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Physical review. E

Critical dynamics of the superfluid phase transition: Multiloop calculation of the microscopic model

Juha Honkonen, Marina Vladimirovna Komarova, Yu. G. Molotkov, Mikhail Yu. Nalimov, Alexander Trenogin

Published Jul 19, 2022• 8 citations
Publisher

Abstract

Results and method of a three-loop renormalization-group calculation in the model of a Bose gas with a local density-density interaction in the formalism of time-dependent Green functions at finite temperature are presented. The results provide support to the recent conjecture [J. Honkonen, M. V. Komarova, Y. G. Molotkov, and M. Y. Nalimov, Nucl. Phys. B 939, 105 (2019)0550-321310.1016/j.nuclphysb.2018.12.015; Y. A. Zhavoronkov, M. V. Komarova, Y. G. Molotkov, M. Y. Nalimov, and J. Honkonen, Theor. Math. Phys. 200, 1237 (2019)0040-577910.1134/S0040577919080142] that the dynamics of the superfluid phase transition is described by a model which belongs to the same universality class as the stochastic model A.

DOI
10.1103/physreve.106.014126

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Results and method of a three-loop renormalization-group calculation in the model of a Bose gas with a local density-density interaction in the formalism of time-dependent Green functions at finite temperature are presented. The results provide support to the recent conjecture [J. Honkonen, M. V. Komarova, Y. G. Molotkov, and M. Y. Nalimov, Nucl. Phys. B 939, 105 (2019)0550-321310.1016/j.nuclphysb.2018.12.015; Y. A. Zhavoronkov, M. V. Komarova, Y. G. Molotkov, M. Y. Nalimov, and J. Honkonen, Theor. Math. Phys. 200, 1237 (2019)0040-577910.1134/S0040577919080142] that the dynamics of the superfluid phase transition is described by a model which belongs to the same universality class as the stochastic model A.
In papers devoted to superfluidity, the generally accepted statement that the dynamics of the corresponding phase transition is described by the stochastic model F or E can be frequently found. Nevertheless, the dynamical critical index has not been found even in the leading order of the perturbation theory. It is also unknown which model, E or F, in fact corresponds to this system. We use two different approaches to study this problem. First, we study the dynamics of the critical behavior in a neighborhood of the λ point using the renormalization group method based on a quantum microscopic model in the formalism of the time-dependent Green’s functions at a finite temperature. Second, we study the stochastic model F to find whether it is stable under compressibility effects. Both approaches lead to the same very unexpected result: the dynamics of the phase transition to the superfluid state are described by the stochastic model A with a known dynamical critical index.
We review the progress made in dynamic bulk critical behaviour in equilibrium in the last 25 years since the review of Halperin and Hohenberg. We unify the presentation of the theoretical background by restricting ourselves to the field-theoretic renormalization group method. The main results obtained in the different universality classes are presented. This contains the critical dynamics near the gas–liquid transition in pure fluids (model H), the plait point and consolute point in mixtures (model H � ), the superfluid transition in 4 He (model F) and 4 He– 3 He mixtures (model F � ), the Curie point (model J) and Neel point (model G) in Heisenberg magnets and the superconducting transition. In comparison with experimental results, it became clear that in most cases one has to consider apart from the universal asymptotic critical behaviour also the non-universal effective behaviour. Either because it turned out to be inevitable due to a small dynamical transient exponent inhibiting the system to reach the asymptotics (e.g., at the superfluid transition) or because one is interested in the region further away from the phase transition like in pure fluids and mixtures at their gas–liquid or demixing transition. The calculation of the critical dynamics is adequate in most cases only in two-loop order. We review these results and present the solution to unreasonable features found for some models. Thus, we consider model C where relaxational and diffusive dynamics are coupled and the scaling properties and the limit to a purely relaxational model (model A) have not been understood. In general for models where the order parameter couples to other conserved densities time scale ratios between the kinetic coefficients of the order parameter and the conserved densities play an important role. Their fixed-point values and the approach to the fixed point are changed considerably in two-loop order compared to their values in one-loop order. These considerations are relevant for the explanation of the dynamical critical shape functions of systems such as superfluid helium (model F) and the isotropic antiferromagnet (model G). As far as possible, the comparison of results obtained by the renormalization group theory with numerical simulations has been made.
Dynamic behaviour of a boson gas near the condensation transition in the symmetric phase is analyzed with the use of an effective large-scale model derived from time-dependent Green functions at finite temperature. A renormalization-group analysis shows that the scaling exponents of critical dynamics of the effective multi-charge model coincide with those of the standard model A. The departure of this result from the description of the superfluid transition by either model E or F of the standard phenomenological stochastic models is corroborated by the analysis of a generalization of model F, which takes into account the effect of compressible fluid velocity. It is also shown that, contrary to the single-charge model A, there are several correction exponents in the effective model, which are calculated at the leading order of the ɛ= 4 − d expansion.
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{s}^{4}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}^{\mathrm{*}}m∗/m\ensuremath{\simeq}1.5, and an effective-scattering-length ratio a∗{a}^{\mathrm{*}}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.
Dynamic critical behavior in superfluid systems is considered in the presence of external stirring and advecting processes. The latter are generated by means of the Gaussian random velocity ensemble with white-noise character in time variable and self-similar spatial dependence. The main focus of this work is to analyze an effect of compressible modes on the critical behavior. The model is formulated through stochastic Langevin equations, which are then recast into the Janssen-De Dominicis response formalism. Employing the field-theoretic perturbative renormalization group method we analyze large-scale properties of the model. Explicit calculations are performed to the leading one-loop approximation in the double (ɛ,y) expansion scheme, where ɛ is a deviation from the upper critical dimension d_{c}=4 and y describes a scaling property of the velocity ensemble. Altogether five distinct universality classes are expected to be macroscopically observable. In contrast to the incompressible case, we find that compressibility leads to an enhancement and stabilization of nontrivial asymptotic regimes.
The phase diagrams and phase transitions of bosons with short-ranged repulsive interactions moving in periodic and/or random external potentials at zero temperature are investigated with emphasis on the superfluid-insulator transition induced by varying a parameter such as the density. Bosons in periodic potentials (e.g., on a lattice) at T=0 exhibit two types of phases: a superfluid phase and Mott insulating phases characterized by integer (or commensurate) boson densities, by the existence of a gap for particle-hole excitations, and by zero compressibility. Generically, the superfluid onset transition in d dimensions from a Mott insulator to superfluidity is ``ideal,'' or mean field in character, but at special multicritical points with particle-hole symmetry it is in the universality class of the (d+1)-dimensional XY model. In the presence of disorder, a third, ``Bose glass'' phase exists. This phase is insulating because of the localization effects of the randomness and analogous to the Fermi glass phase of interacting fermions in a strongly disordered potential.The Bose glass phase is characterized by a finite compressibility, no gap, but an infinite superfluid susceptibility. In the presence of disorder the transition to superfluidity is argued to occur only from the Bose glass phase, and never directly from the Mott insulator. This zero-temperature superfluid-insulator transition is studied via generalizations of the Josephson scaling relation for the superfluid density at the ordinary \ensuremath{\lambda} transition, highlighting the crucial role of quantum fluctuations. The transition is found to have a dynamic critical exponent z exactly equal to d and correlation length and order-parameter correlation exponents \ensuremath{\nu} and \ensuremath{\eta} which satisfy the bounds \ensuremath{\nu}\ensuremath{\ge}2/d and \ensuremath{\eta}\ensuremath{\le}2-d, respectively. It is argued that the superfluid-insulator transition in the presence of disorder may have an upper critical dimension dc{d}_{c}dc​ which is infinite, but a perturbative renormalization-group calculation wherein the critical exponents have mean-field values for weak disorder above d=4 is also discussed. Many of these conclusions are verified by explicit calculations on a model of one-dimensional bosons in the presence of both random and periodic potentials. The general results are applied to experiments on 4He^{4}\mathrm{He}4He absorbed in porous media such as Vycor. Some measurable properties of the superfluid onset are predicted exactly [e.g., the exponent x relating the \ensuremath{\lambda} transition temperature to the zero-temperature superfluid density is found to be d/2(d-1)], while stringent bounds are placed on others. Analysis of preliminary data is consistent with these predictions.