Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9809244 · Physical review. B, Condensed matter

Renormalization group for the large- N strongly commensurate dirty boson model

M. B. Hastings

Published Oct 1, 1999• 6 citations
arXivHTMLPDF

Abstract

The large-$N$ $\ensuremath{\sigma}$ model, in $D<4$ space-time dimensions, with disorder a function of d space dimensions, is analyzed via a renormalization-group treatment. Critical exponents for average quantities are calculated, first to lowest order and then to all orders, in $\ensuremath{\epsilon}=D\ensuremath{-}2\ensuremath{-}d/2.$ In particular, it is found that $\ensuremath{\nu}d=2.$ When $D=d+1,$ this model is equivalent to a large-$N$ limit of the strongly commensurate dirty boson problem.

DOI
10.1103/physrevb.60.9755

Related papers

Bose glass in a large- N commensurate dirty boson model

M. B. Hastings

The large-$N$ commensurate dirty boson model, in both the weakly and strongly commensurate cases, is considered via a perturbative renormalization-group treatment. In the weakly commensurate case, there exists a fixed line under renormalization-group flow, with varying amounts of disorder along the line. Including $1/N$ corrections causes the system to flow to strong disorder, indicating that the model does not have a phase transition perturbatively connected to the Mott insulator-superfluid (MI-SF) transition. I discuss the qualitative effects of instantons on the low-energy density of excitations. In the strongly commensurate case, a fixed point found previously is considered and results are obtained for higher moments of the correlation functions. To lowest order, correlation functions have a log-normal distribution. Finally, I prove two interesting theorems for large-$N$ vector models with disorder, relevant to the problem of replica symmetry breaking and frustration in such systems.

arXiv:cond-mat/9811121•Physical review. B, Condensed matter•Jun 20, 2001•7 citations

Large-Nrenormalization-group study of the commensurate dirty-boson problem

Yong Baek Kim, Xiao-Gang Wen

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.

arXiv:cond-mat/9310022•Physical review. B, Condensed matter•Feb 1, 1994•20 citations

Large-Ntheory of strongly commensurate dirty bosons: absence of a transition in two dimensions

Matthew Case, Igor F. Herbut

The spherical limit of strongly commensurate dirty bosons is studied perturbatively at weak disorder and numerically at strong disorder in two dimensions (2D). We argue that disorder is not perfectly screened by interactions and consequently that the ground state in the effective Anderson localization problem always remains localized. As a result there is only a gapped Mott insulator phase in the theory. Comparisons with other studies and the parallel with disordered fermions in 2D are discussed. We conjecture that while for the physical cases N = 2 (XY) and N = 1 (Ising) the theory should have the ordered phase, it may not for N = 3 (Heisenberg).

arXiv:cond-mat/0105442•Journal of Physics A Mathematical and General•Sep 14, 2001•4 citations

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

Lizeng Zhang, Michael Ma

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.

Physical review. B, Condensed matter•Mar 1, 1992•44 citations

Reply to `Comment on ``Large-Ntheory of strongly commensurate dirty-bosons: absence of a transition in two dimensions'' '

Matthew Case, Igor F. Herbut

We show that the self-consistency requirement on random potentials causes the decrease of correlations with disorder in the large- N theory of commensurate dirty bosons, leaving the possibility that there is no superfluid transition intact.

Journal of Physics A Mathematical and General•Mar 4, 2002•0 citations

Comment on `Large-Ntheory of strongly commensurate dirty bosons: absence of a transition in two dimensions'

M. B. Hastings

We show that the strongly commensurate dirty boson system must have a phase transition in two dimensions, and argue that the low-lying states have a diverging localization length.

Journal of Physics A Mathematical and General•Mar 4, 2002•0 citations
The large-NNN \ensuremathσ\ensuremath{\sigma}\ensuremathσ model, in D<4D<4D<4 space-time dimensions, with disorder a function of d space dimensions, is analyzed via a renormalization-group treatment. Critical exponents for average quantities are calculated, first to lowest order and then to all orders, in \ensuremathϵ=D\ensuremath−2\ensuremath−d/2.\ensuremath{\epsilon}=D\ensuremath{-}2\ensuremath{-}d/2.\ensuremathϵ=D\ensuremath−2\ensuremath−d/2. In particular, it is found that \ensuremathνd=2.\ensuremath{\nu}d=2.\ensuremathνd=2. When D=d+1,D=d+1,D=d+1, this model is equivalent to a large-NNN limit of the strongly commensurate dirty boson problem.
The large-NNN commensurate dirty boson model, in both the weakly and strongly commensurate cases, is considered via a perturbative renormalization-group treatment. In the weakly commensurate case, there exists a fixed line under renormalization-group flow, with varying amounts of disorder along the line. Including 1/N1/N1/N corrections causes the system to flow to strong disorder, indicating that the model does not have a phase transition perturbatively connected to the Mott insulator-superfluid (MI-SF) transition. I discuss the qualitative effects of instantons on the low-energy density of excitations. In the strongly commensurate case, a fixed point found previously is considered and results are obtained for higher moments of the correlation functions. To lowest order, correlation functions have a log-normal distribution. Finally, I prove two interesting theorems for large-NNN vector models with disorder, relevant to the problem of replica symmetry breaking and frustration in such systems.
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 spherical limit of strongly commensurate dirty bosons is studied perturbatively at weak disorder and numerically at strong disorder in two dimensions (2D). We argue that disorder is not perfectly screened by interactions and consequently that the ground state in the effective Anderson localization problem always remains localized. As a result there is only a gapped Mott insulator phase in the theory. Comparisons with other studies and the parallel with disordered fermions in 2D are discussed. We conjecture that while for the physical cases N = 2 (XY) and N = 1 (Ising) the theory should have the ordered phase, it may not for N = 3 (Heisenberg).
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.