤ N, 1 ≤ j ≤ N ,

where P and L ≤ N are arbitrary. Now we formulate the following

PROPOSITION 1 Let the matrix (T)1≤j,k≤N , be dened by the entries

(32) with P2 < N < P . The determinant of (T)1≤j,k≤N is given as:

q−L2 (L−1)(N−L) det(T)1≤j,k≤N

V(qN )V(qL/q)

= q−N2 (P−1)P

(L+N, L+N + 1, . . . L+N + P − 1L, L+ 1, . . . L+ P − 1

)q

=

P∏k=1

L∏j=1

1− qj+k+N−1

1− qj+k−1= Zq(L,N,P) ,

where P ≡ P −N + 1, and Zq(L,N,P) is the generating function of

plane partitions.

The PROPOSITION 1 relates det T to the q-binomial determinant, whichis transformed at q → 1 to the binomial determinant equal, in turn, to thenumber of P-tuples of lattice self-avoiding paths between the end points

Al = (0, N + L+ l − 1) and Bl = (L+ l − 1, L+ l − 1), 1 ≤ l ≤ P.

The Figure gives appropriate picture with end points Al and Bl atP = L = 3 and N = 2.

Ðèñ.: Watermelon conguration (lhs) and plane partition with gradient lines.

The generating function Zq(L,N,P) gives at q → 1 the number of planepartitions A(L,N,P) inside B(L,N,P) :

Zq(L,N,P) =

L∏j=1

N∏k=1

1− qP+j+k−1

1− qj+k−1−→q→1

−→q→1

A(L,N,P) = det

((N + L+ i− 1L+ j − 1

))1≤i,j≤P

.

RHS expresses the fact that number of plane partitions A(L,N,P) isequal to the number of self-avoiding lattice paths. Just the pathsconstituting a P-tuple are in bi-jection with, so-called, gradient linescorresponding to a plane partition inside B(L,N,P).

I.1 FREE FERMION LIMIT OF XXZ HEISENBERG MODEL

• • • Consider XX0 model describing free-fermion limit of XXZ spin- 12Heisenberg chain. The XX0 Hamiltonian in absence of magnetic eldtakes the form:

HXX ≡ −1

2

M∑k=0

(σ−k+1σ+k + σ+

k+1σ−k ) = −1

2

M∑n,m=0

∆nmσ−n σ

+m ,

where ∆nm is the hopping matrix with the entries

∆nm = δ|n−m|,1 + δ|n−m|,M ,

or, less formally,

∆ =

0 1 11 0 1

1 0 1. . .1 0 1

1 1 0

.

The local spin operators σ±k = 12 (σxk ± iσ

yk), k ∈ 0, 1, . . . ,M are

dened on sites of periodic chain of length M + 1, and they are thetensor products:

σ#k = σ0 ⊗ · · · ⊗ σ0 ⊗ σ#︸︷︷︸

k

⊗σ0 ⊗ · · · ⊗ σ0 ,

where σ0 is 2× 2 unit matrix, and σ# at kth site is a Pauli matrix,σ# ∈ su(2) (# is either x, y, z or ±). The commutation rules are:

[σ+k , σ

−l ] = δk,l σ

zl , [σzk, σ

±l ] = ±2 δk,l σ

±l .

We introduce spin up and spin down states, |↑〉 ≡( 1

0

)and

|↓〉 ≡(

01

). The Pauli operators act on |↑〉 and |↓〉 as follows:

σ− |↑〉 = |↓〉, σ− |↓〉 = 0 , σ− =

(0010

),

σ+ |↑〉 = 0, σ+ |↓〉 = |↑〉 , σ+ =

(0100

).

The lattice spin operators dened above act over the the state-vectors⊗Mk=0 |s〉k , where s =↑, ↓. The state-vectors are elements of the

state-space HM+1 =M⊗k=0

(C2)k.

The periodic boundary conditions σ#k+(M+1) = σ#

k are imposed.

**********************************

Let the sites with spin down states are labeled by decreasingcoordinates M ≥ µ1 > µ2 > . . . > µN ≥ 0, which constitute strictpartition µ = (µ1, µ2, . . . , µN ).In the free-fermion limit, we dene N -excitation state-vectors |ΨN (u)〉:

|ΨN (u)〉 =∑

λ⊆MN

Sλ(u2)

(N∏k=1

σ−µk

)|⇑〉 , |⇑〉 ≡

M⊗n=0

|↑〉n ,

where λ = (λ1, λ2, . . . , λN ) is λ = µ− δN , whereδN = (N − 1, N − 2, . . . , 1, 0). Besides,M≡M + 1−N ≥ λ1 ≥ λ2 ≥ · · · ≥ λN ≥ 0.

The parameters u = (u1, u2, . . . , uN ) and u2 ≡ (u21, u22, . . . , u

2N )

correspond to arbitrary complex numbers.

The coecients of the state-vector |ΨN (u)〉 are given by the Schur

functions dened by the Jacobi-Trudi relation:

Sλ(xN ) ≡ Sλ(x1, x2, . . . , xN ) ≡det(xλk+N−kj )1≤j,k≤N

V(xN ),

in which V(xN ) is the Vandermonde determinant

V(xN ) ≡ det(xN−kj )1≤j,k≤N =∏

1≤m<l≤N

(xl − xm) .

There is a natural correspondence between the coordinates of the spindown states µ and the partition λ expressed by the Young diagram:

0 2 3 5 8

µ1µ2µ3µ41 4 6 7

λ1

λ2λ3

λ4

Ðèñ.: Relation of the spin down coordinates µ = (8, 5, 3, 2) and partitionλ = (5, 3, 2, 2) for M = 8, N = 4.

The states |ΨN (u)〉 are the eigen-states,

HXX |ΨN (uN )〉 = EN |ΨN (uN )〉 ,

with eigen-values EN ≡ EXXN (I1, I2, . . . , IN ) = −∑Nl=1 cos

(2πIlM+1

), if

and only if ul (1 ≤ l ≤ N) satisfy the Bethe equations:

u2(M+1)j = (−1)N−1 , u2j = ei

2πM+1 Ij , 1 ≤ j ≤ N .

where Ij are integers or half-integers: M ≥ I1 > I2 > · · · > IN ≥ 0.

The scalar products of the state-vectors are calculated by means of theBinetCauchy formula:

PL/n(y,x) ≡∑

λ⊆(L/n)N

Sλ(x21, . . . , x2N )Sλ(y21 , . . . , y

2N )

=

( N∏l=1

ynl xnl

)det(Tjk)1≤j,k≤N∏

1≤k<j≤N(y2j − y2k

)∏1≤m<l≤N (x2l − x2m)

,

where summation∑

λ⊆(L/n)N is over non-strict partitions λ:L ≥ λ1 ≥ λ2 ≥ · · · ≥ λN ≥ n, n ≥ 0. The entries Tjk take the form:

Tkj =1− (xkyj)

N+L−n

1− xkyj.

**********************************

PROPOSITION 2 The following sums of products of the Schur

functions take place:

∑λ⊆MN−n

Sλ(v−2N )Sλ(u2N−n) =

(N−n∏l=1

u−2nl

)det(Tkj)1≤k,j≤N

V(u2N−n)V(v−2N ),

∑λ⊆MN−n

Sλ(v−2N−n)Sλ(u2N ) =

(N−n∏l=1

v2nl

)det(Tkj)1≤k,j≤N

V(v−2N−n)V(u2N ),

where the entries of the matrices (Tkj)1≤k,j≤N and (Tkj)1≤k,j≤N are:

Tkj = T okj , 1 ≤ k ≤ N − n, 1 ≤ j ≤ N ,

Tkj = v−2(N−k)j , N − n+ 1 ≤ k ≤ N, 1 ≤ j ≤ N ,

and

Tkj = T okj , 1 ≤ k ≤ N, 1 ≤ j ≤ N − n ,

Tkj = u2(N−k)j , 1 ≤ k ≤ N, N − n+ 1 ≤ j ≤ N .

The correlator G (j, l|t) may be considered as the generating function ofrandom walks. Expanding in powers of t one has:

G (j, l|t) =

∞∑K=0

tK

K!〈⇑| σ+

j (−H)Kσ−l |⇑〉 .

This equality may be interpreted as follows. Position of the walker onlattice is labelled by the spin down state, while the spin up statescorrespond to empty sites. The relation enables to enumerate alladmissible trajectories of the walker starting from lth site. The state〈⇑| σ+

j acting on from the left allows to x the ending point of thetrajectories because of orthogonality of the spin states. Hence,

〈⇑| σ+j (−H)Kσ−l |⇑〉 =

(∆K

)jl,

i.e., we obtain the entry of Kth power of the hopping matrix∆nl0≤n,l≤M .

• • • Consider now the multi-particle correlation function

G(j; l|t) = 〈⇑|(

N∏n=1

σ+jn

)e−tH

(N∏k=1

σ−lk

)|⇑〉 ,

which is parametrized by multi-indices j ≡ (j1, j2, . . . , jN ) andl ≡ (l1, l2, . . . , lN ). This correlator is the generation function of Nrandom turns vicious walkers. The average

〈⇑|(

N∏n=1

σ+jn

)(−H)K

(N∏k=1

σ−lk

)|⇑〉 is equal to the number of nests of

trajectories of N random turns walkers being initially located on the sitesl1 > l2 > · · · > lN and arrived at the positions at j1 > j2 > · · · > jNafter K steps. The condition that vicious walkers do not touch eachother up to N steps is guaranteed by the Pauli matrices (σ±k )2 = 0.

Dierentiating by t and using commutators one obtains:

d

dtG(j; l|t) =

N∑k=1

(G(j; l1, l2, . . . , lk + 1, . . . , lN |t)

+G(j; l1, l2, . . . , lk − 1, . . . , lN |t))

(j is xed). The non-intersection condition means that G(j; l|t) = 0 iflk = lp (or jk = jp) for any 1 ≤ k, p ≤ N . The solution is represented inthe determinantal form:

G(j; l|t) = det(G(jr, ls|t)

)1≤r,s≤N ,

where G(j, l|t) is the one-particle correlator. The initial condition is:

G(j; l|0) =∏Nm=1 δjm,lm .

Let |PK(l→ j)| is the number of K-step trajectories traced by N viciouswalkers in the random turns model. A typical conguration of paths forthree walkers is shown in gure:

1 2 3 4 5 6 7

K

Ðèñ.: A conguration of paths of three vicious walkers (l = (2, 5, 7),j = (4, 5, 6)) in the random turns model.

The generating function G(j; l|t) may be expressed in the form:

G(j; l|t) =1

(M + 1)N

∑φN

e−βEN (φN )|VN (eiφN )|2 SλL(eiφN )SλR(e−iφN ) ,

where the parametrization φkm = 2πM+1

(km − M

2

), 1 ≤ m ≤ N , is used.

Summation is over N -tuples φN ≡ (φk1 , φk2 , . . . , φkN ), parametrized byintegers ki, 1 ≤ i ≤ N , respecting M ≥ k1 > k2 · · · > kN ≥ 0. HereEN (φN ) is the energy.

I.3 RANDOM WALKS AS THE CORRELATION FUNCTIONS

The thermal correlation function can be considered which is related to theprojection operator Πn that forbids spin down states on the rst n sitesof the chain and we shall call it the persistence of ferromagnetic string :

T (θ gN , n, t) ≡

〈Ψ(θ gN ) | Πn e

−tHXX Πn |Ψ(θ gN )〉

〈Ψ(θ gN ) | e−tHXX |Ψ(θ g

N )〉, Πn ≡

n−1∏j=0

I + σzj2

,

where t ∈ C. The correlator is calculated on the ground state solution ofthe Bethe equation.We shall consider the nominator separately under arbitraryparametrization:

〈ΨN (v) | Πn e−βHXX Πn |ΨN (u)〉

=∑

λL,λR⊆(M−N−n)NSλL(v−2)SλR(u2)GµL;µR(β) .

To give the combinatorial interpretation, we need a relationship betweenSchur functions and lattice paths.A combinatorial description of the Schur functions may be given in termsof semistandard Young tableaux. A lling of the cells of the Youngdiagram of λ with positive integers n ∈ N+ is called a semistandard

tableau of shape λ provided it is weakly increasing along rows and strictlyincreasing along columns. A denition of the Schur function is given:

Sλ(x1, x2, . . . , xm) =∑T

xT , xT ≡∏i,j

xTij ,

where m ≥ N , the product is over all entries Tij of the tableau T , andthe sum is over all tableaux T of shape λ with the entries being numbers1, 2, . . . ,m.Each semistandard tableau of shape λ with entries not exceeding N is anest of self-avoiding lattice paths with xed start and end points. Let Tijbe an entry in ith row and jth column of the semistandard tableau T .The ith lattice path of the nest C (counted from the top of the nest)encodes the ith row of the tableau (i = 1, . . . , N). It goes fromCi = (N − i+ 1, N − i) to (1, µi = λi +N − i). It makes λi steps to thenorth so that

the step along the line xj corresponds to the occurrences of the letterN − j + 1 in the ith row of T . The power lj of xj in the weight of anyparticular nest of paths is the number of steps to north taken along thevertical line xj . Thus, the Schur function takes the form:

Sλ(x1, x2, . . . , xN ) =∑C

N∏j=1

xljj ,

where summation is over all admissible nests C.

µ1

µ2

µ3

µ4

µ5

С1

С2

С3

С4

С5

С6

x6 x5 x4 x3 x2 x10

1

9

10

6 2 1

1

2

4

5

6

1

3

4

5

6

1

3

5

2

4

4

6

Ðèñ.: A nest of lattice paths C, corresponding to semistandard tableau withweight x36x

25x

24x

43x

32x

31 on the diagram λ = (5, 5, 3, 2, 2, 0) for N = 6.

The scalar product, being the product of two Schur functions, may begraphically expressed as a nest of N self-avoiding lattice paths starting atthe equidistant points Ci and terminating at the equidistant points Bi(1 ≤ i ≤ N). This conguration, known as watermelon. The scalarproduct is given by the sum of all such watermelons. Rotating Fig. by π

4counter-clockwise we see that the watermelon is a particular case ofconguration for lock step random walkers.

Ðèñ.: Watermelon conguration and correspondent plane partition.

The partition function of watermelons is equal to the q-parameterized (Inthe q-parametrization v−2 = q ≡ (q, q2, . . . , qN ),u2 = q/q = (1, q, . . . , qN−1)) scalar product:

〈ΨN (q−12 )|ΨN ((q/q)

12 )〉 =

∑λ⊆MN

Sλ(q)Sλ(q/q)

=∑W

q|ξ|C+|ζ|B = Z(N,N,M) ,

where the sum is taken over all watermelons with the xed endpoints Ci,Bi, 1 ≤ i ≤ N .

Further, we obtain the following transition amplitude:

〈ΨN (vN ) | Πn e−tH Πn |ΨN (u)〉

=∑

λL,λR⊆(M−N−n)N

G(λL + δN ;λR + δN | t)SλL(v−2)SλR(u2)

=

∞∑K=0

tK

K!

∑λL,R\n⊆(M−n)N

SλL(v−2)SλR(u2)

× 〈⇑|

(N∏l=1

σ+µLl

)(−H)K

(N∏p=1

σ−µRp

)|⇑〉 .

Here u and v stand for an arbitrary parametrization.The elements of the expansion in right-hand side may be interpreted interms of nests of self-avoiding lattice paths as follows. For the rst Nsteps the particles are moving to the west (from the right to the left)according the lock step rules. They are doing next K steps according tothe random turns rules. The nal N steps to the west are again accordingthe lock step rules.

The example of the described nest of lattice paths is schematicallydepicted in Figure.

С1

С2

С3

С40

1

8

9B1

B2

B3

B4

L R

K

Ðèñ.: One of the nest of lattice paths that corresponds to the amplitude〈Ψ(vN ) | Π0 (−H)K Π0 |Ψ(uN )〉

II.1 WALKS ON SIMPLICIAL LATTICES

Starting from (M + 1)-dimensional hypercubical lattice with unit spacingZM+1 3m ≡ (m0,m1, . . . ,mM ), let us consider a set of points withcoordinates constrained by the requirement m0 +m1 + . . .+mM = N :

Symp(N)(ZM+1) ≡m ∈ ZM+1

∣∣∣0 ≤ mi, i ∈M,∑i∈M

mi = N

(hereafterM≡ 0, 1, . . .M). We call Symp(N)(ZM+1) simplicial

lattice.Random walks of a walker over sites of Symp(N)(ZM+1) are dened by a

set of admissible steps ΩM such that at each step an ith coordinate mi

increases by unity while a nearest neighboring coordinate decreases byunity. Namely, ΩM is the set of steps with coordinates (e0, e2, . . . , eM )such that for all pairs (i, i+ 1) with 0 ≤ i ≤M and M + 1 = 0(mod 2), ei = ±1 , ei+1 = ∓1 and ej = 0 for all j ∈M and j 6= i, i+ 1.The step-set ΩM ≡ ΩM (m0) ensures that trajectory of a random walkdetermined by the starting point m0 lies in M -dimensional setSymp(N)(ZM+1).

0

N

N

0

N0

1

Ðèñ.: The hopping processes for the one-dimensional nearest-neighbor randomwalk on a segment [0,N].

As an example, the walks on Z2 are dened by a step-setΩ1 = (1,−1), (−1, 1) that ensures that the walks lie on lines

(n0, n1) ∈ Z2 | n0 + n1 = N.

The step-setΩ2 = (−1, 1, 0), (1,−1, 0), (0,−1, 1), (0, 1,−1), (1, 0,−1), (−1, 0, 1) ofthe random walks, or Ω2 = (−1, 1, 0), (0,−1, 1), (1, 0,−1) of thedirected walks, ensures that trajectories of random walks belong toSymp(N)(Z3).

2

1

N

N

N

0

0

Ðèñ.: A two-dimensional triangular simplicial lattice.

Directed random walks on M -dimensional orientated simplicial lattice aredened by a step-set ΓM = (k0, k1, . . . , kM ) such that for all pairs(i, i+ 1) with i ∈M and M + 1 = 0( mod 2), ki = −1, ki+1 = 1, andkj = 0 for all j ∈M\i, i+ 1.

Ðèñ.: The orientated two-dimensional bounded simplicial lattice dened by thestep-set Γ2. The walks are allowed only in the direction of arrows.

The walker's movements should be supplied with appropriate boundaryconditions. The reecting boundary conditions imply that when trajectoryand boundary are intersecting, the corresponding admissible steps are stilltaken from ΩM or ΓM . The retaining boundary conditions imply that thewalker on the boundary continues to move in accordance with theelements of ΩM or ΓM either stays on the boundary. The boundary ofthe simplicial lattice consists of M + 1 faces of highest dimensionalityM − 1. To each component of the boundary a weight gs,s = 0, 1, . . . ,M , is assigned.

Consider the retaining boundary conditions. The exponential generatingfunction of lattice walks is dened as a series

F (N)(l, j|t) =

∞∑k=0

tk

k!G

(N)k (l, j) ,

where G(N)k (l, j) characterize the k-step walks at a node

l = (l0, l1, . . . , lM ) ∈ Simp(N)(ZM+1) when starting at

j = (j0, j1, . . . , jM ) ∈ Simp(N)(ZM+1). The master equation is:

∂tF(N)(l, j|t) =

M∑s=0

F (N)(l, j0, j1, . . . , js − 1, js+1 + 1, . . . jM )

+

M∑s=0

F (N)(l, j0, j1, . . . , js + 1, js+1 − 1, . . . jM )

+

M∑s=0

gsF(N)(l, j0, j1, . . . , js−1, 0, js+1, . . . jM ) δ(N,

∑0≤k≤M

′jk) ,

where δ(n,m) is the Kronecker symbol, and∑′

implies that k = s isomitted. The equation for the directed walks is obtained by removing thesecond sum in right-hand side.

The system of equations for G(N)k (l, j):

G(N)k (l, j) =

M∑s=0

G(N)k−1(l, j0, j1, . . . , js − 1, js+1 + 1, . . . jM )

+

M∑s=0

G(N)k−1(l, j0, j1, . . . , js + 1, js+1 − 1, . . . jM )

+

M∑s=0

gsG(N)k−1(l, j0, j1, . . . , js−1, 0, js+1, . . . jM ) δ(N,

∑0≤k≤M

′jk) ,

where k ≥ 1, while it is natural to impose the condition

G(N)0 (l, j) = δl0j0δl1j1 · · · δlM jM .

II.2 GENERALIZED PHASE MODEL AND ITS SOLUTION

• • • Generalized Phase Model describes N bosonic particles on a cyclicchain of M + 1 nodes. Each conguration of the particles is characterizedby a collection of the occupation numbers (nM , . . . , n1, n0),∑l∈M nl = N . The phase operators φn, φ

†n respect the algebra

[Ni, φj ] = −φiδij , [Ni, φ†j ] = φ†i δij , [φi, φ

†j ] = πiδij ,

where Nj is the number operator, and πi = 1− φ†iφi is the vacuum

projector: φjπj = πjφ†j = 0. The Fock states |nl〉l = (φ†l )

nl |0〉l, where|0〉l is the vacuum state |0〉 at lth site dened by φl|0〉 = 0:

φ†l |nl〉l = |nl + 1〉l , φl |nl〉l = |nl − 1〉l , Nl |nl〉l = nl |nl〉l .

The Fock states |n〉 are generated from |0〉 by the rising operators φ†j :

|n〉 ≡M⊗l=0

|nl〉l =( M∏j=0

(φ†j)nj)|0〉 , |0〉 ≡

M⊗l=0

|0〉l ,

where n ∈ Symp(N)(ZM+1).

As generator of directed walks with the retaining boundary conditions weconsider the non-Hermitian Hamiltonian [N. M. Bogoliubov, T. Nassar,On the spectrum of the non-Hermitian phase-dierence model, Phys.Lett. A 234, 345 (1997)]:

H =

M∑m=0

(φmφ

†m+1 + gmπm

),

where M + 1 is even and periodicity is imposed. The Hamiltoniancommutes with the number operator N .The exponential generating function of the directed walks is expressed asthe correlation function:

F (N)(l, j|t) = 〈l | etH | j〉 ,

where H is the Hamiltonian. Expanding the correlator we obtain the

coecients G(N)k (l, j) characterizing the lattice walks from the node j to

the node l in k steps:

G(N)k (l, j) = 〈l | Hk | j〉 .

• • • Dene L-operator which is 2× 2 matrix with operator entries:

L(n|u) ≡(u−1 + ugnπn φ†n

φn u

),

where u ∈ C is a parameter and gn ∈ R. The intertwining relation

R(u, v) (L(n|u)⊗ L(n|v)) = (L(n|v)⊗ L(n|u))R(u, v) ,

in which R(u, v) is the R-matrix

R(u, v) =

f(v, u) 0 0 0

0 g(v, u) 1 00 0 g(v, u) 00 0 0 f(v, u)

,

where

f(v, u) =u2

u2 − v2, g(v, u) =

uv

u2 − v2, u, v ∈ C .

The monodromy matrix:

T (u) = L(M |u)L(M − 1|u) · · ·L(0|u) =

(A(u) B(u)C(u) D(u)

).

The commutation relations of the matrix elements are given:

R(u, v) (T (u)⊗ T (v)) = (T (v)⊗ T (u))R(u, v) .

The transfer matrix τ(u) is: τ(u) = trT (u) = A(u) +D(u). The relationmeans [τ(u), τ(v)] = 0, u, v ∈ C. The entries of T (u):

uM+1A(u) = 1 + u2(M−1∑m=0

φmφ†m+1 +

M∑m=0

gmπm

)+ u2(M+1)

M∏m=0

gmπm ,

uM+1D(u) = u2φ†0φM + . . .+ u2(M+1) ,

uMB(u) = φ†0 + . . .+ u2MPR ≡ B(u) ,

uMC(u) = φM + . . .+ u2MPL ≡ C(u) ,

where PR =M∑k=0

φ†kgk+1πk+1 . . . gMπM , PL =M∑k=0

g0π0 . . . gk−1πk−1φk.

The Hamiltonian is expressed:

H =∂

∂u2uM+1τ(u)

∣∣∣u=0

=∂

∂u2uM+1(A(u) +D(u))

∣∣∣u=0

.

The N -particle state-vectors are taken in the form

|ΨN (u)〉 =( N∏j=1

B(uj))|0〉,

where u = (u0, u1, . . . , uN ), and B(u) is dened above. The vacuum | 0〉is an eigen-vector of A(u) and D(u),

A(u)|0〉 = α(u)|0〉, D(u)|0〉 = δ(u)|0〉

with the eigen-values α(u) =∏Mj=0(u−1 + gju), δ(u) = uM+1.

The state-vector is the eigen-vectors both of H and τ(u), if and only if:

u−2Nn

M∏j=0

(gj + u−2n

)= (−1)N−1

N∏j=1

u−2j .

From the eigen-value ΘN (v) of τ(v) obtain the spectrum:

EN (u) =∂

∂v2vMΘN (v)

∣∣v=0

=

M∑m=0

gm +

N∑m=1

u−2m .

For the model associated with the R-matrix the scalar product of thestate-vectors is [N. Bogoliubov, Boxed plane partitions as an exactly

solvable boson model, J. Phys. A: Math. Gen. 38, 9415 (2005)]:

〈ΨN (v)|ΨN (u)〉 = V−1N (v2)V−1N (u−2)

N∏j=1

( vjuj

)M+N−1detQ ,

where VN (x) ≡∏

1≤i<k≤N (xk − xi), and Qjk, 1 ≤ j, k ≤ N are:

Qjk =α(vj)δ(uk)(uk/vj)

N−1 − α(uk)δ(vj)(uk/vj)−N+1

uk/vj − vj/uk,

with α(u) and δ(u) as above. The resolution of the identity operator

I =∑u

|ΨN (u)〉〈ΨN (u)|N 2(u)

,

where summation∑u is over all independent solutions of the Bethe

equations.

Let us consider the walker on Simp(N)(ZM+1). His starting point at thenode (N, 0, . . . , 0), and the walk terminates at (0, 0, . . . , N). Thegenerating function of the walks is:

F (N)(t) ≡ 〈0, 0, . . . , N | etH | N, 0, . . . , 0〉 = 〈0 | (φM )NetH(φ†M )N | 0〉 .

Inserting the identity operator obtain:

F (N)(t) =∑u

etEN (u)

N 2(u)〈0 | (φM )N | ΨN (u)〉〈ΨN (u) | (φ†M )N | 0〉 ,

where the summation is over all independent solutions of Betheequations. From B(u) and C(u) obtain:

〈0 | (φM )N | ΨN (u)〉 = limv→0〈ΨN (v)|ΨN (u)〉 = 1 ,

〈ΨN (u) | (φ†M )N | 0〉 = limv→0〈ΨN (u)|ΨN (v)〉 = 1 ,

[N. Bogoliubov, Calculation of correlation functions in totally asymmetric

exactly solvable models on a ring, Theoretical and Mathematical Physics175, 755 (2013)].

Eventually we obtain:

F (N)(t) =∑u

e2t∑Nk=1(gk+u

−2k )

N 2(u).

For instance, at N = 2 and M = 1 we obtain for gk = γ = 1 ∀k:

F (2)(t) = e4tγ∑u2

1,u22

e2t(u−21 +u−2

2 )

N 2(u21, u22)

,

N 2(u21, u22) = 4 +

u21u22

+u22u21

+ 4γ(u22 + u21) + 3γ2u22u21 .

II.3 TOTALLY ASYMMETRIC ZERO RANGE MODEL

When all gi are equal to γ = 1, we obtain the Hamiltonian of TotallyAsymmetric Zero Range Model:

Hzr =

M∑m=0

(φmφ

†m+1 + πm

).

The state-vector enables the representation in the form

|ΨN (u)〉 =∑

λ⊆MN

χRλ (u)( M∏j=0

(φ†j)nj)|0〉 ,

where the function χRλ is equal, up to a multiplicative pre-factor, to

χRλ (x) = χRλ (x1, x2, . . . , xN ) = V−1N (x) det(

1 + x−2i)λj

x2(N−j)i

.

Here λ denotes the partition (λ1, . . . , λN ) of non-increasing non-negativeintegers,

M ≥ λ1 ≥ λ2 ≥ . . . ≥ λN ≥ 0,

and VN (x) is the Vandermonde determinant.

There is a one-to-one correspondence between a sequence of theoccupation numbers (nM , . . . , n1, n0),

∑j∈M nj = N , and the partition

λ = (MnM , (M − 1)nM−1 , . . . , 1n1 , 0n0) ,

where each number S appears nS times (see Fig. 4). The summationgoes over all partitions λ into at most N parts with N ≤M .

0 6

Ðèñ.: A conguration of particles (N = 4) on a lattice (M = 6), thecorresponding partition λ = (61, 50, 40, 32, 20, 11, 00) ≡ (6, 3, 3, 1) and itsYoung diagram.

Acting by the Hamiltonian on the state-vector, we nd:

N∑k=1

χRλ1,...,λk+1,...,λN (u) = EzrN (u)χRλ1,...,λN (u) ,

together with the exclusion condition

χRλ1,...,λl−1=λl+1,λl,...,λN(u) = χRλ1,...,λl−1=λl,λl,...,λN

(u) , 1 ≤ l ≤ N.

The energy EzrN is given with all gi = 1. The state-vector is the

eigen-vector of the Hamiltonian with the periodic boundary conditions ifthe parameters uj satisfy the appropriate Bethe equations.

Expanding the left state-vector, we obtain:

〈ΨN (u) |= V−1N (x)∑

λ⊆MN

det

(xi

xi + x−1i

)λjx2(N−j)i

〈0 |

( M∏i=0

φnii

).

Equations take place:

N∑k=1

χLλ1,...,λk−1,...,λN (u) = EzrN (u)χLλ1,...,λN (u) ,

χLλ1,...,λl−1=λl−1,λl,...,λN (u) = χLλ1,...,λl−1=λl,λl,...,λN(u) , 1 ≤ l ≤ N.

Further one obtains:

〈l0, l1, . . . , lM | ΨN (u)〉 = χRλR(u)

〈ΨN (u) | j0, j1, . . . , jM 〉 = χLλL(u) ,

where λR =(M lM , (M − 1)lM−1 , . . . , 1l1 , 0l0

), and

λL =(M jM , (M − 1)jM−1 , . . . , 1j1 , 0j0

).

The exponential generating function in the considered special case is:

F (N)zr (l, j|t) =

∑u

etEzrN (u)

N 2zr(u)

χRλR(u)χLλL(u) .

Here parameters uj satisfy Bethe equations with gi = 1, and N 2zr(u) is

the squared norm in the same limit:

N 2(u) = 〈ΨN (u)|ΨN (u)〉 =det Q

VN (u2)VN (u−2).

II.4 THE PHASE MODEL

We shall consider the random walks on Simp(N)(ZM+1) which aredened by a step-set ΩM . The Hermitian Hamiltonian Hph of the phasemodel is the generator of the walks :

Hph =

M∑m=0

(φmφ

†m+1 + φ†mφm+1

).

The L-operator is

L(n|u) ≡(u−1 φ†nφn u

),

and the Bethe equations are:

u−2(N+M+1)n = (−1)N−1

N∏j=1

u−2j .

The state-vectors are:

|ΨN (u)〉 =∑

λ⊆MN

Sλ(u2)

(M∏l=0

(φ†l )nl

)|0〉 ,

where Sλ(u2) is the Schur function. The non-strict partition λ isconnected with the coordinates of the Bose particles, as it was explainedabove. On the solutions of the Bethe equations the state-vectors are theeigen-vectors of the Hamiltonian with the eigen-energies

EphN (u) =

N∑k=1

(u2k + u−2k

).

The conjugate state vector is of the form:

〈ΨN (v)| =∑

λ⊆MN

Sλ(v−2) 〈0|(φM )nM . . . (φ1)n1(φ0)n0 .

For arbitrary u and v the scalar product of N -particle state vectors isgiven by the Cauchy-Binet formula:

〈ΨN (v)|ΨN (u)〉 =∑

λ⊆MN

Sλ(v−2)Sλ(u2) =det(Tkj)1≤k,j≤NVN (u2)VN (v−2)

,

Tkj =1− (u2k/v

2j )M+N

1− (u2k/v2j )

.

In the q-parametrization v−2 = q ≡ (q, q2, . . . , qN ),u2 = q/q = (1, q, . . . , qN−1) the scalar product takes the form:

〈ΨN (q−12 )|ΨN ((q/q)

12 )〉 =

det(

1−q(M+N)(j+k−1)

1−q(j+k−1)

)1≤j,k≤N

VN (q)VN (q/q)

=

N∏k=1

N∏j=1

1− qM−1+j+k

1− qj+k−1= Zq(N,N,M) .

As it follows from PROPOSITION, the sum of the Schur functions inq-parametrization may be expressed through the q-binomial determinant:∑

λ⊆MN

Sλ(q)Sλ(q/q) = qNM2 (1−M) det

([2N + i− 1N + j − 1

])1≤i,j≤M

.

The entries are the q-binomial coecients dened as[Rr

]≡ [R]!

[r]! [R− r]!, [n]! ≡ [1] [2] . . . [n], [0]! ≡ 1

where [n] is the q-number being q-analogue of a positive integer n ∈ Z+,

[n] ≡ (1− qn)/(1− q) .

The generating function Zq(L,N, P ) takes the form:

Zq(L,N, P ) =

L∏j=1

N∏k=1

P∏i=1

1− qi+j+k−1

1− qi+j+k−2=

L∏j=1

N∏k=1

1− qP+j+k−1

1− qj+k−1.

R.H.S. tends to A(L,N, P ) at q → 1:

A(L,N, P ) =

L∏j=1

N∏k=1

P∏i=1

i+ j + k − 1

i+ j + k − 2=

L∏j=1

N∏k=1

P + j + k − 1

j + k − 1,

i.e. there are A(L,N, P ) plane partitions in box B(L,N, P ) (theMcMahon's formula).

In other words we obtain that the scalar product 〈ΨN (q−12 )|ΨN ((q/q)

12 )〉

for the phase model in question is equal to the generating functionZq(N,N,M) of plane partitions in the box B(N,N,M).

The generating function of random walks on a Simp(N)(ZM+1) is similarto that of the Totally Asymmetric Zero Range Model:

F(N)ph (l, j|t) =

∑u

etEphN (u)

N 2ph(u)

SλR(u2)SλL(u2)

with the squared norm N 2ph(u):

N 2ph(u) =

∑λ⊆MN

Sλ(u2)Sλ(u2) .

III. CONCLUDING REMARKS

We have presented an approach to the calculation of thermal correlationfunctions of quantum integrable models based on the theory ofsymmetric functions. We considered XX0 model and two modicationsof the phase model: the modication described by non-HermitianHamiltonian and retaining boundary conditions and the modicationdescribed by Hermitian Hamiltonain and by reecting boundaryconditions. The approach is based on representation of the state-vectorswith the use of symmetric functions (apart from the symmetric functionsfor the phase model with retaining conditions, we should mention theShur functions in the case of XXZ model for innite anisotropy, as wellas for four-vertex model).Such notions of enumerative combinatorics as partitions, plane partitions,self-avoiding lattice paths (the limiting cases of the XXZ model) appearnaturally in the investigation of form factors and in the study of theasymptotic behaviour of the correlation functions. The problem ofcomputing the formfactors of operators in the special q-parametrization isreduced to the computation of q-binomial determinants which are thegenerating functions for plane partitions in a box of nite size.

Binomial determinants appear in the limit q → 1, thus giving aninterpretation of the resulting form factors in terms of self-avoidinglattice paths.

THANKS !

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