The crossover from ideal-Bose-gas behavior to nonclassical, XY-like criticality in a dilute, interacting Bose fluid is considered in detail with emphasis on the superfluid density, ${\ensuremath{\rho}}_{s}$(T), below the transition. A general discussion of crossover scaling in constrained systems shows that ${\ensuremath{\rho}}_{s}$ and other temperature-dependent thermodynamic functions at controlled overall density can be expressed in terms of universal scaling functions. The finite-temperature Green's function formalism is used to calculate the associated linear scaling fields and the crossover exponent exactly for all dimensionalities d>2, scaling predictions, including logarithmic singularities just at d=3, being checked via second-order computations. A mapping of the Bose problem onto the classical
s4 spin model is achieved which is asymptotically exact in the critical region. In this way, previously known renormalization-group results are used to calculate the scaling functions for ${\ensuremath{\rho}}_{s}$, for the off-diagonal susceptibility, etc., to first order in \ensuremath{\epsilon}=4-d. Good agreement is found with the experimental measurements by Reppy and co-workers of ${\ensuremath{\rho}}_{s}$(T) for helium absorbed at low coverages in Vycor glass (of pore size 50--80 A\r{}). An effective-mass ratio
m∗/m\ensuremath{\simeq}1.5, and an effective-scattering-length ratio
a∗/a\ensuremath{\simeq}1.3 are indicated for mobile helium in the Vycor (although the irregular character of Vycor has not, at this point, been fully allowed for in the theory). Other approaches in the literature to computing the critical behavior of quantum-mechanical systems are reviewed and related to the present analysis.
A renormalizable parameter \ensuremath{\xi} (0\ensuremath{\le}\ensuremath{\xi}\ensuremath{\le}1) is introduced into the interacting Bose-system Hamiltonian such that the zero-mode amplitudes satisfy the commutation relation [
b0,${b}_{0}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$]=1-\ensuremath{\xi}. The different phases of the system correspond to specific values of \ensuremath{\xi}. The normal component (phase I) is obtained when \ensuremath{\xi}=0 and there are two ordered states according to the strength of the interaction: Phase II' (0<\ensuremath{\xi}\ensuremath{\ll}1) is found in the strongly interacting system and is characterized by a broken gauge symmetry without Bose-Einstein condensation, while phase II'' (\ensuremath{\xi}=1), in the weakly interacting system, displays Bose-Einstein condensation. The continuity equation for the particle density holds in both ordered phases to within an infinitesimal term. A gauge transformation on the field operators shows that \ensuremath{\xi} ''twists'' the phase angle related to the zero mode, and a response to this twisting field is defined in terms of the generalized force \ensuremath{\chi}(T) associated with \ensuremath{\xi}. For the ideal Bose gas \ensuremath{\chi}(T) equals the condensate density and hence is simply related to the helicity modulus introduced by Fisher et al. A renormalization-group transformation shows that the dense system transition (I-II') gives rise to a two-thirds correlation-length critical exponent, whereas a unity crossover exponent appears in the dilute system transition (I-II''), in agreement with the experimental findings.
The behavior of a dilute Bose superfluid at low temperatures, $T<<{T}_{\ensuremath{\lambda}}$, is considered with emphasis on the superfluid density, ${\ensuremath{\rho}}_{s}(T)$. Many of the predictions of the Bogoliubov model are exact in this regime and are shown to lead to a scaling description of the crossover from ideal to interacting Bose gas behavior as the density is reduced. Combined with previous work on scaling near criticality, $T\ensuremath{\simeq}{T}_{\ensuremath{\lambda}}$, these results complete the crossover description of the dilute superfluid over the full temperature range. Diagrammatic perturbation theory in the superfluid phase is outlined and previous results from the helium literature are summarized. The various divergences encountered have previously been shown to arise from nonanalyticities in the self-energy functions at zero momentum, and these are related to the spin-wave nature of the superfluid phase. A full renormalization-group fixed-point picture is presented summarizing the zero-temperature and finite-temperature regimes and the flows connecting them.
We present a detailed investigation of the momentum-dependent self-energy $\ensuremath{\Sigma}(k)$ at zero frequency of weakly interacting bosons at the critical temperature
Tc of Bose-Einstein condensation in dimensions $3\ensuremath{\leqslant}D<4$. Applying the functional renormalization group, we calculate the universal scaling function for the self-energy at zero frequency but at all wave vectors within an approximation which truncates the flow equations of the irreducible vertices at the four-point level. The self-energy interpolates between the critical regime
k⪡kc and the short-wavelength regime
k⪢kc, where
kc is the crossover scale. In the critical regime, the self-energy correctly approaches the asymptotic behavior $\ensuremath{\Sigma}(k)\ensuremath{\propto}{k}^{2\ensuremath{-}\ensuremath{\eta}}$, and in the short-wavelength regime the behavior is $\ensuremath{\Sigma}(k)\ensuremath{\propto}{k}^{2(D\ensuremath{-}3)}$ in
D>3. In
D=3, we recover the logarithmic divergence $\ensuremath{\Sigma}(k)\ensuremath{\propto}\mathrm{ln}(k∕{k}_{c})$ encountered in perturbation theory. Our approach yields the crossover scale
kc as well as a reasonable estimate for the critical exponent $\ensuremath{\eta}$ in
D=3. From our scaling function we find for the interaction-induced shift in
Tc in three dimensions, $\ensuremath{\Delta}{T}_{c}∕{T}_{c}=1.23a{n}^{1∕3}$, where
a is the
s-wave scattering length and
n is the density, in excellent agreement with other approaches. We also discuss the flow of marginal parameters in
D=3 and extend our truncation scheme of the renormalization group equations by including the six- and eight-point vertex, which yields an improved estimate for the anomalous dimension $\ensuremath{\eta}\ensuremath{\approx}0.0513$. We further calculate the constant ${\mathrm{lim}}_{k\ensuremath{\rightarrow}0}\phantom{\rule{0.2em}{0ex}}\ensuremath{\Sigma}(k)∕{k}^{2\ensuremath{-}\ensuremath{\eta}}$ and find good agreement with recent Monte Carlo data.
We analyze the theoretical derivation of the beyond-mean-field equation of state for two-dimensional gas of dilute, ultracold alkali-metal atoms in the Bardeen–Cooper–Schrieffer (BCS) to Bose–Einstein condensate (BEC) crossover. We show that at zero temperature our theory — considering Gaussian fluctuations on top of the mean-field equation of state — is in very good agreement with experimental data. Subsequently, we investigate the superfluid density at finite temperature and its renormalization due to the proliferation of vortex–antivortex pairs. By doing so, we determine the Berezinskii–Kosterlitz–Thouless (BKT) critical temperature — at which the renormalized superfluid density jumps to zero — as a function of the inter-atomic potential strength. We find that the Nelson–Kosterlitz criterion overestimates the BKT temperature with respect to the renormalization group equations, this effect being particularly relevant in the intermediate regime of the crossover.
Motivated by the collective behavior of biological swarms, we study the critical dynamics of field theories with coupling between order parameter and conjugate momentum in the presence of dissipation. Under a fixed-network approximation, we perform a dynamical renormalization group calculation at one loop in the near-critical disordered region, and we show that the violation of momentum conservation generates a crossover between an unstable fixed point, characterized by a dynamic critical exponent z=d/2, and a stable fixed point with z=2. Interestingly, the two fixed points have different upper critical dimensions. The interplay between these two fixed points gives rise to a crossover in the critical dynamics of the system, characterized by a crossover exponent κ=4/d. The crossover is regulated by a conservation length scale R_{0}, given by the ratio between the transport coefficient and the effective friction, which is larger as the dissipation is smaller: Beyond R_{0}, the stable fixed point dominates, while at shorter distances dynamics is ruled by the unstable fixed point and critical exponent, a behavior which is all the more relevant in finite-size systems with weak dissipation. We run numerical simulations in three dimensions and find a crossover between the exponents z=3/2 and z=2 in the critical slowdown of the system, confirming the renormalization group results. From the biophysical point of view, our calculation indicates that in finite-size biological groups mode coupling terms in the equation of motion can significantly change the dynamical critical exponents even in the presence of dissipation, a step toward reconciling theory with experiments in natural swarms. Moreover, our result provides the scale within which fully conservative Bose-Einstein condensation is a good approximation in systems with weak symmetry-breaking terms violating number conservation, as quantum magnets or photon gases.