[Paper Review] Multiparameter statistical models from $N^2 imes N^2$ braid matrices: Explicit eigenvalues of transfer matrices ${\bf T}^{(r)}$, spin chains, factorizable scatterings for all $N$

B. Abdesselam, Amlan Chakrabarti|ArXiv.org|Jun 14, 2008

Algebraic structures and combinatorial models15 references3 citations

TL;DR

This paper presents a systematic construction of multiparameter statistical models based on $N^2 \times N^2$ braid matrices for all $N$, enabling explicit computation of eigenvalues of transfer matrices $\mathbf{T}^{(r)}$ via linear equations with constant coefficients. The key contribution is a general solution for eigenvalues and eigenstates across all $N$ and $r$, rooted in nested projectors and zero-sum multiplets from index permutations, with $N^2$ free parameters and applications to spin chains and factorizable $S$-matrices via inverse Cayley transforms.

ABSTRACT

For a class of multiparameter statistical models based on $N^2 imes N^2$ braid matrices the eigenvalues of the transfer matrix ${\bf T}^{(r)}$ are obtained explicitly for all $(r,N)$. Our formalism yields them as solutions of sets of linear equations with simple constant coefficients. The role of zero-sum multiplets constituted in terms of roots of unity is pointed out and their origin is traced to circular permutations of the indices in the tensor products of basis states induced by our class of ${\bf T}^{(r)}$ matrices. The role of free parameters, increasing as $N^2$ with $N$, is emphasized throughout. Spin chain Hamiltonians are constructed and studied for all $N$. Inverse Cayley transforms of Yang-Baxter matrices corresponding to our braid matrices are obtained for all $N$. They provide potentials for factorizable $S$-matrices. Main results are summarized and perspectives are indicated in the concluding remarks.

Motivation & Objective

• To develop a general framework for constructing multiparameter statistical models based on $N^2 \times N^2$ braid matrices for all $N$.
• To derive explicit eigenvalues of transfer matrices $\mathbf{T}^{(r)}$ for all $r$ and $N$ using linear systems with constant coefficients.
• To explore the role of zero-sum multiplets formed by roots of unity arising from circular permutations of tensor product indices.
• To construct spin chain Hamiltonians and factorizable $S$-matrices for all $N$ via inverse Cayley transforms of Yang-Baxter matrices.
• To emphasize the increase in free parameters scaling as $N^2$, enabling richer model families than previous approaches.

Proposed method

• Construction of $N^2 \times N^2$ braid matrices using a nested sequence of projectors defined via indices $i,j$ and their duals $\bar{i}, \bar{j}$ for even and odd $N$.
• Definition of transfer matrices $\mathbf{T}^{(r)} = \sum_{a=1}^N T_{aa}^{(r)}$ via coproduct rules, with eigenvalues derived from linear equations with constant coefficients.
• Use of parameter sets $m_{ij}^{(\epsilon)}$ satisfying $m_{ij}^{(\epsilon)} = m_{i\bar{j}}^{(\epsilon)}$ to ensure braid matrix consistency and $N^2$-parameter freedom.
• Identification of zero-sum multiplets through circular permutations of indices in tensor product states, linked to roots of unity in eigenvalue spectra.
• Application of inverse Cayley transformation to Yang-Baxter matrices to generate potentials for factorizable $S$-matrices.
• Systematic derivation of eigenvalues and eigenstates by decomposing the $N^r$-dimensional space into subspaces closed under $\mathbf{T}^{(r)}$ action.

Experimental results

Research questions

• RQ1 How can eigenvalues of transfer matrices $\mathbf{T}^{(r)}$ be computed explicitly for all $N$ and $r$ using a unified formalism?
• RQ2 What is the origin and role of zero-sum multiplets in the spectrum of $\mathbf{T}^{(r)}$, and how are they related to index permutations in tensor products?
• RQ3 How do the number and structure of free parameters scale with $N$, and what impact does this have on model richness?
• RQ4 Can spin chain Hamiltonians be systematically constructed from these braid matrices for all $N$?
• RQ5 How can inverse Cayley transforms of Yang-Baxter matrices be derived to yield factorizable $S$-matrices for all $N$?

Key findings

• Eigenvalues of $\mathbf{T}^{(r)}$ are obtained as solutions to sets of linear equations with constant coefficients, valid for all $N$ and $r$.
• The number of free parameters in the model increases as $N^2$, significantly enriching the parameter space compared to standard models.
• Zero-sum multiplets in the spectrum arise from circular permutations of indices in tensor product states, parametrized by roots of unity.
• For $N=2n$, $\mathrm{Tr}(\mathbf{T}^{(r)}) = 2\sum_{i=1}^n e^{r m_{ii}^{(+)}\theta}$; for $N=2n-1$, $\mathrm{Tr}(\mathbf{T}^{(r)}) = 2\sum_{i=1}^{n-1} e^{r m_{ii}^{(+)}\theta} + 1$.
• Inverse Cayley transforms of Yang-Baxter matrices yield $S$-matrix potentials compatible with factorizability for all $N$.
• The formalism allows explicit construction of eigenvalues and eigenstates up to $r=5$, with clear generalization to higher $r$.