## Abstract

A quantum phase model is introduced as a limit for very strong interactions of a strongly correlated *q*-boson hopping model. The exact solution of the phase model is reviewed, and solutions are also provided for two correlation functions of the model. Explicit expressions, including both amplitude and scaling exponent, are derived for these correlation functions in the low temperature limit. The amplitudes were found to be related to the number of plane partitions contained in boxes of finite size.

## 1. Introduction

When quantum groups and strongly interacting boson and fermion systems were topical problems in physics and applied mathematics, Prof. R. K. Bullough and the present authors introduced and solved exactly a special *q*-boson model [1,2] in which the nonlinear (i.e. strong) interactions between bosons can be hidden in their commutation relations (in the related *q*-algebra). Results for the eigenstates and eigenvalues of the model were reported in Bogoliubov *et al*. [1,2] and Bullough *et al*. [3]. The thermodynamics of nonlinear systems was of special interest to all of us, and for this model in the strong coupling limit it was studied already in Bullough *et al*. [3]. Somewhat later also correlation function results in the same limit were reported [4].

The *q*-boson model had an obvious limit of ordinary non-interacting bosons for vanishing interactions, but in the limit of very strong interactions it could be expressed in terms of non-interacting phase operators. It turned out, however, that the closed operator algebra now contained, in addition to the phase and its hermitian conjugate, also the number operator as an independent variable. The proper formulation of the quantum phase problem had long been debated [5], and the new phase model introduced avoided the typical difficulties which had plagued the previous versions [5]. The phase model was solved and used as a model for atomic beam-splitters [6].

Fairly soon it also became evident that *q*-boson type integrable models are related to many mathematical problems. They are related to the theory of symmetric functions [7] and to the theory of plane partitions [8–13]. Plane partitions (three-dimensional Young diagrams) [7,14,15] were then discovered to be connected with amazingly wide ranging problems in mathematics as well as theoretical physics. They are intensively studied, e.g. in probability theory [16,17], enumerative combinatorics [18], theory of faceted crystals [19,20], directed percolation [21], theory of random walks on lattices [9,10,15,22] and topological string theory [23]. Prof. Bullough would certainly have loved to include all these various connections in his famous ‘diagram of inter-related nonlinear problems’.

In the present article, we give a review of the phase model and its solution, and consider then its temperature-dependent correlation functions in a system of finite size. Exact answers for correlation functions are reported in determinant form. We then determine the exact asymptotics, including the amplitudes, of these correlation functions in the low temperature limit, and find the related critical exponents. The amplitudes of the leading asymptotics are found to depend on the number of plane partitions confined into a finite box. The results reported for the correlation functions, at the end of §3 and in §5, are new.

## 2. A *q*-boson lattice

The *q*-boson hopping model is defined by the Hamiltonian [1–3]
2.1
Here periodic boundary conditions *n*+*M*=*n* are assumed, and and are the three independent operators of a *q*-boson algebra:
2.2
The *c*-number *q* is expressed in the form *q*=e^{γ} with a real *γ*>0. The apparent simplicity of the model is deceptive; nonlinear interactions between bosons are hidden in the commutation relations of the *q*-boson operators.

The *q*-boson algebra has a representation in the Fock space such that
2.3
in which the Fock vectors (states) |*n*_{1},…,*n*_{M}〉 are orthogonal, 〈*p*_{1},…,*p*_{M}|*n*_{1},…,*n*_{M}〉=*δ*_{p1n1},…,*δ*_{pMnM}, and are generated from the vacuum state,
2.4
by acting on it with operators: ; 0≤*n*_{j}≤*N*, . The ‘box’ variable is defined as [*n*]=(1−*q*^{−2n})/(1−*q*^{−2}), and .

The *q*-boson operators can be expressed in terms of canonical bosons described by operators with the ordinary commutation relations , such that

For () the *q*-bosons become ordinary bosons, , and the *q*-boson model becomes the linear boson hopping model with Hamiltonian

In the strong coupling limit () the *q*-boson operators are transformed into *ϕ*,*ϕ*^{†},*N* with the commutation relations
2.5
in which *π*_{j} is the vacuum projector *π*_{j}=(|0〉〈0|)_{j}. In this limit the *q*-boson Hamiltonian becomes
2.6
which will be called the *phase model* Hamiltonian in the following. The introduced operator *ϕ* is ‘one-sidedly unitary’ or isometric as it satisfies , . In fact are exponential phase operators [24], and can be expressed in terms of ordinary bosons such that

In the continuum limit in which as the lattice spacing , the *q*-boson Hamiltonian becomes the continuum limit Bose gas Hamiltonian

## 3. The phase model

In this section, we describe the solution of the phase model Hamiltonian equation (2.6). A representation for the algebra of phase operators (2.5) in the Fock space follows from equation (2.3):
3.1
where ; 0≤*n*_{j}≤*N*, and |0〉 is given by equation (2.4).

Consider a system of *N* particles on a periodic one-dimensional lattice of length *M*+1, labelled by *i*=*M*,*M*−1,…,1,0. Each lattice site can contain an arbitrary number of particles that evolve with the following dynamic rule: a randomly chosen particle can jump to an adjacent site on the left or right with an equal probability of 1/2 (figure 1). Any configuration of particles can be represented by a sequence {*n*}≡(*n*_{0},*n*_{1},…,*n*_{M}) of occupation numbers *n*_{j}, which satisfy the condition 0≤*n*_{0},*n*_{1},…,*n*_{M}≤*N*, *n*_{0}+*n*_{1}+⋯+*n*_{M}=*N*. Denoting the particle configurations as Fock vectors |*n*_{0},…,*n*_{M}〉, we can express the generator of this particle hopping process as the Hamiltonian of the phase model equation (2.6).

The solution of the phase model for its eigenvectors and eigenstates by the quantum inverse method [25–27] was reported in Bogoliubov *et al*. [1,2] and Bullough *et al*. [3]. The *L* operator of the phase model was found to be
3.2
where parameter . This *L* operator satisfies the intertwining relation
with the *R* matrix
3.3
where *f*(*v*,*u*)=*u*^{2}/(*u*^{2}−*v*^{2}) and *g*(*v*,*u*)=*uv*/(*u*^{2}−*v*^{2}). The monodromy matrix of the phase model can be expressed as a product of *L* operators:
3.4
and the commutation relations of the entries of this matrix, *A*,*B*,*C* and *D*, are given by the *R* matrix of the phase model such that
3.5
The most important of these relations are
The transfer matrix *τ*(*u*) of the problem is the matrix trace of the monodromy matrix equation (3.4),
3.6
From the intertwining relation equation (3.5) it follows that [*τ*(*u*),*τ*(*v*)]=0 for arbitrary values of parameters *u*,*v*, and the transfer matrix can be considered as the generating function of the integrals of motion of the model.

The Hamiltonian of the phase model equation (2.6) can be expressed in terms of the transfer matrix such that
The *N*-particle state vectors can be expressed in the form
3.7
where |0〉 is given by equation (2.4), and the conjugated state vectors are thus given by
3.8
The parameters *u*_{i} above are arbitrary complex numbers.

Using the algebraic Bethe ansatz method [25–27], we find that the state vectors of equation (3.7) are eigenstates of the transfer matrix equation (3.6), and hence of the Hamiltonian equation (2.6), if the parameters *u*_{1},…,*u*_{N} satisfy the Bethe equations
3.9
Using the parametrization in which *p* appears as a momentum variable, the Bethe equations take the form
3.10
where *P* is the total momentum of the system. These equations are invariant under the transformation , and we can thus restrict the consideration into the domain −*π*≤*p*_{n}<*π*. The solutions *p*_{n} to the Bethe equations (3.10) are real numbers and can be parametrized such that *p*_{n}=(2*πI*_{n}+*P*)/(*M*+*N*+1). Here, *I*_{n} are integers or half-integers depending on whether *N* is odd or even, and satisfy the condition *M*+*N*≥*I*_{1}>*I*_{2}>⋯>*I*_{N}≥0. The eigenstates can be shown to form a complete and orthogonal set of states [4].

The eigenenergies of the Hamiltonian equation (2.6) are given by
3.11
The ground state of the model is the eigenstate that corresponds to the lowest eigenenergy, . It is determined by the solution to the Bethe equations (3.10) for *P*=0:
3.12

To distinguish arbitrary complex parameters from solutions to the Bethe equations, we denote the former by *u*,*v* and the latter by *p*. By bold letters, we denote sets of *K* numbers: **u**≡(*u*_{1},*u*_{2},…,*u*_{K}) and e^{ip}≡(e^{ip1},e^{ip2},…,e^{ipK}).

Using the explicit expressions for the *B*(*u*) and *C*(*u*) operators reported in Bogoliubov [8], we can express the state vectors equations (3.7) and (3.8) in the ‘coordinate’ form,
3.13
in which
is the Schur function, ** λ** denotes the partition (

*λ*

_{1},…,

*λ*

_{N}) of non-increasing non-negative integers,

*M*≥

*λ*

_{1}≥

*λ*

_{2}≥⋯≥

*λ*

_{N}≥0, and is the Vandermonde determinant. There is a one-to-one correspondence between the configuration of occupation numbers (

*n*

_{M},…,

*n*

_{1},

*n*

_{0}), 0≤

*n*

_{0},

*n*

_{1},…,

*n*

_{M}≤

*N*;

*n*

_{0}+

*n*

_{1}+⋯+

*n*

_{M}=

*N*, and the partition

**=(**

*λ**M*

^{nM},(

*M*−1)

^{nM−1},…,1

^{n1},0

^{n0}), where each number

*S*appears

*n*

_{S}times. The sum in equation (3.13) is taken over all partitions

**into at most**

*λ**N*parts with

*N*≤

*M*.

The correspondence between the coordinates of the particles and the entries *λ*_{j} of the partition ** λ** is demonstrated in figure 2. In order to calculate correlation functions we also need the conjugated state vector, which is given by
3.14

Calculation of correlation functions for the phase model can be based on the Cauchy relation of the Schur functions such that
3.15
in which the entries of the *N*×*N* matrix *H*_{jk} are given by
and summation is over all partitions ** λ** into at most

*N*parts with

*N*≤

*M*.

The scalar product of *N*-particle state vectors for arbitrary **u** and **v** is given by
3.16
For **u**=**v**=e^{ip} this scalar product gives the normalization of the eigenvectors of the Bethe equations (3.10):
3.17
If **p** and are different sets of solutions to the Bethe equations the related eigenvectors are orthogonal: . Being a complete and orthogonal set of states [4], these eigenvectors provide the resolution of the identity operator,
3.18
where summation is over all different solutions to the Bethe equations (3.10).

In the calculation of the *k*th order transition element of the annihilation operator *ϕ*_{0}, we use an interesting property of the *C*(*v*) operator reported in Bogoliubov [9,10]: . Now the transition element can be deduced from equation (3.16):
3.19
The limit in the above expression has been evaluated in Bogoliubov [9,10], and we find that
3.20
where the elements of the *N*×*N* matrix *Q* are given by
3.21

After having determined the transition elements, we can now calculate various correlation functions of the model. The *K*th order temperature-dependent correlation function in the *N*−*K* particle sector, for example, can be defined such that
3.22
where *t*=1/*T* is the inverse temperature (*k*_{B}=1), are the ground state solutions (see equation (3.12))
3.23
of the Bethe equations
3.24
and
3.25
Here is the ground state energy of the system with *N*−*K* particles, and equations (3.17) and (3.23) were used. Inserting the identity operator equation (3.18) into equation (3.22) and using the translational invariance of the model on the periodic chain, which means that
we find that
3.26
Here summation is over all solutions of the Bethe equations (3.10), *E*_{N} are the eigenenergies of equation (3.11), and are the normalizations of equation (3.17).

For example, the ‘recurrence’ correlation function [6] is thus given by where matrix is given by equation (3.21) with solutions of the Bethe equations: and

Let us consider then the temperature-dependent correlation function for such eigenstates that have no particles at the last *n* sites of the lattice. We thus consider the ‘persistence of empty sites’ correlation function
3.27
in which the average is over the ground state solutions of the Bethe equations (3.12), projection operator *Π*_{n} can be expressed in terms of the vacuum projection operators *π*_{j}, such that , and
Using the result *π*|*n*>=*δ*_{0n}|*n*> and the coordinate form of the eigenstate equation (3.13), we find that
3.28
where summation is over all partitions ** λ** into at most

*N*parts such that

*N*≤

*M*−

*n*. Using the Cauchy relation equation (3.15), we find further that 3.29 This result facilitates the solution to the correlation function equation (3.27) in the form 3.30 where summation is over all solutions of the Bethe equations (3.10).

## 4. Boxed plane partitions

Partition *λ*=(*λ*_{1},*λ*_{2},…,*λ*_{N}) is a non-increasing sequence *λ*_{1}≥*λ*_{2}≥⋯≥*λ*_{N} of non-negative integers *λ*_{j} called the parts of partition *λ*. A plane partition *π* is an array *π*_{ij} of non-negative integers that satisfy *π*_{ij}≥*π*_{i+1j} and *π*_{ij}≥*π*_{ij+1} for all *i*,*j*≥1. Each column or row of a plane partition is in itself an ordinary partition. A plane partition is said to be contained in an *L*×*N*×*M* rectangular box if *π*_{ij}≤*M* for all *i* and *j*, and *π*_{ij}=0 whenever *i*>*L* or *j*>*N*. Such a plane partition is also called a boxed plane partition. Plane partitions are usually interpreted as stacks of unit cubes—three-dimensional Young diagrams. The height of a stack at point *ij* is *π*_{ij}. The volume of partition |*π*| equals the total number of cubes of the diagram, . Allowing some of the partitions to be empty we can express the plane partition contained in an *L*×*N*×*M* box (*N*>*L*) as an *N*×*N* matrix with vanishing matrix elements in the last *K*=*N*−*L* rows (*π*_{jm}≡0 for *j*≥*L*). A particular example of the plane partitions contained in a 4×6×5 box is shown in figure 3.

It was shown in Bogoliubov [9,10] that the transition element is the generating function of the plane partitions contained in an *L*×*N*×*M* box if the parameters *u*_{k} and *v*_{k} are chosen so that *v*_{j}=*q*^{−j/2}, *u*_{j}=*q*^{(j−1)/2}. From equation (3.20), we thus find that
4.1
in which the entries of the *N*×*N* matrix *S* are given by
and
The determinant of matrix *S* can also be evaluated [9–11], and we find that
This result is the partition function for the three-dimensional Young diagrams confined in an *L*×*N*×*M* box with the Boltzmann weight *q*=e^{−1/T}. For it is reduced to the classical result of MacMahon for the number of plane partitions contained in an *L*×*N*×*M* box [15]:
4.2

## 5. The low temperature asymptotics

Let us consider finally the case when the number of lattice sites is very large, *M*≫1, while the number *N* is assumed to satisfy *M*≫*N*≫1. In this limit summation over the solutions of the Bethe equations in the recurrence correlation function equation (3.26) can be replaced by integration, and we thus find that
5.1
In this limit, we can also approximate the ground state solutions equation (3.12) to the Bethe equations, which appear in the integrand of equation (5.1), such that . When the inverse temperature (temperature ), the correlation function can thus be expressed in the form
5.2
The integral remaining in the correlation function above is the Mehta integral for the Gaussian unitary ensemble of random matrices, and it can be explicitly evaluated [28]:

Using the classical MacMahon result of equation (4.2), we now find an explicit expression for the leading order term in *t*^{−1} of the recurrence correlation function:
5.3
with *A*(*N*−*K*,*N*,*M*) the number of plane partitions contained in an (*N*−*K*)×*N*×*M* box.

Similarly, we find for the persistence of the empty sites correlation function equation (3.27) that
5.4
with *A*(*N*,*N*,*M*−*n*) the number of plane partitions contained in an *N*×*N*×(*M*−*n*) box.

Notice that both correlation functions above decay as a power law and have the same critical exponent *γ*=*N*^{2}/2, but their amplitudes are different: they depend on the squared number of plane partitions contained in a box of different size. These amplitudes are observable quantities.

The matrix element equations (3.19) and (3.29) which are the basic quantities in the above correlation functions, can be shown (details not reported here) to be related to the generating function [15] of random vicious walkers (or continuous time ‘random turns’ walkers according to Fisher’s classification [29]) in a similar manner as they are shown in §4 to be related to the generating function (partition function) of boxed plane partitions. It is thus understandable that the scaling exponents reported in Krattenthaler *et al*. [22] and Katori *et al*. [30] for the asymptotic decay of random vicious walks at large times *t* are similar to the critical exponent *γ* obtained here.

## 6. Conclusions

The phase model considered here is a rather generic description of interacting quantum particles on a lattice. As discussed above, it can be related to interacting boson systems, quantum optics systems and tight-binding particles hopping on a lattice. It can as well be transformed into a model of non-interacting fermions and an *XX* spin chain. In a sense it describes particles that have a pseudo-fermionic nature. Because of having an independent number operator in the operator algebra, an action by the particle exchange operator of the phase model does not involve a prefactor including a square root of site occupation numbers as in the case of ordinary bosons, and the relation to fermion variables is thus more direct. Mapping of the boson hopping model in this case to a problem of free fermions is illustrated in figure 4. Correlation functions of the phase model and all the other related models are of practical and theoretical interest as they are related to directly observable quantities. Therefore, exact new results were reported here for two correlations functions, and for their low temperature asymptotics, in particular. It is noteworthy that these results are for systems of finite size. Explicit expressions were given for the amplitudes and scaling exponents of the asymptotically leading terms. No particular physical system was used here as an example; the results obtained can be applied to many different systems as already emphasized.

## Acknowledgements

N.B. would like to thank the University of Jyväskylä for hospitality. This work was partially supported by the RFBR project 10-01-00600.

## Footnotes

One contribution of 13 to a Theme Issue ‘Nonlinear phenomena, optical and quantum solitons’.

- This journal is © 2011 The Royal Society

This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.