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arXiv:1610.01092 · Physical Review A

Detecting dimensional crossover and finite Hilbert space through entanglement entropies

Mariano Garagiola, Eloisa Cuestas, Federico M. Pont, Pablo Serra, Omar Osenda

Published Oct 19, 2016• 8 citations
arXivHTMLPDF

Abstract

The information content of the two-particle one- and two-dimensional Calogero model is studied by using the von Neumann and R\'enyi entropies. The one-dimensional model is shown to have nonmonotonic entropies with finite values in the large-interaction-strength limit. On the other hand, the von Neumann entropy of the two-dimensional model with isotropic confinement is a monotone increasing function of the interaction strength which diverges logarithmically. By considering an anisotropic confinement in the two-dimensional case we show that the one-dimensional behavior is eventually reached when the anisotropy increases. The crossover from two to one dimensions is demonstrated by using the harmonic approximation and it is shown that the von Neumann divergence only occurs in the isotropic case. The R\'enyi entropies are used to highlight the structure of the model spectrum. In particular, it is shown that these entropies have a nonmonotonic and nonanalytical behavior in the neighborhood of the interaction strength parameter values where the Hilbert space and, consequently, the spectrum of the reduced density matrix are both finite.

DOI
10.1103/physreva.94.042115

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The information content of the two-particle one- and two-dimensional Calogero model is studied by using the von Neumann and R'enyi entropies. The one-dimensional model is shown to have nonmonotonic entropies with finite values in the large-interaction-strength limit. On the other hand, the von Neumann entropy of the two-dimensional model with isotropic confinement is a monotone increasing function of the interaction strength which diverges logarithmically. By considering an anisotropic confinement in the two-dimensional case we show that the one-dimensional behavior is eventually reached when the anisotropy increases. The crossover from two to one dimensions is demonstrated by using the harmonic approximation and it is shown that the von Neumann divergence only occurs in the isotropic case. The R'enyi entropies are used to highlight the structure of the model spectrum. In particular, it is shown that these entropies have a nonmonotonic and nonanalytical behavior in the neighborhood of the interaction strength parameter values where the Hilbert space and, consequently, the spectrum of the reduced density matrix are both finite.
Abstract The Calogero model can be regularised through the introduction of a cutoff parameter which removes the divergence in the interaction term. In this work we show that the one-dimensional two-particle regularised Calogero model is quasi-exactly solvable and that for certain values of the Hamiltonian parameters the eigenfunctions can be written in terms of Heun’s confluent polynomials. These eigenfunctions are such that the reduced density matrix of the two-particle density operator can be obtained exactly as well as its entanglement spectrum. We found that the number of non-zero eigenvalues of the reduced density matrix is finite in these cases. The limits for the cutoff distance going to zero (Calogero) and infinity are analysed and all the previously obtained results for the Calogero model are reproduced. Once the exact eigenfunctions are obtained, the exact von Neumann and Rényi entanglement entropies are studied to characterise the physical traits of the model. The quasi-exactly solvable character of the model is assessed studying the numerically calculated Rényi entropy and entanglement spectrum for the whole parameter space.
The information content of continuous quantum variables systems is usually studied using a number of well known approximation methods. The approximations are made to obtain the spectrum, eigenfunctions or the reduced density matrices that are essential to calculate the entropy-like quantities that quantify the information. Even in the sparse cases where the spectrum and eigenfunctions are exactly known the entanglement spectrum, {\em i.e.} the spectrum of the reduced density matrices that characterize the problem, must be obtained in an approximate fashion. In this work, we obtain analytically a finite representation of the reduced density matrices of the fundamental state of the N-particle Calogero model for a discrete set of values of the interaction parameter. As a consequence, the exact entanglement spectrum and von Neumann entropy is worked out.
We study three different measures of quantum correlations---entanglement spectrum, entanglement entropy, and logarithmic negativity---for a (1+11+11+1)-dimensional massive scalar field in a flat space-time. The entanglement spectrum for the discretized scalar field in the ground state indicates a crossover in the zero-mode regime, which is further substantiated by an analytical treatment of both entanglement entropy and logarithmic negativity. The exact nature of this crossover depends on the boundary conditions used---the leading order term switches from a log to log-log behavior for the periodic and Neumann boundary conditions. In contrast, for Dirichlet, it is the parameters within the leading log-log term that are switched. We show that this crossover manifests as a change in the behavior of the leading order divergent term for entanglement entropy and logarithmic negativity close to the zero-mode limit. We thus show that the two regimes have fundamentally different information content. Furthermore, an analysis of the ground-state fidelity shows us that the region between critical point $\mathrm{\ensuremath{\Lambda}}=0$ and the crossover point is dominated by zero-mode effects, featuring an explicit dependence on the IR cutoff of the system. For the reduced state of a single oscillator, we show that this crossover occurs in the region $Na{m}_{f}\ensuremath{\sim}\ensuremath{\mathcal{O}}(1)$.
We investigate boundary critical phenomena from a quantum-information perspective. Bipartite entanglement in the ground state of one-dimensional quantum systems is quantified using the R\'enyi entropy ${S}_{\ensuremath{\alpha}}$, which includes the von Neumann entropy $(\ensuremath{\alpha}\ensuremath{\rightarrow}1)$ and the single-copy entanglement $(\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\infty})$ as special cases. We identify the contribution of the boundaries to the R\'enyi entropy, and show that there is an entanglement loss along boundary renormalization group (RG) flows. This property, which is intimately related to the Affleck-Ludwig ggg theorem, is a consequence of majorization relations between the spectra of the reduced density matrix along the boundary RG flows. We also point out that the bulk contribution to the single-copy entanglement is half of that to the von Neumann entropy, whereas the boundary contribution is the same.
The constituents of a quantum many-body system can be inextricably linked, a phenomenon known as quantum entanglement. Entanglement can be used as a resource for quantum computing, quantum communication and detecting phase transitions, among others. The amount of entanglement can be quantified via the von Neumann and Rényi entropies, which have their origins in information theory. In this work, the quantum entanglement between subsystems of a one dimen- sional lattice model of fermions is quantified. The von Neumann and Rényi entropies were calculated for two types of subsystems. In the first study, the subsystems were treated as two subsets of particles, and in the second, as two spatial subregions. Finally, by considering particle superselection rules, the amount of entanglement that can actually be accessed as a resource was calcu- lated. In all cases, the quantum entanglement served to detect phase transitions in the model.
The interplay of quantum and classical fluctuations in the vicinity of a quantum critical point (QCP) gives rise to various regimes or phases with distinct quantum character. In this work, we show that the R\'enyi entropy is a precious tool to characterize the phase diagram of critical systems not only around the QCP but also away from it, thanks to its capability to detect the emergence of local moments at finite temperature. For an efficient evaluation of the R\'enyi entropy, we introduce an algorithm based on a path-integral Langevin dynamics combined with a previously proposed thermodynamic integration method built on regularized paths. We apply this framework to study the critical behavior of a linear chain of anharmonic oscillators, a particular realization of the ${\ensuremath{\phi}}^{4}$ model. We fully resolved its phase diagram, as a function of both temperature and interaction strength. At finite temperature, we find a sequence of three regimes---para, disordered, and antiferro---met as the interaction is increased. The R\'enyi entropy divergence coincides with the crossover between the para and disordered regimes, which shows no temperature dependence. The occurrence of the antiferro regime, on the other hand, is temperature dependent. The two crossover lines merge in proximity of the QCP, at zero temperature, where the R\'enyi entropy is sharply peaked. Via its subsystem-size scaling, we confirm that the transition belongs to the two-dimensional Ising universality class. This phenomenology is expected to happen in all ${\ensuremath{\phi}}^{4}$-like systems, as well as in the elusive water-ice transition across phases VII, VIII, and X.