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arXiv:1402.5417 · Physical Review Letters

Critical Exponents of the Superfluid–Bose-Glass Transition in Three Dimensions

Zhiyuan Yao, Karine Piacentini Coelho da Costa, Mikhail N. Kiselev, Nikolay V. Prokof'ev

Published Jun 3, 2014• 44 citations
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Abstract

Recent experimental and numerical studies of the critical-temperature exponent ϕ for the superfluid-Bose-glass universality in three-dimensional systems report strong violations of the key quantum critical relation, ϕ=νz, where z and ν are the dynamic and correlation-length exponents, respectively; these studies question the conventional scaling laws for this quantum critical point. Using Monte Carlo simulations of the disordered Bose-Hubbard model, we demonstrate that previous work on the superfluid-to-normal-fluid transition-temperature dependence on the chemical potential (or the magnetic field, in spin systems), T_{c}∝(μ-μ_{c})^{ϕ}, was misinterpreting transient behavior on approach to the fluctuation region with the genuine critical law. When the model parameters are modified to have a broad quantum critical region, simulations of both quantum and classical models reveal that the ϕ=νz law [with ϕ=2.7(2), z=3, and ν=0.88(5)] holds true, resolving the ϕ-exponent "crisis."

DOI
10.1103/physrevlett.112.225301

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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.

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Recent experimental and numerical studies of the critical-temperature exponent ϕ for the superfluid-Bose-glass universality in three-dimensional systems report strong violations of the key quantum critical relation, ϕ=νz, where z and ν are the dynamic and correlation-length exponents, respectively; these studies question the conventional scaling laws for this quantum critical point. Using Monte Carlo simulations of the disordered Bose-Hubbard model, we demonstrate that previous work on the superfluid-to-normal-fluid transition-temperature dependence on the chemical potential (or the magnetic field, in spin systems), T_{c}∝(μ-μ_{c})^{ϕ}, was misinterpreting transient behavior on approach to the fluctuation region with the genuine critical law. When the model parameters are modified to have a broad quantum critical region, simulations of both quantum and classical models reveal that the ϕ=νz law [with ϕ=2.7(2), z=3, and ν=0.88(5)] holds true, resolving the ϕ-exponent "crisis."
We present a Monte Carlo study of the Bose glass to superfluid transition in the three-dimensional Bose-Hubbard model. Simulations are performed on the classical (3+1)(3+1)(3+1) dimensional link-current representation using the geometrical worm algorithm. Finite-size scaling analysis (on lattices as large as $16\ifmmode\times\else\texttimes\fi{}16\ifmmode\times\else\texttimes\fi{}16\ifmmode\times\else\texttimes\fi{}512$ sites) of the superfluid stiffness and the compressibility is consistent with a value of the dynamical critical exponent z=3z=3z=3, in agreement with existing scaling and renormalization group arguments that z=dz=dz=d. We find also a value of $\ensuremath{\nu}=0.70(12)$ for the correlation length exponent, satisfying the relation $\ensuremath{\nu}\ensuremath{\ge}2∕d$. However, a detailed study of the correlation functions, $C(r,\ensuremath{\tau})$, at the quantum critical point are not consistent with this value of zzz. We speculate that this discrepancy could be due to the fact that the correlation functions have not reached their true asymptotic behavior because of the relatively small spatial extent of the lattices used in the present study.
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.
In this comment we address the preprint Z. Yao et al. (arXiv:1402.5417v1) concerning the conditions upon which the correct critical temperature exponent ϕϕϕ can be observed (experimentally or numerically) close to the superfluid-Bose glass quantum phase transition in three dimensions. Yao et al. announce to have resolved the contradiction between the predictions of Fisher et al. [Phys. Rev. B 40, 546 (1989)] that ϕ=νz≥2ϕ= νz \geq 2ϕ=νz≥2 (where ννν and zzz are critical exponents at the quantum phase transition) and recent experiments and simulations, showing an exponent ϕ∼1.1ϕ\sim 1.1ϕ∼1.1. Yao et al's resolution hinges on claiming that the ϕ∼1.1ϕ\sim 1.1ϕ∼1.1 observations, which were conducted by varying the chemical potential, were not made to sufficiently close to the critical point to observe universal behavior. Here we critically examine their criteria for observing universal behavior. We show that past simulations were in fact conducted closer to the quantum critical point than the new results of Yao et al.; and that Yao et al.'s own results for varying chemical potential are consistent with previous results showing ϕ∼1.1ϕ\sim 1.1ϕ∼1.1, and do not show a clear crossover to a different ϕϕϕ value.
The superfluid to insulator quantum phase transition of a three-dimensional particle-hole symmetric system of disordered bosons is studied. To this end, a site-diluted quantum rotor Hamiltonian is mapped onto a classical (3+1)-dimensional XYXYXY model with columnar disorder and analyzed by means of large-scale Monte Carlo simulations. The superfluid-Mott insulator transition of the clean, undiluted system is in the four-dimensional XYXYXY universality class and shows mean-field critical behavior with logarithmic corrections. The clean correlation length exponent $\ensuremath{\nu}=1/2$ violates the Harris criterion, indicating that disorder must be a relevant perturbation. For nonzero dilutions below the lattice percolation threshold of pc=0.688392{p}_{c}=0.688\phantom{\rule{0.16em}{0ex}}392pc​=0.688392, our simulations yield conventional power-law critical behavior with dilution-independent critical exponents z=1.67(6)z=1.67(6)z=1.67(6), $\ensuremath{\nu}=0.90(5)$, $\ensuremath{\beta}/\ensuremath{\nu}=1.09(3)$, and $\ensuremath{\gamma}/\ensuremath{\nu}=2.50(3)$. The critical behavior of the transition across the lattice percolation threshold is controlled by the classical percolation exponents. Our results are discussed in the context of a classification of disordered quantum phase transitions, as well as experiments in superfluids, superconductors, and magnetic systems.
The nature of the superfluid-to-Bose-glass (SF-BG) quantum phase transition, occurring in systems of interacting bosons immersed in a disordered environment, remains elusive. One fundamental open question is whether or not the transition obeys conventional scaling at quantum critical points (QCPs): this scaling would lock the value of the crossover exponent $\ensuremath{\phi}$---dictating the vanishing of the superfluid critical temperature upon approaching the QCP---to the value of quantum critical exponents for the ground-state transition. Yet such a relation between exponents has been called into question by several numerical as well as experimental results on the SF-BG transition. Here we revisit this issue in the case of the S=1/2S=1/2S=1/2 Heisenberg antiferromagnet on a site-diluted cubic lattice, which lends itself to efficient quantum Monte Carlo simulations. Our results show that the model exhibits a percolation transition in zero applied field, with the correlation length exponent $\ensuremath{\nu}=0.87(8)$ and $\ensuremath{\phi}=1.1(1)$ consistent with 3d3d3d percolation. When applying a sufficiently strong magnetic field, the dilution-induced transition decouples from geometric percolation, and it becomes a SF-BG transition; nonetheless, the $\ensuremath{\nu}$ and $\ensuremath{\phi}$ exponents maintain values consistent with those of the percolation transition. These results contradict the conventional scaling, which predicts $\ensuremath{\phi}\ensuremath{\geqslant}2$; and they suggest a possible relationship between the SF-BG transition and percolation of phase coherence.
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.