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arXiv:0705.1661 · Physical Review B

Functional renormalization for quantum phase transitions with nonrelativistic bosons

CHRISTOF WETTERICH

Published Feb 11, 2008• 54 citations
arXivHTMLPDF

Abstract

Functional renormalization yields a simple unified description of bosons at zero temperature, in arbitrary space dimension $d$ and for $M$ complex fields. We concentrate on nonrelativistic bosons and an action with a linear time derivative. The ordered phase can be associated with a nonzero density of (quasi)particles $n$. The behavior of observables and correlation functions in the ordered phase depends crucially on the momentum ${k}_{\mathit{ph}}$, which is characteristic for a given experiment. For the dilute regime ${k}_{\mathit{ph}}\ensuremath{\gtrsim}{n}^{1∕d}$, the quantum phase transition is simple, with the same ``mean field'' critical exponents for all $d$ and $M$. On the other hand, the dense regime ${k}_{\mathit{ph}}⪡{n}^{1∕d}$ reveals a rather rich spectrum of features, depending on $d$ and $M$. In this regime, one observes for $d\ensuremath{\leqslant}3$ a crossover to a relativistic action with second time derivatives. This admits order for $d>1$, whereas $d=1$ shows a behavior similar to the low temperature phase of the classical two-dimensional $\mathrm{O}(2M)$ models.

DOI
10.1103/physrevb.77.064504

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Functional renormalization yields a simple unified description of bosons at zero temperature, in arbitrary space dimension ddd and for MMM complex fields. We concentrate on nonrelativistic bosons and an action with a linear time derivative. The ordered phase can be associated with a nonzero density of (quasi)particles nnn. The behavior of observables and correlation functions in the ordered phase depends crucially on the momentum kph{k}_{\mathit{ph}}kph​, which is characteristic for a given experiment. For the dilute regime kph\ensuremath≳n1∕d{k}_{\mathit{ph}}\ensuremath{\gtrsim}{n}^{1∕d}kph​\ensuremath≳n1∕d, the quantum phase transition is simple, with the same ``mean field'' critical exponents for all ddd and MMM. On the other hand, the dense regime kph⪡n1∕d{k}_{\mathit{ph}}⪡{n}^{1∕d}kph​⪡n1∕d reveals a rather rich spectrum of features, depending on ddd and MMM. In this regime, one observes for d\ensuremath⩽3d\ensuremath{\leqslant}3d\ensuremath⩽3 a crossover to a relativistic action with second time derivatives. This admits order for d>1d>1d>1, whereas d=1d=1d=1 shows a behavior similar to the low temperature phase of the classical two-dimensional O(2M)\mathrm{O}(2M)O(2M) models.
We study the critical behavior and phase diagram of the ddd-dimensional random field O(N)O(N)O(N) model by means of the nonperturbative functional renormalization group approach presented in the preceding paper. We show that the dimensional-reduction predictions, which are obtained from conventional perturbation theory, break down below a critical dimension dDR(N){d}_{\text{DR}}(N)dDR​(N) and we provide a description of criticality, ferromagnetic ordering, and quasi-long range order in the whole (N,d)(N,d)(N,d) plane. Below dDR(N){d}_{\text{DR}}(N)dDR​(N), our formalism gives access to both the typical behavior of the system, which is controlled by zero-temperature fixed points at which the effective action has a nonanalytic field dependence, and to the physics of rare low-energy excitations (``droplets''), which is described at nonzero temperature by the rounding of the nonanalyticity in a thermal boundary layer.
This paper proposes an approach to the study of critical phenomena in quantum-mechanical systems at zero or low temperatures, where classical free-energy functionals of the Landau-Ginzburg-Wilson sort are not valid. The functional integral transformations first proposed by Stratonovich and Hubbard allow one to construct a quantum-mechanical generalization of the Landau-Ginzburg-Wilson functional in which the order-parameter field depends on (imaginary) time as well as space. Since the time variable lies in the finite interval [$0,\ensuremath{-}i\ensuremath{\beta}$], where $\ensuremath{\beta}$ is the inverse temperature, the resulting description of a ddd-dimensional system shares some features with that of a (d+1d+1d+1)-dimensional classical system which has finite extent in one dimension. However, the analogy is not complete, in general, since time and space do not necessarily enter the generalized free-energy functional in the same way. The Wilson renormalization group is used here to investigate the critical behavior of several systems for which these generalized functionals can be constructed simply. Of these, the itinerant ferromagnet is studied in greater detail. The principal results of this investigation are (i) at zero temperature, in situations where the ordering is brought about by changing a coupling constant, the dimensionality which separates classical from nonclassical critical-exponent behavior is not 4, as is usually the case in classical statistics, but $4\ensuremath{-}z$ dimensions, where zzz depends on the way the frequency enters the generalized free-energy functional. When it does so in the same way that the wave vector does, as happens in the case of interacting magnetic excitons, the effective dimensionality is simply increased by 1; z=1z=1z=1. It need not appear in this fashion, however, and in the examples of itinerant antiferromagnetism and clean and dirty itinerant ferromagnetism, one finds z=2,3,and4z=2, 3, \mathrm{and} 4z=2,3,and4, respectively. (ii) At finite temperatures, one finds that a classical statistical-mechanical description holds (and nonclassical exponents, for d<4d<4d<4) very close to the critical value of the coupling Uc{U}_{c}Uc​, when $\frac{(U\ensuremath{-}{U}_{c})}{{U}_{c}}\ensuremath{\ll}{(\frac{T}{{U}_{c}})}^{\frac{2}{z}}$. z2\frac{z}{2}2z​ is therefore the quantum-to-classical crossover exponent.
This paper proposes an approach to the study of critical phenomena in quantum-mechanical systems at zero or low temperatures, where classical free-energy functionals of the Landau-Ginzburg-Wilson sort are not valid. The functional integral transformations first proposed by Stratonovich and Hubbard allow one to construct a quantum-mechanical generalization of the Landau-Ginzburg-Wilson functional in which the order-parameter field depends on (imaginary) time as well as space. Since the time variable lies in the finite interval [0,- iβ ], where β is the inverse temperature, the resulting description of a d -dimensional system shares some features with that of a ( d + 1)-dimensional classical system which has finite extent in one dimension. However, the analogy is not complete, in general, since time and space do not necessarily enter the generalized free-energy functional m the same way. The Wilson renormalization group is used here to investigate the critical behavior of several systems for which these generalized functionals can be constructed simply. Of these, the itinerant ferromagnet is studied in greater detail. The principal results of this investigation are (i) at zero temperature, in situations where the ordering is brought about by changing a coupling constant, the dimensionality which separates classical from nonclassical critical-exponent behavior is not 4, as is usually the case in classical statistics, but 4 − z dimensions, where z depends on the way the frequency enters the generalized free-energy functional. When it does so m the same way that the wave vector does, as happens in the case of interacting magnetic excitons, the effective dimensionality is simply increased by 1; z = 1. It need not appear in this fashion, however, and in the examples of itinerant antiferromagnetism and clean and dirty itinerant ferromagnetism, one finds z = 2, 3, and 4, respectively, (ii) At finite temperatures, one finds that a classical statistical-mechanical description holds (and nonclassical exponents, for d < 4) very close to the critical value of the coupling U c , when ( U − U c ) / U c ≪ ( T / U c ) 2 / z z / 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429494116/afa305c7-7e11-4618-94aa-661e4acf5b1d/content/eq2970.tif"/> is therefore the quantum-to-classical crossover exponent.
At temperatures low compared with characteristic atomic frequencies, classical free energy functionals where the order parameter field ψ which depends only on position are inadequate to describe fluctuation phenomena; the noncommutativity of quantum‐mechanical operators leads to a ψ which also depends on time. We show how to generalize the renormalization group to such functionals. For phase transitions at T=0 caused by variation of a coupling parameter, the boundary between classical and nonclassical exponent behavior is at 4‐z, not 4 dimensions, where z≳1 depends on the dynamics of the system: z=1 for van Vleck ferromagnets, z=2 for itinerant antiferomagnets, z=3 for clean and z=4 for dirty itinerant ferromagnets. For small Tc there is a crossover region where the renormalization group equations go from their T=0 form to a Wilson form, implying a large precritical region where classical exponents are correct and a small true critical region. One also has an explicit construction of the Wilson free energy functional valid in the critical region.
We study weakly interacting Bose gases using the functional renormalization group with a hydrodynamic effective action. We use a scale-dependent parametrization of the boson fields that interpolates between a Cartesian representation at high momenta and an amplitude-phase one for low momenta. We apply this to Bose gases in two and three dimensions near the superfluid phase transition where they can be described by statistical O(2) models. We are able to give consistent physical descriptions of the infrared regime in both two and three dimensions. In particular, and in contrast to previous studies using the functional renormalization group, we find a stable superfluid phase at finite temperatures in two dimensions. We compare our results for the superfluid and boson densities with Monte-Carlo simulations, and we find they are in reasonable agreement.
Pairing and topological phases in cold atoms with long range interactions Chapter 1 Introduction The systematic study and classification of phase transitions has become a popular research area in physics over the past decades. The most familiar phenomena associated to the concept of phase transition are the ones giving rise to macroscopic changes in the properties of strongly correlated many-body systems due to thermal fluctuations. The most prominent examples of this kind of behaviour are the ferromagnetic-paramagnetic transition in lattice spin models and the liquid-gas transition. On the other hand, when dealing with quantum-mechanical systems changes of matter from one state to another can arise as the result of the variation of parameters other than temperature. In particular, when a given many-body quantum problem is studied at zero temperature, thereby suppressing thermal fluctuations, it is the interplay between kinetic energy and interactions that can cause the system to jump into different phases as the model parameters are varied. Phase transitions occurring at zero temperature are widely known as quantum phase transitions. In the framework of the physics of phase transitions, the role of dimensionality is a noticeable one. The major challenge in this perspective is the treatment of low dimensional systems, where standard mean-field treatments and ordinary perturbation theory are known to fail due to the enhancement of fluctuations. The goal of describing collective behaviours in reduced dimensionality is usually pursued in one-dimensional (1D) systems, which turn out to be simple enough to be studied in an effective way through both numerical and analytical techniques, while exhibiting a rich and deeply fascinating phenomenology. As a matter of fact, since particles confined to one dimension cannot avoid interactions with each other, the typical low energy excitations of the system are represented by collective density waves, which substitute nearly free quasiparticle excitations characterizing the behaviour of three-dimensional Fermi liquids in the description of the effects of the interplay between kinetic fluctuations and interactions. On the other hand, the tools employed in the activity of tackling 1D manybody problems are efficient and well developed. The analytical techniques aremainly based on the bosonization technique, which allows for the reformulation of strongly correlated 1D systems onto free bosonic theories, which can then be analyzed by means of standard path integral techniques. Meanwhile, the density matrix renormalization group (DMRG) algorithm represents the state-of-theart numerical machinery in the study of 1D physics, since it allows to efficiently extract the low energy properties of the model Hamiltonian and consequently characterize the corresponding phase diagram by means of the behaviour of properly chosen order parameters. As a final remark pointing out the relevance of the research effort in the direction of a better understanding of 1D many-body quantum systems, it is worth noticing that 1D systems do not represent only a playground for theorists, but have been experimentally realized by means of setups with cold atoms trapped in optical lattices, where the experimentalist is able to tune the model parameters by varying the characteristic features of the trapping laser beams and meanwhile has a comfortable access to the most relevant observables.