Symmetry resolved entanglement in gapped integrable systems: a corner  transfer matrix approach

We study the symmetry resolved entanglement entropies in gapped integrable lattice models. We use the corner transfer matrix to investigate two prototypical gapped systems with a U(1) symmetry: the complex harmonic chain and the XXZ spin-chain. While the former is a free bosonic system, the latter is genuinely interacting. We focus on a subsystem being half of an infinitely long chain. In both models, we obtain exact expressions for the charged moments and for the symmetry resolved entropies. While for the spin chain we found exact equipartition of entanglement (i.e. all the symmetry resolved entropies are the same), this is not the case for the harmonic system where equipartition is effectively recovered only in some limits. Exploiting the gaussianity of the harmonic chain, we also develop an exact correlation matrix approach to the symmetry resolved entanglement that allows us to test numerically our analytic results.

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Multifractal amplitude fluctuations in the transfer-matrix approach to the statistics of disordered lattice models

Journal of Physics A General Physics ◽

10.1088/0305-4470/20/5/011 ◽

1987 ◽

Vol 20 (5) ◽

pp. L319-L324 ◽

Cited By ~ 2

Author(s):

G Jug

Keyword(s):

Transfer Matrix ◽

Lattice Models ◽

Matrix Approach ◽

Disordered Lattice ◽

Amplitude Fluctuations ◽

Transfer Matrix Approach

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Complete spectrum of quantum integrable lattice models associated to Y(gl(n)) by separation of variables

SciPost Physics ◽

10.21468/scipostphys.6.6.071 ◽

2019 ◽

Vol 6 (6) ◽

Cited By ~ 6

Author(s):

Jean Michel Maillet ◽

Giuliano Niccoli

Keyword(s):

Transfer Matrix ◽

Separation Of Variables ◽

Lattice Models ◽

Complete Characterization ◽

Simple Spectrum ◽

Transfer Matrices ◽

Integrable Lattice ◽

Transfer Matrix Spectrum ◽

Integrable Lattice Models ◽

Matrix Spectrum

We apply our new approach of quantum Separation of Variables (SoV) to the complete characterization of the transfer matrix spectrum of quantum integrable lattice models associated to \bm{gl_n}𝐠𝐥𝐧-invariant \bm{R}𝐑-matrices in the fundamental representations. We consider lattices with \bm{N}𝐍-sites and general quasi-periodic boundary conditions associated to an arbitrary twist matrix \bm{K}𝐊 having simple spectrum (but not necessarily diagonalizable). In our approach the SoV basis is constructed in an universal manner starting from the direct use of the conserved charges of the models, e.g. from the commuting family of transfer matrices. Using the integrable structure of the models, incarnated in the hierarchy of transfer matrices fusion relations, we prove that our SoV basis indeed separates the spectrum of the corresponding transfer matrices. Moreover, the combined use of the fusion rules, of the known analytic properties of the transfer matrices and of the SoV basis allows us to obtain the complete characterization of the transfer matrix spectrum and to prove its simplicity. Any transfer matrix eigenvalue is completely characterized as a solution of a so-called quantum spectral curve equation that we obtain as a difference functional equation of order \bm{n}𝐧. Namely, any eigenvalue satisfies this equation and any solution of this equation having prescribed properties that we give leads to an eigenvalue. We construct the associated eigenvector, unique up to normalization, of the transfer matrices by computing its decomposition on the SoV basis that is of a factorized form written in terms of the powers of the corresponding eigenvalues. Finally, if the twist matrix \bm{K}𝐊 is diagonalizable with simple spectrum we prove that the transfer matrix is also diagonalizable with simple spectrum. In that case, we give a construction of the Baxter \bm{Q}𝐐-operator and show that it satisfies a \bm{T}𝐓-\bm{Q}𝐐 equation of order \bm{n}𝐧, the quantum spectral curve equation, involving the hierarchy of the fused transfer matrices.

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