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arXiv:1712.01844 · Physical Review A

Information measures for a local quantum phase transition: Lattice fermions in a one-dimensional harmonic trap

Yicheng Zhang, Lev Vidmar, Marcos Rigol

Published Feb 5, 2018• 17 citations
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Abstract

A local quantum phase transition is studied with quantum information measures for trapped spinless fermions in one-dimensional lattices. It is found that ground-state bipartite entanglement entropies, as well as the low-temperature mutual information, are excellent order parameters for describing the characteristic-density-driven local quantum phase transition.

DOI
10.1103/physreva.97.023605

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A local quantum phase transition is studied with quantum information measures for trapped spinless fermions in one-dimensional lattices. It is found that ground-state bipartite entanglement entropies, as well as the low-temperature mutual information, are excellent order parameters for describing the characteristic-density-driven local quantum phase transition.
We study ground-state quantum entanglement in the one-dimensional Bose-Hubbard model in the presence of a harmonic trap. We focus on two transitions that occur upon increasing the characteristic particle density: the formation of a Mott-insulating domain with site occupation one at the center of the trap (lower transition) and the emergence of a superfluid domain at the center of the Mott-insulating one (upper transition). These transitions generate discontinuities in derivatives of the total energy and have been characterized by local (nonextensive) order parameters, so we refer to them as local quantum phase transitions. We show that a second derivative of the total energy is continuous with a kink at the lower transition, and that it is discontinuous at the upper transition. We also show that bipartite entanglement entropies are order parameters for those local quantum phase transitions. We use the density-matrix renormalization group and show that the transition points extracted from entanglement measures agree with the predictions of the local density approximation in the thermodynamic limit. We discuss how to determine the transition points from results in small systems, such as the ones realized in recent optical lattice experiments that measured the second-order Renyi entanglement entropy.
We study ground-state quantum entanglement in the one-dimensional Bose-Hubbard model in the presence of a harmonic trap. We focus on two transitions that occur upon increasing the characteristic particle density: the formation of a Mott-insulating domain with site occupation one at the center of the trap (lower transition) and the emergence of a superfluid domain at the center of the Mott-insulating one (upper transition). These transitions generate discontinuities in derivatives of the total energy and have been characterized by local (nonextensive) order parameters, so we refer to them as local quantum phase transitions. We show that a second derivative of the total energy is continuous with a kink at the lower transition, and that it is discontinuous at the upper transition. We also show that bipartite entanglement entropies are order parameters for those local quantum phase transitions. We use the density-matrix renormalization group and show that the transition points extracted from entanglement measures agree with the predictions of the local density approximation in the thermodynamic limit. We discuss how to determine the transition points from results in small systems, such as the ones realized in recent optical lattice experiments that measured the second-order Renyi entanglement entropy.
The entanglement properties of phase transition in a two-dimensional harmonic lattice, similar to the one observed in recent ion trap experiments, are discussed for both finite number of particles and thermodynamical limit. We show that for the ground state at the critical value of the trapping potential, two entanglement measures, the negativity between two neighbouring sites and the block entropy for blocks of size 1, 2 and 3, change abruptly. Entanglement thus indicates quantum phase transitions in general, not only in the finite-dimensional case considered in Wu et al (2004 Phys. Rev. Lett. 93 250404). Finally, we consider the thermal state and compare its exact entanglement with a temperature entanglement witness introduced in Anders (2008 Phys. Rev. A 77 062102).
The entanglement properties of the phase transition in a two dimensional harmonic lattice, similar to the one observed in recent ion trap experiments, are discussed both, for finite number of particles and thermodynamical limit. We show that for the ground state at the critical value of the trapping potential two entanglement measures, the negativity between two neighbouring sites and the block entropy for blocks of size 1, 2 and 3, change abruptly. Entanglement thus indicates quantum phase transitions in general; not only in the finite dimensional case considered in [Phys. Rev. Lett. {\bf 93}, 250404 (2004)]. Finally, we consider the thermal state and compare its exact entanglement with a temperature entanglement witness introduced in [Phys. Rev. A {\bf 77} 062102 (2008)].
For indistinguishable itinerant particles subject to a superselection rule fixing their total number, a portion of the entanglement entropy under a spatial bipartition of the ground state is due to particle fluctuations between subsystems and thus is inaccessible as a resource for quantum information processing. We quantify the remaining operationally accessible entanglement in a model of interacting spinless fermions on a one-dimensional lattice via exact diagonalization and the density matrix renormalization group. We find that the accessible entanglement exactly vanishes at the first-order phase transition between a Tomonaga-Luttinger liquid and phase separated solid for attractive interactions and is maximal at the transition to the charge density wave for repulsive interactions. Throughout the phase diagram, we discuss the connection between the accessible entanglement entropy and the variance of the probability distribution describing intrasubregion particle-number fluctuations.
In recent years, the presence of local potentials has significantly enriched and diversified the entanglement patterns in monitored free fermion systems. In our approach, we employ the stochastic Schrödinger equation to simulate a one-dimensional spinless fermion system under continuous measurement and local potentials. By averaging the steady-state entanglement entropy over many quantum trajectories, we investigate its dependence on measurement and localization parameters. We used a phenomenological model to interpret the numerical results, and the results show that the introduction of local potentials does not destroy the universality class of the entanglement phase transition, and that the phase boundary is jointly characterized by the measurement process and the localization mechanism. This work offers a new perspective on the characterization of the entanglement phase boundary arising from the combined effects of measurement and localization, and provides criteria for detecting this novel phase transition in cold atom systems, trapped ions, and quantum dot arrays.