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arXiv:2211.04645 · Physical review. B./Physical review. B

Quantum-critical scaling at the Bose-glass transition of the 3d diluted Heisenberg antiferromagnet in a field

Yuchen Fan, Rong Yu, Tommaso Roscilde

Published Jan 19, 2024• 0 citations
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Abstract

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 $S=1/2$ 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 $3d$ 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.

DOI
10.1103/physrevb.109.l020405

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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 \ensuremathϕ\ensuremath{\phi}\ensuremathϕ---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 \ensuremathν=0.87(8)\ensuremath{\nu}=0.87(8)\ensuremathν=0.87(8) and \ensuremathϕ=1.1(1)\ensuremath{\phi}=1.1(1)\ensuremathϕ=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 \ensuremathν\ensuremath{\nu}\ensuremathν and \ensuremathϕ\ensuremath{\phi}\ensuremathϕ exponents maintain values consistent with those of the percolation transition. These results contradict the conventional scaling, which predicts \ensuremathϕ\ensuremath⩾2\ensuremath{\phi}\ensuremath{\geqslant}2\ensuremathϕ\ensuremath⩾2; and they suggest a possible relationship between the SF-BG transition and percolation of phase coherence.
We discuss a magnetically ordered (``superfluid'') phase near quantum transition to the Bose-glass phase in a simple modeling system, a Heisenberg antiferromagnet with spatial dimension d>2d>2d>2 in an external magnetic field with disorder in exchange coupling constants. Our analytical consideration is based on hydrodynamic description of long-wavelength excitations. Results obtained are valid in the entire critical region near the quantum critical point (QCP), allowing us to describe a possible crossover from one critical behavior to another. We demonstrate that the system behaves in full agreement with predictions by M. P. Fisher et al. [Phys. Rev. B 40, 546 (1989)] in close vicinity to the QCP. We find as an extension to that analysis that the anomalous dimension $\ensuremath{\eta}=2\ensuremath{-}d$ and $\ensuremath{\beta}=\ensuremath{\nu}d/2$, where $\ensuremath{\beta}$ and $\ensuremath{\nu}$ are critical exponents of the order parameter and the correlation length, respectively. The density of states per spin of low-energy localized excitations is found to be independent of ddd (``superuniversal''). We show that many recent experimental and numerical results obtained in various three-dimensional (3D) systems can be described by our formulas using percolation critical exponents. Then, it is a possibility that a percolation critical regime arises in the ordered phase in some 3D systems not very close to the QCP.
We investigate the superfluid (SF) to Bose-glass (BG) quantum phase transition using extensive quantum Monte Carlo simulations of two-dimensional hard-core bosons in a random box potential. T=0 critical properties are studied by thorough finite-size scaling of condensate and SF densities, both vanishing at the same critical disorder Wc=4.80(5). Our results give the following estimates for the critical exponents: z=1.85(15), ν=1.20(12), η=-0.40(15). Furthermore, the probability distribution of the SF response P(lnρSF) displays striking differences across the transition: while it narrows with increasing system sizes L in the SF phase, it broadens in the BG regime, indicating an absence of self-averaging, and at the critical point P(lnρSF+zlnL) is scale invariant. Finally, high-precision measurements of the local density rule out a percolation picture for the SF-BG transition.
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.
We investigate the zero-temperature superfluid to insulator transitions in a diluted two-dimensional quantum rotor model with particle-hole symmetry. We map the Hamiltonian onto a classical (2+1)(2+1)(2+1)-dimensional XYXYXY model with columnar disorder which we analyze by means of large-scale Monte Carlo simulations. For dilutions below the lattice percolation threshold, the system undergoes a generic superfluid-Mott glass transition. In contrast to other quantum phase transitions in disordered systems, its critical behavior is of conventional power-law type with universal (dilution-independent) critical exponents $z=1.52(3), \ensuremath{\nu}=1.16(5), \ensuremath{\beta}/\ensuremath{\nu}=0.48(2), \ensuremath{\gamma}/\ensuremath{\nu}=2.52(4)$, and $\ensuremath{\eta}=\ensuremath{-}0.52(4)$. These values agree with and improve upon earlier Monte Carlo results [Phys. Rev. Lett. 92, 015703 (2004)] while (partially) excluding other findings in the literature. As a further test of universality, we also consider a soft-spin version of the classical Hamiltonian. In addition, we study the percolation quantum phase transition across the lattice percolation threshold; its critical behavior is governed by the lattice percolation exponents in agreement with recent theoretical predictions. We relate our results to a general classification of phase transitions in disordered systems, and we briefly discuss experiments.
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."
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.