arXiv subjects

Rodrigo A. Pimenta

Publications and source records attributed to Rodrigo A. Pimenta.

At least 19 recordsLinked to original sources

arXiv2025

The $q$-Racah polynomials from scalar products of Bethe states II

The theory of Leonard triples is applied to the derivation of normalized scalar products of on-shell and off-shell Bethe states generated from a Leonard pair. The scalar products take the form of linear combinations of $q$-Racah polynomials with coefficients depending on the off-shell parameters. Upon specializations, explicit solutions for the corresponding Belliard-Slavnov linear systems are obtained. It implies the existence of a determinant formula in terms of inhomogeneous Bethe roots for the $q$-Racah polynomials. Also, a set of relations that determines solutions (Bethe roots) of the corresponding Bethe equations of inhomogeneous type in terms of solutions of Bethe equations of homogenous type is obtained.

math-ph↗

arXiv2024

Exceptional Points, Bulk-Boundary Correspondence, and Entanglement Properties for a Dimerized Hatano-Nelson Model with Staggered Potentials

It is well-known that the standard bulk-boundary correspondence does not hold for non-Hermitian systems in which also new phenomena such as exceptional points do occur. Here we study, mostly by analytical means, a paradigmatic one-dimensional non-Hermitian model with dimerization, asymmetric hopping, and imaginary staggered potentials. We present analytical solutions for the singular-value and the eigensystem of this model with both open and closed boundary conditions. We explicitly demonstrate that the proper bulk-boundary correspondence is between topological winding numbers in the periodic case and singular values, {\it not eigenvalues}, in the open case. These protected singular values are connected to hidden edge modes which only become exact zero-energy eigenmodes in the semi-infinite chain limit. We also show that a non-trivial topology leads to protected eigenvalues in the entanglement spectrum. In the $\mathcal{PT}$-symmetric case, we find that the model has a so far overlooked phase where exceptional points become dense in the thermodynamic limit. This phase shows unusual hyper-ballistic transport properties with a dynamical critical exponent $z=1/2$.

cond-mat.stat-mech↗

arXiv2023

Random free-fermion quantum spin chain with multi-spin interactions

We study the effects of quenched disorder in a class of quantum chains with (p+1)-multispin interactions exhibiting a free fermionic spectrum, paying special attention to the case p=2. Depending if disorder couples to (i) all the couplings or just to (ii) some of them, we have two distinct physical scenarios. In case (i), we find that the transitions of the model are governed by a universal infinite-randomness critical point surrounded by quantum Griffiths phases similarly as happens to the random transverse-field Ising chain. In case (ii), we find that quenched disorder becomes an irrelevant perturbation: the clean critical behavior is stable and Griffiths phases are absent. Beyond the perturbative regime, disorder stabilizes a line of finite-randomness critical points (with nonuniversal critical exponents), that ends in a multicritical point of infinite-randomness type. In that case, quantum Griffiths phases also appear surrounding the finite-disorder transition point. We have characterized the correlation functions and the low-temperature thermodynamics of these chains. Our results are derived from a strong-disorder renormalization-group technique and from finite-size scaling analysis of the spectral gap computed exactly (up to sizes ~10^{7}) via an efficient new numerical method recently introduced in the literature [Phys. Rev. B 104, 174206 (2021)].

cond-mat.dis-nn↗

arXiv2023

Ising analogues of quantum spin chains with multispin interactions

A new family of free fermionic quantum spin chains with multispin interactions was recently introduced. Here we show that it is possible to build standard quantum Ising chains -- but with inhomogeneous couplings -- which have the same spectra as the novel spin chains with multispin interactions. The Ising models are obtained by associating an antisymmetric tridiagonal matrix to the polynomials that characterize the quasienergies of the system via a modified Euclidean algorithm. For the simplest non-trivial case, corresponding to the Fendley model, the phase diagram of the inhomogeneous Ising model is investigated numerically. It is characterized by gapped phases separated by critical lines with order-disorder transitions depending on the parity of the total number of energy density operators in the Hamiltonian.

cond-mat.stat-mech↗

arXiv2023

Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz

An operator of Heun-Askey-Wilson type is diagonalized within the framework of the algebraic Bethe ansatz using the theory of Leonard pairs. For different specializations and the generic case, the corresponding eigenstates are constructed in the form of Bethe states, whose Bethe roots satisfy Bethe ansatz equations of homogeneous or inhomogenous type. For each set of Bethe equations, an alternative presentation is given in terms of `symmetrized' Bethe roots. Also, two families of on-shell Bethe states are shown to generate two explicit bases on which a Leonard pair acts in a tridiagonal fashion. In a second part, the (in)homogeneous Baxter T-Q relations are derived. Certain realizations of the Heun-Askey-Wilson operator as second q-difference operators are introduced. Acting on the Q-polynomials, they produce the T-Q relations. For a special case, the Q-polynomial is identified with the Askey-Wilson polynomial, which allows one to obtain the solution of the associated Bethe ansatz equations. The analysis presented can be viewed as a toy model for studying integrable models generated from the Askey-Wilson algebra and its generalizations. As examples, the q-analog of the quantum Euler top and various types of three-sites Heisenberg spin chains in a magnetic field with inhomogeneous couplings, three-body and boundary interactions are solved. Numerical examples are given. The results also apply to the time-band limiting problem in signal processing.

math-ph↗

arXiv2022

The $q$-Racah polynomials from scalar products of Bethe states

The $q$-Racah polynomials are expressed in terms of certain ratios of scalar products of Bethe states associated with Bethe equations of either homogeneous or inhomogeneous type. This result is obtained by combining the theory of Leonard pairs and the modified algebraic Bethe ansatz.

math-ph↗

arXiv2022

Free-parafermionic $Z(N)$ and free-fermionic $XY$ quantum chains

The relationship between the eigenspectrum of Ising and XY quantum chains is well known. Although the Ising model has a $Z(2)$ symmetry and the XY model a $U(1)$ symmetry, both models are described in terms of free-fermionic quasi-particles. The fermionic quasi-energies are obtained by means of a Jordan-Wigner transformation. On the other hand, there exist in the literature a huge family of $Z(N)$ quantum chains whose eigenspectra, for $N>2$, are given in terms of free parafermions and they are not derived from the standard Jordan-Wigner transformation. The first members of this family are the $Z(N)$ free-parafermionic Baxter quantum chains. In this paper we introduce a family of XY models that beyond two-body also have $N$-multispin interactions. Similarly to the standard XY model they have a $U(1)$ symmetry and are also solved by the Jordan-Wigner transformation. We show that with appropriate choices of the $N$-multispin couplings, the eigenspectra of these XY models are given in terms of combinations of $Z(N)$ free-parafermionic quasi-energies. In particular all the eigenenergies of the $Z(N)$ free-parafermionic models are also present in the related free-fermionic XY models. The correspondence is established via the identification of the characteristic polynomial which fixes the eigenspectrum. In the $Z(N)$ free-parafermionic models the quasi-energies obey an exclusion circle principle that is not present in the related $N$-multispin XY models.

cond-mat.stat-mech↗

arXiv2021

Powerful method to evaluate the mass gaps of free-particle quantum critical systems

We present a numerical method for the evaluation of the mass gap, and the low-lying energy gaps, of a large family of free-fermionic and free-parafermionic quantum chains. The method is suitable for some generalizations of the quantum Ising and XY models with multispin interactions. We illustrate the method by considering the Ising quantum chains with uniform and random coupling constants. The mass gaps of these quantum chains are obtained from the largest root of a characteristic polynomial. We also show that the Laguerre bound, for the largest root of a polynomial, used as an initial guess for the largest root in the method, gives us estimates for the mass gaps sharing the same leading finite-size behavior as the exact results. This opens an interesting possibility of obtaining precise critical properties very efficiently which we explore by studying the critical point and the paramagnetic Griffiths phase of the quantum Ising chain with random couplings. In this last phase, we obtain the effective dynamical critical exponent as a function of the distance-to-criticality. Finally, we compare the mass gap estimates derived from the Laguerre bound and the strong-disorder renormalization-group method. Both estimates require comparable computational efforts, with the former having the advantage of being more accurate and also being applicable away from infinite-randomness fixed points. We believe this method is a relevant tool for tackling critical quantum chains with and without quenched disorder.

cond-mat.stat-mech↗

arXiv2021

Scalar product for the XXZ spin chain with general integrable boundaries

We calculate the scalar product of Bethe states of the XXZ spin-$\frac{1}{2}$ chain with general integrable boundary conditions. The off-shell equations satisfied by the transfer matrix and the off-shell Bethe vectors allow one to derive a linear system for the scalar product of off-shell and on-shell Bethe states. We show that this linear system can be solved in terms of a compact determinant formula that involves the Jacobian of the transfer matrix eigenvalue and certain q-Pochhammer polynomials of the boundary couplings.

math-ph↗

arXiv2020

Integrable quantum spin chains with free fermionic and parafermionic spectrum

We present a general study of the large family of exact integrable quantum chains with multispin interactions introduced recently in \cite{AP2020}. The exact integrability follows from the algebraic properties of the energy density operators defining the quantum chains. The Hamiltonians are characterized by a parameter $p=1,2,\dots$ related to the number of interacting spins in the multispin interaction. In the general case the quantum spins are of infinite dimension. In special cases, characterized by the parameter $N=2,3,\ldots$, the quantum chains describe the dynamics of $Z(N)$ quantum spin chains. The simplest case $p=1$ corresponds to the free fermionic quantum Ising chain ($N=2$) or the $Z(N)$ free parafermionic quantum chain. The eigenenergies of the quantum chains are given in terms of the roots of special polynomials, and for general values of $p$ the quantum chains are characterized by a free fermionic ($N=2$) or free parafermionic ($N>2$) eigenspectrum. The models have a special critical point when all coupling constants are equal. At this point the ground-state energy is exactly calculated in the bulk limit, and our analytical and numerical analyses indicate that the models belong to universality classes of critical behavior with dynamical critical exponent $z = (p+1)/N$ and specific-heat exponent $α= \max\{0,1-(p+1)/N\}$.

cond-mat.stat-mech↗

arXiv2020

Free fermionic and parafermionic quantum spin chains with multispin interactions

We introduce a new a family of $Z(N)$ multispins quantum chains with a free-fermionic ($N=2$) or free-parafermionic ($N>2$) eigenspectrum. The models have $(p+1)$ interacting spins ($p=1,2,\dots$), being Hermitian in the $Z(2)$ (Ising) case and non-Hermitian for $N>2$. We construct a set of mutually commuting charges that allows us to derive the eigenenergies in terms of the roots of polynomials generated by a recurrence relation of order $(p+1)$. In the critical limit we identify these polynomials with certain hypergeometric polynomials ${}_{p+1}F_p$. Also in the critical regime, we calculate the ground state energy in the bulk limit and verify that they are given in terms of the Lauricella hypergeometric series. The models with special couplings are self-dual and at the self-dual point show a critical behavior with dynamical critical exponent $z_c=\frac{p+1}{N}$.

cond-mat.stat-mech↗

arXiv2019

Towards the solution of an integrable $D_2^{(2)}$ spin chain

Two branches of integrable open quantum-group invariant $D_{n+1}^{(2)}$ quantum spin chains are known. For one branch (epsilon=0), a complete Bethe ansatz solution has been proposed. However, the other branch (epsilon=1) has so far resisted solution. In an effort to address this problem, we consider here the simplest case n=1. We propose a Bethe ansatz solution, which however is not complete, as it describes only the transfer-matrix eigenvalues with odd degeneracy. We also consider a proposal for the missing eigenvalues.

hep-th↗

arXiv2018

Higher rank classical analogs of the Askey-Wilson algebra from the $sl_N$ Onsager algebra

The $sl_N$-Onsager algebra has been introduced by Uglov and Ivanov in 1995. In this letter, a FRT presentation of the $sl_N$-Onsager algebra is given, its current algebra and commutative subalgebra are constructed. Certain quotients of the $sl_N$-Onsager algebra are then considered, which produce `classical' analogs of higher rank extensions of the Askey-Wilson algebra. As examples, the cases $N=3$ and $N=4$ are described in details.

math-ph↗

arXiv2018

New D_{n+1}^(2) K-matrices with quantum group symmetry

We find new families of solutions of the $D_{n+1}^{(2)}$ boundary Yang-Baxter equation. The open spin-chain transfer matrices constructed with these K-matrices have quantum group symmetry corresponding to removing one node from the $D_{n+1}^{(2)}$ Dynkin diagram, namely, $U_{q}(B_{n-p}) \otimes U_{q}(B_{p})$, where $p=0, \ldots, n$. These transfer matrices also have a $p \leftrightarrow n-p$ duality symmetry. These symmetries help to account for the degeneracies in the spectrum of the transfer matrix.

hep-th↗

arXiv2018

A tale of two Bethe ansätze

We revisit the construction of the eigenvectors of the single and double-row transfer matrices associated with the Zamolodchikov-Fateev model, within the algebraic Bethe ansatz method. The left and right eigenvectors are constructed using two different methods: the fusion technique and Tarasov's construction. A simple explicit relation between the eigenvectors from the two Bethe ansätze is obtained. As a consequence, we obtain the Slavnov formula for the scalar product between on-shell and off-shell Tarasov-Bethe vectors.

math-ph↗

arXiv2017

The integrable quantum group invariant A_{2n-1}^(2) and D_{n+1}^(2) open spin chains

A family of A_{2n}^(2) integrable open spin chains with U_q(C_n) symmetry was recently identified in arXiv:1702.01482. We identify here in a similar way a family of A_{2n-1}^(2) integrable open spin chains with U_q(D_n) symmetry, and two families of D_{n+1}^(2) integrable open spin chains with U_q(B_n) symmetry. We discuss the consequences of these symmetries for the degeneracies and multiplicities of the spectrum. We propose Bethe ansatz solutions for two of these models, whose completeness we check numerically for small values of n and chain length N. We find formulas for the Dynkin labels in terms of the numbers of Bethe roots of each type, which are useful for determining the corresponding degeneracies. In an appendix, we briefly consider D_{n+1}^(2) chains with other integrable boundary conditions, which do not have quantum group symmetry.

math-ph↗

arXiv2016

Algebraic Bethe ansatz for the Temperley-Lieb spin-1 chain

We use the algebraic Bethe ansatz to obtain the eigenvalues and eigenvectors of the spin-1 Temperley-Lieb open quantum chain with "free" boundary conditions. We exploit the associated reflection algebra in order to prove the off-shell equation satisfied by the Bethe vectors.

math-ph↗

arXiv2016

Universal Bethe ansatz solution for the Temperley-Lieb spin chain

We consider the Temperley-Lieb (TL) open quantum spin chain with "free" boundary conditions associated with the spin-$s$ representation of quantum-deformed $sl(2)$. We construct the transfer matrix, and determine its eigenvalues and the corresponding Bethe equations using analytical Bethe ansatz. We show that the transfer matrix has quantum group symmetry, and we propose explicit formulas for the number of solutions of the Bethe equations and the degeneracies of the transfer-matrix eigenvalues. We propose an algebraic Bethe ansatz construction of the off-shell Bethe states, and we conjecture that the on-shell Bethe states are highest-weight states of the quantum group. We also propose a determinant formula for the scalar product between an off-shell Bethe state and its on-shell dual, as well as for the square of the norm. We find that all of these results, except for the degeneracies and a constant factor in the scalar product, are universal in the sense that they do not depend on the value of the spin. In an appendix, we briefly consider the closed TL spin chain with periodic boundary conditions, and show how a previously-proposed solution can be improved so as to obtain the complete (albeit non-universal) spectrum.

math-ph↗