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arXiv:2109.00631 · Physical Review A

Stability of the Fulde-Ferrell-Larkin-Ovchinnikov states in anisotropic systems and critical behavior at thermal m -axial Lifshitz points

Piotr Zdybel, Mateusz Homenda, Andrzej Chlebicki, Paweł Jakubczyk

Published Dec 27, 2021• 8 citations
arXivHTMLPDF

Abstract

We revisit the question concerning the stability of nonuniform superfluid states of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type to thermal and quantum fluctuations. On general grounds, we argue that the mean-field phase diagram hosting a Lifshitz point cannot be stable to fluctuations for isotropic, continuum systems, at any temperature $T>0$ in any dimensionality $d<4$. In contrast, in layered unidirectional systems, the lower critical dimension for the onset of FFLO-type long-range order accompanied by a Lifshitz point at $T>0$ is $d=5/2$. In consequence, its occurrence is excluded in $d=2$, but not in $d=3$. We propose a relatively simple method, based on nonperturbative renormalization group, to compute the critical exponents of the thermal $m$-axial Lifshitz point continuously varying $m$, spatial dimensionality $d$, and the number of order parameter components, $N$. We point out the possibility of a robust, fine-tuning free occurrence of a quantum Lifshitz point in the phase diagram of imbalanced Fermi mixtures.

DOI
10.1103/physreva.104.063317

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We revisit the question concerning the stability of nonuniform superfluid states of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type to thermal and quantum fluctuations. On general grounds, we argue that the mean-field phase diagram hosting a Lifshitz point cannot be stable to fluctuations for isotropic, continuum systems, at any temperature T>0T>0T>0 in any dimensionality d<4d<4d<4. In contrast, in layered unidirectional systems, the lower critical dimension for the onset of FFLO-type long-range order accompanied by a Lifshitz point at T>0T>0T>0 is d=5/2d=5/2d=5/2. In consequence, its occurrence is excluded in d=2d=2d=2, but not in d=3d=3d=3. We propose a relatively simple method, based on nonperturbative renormalization group, to compute the critical exponents of the thermal mmm-axial Lifshitz point continuously varying mmm, spatial dimensionality ddd, and the number of order parameter components, NNN. We point out the possibility of a robust, fine-tuning free occurrence of a quantum Lifshitz point in the phase diagram of imbalanced Fermi mixtures.
Within the renormalization-group framework we study the stability of superfluid density wave states, known as Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phases, with respect to thermal order-parameter fluctuations in two- and three-dimensional ($d\ensuremath{\in}{2,3}$) systems. We analyze the renormalization-group flow of the relevant ordering wave vector $\stackrel{P\vec}{{Q}_{0}}$. The calculation indicates an instability of the FFLO-type states towards either a uniform superfluid or the normal state in $d\ensuremath{\in}{2,3}$ and T>0T>0T>0. In d=2d=2d=2 this is signaled by $\stackrel{P\vec}{{Q}_{0}}$ being renormalized towards zero, corresponding to the flow being attracted either to the usual Kosterlitz-Thouless fixed point or to the normal phase. We supplement a solution of the RG flow equations by a simple scaling argument, supporting the generality of the result. The tendency to reduce the magnitude of $\stackrel{P\vec}{{Q}_{0}}$ by thermal fluctuations persists in d=3d=3d=3, where the very presence of long-range order is immune to thermal fluctuations, but the effect of attracting $\stackrel{P\vec}{{Q}_{0}}$ towards zero by the flow remains observed at T>0T>0T>0.
In ultracold Fermi gases, the effect of spin imbalance on superfluidity has been the subject of intense study. One of the reasons for this is that spin imbalance frustrates the Bardeen-Cooper-Schrieffer (BCS) superfluid pairing mechanism, in which fermions in different spin states combine into Cooper pairs with zero momentum. In 1964, it was proposed that an exotic superfluid state called the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, in which the Cooper pairs have nonzero momentum, could exist in a spin-imbalanced Fermi gas. At the saddle-point (mean-field) level, it has been shown that the FFLO state only occupies a very small sliver in the ground-state phase diagram of a three-dimensional (3D) Fermi gas. However, a question that remains to be investigated is as follows: What is the influence of phase fluctuations on the FFLO state? In this work, we show that phase fluctuations only lead to relatively small quantitative corrections to the presence of the FFLO state in the saddle-point phase diagram of a 3D spin-imbalanced Fermi gas. Starting from the partition function of the system, we calculate the effective action within the path-integral adiabatic approximation. The action is then expanded up to second order in the fluctuation field around the saddle point, leading to the fluctuation free energy. Using this free energy, we calculate corrections due to phase fluctuations to the BCS-FFLO transition in the saddle-point phase diagram. At temperatures at which the FFLO state exists, we find only small corrections to the size of the FFLO area. Our results suggest that fluctuations of the phase of the FFLO order parameter, which can be interpreted as an oscillation of its momentum vector, do not cause an instability of the FFLO state with respect to the BCS state.
We investigate exotic paired states of spin-imbalanced Fermi gases in anisotropic lattices, tuning the dimension between one and three. We calculate the finite temperature phase diagram of the system using real-space dynamical mean-field theory in combination with the quantum Monte Carlo method. We find that regardless of the intermediate dimensions examined, the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state survives to reach about one third of the BCS critical temperature of the spin-density balanced case. We show how the gapless nature of the state found is reflected in the local spectral function. While the FFLO state is found at a wide range of polarizations at low temperatures across the dimensional crossover, with increasing temperature we find out strongly dimensionality-dependent melting characteristics of shell structures related to harmonic confinement. Moreover, we show that intermediate dimension can help to stabilize an extremely uniform finite temperature FFLO state despite the presence of harmonic confinement.
The question whether an imbalanced Fermi gas can accommodate polarized superfluidity has been the subject of intense study. One prominent candidate for a polarized superfluid state is the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state. In a one-dimensional (1D) Fermi gas, indirect evidence for this state has recently been found, but in the three-dimensional (3D) case, this exotic state has hitherto remained elusive. In this brief review, we study the influence of a 1D periodic potential on the FFLO state in an imbalanced 3D Fermi gas. We present a path-integral derivation of the free energy of the system, where the possibility of the FFLO state is included through the saddle-point approximation. Using the free energy, the phase diagram of the system, both at fixed chemical potentials and at fixed densities, is constructed, showing that the stability region of the FFLO state is significantly enlarged by the 1D potential. Subsequently, the influence of the potential on the FFLO pairing mechanism is investigated, leading to a qualitative understanding of the shape of the FFLO region in the different phase diagrams. We conclude by investigating the possibility of an FFLO state with a wave vector that lies skewed with respect to the direction along which the periodic potential varies. Our results provide an additional way to facilitate the experimental observation of the FFLO state in a 3D Fermi gas.
The ground-state phase diagram of attractively interacting Fermi gases in two dimensions with a population imbalance is investigated. We find the regime of stability for the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, in which pairing occurs at a finite wave vector, and determine the magnitude of the pairing amplitude $\mathrm{\ensuremath{\Delta}}$ and FFLO wave vector qqq in the ordered phase, finding that $\mathrm{\ensuremath{\Delta}}$ can be of the order of the two-body binding energy. Our results rely on a careful analysis of the zero-temperature gap equation for the FFLO state, which possesses nonanalyticities as a function of $\mathrm{\ensuremath{\Delta}}$ and qqq, invalidating a Ginzburg-Landau expansion in small $\mathrm{\ensuremath{\Delta}}$.
The exotic Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states have been actively searched for experimentally since the mean-field based FFLO theories were put forward half a century ago. Here, we investigate the stability of FFLO states in the presence of pairing fluctuations. We conclude that FFLO superfluids cannot exist in continuum in three and two dimensions, due to their intrinsic instability, associated with infinite quantum degeneracy of the pairs. These results address the absence of convincing experimental observations of FFLO phases in both condensed matter and in ultracold atomic Fermi gases with a population imbalance. We predict that the true ground state has a pair momentum distribution highly peaked on an entire constant energy surface.