Revisiting the critical velocity of a clean one-dimensional superconductor

# Revisiting the critical velocity of a clean one-dimensional superconductor

Tzu-Chieh Wei Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada [    Paul M. Goldbart Department of Physics, Institute for Condensed Matter Theory, and Federick Seitz Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, U.S.A.
April 15, 2009
###### Abstract

We revisit the problem of the critical velocity of a clean one-dimensional superconductor. At the level of mean-field theory, we find that the zero-temperature value of the critical velocity—the uniform velocity of the superfluid condensate at which the superconducting state becomes unstable—is a factor of smaller than the Landau critical velocity. This is in contrast to a prior finding, which held that the critical velocity is equal to the Landau critical velocity. The smaller value of the critical velocity, which our analysis yields, is the result of a pre-emptive Clogston-Chandrasekhar–like discontinuous phase transition, and is an analog of the threshold value of the uniform exchange-field of a superconductor, previously investigated by Sarma and by Maki and Tsuneto. We also consider the impact of nonzero temperature, study critical currents, and examine metastability and its limits in the temperature versus flow-velocity phase diagram. In addition, we comment on the effects of electron scattering by impurities.

###### pacs:
74.25.Sv, 74.62.-c, 74.25.Dw

Present address: ]Department of Physics and Astronomy, University of British Columbia, Vancouver, BC V6T 1Z1, Canada

## I Introduction

The Landau criterion Landau () concerns the threshold velocity of an obstacle moving through stationary superfluids at zero temperature, beyond which excitations are created and superfluidity is lost. By Galilean invariance, the criterion also applies to the threshold velocity of a uniformly flowing superfluid; see, e.g., Ref. Varoquaux06 (). The value of the threshold velocity is determined by the following formula

 vL=minp(Epp), (1)

where is the excitation spectrum. For superconductors the Landau criterion gives , where is the pairing gap at zero temperature and zero flow, and is the Fermi momentum of the entities that are paired (with being the associated angular wavenumber). However, it was found by Rogers Rogers (); Bardeen (); Zagoskin () that in three spatial dimensions the critical velocity —the uniform velocity of the superfluid condensate at which the superconducting state becomes unstable—of a clean superconductor exceeds . Furthermore, as discussed, e.g., in Ref Zagoskin (), for superflow velocities in the range , gapless excitations occur in clean superconductors. Moreover, the ratio has been found to be (). In contrast, it was found Zagoskin () that in two dimensions . As for the case of one dimension, Bagwell reported Bagwell () that , as found in two dimensions. In contrast to the case of three dimensions, in neither one nor two dimensions is gapless superconductivity predicted to occur in the presence of flow.

In this Paper, we re-analyze the critical velocity of a clean superconductor in one spatial dimension via a mean-field treatment, and obtain the dependence of this velocity on the temperature , along with the associated temperature-velocity phase diagram. In particular, we find that at zero temperature the critical velocity is smaller by a factor of than the Landau critical velocity . Even though for a gap in the quasiparticle excitation spectrum remains, the superconducting state first becomes unstable there. This is in contrast to a previous report Bagwell (), which held that the critical velocity is , i.e., the velocity at which the gap in the quasiparticle spectrum closes and excitations proliferate so as to destroy superconductivity. The smaller value of the critical velocity, which we obtain here, is the result of a pre-emptive Clogston-Chandrasekhar–like discontinuous phase transition ClogstonChandrasekhar (), and is analogous to the threshold uniform exchange-field in a superconductor, previously investigated by Sarma Sarma () and by Maki and Tsuneto MakiTsuneto (). At low temperatures (i.e., for , where is the critical temperature in the absence of flow), the transition from the superconducting to the normal state, occurring due to the presence of flow, remains discontinuous. In contrast, for the transition is continuous, i.e., it is associated with a continuously vanishing order parameter, e.g., as approaches the flow-velocity–dependent transition temperature from below.

From the experimental standpoint, it is usually more convenient to control the condensate current rather than its velocity. (An exception is provided by a closed loop threaded by magnetic flux.) The more widely appropriate physical observable is thus the critical current, i.e., the maximum equilibrium current that a superconductor can sustain, and this is what experiments on superconductors frequently measure. It should, however, be remarked that recent experiments on trapped condensates of atomic gases make it possible to probe the critical velocity directly KetterleGroup ().

Langer and Fisher LangerFisher67 () introduced a fresh perspective on the issue of the critical velocity of a sueprfluid. They explained that—as a matter of principle—superflow is inherently unstable, even at velocities below those that allow quasiparticle excitations to proliferate. This instability results from the possible occurrence of intrinsic, topologically allowed, collective excitation events, in which, e.g., a vortex ring nucleates locally, traverses an Arrhenius energy barrier, and grows without bound, thus eradicating a quantum of flow velocity. In the case of narrow channels, such as the one-dimensional superconductors discussed here, the fluctuations take the form of phase slip events, which can be either thermal LA () or quantum QPS (). But, before one takes into account the effects of fluctuations, it is important to understand and clarify the behavior at the mean-field level, which is mainly what we consider in the present Paper. In other words, the work reported here assumes that the rate of such toplogical fluctuations is negligibly small, so that our critical-velocity results, established on the basis of thermodynamic stability and the possibility of quasiparticle production, retain a use. In the present Paper we focus on issues of phases of thermodynamic equilibrium, competitions between them, and metastability; we do not attempt to address issues of kinetics, such as the rates at which phase transitions proceed and metastable states evolve into stable ones.

The remainder of the Paper is organized as follows. In Sec. II we diagonalize the Hamiltonian of a superconductor in the presence of flow, and in Sec. III we calculate the free energy of this system. In Sec. IV we use the self-consistency of the order parameter to calculate the dependence of the order parameter on the flow velocity at various temperatures. In Sec. V we calculate the dependence of the transition temperature on the flow velocity, assuming the transition to be continuous, i.e., in the limit of linear instability. Double solutions for the transition temperature turn out to exist for larger values of the flow velocity. We resolve this issue in Sec. VI by identifying the globally stable solution, i.e., the one that corresponds to the lower free energy. In Sec. VII we address the existence of metastable solutions, and discuss the extent to which these provide a superflow-based analog of the phenomena of supercooling and superheating. We also obtain the analog of the superheating limit, together with the equilibrium phase boundary and the supercooling limit. Having ascertained the true, equilibrium, velocity-dependent order parameter, we calculate in Sec. VIII the dependence of the supercurrent on the flow velocity, and hence determine the critical current and superfluid density. In Sec. IX we briefly discuss the effect of disorder on the nature of the transition. In Sec. X we contrast the critical-velocity results for one dimension with those for two and three dimensions. We conclude in Sec. XI.

## Ii Hamiltonian

We shall use a microscopic approach to discuss an effectively one-dimensional superconducting system, in which there is spatially uniform flow at velocity . Here, is the electron mass and is the corresponding momentum associated with the flow (or equivalently, the angular wavenumber, as we shall set ). (What we mean by “effectively one-dimensional” is that the transverse dimensions are much smaller than the zero-temperature coherence length .) We begin by writing down a Hamiltonian that is equivalent to Eq. (III.4) of Rogers Rogers () and Eq. (1) of Nozières and Schmitt-Rink NozieresSchmitt-Rink ():

 H=∑k,σ((k+q)22m−μ)c†k+q,σck+q,σ+∑k1,k2Vk1,k2c†k1+q,↑c†−k1+q,↓c−k2+q,↓ck2+q,↑, (2)

where and are respectively the annihilation and creation operators of electrons of spin-projection at (one-dimensional) momentum , is the BCS pairing potential BCS (), and is the chemical potential. Following Rogers Rogers (), we assume that depends on the difference between and , and hence is independent of . To justify our ignoring any dependence, we note that the critical momentum is no bigger than roughly , and this is roughly , i.e., several orders of magnitude smaller than , where is the atomic spacing and is the zero-temperature superconducting coherence length. This form of Hamiltonian anticipates that any Cooper pairs are formed via the pairing of electrons in states and , and that the resulting pairs have center-of-mass momentum .

Next, we make the mean-field approximation and adopt the BCS form for  BCS (), thus arriving at the Hamiltonian

 H = |Δ|2g+∑k(εq(k)c†k+q,↑ck+q,↑+εq(−k)c†−k+q,↓c−k+q,↓ (3) −(Δ∗c−k+q,↓ck+q,↑+Δc†k+q,↑c†−k+q,↓)),

in which is the magnitude of in the momentum range for which the pairing potential is nonzero, , and is the self-consistency condition on the order parameter (and similarly for ). This mean-field Hamiltonian can be solved via the Bogoliubov-Valatin transformation

 ck+q,↑=ukγ1;k+vkγ†2;kandc−k+q,↓=ukγ2;k−vkγ†1;k, (4)

with

 u2k=12(1+¯¯¯εkEk)andv2k=12(1−¯¯¯εkEk), (5)

where, for convenience, we have made the definitions and . We choose to be real and non-negative, and thus may drop the absolute value on in the definition of . With this procedure, the Hamiltonian becomes diagonal:

 H=Δ2g+∑k((Ek+kv)γ†1;kγ1;k+(Ek−kv)γ†2;kγ2;k)+∑k(¯¯¯εk−Ek). (6)

This expression also holds for higher dimensions, provided we replace by , by , and by .

## Iii Free energy

As the Hamiltonian has been diagonalized, we can readily evaluate the Helmholtz free energy of the system:

 F≡⟨H⟩−TS=⟨H⟩+T∑k[f1;klnf1;k+(1−f1;k)ln(1−f1;k)+f2;klnf2;k+(1−f2;k)ln(1−f2;k)], (7)

where and are Fermi distribution functions, defined via

 f1;k≡⟨γ†1;kγ1;k⟩=1eβ(Ek+kv)+1andf2;k≡⟨γ†2;kγ2;k⟩=1eβ(Ek−kv)+1, (8)

indicates an average, weighted by the equilibrium density matrix, and is the inverse temperature (with Boltzmann’s constant set to unity). We therefore arrive at the result:

 F=Δ2g+∑k¯¯¯εk−T∑kln(2coshβEk+2coshβkv). (9)

One may equally well calculate the partition function , so as to obtain the free energy via . The order parameter is to be determined self-consistently, via

 Δ=g∑k⟨c−k+q,↓ck+q,↑⟩=g∑kΔ2Ek(1−f1;k−f2;k). (10)

We note that because the momentum sum runs over both positive and negative values one, can safely replace by (or vice versa) in this equation, and thus arrives at the result:

 Δ=g∑kΔ2Ek(1−2f2;k), (11)

which is identical to Eq. (21) of Ref. Bagwell (). This equation can also be derived by demanding stationarity of the free energy, i.e., . The above results, Eqs. (7) to (11) hold for two- and three-dimensional systems as well, provided we replace by , by , and by , but in the following we shall focus mainly on the case of one dimension.

## Iv Order parameter

From Eq. (10), or equivalently Eq. (11), we can determine the values of the order parameter at any temperature and superflow velocity. Of course, there is always the trivial solution , which corresponds to the normal state. However, in this section our focus is on nontrivial solutions.

### iv.1 Zero temperature

Let us first examine the limit of zero temperature. In this limit, the self-consistency condition (11) becomes

 Δ(T,v)∣∣T=0=g∑k>0ΔEk(1−Θ(kv−Ek)), (12)

where is the Heaviside step function. By linearizing the -dependent spectrum around , and exchanging for the value of the order parameter , we arrive at the following result for :

 ln(ΔΔ0)=−∫ωD0dξ1√ξ2+Δ2Θ(kFv−√ξ2+Δ2). (13)

This condition yields two branches of solutions for the order parameter: (1) If , we obtain ; on the other hand, (2) if , the condition becomes

 ln(ΔΔ0)=−∫√k2Fv2−Δ20dξ√ξ2+Δ2=−sinh−1⎛⎜ ⎜⎝√k2Fv2−Δ2Δ⎞⎟ ⎟⎠, (14)

which leads to . These two zero-temperature solutions were first obtained by Sarma Sarma () in connection with the exchange-field effect in superconductors. In Fig. 1 we show these two zero-temperature solutions [i.e., the horizontal and parabolic curves emanating from the upper right point ], together with the corresponding solutions at nonzero temperatures. By continuity, it is not surprising that there is a low-temperature regime in which there is multivaluedness in the solutions for , as we shall see in the following subsection and in Fig. 1.

### iv.2 Nonzero temperatures

For these, we can solve Eq. (10), or equivalently Eq. (11), numerically, and obtain the velocity-dependent order parameter at arbitrary temperatures, as illustrated in Fig. 1 for several values of the temperature. We remark that in solving for the order parameter we have linearized the spectrum about Fermi momentum ; this linearization is valid as long as . From Fig. 1 it is evident that over a certain (higher) range of velocities there are two solutions for , these solutions differing from those obtained by Bagwell Bagwell (), by whom multiple solutions were not found. This type of multiplicity feature was first observed by Sarma Sarma () in the context of the exchange field in superconductors. In fact, the self-consistency conditions holding in the exchange-field case are identical to the ones holding here, provided one makes an identification between and the exchange-field energy . The two situations appear to be essentially equivalent.

The feature of double solutions that we have found for the order parameter, as shown in Fig. 1, is in contrast with previous findings Bagwell () (see, in particular, Fig. 3 therein). We suspect that this discrepancy results from the use of an iterative scheme for solving the self-consistency equation numerically; when multiple solutions exist, such scheme yields results that depend sensitively on the initial conditions. We have instead used the bisection method, which locates all possible solutions.

## V Transition temperatures: linear instability

If one assumes that the transition between the superconducting and normal states is continuous, one can determine the transition temperature via the self-consistency condition taken in the limit of vanishing . We note that, owing to the relation of the Fermi function to the Matsubara sum (see, e.g., Ref. Zagoskin ()), Eq. (10) is equivalent to

 Δ=g∑kT∑ωn−Δ(iωn−kv−Ek)(iωn−kv+Ek), (15)

where with are the Matsubara frequencies. We denote by the value of the temperature that solves Eq. (15) in the limit . In this limit, imposing a cutoff on the Mastubara frequencies, and integrating out the momentum , we arrive at the condition

 (16)

in which we have redefined to mean . By exchanging for the zero-flow critical temperature and performing the summation, Eq. (16) becomes

 ln(TcT0c)=ψ(12)−Reψ(12+ikFv2πTc), (17)

where is the di-gamma function DiGamma ().

The solutions of Eq. (17) for are shown in Fig. 2, which, like Fig. 1, exhibits a double-solution feature for a range of velocities. The solution at and corresponds to the branch . As we shall see below, this branch corresponds to an unstable solution; a correct description of the vs.  phase diagram requires the consideration of free energies.

## Vi Velocity-temperature phase diagram

In the light of the results obtained in Sec. V, we ask the question: What is the true transition temperature? To determine this, one has to take into account the multiple solutions of the order-parameter self-consistency condition and compare their free energies (e.g., see Ref. Bardeen ()). With this prescription we ask: Will our transition temperature have the same functional form as the of Sarma Sarma () and of Maki and Tsuneto MakiTsuneto () (SMT), in which is the magnetic exchange field?

In the case considered by SMT, a comparison is made between the free energies of a spin-unpolarized superconducting state and a normal state that is partially spin-polarized, both states being subject to an exchange field. In the present case the comparison is between a flowing superconducting state and a stationary normal state. Viewed from a reference frame that flows with the superconducting state, the normal state flows and can be regarded as the analog of SMT’s polarized state. Furthermore, our superconducting state is the analog of SMT’s superconducting state. This perspective suggests that the correct procedure for our purposes is to compare the free energies of the flowing superconducting state and the stationary normal state, as was first done by Bagwell Bagwell ().

The physical reason for the equivalence between the exchange-field case considered by Sarma and the superflow case considered here is as follows. In the case of Sarma, due to the exchange field, the up and down electrons in a Cooper pair have an energy difference of . In the case of flow, the electron pairing is between states and , and which have an energy difference of near the Fermi surface. In the former case, the normal state can be polarized, and thus can reduce its energy by an amount proportional to . In the latter case, the stationary normal state has an energy that is lower than the flowing normal state by an amount proportional to . Hence, the two scenarios appear to be equivalent.

To understand the root of our state-selection procedure, we begin with an analogy. Consider the liquid and crystalline states of a particular material. Following Callen callen (), we observe that the information sufficient to specify uniquely an equilibrium state is greater for the crystalline state than for the liquid state, owing to the spontaneously broken translational symmetry of the crystal and the resulting low-energy Goldstone field, i.e., the displacement field. Thus, when assessing the relative thermodynamic stability of a liquid and a crystalline state, one must specify a displacement field for the crystal but one must not (and indeed cannot) do so for the liquid state. It is meaningful to speak of an equilibrium crystalline state that supports a specified static shear stress, but not so for a liquid, as stress would induce a nonequilibrium dissipative steady state of shear flow, in which entropy would constantly be being produced. In summary, there is a unique equilibrium state for the liquid, but there is a family of equilibrium states for the crystal, differing in their displacement field (or strain field, or its thermodynamic conjugate, the stress field). One is entitled to consider the relative thermodynamic stability of a liquid and any one of this family of crystalline states. Returning now to the case of superconductivity, the precise analogy is on the one hand between the crystal and the superconductor, and on the other hand between the liquid and the normal metal.

One is thus entitled to consider the relative thermodynamic stability of a stationary normal state and any one of the family of the flowing superconducting states. That there is a family of flowing superconducting states originates in the spontaneously broken gauge symmetry of the superconducting state, which amounts to a controllable thermodynamic field, (i.e., the phase field), which is the analog of the displacement field of the crystal. (The analogy is: phase displacement ; velocity strain; current density stress.)  By contrast, the normal state is unique.

To obtain the phase diagram, one should then compare the free energy of a flowing superconducting state to the free energy of a stationary (i.e., not flowing) normal state Bagwell (). The energy difference between the flowing and stationary normal states plays the role of the paramagnetic energy. We shall see that the true equilibrium transition temperature then has exactly the functional form obtained by Sarma Sarma () and by Maki and Tsuneto MakiTsuneto ().

The free energy (9) of the superconducting state at a fixed chemical potential is given explicitly by

 Fs(q,μ)=Δ2g+∑k(k22m+q22m−μ)−T∑kln⎡⎣2coshβ√(k22m+q22m−μ)2+Δ2+2coshβkv⎤⎦. (18)

The stationary, normal state at chemical potential thus has free energy

 Fn(0,μ−q22m)=∑k(k22m+q22m−μ)−T∑kln[2coshβ(k22m+q22m−μ)+2]. (19)

Our goal is to find the difference between two energies at the same chemical potential, which is given by

 Fs(q,μ)−Fn(0,μ)=[Fs(q,μ)−Fn(0,μ−q22m)]−[Fn(0,μ)−Fn(0,μ−q22m)], (20)

where the term in the second pair of square brackets equals (i.e., the analog of the paramagnetic energy), with being the total number of electrons. Hence, from Eqs. (19) and (20) we arrive at the result that

 Fs(q,μ)−Fn(0,μ) = Nq22m+Δ2g−T∑kln⎡⎢ ⎢ ⎢ ⎢⎣coshβ√(k22m+q22m−μ)2+Δ2+coshβkvcoshβ(k22m+q22m−μ)+1⎤⎥ ⎥ ⎥ ⎥⎦ (21) = Nq22m+Δ2g−T∑kln[coshβEk+coshβkvcoshβ¯¯¯εk+1].

What we should do to obtain the phase diagram is to compare the possible nonzero solution(s) for the order parameter to determine which solution possesses the lower free energy and whether this free energy is lower than that of a stationary (i.e., nonmoving) normal state, i.e., whether . For convenience, let us make the definition

 δF(v,μ)≡Nq22m−Fs(q,μ)+Fn(0,μ)=−Δ2g+T∑kln[coshβEk+coshβkvcoshβ¯¯¯εk+1]. (22)

The solution for that has the largest value of , provided this difference is greater than zero, is the true equilibrium solution; otherwise the normal state will be the true equilibrium state. The kinetic flow term can be re-expressed (by relating the total electron number to the density of state) as , where is the density of states per spin, and we have, for convenience, set the total “volume” of the system to be unity. Note that from Eq. (22) we have at zero temperature

 δF(v,μ)=N0Δ22+2∑k(kv−Ek+Δ22Ek)Θ(kv−Ek). (23)

This expression also holds for higher-dimensional systems, provided we replace by , by , and by . The effective condensation energy for the solution [see the discussion below Eq. (13)] is . Balancing against the additional energy due to the flow, we obtain . This corresponds to the Clogston-Chandrasekhar limit in the exchange-field case ClogstonChandrasekhar ().

In Fig. 3, we plot versus the momentum for the two superconducting solutions at several temperatures, as well as . The crossing of the curve representing by the curve representing the superconducting state signifies a discontinuous transition. By finding such transitions at various values of , we arrive at the equilibrium phase boundary , shown as the curve ABD in Fig. 4.

## Vii Supercooling and superheating

As discussed in Sec. VI, we have discussed the true equilibrium phase transition curve ABD; see Fig. 4. But what is the physical meaning of the branch BC? It would be a limit of metastability if a normal state with flow were a true equilibrium state. Suppose that the system were in an equilibrium normal state with (i.e., lying to the right of the equilibrium phase boundary ABD), and were then ‘quenched’ by the rapid change of and/or into the metastable region BCD. The system would remain in the normal state, but metastably so, until a fluctuation were to occur that would nucleate a droplet of the superconducting phase large enough to grow and complete the conversion of the state to the true equilibrium state, i.e., the superconducting state. The metastability limit (which is also known as a spinodal line) is formally obtained by solving for the temperature obeying the equation [c.f. Eqs. (10), (15) and (16)]

 limΔ→01ΔδFδΔ=0. (24)

However, as we argue in Sec. VI, a normal state with a constant flow is not a true equilibrium state; therefore, strictly speaking, the curve BC does not represent a meaningful supercooling curve.

On the other hand, the branch BE is a true limit of metastability for the superconducting state. The system can be ‘quenched’ into a metastable superconducting state from an equilibrium superconducting state [i.e., from to the left of the equilibrium phase boundary] by rapidly changing to a value in the region BED. The system will remain superconducting, but metastably so, until a normal-phase nucleation event carries it to the equilibrium (i.e., normal) state. In this case, the metastability limit is determined by simultaneously solving the equations and (but ) numerically; see Fig. 4.

We note that a corresponding diagram in the case of exchange-field effect in superconductors was first obtained by Maki and Tsuneto MakiTsuneto (). To answer the question of how long it would take for system in a metastable state to find its equilibrium state is a kinetic one that would require further investigation.

## Viii Supercurrent

In this section we examine the dependence of the supercurrent on the flow velocity, and thereby determine the critical current, i.e., the maximum equilibrium supercurrent that the system can sustain at various temperatures. In general, the charge current carried by the system is defined via

 IQ≡e∑k,σq+km⟨c†k+q,σck+q,σ⟩. (25)

As is the total number of electrons , we can re-write Eq. (25) as

 IQ=emNv+e∑k,σkm⟨c†k+q,σck+q,σ⟩, (26)

where . By using Eqs. (4) and (8), the second term can be simplified to , and thus the expression for becomes

 IQ(T,v)=−|e|m{Nmv+2∑kkf1;k}, (27)

where the electron charge is and the temperature is implicit in the Fermi function ; see Eq. (8).

We shall compute the current in the superconducting state as a function of and , and then, by choosing the value of in the range that maximizes at fixed , we shall obtain the critical current, denoted by , at that temperature. Equation (27), applied at , tells us that the zero-temperature current is given by

 IQ(T=0,v) = −|e|m[Nmv−2N0kF√k2Fv2−Δ20Θ(kFv−Δ0)], (28) = I0Q⎡⎣vvL−√v2v2L−1Θ(v−vL)⎤⎦, (29)

where . Although, formally, achieves its maximum value (viz., ) at , this maximum value is unattainable, because the superconducting state gives way to the normal state via a pre-emptive transition at (unless the system falls out of equilibrium and remains metabstably in the superconducting state). Therefore, the maximum equilibrium critical current at zero temperature occurs at , and has the value . At the superconducting state first becomes unstable, so the critical current that is attainable in practice is slightly less than . This is unlike the behavior in three dimensions, as well as the higher of the nonzero temperatures; for these situations the maximum current is achieved at a value of superfluid velocity for which the superconducting state is still globally stable.

In Fig. 5 we show the equilibrium supercurrent vs. the superfluid velocity  for various values of the temperature. The value of that corresponds to a maximum in the current (i.e., the critical current ) is not necessarily . (It is for lower temperatures, but not for higher ones.) The temperature dependence of the critical current is shown in Fig. 6.

A related quantity of interest is the superfluid density , defined as

 ns=∂js(v)∂v∣∣v=0, (30)

i.e., the response of the current density (whose expression is identical to because we have set the system ‘volume’ to unity) to an infinitesimal change in the ‘driving’ velocity . The temperature dependence of is shown in Fig. 7. The vanishing of near is, as expected, linear, and the departure of from its zero-temperature value has, at low temperatures, a thermally activated form.

## Ix Effect of disorder

So far, we have discussed clean systems, i.e., systems not disordered by any scattering electrons by impurities. What effect will disorder have? In particular, would the disorder change the nature of the superconducting-to-normal transition? In this section we briefly discuss the latter issue. In the case of disorder, the Green function technique is better suited for dealing with disorder than is trying to diagonalize the Hamiltonian with inclusion of an arbitrary configuration of the impurities. An equivalent calculation in the context of the exchange field was done by Maki and Tsuneto MakiTsuneto () and, more recently, by Wei and Goldbart WeiGoldbart () in the context of Little-Parks effect in small rings. Here, in contrast with Ref. WeiGoldbart (), we are not concerned with the finiteness of the system size, and we simply quote the relevant results for the equation obeyed by the critical temperature, i.e., Eq. (B39) of Ref. WeiGoldbart ():

 ln⎛⎝Tc(v)T0c⎞⎠ = ψ(12)−1√α2−χ2[−α+√α2−χ22ψ(1+α+√α2−χ22) (31) +α+√α2−χ22ψ(1+α−√α2−χ22)],

where, in the present case, and , the parameter is the elastic mean-free time, and is the di-gamma function DiGamma ().

In the clean limit (i.e., ), Eq. (31) reduces to Eq. (17), for which multiple solutions for exist. For strong disorder (), Eq. (31) reduces to

 ln⎛⎝Tc(v)T0c⎞⎠=ψ(12)−ψ(12+χ24α), (32)

for which no multiple solutions for exist. This suggests that the assumption of a vanishing order parameter in the search for the critical temperature is valid and, hence, that the transition is continuous. Moreover, from Eq. (32), in the strong-disorder limit the critical velocity is determined via

 kFvc/Δ0=1/√4τ0Δ0. (33)

Numerically, we have found that for multiple solutions do not occur. This, then, gives the threshold for the disorder strength that divides the discontinuous and continuous transition regimes. In Fig. 8 we show the solutions of Eq. (31) for various values of . In the strong-disorder limit we obtain from Eq. (33) that (where is the elastic mean-free path and is zero-temperature coherence length), in particular that the critical velocity exceeds . But will the critical current also exceed its clean-limit value? According to Bardeen Bardeen (), the superfluid density in the strong-disorder limit is reduced by a factor of roughly . An order-of-magnitude estimate gives for the critical current the product of the superfluid density (where denotes the superfluid density in the clean limit) and the critical velocity . This, in turn, gives that the critical supercurrent is reduced by roughly a factor , which is much less than unity in the strong-disorder limit. Therefore, even though the critical velocity is increased by disorder, the critical current is reduced by it.

We now digress to discuss the scenario of a closed superconducting loop threaded by magnetic flux, which is pertinent to the experiment performed by Liu et al. PennState () and the situation considered theoretically in Ref. WeiGoldbart (). There, the boosted velocity is induced by the magnetic flux. The measurement of vs. flux (with ) presented in Fig. 4 of Ref. PennState () shows non--periodic behavior, with the first dome (near ) being higher than the second dome (near ). Naturally, the data explore only a limited range of flux, so it is not possible to determine with certainty whether higher flux values would yield dome heights that oscillate with flux or decay. Oscillations would be a consequence of flux through the hole; decay would be a consequence of flux through the sample, which would cause pair breaking. To ascertain which of these two effects (oscillation or decay) is likely to dominate in the range of fluxes probed by the experiment, it is useful to compare several lengthscales. First, the sample radius is smaller than the coherence length , by a factor of roughly . Following Ref. WeiGoldbart (), this suggests the smallness of the radius as the origin of the dome-height mismatch. On the other hand, a more complete characterization of the system would take into account the elastic mean-free path , which for the sample studied in Ref. PennState () is shorter than the radius, by a factor of approximately . This puts the sample in the dirty regime, and as a result, the impact of the smallness of the radius is suppressed and tends to restore the periodicity of discussed in Ref. WeiGoldbart () and given by

 ln(Tc(ϕ)T0c)=lnt(ϕ)=ψ(12)−ψ(12+Γleξ0x2m(ϕ)t(ϕ)R2), (34)

where and . What else could cause the lack of periodicity? Because the magnetic flux is not confined inside the hole in the loop, but is also present throughout the sample, owing to its finite thickness, the orbital pair-breaking effect comes into play. To account for this effect, we use the scenario of the orbital pair-breaking effect of a parallel magnetic field applied to a film of finite thickness (which is about in the experiment), which leads us to add to the argument of the second term on the r.h.s. of Eq. (32) the term

 16De2B2d22πkBTc(ϕ)ℏc2=Γϕ2ξ0led236t(ϕ)R4; (35)

see, e.g., Ref. Tinkham (). (The term should be correct up to a numerical factor due to geometry.)  As we see from Fig. 9, the resulting dependence of the critical temperature on the flux seems to reproduce, at least qualitatively, the features reported by Liu et al. PennState ().

## X Higher dimensions

We have seen that in the clean limit the discontinuous transition at flow velocity occurs due to a competition between the condensation energy and the flow kinetic energy (for one dimension; see footnote footnoteDOS ()), where . The principle that we employed in Sec. VI is to compare the free energies of the flowing superconducting states and of the stationary normal state. As further confirmation of the validity of this principle, we now apply it to higher-dimensional settings, and compare the results with the known one mentioned in the Introduction.

In two dimensions, the solution of the self-consistency equation (11) gives for , and for . This indicates a discontinuous transition at . Indeed, this result is consistent with free-energy considerations: by using the condensation energy and the flow kinetic energy (the difference from the one-dimensional case arising from the density of states; see footnote footnoteDOS ()), we conclude that .

In three dimensions the solution of the self-consistency equation (11) gives for , and for (see Ref. Zagoskin ()) gives the following implicit equation for :

 √1−(ΔkFv)2−ln[1+√1−(ΔkFv)2]=ln(vvL), (36)

where we have ignored terms of order or smaller (with denoting the Debye frequency). Equation (36) gives that at . Is this linear-instability result consistent with free-energy considerations (i.e., the balancing of the two energies)? If one were to naïvely use the condensation energy to balance the flow energy  footnoteDOS (), one would obtain ! Does this mean that the superconducting state really becomes unstable at ? Such a result would be in conflict with the result, mentioned in Sec. I, that ? How is this apparent conflict resolved? For , the effective condensation energy is indeed . But for , , so it would be incorrect to equate and in order to deduce . Instead, one should use the expression for appropriate to the range . In fact, an evaluation of Eq. (23) in three dimensions gives (for )

 δF = N02Δ2+N03(kFv)2[√1−(ΔkFv)2]3 (37) +N0Δ2sinh−1√(kFv)2−Δ2Δ−N0Δ2ln√(kFv)2−Δ2+kFvΔ,

where the dependence of on is given by Eq. (36). From Eqs. (23) and (37) we have that for , and . Only when increases to (i.e., when ) do we have . Thus, the transition at is continuous, as the order parameter vanishes continuously there, and, hence, the conflict is resolved. The dependence of the quantity on flow velocity is plotted in Fig. 10, which clearly shows that the superconducting state is stable for flow velocities in the range .

## Xi Concluding remarks

We have revisited the issue of the critical velocity of an effectively one-dimensional superconductor using mean-field theory. As we have discussed, this issue is equivalent to that of the threshold uniform exchange-field in a superconductor, first investigated by Sarma Sarma () and by Maki and Tsuneto MakiTsuneto (). This equivalence is due to the correspondence of (i) the Zeeman frustration energy between the two electrons of a Cooper pair (in the case of exchange field), and (ii) the kinetic frustration energy (in the case of flow). In particular, we have found that at zero temperature the critical velocity is a factor of smaller than the Landau critical velocity, in contrast to a previous finding Bagwell (). Our result originates in a Clogston-Chandrasekhar–like discontinuous transition between the superconducting and the normal state. The physical reason for the discontinuity of the transition is that it results from a balance between the condensation and flow energies. (Recall that the bulk superconducting phase transition in magnetic field results from a balance between the condensation () and magnetic field () energies, and that the transition is discontinuous.)  The transition remains discontinuous for temperatures below approximately ; above that, it is continuous.

We have also studied the issue of critical supercurrents, determined the current-temperature phase diagram, and examined metastability (and its limits) in the temperature versus flow-velocity phase diagram. As a test of our underlying principle, namely the comparison of the free energies of flowing superconducting and stationary normal states, we have also examined the two- and three-dimensional cases, and have thus obtained results that are in agreement with previous findings. In addition, we have commented on the effects of electron scattering by impurities and, in particular, we have argued that strong disorder renders continuous the aforementioned discontinuous phase transition. Even though disorder can increase the value of the critical velocity, the physically measurable quantity, i.e., the critical current, is still reduced by the disorder.

Throughout this Paper, we have adopted a mean-field approach, thus ignoring the effects of fluctuations such as phase slips. Furthermore, we have limited ourselves to the analysis of equilibrium properties as well as issues of metastability. The intriguing issue of how a flowing superconducting state undergoes the transition to the stationary normal state, which essential dynamical processes take place, and on what timescale all remain as research directions for the future.

Acknowledgments. The authors acknowledge useful discussions with Anton Burkov, Tony Leggett and Frank Wilhelm. This work was supported by IQC, NSERC, and ORF (TCW) and by DOE Grant No. DEFG02-07ER46453 through the Frederick Seitz Materials Research Laboratory at the University of Illinois (PMG).

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