Temperley–Lieb algebra

In statistical mechanics, the Temperley–Lieb algebra is an algebra from which are built certain transfer matrices, invented by Neville Temperley and Elliott Lieb. It is also related to integrable models, knot theory and the braid group, quantum groups and subfactors of von Neumann algebras.

Definition

Let ${\displaystyle R}$ be a commutative ring and fix ${\displaystyle \delta \in R}$. The Temperley–Lieb algebra ${\displaystyle TL_{n}(\delta )}$ is the ${\displaystyle R}$-algebra generated by the elements ${\displaystyle U_{1},U_{2},\ldots ,U_{n-1}}$, subject to the Jones relations:

• ${\displaystyle U_{i}^{2}=\delta U_{i}}$ for all ${\displaystyle 1\leq i\leq n-1}$
• ${\displaystyle U_{i}U_{i+1}U_{i}=U_{i}}$ for all ${\displaystyle 1\leq i\leq n-2}$
• ${\displaystyle U_{i}U_{i-1}U_{i}=U_{i}}$ for all ${\displaystyle 2\leq i\leq n-1}$
• ${\displaystyle U_{i}U_{j}=U_{j}U_{i}}$ for all ${\displaystyle 1\leq i,j\leq n-1}$ such that ${\displaystyle |i-j|\neq 1}$

${\displaystyle TL_{n}(\delta )}$ 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 ${\displaystyle TL_{3}(\delta )}$ 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 ${\displaystyle \delta }$, for example:

× = = ${\displaystyle \delta }$ .

The identity element is the diagram in which each point is connected to the one directly across the rectangle from it, and the generator ${\displaystyle U_{i}}$ is the diagram in which the ${\displaystyle i}$-th point is connected to the ${\displaystyle (i+1)}$-th point, the ${\displaystyle (2n-i+1)}$-th point is connected to the ${\displaystyle (2n-i)}$-th point, and all other points are connected to the point directly across the rectangle. The generators of ${\displaystyle TL_{5}(\delta )}$ are:

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

The Jones relations can be seen graphically:

= ${\displaystyle \delta }$

=

=

The Temperley–Lieb Hamiltonian

Consider an interaction-round-a-face model e.g. a square lattice model and let ${\displaystyle L}$ be the number of sites on the lattice. Following Temperley and Lieb[1] we define the Temperley–Lieb Hamiltonian (the TL Hamiltonian) as

${\displaystyle {\mathcal {H}}=\sum _{j=1}^{L-1}(\delta -U_{j})}$

Applications

In what follows we consider the special case ${\displaystyle \delta =1}$.

We will firstly consider the case ${\displaystyle L=3}$. The TL Hamiltonian is ${\displaystyle {\mathcal {H}}=2-e_{1}-e_{2}}$, namely

${\displaystyle {\mathcal {H}}}$ = 2 - - .

We have two possible states,

and .

In acting by ${\displaystyle {\mathcal {H}}}$ on these states, we find

${\displaystyle {\mathcal {H}}}$ = 2 - - = - ,

and

${\displaystyle {\mathcal {H}}}$ = 2 - - = - + .

Writing ${\displaystyle {\mathcal {H}}}$ as a matrix in the basis of possible states we have,

${\displaystyle {\mathcal {H}}=\left({\begin{array}{rr}1&-1\\-1&1\end{array}}\right)}$

The eigenvector of ${\displaystyle {\mathcal {H}}}$ with the lowest eigenvalue is known as the ground state. In this case, the lowest eigenvalue ${\displaystyle \lambda _{0}}$ for ${\displaystyle {\mathcal {H}}}$ is ${\displaystyle \lambda _{0}=0}$. The corresponding eigenvector is ${\displaystyle \psi _{0}=(1,1)}$. As we vary the number of sites ${\displaystyle L}$ we find the following table[2]

${\displaystyle L}$	${\displaystyle \psi _{0}}$	${\displaystyle L}$	${\displaystyle \psi _{0}}$
2	(1)	3	(1, 1)
4	(2, 1)	5	${\displaystyle (3_{3},1_{2})}$
6	${\displaystyle (11,5_{2},4,1)}$	7	${\displaystyle (26_{4},10_{2},9_{2},8_{2},5_{2},1_{2})}$
8	${\displaystyle (170,75_{2},71,56_{2},50,30,14_{4},6,1)}$	9	${\displaystyle (646,\ldots )}$
${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$

where we have used the notation ${\displaystyle m_{n}=(m,\ldots ,m)}$ ${\displaystyle n}$-times e.g., ${\displaystyle 5_{2}=(5,5)}$.

Combinatorial Properties

An interesting observation is that the largest components of the ground state of ${\displaystyle {\mathcal {H}}}$ have a combinatorial enumeration as we vary the number of sites,[3] as was first observed by Murray Batchelor, Jan de Gier and Bernard Nienhuis.[2] Using the resources of the on-line encyclopedia of integer sequences, Batchelor et al. found, for an even numbers of sites

${\displaystyle 1,2,11,170,\ldots =\prod _{j=0}^{n-1}\left(3j+1\right){\frac {(2j)!(6j)!}{(4j)!(4j+1)!}}}$

and for an odd numbers of sites

${\displaystyle 1,3,26,646,\ldots =\prod _{j=0}^{n-1}(3j+2){\frac {(2j+2)!(6j+3)!}{(4j+2)!(4j+3)!}}.}$

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

References

1. Temperley, Neville; Lieb, Elliott (1971). "Relations between the 'percolation' and 'colouring' problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the 'percolation' problem". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 322 (1549): 251–280. doi:10.1098/rspa.1971.0067. JSTOR 77727. MR 0498284.
2. Batchelor, Murray; de Gier, Jan; Nienhuis, Bernard (2001). "The quantum symmetric ${\displaystyle XXZ}$ chain at ${\displaystyle \Delta =-1/2}$, alternating-sign matrices and plane partitions". Journal of Physics A. 34 (19): L265–L270. arXiv:cond-mat/0101385. doi:10.1088/0305-4470/34/19/101. MR 1836155.
3. de Gier, Jan (2005). "Loops, matchings and alternating-sign matrices". Discrete Mathematics. 298 (1–3): 365–388. arXiv:math/0211285. doi:10.1016/j.disc.2003.11.060. MR 2163456.

Further reading

• Kauffman, Louis H. (1987). "State Models and the Jones Polynomial". Topology. 26 (3): 395–407. doi:10.1016/0040-9383(87)90009-7. MR 0899057.
• Baxter, Rodney J. (1982). Exactly solved models in statistical mechanics. London: Academic Press Inc. ISBN 0-12-083180-5. MR 0690578.