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.
The weakly interacting Bose gas in two dimensions is considered in the dilute limit
n1/2a\ensuremath{\ll}1, where n is the particle density and a is the range of the potential. The standard many-body perturbation theory for this system has two separate divergences: the first, associated with classical phase fluctuations, is responsible for the vanishing of the long-range order; the second is quantum mechanical and is connected with the vanishing of the scattering t matrix at long wavelengths and low energies. An earlier diagrammatic theory of Popov, which provides a consistent description of the system in the dilute limit, is rederived heuristically from a quasiparticle picture, and also using the renormalization group. It is shown that the superfluid transition temperature is
Tc\ensuremath{\approxeq}4\ensuremath{\pi}(${\ensuremath{\Elzxh}}^{2}$/2m)n/[ln ln(1/
na2)], and the condition of validity of the dilute limit is ln ln(1/
na2)\ensuremath{\gg}1. The connection to the dilute Bose gas in dimensions d>2 and the universal behavior beyond the extreme asymptotic low-density domain are also discussed.
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 derive via diagrammatic perturbation theory the scaling behavior of the condensate and superfluid mass density of a dilute Bose gas just below the condensation temperature, T(c). Sufficiently below T(c) particle excitations are described by mean field (Bogoliubov). Near T(c), however, mean field fails, and the system undergoes a second order phase transition, rather than first order as predicted by Bogoliubov theory. Both condensation and superfluidity occur at the same T(c), and have similar scaling functions below T(c), but different finite size scaling at T(c) to leading order in the system size. A self-consistent two-loop calculation yields the condensate fraction critical exponent, 2beta approximately 0.66.
We discuss some problems related to quantum critical phenomena in superfluids and superconductors. In Ch. 1, we apply generalizations of hyperuniversality to quantum phase transitions at zero temperature. We find new universal amplitude combinations involving the superfluid density in Bose systems, as well as confirm and extend previous proposals for universal transport coefficients in two-dimensional superconducting films and magnetic-field-induced metal-insulator transitions. In Ch. 2, we apply the double-dimensionality expansion of Dorogovstev to derive renormalization-group recursion relations for a Bose fluid in a random external potential. We find a nontrivial fixed point. The onset of mean-field behavior for dimensions d > d[subscript]c = 4 is unconventional, yielding discontinuous exponents. Including positive temperatures, we give a clear picture of various crossover regimes. In Ch. 3, We give a detailed derivation of the critical thermodynamics of O(n) spin models, correct to O([epsilon] = 4 - d), using a generalization of the renormalization-group trajectory integral and noncritical matching technique introduced by Rudnick and Nelson. We especially emphasize the coexistence-curve behavior for spins with a continuous symmetry (n [...] 2), deriving detailed expressions for the renormalized spin-wave stiffness and longitudinal susceptibility.
Using the exact renormalization group we calculate 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 $3\ensuremath{\leqslant}D<4$. We obtain the complete crossover function interpolating between the critical regime
k⪡kc, where $\ensuremath{\Sigma}(k)\ensuremath{\propto}{k}^{2\ensuremath{-}\ensuremath{\eta}}$, and the short-wavelength regime
k⪢kc, where $\ensuremath{\Sigma}(k)\ensuremath{\propto}{k}^{2(D\ensuremath{-}3)}$ in
D>3 and $\ensuremath{\Sigma}(k)\ensuremath{\propto}\mathrm{ln}(k∕{k}_{c})$ in
D=3. Our approach yields the crossover scale
kc on the same footing with a reasonable estimate for the critical exponent $\ensuremath{\eta}$ in
D=3. From our $\ensuremath{\Sigma}(k)$ we find for the interaction-induced shift of
Tc in three dimensions $\ensuremath{\Delta}{T}_{c}∕{T}_{c}\ensuremath{\approx}1.23a{n}^{1∕3}$, where
a is the
s-wave scattering length and
n is the density.