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 [ ] = 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 2 + β 2 + γ 4 + δ 2 2+ mogeneous state in the presence of a strong magnetic F j j jr jµ j j jr j (1) µ 2 2 + ( ∗)2( )2 + c.c. + ν 6; field, where Zeeman splitting of the Fermi surfaces leads j j jr j 8 r j j to the formation of Cooper pairs with nonzero total mo- mentum. Similar inhomogeneous states can form and are where 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 supercon- ductors [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 ^ 2 ^ 4 2 In this Letter, we report that on the phase diagrams of as β = βvFγ, δ = δvFν, and µ =µv ^ Fν, where vF is the ^ ^ superconductors featuring the FFLO state should rather Fermi velocity and β; δ; µ^ 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 β^ = 1, δ^ = 1=2, andµ ^ = 4 and in two dimensions we have as the Zeeman splitting field or temperature is increased, β^ = 1=2, δ^ = 3=16, andµ ^ = 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 FF = rescaled theory to vary, which parametrizes changes in iqx FF e and the Larkin-Ovchinnikov (LO) state LO = both temperature and Zeeman splitting field. j j LO cos qx. For an infinite system, assuming that the Having derived boundary conditions, we will now nu- j j 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 + 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 [ + v] = F [ ] + 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 α β + 2γ + δ + Ω − r r is the following: besides the inhomogeneous order pa- (8) rameter, the bulk FFLO state has inhomogeneous en- 5µ 2 2 2 5 d ( ) + 3ν vd x ergy density. The numerical solutions for one- and two- 4 r − r 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 + δ v @Ω 4 r − r areas with positive energy density. However, if the sys- ) (9) tem has a boundary, a solution can be found where the + δ 2 v ndS; boundary cuts off a segment of inhomogeneous solutions r r · 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 + 2 δ 3 n = 0; (10) tions clearly show that the boundaries are characterized 4 r − r · by negative energy density as seen in Fig.1, resulting in δ 2 = 0: (11) the stability of the surface PDW state for largeα ~.