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Physical review. B, Condensed matter

Real-space renormalization-group study of hard-core dirty bosons

Lizeng Zhang, Michael Ma

Published Mar 1, 1992• 44 citations
Publisher

Abstract

We investigate the critical phenomena of hard-core bosons in a disordered medium. Such a system is mapped onto a quantum spin-1/2 XY model with transverse random field. The system then is studied through a quantum real-space renormalization-group method. We find that randomness is always relevant in a one-dimensional (1D) system, in agreement with exact results. In two and three dimensions, there is a critical amount of disorder, below which the superfluid phase is stable. In 2D, the dynamic exponent z=1.7 for compressible states, and is close to the value of z=d as predicted by Fisher et al. $z--- is smaller for incompressible states. The correlation length exponent \ensuremath{\nu} is insensitive to z, and roughly equals 1.4. Unlike the superfluid--Mott-insulator transition without disorder, which has two distinct universality classes, we find there is only one universality class for the superfluid--Bose-glass transition.

DOI
10.1103/physrevb.45.4855

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We investigate the critical phenomena of hard-core bosons in a disordered medium. Such a system is mapped onto a quantum spin-1/2 XY model with transverse random field. The system then is studied through a quantum real-space renormalization-group method. We find that randomness is always relevant in a one-dimensional (1D) system, in agreement with exact results. In two and three dimensions, there is a critical amount of disorder, below which the superfluid phase is stable. In 2D, the dynamic exponent z=1.7 for compressible states, and is close to the value of z=d as predicted by Fisher et al. $z--- is smaller for incompressible states. The correlation length exponent \ensuremath{\nu} is insensitive to z, and roughly equals 1.4. Unlike the superfluid--Mott-insulator transition without disorder, which has two distinct universality classes, we find there is only one universality class for the superfluid--Bose-glass transition.
The critical phenomenon of the zero-temperature superfluid--Bose-glass phase transition for hard-core bosons on a three-dimensional disordered lattice is studied using a quantum real-space renormalization-group method. The correlation-length exponent \ensuremath{\nu} and the dynamic exponent z are computed. The critical exponent z is found to be 2.5 for compressible states and 1.3 for incompressible states. The exponent \ensuremath{\nu} is shown to be insensitive to z as that in the two-dimensional case, and has a value roughly equal to 1.
We use a large-N renormalization-group (RG) method to study a model of interacting boson system with a quenched random potential. In the absence of impurities, the pure boson system has a critical point that describes the superfluid--Mott-insulator (SF--MI) transition. The SF--MI transition of d-dimensional bosons belongs to the (d+1)-dimensional XY model universality class. In this paper, we study the dirty-boson critical points in the neighborhood of this pure SF--MI critical point. In general, the on-site random potential in the original lattice model gives two types of randomness in the effective-field-theoretic action. One is the randomness in the effective on-site repulsion w(x) and the other is the randomness of the chemical potential u(x). It turns out that d=2 is the critical dimension for both types of disorder but the roles of these two types of disorder are reversed as d=2 is crossed. Applying \ensuremath{\epsilon}=d-2 expansion, we found coupled RG equations for both kinds of randomness which reveal several nontrivial critical points. All the weak random fixed points we found have three or more relevant directions. We conclude that the direct SF--MI transition is unlikely to occur near two dimensions.
The ground-state of hard-core bosons in a random one-body potential is studied numerically to observe the transition from superfluid to insulating ``Bose glass'' phase. Finite-size scaling analyses are performed to estimate the transition point and critical exponents. The dynamical critical exponent is z=1.95\ifmmode\pm\else\textpm\fi{}0.25 and is consistent with the prediction of Fisher et al. [Phys. Rev. B 40, 546 (1989)]. Values for other exponents and preliminary results regarding the universal conductivity are presented.
In this dissertation, we study the superfluid-insulator quantum phase transition in disordered boson systems. Recently, there has been considerable controversy over the validity of the scaling relations of the superfluid--Bose-glass quantum phase transition in three dimensions. Results from experimental and numerical studies on disordered quantum magnets contradict the scaling relations and the associated conventional scaling hypothesis for the singular part of the free energy. We determine various critical exponents of the superfluid--Bose-glass quantum phase transition in three-dimensional disordered Bose-Hubbard model through extensive Monte Carlo simulations. Our numerical study shows the previous studies on disordered quantum magnets were performed outside the quantum critical region, and our results for the critical exponents are in perfect agreement with previous theoretical predictions. We next move to study theoretically and numerically the superfluid-insulator quantum phase transition in one-dimensional disordered boson systems. While the superfluid-insulator transition in the weak disorder limit is well understood through the perturbative renormalization group study by Giamarchi and Schulz, transitions in the strong disorder regime are beyond the reach of this method. This problem was recently attacked by Altman et al. with the real space renormalization group method. They reached the conclusion that the superfluid-insulator transition in the strong disorder regime can be explained by the Coulomb blockade physics of weak links. However, their method is actually uncontrolled. Taking account of the crucial role of the hydrodynamic renormalization of weak links, finally, Pollet et al. put forward an asymptotically exact renormalization group theory of the superfluid-insulator transition. Based on this theory, we are able to provide an accurate description of the interplay between the well-known Giamarchi-Schulz criticality and the new weak-link criticality. A significant part of the ground-state phase diagram of one-dimensional disordered Bose-Hubbard model at unit filling is also determined numerically. In particular, we established the position of the multicritical point beyond which the new weak-link criticality takes over of the Giamarchi–Schulz criticality.
We determine the dynamical critical exponent z appearing at the Bose glass to superfluid transition in two dimensions by performing large scale numerical studies of two microscopically different quantum models within the universality class: The hard-core boson model and the quantum rotor (soft core) model, both subject to strong on-site disorder. By performing many simulations at different system size L and inverse temperature β close to the quantum critical point, the position of the critical point and the critical exponents, z, ν, and η can be determined independently of any implicit assumptions of the numerical value of z, in contrast to most prior studies. This is done by a careful scaling analysis close to the critical point with a particular focus on the temperature dependence of the scaling functions. For the hard-core boson model we find z=1.88(8), ν=0.99(3), and η=-0.16(8) with a critical field of h(c)=4.79(3), while for the quantum rotor model we find z=1.99(5), ν=1.00(2), and η=-0.3(1) with a critical hopping parameter of t(c)=0.0760(5). In both cases do we find a correlation length exponent consistent with ν=1, saturating the bound ν≥2/d as well as a value of z significantly larger than previous studies, and for the quantum rotor model consistent with z=d.
I argue that the system of interacting bosons at zero temperature and in random external potential possesses a simple critical point that describes the proliferation of disorder-induced defects in the superfluid ground state, and that is located at weak disorder close to and above one dimension. This makes it possible to address the critical behavior at the superfluid-Bose glass transition in dirty boson systems by expanding around the lower critical dimension of d=1.d=1.d=1. Within the formulated renormalization procedure near d=1,d=1,d=1, the dynamical critical exponent is obtained exactly and the correlation length critical exponent is calculated as a Laurent series in the parameter $\sqrt{\ensuremath{\epsilon}},$ with $\ensuremath{\epsilon}=d\ensuremath{-}1:$ z=d,z=d,z=d, $\ensuremath{\nu}=1/\sqrt{3\ensuremath{\epsilon}}+O(\sqrt{\ensuremath{\epsilon}})$ for the short-range, and z=1,z=1,z=1, $\ensuremath{\nu}=\sqrt{2/3\ensuremath{\epsilon}}+O(\sqrt{\ensuremath{\epsilon}}),$ for the long-range Coulomb interaction between bosons. The identified critical point should be stable against the residual perturbations in the effective action for the superfluid, at least in dimensions 1< d< 2,1<~d<~2,1< d< 2, for both short-range and Coulomb interactions. For the superfluid-Mott insulator transition in the system in a periodic potential and at a commensurate density of bosons, I find $\ensuremath{\nu}=(\frac{1}{2}\sqrt{\ensuremath{\epsilon}})+\frac{1}{4}+O(\sqrt{\ensuremath{\epsilon}}),$ which yields a result in reasonable agreement with the known XY\mathrm{XY}XY critical exponent in d=2.d=2.d=2. The critical behavior of the superfluid density, phonon velocity, and the compressibility in the system with short-range interactions is discussed.