Surface Pair-Density-Wave Superconducting and Superfluid States

Surface Pair-Density-Wave Superconducting and Superﬂuid States

Mats Barkman,1 Albert Samoilenka,1 and Egor Babaev1 1Department of Physics, Royal Institute of Technology, SE-106 91 Stockholm, Sweden Fulde, Ferrell, Larkin, and Ovchinnikov (FFLO) predicted inhomogeneous superconducting and superfluid ground states, spontaneously breaking translation symmetries. In this Letter, we demon- strate that the transition from the FFLO to the normal state as a function of temperature or increased Fermi surface splitting is not a direct one. Instead the system has an additional phase transition to a different state where pair-density-wave superconductivity (or superfluidity) exists only on the boundaries of the system, while the bulk of the system is normal. The surface pair- density-wave state is very robust and exists for much larger fields and temperatures than the FFLO state.

In regular BCS theory, the formation of Cooper pairs The Ginzburg-Landau description of superconductors binding together two electrons with opposite spin and op- in the presence of Zeeman splitting was derived from mi- posite momentum results in a uniform superconducting croscopic theory in Ref. [31]. The free energy functional state [1,2]. In 1964, Fulde and Ferrell [3] and Larkin and R d reads F [ψ] = Ω d x where the free energy density is Ovchinnikov [4] (FFLO) independently predicted that F F under certain conditions there should appear an inho- =α ψ 2 + β ψ 2 + γ ψ 4 + δ 2ψ 2+ mogeneous state in the presence of a strong magnetic F | | |∇ |µ | | |∇ | (1) µ ψ 2 ψ 2 + (ψ∗)2( ψ)2 + c.c. + ν ψ 6, field, where Zeeman splitting of the Fermi surfaces leads | | |∇ | 8 ∇ | | to the formation of Cooper pairs with nonzero total momentum. Similar inhomogeneous states can form and are where ψ is a complex field referred to as the supercon- of great interest in cold-atom gases [5–8] and in color su- ducting order parameter and c.c. denotes complex conju- perconducting states of quarks that are expected to form gation. The coefficients α, γ, and ν depend on the applied in cores of neutron stars [9]. Various predictions indi- Zeeman splitting field H and temperature T accordingly cate that the FFLO state may host many rich physical phenomena including topological defects and phase tran- α = πN(0) K1(H,T ) K1(H0(T ),T ) − − sitions associated with these defects [10–12]. Other inter- H H0(T ) 1 H0(T ) N(0) − Im Ψ(1) i , (2) esting studies include the orbital third critical magnetic ≈ 2πT 2 − 2πT field [13] as well as states in samples with nontrivial ge- πN(0)K3(H0(T ),T ) ometries [14, 15] and multiple competing inhomogeneous γ , (3) 4 states in two-dimensional systems [16]. For a more de- ≈ πN(0)K (H (T ),T ) tailed review of the FFLO state, see Refs. [17–19]. ν 5 0 , (4) ≈ − 8 The anticipated interesting properties made this state highly sought after, yet there is still no universally ac- where N(0) is the electron density of states at the Fermi cepted experimental proof. The orbital effect is signifi- surface and we have defined the functions cantly stronger than the paramagnetic effect in most su- n 2T ( 1) h (n−1) i perconductors, hindering observations of the FFLO state. Kn(H,T ) = − Re Ψ (z) , (5) (2πT )n (n 1)! More specifically the upper critical orbital field must − be significantly larger than the Chandrasekhar-Clogston where z = 1/2 iH/2πT and Ψ(n) is the polygamma limit [20, 21] for a FFLO regime to exist. Materials − function of order n. The function H0(T ) indicates where where possible FFLO states were discussed are heavy α changes sign and is defined implicitly by the equation fermions superconductors [22], layered organic superconductors [23] such as λ-(BETS)2FeCl4 [24, 25] and β” salt T 1 H (T ) 1 ln c = Re Ψ(0) i 0 Ψ(0) , (6) [26, 27], and iron-based superconductors [28]. Among T 2 − 2πT − 2 the direct experimental probes to identify this state, it has been suggested to study the Josephson effect [29] and where Tc is the critical temperature above which the nor- arXiv:1811.09590v3 [cond-mat.supr-con] 13 May 2019 Andreev bound states [30]. mal state is entered. The remaining coefficients are given ˆ 2 ˆ 4 2 In this Letter, we report that on the phase diagrams of as β = βvFγ, δ = δvFν, and µ =µv ˆ Fν, where vF is the ˆ ˆ superconductors featuring the FFLO state should rather Fermi velocity and β, δ, µˆ are positive constants that de- generically appear another state that has a form of sur- pend on the dimensionality d. In one dimension we have face pair-density-wave superconductivity. We find that βˆ = 1, δˆ = 1/2, andµ ˆ = 4 and in two dimensions we have as the Zeeman splitting field or temperature is increased, βˆ = 1/2, δˆ = 3/16, andµ ˆ = 2. In the parameter regime superconductivity disappears only in the bulk of the sys- in which β is negative, inhomogeneous order parameters tem but a sample should transition into a state with a may be energetically favorable. For a derivation of the superconducting surface. Ginzburg-Landau expansion in cold atoms, see Ref. [6]. 2

Typically considered structures of the order param- Consequently, there is only one free parameterα ˜ in the eter are the so-called Fulde-Ferrell (FF) state ψFF = rescaled theory to vary, which parametrizes changes in iqx ψFF e and the Larkin-Ovchinnikov (LO) state ψLO = both temperature and Zeeman splitting field. | | ψLO cos qx. For an infinite system, assuming that the Having derived boundary conditions, we will now nu- | | order parameter vanishes close to the tricritical point, merically minimize the free energy for a superconductor the average free energy density of these states can be in both one- and two-dimensional domains, while vary- minimized analytically by neglecting higher order terms, ingα ˜. The associated free energy is calculated in or- resulting in the conclusion that the LO state is energeti- der to locate phase transitions. Two different numer- cally preferred over the FF state. The second-order phase ical approaches were used. The solutions in Fig.1 bulk transition into the normal state occurs at α = αc = were obtained using a finite difference scheme and nonlin- β2/4δ . In general, the optimal order parameter struc- ear conjugate gradient method, parallelized on a graph- ture is found by solving the equation of motion associated ical processing unit. These results were also supported with the free energy functional (called Ginzburg-Landau by calculations using the finite element method within equations in this context). This was done analytically FreeFem++ framework [32]. in the one-dimensional case for an infinite sized super- We find that the free energy remains negative forα ˜ conductor, resulting in elliptic sine solutions [31]. The bulk larger than the critical valueα ˜c , where in one dimen- sinusoidal oscillations are recovered by approaching the bulk bulk sionα ˜c = 2 and in two dimensionsα ˜c = 4/3. The transition into the normal state. We solve the equation origin of it is the formation of a distinct surface pair- in a superconductor with boundaries. We consider the density-wave (PDW) superconducting state, which has a case of the real order parameter. The equation of mo- superconducting gap on the boundaries of the system but tion can be derived through variational principles. By not in its bulk. The obtained order parameter structure mapping ψ ψ + v in the free energy functional, where 7→ has the form of a damped oscillation with an amplitude v is some small arbitrary perturbation, we find to linear that vanishes in the bulk but remains nonzero close to order in v using Eq.1 the boundaries. The boundary states are found in both one- and two-dimensional systems. The results have been F [ψ + v] = F [ψ] + δFbulk + δFsurface (7) verified by altering the system size and it was found that, for a sufficiently large system, the surface state is inde- where pendent of system size. Z 2 3 4 The origin of the appearance of the surface PDW state δFbulk = 2 αψ β ψ + 2γψ + δ ψ+ Ω − ∇ ∇ is the following: besides the inhomogeneous order pa- (8) rameter, the bulk FFLO state has inhomogeneous en- 5µ 2 2 2 5 d ψ( ψ) ψ ψ + 3νψ vd x ergy density. The numerical solutions for one- and two- 4 ∇ − ∇ dimensional cases are plotted in Fig.1. As the system and approaches the phase transition from the bulk FFLO to the bulk normal state, the energy gain from the areas Z ( h 5µ 2i 3 with negative energy density becomes balanced by the δFsurface = 2 β + ψ ψ δ ψ v ∂Ω 4 ∇ − ∇ areas with positive energy density. However, if the sys- ) (9) tem has a boundary, a solution can be found where the + δ 2ψ v ndS, boundary cuts off a segment of inhomogeneous solutions ∇ ∇ · with positive energy and has a decaying oscillatory con- figuration of the order parameter extending to a certain where n is the normal vector to the boundary ∂Ω. By length scale in the bulk. That is, a decaying solution near setting δFbulk = 0 we find the equation of motion and by the boundary starting with a negative energy segment setting δFsurface = 0 we find the two boundary conditions should be stable even when the system does not support the FFLO state in the bulk. Indeed, the numerical solu- h 5µ i β + ψ2 ψ δ 3ψ n = 0, (10) tions clearly show that the boundaries are characterized 4 ∇ − ∇ · by negative energy density as seen in Fig.1, resulting in δ 2ψ = 0. (11) the stability of the surface PDW state for largeα ˜. The ∇ generality of the argument implies that these surface su- It is convenient to rescale the theory in the regime perconducting states of nontopological origin should be where β is negative by defining the dimensionless quan- present for all states where free energy density is oscil- ˜ 2 tities ψ = ψ/ ψU ,α ˜ = α/αU,x ˜ = q0x, where ψU = lating in space. This includes generalizations of FFLO | 2| 2 | | γ/2ν, αU = γ /4ν, and q0 = β/2δ. The free en- states to systems with unconventional pairing [34]. − − 2 d ˜ ˜ ergy can thus be written F [ψ] = αU ψU /q0 F [ψ], where The existence of a surface PDW state in a semi-infinite ˜ ˜ R ˜ d | | F [ψ] = Ω d x˜, in which the rescaled free energy den- system Ω = [0, ) can be proven analytically. To that sity is identicalF to Eq.1, where the coefficients have been end, we use an variational∞ ansatz of the form replaced accordingly α α,˜ β β˜, and so on, where γ˜ = 2˜ν = 2, β˜ = 7→2δ˜ = 7→2βˆ2/δˆ, andµ ˜ = βˆµ/ˆ δˆ. ψ(x) = ∆e−kx cos(qx + φ), (12) − − − − 3

5.0 15 2.5 α˜ = 1.0 α˜ = 1.9 α˜ = 1.0 α˜ = 1.2 0.6 0.0 ˜ 0 y 2.5 0.4 − 5.0 15 − − 5.0 15 0.2 2.5 α˜ = 2.1 α˜ = 5.0 α˜ = 1.3 α˜ = 1.4 ˜ 0.0 ˜ 0 0.0 y ψ 2.5 − 5.0 15 0.2 − − − 5.0 15 α˜ = 7.5 α˜ = 8.0 α˜ = 5.0 α˜ = 10 2.5 0.4 −

0.0 ˜ 0 y 2.5 − 0.6 − 5.0 15 − 30 15 0 15 30 30 15 0 15 30 − 15 0 15 15 0 15 − − x˜ − − x˜ − x˜ − x˜

FIG. 1: Numerically calculated order parameters ψ˜ in one- and two-dimensional system domains for various temperatures or, equivalently, various Zeeman splitting fields, parametrized byα ˜. In the one-dimensional domain, the blue curve is the order parameter and the red curve is the energy density. One can clearly observe a nontrivial oscillatory pattern of the energy density in the bulk of the system. The images show the sequence for going from superconducting to normal state by increasing the parameterα ˜. In the first panel, the system is in the FFLO state, and in the last panel the system is close to the normal state. For intermediateα ˜, the order parameter first vanishes in the bulk of the system, and the system enters the surface pair-density-wave state that persists for a much wider range of parameters than the FFLO state. In calculations of the two-dimensional system, the term 0.5 ˜ ψ˜ 4 was retained in the Ginzburg-Landau expansion, for details see Ref. [33]. |∇ | where the parameters ∆, k, q and φ should be found such ple, in two dimensions the situation corresponds to an in that the free energy is minimized, subject to the bound- plane field and there is no external field perpendicular to ary conditions. The surface PDW to normal transition the plane. In the context of cold atoms, it corresponds to occurs when the energy is minimal with ∆ = 0. When the fermionic population imbalance in the absence of ro- the transition into the normal state is of second order, tation. Therefore, the physical origin and the structure of it is sufficient to consider terms proportional to ∆ 2 in the solution is very different from that of the third upper | | the free energy. In addition, the boundary condition in critical magnetic field Hc3 [35] that was recently studied Eq. 10 takes the simpler form (β ψ δ 3ψ) n = 0 in FFLO systems [13]. The results also have implications close to the transition into the normal∇ − state.∇ Carrying· for the problem of FFLO states in mesoscopic systems, out the minimization analytically, we find that the free where the literature focuses on commensurateness effects energy associated with the surface PDW state remains [14, 15, 36]. Consider the parameter regime where the surface bulk negative until α = αc = 4αc . The numerical cal- surface PDW state exists. By gradually decreasing the culation shows that the phase transition indeed occurs at system size, the areas with the PDW state that live on op- this value of α, which implies that the variational ansatz posite surfaces will eventually start to overlap with each in Eq. 12 captures very well the solution for the surface other. Thus, the above results imply that, for a range of PDW state in one dimension. mesoscopic system sizes of the order of 1/k in Eq. 12, We can draw the phase diagram with respect to H the most robust solutions will have a form very different and T for the one-dimensional system, as shown in Fig. from a periodically oscillating function, which warrants 2. The inhomogeneous regime is split into two parts: further investigation. the bulk FFLO state and the surface PDW state. The In conclusion, we have studied superconducting or su- regime of surface PDW state on the phase diagram is perfluid systems supporting the FFLO state. We found significantly larger than the bulk FFLO regime. that, at elevated temperatures or for strong splitting of Note that the only role of the external magnetic field is Fermi surfaces, such systems support a state where the to create an electronic population imbalance. For exam- surface of the sample is a PDW superconductor or 4

3.5 main superconducting. Secondly, the system transitions Surface PDW into the fully normal state only at a higher temperature. 3.0 FFLO In the considered regime, we found that the surface PDW Uniform state is more robust than the FFLO state and extends 2.5 Normal to much larger values of Fermi surface splitting and temperatures. H 2.0 The effect can be used to experimentally prove the elu- sive FFLO state as follows: The main specific heat fea- 1.5 ture should be detectable well below the superconducting phase transition. This is because the contribution to the 1.0 specific heat should be dominated by the bulk, where the 0.3 0.4 0.5 0.6 0.7 0.8 gap disappears earlier than on the surface. The transport T/T measurements should at the same time indicate that sys- c tem retains superconductivity due to the surface PDW state. Because of superconductivity existing only in a FIG. 2: Phase diagram for a superconductor with bound- thin layer in the surface PDW state, another expected aries in the presence of Zeeman splitting of Fermi surfaces experimental signature is a concomitant increase of mag- caused by the in plane external field H. The role of the netic field penetration lengths for fields perpendicular to external field here is to create imbalance of electronic the superconducting layer. Very sensitive specific heat populations with different spins. In the context of cold measurements should see two features associated with atoms, this corresponds to fermionic population imbal- the bulk and surface phase transitions and yield the phase ance. At low H the superconductor is in the uniform su- diagram shown in Fig.2. In cold atoms [6], the surface perconducting state. At lower temperatures and high H, PDW state can be directly observed for experiments re- the inhomogeneous regime is entered. There exists a nar- alizing sharp potential walls [37, 38]. Finally, the results row regime in which the FFLO state exists in the bulk of have implications for the models of color superconduc- the superconductor. One finds a large surface PDW state tivity in the neutron stars cores, at the interface between regime where the system has superconducting boundaries nuclear and quark matter, which is widely believed to be but normal bulk. The figure shows calculations based on of the FFLO type [9]. a one dimensional model. The work was supported by the Swedish Research Council Grants No. 642-2013-7837 and No. VR2016- 06122 and Goran Gustafsson Foundation for Research in superfluid while the bulk of the system is normal. Corre- Natural Sciences and Medicine. The computations were spondingly, when the temperature is increased, the sys- performed on resources provided by the Swedish National tem has two phase transitions: Firstly, superconductivity Infrastructure for Computing (SNIC) at the National Su- disappears in the bulk of the system, while surfaces re- percomputer Center in Linkoping, Sweden.

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