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Physical review. E

Bose-Einstein condensation in chains with power-law hoppings: Exact mapping on the critical behavior in d -dimensional regular lattices

Wandearley S. Dias, D. Bertrand, M. L. Lyra

Published Jun 5, 2017• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.95.062105

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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.
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.
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.
Through exact numerical solutions we show Bose-Einstein condensation (BEC) for the one-dimensional (1D) bosons with repulsive short-range interactions at zero temperature by taking a particular large size limit. Following the Penrose-Onsager criterion of BEC, we define condensate fraction by the fraction of the largest eigenvalue of the one-particle reduced density matrix. % We show the finite-size scaling behavior such that condensate fraction is given by a scaling function of one-variable: interaction parameter multiplied by a power of particle number. Condensate fraction is nonzero and constant for any large value of particle number or system size, if the interaction parameter is proportional to the negative power of particle number. %Here the interaction parameter is defined by the coupling constant of the delta-function potentials devided by the density. %With the scaling behavior we derive various themodynamic limits where condensate fraction is constant for any large system size; for instance, it is the case even in the system of a finite particle number.
Abstract Phase transitions and critical phenomena are among the most intriguing phenomena in nature and the renormalization-group theory for them is one of the greatest achievements of theoretical physics. However, the predictions of the theory above an upper critical dimension d c seriously disagree with reality. In addition to its fundamental significance, the problem is also of practical importance because both complex classical systems with long-range spatial or temporal interactions and quantum phase transitions with long-range interactions can substantially lower d c . The extant scenarios built on a dangerous irrelevant variable (DIV) to resolve the problem introduce two sets of critical exponents and even two sets of scaling laws whose origin is obscure. Here, we consider the DIV from a different perspective and clearly unveil the origin of the two sets of exponents and hence the intrinsic inconsistency in those scenarios. We then develop an effective-dimension theory in which critical fluctuations and system volume are fixed at an effective dimension by the DIV. This enables us to account for all the extant results consistently. A novel asymptotic finite-size scaling behavior for a correlation function together with a new anomalous dimension and its associated scaling law is also predicted. We also apply the theory to quantum phase transitions.
Ideal Bose-Einstein condensation (BEC) remains a paradigmatic example of a continuous phase transition and a cornerstone for understanding quantum degenerate bosonic matter. We demonstrate that critical behavior of the ideal Bose gas near the BEC phase transition falls into three distinct classes, determined exclusively by the low-energy scaling of the density of states. Depending on its scaling exponent, which is controlled by dimensionality and confinement, the transition displays either the usual algebraic divergences of thermodynamic susceptibilities, divergent behavior with marginal logarithmic corrections, or a more subtle form of criticality, where only the correlation length diverges. Our work provides a unified framework for criticality in noninteracting bosonic systems. This classification applies broadly to atomic, photonic, polaritonic, and magnonic condensates, where dimensionality, confinement, and spectral engineering can strongly reshape the density of states.