We demonstrate the existence of generalized Aubry-André self-duality in a class of non-Hermitian quasiperiodic lattices with complex potentials. From the self-duality relations, the analytical expression of mobility edges is derived. Compared to Hermitian systems, mobility edges in non-Hermitian ones not only separate localized from extended states but also indicate the coexistence of complex and real eigenenergies, making possible a topological characterization of mobility edges. An experimental scheme, based on optical pulse propagation in synthetic photonic mesh lattices, is suggested to implement a non-Hermitian quasicrystal displaying mobility edges.
We address the importance of symmetry and symmetry breaking on linear response theories of fermionic BCS superfluids. The linear theory of a noninteracting Fermi gas is reviewed and several consistency constraints are verified. The challenge to formulate linear response theories of BCS superfluids consistent with density and spin conservation laws comes from the presence of a broken U(1)EM symmetry associated with electromagnetism (EM) and we discuss two routes for circumventing this. The first route follows Nambu's integral-equation approach for the EM vertex function, but this method is not specific for BCS superfluids. We focus on the second route based on a consistent-fluctuation-of-the order-parameter (CFOP) approach where the gauge transformation and the fluctuations of the order parameter are treated on equal footing. The CFOP approach allows one to explicitly verify several important constraints: The EM vertex satisfies not only a Ward identity which guarantees charge conservation but also a Q-limit Ward identity associated with the compressibility sum rule. In contrast, the spin degrees of freedom associated with another U(1)z symmetry are not affected by the Cooper-pair condensation that breaks only the U(1)EM symmetry. As a consequence the collective modes from the fluctuations of the order parameter only couple to the density response function but decouple from the spin response function, which reflects the different fates of the two U(1) symmetries in the superfluid phase. Our formulation lays the ground work for application to more general theories of BCS-Bose Einstein Condensation (BEC) crossover both above and below Tc. PACS numbers: 74.20.Fg,03.75.Ss Here S ↑,↓ = ±1 as in (3). The external field A µ has a different physical meaning from that in the density channel. Since n S is the z component of spin density, the field φ coupled to n S corresponds to B z . J S is the difference between the spin-up and the spin-down currents, i.e., the magnetization current. Therefore the field which couples to it is the magnetizing field A ≡ m. The effective external vector field is thus A µ ≡ (B z , m). The resulting Hamiltonian then describes a generalized spin-magnetic field interaction. The Noether current for the global U(1) z symmetry is J µ S = (n S , J S ), which also satisfies the conservation law ∂ µ J µ S = 0.
This article presents a comparison of two finite-temperature BCS-Bose-Einstein condensation (BEC) crossover theories above the transition temperature: Nozieres-Schmitt-Rink (NSR) theory and finite-T extended BCS-Leggett theory. The comparison is cast in the form of numerical studies of the behavior of the fermionic spectral function both theoretically and as constrained by (primarily) radio frequency (rf) experiments. Both theories include pair fluctuations and exhibit pseudogap effects, although the nature of this pseudogap is very different. The pseudogap in finite-T extended BCS-Leggett theory is found to follow a BCS-like dispersion which, in turn, is associated with a broadened BCS-like self-energy, rather more similar to what is observed in high-temperature superconductors (albeit, for a d-wave case). The fermionic quasiparticle dispersion is different in NSR theory and the damping is considerably larger. We argue that the two theories are appropriate in different temperature regimes with the BCS-Leggett approach being more suitable nearer to condensation. There should, in effect, be little difference at higher T as the pseudogap becomes weaker and where the simplifying approximations used in the BCS-Leggett approach break down. On the basis of momentum-integrated rf studies of unpolarized gases, it would be difficult to distinguish which theory is the better one. A full comparison for polarized gases is not possible since it is claimed that there are inconsistencies in the NSR approach (not found in the BCS-Leggett scheme). Future experiments along the lines of momentum-resolved experiments look to be very promising in distinguishing the two theories. A. Analysis of different crossover theories
We show how in ultracold Fermi gases the difference between the finite temperature T structure factors, called S_(ω,q), associated with spin and density, reflects coherent order at all ω, q, k(F)a, and T. This observation can be exploited in two photon Bragg scattering experiments on gases which are subject to variable attractive interactions. Our calculations incorporate spin and particle number conservation laws which lead to compatibility at general T with two f-sum rules. Because of its generality a measurement of S_(ω,q) can be a qualitative, direct, in situ approach for establishing superfluid order.
Recent experiments on the shear viscosity η in a unitary Fermi gas fail to see the theoretically predicted upturn in η at the lower T. In this Letter, we compute η in a fashion which is demonstrably consistent with conservation laws and, in the process, provide an understanding of recent experiments. We show that this disagreement with prior theories cannot be readily attributed to the trap, since (via edge effects) trap-averaged viscosities will be larger than their homogeneous counterparts. The small values of η we find can be simply understood; they reflect the fact that the Goldstone bosons (phonons) do not couple to transverse probes such as η, and fermionic excitations, which determine the viscosity, are necessarily absent in the ground state.
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