We describe the effects of disorder on the critical temperature of s-wave superfluids from the BCS to the BEC regime, with direct application to ultracold Fermi atoms. In the BCS regime the pair breaking and phase coherence temperature scales are essentially the same allowing strong correlations between the amplitude and phase of the order parameter. As non-pair breaking disorder is introduced the largely overlapping Cooper pairs conspire to maintain phase coherence such that the critical temperature remains essentially unchanged. However, in the BEC regime the pair breaking and phase coherence temperature scales are very different such that non-pair breaking disorder can affect dramatically phase coherence, and thus the critical temperature, without the requirement of breaking tightly-bound fermion pairs simultaneously. Finally, we find that the superfluid is more robust against weak disorder in the intermediate region between the two regimes.
Deep Dive into Disorder effects during the evolution from BCS to BEC superfluidity.
We describe the effects of disorder on the critical temperature of s-wave superfluids from the BCS to the BEC regime, with direct application to ultracold Fermi atoms. In the BCS regime the pair breaking and phase coherence temperature scales are essentially the same allowing strong correlations between the amplitude and phase of the order parameter. As non-pair breaking disorder is introduced the largely overlapping Cooper pairs conspire to maintain phase coherence such that the critical temperature remains essentially unchanged. However, in the BEC regime the pair breaking and phase coherence temperature scales are very different such that non-pair breaking disorder can affect dramatically phase coherence, and thus the critical temperature, without the requirement of breaking tightly-bound fermion pairs simultaneously. Finally, we find that the superfluid is more robust against weak disorder in the intermediate region between the two regimes.
Ultracold atoms are special systems for studying superfluid phases of fermions or bosons at very low temperatures, because of unprecedented tunability. In particular, ultracold fermions with tunable interactions were used to study experimentally the so-called BCS-to-BEC evolution, and population imbalanced systems. In addition, there are other interesting directions to be pursued, including studies of the BCS-to-BEC evolution in optical lattices [1,2], and the effects of disorder during the BCS-to-BEC evolution, which would allow the very important study of the simultaneous effects of interactions and disorder at zero [3] and finite temperatures [4].
In ordinary condensed matter (CM) systems the control of interactions is not possible, and the control of disorder is very limited, because the disorder potential is not known and can not be changed at the turn of a knob. Thus, in standard CM the disorder is usually described in terms of defects or impurities, whose positions in the solid are assumed to be random. In ultracold atoms it is now possible to create controlled disorder using laser speckles or lasers with incommensurate wavelengths, which were used to study the phenomenon of Anderson localization in ultracold Bose atoms [5,6], but that could also be used to study disorder effects in ultracold fermions.
Thus, here, we describe the finite temperature phase diagram of three dimensional (3D) s-wave Fermi superfluids from the BCS to the BEC limit as a function of disorder, which is independent of the hyperfine states of the atoms and is created by a Gaussian-correlated laser speckle potential. Our main results are as follows. First, in the BCS limit the amplitude and phase of the order parameter are strongly coupled, such that pair breaking and loss of phase coherence occur simultaneously. In this case, the critical temperature is essentially unaffected by weak disorder, since the disorder potential is not pairbreaking and phase coherence is not easily destroyed in accordance with Anderson’s theorem [7]. Second, in the BEC limit the breaking of local pairs and the loss of phase coherence occur at very different temperature scales. In this case, the critical temperature is strongly affected by weak disorder, since phase coherence is more easily destroyed without the need to break local pairs simultaneously, and Anderson’s theorem does not apply. Third, we find that superfluidity is more robust to disorder in the intermediate region between the BCS and BEC regimes.
To investigate the physics described above, we start with the real space Hamiltonian (h = 1) density for three dimensional s-wave superfluids
where
represents a term containing the interaction potential V (x, x ′ ) = -g δ(xx ′ ), and ψ † σ (x) represents the creation of fermions with mass m and hyperfine state (spin) σ. In addition, V dis (x) represents the disorder potential, and µ, the chemical potential. We choose V dis (x) to be independent of the hyperfine state, a choice that can be easily relaxed.
Although there are many ways to introduce disorder, we will make a particular choice that the disorder potential is governed by a Gaussian distribution
To derive the effective action for a fixed configuration of disorder, we define the local chemical potential µ(x) = µ -V dis (x) and follow the functional integral formulation of the evolution from BCS to BEC superfluidity [8] to obtain S eff = S 0 + S G , where S 0 is the action of unbound fermions in the presence of weak disorder and is given by
with ξ(k, x) = ǫ k -µ(x), and ǫ k = k 2 /2m. The second contribution to S eff corresponds to Gaussian pairing fluctuations
where ∆(q) is the pairing field, and
is the pair correlation function in the presence of disorder, where C = -(mV /4πa s ) + k (2ǫ k ) -1 , q = (q, iq ℓ ) represents the four-momentum, the function
] describes the occupation of fermions with energy ξ 1 = ξ(k-q/2), and the function
] describes the occupation of fermions with energy ξ 2 = ξ(k + q/2).
The description above corresponds to a semiclassical approximation which is valid for weak disorder potentials. However, quenched (as opposed to annealed) disorder is characterized by a static disorder potential and the necessity to average the thermodynamic potential Ω(V dis ) rather than the partition function [9]. The configurationally averaged thermodynamic potential is Ω av (κ) = Ω(V dis ) , and depends on disorder via the parameter κ. The thermodynamic potential for a fixed configuration is Ω(V dis ) = -T ln Z(V dis ) when expressed in terms of the partition function
is the partition function for unbound fermions, and
is the partition function for the pairing field. Thus, Ω(
where the unbound fermion thermodynamic potential is
Expanding in V dis and taking the configurational average leads to Ω 0,av (κ) = T S 0 [V dis ] = Ω 0,av (0) + ∆Ω 0,av (κ) for the thermodynamic potential of unbound fermions, and Ω G,av (κ) = Ω G,av (0) + ∆Ω G,av (κ) for the thermodynamic poten
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