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Bose-Einstein condensation in chains with long-reach couplings

Delvis Bertrand Jorge de Barros

Published Feb 20, 2014• 0 citations
OpenAlex

Abstract

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.

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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.
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.
The Bose-Einstein condensation of noninteracting particles restricted to move on the sites of hierarchical diamond lattices is investigated. Using a tight-binding single-particle Hamiltonian with properly rescaled hopping amplitudes, we are able to employ an orthogonal basis transformation to exactly map it on a set of decoupled linear chains with sizes and degeneracies written in terms of the network branching parameter q and generation number n. The integrated density of states is shown to have a fractal structure of gaps and degeneracies with a power-law decay at the band bottom. The spectral dimension d(s) coincides with the network topological dimension d(f) = ln(2q)/ln(2). We perform a finite-size scaling analysis of the fraction of condensed particles and specific heat to characterize the critical behavior of the BEC transition that occurs for q > 2 (d(s) > 2). The critical exponents are shown to follow those for lattices with a pure power-law spectral density, with non-mean-field values for q < 8 (d(s) < 4). The transition temperature is shown to grow monotonically with the branching parameter, obeying the relation 1/T(c) = a + b/(q - 2).
We investigate the zero-temperature phase diagram of the one-dimensional Bose-Hubbard model with power-law hopping decaying with distance as 1/r^{α} using exact large scale quantum Monte Carlo simulations. For all 1<α≤3 the quantum phase transition from a superfluid and a Mott insulator at unit filling is found to be continuous and scale invariant, in marked contrast with the Berezinskii-Kosterlitz-Thouless (BKT) scenario that is recovered only for α>3. By performing finite-size scaling collapses of the superfluid stiffness and extracting dynamical and correlation-length exponents from the low-energy spectrum, we establish that these transitions define a distinct universality class throughout the long-range regime 1<α≤3. Analysis of the single-particle correlation functions and grand canonical phase diagram further reveals a sequence of ordering regimes within the superfluid phase: true long-range order for α≤2, anomalous quasi-long-range order for 2<α≤3, and conventional algebraic decay for α>3. Our exact numerical results provide a benchmark to compare theories of long-range quantum models and are relevant for experiments with cold neutral atom, molecules and ion chains.
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We study an anisotropic variant of the ddd-dimensional imperfect Bose gas on a lattice, where the asymptotic behavior of the dispersion ϵk\epsilon_{\bf k}ϵk​ at vanishing momentum k\bf{k}k may differ from the standard quadratic form. We derive a general expression for the lower critical dimension dLd_LdL​ of the model and show that it may be pushed to arbitrarily high values by engineering the hopping parameters. The same applies to the upper critical dimension dUd_UdU​. For d>dLd>d_Ld>dL​ we find a general, exact expression for the Bose-Einstein condensation temperature Tc(μ)T_c(\mu)Tc​(μ) as a function of the chemical potential μ\muμ. The asymptotic behavior of Tc(μ)T_c(\mu)Tc​(μ) at Tc→0T_c\to 0Tc​→0 is governed by the shift exponent ψ\psiψ, whose value turns out to fully determine whether the system is below or above the upper critical dimension. The critical exponents at condensation as well as the exponent ψ\psiψ may attain non-standard values for specific choices of the lattice kinetic couplings. Finally, we study the mechanism leading to the affinity between the imperfect Bose gas and the classical spherical model of spins in the critical regime.