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Rafael I. Nepomechie

Publications and source records attributed to Rafael I. Nepomechie.

At least 55 records · Page 3Linked to original sources
arXiv2018

The spin-s homogeneous central spin model: exact spectrum and dynamics

We consider the problem of a central spin with arbitrary spin s that interacts pairwise and uniformly with a bath of N spins with s=1/2. We present two approaches for determining the exact spectrum of this model, one based on properties of SU(2), and the other based on integrability. We also analyze the exact time evolution of a spin coherent state, and compute the time evolution of various quantities of physical interest, including the entanglement entropy, spin polarization and Loschmidt echo.

quant-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↗
arXiv2018

Surveying the quantum group symmetries of integrable open spin chains

Using anisotropic R-matrices associated with affine Lie algebras $\hat g$ (specifically, $A_{2n}^{(2)}, A_{2n-1}^{(2)}, B_n^{(1)}, C_n^{(1)}, D_n^{(1)}$) and suitable corresponding K-matrices, we construct families of integrable open quantum spin chains of finite length, whose transfer matrices are invariant under the quantum group corresponding to removing one node from the Dynkin diagram of $\hat g$. We show that these transfer matrices also have a duality symmetry (for the cases $C_n^{(1)}$ and $D_n^{(1)}$) and additional $Z_2$ symmetries that map complex representations to their conjugates (for the cases $A_{2n-1}^{(2)}, B_n^{(1)}, D_n^{(1)}$). A key simplification is achieved by working in a certain "unitary" gauge, in which only the unbroken symmetry generators appear. The proofs of these symmetries rely on some new properties of the R-matrices. We use these symmetries to explain the degeneracies of the transfer matrices.

hep-th↗
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↗
arXiv2017

Quantum group symmetries and completeness for A_{2n}^(2) open spin chains

We argue that the Hamiltonians for A_{2n}^(2) open quantum spin chains corresponding to two choices of integrable boundary conditions have the symmetries U_q(B_n) and U_q(C_n), respectively. We find a formula for the Dynkin labels of the Bethe states (which determine the degeneracies of the corresponding eigenvalues) in terms of the numbers of Bethe roots of each type. With the help of this formula, we verify numerically (for a generic value of the anisotropy parameter) that the degeneracies and multiplicities of the spectra implied by the quantum group symmetries are completely described by the Bethe ansatz.

math-ph↗
arXiv2016

An inhomogeneous Lax representation for the Hirota equation

Motivated by recent work on quantum integrable models without U(1) symmetry, we show that the sl(2) Hirota equation admits a Lax representation with inhomogeneous terms. The compatibility of the auxiliary linear problem leads to a new consistent family of Hirota-like equations.

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↗
arXiv2016

Algebraic Bethe ansatz for the quantum group invariant open XXZ chain at roots of unity

For generic values of q, all the eigenvectors of the transfer matrix of the U_q sl(2)-invariant open spin-1/2 XXZ chain with finite length N can be constructed using the algebraic Bethe ansatz (ABA) formalism of Sklyanin. However, when q is a root of unity (q=exp(i pi/p) with integer p>1), the Bethe equations acquire continuous solutions, and the transfer matrix develops Jordan cells. Hence, there appear eigenvectors of two new types: eigenvectors corresponding to continuous solutions (exact complete p-strings), and generalized eigenvectors. We propose general ABA constructions for these two new types of eigenvectors. We present many explicit examples, and we construct complete sets of (generalized) eigenvectors for various values of p and N.

math-ph↗
arXiv2015

Wrapping corrections for non-diagonal boundaries in AdS/CFT

We consider an open string stretched between a Y=0 brane and a Y_theta=0 brane. The latter brane is rotated with respect to the former by an angle theta, and is described by a non-diagonal boundary S-matrix. This system interpolates smoothly between the Y-Y (theta =0) and the Y-bar Y (theta = pi/2) systems, which are described by diagonal boundary S-matrices. We use integrability to compute the energies of one-particle states at weak coupling up to leading wrapping order (4, 6 loops) as a function of the angle. The results for the diagonal cases exactly match with those obtained previously.

hep-th↗
arXiv2015

Bethe ansatz for an AdS/CFT open spin chain with non-diagonal boundaries

We consider the integrable open-chain transfer matrix corresponding to a Y=0 brane at one boundary, and a Y_theta=0 brane (rotated with the respect to the former by an angle theta) at the other boundary. We determine the exact eigenvalues of this transfer matrix in terms of solutions of a corresponding set of Bethe equations.

hep-th↗
arXiv2015

Fusion for AdS/CFT boundary S-matrices

We propose a fusion formula for AdS/CFT worldsheet boundary S-matrices. We show that, starting from the fundamental Y=0 boundary S-matrix, this formula correctly reproduces the two-particle bound-state boundary S-matrices.

hep-th↗
arXiv2015

Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity

We consider the sl(2)_q-invariant open spin-1/2 XXZ quantum spin chain of finite length N. For the case that q is a root of unity, we propose a formula for the number of admissible solutions of the Bethe ansatz equations in terms of dimensions of irreducible representations of the Temperley-Lieb algebra; and a formula for the degeneracies of the transfer matrix eigenvalues in terms of dimensions of tilting sl(2)_q-modules. These formulas include corrections that appear if two or more tilting modules are spectrum-degenerate. For the XX case (q=exp(i pi/2)), we give explicit formulas for the number of admissible solutions and degeneracies. We also consider the cases of generic q and the isotropic (q->1) limit. Numerical solutions of the Bethe equations up to N=8 are presented. Our results are consistent with the Bethe ansatz solution being complete.

math-ph↗
arXiv2014

Twisting singular solutions of Bethe's equations

The Bethe equations for the periodic XXX and XXZ spin chains admit singular solutions, for which the corresponding eigenvalues and eigenvectors are ill-defined. We use a twist regularization to derive conditions for such singular solutions to be physical, in which case they correspond to genuine eigenvalues and eigenvectors of the Hamiltonian.

math-ph↗
arXiv2014

The spectrum of tachyons in AdS/CFT

We analyze the spectrum of open strings stretched between a D-brane and an anti-D-brane in planar AdS/CFT using various tools. We focus on open strings ending on two giant gravitons with different orientation in $AdS_5 \times S^5$ and study the spectrum of string excitations using the following approaches: open spin-chain, boundary asymptotic Bethe ansatz and boundary thermodynamic Bethe ansatz (BTBA). We find agreement between a perturbative high order diagrammatic calculation in ${\cal N}=4$ SYM and the leading finite-size boundary Luscher correction. We study the ground state energy of the system at finite coupling by deriving and numerically solving a set of BTBA equations. While the numerics give reasonable results at small coupling, they break down at finite coupling when the total energy of the string gets close to zero, possibly indicating that the state turns tachyonic. The location of the breakdown is also predicted analytically.

hep-th↗
arXiv2014

Singular solutions, repeated roots and completeness for higher-spin chains

We investigate the completeness of the solutions of the Bethe equations for the integrable spin-s isotropic (XXX) spin chain with periodic boundary conditions. Solutions containing the exact string i s, i (s-1), ..., -i(s-1), -is are singular. For s>1/2, there exist also "strange" solutions with repeated roots, which nevertheless are physical (i.e., correspond to eigenstates of the Hamiltonian). We derive conditions for the singular solutions and the solutions with repeated roots to be physical. We formulate a conjecture for the number of solutions with pairwise distinct roots in terms of the numbers of singular and strange solutions. Using homotopy continuation, we solve the Bethe equations numerically for s=1 and s=3/2 up to 8 sites, and find some support for the conjecture. We also exhibit several examples of strange solutions.

math-ph↗