Physical Review Letters

Critical Behavior for Correlated Strongly Coupled Boson Systems in 1 + 1 Dimensions

N. M. Bogoliubov, R. Bullough, JUSSI TIMONEN

Published Jun 20, 1994• 46 citations

Abstract

The natural integrable correlated strongly coupled boson system in 1 + 1 dimensions is the $q$-boson hopping model; we calculate its critical exponent $\ensuremath{\theta}$ and determine its correlation functions. For small couplings the $q$-boson model has natural connections with the Bose gas and the $\mathrm{XY}$ models of very large spin for which $\ensuremath{\theta}'\mathrm{s}$ and correlators are reported. For large couplings the hopping model is a new phase of interacting bosons substantially different from the impenetrable Bose gas.

DOI: 10.1103/physrevlett.72.3933

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Physical Review Letters

Critical Behavior for Correlated Strongly Coupled Boson Systems in 1 + 1 Dimensions

N. M. Bogoliubov, R. Bullough, JUSSI TIMONEN

Published Jun 20, 1994• 46 citations

Publisher

Abstract

The natural integrable correlated strongly coupled boson system in 1 + 1 dimensions is the

q

-boson hopping model; we calculate its critical exponent

\ensuremath{\theta}

and determine its correlation functions. For small couplings the

q

-boson model has natural connections with the Bose gas and the

\mathrm{XY}

models of very large spin for which

\ensuremath{\theta}'\mathrm{s}

and correlators are reported. For large couplings the hopping model is a new phase of interacting bosons substantially different from the impenetrable Bose gas.

DOI: 10.1103/physrevlett.72.3933

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Critical behavior at superconductor-insulator phase transitions near one dimension

Igor F. Herbut

I argue that the system of interacting bosons at zero temperature and in random external potential possesses a simple critical point that describes the proliferation of disorder-induced defects in the superfluid ground state, and that is located at weak disorder close to and above one dimension. This makes it possible to address the critical behavior at the superfluid-Bose glass transition in dirty boson systems by expanding around the lower critical dimension of $d=1.$ Within the formulated renormalization procedure near $d=1,$ the dynamical critical exponent is obtained exactly and the correlation length critical exponent is calculated as a Laurent series in the parameter $\sqrt{\ensuremath{\epsilon}},$ with $\ensuremath{\epsilon}=d\ensuremath{-}1:$ $z=d,$ $\ensuremath{\nu}=1/\sqrt{3\ensuremath{\epsilon}}+O(\sqrt{\ensuremath{\epsilon}})$ for the short-range, and $z=1,$ $\ensuremath{\nu}=\sqrt{2/3\ensuremath{\epsilon}}+O(\sqrt{\ensuremath{\epsilon}}),$ for the long-range Coulomb interaction between bosons. The identified critical point should be stable against the residual perturbations in the effective action for the superfluid, at least in dimensions $1<~d<~2,$ for both short-range and Coulomb interactions. For the superfluid-Mott insulator transition in the system in a periodic potential and at a commensurate density of bosons, I find $\ensuremath{\nu}=(\frac{1}{2}\sqrt{\ensuremath{\epsilon}})+\frac{1}{4}+O(\sqrt{\ensuremath{\epsilon}}),$ which yields a result in reasonable agreement with the known $\mathrm{XY}$ critical exponent in $d=2.$ The critical behavior of the superfluid density, phonon velocity, and the compressibility in the system with short-range interactions is discussed.I argue that the system of interacting bosons at zero temperature and in random external potential possesses a simple critical point that describes the proliferation of disorder-induced defects in the superfluid ground state, and that is located at weak disorder close to and above one dimension. This makes it possible to address the critical behavior at the superfluid-Bose glass transition in dirty boson systems by expanding around the lower critical dimension of $d=1.$ Within the formulated renormalization procedure near $d=1,$ the dynamical critical exponent is obtained exactly and the correlation length critical exponent is calculated as a Laurent series in the parameter $\sqrt{\ensuremath{\epsilon}},$ with $\ensuremath{\epsilon}=d\ensuremath{-}1:$ $z=d,$ $\ensuremath{\nu}=1/\sqrt{3\ensuremath{\epsilon}}+O(\sqrt{\ensuremath{\epsilon}})$ for the short-range, and $z=1,$ $\ensuremath{\nu}=\sqrt{2/3\ensuremath{\epsilon}}+O(\sqrt{\ensuremath{\epsilon}}),$ for the long-range Coulomb interaction between bosons. The identified critical point should be stable against the residual perturbations in the effective action for the superfluid, at least in dimensions $1<~d<~2,$ for both short-range and Coulomb interactions. For the superfluid-Mott insulator transition in the system in a periodic potential and at a commensurate density of bosons, I find $\ensuremath{\nu}=(\frac{1}{2}\sqrt{\ensuremath{\epsilon}})+\frac{1}{4}+O(\sqrt{\ensuremath{\epsilon}}),$ which yields a result in reasonable agreement with the known $\mathrm{XY}$ critical exponent in $d=2.$ The critical behavior of the superfluid density, phonon velocity, and the compressibility in the system with short-range interactions is discussed.

arXiv:cond-mat/9802141Physical review. B, Condensed matterJul 1, 199827 citations

Nonstandard critical behavior of the imperfect Bose gas on a lattice

Paweł Jakubczyk, Jacek Wojtkiewicz

We study an anisotropic variant of the $d$ -dimensional imperfect Bose gas on a lattice, where the asymptotic behavior of the dispersion $\epsilon_{\bf k}$ at vanishing momentum $\bf{k}$ may differ from the standard quadratic form. We derive a general expression for the lower critical dimension $d_L$ of the model and show that it may be pushed to arbitrarily high values by engineering the hopping parameters. The same applies to the upper critical dimension $d_U$ . For $d>d_L$ we find a general, exact expression for the Bose-Einstein condensation temperature $T_c(\mu)$ as a function of the chemical potential $\mu$ . The asymptotic behavior of $T_c(\mu)$ at $T_c\to 0$ is governed by the shift exponent $\psi$ , whose value turns out to fully determine whether the system is below or above the upper critical dimension. The critical exponents at condensation as well as the exponent $\psi$ may attain non-standard values for specific choices of the lattice kinetic couplings. Finally, we study the mechanism leading to the affinity between the imperfect Bose gas and the classical spherical model of spins in the critical regime.We study an anisotropic variant of the $d$-dimensional imperfect Bose gas on a lattice, where the asymptotic behavior of the dispersion $\epsilon_{\bf k}$ at vanishing momentum $\bf{k}$ may differ from the standard quadratic form. We derive a general expression for the lower critical dimension $d_L$ of the model and show that it may be pushed to arbitrarily high values by engineering the hopping parameters. The same applies to the upper critical dimension $d_U$. For $d>d_L$ we find a general, exact expression for the Bose-Einstein condensation temperature $T_c(\mu)$ as a function of the chemical potential $\mu$. The asymptotic behavior of $T_c(\mu)$ at $T_c\to 0$ is governed by the shift exponent $\psi$, whose value turns out to fully determine whether the system is below or above the upper critical dimension. The critical exponents at condensation as well as the exponent $\psi$ may attain non-standard values for specific choices of the lattice kinetic couplings. Finally, we study the mechanism leading to the affinity between the imperfect Bose gas and the classical spherical model of spins in the critical regime.

Sep 27, 20160 citations

The natural integrable correlated strongly coupled boson system in 1 + 1 dimensions is the

q

-boson hopping model; we calculate its critical exponent

\ensuremath{\theta}

and determine its correlation functions. For small couplings the

q

-boson model has natural connections with the Bose gas and the

\mathrm{XY}

models of very large spin for which

\ensuremath{\theta}'\mathrm{s}

and correlators are reported. For large couplings the hopping model is a new phase of interacting bosons substantially different from the impenetrable Bose gas.

XY

model with power-law couplings---undergo a non-BKT continuous phase transition instead. We use exact quantum Monte Carlo methods to determine the phase diagram for different values of the exponent $\ensuremath{\alpha}$, focusing on the regime $\ensuremath{\alpha}>2$. We find that the scaling of the superfluid stiffness with the system size is scale invariant at the transition point for any $\ensuremath{\alpha}\ensuremath{\le}3$---a behavior incompatible with the BKT scenario and typical of continuous phase transitions in higher dimension. By a scaling analysis near the transition point, we find that our data are consistent with a correlation length exponent satisfying the Harris bound $\ensuremath{\nu}\ensuremath{\ge}2$ and demonstrate a universal behavior of disordered bosons in one dimension. For $\ensuremath{\alpha}>3$ our data are consistent with a BKT scenario where the liquid is pinned by infinitesimal disorder.

d=1.

Within the formulated renormalization procedure near

d=1,

the dynamical critical exponent is obtained exactly and the correlation length critical exponent is calculated as a Laurent series in the parameter $\sqrt{\ensuremath{\epsilon}},$ with $\ensuremath{\epsilon}=d\ensuremath{-}1:$

z=d,

$\ensuremath{\nu}=1/\sqrt{3\ensuremath{\epsilon}}+O(\sqrt{\ensuremath{\epsilon}})$ for the short-range, and

z=1,

$\ensuremath{\nu}=\sqrt{2/3\ensuremath{\epsilon}}+O(\sqrt{\ensuremath{\epsilon}}),$ for the long-range Coulomb interaction between bosons. The identified critical point should be stable against the residual perturbations in the effective action for the superfluid, at least in dimensions

1<~d<~2,

for both short-range and Coulomb interactions. For the superfluid-Mott insulator transition in the system in a periodic potential and at a commensurate density of bosons, I find $\ensuremath{\nu}=(\frac{1}{2}\sqrt{\ensuremath{\epsilon}})+\frac{1}{4}+O(\sqrt{\ensuremath{\epsilon}}),$ which yields a result in reasonable agreement with the known

\mathrm{XY}

critical exponent in

d=2.

The critical behavior of the superfluid density, phonon velocity, and the compressibility in the system with short-range interactions is discussed.

We study an anisotropic variant of the

d

-dimensional imperfect Bose gas on a lattice, where the asymptotic behavior of the dispersion

\epsilon_{\bf k}

at vanishing momentum

\bf{k}

may differ from the standard quadratic form. We derive a general expression for the lower critical dimension

d_L

of the model and show that it may be pushed to arbitrarily high values by engineering the hopping parameters. The same applies to the upper critical dimension

d_U

. For

d>d_L

we find a general, exact expression for the Bose-Einstein condensation temperature

T_c(\mu)

as a function of the chemical potential

\mu

. The asymptotic behavior of

T_c(\mu)

T_c\to 0

is governed by the shift exponent

\psi

, whose value turns out to fully determine whether the system is below or above the upper critical dimension. The critical exponents at condensation as well as the exponent

\psi

may attain non-standard values for specific choices of the lattice kinetic couplings. Finally, we study the mechanism leading to the affinity between the imperfect Bose gas and the classical spherical model of spins in the critical regime.