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arXiv:1305.6984 · Physical Review A

Stability criterion for superfluidity based on the density spectral function

Shohei Watabe, Yusuke S. Kato

Published Dec 6, 2013• 4 citations
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Abstract

We study a stability criterion hypothesis for superfluids expressed in terms of the local density spectral function ${\mathcal{I}}_{n}(\mathbf{r},\ensuremath{\omega})$ that is applicable to both homogeneous and inhomogeneous systems. We evaluate the local density spectral function in the presence of a one-dimensional repulsive or attractive external potential within Bogoliubov theory, using solutions for the tunneling problem. We also evaluate the local density spectral function using an orthogonal basis, and calculate the autocorrelation function ${C}_{n}(\mathbf{r},t)$. When superfluids in a $d$-dimensional system flow below a threshold, ${\mathcal{I}}_{n}(\mathbf{r},\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{d}$ holds in the low-energy regime and ${C}_{n}(\mathbf{r},t)\ensuremath{\propto}1/{t}^{d+1}$ holds in the long-time regime. However, when superfluids flow with the critical current, ${\mathcal{I}}_{n}(\mathbf{r},\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{\beta}}$ holds in the low-energy regime and ${C}_{n}(\mathbf{r},t)\ensuremath{\propto}1/{t}^{\ensuremath{\beta}+1}$ holds in the long-time regime with $\ensuremath{\beta}<d$. These results support the stability criterion hypothesis recently proposed.

DOI
10.1103/physreva.88.063612

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We study a stability criterion hypothesis for superfluids expressed in terms of the local density spectral function In(r,\ensuremathω){\mathcal{I}}_{n}(\mathbf{r},\ensuremath{\omega})In​(r,\ensuremathω) that is applicable to both homogeneous and inhomogeneous systems. We evaluate the local density spectral function in the presence of a one-dimensional repulsive or attractive external potential within Bogoliubov theory, using solutions for the tunneling problem. We also evaluate the local density spectral function using an orthogonal basis, and calculate the autocorrelation function Cn(r,t){C}_{n}(\mathbf{r},t)Cn​(r,t). When superfluids in a ddd-dimensional system flow below a threshold, In(r,\ensuremathω)\ensuremath∝\ensuremathωd{\mathcal{I}}_{n}(\mathbf{r},\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{d}In​(r,\ensuremathω)\ensuremath∝\ensuremathωd holds in the low-energy regime and Cn(r,t)\ensuremath∝1/td+1{C}_{n}(\mathbf{r},t)\ensuremath{\propto}1/{t}^{d+1}Cn​(r,t)\ensuremath∝1/td+1 holds in the long-time regime. However, when superfluids flow with the critical current, In(r,\ensuremathω)\ensuremath∝\ensuremathω\ensuremathβ{\mathcal{I}}_{n}(\mathbf{r},\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{\beta}}In​(r,\ensuremathω)\ensuremath∝\ensuremathω\ensuremathβ holds in the low-energy regime and Cn(r,t)\ensuremath∝1/t\ensuremathβ+1{C}_{n}(\mathbf{r},t)\ensuremath{\propto}1/{t}^{\ensuremath{\beta}+1}Cn​(r,t)\ensuremath∝1/t\ensuremathβ+1 holds in the long-time regime with \ensuremathβ<d\ensuremath{\beta}<d\ensuremathβ<d. These results support the stability criterion hypothesis recently proposed.
We propose a stability criterion of superfluids in condensed Bose-Einstein systems, which incorporates the spectral function or the autocorrelation function of the local density. Within the Gross-Pitaevskii-Bogoliubov theory, we demonstrate the validity of our criterion for the soliton-emission instability, with use of explicit forms of zero modes of the Bogoliubov equation and a dynamical scaling near the saddle-node bifurcation. We also show that the criterion is applicable to the Landau phonon instability and the Landau roton instability within the single-mode approximation.
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In the context of ultracold fermionic atoms with effective spin S confined to an elongated trap we study the one-dimensional gas interacting via an attractive δ-function potential using the Bethe ansatz solution. There are N = 2 S + 1 fundamental states: The particles can either be unpaired or clustered in bound states of 2, 3, ..., 2 S and 2S + 1 atoms. In a magnetic field, the rich ground state phase diagram consists of these N states and various mixed phases in which combinations of the fundamental states coexist. The phase diagram simplifies considerably in zero-field, where only bound states of N atoms can exist. Due to the harmonic confinement and within the local density approximation, the density profile of bound states decreases along the tube from the center of the trap to its boundaries. In an array of tubes with weak Josephson tunneling superfluid order may arise. In zero-field the response functions determining the superfluid and density wave order are calculated using conformal field theory and the exact Bethe ansatz solution. The response function for superfluidity consists of a power law with distance, while the correlation function for density waves is a power law of distance times a sinusoidal factor oscillating with distance with a period given by two times the Fermi momentum. For S = 1/2 superfluidity is a possibility for all densities and density waves can be excluded. For S ≥ 3/2 superfluidity may occur at low densities but at high densities it gives way to density waves. We discuss the scenario of phase separation where for S ≥ 3/2 the system has superfluid long-range order toward the trap boundaries and density waves at its center.
The dynamics of a confined fluid of Bose atoms is treated within the linear response regime, with a view to establishing a current-density functional formalism for an inhomogeneous superfluid state. After evaluating in full detail a simplified case of an external coupling to the density and phase of the condensate, the theory is extended to include the coupling to the total current density. The Kohn-Sham response functions of the condensate and all the exchange-correlation kernels for the superfluid are introduced from the microscopic equations of motion and are expressed in a physically transparent way through functional derivatives of correlation functions. A microscopic formula for the superfluid density is derived and used to introduce a generalized hydrodynamic approach for a weakly inhomogeneous two-fluid model in isothermal conditions. Local-density expressions are thereby derived for the velocities of first and second sound in the weakly inhomogeneous superfluid and for visco-elastic functions describing the transition from the hydrodynamic to the collisionless regime. Landau's hydrodynamic theory and known results in Green's functions language are recovered in the limiting case of a homogeneous superfluid.
We investigate the problem of an ultracold atomic gas in the superfluid phase flowing in the presence of a potential barrier or a periodic potential. We use a hydrodynamic scheme in the local density approximation (LDA) to obtain an analytic expression for the critical current as a function of the barrier height or the lattice intensity, which applies to both Bose and Fermi superfluids. In this scheme, the stationary flow becomes energetically unstable when the local superfluid velocity is equal to the local sound velocity at the point where the external potential is maximum. We compare this prediction with the results of the numerical solutions of the Gross-Pitaevskii and Bogoliubov-de Gennes equations. We discuss the role of long-wavelength excitations in determining the critical velocity. Our results allow one to identify the different regimes of superfluid flow, namely, the LDA hydrodynamic regime, the regime of quantum effects beyond LDA for weak barriers and the regime of tunneling between weakly coupled superfluids for strong barriers. We finally discuss the relevance of these results in the context of current experiments with ultracold gases.
Abstract A density-functional theory for superfluid systems is developed in the framework of the functional renormalization group based on the effective action formalism. We introduce the effective action for the particle-number and non-local pairing densities and demonstrate that the Hohenberg–Kohn theorem for superfluid systems is established in terms of the effective action. The flow equation for the effective action is then derived, where the flow parameter runs from 000 to 111, corresponding to the non-interacting and interacting systems. From the flow equation and the variational equation that the equilibrium density satisfies, we obtain the exact expression for the Kohn–Sham potential generalized to include the pairing potentials. The resultant Kohn–Sham potential has a nice feature in that it expresses the microscopic formulae of the external, Hartree, pairing and exchange–correlation terms separately. It is shown that our Kohn–Sham potential gives the ground-state energy of the Hartree–Fock–Bogoliubov theory by neglecting the correlations. An advantage of our exact formalism lies in the fact that it provides ways to improve the correlation part systematically.