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arXiv:1507.07542 · Physical review. B./Physical review. B

Finite-temperature quantum fluctuations in two-dimensional Fermi superfluids

Giacomo Bighin, Luca Salasnich

Published Jan 28, 2016• 42 citations
arXivHTMLPDF

Abstract

In two-dimensional systems with a continuous symmetry, the Mermin-Wagner-Hohenberg theorem precludes spontaneous symmetry breaking and condensation at finite temperature. The Berezinskii-Kosterlitz-Thouless critical temperature marks the transition from a superfluid phase characterized by quasicondensation and algebraic long-range order, to a normal phase in which vortex proliferation completely destroys superfluidity. As opposed to conventional off-diagonal long-range order typical of three-dimensional superfluid systems, algebraic long-range order is driven by quantum and thermal fluctuations strongly enhanced in reduced dimensionality. Motivated by this unique scenario and by the very recent experimental realization of trapped quasi-two-dimensional fermionic clouds, we include one-loop Gaussian fluctuations in the theoretical description of resonant Fermi superfluids in two dimensions demonstrating that first sound, second sound, and also critical temperature are strongly renormalized, away from their mean-field values. In particular, we prove that in the intermediate- and strong-coupling regimes, these quantities are radically different when Gaussian fluctuations are taken into account. Our one-loop theory shows good agreement with very recent experimental data on the Berezinskii-Kosterlitz-Thouless critical temperature [Phys. Rev. Lett. 115, 010401 (2015)] and on the first sound velocity, giving predictions for the second sound as a function of interaction strength and temperature that are open for experimental verification.

DOI
10.1103/physrevb.93.014519

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In two-dimensional systems with a continuous symmetry, the Mermin-Wagner-Hohenberg theorem precludes spontaneous symmetry breaking and condensation at finite temperature. The Berezinskii-Kosterlitz-Thouless critical temperature marks the transition from a superfluid phase characterized by quasicondensation and algebraic long-range order, to a normal phase in which vortex proliferation completely destroys superfluidity. As opposed to conventional off-diagonal long-range order typical of three-dimensional superfluid systems, algebraic long-range order is driven by quantum and thermal fluctuations strongly enhanced in reduced dimensionality. Motivated by this unique scenario and by the very recent experimental realization of trapped quasi-two-dimensional fermionic clouds, we include one-loop Gaussian fluctuations in the theoretical description of resonant Fermi superfluids in two dimensions demonstrating that first sound, second sound, and also critical temperature are strongly renormalized, away from their mean-field values. In particular, we prove that in the intermediate- and strong-coupling regimes, these quantities are radically different when Gaussian fluctuations are taken into account. Our one-loop theory shows good agreement with very recent experimental data on the Berezinskii-Kosterlitz-Thouless critical temperature [Phys. Rev. Lett. 115, 010401 (2015)] and on the first sound velocity, giving predictions for the second sound as a function of interaction strength and temperature that are open for experimental verification.
The role of fluctuations is enhanced in lower dimensionality systems: in a two dimensions offdiagonal long-range order is destroyed by the fluctuations at any finite temperature, drastically modifying the critical properties with respect to the three-dimensional counterpart.Recently two-dimensional systems of interacting fermions have been the subject of Montecarlo studies and experimental investigations, in particular an ultracold gas of attractive fermions with a widely tunable interaction due to a Feshbach resonance has been realized and the Berezinskii-Kosterlitz-Thouless transition has been observed.The present work deals with the theoretical description of an ultracold Fermi gas: we discuss the role of Gaussian fluctuations of the order parameter in the equation of state, in particular we take into account the first sound velocity, showing that the inclusion of order parameter fluctuations is needed in order to get the correct composite-boson limit in the strong-coupling regime.The theory is also compared with experimental data.Finally we focus on the superfluid density in the weak-coupling, intermediate and strong-coupling regimes at finite temperature, through which the Berezinskii-Kosterlitz-Thouless critical temperature is obtained.
We review our theoretical results of the sound propagation in two-dimensional (2D) systems of ultracold fermionic and bosonic atoms. In the superfluid phase, characterized by the spontaneous symmetry breaking of the U(1) symmetry, there is the coexistence of first and second sound. In the case of weakly-interacting repulsive bosons, we model the recent measurements of the sound velocities of 39K atoms in 2D obtained in the weakly-interacting regime and around the Berezinskii–Kosterlitz–Thouless (BKT) superfluid-to-normal transition temperature. In particular, we perform a quite accurate computation of the superfluid density and show that it is reasonably consistent with the experimental results. For superfluid attractive fermions, we calculate the first and second sound velocities across the whole BCS-BEC crossover. In the low-temperature regime, we reproduce the recent measurements of first-sound speed with 6Li atoms. We also predict that there is mixing between sound modes only in the finite-temperature BEC regime.
Abstract The physics of two-dimensional (2D) quantum gases can be revealed in strongly confining optical lattices. Upon cooling, 2D bosonic quantum gases as well as Fermi gases where pairing is present, become superfluid. The superfluid-to-normal transition is no longer governed by the presence or absence of a condensate, but by the Berezinskii–Kosterlitz–Thouless (BKT) mechanism: above a critical temperature, vortices and antivortices proliferate and destroy phase coherence. We investigate the BKT transition for superfluid 2D Fermi gases in the whole range of the BCS–BEC crossover, from weakly bound Cooper pairing (the BCS state), up to strongly bound molecules (the BEC state). Using a path-integral description, we then focus on the case of imbalanced gases: when the number of ‘spin-up’ and ‘spin-down’ fermions that form the pair is no longer equal. When an excess of one spin species exists, pairing is frustrated and the vortex energetics is strongly affected, influencing the KT mechanism. In the present work we are concentrated on the effect of both phase and amplitude fluctuations on phase diagrams of the fermion system. The amplitude fluctuations only slightly influence the BKT phase transition temperature. However, they lead to a substantial modification of the complete phase diagram for the Fermi gas in 2D with respect to that obtained taking into account only phase fluctuations.
This paper addresses the transition from the normal to the superfluid state in strongly correlated two dimensional fermionic superconductors and Fermi gases. We arrive at the Berezinskii-Kosterlitz-Thouless (BKT) temperature TBKTT_{\text{BKT}}TBKT​ as a function of \emph{attractive} pairing strength by associating it with the onset of quasi-condensation in the normal phase. Our approach builds on a criterion for determining the BKT transition temperature for atomic gases which is based on a well established Quantum Monte Carlo analysis of the phase space density. This latter quantity, when derived from BCS-BEC crossover theory for fermions, leads to non-monotonic behavior for TBKTT_{\text{BKT}}TBKT​ as a function of the attractive interaction or inverse scattering length. In Fermi gases, this implies a robust superconducting dome followed by a long tail from the flat BEC asymptote, rather similar to what is observed experimentally. For lattice systems we find that TBKTT_{\text{BKT}}TBKT​ has an absolute maximum of the order of 0.1EF0.1 E_F0.1EF​. We discuss how our results compare with those derived from the Nelson Kosterlitz criterion based on the mean field superfluid density and the approach to the transition from below. While there is agreement in the strict mean-field BCS regime at weak coupling, we find that at moderate pairing strength bosonic excitations cause a substantial increase in TBKTT_\text{BKT}TBKT​ followed by an often dramatic decrease before the system enters the BEC regime.
In local theories with short-range interactions, continuous symmetries cannot be spontaneously broken in two spatial dimensions at nonzero temperature. U(1)\mathrm{U}(1)U(1)-symmetric models instead admit a Berezinskii--Kosterlitz--Thouless (BKT) phase with quasi-long-range order and nonzero phase stiffness, which is normally destroyed at high temperatures by vortex proliferation. Here, we show that a BKT phase, and with it superfluidity, can persist to arbitrarily high temperatures. Using the large-NNN expansion, we study a (2+1)(2+1)(2+1)-dimensional quantum field theory in which a complex scalar is coupled to the critical O(N)\mathrm{O}(N)O(N) model. At the biconical fixed point, where the inter-sector coupling is negative, the thermal effective potential has a circle of minima with nonzero U(1)\mathrm{U}(1)U(1) amplitude. Integrating out the gapped fluctuations yields an effective theory for the compact phase mode at distances large compared to 1/T1/T1/T. Since temperature is the only scale at the fixed point, the phase stiffness grows linearly with temperature, with a proportionality factor that scales linearly with NNN and is fixed entirely by the fixed point. This renders vortex operators irrelevant for sufficiently large NNN. We further show that the vortex fugacity is exponentially small in NNN.
Using the Gaussian pair fluctuation theory, we investigate quantum fluctuations of a strongly interacting two-dimensional chiral ppp-wave Fermi superfluid at the transition from a Bose-Einstein condensate (BEC) to a topologically nontrivial Bardeen-Cooper-Schrieffer superfluid. Near the topological phase transition at zero chemical potential, $\ensuremath{\mu}=0$, we observe that quantum fluctuations strongly renormalize the zero-temperature equations of state, sound velocity, pair-breaking velocity, and Berezinskii-Kosterlitz-Thouless (BKT) critical temperature of the Fermi superfluid, all of which can be nonanalytic functions of the interaction strength. The indication of nonanalyticity is particularly evident in the BKT critical temperature, which also exhibits a pronounced peak near the topological phase transition. Across the transition and towards the BEC limit we find that the system quickly becomes a trivial interacting Bose liquid, whose properties are less dependent on the interparticle interaction. The qualitative behavior of composite bosons in the BEC limit remains to be understood.