We address computational issues relevant to the study of disordered quantum mechanical systems at very low temperatures. As an example we consider the disordered Bose- Hubbard model in three dimensions directly at the Bose-glass to superfluid phase transition. The universal aspects of the critical behaviour are captured by a (3 + 1) dimensional link-current model for which an efficient 'worm' algorithm is known. We present a calculation of the distribution of the superfluid stiffness over the disorder realizations, outline a number of important considerations for performing such estimates, and suggest a modification of the link-current Hamiltonian that improves the numerical efficiency of the averaging procedure without changing the universal properties of the model.We address computational issues relevant to the study of disordered quantum mechanical systems at very low temperatures. As an example we consider the disordered Bose- Hubbard model in three dimensions directly at the Bose-glass to superfluid phase transition. The universal aspects of the critical behaviour are captured by a (3 + 1) dimensional link-current model for which an efficient 'worm' algorithm is known. We present a calculation of the distribution of the superfluid stiffness over the disorder realizations, outline a number of important considerations for performing such estimates, and suggest a modification of the link-current Hamiltonian that improves the numerical efficiency of the averaging procedure without changing the universal properties of the model.

arXiv:cond-mat/0601180Jan 10, 20060 citations

Bose-glass to superfluid transition in the three-dimensional Bose-Hubbard model

Peter B. Hitchcock, Erik Schwartz Sørensen

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 $(3+1)$ 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 $z=3$ , in agreement with existing scaling and renormalization group arguments that $z=d$ . 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 $z$ . 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 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 $(3+1)$ 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 $z=3$, in agreement with existing scaling and renormalization group arguments that $z=d$. 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 $z$. 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.

arXiv:cond-mat/0601182Physical Review BMay 24, 200636 citations

QUANTUM PHASE TRANSITIONS IN DISORDERED BOSON SYSTEMS

Zhiyuan Yao

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.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.

Apr 12, 20210 citations

Superfluid Phase Transitions in Disordered Systems

Hannes Meier

This thesis presents results from large scale Monte Carlo simulations of systems subject to a superfluid phase transition in the presence of disorder. The simulations are performed by state-of-the-art, collective Monte Carlo algorithms treating phase degrees of freedom in effective models with amplitude fluctuations integrated out. In Paper I a model system for the possible solid to supersolid transition in 4He is presented.The Wolff cluster algorithm is used to study how the presence of linearly correlated random defects is able to alter the universality class of the 3-dimensional XY-model. In the pure case the superfluid density and heat capacity have singular onsets, which are not seen in the supersolid experiments where instead a smooth onset is obtained. Using finite size scaling of Monte Carlo data, we find a similar smooth onset in our simulations, governed by exponents ν=1 for the superfluid density and α=-1 for the heat capacity. These results are in qualitative agreement with experiments for the observed transition in solid 4He. In Paper II a systematic investigation of the scaling result z=d for the dynamic critical exponentat the Bose glass to superfluid quantum phase transition is performed. The result z=d has been believed to be exact for about 20 years, but although it has been questioned lately no accurate estimate of z has been available. An effective link current model of quantum bosons at T=0 with disorder in 2D is simulated using highly effective worm Monte Carlo simulations.The data analysis is based on a finite size scaling approach todetermine the quantum correlation time from simulationdata for boson world lines without any a priori assumption on the critical parameters. The resulting critical exponents are z=1.8 \pm 0.05, ν=1.15 \pm 0.03, and η=-0.3 \pm 0.1. This suggests that z=d is not satisfied.This thesis presents results from large scale Monte Carlo simulations of systems subject to a superfluid phase transition in the presence of disorder. The simulations are performed by state-of-the-art, collective Monte Carlo algorithms treating phase degrees of freedom in effective models with amplitude fluctuations integrated out. In Paper I a model system for the possible solid to supersolid transition in 4He is presented.The Wolff cluster algorithm is used to study how the presence of linearly correlated random defects is able to alter the universality class of the 3-dimensional XY-model. In the pure case the superfluid density and heat capacity have singular onsets, which are not seen in the supersolid experiments where instead a smooth onset is obtained. Using finite size scaling of Monte Carlo data, we find a similar smooth onset in our simulations, governed by exponents ν=1 for the superfluid density and α=-1 for the heat capacity. These results are in qualitative agreement with experiments for the observed transition in solid 4He. In Paper II a systematic investigation of the scaling result z=d for the dynamic critical exponentat the Bose glass to superfluid quantum phase transition is performed. The result z=d has been believed to be exact for about 20 years, but although it has been questioned lately no accurate estimate of z has been available. An effective link current model of quantum bosons at T=0 with disorder in 2D is simulated using highly effective worm Monte Carlo simulations.The data analysis is based on a finite size scaling approach todetermine the quantum correlation time from simulationdata for boson world lines without any a priori assumption on the critical parameters. The resulting critical exponents are z=1.8 \pm 0.05, ν=1.15 \pm 0.03, and η=-0.3 \pm 0.1. This suggests that z=d is not satisfied.

Jan 1, 20110 citations

Numerical evidence for strong randomness scaling at a superfluid-insulator transition of one-dimensional bosons

Susanne Pielawa, Ehud Altman

We present numerical evidence from Monte Carlo simulations that the superfluid-insulator quantum phase transition of interacting bosons subject to strong disorder in one dimension is controlled by the strong-randomness critical point. At this critical point the distribution of superfluid stiffness over disorder realizations develops a power-law tail reflecting a universal distribution of weak links. The Luttinger parameter on the other hand does not take on a universal value at this critical point, in marked contrast to the known Berezinskii-Kosterlitz-Thouless-like superfluid-insulator transition in weakly disordered systems. We develop a finite-size scaling procedure which allows us to directly compare the numerical results from systems of linear size up to 1024 sites with theoretical predictions obtained by Altman et al. [Phys. Rev. Lett. 93, 150402 (2004)] using a strong disorder renormalization group approach. The data shows good agreement with the scaling expected at the strong-randomness critical point.We present numerical evidence from Monte Carlo simulations that the superfluid-insulator quantum phase transition of interacting bosons subject to strong disorder in one dimension is controlled by the strong-randomness critical point. At this critical point the distribution of superfluid stiffness over disorder realizations develops a power-law tail reflecting a universal distribution of weak links. The Luttinger parameter on the other hand does not take on a universal value at this critical point, in marked contrast to the known Berezinskii-Kosterlitz-Thouless-like superfluid-insulator transition in weakly disordered systems. We develop a finite-size scaling procedure which allows us to directly compare the numerical results from systems of linear size up to 1024 sites with theoretical predictions obtained by Altman et al. [Phys. Rev. Lett. 93, 150402 (2004)] using a strong disorder renormalization group approach. The data shows good agreement with the scaling expected at the strong-randomness critical point.

arXiv:1308.1100Physical Review BDec 3, 201319 citations

Phase diagram of the disordered Bose-Hubbard model

Victor Gurarie, Lode Pollet, Nikolay V. Prokof'ev +2 more

We establish the phase diagram of the disordered three-dimensional Bose-Hubbard model at unity filling which has been controversial for many years. The theorem of inclusions, proven by Pollet et al. [Phys. Rev. Lett. 103, 140402 (2009)] states that the Bose-glass phase always intervenes between the Mott insulating and superfluid phases. Here, we note that assumptions on which the theorem is based exclude phase transitions between gapped (Mott insulator) and gapless phases (Bose glass). The apparent paradox is resolved through a unique mechanism: such transitions have to be of the Griffiths type when the vanishing of the gap at the critical point is due to a zero concentration of rare regions where extreme fluctuations of disorder mimic a regular gapless system. An exactly solvable random transverse field Ising model in one dimension is used to illustrate the point. A highly nontrivial overall shape of the phase diagram is revealed with the worm algorithm. The phase diagram features a long superfluid finger at strong disorder and on-site interaction. Moreover, bosonic superfluidity is extremely robust against disorder in a broad range of interaction parameters; it persists in random potentials nearly 50 (!) times larger than the particle half-bandwidth. Finally, we comment on the feasibility of obtaining this phase diagram in cold-atom experiments, which work with trapped systems at finite temperature.We establish the phase diagram of the disordered three-dimensional Bose-Hubbard model at unity filling which has been controversial for many years. The theorem of inclusions, proven by Pollet et al. [Phys. Rev. Lett. 103, 140402 (2009)] states that the Bose-glass phase always intervenes between the Mott insulating and superfluid phases. Here, we note that assumptions on which the theorem is based exclude phase transitions between gapped (Mott insulator) and gapless phases (Bose glass). The apparent paradox is resolved through a unique mechanism: such transitions have to be of the Griffiths type when the vanishing of the gap at the critical point is due to a zero concentration of rare regions where extreme fluctuations of disorder mimic a regular gapless system. An exactly solvable random transverse field Ising model in one dimension is used to illustrate the point. A highly nontrivial overall shape of the phase diagram is revealed with the worm algorithm. The phase diagram features a long superfluid finger at strong disorder and on-site interaction. Moreover, bosonic superfluidity is extremely robust against disorder in a broad range of interaction parameters; it persists in random potentials nearly 50 (!) times larger than the particle half-bandwidth. Finally, we comment on the feasibility of obtaining this phase diagram in cold-atom experiments, which work with trapped systems at finite temperature.

arXiv:0909.4593Physical Review BDec 17, 2009121 citations

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)

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=3

, in agreement with existing scaling and renormalization group arguments that

z=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

z

. 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.