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Physical Review E

Bose-Einstein condensation in diamond hierarchical lattices

M. L. Lyra, F. A. B. F. de Moura, I. N. de Oliveira, Maurizio Serva

Published May 20, 2014• 18 citations
Publisher

Abstract

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).

DOI
10.1103/physreve.89.052133

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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).
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 demonstrate that a topology-induced Bose-Einstein condensation (BEC) takes place in a complex network. As a model topology, we consider the deterministic Apollonian network which exhibits scale-free, small-world, and hierarchical properties. Within a tight-binding approach for noninteracting bosons, we report that the BEC transition temperature and the gap between the ground and first excited states follow the same finite-size scaling law. An anomalous density dependence of the transition temperature is reported and linked to the structure of gaps and degeneracies of the energy spectrum. The specific heat is shown to be discontinuous at the transition, with low-temperature modulations as a consequence of the fragmented density of states.
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.
Modern quantum engineering techniques allow for synthesizing quantum systems in exotic lattice geometries, from self-similar fractal networks to negatively curved hyperbolic graphs. We demonstrate that these structures profoundly reshape Bose-Einstein condensation. Fractal lattices dramatically lower the condensation temperature and enhance condensation fluctuations. In a Sierpiński carpet, quasi-degeneracies in the tight-binding spectrum fragment the condensate. Hyperbolic lattices, on the other hand, exhibit condensation features similar to regular three-dimensional lattices, despite their embedding in only two dimensions: The critical temperature increases as the system grows, and the temperature-dependence of the condensate fraction follows the same power-law as for cubic lattices. We explain these similarities through the similarity of the densities of state at low energies. When strong repulsive interactions are included, the gas enters a Mott insulating state. Using a multi-site Gutzwiller approach as well as a simple strong-coupling expansion, for the Sierpiński triangle we find a smooth interpolation between the characteristic insulating lobes of one-dimensional and two-dimensional systems. Our findings establish lattice geometry as a powerful tuning knob for quantum phase phenomena and pave the way for experimental exploration in photonic waveguide arrays and Rydberg-atom tweezer arrays.
A perfect Bose gas can condensate in one dimension in the presence of a random potential due to the presence of Lifshitz tails in the one-particle density of states. Here, we show that scale-free correlations in the random potential suppress the disorder induced Bose-Einstein condensation (BEC). Within a tight-binding approach, we consider free Bosons moving in a scale-free correlated random potential with spectral density decaying as $1/{k}^{\ensuremath{\alpha}}$. The critical temperature for BEC is shown to vanish in chains with a binary nonstationary potential $(\ensuremath{\alpha}>1)$. On the other hand, a weaker suppression of BEC takes place in nonbinarized scale-free potentials. After a slightly increase in the stationary regime, the BEC transition temperature continuously decays as the spectral exponent $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\infty}$.
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.