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

###### Abstract

Recent experimental and numerical studies of the critical-temperature exponent for the superfluid-Bose glass universality in three-dimensional systems report strong violations of the key quantum critical relation, , where and are the dynamic and correlation length exponents, respectively, and 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 chemical potential (or magnetic field, in spin systems), , 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 law [with , , and ] holds true, resolving the -exponent “crisis”.

###### pacs:

67.85.Hj, 67.85.-d,64.70.TgDisordered Bose-Hubbard (DBH) model is frequently employed as a key prototype system
to discuss and understand a number of important experimental cases, such as He in porous media and on various substrates, thin superconducting films, cold atoms in disordered optical lattice potentials, and disordered magnets (see Yu1 ; Zhelud and references therein), *etc*.

The pioneering work Giamarchi ; Fisher on the DBH model has established that at an insulating Bose glass (BG) phase will emerge as a result of localization effects in disordered potentials. On a lattice, this phase will intervene between the Mott-insulator (MI) and superfluid (SF) phases at arbitrary weak disorder strength Fisher ; theorem and completely destroy the MI phase at strong disorder. In contrast with the gapped incompressible MI phase, the BG phase has finite compressibility, , due to finite density of localized gapless quasiparticle and quasihole excitations. Using scaling arguments, and the fact that at the critical point of the quantum SF-BG transition, it was predicted that the dynamic critical exponent, , always equals the dimension of space; i.e., Fisher . The decrease of the normal-to-superfluid transition temperature, , on approach to the quantum critical point (QCP) is characterized by the exponent: , where is the control parameter used to reach the QCP. Standard scaling analysis of the quantum-critical free-energy density predicts that has to satisfy the relation . Therefore, taking into account Harris criterion Harris for the correlation length exponent in disordered systems, it is expected that , within the standard picture of quantum critical phenomena.

Despite substantial research efforts in the last two decades, some aspects of the universal critical behavior described above remain controversial (see, e.g., Ref. Zheludev ). For instance, Ref. Weichman argues that finite at the SF-BG critical point might come from the regular analytic (rather than singular critical) part of the free energy, and, thus, should be considered as an undetermined critical exponent. Moreover, recent experiments on magnetic systems Yu1 , as well as quantum Monte Carlo simulations of related disordered antiferromagnets with single-ion anisotropy Yu2 , which use magnetic field (equivalent to the chemical potential in the bosonic system) as a control parameter to drive the system to quantum criticality, report compelling evidence that the values and are in strong violation of the key relation and the bound . As a result, finite-temperature scaling relations used to describe SF-BG criticality for decades, are challenged.

In this Letter, we address the -exponent “crisis” in the three-dimensional SF-BG universality class by performing accurate studies of quantum and classical models using Monte Carlo simulations based on Worm Algorithm Worm ; easyworm and established protocols of measuring critical points
using finite-size scaling (FSS) plots of mean-square winding number fluctuations (see, e.g., Ref. soyler ) averaged over disorder realizations (typically 5000-20000 realizations).
With regard to previous studies, we find that they were performed
away from the quantum critical region, and the genuine critical behavior was simply out of reach—the transition temperature drops below the detection limit before the data become suitable for extraction of . However, the low- problem is avoided when the SF-BG transition is approached by increasing disorder strength at constant particle density. In this regime,
simulations of the -dimensional classical *J*-current model
(in the same universality class) reveal that , ,
are fully consistent with the relation. This conclusion is further confirmed by
quantum Monte Carlo simulations of the hard-core DBH, putting an end to the controversy.

Consider the hard-core DBH on the simple cubic lattice (equivalent to the spin-1/2 -ferromagnet in magnetic field) with the Hamiltonian

(1) |

where is the bosonic annihilation operator, is the hopping amplitude, is the particle number operator with the hard-core constraint , stands for summation over the nearest-neighbor sites, and . Here is the chemical potential and is a bounded random potential with uniform distribution on the interval and un-correlated in space. The SF-BG transition is induced by fixing disorder strength at and decreasing the chemical potential, similarly to the protocol employed in Refs. Yu1 ; Yu2 ; Zheludev . Our data for are shown in Fig. 1. They feature an extended region in the parameter space where is decreasing by closely following the reported law. However, with highly accurate data for (our system sizes are at least an order of magnitude larger than in previous work) we observe that the last point is deviating from this power-law well outside of its error bar, see inset in Fig. 1, indicating that most of the points in Fig. 1 might not be in the critical regime yet. This observation is confirmed by revealing the dependence in Fig. 2. Since density remains finite at the QCP, one requirement of being in the quantum critical region is to have . This condition is clearly violated for most of the points used to establish the law in previous studies at low fields.

Since current problems with scaling relations are likely originating from strong dependence when is used as a control parameter (leading to the critical region with extremely small values), we radically change the strategy and study the SF-BG criticality as a function of disorder strength at constant density. Universal properties of QCPs in -dimensions can
be equally well studied using -dimensional classical mappings which are algorithmically superior from the numerical point of view. The simplest classical counterpart of the hard-core DBH in is the -dimensional *J*-current model Jcurrent

(2) |

with the and constraints.
Here index enumerates space-time directions , is the site index in the hyper-cubic
space-time lattice, is the chemical potential plus bounded random potential energy that depends on space coordinate only.
The random potential is uncorrelated in space and is uniformly
distributed on the interval. An integer valued current is
defined on lattice bonds and satisfies the divergence-free condition;
i.e., , where it is understood that . Graphically, the configuration space
is composed of *J*-current loops mimicking path-integral trajectories of bosonic particles.
In terms of the underlying bosonic system, and
represent the on-site occupation
numbers and hopping transitions, respectively, while .

Accurate determination of the critical exponent ultimately rests on precise location
of the QCP, or critical disorder strength , where the power law originates.
[Otherwise, one can be easily
mislead by the transient behavior (similarly to one shown in Fig. 1). Likewise,
all data points for the *J*-current model can be fit nearly perfectly with the power law
based on if is kept as a free parameter.]
To determine along with the correlation length exponent , we employ
FSS of scale-invariant mean-square winding number fluctuations,

(3) |

where is the winding number in direction. If small detuning from the QCP is characterized by , then the correlation lengths in space and time directions, and , diverge as , and is a universal function of length scale ratios

(4) |

In the last equality we assume that the ratio is fixed. By plotting for different system sizes, one determines the critical parameter from the crossing point of curves (if was guessed correctly). We argue that is an exact relation. Indeed, in the vicinity of QCP the compressibility can be formally decomposed into critical and regular (non-singular) parts with Fisher . One may speculate that finite is due to regular part, while the critical part vanishes at . However, this possibility is immediately ruled out by observation that finite in the BG phase is due to localized single-particle modes, while such modes do not exist in the superfluid phase. Thus, finite is entirely due to critical modes and (our FSS data are in perfect agreement with this conclusion, see Fig. 3).

Our simulations of model (2) were done with at half-integer filling factor, when . For FSS at the QCP we fix and consider only large system sizes from to sites (we hit the limit of what modern computer cluster can handle in reasonable time, given that every parameter point has to be averaged over disorder realizations). The crossing of -curves shown in Fig. 3 pinpoints the critical disorder strength to be at .

From Eq. (4), it follows that at the critical point

(5) |

enabling one to determine the correlation length exponent from the slopes of universal curves at the crossing point. The corresponding analysis is shown in Fig. 4 where is deduced from the log-log plot of derivatives. This result is in full agreement with previous findings Yu2 ; sorensen .

We now proceed to the evaluation of the critical-temperature exponent from accurate measurements of (using similar FSS analysis) and the power-law fit to the lowest transition temperatures, see Fig. 5. In striking contrast to Fig. 1 and previously reported results Yu1 ; Yu2 , all data points nicely follow the power-law curve as decreases nearly two orders in magnitude! If were left undetermined we would have to conclude that . However, if the power-law fit is performed with the known value of QCP (i.e., with ), the prediction is different: The exponent decreases from to as we reduce the number of the lowest-temperature points to be included in the fit from to . We thus claim our final result as , which is in good agreement with the prediction based on the quantum critical relation with and . [The order parameter exponent deduced from the constant-density approach, , also differs significantly from the value characteristic of the transient interval.]

To verify the universality of our findings and to shed light on what to expect if a similar study is attempted experimentally using magnetic or cold-atom systems, we performed quantum Monte Carlo simulation of model (1) at half-integer filling factor (i.e., at , or zero external magnetic field in the case of spin-1/2 -ferromagnet). Our data for normal-to-superfluid transition temperature as a function of disorder strength are shown in Fig. 6 ( was determined from FSS analysis of plots with ). Given that simulations of quantum models are more challenging numerically, we did not attempt to determine and averaged results over smaller number of disorder realizations, from at high temperature to at low temperature. The lowest transition temperatures can be perfectly fitted to the law with . This critical behavior starts at temperatures as high as and we were able to verify it down to , see Fig. 6 inset. There is no doubt that the condition is satisfied at the SF-BG transition.

In summary, we addressed the current -exponent “crisis” for the superfluid-to-Bose Glass
universality class in three dimensions. Previous work questioned conventional scaling relations
and with for the SF-BG quantum critical point.
Using extensive Monte Carlo simulations of the hard-core DBH and its
classical *J*-current counterpart we were able to identify problems with previous analysis
(strong dependence of density/magnetization on chemical potential/external magnetic field on approach to quantum criticality). We argued that is an exact relation, and used
it to determine the critical-temperature exponent from simulations of the
*J*-current model. Our final result is in good agreement with the
quantum critical prediction based on , putting the
controversy to an end. We verified universality of our findings and determined under
what conditions the exponent can be studied experimentally.

We thank Y. Deng for help with simulations. M. K. appreciates fruitful discussions with A. Zheludev. This work was supported by the National Science Foundation under the grant PHY-1314735, the MURI Program “New Quantum Phases of Matter” from AFOSR; the work of K. P. C. da C. was supported by FAPESP. We also thank ICTP (Trieste), the Aspen Center for Physics and the NSF Grant # 1066293 for hospitality during the crucial stages of this work.

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