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arXiv:cond-mat/0609106 · Physical Review A

Renormalization-group fixed points, universal phase diagram, and 1∕N expansion for quantum liquids with interactions near the unitarity limit

Predrag Nikolić, Subir Sachdev A

Published Mar 16, 2007• 181 citations
arXivHTMLPDF

Abstract

It has long been known that particles with short-range repulsive interactions in spatial dimension $d=1$ form universal quantum liquids in the low density limit: all properties can be related to those of the spinless free Fermi gas. Previous renormalization-group (RG) analyses demonstrated that this universality is described by a RG fixed point, infrared stable for $d<2$, of the zero density gas. We show that for $d>2$ the same fixed point describes the universal properties of particles with short-range attractive interactions near a Feshbach resonance; the fixed point is now infrared unstable, and the relevant perturbation is the detuning of the resonance. Some exponents are determined exactly, and the same expansion in powers of $(d\ensuremath{-}2)$ applies for scaling functions for $d<2$ and $d>2$. A separate exact RG analysis of a field theory of the particles coupled to ``molecules'' finds an alternative description of the same fixed point, with identical exponents; this approach yields a $(4\ensuremath{-}d)$ expansion which agrees with the recent results of Nishida and Son [Phys. Rev. Lett. 97, 050403 (2006)]. The existence of the RG fixed point implies a universal phase diagram as a function of density, temperature, population imbalance, and detuning; in particular, this applies to the crossover between the Bose-Einstein condensate (BEC) and Bardeen-Cooper-Schrieffer (BCS) superfluid of $s$-wave paired fermions. Our results open the way towards computation of these universal properties using the standard field-theoretic techniques of critical phenomena, along with a systematic analysis of corrections to universality. We also propose a $1∕N$ expansion [based upon models with $\mathrm{Sp}(2N)$ symmetry] of the fixed point and its vicinity, and use it to obtain results for the phase diagram.

DOI
10.1103/physreva.75.033608

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We explore the universal properties of interacting fermionic lattice systems, mostly focusing on the development of pairing correlations from attractive interactions. Using renormalization group, we identify a large number of fixed points and show that they correspond to resonant scattering in multiple channels. Pairing resonances in finite-density band insulators occur between quasiparticles and quasiholes living at different symmetry-related wave vectors in the Brillouin zone. This allows a BCS-BEC crossover interpretation of both Cooper and particle-hole pairing. We show that, in two dimensions, the runaway flows of relevant attractive interactions lead to charged-boson-dominated low-energy dynamics in the insulating states and superfluid transitions in the bosonic mean-field or XY\mathit{XY}XY universality classes. Analogous phenomena in higher dimensions are restricted to the strong-coupling limit, while, at weak couplings, the transition is in the pair-breaking BCS class. The models discussed here can be realized with ultracold gases of alkali atoms tuned to a broad Feshbach resonance in an optical lattice, enabling experimental studies of pairing correlations in insulators, especially in their universal regimes. In turn, these simple and tractable models capture the emergence of fluctuation-driven superconducting transitions in fermionic systems, which is of interest in the context of high-temperature superconductors.
We consider a low-density system of Bose particles in general dimensionality interacting through a potential having a short-range part and a long-range tail that falls off like 1/rs{\mathit{r}}^{\mathit{s}}rs, focusing mainly on the special case s=2. We use a renormalization-group approach for the computation of the ground-state properties, which we argue is asymptotically exact in the dilute limit. The strength of the long-range part determines fixed-point values for the amplitude of the short-range part; several phases can occur, depending on the relative strengths of the short-range part and the fixed-point amplitudes. For the case in which the physics is determined by the flow away from the unstable fixed point (or when there are no fixed points at all), there results a collapsed state. When the flow is towards the stable fixed point, the outcome depends on whether the fixed-point value is negative or positive: In the former case we predict a kind of bound state, which is characterized by nontrivial correlations, even though the particles are not pairwise bound; in the latter case, a density expansion for the ground-state properties is given. The leading term of this expansion gives an exact result (a generalization of the one-dimensional result previously given by Sutherland) whenever the renormalization flow ends at the stable fixed point (including in particular the case that the flow starts at the fixed point). For the case that the renormalization flow stops before reaching the fixed point, the expansion parameter is proportional to the density raised to a nonuniversal power. The case s>2 can be reduced to the treatment of purely short-range interactions, whereas the repulsive case of s2 cannot be handled without additionally introducing a background of opposite charge. The case of attraction for s2 always leads to a collapse.
The stability or lack thereof of nonrelativistic fermionic systems to interactions is studied within the renormalization-group (RG) framework, in close analogy with the study of critical phenomena using ${\ensuremath{\varphi}}^{4}$ scalar field theory. A brief introduction to ${\ensuremath{\varphi}}^{4}$ theory in four dimensions and the path-integral formulation for fermions is given before turning to the problem at hand. As for the latter, the following procedure is used. First, the modes on either side of the Fermi surface within a cutoff $\ensuremath{\Lambda}$ are chosen for study, in analogy with the modes near the origin in ${\ensuremath{\varphi}}^{4}$ theory, and a path integral is written to describe them. Next, an RG transformation that eliminates a part of these modes, but preserves the action of the noninteracting system, is identified. Finally the possible perturbations of this free-field fixed point are classified as relevant, irrelevant or marginal. A d=1d=1d=1 warmup calculation involving a system of fermions shows how, in contrast to mean-field theory, which predicts a charge-density wave for arbitrarily weak repulsion, and superconductivity for arbitrarily weak attraction, the renormalization-group approach correctly yields a scale-invariant system (Luttinger liquid) by taking into account both instabilities. Application of the renormalization group in d=2and3d=2 \mathrm{and} 3d=2and3, for rotationally invariant Fermi surfaces, automatically leads to Landau's Fermi-liquid theory, which appears as a fixed point characterized by an effective mass and a Landau function FFF, with the only relevant perturbations being of the superconducting (BCS) type. The functional flow equations for the BCS couplings are derived and separated into an infinite number of flows, one for each angular momentum. It is shown that similar results hold for rotationally noninvariant (but time-reversal-invariant) Fermi surfaces also, with obvious loss of rotational invariance in the parametrization of the fixed-point interactions. A study of a nested Fermi surface shows an additional relevant flow leading to charge-density-wave formation. It is pointed out that, for small $\frac{\ensuremath{\Lambda}}{{K}_{F}}$, a 1N\frac{1}{N}N1​ expansion emerges, with $N=\frac{{K}_{F}}{\ensuremath{\Lambda}}$, which explains why one is able to solve the narrow-cutoff theory. The search for non-Fermi liquids in d=2d=2d=2 using the RG is discussed. Bringing a variety of phenomena (Landau theory, charge-density waves, BCS instability, nesting, etc.) under the one unifying principle of the RG not only allows us to better understand and unify them, but also paves the way for generalizations and extensions. The article is pedagogical in nature and is expected to be accessible to any serious graduate student. On the other hand, its survey of the vast literature is mostly limited to the RG approach.
The physics of interacting Fermi systems is extremely sensitive to the energy scale. Of particular interest is the low energy regime where correlation induced collective behavior emerges. The theory of interacting Fermi systems is confronted with the occurrence of very different phenomena along a continuum of scales calling for methods capable of computing physical observables as a function of energy scale. In this thesis, we perform a comprehensive renormalization group analysis of two and three-dimensional Fermi systems at low and zero temperature. We examine systems with spontaneous symmetry-breaking and quantum critical behavior by deriving and solving flow equations within the functional renormalization group framework. We extend the Hertz-Millis theory of quantumphase transitions in itinerant fermion systems to phases with discrete and continuous symmetry-breaking, and to quantum critical points where the zero temperature theory is associated with a non-Gaussian fixed point. The order parameter is implemented by a bosonic Hubbard-Stratonovich field, which - for continuous symmetry-breaking - splits into two components corresponding to longitudinal and transversal Goldstone fluctuations. We compute the finite temperature phase boundary near the quantum critical point explicitly including non-Gaussian fluctuations. We then set up a coupled fermion-boson renormalization group theory that captures the mutual interplay of gapless fermions with massless order parameter fluctuations when approaching a quantum critical point. As a first application, we compute the complete set of quantum critical exponents at the semimetal-to-superfluid quantum phase transition of attractively interacting Dirac fermions in two dimensions. Both, the order parameter propagator and the fermion propagator become nonanalytic functions of momenta destroying the Fermi liquid behavior. We finally compute the effects of quantum fluctuations in the superfluid ground state of an attractively interacting Fermi system, employing the attractive Hubbard model as a prototype. The flow equations capture the influence of longitudinal and Goldstone order parameter fluctuations on non-universal quantities such as the fermionic gap and the fermion-boson vertex, as well as the exact universal infrared asymptotics present in every fermionic superfluid.
Strongly interacting Fermi gas systems play an important role in many areas of physics. Their systematic analytical study has been a difficult theoretical endeavour as it is outside the reach of ordinary perturbation theory. Much progress has been made with the development of the large-N expansion, where a fermion is generalized to have N (or 2N) components and the Lagrangian’s global symmetry is extended to the unitary group U(N) (or the compact symplectic group USp(2N)). The chosen symmetry group dictates the system’s properties as U(N) enhances the fermion-hole interaction channel and suppresses the fermion-fermion channel, whereas USp(2N) treats both channels equally by allowing two independent interaction terms in the Lagrangian density, the density-density interaction which enhances the fermion-hole channel and the pairing interaction which enhances the fermion-fermion channel. Literature has been either only focusing on the pairing interaction term or including both but restricting their respective coupling constants to be the same. The most general renormalizable quantum field theory (QFT) compatible with a global USp(2N)-symmetry contains both interaction terms with independent, dimensionless coupling constants. By studying such a system at zero temperature, we want to understand how the two interaction terms influence each other by independently tuning the coupling constants from the attractive to the repulsive regime. To this end, we calculate the ground state energy density to leading order in the large-N expansion in four different cases which are classified by the coupling constant regime and map out the phase diagram. We find that the density-density interaction renders the theory unstable if its coupling constant g₁ is tuned above 2π and that the Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensate (BEC) crossover extends to 𝜇̃ /(ϵB) = −g₁/(2π) for g₁ < 2π, where ϵB is the Cooper pair binding energy in vacuum and 𝜇̃ the renormalized chemical potential. Furthermore, we find an exotic phase at the phase boundary (g₁= 2π, 𝜇̃ = −ϵB ) resulting from a flat direction in the leading-order effective potential, where the degenerate ground states span the entire BCS–BEC crossover.
The large distance behaviour, near the critical temperature, of continuous phase transitions with short range interactions can be reproduced by an effective local field theory. Within this framework, the universal properties of the large distance can be determined by finding fixed points of general, functional renormalization group (RG) equations (Wilson, Wegner). For dimension D > 4, the large scale behavior is governed by the Gaussian fixed point (related to mean field theory), while for D < 4 the Gaussian fixed point is unstable. Near D = 4, the RG equations can be solved in the form of an ǫ = 4 − D expansion, a new stable fixed point is found and this leads to the famous Wilson-Fisher expansion for critical exponents and other universal quantities. Moreover, one realizes that, in this framework, the effective field theory can be reduced to a renormalizable field theory and, at leading order in the critical domain, the much simpler quantum field theory renormalization group can be used. A number of physicists, including the Saclay group (Brezin, Le Guillou, Zinn-Justin) have developed sophisticated field theory techniques, to prove general scaling laws and to calculate exponents to higher orders [1].