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arXiv:1706.01524 · Physical Review A

Renormalization theory for the Fulde-Ferrell-Larkin-Ovchinnikov states at T>0

Paweł Jakubczyk

Published Jun 29, 2017• 20 citations
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Abstract

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 $T>0$. In $d=2$ 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 $d=3$, 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 $T>0$.

DOI
10.1103/physreva.95.063626

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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∈2,3d\ensuremath{\in}{2,3}d\ensuremath∈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∈2,3d\ensuremath{\in}{2,3}d\ensuremath∈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.
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.
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.
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 $\mathrm{\ensuremath{\Delta}}$.
We investigate, in three spatial dimensions, the transition from the normal state to the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid phases. We make use of a Fourier expansion for the order parameter and the Green's functions to handle the quasiclassical equations in the vicinity of the transition. We show that, below the tricritical point, the transition is always first order. We find that, at the transition, the higher Fourier components in the order parameter are always essentially negligible. Below the tricritical point we have the already known result that the order parameter has a spatial dependence which is essentially cos(q∙r)\mathrm{cos}(\mathbf{q}∙\mathbf{r})cos(q∙r). However, when the temperature is lowered the order parameter switches to a sum of two cosines, with equal weights and wave vectors with the same length, but orthogonal directions. Finally, by further lowering the temperature, and down to T=0T=0T=0, one finds another transition toward an order parameter that is the sum of three cosines with, again, equal weights and orthogonal directions. Hence the structure of the order parameter gets more complex as the temperature is lowered. On the other hand, the resulting critical temperatures are found to be only slightly higher than the ones corresponding to the standard second-order FFLO transition. We apply our results to the specific case of ultracold Fermi gases and show that the differences in atomic populations of the two hyperfine states involved in the BCS condensation display sizeable variations when one goes from the normal state to the superfluid FFLO phases, or from one FFLO phase to another. Experimentally this should allow us to clearly identify the various phase transitions.
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.
We explore the pairing properties of the one-dimensional attractive Hubbard model in the presence of finite spin polarization. The correlation exponents for the most important fluctuations are determined as a function of the density and the polarization. We find that in a system with spin population imbalance, Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-type pairing at wave vector $Q=\ensuremath{\mid}{k}_{F,\ensuremath{\uparrow}}\ensuremath{-}{k}_{F,\ensuremath{\downarrow}}\ensuremath{\mid}$ is always dominant and there is no Chandrasekhar-Clogston limit. We then investigate the case of weakly coupled one-dimensional (1D) systems and determine the region of stability of the 1D FFLO phase. This picture is corroborated by density-matrix-renormalization-group simulations of the spatial noise correlations in uniform and trapped systems, unambiguously revealing the presence of fermion pairs with nonzero momentum QQQ. This opens up an interesting possibility for experimental studies of FFLO states.