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EXPLORING ENTANGLEMENT IN QUANTUM SYSTEMS WITH DEGENERATE GROUND STATE

V. S. Okatev, O. M. Sotnikov, В. В. Мазуренко

Published Jan 1, 2025• 0 citations
OpenAlex

Abstract

We develop an approach to characterize correlations in quantum systems with degenerate ground states. It includes generation of the random linear combinations of the eigenfunctions obtained with exact diagonalization and calculation of their entanglement entropy. For tests the Ising model was used.

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We develop an approach for characterizing nonlocal quantum correlations in spin systems with exactly or nearly degenerate ground states. Starting with linearly independent degenerate eigenfunctions calculated with exact diagonalization we generate a finite set of their random linear combinations with Haar measure, which guarantees that these combinations are uniformly distributed in the space spanned by the initial eigenstates. Estimating the von Neumann entropy of the random wave functions helps to reveal previously unknown features of the quantum correlations in the phases with degeneracy of the ground state. For instance, spin spiral phase of the quantum magnet with Dzyaloshinskii-Moriya interaction is characterized by the enhancement of the entanglement entropy, which can be qualitatively explained by the changes in behavior of two- and three-spin correlation functions. To establish the connection between our theoretical findings and real experiments we elaborate on the problem of estimating observables on the basis of the single-shot measurements of numerous degenerate eigenstates. By the example of solving a simple Ising model it is shown that digital quantum simulations performed with quantum computers can provide accurate description of the entanglement of the degenerate systems even in the presence of noise. We also discuss the ground-state properties of degenerate fermionic and bosonic models, which can be useful for theoretical analysis of the experiments with ultracold atoms.
We consider a sublattice-symmetric free-fermion model on a one-dimensional lattice with random hopping amplitudes decaying with the distance as $|{t}_{l}|\ensuremath{\sim}{l}^{\ensuremath{-}\ensuremath{\alpha}}$, and address the question how far an analog of the random-singlet state (RSS) conceived originally for describing the ground state of certain random spin chains is valid for this model. For this purpose, we study the effective central charge characterizing the logarithmic divergence of the entanglement entropy (EE) and the prefactor of the distribution of distances between localization centers on the two sublattices, which must fulfill a consistency relation for a RSS. For $\ensuremath{\alpha}>1$, we find by exact diagonalization an overall logarithmic divergence of the entanglement entropy with an effective central charge varying with $\ensuremath{\alpha}$. The large-$\ensuremath{\alpha}$ limit of the effective central charge is found to be different from that of the nearest-neighbor hopping model. The consistency relation of RSS is violated for $\ensuremath{\alpha}\ensuremath{\le}2$, while for $\ensuremath{\alpha}>2$ it is possibly valid, but this conclusion is hampered by a crossover induced by the short-range fixed point. The EE is also calculated by the strong-disorder renormalization group (SDRG) method numerically. Aside from the traditional scheme, we construct and apply a more efficient minimal SDRG scheme having a linear (nearest-neighbor) structure, which turns out to be an accurate approximation of the full SDRG scheme for not too small $\ensuremath{\alpha}$. The SDRG method is found to provide systematically lower effective central charges than exact diagonalization does, nevertheless, it becomes more and more accurate for increasing $\ensuremath{\alpha}$. Furthermore, as opposed to nearest-neighbor models, it indicates a weak dependence on the disorder distribution.
Motivated by the fact that random quantum states are generic, i. e., that they occur as eigenvectors or during time evolution of sufficiently complex quantum system, we are going to address three questions: (i) what is the entanglement content of random quantum states, (ii) how can we efficiently generate them, and (iii) how come that there is apparently no entanglement in macroscopic world. To explain the later we will use properties of random quantum states. To answer the first question we are going to calculate average values of all Schmidt coefficient and to address the second one we are going to explicitly calculate convergence times of various random protocols.