Temperley–Lieb algebra

Definition

Let R be a commutative ring and fix δ ∈ R . The Temperley–Lieb algebra T L n ( δ ) is the R -algebra generated by the elements U 1 , U 2 , … , U n − 1 , subject to the Jones relations:

T L n ( δ ) may be represented diagrammatically as the vector space over noncrossing pairings on a rectangle with n points on two opposite sides. The five basis elements of T L 3 ( δ ) are the following:

.

Multiplication on basis elements can be performed by placing two rectangles side by side, and replacing any closed loops by a factor of δ, for example:

× = = δ .

The identity element is the diagram in which each point is connected to the one directly across the rectangle from it, and the generator U i is the diagram in which the ith point is connected to the i+1th point, the 2n − i + 1th point is connected to the 2n − ith point, and all other points are connected to the point directly across the rectangle. The generators of T L 5 ( δ ) are:

From left to right, the unit 1 and the generators U₁, U₂, U₃, U₄.

The Jones relations can be seen graphically:

= δ

=

=

The Temperley-Lieb Hamiltonian

Consider an interaction-round-a-face model e.g. a square lattice model and let L be the number of sites on the lattice. Following Temperley and Lieb we define the Temperley-Lieb hamiltonian (the TL hamiltonian) as

H = ∑ j = 1 L − 1 ( 1 − e j )

where e j = U ( λ ) / sin ⁡ λ , for some spectral parameter λ ∈ R .

Applications

We will firstly consider the case L = 3 . The TL hamiltonian is H = 2 − e 1 − e 2 , namely

H = 2 - - .

We have two possible states,

and .

In acting by H on these states, we find

H = 2 - - = - ,

and

H = 2 - - = - + .

Writing H as a matrix in the basis of possible states we have,

H = ( 1 − 1 − 1 1 )

The eigenvector of H with the lowest eigenvalue is known as the ground state. In this case, the lowest eigenvalue λ 0 for H is λ 0 = 0 . The corresponding eigenvector is ψ 0 = ( 1 , 1 ) . As we vary the number of sites L we find the following table

where we have used the notation m n = ( m , … , m ) n -times e.g. 5 2 = ( 5 , 5 ) .

Combinatorial Properties

An interesting observation is that the largest components of the ground state of H have a combinatorial enumeration as we vary the number of sites, as was first observed by Murray Batchelor, Jan de Gier and Bernard Nienhuis. Using the resources of the on-line encyclopedia of integer sequences, Batchelor et al. found, for an even numbers of sites

1 , 2 , 11 , 170 , … = ∏ j = 0 n − 1 ( 3 j + 1 ) ( 2 j ) ! ( 6 j ) ! ( 4 j ) ! ( 4 j + 1 ) !

and for an odd numbers of sites

1 , 3 , 26 , 646 , … = ∏ j = 0 n − 1 ( 3 j + 2 ) ( 2 j + 2 ) ! ( 6 j + 3 ) ! ( 4 j + 2 ) ! ( 4 j + 3 ) ! .

Surprisingly, these sequences corresponded to well known combinatorial objects. For L even, this sequence (see A051255) corresponds to cyclically symmetric transpose complement plane partitions and for L odd (see A005156) these correspond to ( 2 n + 1 ) × ( 2 n + 1 ) alternating sign matrices symmetric about the vertical axis.