arXiv subjects

M. J. Martins

Publications and source records attributed to M. J. Martins.

At least 19 recordsLinked to original sources

arXiv2026

Bethe equations for the critical three-state Potts spin chain with toroidal boundary conditions

In this paper, we consider the parameterization of the spectra of the three-state critical Potts quantum chain with integrable twisted boundary conditions in terms of Bethe ansatz type equations. The Bethe equations are found by investigating the structure of the eigenvalues of the respective twisted transfer matrices, and with the help of certain identities satisfied by the product of transfer matrix operators. We have studied the completeness of the spectrum in terms of the Bethe roots for small lattice sizes and have computed the eigenstate momenta. We found that the spins of the low-lying excitations can have fractional values in accordance with predictions of the underlying conformal field theory. We argue that our framework can be used to build integrable Hamiltonians whose spectra are determined by mixing different toroidal boundary conditions.

math-ph↗

arXiv2025

On the equivalence between n-state spin and vertex models on the square lattice

In this paper we investigate a correspondence among spin and vertex models with the same number of local states on the square lattice with toroidal boundary conditions. We argue that the partition functions of an arbitrary $n$-state spin model and of a certain specific $n$-state vertex model coincide for finite lattice sizes. The equivalent vertex model has $n^3$ non-null Boltzmann weights and their relationship with the edge weights of the spin model is explicitly presented. In particular, the Ising model in a magnetic field is mapped to an eight-vertex model whose weights configurations combine both even and odd number of incoming and outcoming arrows at a vertex. We have studied the Yang-Baxter algebra for such mixed eight-vertex model when the weights are invariant under arrows reversing. We find that while the Lax operator lie on the same elliptic curve of the even eight-vertex model the respective $\mathrm{R}$-matrix can not be presented in terms of the difference of two rapidities. We also argue that the spin-vertex equivalence may be used to imbed an integrable spin model in the realm of the quantum inverse scattering framework. As an example, we show how to determine the $\mathrm{R}$-matrix of the 27-vertex model equivalent to a three-state spin model devised by Fateev and Zamolodchikov.

math-ph↗

arXiv2025

Embedding integrable spin models in solvable vertex models on the square lattice

Exploring a mapping among $n$-state spin and vertex models on the square lattice we argue that a given integrable spin model with edge weights satisfying the rapidity difference property can be formulated in the framework of an equivalent solvable vertex model. The Lax operator and the $\mathrm{R}$-matrix associated to the vertex model are built in terms of the edge weights of the spin model and these operators are shown to satisfy the Yang-Baxter algebra. The unitarity of the $\mathrm{R}$-matrix follows from an assumption that the vertical edge weights of the spin model satisfy certain local identity known as inversion relation. We apply this embedding to the scalar $n$-state Potts model and we argue that the corresponding $\mathrm{R}$-matrix can be written in terms of the underlying Temperley-Lieb operators. We also consider our construction for the integrable Ashkin-Teller model and the respective $\mathrm{R}$-matrix is expressed in terms of sixteen distinct weights parametrized by theta functions. We comment on the possible extention of our results to spin models whose edge weights are not expressible in terms of the difference of spectral parameters.

math-ph↗

arXiv2023

The spectrum properties of an integrable $G_2$ invariant vertex model

This paper is concerned with the study of properties of the exact solution of the fundamental integrable $G_2$ vertex model. The model $R$-matrix and respective spin chain are presented in terms of the basis generators of the $G_2$ Lie algebra. This formulation permits us to related the number of the Bethe roots of the respective Bethe equations with the eigenvalues of the $U(1)$ conserved charges from the Cartan subalgebra of $G_2$. The Bethe equations are solved by a peculiar string structure which combines complex three-strings with real roots allowing us determine the bulk properties in the thermodynamic limit. We argue that $G_2$ spin chain is gapless but the low-lying excitations have two different speeds of sound and the underlying continuum limit is therefore not strictly Lorentz invariant. We have investigate the finite-size corrections to the ground state energy and proposed that the critical properties of the system should be governed by the product of two $c=1$ conformal field theories. By combining numerical and analytical methods we have computed the bulk free-energy of the $G_2$ vertex model. We found that there are three regimes in the spectral parameter in which the free-energy is limited and continuous. There exists however at least two sharp corner points in which the bulk-free energy is not differentiable.

math-ph↗

arXiv2020

Integrability of the spin-1/2 fermions with charge pairing and Hubbard interaction

In this paper we study the exact solution of a one-dimensional model of spin-$\frac{1}{2}$ electrons composed by a nearest-neighbor triplet pairing term and the on-site Hubbard interaction. We argue that this model admits a Bethe ansatz solution through a mapping to a Hubbard chain with imaginary kinetic hopping terms. The Bethe equations are similar to that found by Lieb and Wu \cite{LW} but with additional twist phases which are dependent on the ring size. We have studied the spectrum of the model with repulsive interaction by exact diagonalization and through the Bethe equations for large lattice sizes. One feature of the model is that it is possible to define the charge gap for even and odd lattice sites and both converge to the same value in the infinite size limit. We analyze the finite-size corrections to the low-lying spin excitations and argue that they are equivalent to that of the spin-$\frac{1}{2}$ isotropic Heisenberg model with a boundary twist depending on the lattice parity. We present the classical statistical mechanics model whose transfer matrix commutes with the model Hamiltonian. To this end we have used the construction employed by Shastry \cite{SHA1,SHA2} for the Hubbard model. In our case, however, the building block is a free-fermion eight-vertex model with a particular null weight.

math-ph↗

arXiv2017

Integrability of the odd eight-vertex model with symmetric weights

In this paper we investigate the integrability properties of a two-state vertex model on the square lattice whose microstates at a vertex has always an odd number of incoming or outcoming arrows. This model was named odd eight-vertex model by Wu and Kunz \cite{WK} to distinguish it from the well known eight-vertex model possessing an even number of arrows orientations at each vertex. When the energy weights are invariant under arrows inversion we show that the integrable manifold of the odd eight-vertex model coincides with that of the even eight-vertex model. The form of the $\mathrm{R}$-matrix for the odd eight-vertex model is however not the same as that of the respective Lax operator. Altogether we find that these eight-vertex models give rise to a generic sheaf of $\mathrm{R}$-matrices satisfying the Yang-Baxter equations resembling intertwiner relations associated to equidimensional representations.

math-ph↗

arXiv2017

The spectrum of a vertex model and related spin one chain sitting in a genus five curve

We derive the transfer matrix eigenvalues of a three-state vertex model whose weights are based on a $\mathrm{R}$-matrix not of difference form with spectral parameters lying on a genus five curve. We have shown that the basic building blocks for both the transfer matrix eigenvalues and Bethe equations can be expressed in terms of meromorphic functions on an elliptic curve. We discuss the properties of an underlying spin one chain originated from a particular choice of the $\mathrm{R}$-matrix second spectral parameter. We present numerical and analytical evidences that the respective low-energy excitations can be gapped or massless depending on the strength of the interaction coupling. In the massive phase we provide analytical and numerical evidences in favor of an exact expression for the lowest energy gap. We point out that the critical point separating these two distinct physical regimes coincides with the one in which the weights geometry degenerate into union of genus one curves.

math-ph↗

arXiv2016

Algebro-Geometric approach for a centrally extended U_q[sl(2|2)] R-matrix

In this paper we investigate the algebraic geometric nature of a solution of the Yang-Baxter equation based on the quantum deformation of the centrally extended $sl(2|2)$ superalgebra proposed by Beisert and Koroteev \cite{BEKO}. We derive an alternative representation for the $\mathrm{R}$-matrix in which the matrix elements are given in terms of rational functions depending on weights sited on a degree six surface. For generic gauge the weights geometry are governed by a genus one ruled surface while for a symmetric gauge choice the weights lie instead on a genus five curve. We have written down the polynomial identities satisfied by the $\mathrm{R}$-matrix entries needed to uncover the corresponding geometric properties. For arbitrary gauge the $\mathrm{R}$-matrix geometry is argued to be birational to the direct product $\mathbb{CP}^1 \times \mathbb{CP}^1 \times \mathrm{A}$ where $\mathrm{A}$ is an Abelian surface. For the symmetric gauge we present evidences that the geometric content is that of a surface of general type lying on the so-called Severi line with irregularity two and geometric genus nine. We discuss potential geometric degenerations when the two free couplings are restricted to certain one-dimensional subspaces.

math-ph↗

arXiv2016

Algebraic Geometry methods associated to the one-dimensional Hubbard model

In this paper we study the covering vertex model of the one-dimensional Hubbard Hamiltonian constructed by Shastry in the realm of algebraic geometry. We show that the Lax operator sits in a genus one curve which is not isomorphic but only isogenous to the curve suitable for the AdS/CFT context. We provide an uniformization of the Lax operator in terms of ratios of theta functions allowing us to establish relativistic like properties such as crossing and unitarity. We show that the respective $\mathrm{R}$-matrix weights lie on an Abelian surface being birational to the product of two elliptic curves with distinct $\mathrm{J}$-invariants. One of the curves is isomorphic to that of the Lax operator but the other is solely fourfold isogenous. These results clarify the reason the $\mathrm{R}$-matrix can not be written using only difference of spectral parameters of the Lax operator.

math-ph↗

arXiv2015

The symmetric six-vertex model and the Segre cubic threefold

In this paper we investigate the mathematical properties of the integrability of the symmetric six-vertex model towards the view of Algebraic Geometry. We show that the algebraic variety originated from Baxter's commuting transfer method is birationally isomorphic to a ubiquitous threefold known as Segre cubic primal. This relation makes it possible to present the most generic solution for the Yang-Baxter triple associated to this lattice model. The respective $\mathrm{R}$-matrix and Lax operators are parametrized by three independent affine spectral variables.

math-ph↗

arXiv2014

An Integrable Nineteen Vertex Model Lying on a Hypersurface

We have found a family of solvable nineteen vertex model with statistical configurations invariant by the time reversal symmetry within a systematic study of the respective Yang-Baxter relation. The Boltzmann weights sit on a degree seven algebraic threefold which is shown birationally equivalent to the three-dimensional projective space. This permits to write parameterized expressions for both the transition operator and the $\mathrm{R}$-matrix depending on three independent affine spectral parameters. The Hamiltonian limit tells us that the azimuthal magnetic field term is connected with the asymmetry among two types of spectral variables. The absence of magnetic field defines a physical submanifold whose geometrical properties are remarkably shown to be governed by a quartic $\mathrm{K}3$ surface. This expands considerably the class of irrational manifolds that could emerge in the theory of quantum integrable models.

math-ph↗

arXiv2013

Integrable three-state vertex models with weights lying on genus five curves

We investigate the Yang-Baxter algebra for $\mathrm{U}(1)$ invariant three-state vertex models whose Boltzmann weights configurations break explicitly the parity-time reversal symmetry. We uncover two families of regular Lax operators with nineteen non-null weights which ultimately sit on algebraic plane curves with genus five. We argue that these curves admit degree two morphisms onto elliptic curves and thus they are bielliptic. The associated $\mathrm{R}$-matrices are non-additive in the spectral parameters and it has been checked that they satisfy the Yang-Baxter equation. The respective integrable quantum spin-1 Hamiltonians are exhibited.

math-ph↗

arXiv2011

The factorized F-matrices for arbitrary U(1)^(N-1) integrable vertex models

We discuss the $F$-matrices associated to the $R$-matrix of a general $N$-state vertex model whose statistical configurations encode $N-1$ U(1) symmetries. The factorization condition is shown for arbitrary weights being based only on the unitarity property and the Yang-Baxter relation satisfied by the $R$-matrix. Focusing on the N=3 case we are able to conjecture the structure of some relevant twisted monodromy matrix elements for general weights. We apply this result providing the algebraic expressions of the domain wall partition functions built up in terms of the creation and annihilation monodromy fields. For N=3 we also exhibit a $R$-matrix whose weights lie on a del Pezzo surface and have a rather general structure.

math-ph↗

arXiv2011

The monodromy matrix in the F-basis for arbitrary six-vertex models

We present the expressions for the monodromy matrix elements of the six-vertex model in the F-basis for arbitrary Boltzmann weights. The results rely solely on the property of unitarity and Yang-Baxter relations, avoiding any specific parameterization of the weights. This allows us to write complete algebraic expressions for the inner products and the underlying domain wall partition functions in the case of arbitrary rapidities. We then apply our results for the trigonometric six-vertex model in the presence of inhomogeneous electric fields and obtain a determinant formula for the respective on-shell scalar products.

math-ph↗

arXiv2010

The Yang-Baxter equation for PT invariant nineteen vertex models

We study the solutions of the Yang-Baxter equation associated to nineteen vertex models invariant by the parity-time symmetry from the perspective of algebraic geometry. We determine the form of the algebraic curves constraining the respective Boltzmann weights and found that they possess a universal structure. This allows us to classify the integrable manifolds in four different families reproducing three known models besides uncovering a novel nineteen vertex model in a unified way. The introduction of the spectral parameter on the weights is made via the parameterization of the fundamental algebraic curve which is a conic. The diagonalization of the transfer matrix of the new vertex model and its thermodynamic limit properties are discussed. We point out a connection between the form of the main curve and the nature of the excitations of the corresponding spin-1 chains.

math-ph↗

arXiv2009

The spectrum of an open vertex model based on the U_q[SU(2)] at roots of unity

We study the exact solution of an $N$-state vertex model based on the representation of the $U_q[SU(2)]$ algebra at roots of unity with diagonal open boundaries. We find that the respective reflection equation provides us one general class of diagonal $K$-matrices having one free-parameter. We determine the eigenvalues of the double-row transfer matrix and the respective Bethe ansatz equation within the algebraic Bethe ansatz framework. The structure of the Bethe ansatz equation combine a pseudomomenta function depending on a free-parameter with scattering phase-shifts that are fixed by the roots of unity and boundary variables.

math-ph↗

arXiv2009

Critical Properties of an Integrable Supersymmetric Eletronic Model

We investigate the physical properties of an integrable extension of the Hubbard model with a free parameter $γ$ related to the quantum deformation of the superalgebra $sl(2|2)^{(2)}$. The Bethe ansatz solution is used to determine the nature of the spin and charge excitations. The dispersion relation of the charge branch is given by a peculiar product between energy-momenta functions exhibiting massless and massive behaviors. The study of the finite-size corrections to the spectrum reveals us that the underlying conformal theory has central charge $c=-1$ and critical exponents depending on the parameter $γ$. We note that exact results at the isotropic point $γ=0$ can be established without recourse to the Bethe ansatz solution.

cond-mat.stat-mech↗

arXiv2009

Algebraic Bethe ansatz for U(1) Invariant Integrable Models: Compact and non-Compact Applications

We apply the algebraic Bethe ansatz developed in our previous paper \cite{CM} to three different families of U(1) integrable vertex models with arbitrary $N$ bond states. These statistical mechanics systems are based on the higher spin representations of the quantum group $U_q[SU(2)]$ for both generic and non-generic values of $q$ as well as on the non-compact discrete representation of the $SL(2,{\cal R})$ algebra. We present for all these models the explicit expressions for both the on-shell and the off-shell properties associated to the respective transfer matrices eigenvalue problems. The amplitudes governing the vectors not parallel to the Bethe states are shown to factorize in terms of elementary building blocks functions. The results for the non-compact $SL(2,{\cal R})$ model are argued to be derived from those obtained for the compact systems by taking suitable $N \to \infty$ limits. This permits us to study the properties of the non-compact $SL(2,{\cal R})$ model starting from systems with finite degrees of freedom.

math-ph↗