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arXiv:1403.1739 · Physical Review A

Destruction of long-range order by quenching of the hopping range in one dimension

Masaki Tezuka, Antonio M. Garcı́a-Garcı́a, Miguel A. Cazalilla

Published Nov 17, 2014• 9 citations
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Abstract

We study the dynamics in a one-dimensional hard-core Bose gas with power-law hopping after an abrupt reduction of the hopping range using the time-dependent density-matrix renormalization-group and bosonization techniques. In particular, we focus on the destruction of the Bose-Einstein condensate (BEC), which is present in the initial state in the thermodynamic limit. We argue that this type of quench is akin to a sudden reduction in the effective dimensionality $d$ of the system (from $d>1$ to $d=1)$. We identify two regimes in the evolution of the BEC fraction. For short times the decay of the BEC fraction is Gaussian while for intermediate to long times it is well described by a stretched exponential with an exponent that depends on the initial effective dimensionality of the system. These results are potentially relevant for cold trapped-ion experiments which can simulate an equivalent of hard-core bosons, i.e., spins, with tunable long-range interactions.

DOI
10.1103/physreva.90.053618

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We study the dynamics in a one-dimensional hard-core Bose gas with power-law hopping after an abrupt reduction of the hopping range using the time-dependent density-matrix renormalization-group and bosonization techniques. In particular, we focus on the destruction of the Bose-Einstein condensate (BEC), which is present in the initial state in the thermodynamic limit. We argue that this type of quench is akin to a sudden reduction in the effective dimensionality ddd of the system (from d>1d>1d>1 to d=1)d=1)d=1). We identify two regimes in the evolution of the BEC fraction. For short times the decay of the BEC fraction is Gaussian while for intermediate to long times it is well described by a stretched exponential with an exponent that depends on the initial effective dimensionality of the system. These results are potentially relevant for cold trapped-ion experiments which can simulate an equivalent of hard-core bosons, i.e., spins, with tunable long-range interactions.
We investigate the response of a one-dimensional Bose gas to a slow increase of its interaction strength. We focus on the rich dynamics of equal-time single-particle correlations treating the Lieb-Liniger model within a bosonization approach and the Bose-Hubbard model using the time-dependent density-matrix renormalization group method. For short distances, correlations follow a power law with distance with an exponent given by the adiabatic approximation. In contrast, for long distances, correlations decay algebraically with an exponent understood within the sudden quench approximation. This long distance regime is separated from an intermediate distance one by a generalized Lieb-Robinson criterion. At long times, in this intermediate regime, bosonization predicts that single-particle correlations decay following a stretched exponential, an unconventional behavior. We develop here an intuitive understanding for the propagation of correlations, in terms of a generalized light cone, applicable to a large variety of systems and quench forms.
We study the superfluid--Bose-glass transition in a one-dimensional lattice boson model with power-law decaying hopping amplitude ${t}_{i\ensuremath{-}j}\ensuremath{\sim}1/{|i\ensuremath{-}j|}^{\ensuremath{\alpha}}$, using bosonization and the nonperturbative functional renormalization group (FRG). When $\ensuremath{\alpha}$ is smaller than a critical value ${\ensuremath{\alpha}}_{c}<3$, the U(1) symmetry is spontaneously broken, which leads to a density mode with nonlinear dispersion and dynamical exponent $z=(\ensuremath{\alpha}\ensuremath{-}1)/2$; the superfluid phase is then stable for sufficiently weak disorder, contrary to the case of short-range hopping where disorder is a relevant perturbation when the Luttinger parameter is smaller than 3/23/23/2. In the presence of disorder, however, long-range hopping has no effect in the infrared limit and the FRG flow eventually becomes similar to that of a boson system with short-range hopping. This implies that the superfluid phase, when stable, exhibits a density mode with linear dispersion (z=1z=1z=1) and the superfluid--Bose-glass transition remains in the Berezinskii-Kosterlitz-Thouless universality class, while the Bose-glass fixed point is insensitive to long-range hopping. We compare our findings with a recent numerical study.
Recent experimental progress on the realization of quantum systems with highly controllable long-range interactions has impelled the study of quantum phase transitions in low-dimensional systems with power-law couplings. Long-range couplings mimic higher-dimensional effects in several physical contexts. Here, we provide the exact relation between the spectral dimension d at the band bottom and the exponent α that tunes the range of power-law hoppings of a one-dimensional ideal lattice Bose gas. We also develop a finite-size scaling analysis to obtain some relevant critical exponents and the critical temperature of the BEC transition. In particular, an irrelevant dangerous scaling field has to be taken into account when the hopping range is sufficiently large to make the effective dimensionality d>4.
We studied the non-equilibrium quench dynamics from free to hard-core 1D bosons in the presence of a hard-wall confining potential. The density profile and the two-point fermionic correlation function in the stationary state as well as their full time evolution was characterised. It was found that for long times the system relaxes to a uniform density profile, but the correlation function memorises the initial state with a stationary algebraic long-distance decay, which is opposite to the exponential behaviour found for the same quench in the periodic setup. We also compute the stationary bosonic two-point correlator which was found to decay exponentially for large distances. A two-step mechanism was shown to govern the time evolution; a quick approach to an almost stationary value was followed by a slow algebraic relaxation to the true stationary state.
Quantum fluctuations destroy phase coherence and thus long-range order in one-dimensional (1D) superconductivity as long as all interactions are short-ranged.Here we discuss the possibility of restoring phase coherence by power-law hopping.A 1D attractive-U Hubbard model with powerlaw hopping having the power-law exponent α is studied by Abelian bosonization and density-matrix renormalization group (DMRG).For 1/2 < α < 3/2, the system has an effective spatial dimensionality d eff > 1.We demonstrate that long-range superconducting order is restored for 1/2 < α < α c , with 5/4 < α c < 3/2.This conclusion is supported by a DMRG analysis of the model.
In this work, we investigate the critical properties of the Bose-Einstein condensation transition in a model of non-interacting bosons whose one-particle hamiltonian in the tight-binding approximation includes long-range couplings in a linear chain with periodic boundary conditions. The couplings along the chain are described by a kinetic hopping term which is responsible by the quantum mobility of the particles and whose amplitude decays with the distance between the sites as a power-law with a characteristic exponent a. We show that the density of states in the vicinity of the lower band edge of the energy spectrum strongly depends on a. In the re- gime of 1 < α < 2, the density of states vanishes as a power-law, similar to the behavior found for bosons in hypercubic lattices with dimension d = 2/(α -1). We determine the critical properties of the Bose-Einstein condensation transition that takes place in this regime using a hypothesis of a single characteristic length scale and a finite-size scaling analysis. In the regime of 3/2 < α < 2, the critical exponents are quite similar to those that characterize the transition in the corresponding hypercubic lattice. On the other hand, in the regime of 1 < α < 3/2, the exponents obtained from the single length scale hypothesis are quite distinct form those on the hypercubic lattice. This result indicates that the single length scale hypothesis can not correctly capture the critical exponents in the regime where the effective dimensionality of the model is above the upper critical dimension above which the transition has a mean-field critical behavior.