On Hamiltonians for six-vertex models
In this paper, we explain a connection between a family of free-fermionic six-vertex models and a discrete time evolution operator on one-dimensional Fermionic Fock space. The family of ice models generalize those with domain wall boundary, and we focus on two sets of Boltzmann weights whose partition functions were previously shown to generalize a generating function identity of Tokuyama. We produce associated Hamiltonians that recover these Boltzmann weights, and furthermore calculate the partition functions using commutation relations and elementary combinatorics. We give an expression for these partition functions as determinants, akin to the Jacobi-Trudi identity for Schur polynomials.
Key words and phrases:Six-vertex model, Fock space, discrete time evolution, -functions, partition function
Hamiltonians arising from Fock representations of Clifford algebras were explored by the Kyoto school, for example in [5, 6, 12]. The Boson-Fermion correspondence gives an explicit isomorphism between this Fermionic Fock representation and a polynomial algebra, the Bosonic Fock space. The image of elements under this correspondence are commonly called “-functions.” The Kyoto school papers show that -functions are solutions to integrable hierarchies of nonlinear differential equations. Moreover, the Bosonic Fock space may be identified with the ring of symmetric functions over a field. In particular if the Clifford algebra is , then there exists a simple family of -functions equal to Schur polynomials.
Thinking of each application of the Hamiltonian operator as a step in discrete time, the evolution of a one-dimensional model gives rise to a two-dimensional lattice model. In the case above of the Hamiltonian for , the resulting two-dimensional model is the five-vertex model; a nice exposition of this fact may be found in Zinn-Justin . Our first result is a generalization of this fact, using a deformation of the above Hamiltonian operator for whose evolution produces the six-vertex model studied in . These models have boundary conditions generalizing the more familiar domain wall boundary conditions.
As a consequence, the partition functions for these six-vertex models may be studied using methods for evaluating -functions, e.g., the time evolution of fermionic fields (given in Proposition 4) and Wick’s theorem. This theme is also present in , and we borrow many of the same techniques for analyzing our more complicated six-vertex model. The partition function of our family of six-vertex models was computed (upon using a combinatorial bijection with Gelfand-Tsetlin patterns) by Tokuyama  and subsequently reproved using the Yang-Baxter equation in . Here we give an alternate proof of Tokuyama’s result using only commutation relations for Hamiltonians and some elementary combinatorial facts. Moreover, we provide new explicit expressions for partition functions as a determinant, akin to the Jacobi-Trudi identity for ordinary Schur polynomials.
The modified -functions we study involve a Hamiltonian arising from the theory of super Clifford algebras. Super Clifford algebras and their (super) Boson-Fermion correspondence were studied by Kac and Van de Leur in , and the Hamiltonian operators we present in Definition 1 are their in Proposition 4.4 of . They did not pursue explicit formulas for the resulting -function, but combining Wick’s theorem and Jacobi-Trudi type formulas identifies them as (skew) supersymmetric Schur polynomials, which we detail in an appendix at the end of the paper.
Our study of partition functions for lattice models using Hamiltonian operators was motivated by results in the representation theory of -adic groups. The Shintani-Casselman-Shalika formula is an explicit expression for the spherical Whittaker function of an unramified principal series on a (quasi-split) reductive group over a -adic field. While we refrain from fully defining it here, it is a matrix coefficient on the space of the principal series of particular importance in automorphic forms. Most notably, for the group , its special values are precisely the partition function of the six vertex model in [3, 9]. The approach to using Hamiltonians to study Whittaker functions over real reductive groups was used previously by Kazhdan and Kostant [15, 7], who observed that the Laplacian on the group, restricted to the space of Whittaker functions, is the quantum Toda Hamiltonian. This resulted in a proof of the total integrability of the Toda lattice. In the theory of automorphic forms the role of the Laplacian at real places is replaced by Hecke operators at the -adic places, and a approach to these via Hamiltonians has been put forward in papers of Gerasimov, Lebedev, and Oblezin, e.g., . Viewed in terms of two-dimensional lattice models, these operators act on the column parameters of the model, while ours act on the rows. Given these connections, we hope the results of this paper are a first step toward a Hamiltonian realization of a natural class of operators for representations of -adic groups.
This work was supported by NSF grant DMS-1406238.
2. Preliminaries and Statement of results
2.1. The Boson-Fermion correspondence and -functions
To give a precise statement of our results, we first define the relevant Hamiltonian operators from the Clifford algebra and recall the Boson-Fermion correspondence in this case. Further information can be found in , , and . Our notational conventions more closely follow this latter reference – in particular we locate fermions at half-integers.
Let denote the Clifford algebra generated by creation and annilation operators and with and satisfying relations
where . We create generating functions on these operators:
Then let be the set of “free-fermions” defined by
Let denote the left -module and let denote the right -module ; these are termed the “Fock representations of .” They are cyclic -modules whose generators are typically denoted with bra-ket notation mod and mod , respectively, for some choice of “vacuum vector” 0. There is a symmetric bilinear form on denoted by .
The space is also a module, where we define
with regarded as abstract symbols satisfying a Lie bracket defined as in (1.6) of  and is a central element. The notation is used elsewhere to mean the “normal ordering” with respect to the vacuum vector defined by
which is useful in eliminating trivial infinite quantities from the definitions. The action of on an element is given by
For each define the elements
which satisfy the commutation relations where denotes the Kronecker delta symbol. This allows us to define, associated to infinitely many parameters an action by the element
The element is central in and so we may decompose as a direct sum of -eigenspaces indexed by integer eigenvalues . These are irreducible representations of with highest weight vector defined by
There is an analogous story for the the Fock space with highest weight representations indexed by .
The following map is an isomorphism of vector spaces from to , where each :
Jimbo and Miwa  justify this by noting that the image consists of Schur functions, which give a basis for the polynomial ring in infinitely many variables . In particular, given and a non-increasing sequence , we define
(Note that if we pad the partition with additional parts, the state of free fermions is unchanged.) Then the Schur function is realized as
where the variables in are (up to a simple factor of ) equal to the power sum symmetric functions. Thus making the change of variables:
gives the symmetric function as a symmetric polynomial in the variables . Think of this change of variables as for the matrix with eigenvalues ; this substitution appears in this form as Equation (1.3) in Miwa .
2.2. -functions for superalgebra Hamiltonians
In order to describe our results on -functions, first consider the following generalizations of the transformation in (5), and the subsequent Hamiltonian operators that arise from them.
For and , define
Let . Then define by
Similarly, to an infinite set of variables , let
Let , and define by
Just as in (5) could be regarded as a trace, the elements may be viewed as supertraces (as defined, for example, in Section I.1.5 of ). Indeed if is a graded vector space with basis given by a union of bases for and , then an operator , viewed as a Lie superalgebra, has block form
Thus, for example, in where the eigenvalues of are in the component and in the component. As noted in the Overview, these Hamiltonians arise in the study of superalgebras and are denoted in Proposition 4.4 of , where we have set .
By combining Wick’s theorem and the Jacobi-Trudi formula for (skew) supersymmetric Schur functions, we obtain the following formula for the -function of the superalgebra Hamiltonian in Definition 1:
with as defined in (21). The first equality above, using Wick’s theorem, is no harder than for the Hamiltonian in (2) as it is obtained from a formal change of variables. The second equality may be found on p. 22 of Macdonald  or, in the non-skew case, Equation (1.7) in Moens and Van der Jeugt . Details for the first equality above are presented in an appendix at the end of the paper.
In what follows, we make a small but very important change in the -functions in (6). We replace by in the bra-ket, inserting the annihilation operator between each application of ; we make a similar adjustment to . This radically changes the analysis for the resulting -functions. Perhaps they should no longer be referred to as “-functions” since these are typically expectation values of multiplied by group-like elements – exponentiated linear combinations of elements (see for example Section 3 of ). By contrast, the operators destroy one particle with each application. Remarkably, this change is precisely what is needed for the bra-ket to match the partition function of two-dimensional ice-type lattice models studied in .
To state our results, we introduce one further shorthand notation. In the event that is strictly decreasing, we have that is non-decreasing, in which case we write
Observe that one can also view as the coefficient of in .
Our main results are the following.
Let and be strictly decreasing, and let . Then
where ; ; and . The notation means the sum of the parts of the partition .
The previous result tells us how to compute the weight associated with the evolution of to from one application of our (shifted) Hamiltonians. If has parts, it follows that the only state which can appear after successive applications of, for example, the operator is one of the form for some . (An analogous result holds in the other case.) In the following theorem we compute the weight associated to ; i.e., when the result of applications yields the vacuum. When we wish to use solely bra-kets made from partitions, the empty partition (which may be padded with ’s) is used to denote the vacuum state.
Let . Then with we have
where the product in the bra-ket is taken so that is rightmost in the first product, and is rightmost in the second product.
Notice that the left side of the first expression can be thought of as
as the vary over all partitions with parts (and the only partition with parts is ). Theorem 1 tells us that the only sequences of partitions that appear in this product are ones which satisfy the interleaving condition from the theorem. (Of course, the same holds true when considering the second expression.)
3. Pictorial representations for free-fermions
Before moving towards the proofs of our main results, we pause briefly to carry out a few illustrative computations. For this, it will be useful to have a pictorial representation for Fermion states. We represent the vacuum state with a sea of particles (black dots) occupying each negative half-integer position:
Given any integer , the state is depicted simply by
and for a strictly decreasing partition , one depicts by taking the vacuum state and creating particles at the positions . These diagrams are useful, but caution is required. The state refers to a precise order of creation or annihilation operators applied to the vacuum as in (3). If one wants to perform a creation or annihilation operation and express the result as a linear combination of , one must keep track of the associated signs from commutation relations. This is achieved by counting the number of particles to the right of the position where the creation or annihilation is taking place; the resulting sign is simply to this power.
For the purposes of illustration, we examine the action of our shifted Hamiltonians on with . Recall that we assigned variables corresponding to each non-zero part in and factored the Hamiltonian as a product of operators or for each pair of variables as in Definition 1. For the operator, we compute
The action of operator gives all ways of moving a single particle units to the left. So we must analyze which moves are non-zero on the state :
Our relations (1) imply that creating a particle where one already exists annihilates the state. Thus in our example, the action by is 0 for any . In fact, the following are the products of operators which yield nonzero states: , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . Obviously there are far too many here to compute explicitly, so we’ll content ourselves with a slightly different question: what is the coefficient of in the ket appearing in equation (7)? That is, what is the value of
Note that since the operators from only move particles to the left, the particle which begins at position must move to , the particle which begins at position must stay fixed, and the particle which begins at position must move to . The possible migrations (coming from various summands in the operators , and ) are given in Figure 1.
Analyzing the contribution from each of these and summing yields a coefficient of
Note that for the Hamiltonian, the action of the associated on moves particles to the right. Hence analyzing all possible rightward motions is even more intractable, since there is a sea of antiparticles on the right which allows for particles to move as far as they like. We will compute
Since our operator finishes by annihilating at position , it suffices for us to determine those rightward migrations that carry to . The possible migrations are given in Figure 1. Analyzing the contribution from each yields
While none of our results rely on this pictorial representation of fermions and their images under Hamiltonian operators, we find it useful to see directly how the Hamiltonians are behaving and to have tools for explicit computation of special cases of the results above.
4. A connection to ice models
The value of the bra-ket from Theorem 1 has previously appeared in a generating function identity of Tokuyama  phrased in the language of “strict” Gelfand-Tsetlin patterns. A Gelfand-Tsetlin pattern (of size ) is a sequence of partitions (for ) that satisfy the interleaving condition
Such a pattern is called strict if each is strictly decreasing. A Gelfand-Tsetlin pattern is often depicted as a triangular array
and so the are often called the rows of the pattern. The set of all Gelfand-Tsetlin patterns with fixed top row arise naturally in representation theory, as they parametrize bases for highest weight representations of according to the multiplicity-free branching rules from to .
Strict Gelfand-Tsetlin patterns are perhaps less natural, but it was noticed by Hamel and King  that they are in bijection with admissible states of certain two-dimensional lattice models (or “ice” models). This bijection was exploited in  to give a proof of Tokuyama’s formula using the Yang-Baxter equation. We now describe this bijection more precisely. Consider the set of balanced directed graphs with -columns (labeled from left to right as through ) and -rows (labeled from top to bottom as through ), subject to the following boundary conditions: inward pointing boundary edges along each row, downward pointing boundary edges along the bottom boundary, and upward pointing edges at the top boundary of a column if and only if the column index corresponds to a part of . Such a graph is called an admissible state of ice, and we write for the set of all such admissible states. To create a state of ice from a pattern , one labels the northern edge of the vertex at upward if and only if is a part of ; this determines the direction along any vertical edge, and the horizontal edges are filled in inductively so as to produce a state of . For example, under this bijection we make the following identification: