We study the low-temperature limit of the d -dimensional imperfect Bose gas. Relying on an exact analysis of the microscopic model, we establish the existence of a second-order quantum phase transition to a phase involving the Bose–Einstein condensate. The transition is triggered by varying the chemical potential and persists at non-zero temperatures T for d > 2. We extract the exact phase diagram and identify the scaling regimes in the vicinity of the quantum-critical point focusing on the behavior of the correlation length ξ. The length ξ develops an essential singularity exclusively for d = 2. We follow the evolution of the phase diagram varying d . For d > 2 our results agree with renormalization-group-based analysis of the effective bosonic order-parameter models with the dynamical exponent z = 2.
We describe the nonzero temperature
(T), low frequency $(\ensuremath{\omega})$ dynamics of the order parameter near quantum critical points in two spatial dimensions
(d), with a special focus on the regime $\ensuremath{\Elzxh}\ensuremath{\omega}\ensuremath{\ll}{k}_{B}T$. For the case of a ``relativistic,''
O(n)-symmetric, bosonic quantum field theory we show that, for small $\ensuremath{\epsilon}=3\ensuremath{-}d$, the dynamics is described by an effective classical model of waves with a quartic interaction. We provide analytical and numerical analyses of the classical wave model directly in
d=2. We describe the crossover from the finite frequency, ``amplitude fluctuation,'' gapped quasiparticle mode in the quantum paramagnet (or Mott insulator), to the zero frequency ``phase''
(n> 2) or ``domain wall''
(n=1) relaxation mode near the ordered state. For static properties, we show how a surprising, duality-like transformation allows an exact treatment of the strong-coupling limit for all n. For
n=2, we compute the universal T dependence of the superfluid density below the Kosterlitz-Thouless temperature, and discuss implications for the high temperature superconductors. For
n=3, our computations of the dynamic structure factor relate to neutron scattering experiments on
La1.85Sr0.15CuO4, and to light scattering experiments on double layer quantum Hall systems. We expect that closely related effective classical wave models will apply also to other quantum critical points in
d=2. Although computations in appendixes do rely upon technical results on the $\ensuremath{\epsilon}$-expansion of quantum critical points obtained in earlier papers, the physical discussion in the body of the paper is self-contained, and can be read without consulting these earlier works.
We study the Berezinskii-Kosterlitz-Thouless transition in a weakly interacting 2D quantum Bose gas using the concept of universality and numerical simulations of the classical $|\ensuremath{\psi}{|}^{4}$ model on a lattice. The critical density and chemical potential are given by relations ${n}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}(mT/2\ensuremath{\pi}{\ensuremath{\Elzxh}}^{2})\mathrm{ln}(\ensuremath{\xi}{\ensuremath{\Elzxh}}^{2}/mU)$ and ${\ensuremath{\mu}}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}(mTU/\ensuremath{\pi}{\ensuremath{\Elzxh}}^{2})\mathrm{ln}({\ensuremath{\xi}}_{\ensuremath{\mu}}{\ensuremath{\Elzxh}}^{2}/mU)$, where
T is the temperature,
m is the mass, and
U is the effective interaction. The dimensionless constant $\ensuremath{\xi}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}380\ifmmode\pm\else\textpm\fi{}3$ is very large and thus any quantitative analysis of the experimental data crucially depends on its value. For ${\ensuremath{\xi}}_{\ensuremath{\mu}}$ our result is ${\ensuremath{\xi}}_{\ensuremath{\mu}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}13.2\ifmmode\pm\else\textpm\fi{}0.4$. We also report the study of the quasicondensate correlations at the critical point.
We use the functional renormalization group (FRG) to derive analytical expressions for thermodynamic observables (density, pressure, entropy, and compressibility) as well as for single-particle properties (wave-function renormalization and effective mass) of interacting bosons in two dimensions as a function of temperature
T and chemical potential $\ensuremath{\mu}$. We focus on the quantum disordered and the quantum critical regime close to the dilute Bose gas quantum critical point. Our approach is based on a truncated vertex expansion of the hierarchy of FRG flow equations and the decoupling of the two-body contact interaction in the particle-particle channel using a suitable Hubbard-Stratonovich transformation. Our analytic FRG results extend previous analytical renormalization-group calculations for thermodynamic observables at $\ensuremath{\mu}=0$ to finite values of $\ensuremath{\mu}$. To confirm the validity of our FRG approach, we have also performed quantum Monte Carlo simulations to obtain the magnetization, susceptibility, and correlation length of the two-dimensional spin-
1/2 quantum
XY model with coupling
J in a regime where its quantum critical behavior is controlled by the dilute Bose gas quantum critical point. We find that our analytical results describe the Monte Carlo data for $\ensuremath{\mu}\ensuremath{\le}0$ rather accurately up to relatively high temperatures $T\ensuremath{\lesssim}0.1J$.
A paramagnetic-ferromagnetic quantum phase transition is known to occur at zero temperature in a two-dimensional coherently coupled Bose mixture of dilute ultracold atomic gases provided the interspecies interaction strength is large enough. Here we study the fate of such a transition at finite temperature by performing numerical simulations with the stochastic (projected) Gross-Pitaevskii formalism, which includes both thermal and beyond mean-field effects. By extracting the average magnetization, the magnetic fluctuations and characteristic relaxation frequency (or critical slowing down), we identify a finite-temperature critical line for the transition. We find that the critical point shifts linearly with temperature and, in addition, the three quantities used to probe the transition exhibit a temperature power-law scaling. The scaling of the critical slowing down is found to be consistent with thermal critical exponents and is very well approximated by the square of the spin excitation gap at zero temperature.
The equation of state of a weakly interacting two-dimensional Bose gas is studied at zero temperature by means of quantum Monte Carlo methods. Going down to as low densities as $n{a}^{2}\ensuremath{\propto}{10}^{\ensuremath{-}100}$ permits us to obtain agreement on beyond mean-field level between predictions of perturbative methods and direct many-body numerical simulation, thus providing an answer to the fundamental question of the equation of state of a two-dimensional dilute Bose gas in the universal regime (i.e., entirely described by the gas parameter
na2). We also show that the measure of the frequency of a breathing collective oscillation in a trap at very low densities can be used to test the universal equation of state of a two-dimensional Bose gas.