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# Bracket polynomial

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 R be a commutative ring and fix \( \delta \in R \). The Temperley–Lieb algebra TL_n(\delta) is the R-algebra generated by the elements \( U_1, U_2, \ldots, U_{n-1} \), subject to the Jones relations:

\(U_i^2 = \delta U_i for all 1 \leq i \leq n-1 \)

\(U_i U_{i+1} U_i = U_i for all 1 \leq i \leq n-2 \)

\(U_i U_{i-1} U_i = U_i for all 2 \leq i \leq n-1 \)

\(U_i U_j = U_j U_i for all 1 \leq i,j \leq n-1 \) such that \( |i-j| \neq 1 \)

\(TL_n(\delta) \) 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 \(TL_3(\delta) \) are the following:

Basis of the Temperley–Lieb algebra .

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 \( TL_5(\delta) \) are:

Generators of the Temperley–Lieb algebra

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[1] we define the Temperley-Lieb hamiltonian (the TL hamiltonian) as

\(\mathcal{H} = \sum_{j=1}^{L-1} (1 - e_j) \)

where \(e_j = U(\lambda)/\sin\lambda \), for some spectral parameter \(\lambda \in R \).

Applications

We will firstly consider the case L = 3. The TL hamiltonian is \(\mathcal{H} = 2 - e_1 - e_2 , namely

\( \mathcal{H} \) = 2 - E - E .

We have two possible states,

and

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

\( \mathcal{H} \) = 2 - - = -

and

\( \mathcal{H} \) = 2 - - = -

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

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

The eigenvector of \mathcal{H} with the lowest eigenvalue is known as the ground state. In this case, the lowest eigenvalue \lambda_0 for \mathcal{H} is \lambda_0 = 0. The corresponding eigenvector is \( \psi_0 = (1, 1) \). As we vary the number of sites L we find the following table[2]

\( L \) | \( \psi_0\) | \( L\) | \( \psi_0 \) |
---|---|---|---|

2 | (1) | 3 | (1, 1) |

4 | (2, 1) | 5 | \( \) |

6 | \( 11, 5_2,4, 1) \) | 7 | \( (26_4, 10_2, 9_2, 8_2, 5_2, 1_2) \) |

8 | \( (170, 75_2, 71, 56_2, 50, 30, 14_4, 6, 1)\) | 9 | \( (646, \ldots)\) |

\( \vdots\) | \( \vdots \) | \( \vdots\) | \( \vdots\) |

where we have use the notation \( m_n = (m, \ldots, m) n-times i.e. 5_2 = (5, 5) \).

Combinatorial Properties

An interesting observation is that the largest components of the ground state of \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

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

and for an odd numbers of sites

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

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

References

Temperley N. and Lieb E., (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, Proc. R. Soc. A 322 251.

Batchelor M., de Gier J. and Nienhuis B., (2001), The quantum symmetric XXZ chain at \Delta = -1/2, alternating-sign matrices and plane partitions, J. Phys. A 34, L265-L270.

de Gier J., (2005), Loops, matchings and alternating-sign matrices, Discrete Mathematics Volume 298, Issues 1-3, Pages 365-388.

Further reading

Louis H. Kauffman, State Models and the Jones Polynomial. Topology, 26(3):395-407, 1987.

R.J. Baxter, Exactly solved models in statistical mechanics Academic Press Inc. (1982)

N. Temperley, E. Lieb, 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 Series A 322 (1971), 251-280.

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