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arXiv:1005.3150 · Journal of Statistical Mechanics Theory and Experiment

Scaling of bipartite entanglement in one-dimensional lattice systems with a trapping potential

Massimo Campostrini B, Ettore Vicari

Published Aug 20, 2010• 35 citations
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Abstract

We study the effects of a power-law trapping potential on the scaling behaviour of the entanglement at the quantum critical point of one-dimensional (1D) lattice particle systems. We compute bipartite von Neumann and Rényi entropies in the presence of the trap and analyse their scaling behaviour with increasing trap size. As a theoretical laboratory, we consider the quantum XY chain in an external transverse field acting as a trap for the spinless fermions of its quadratic Hamiltonian representation. We then investigate confined-particle systems described by the 1D Bose–Hubbard model in the superfluid phase (around the centre of the trap). In both cases conformal field theory predicts logarithmically divergent bipartite entanglement entropies for the homogeneous systems without a trap. The presence of the trapping potential breaks conformal invariance, affecting the critical behaviour of the homogeneous system. Our results show that the bipartite entanglement entropies diverge logarithmically with increasing trap size and present notable scaling behaviours in the trap-size scaling limit.

DOI
10.1088/1742-5468/2010/08/p08020

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We study the quantum (zero-temperature) critical behaviors of confined particle systems described by the one-dimensional (1D) Bose-Hubbard model in the presence of a confining potential, at the Mott insulator to superfluid transitions, and within the gapless superfluid phase. Specifically, we consider the hard-core limit of the model, which allows us to study the effects of the confining potential by exact and very accurate numerical results. We analyze the quantum critical behaviors in the large trap-size limit within the framework of the trap-size scaling (TSS) theory, which introduces a new trap exponent $\ensuremath{\theta}$ to describe the dependence on the trap size. This study is relevant for experiments of confined quasi-1D cold atom systems in optical lattices. At the low-density Mott transition, TSS can be shown analytically within the spinless fermion representation of the hard-core limit. The trap-size dependence turns out to be more subtle in the other critical regions, when the corresponding homogeneous system has a nonzero filling fff, showing an infinite number of level crossings of the lowest states when increasing the trap size. At the n=1n=1n=1 Mott transition, this gives rise to a modulated TSS: The TSS is still controlled by the trap-size exponent $\ensuremath{\theta}$, but it gets modulated by periodic functions of the trap size. Modulations of the asymptotic power-law behavior is also found in the gapless superfluid region, with additional multiscaling behaviors.
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We investigate the critical properties of cold bosonic gases in three dimensions, confined by an external quadratic potential coupled to the particle density, and realistically described by the Bose-Hubbard (BH) model. The trapping potential is often included in experiments with cold atoms and modifies the critical finite-size scaling of the homogeneous system in a nontrivial way. The trap-size scaling (TSS) theory accounts for this effect through the exponent $\ensuremath{\theta}$. We perform extensive simulations of the BH model at the critical temperature, in the presence of harmonic traps. We find that the TSS predictions are universal once we account for the effective way in which the trap locally modifies the chemical potential $\ensuremath{\mu}$ of the system. The trap exponent for the BH model at $\ensuremath{\mu}=0$ is the one corresponding to an effective quartic potential. At positive $\ensuremath{\mu}$, evidence suggests that TSS breaks down sufficiently far from the center of the trap, as the system encounters an effective phase boundary.