Charge-4e superconductivity from nematic superconductors in 2D and 3D
CCharge- e superconductivity from nematic superconductors in 2D and 3D Shao-Kai Jian, Yingyi Huang, and Hong Yao
2, 3, ∗ Condensed Matter Theory Center, Department of Physics,University of Maryland, College Park, Maryland 20742, USA Institute for Advanced Study, Tsinghua University, Beijing 100084, China State Key Laboratory of Low Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China (Dated: February 8, 2021)Charge-4 e superconductivity as a novel phase of matter remains elusive so far. Here we showthat charge-4e phase can arise as a vestigial order above the nematic superconducting transitiontemperature in time-reversal-invariant nematic superconductors. On the one hand, the nontrivialtopological defect—nematic vortex—is energetically favored over the superconducting phase vortexwhen the nematic stiffness is less than the superfluid stiffness; consequently the charge-4 e phaseemerges by proliferation of nematic vortices upon increasing temperatures. On the other hand, theGinzburg-Landau theory of the nematic superconductors has two distinct decoupling channels toeither charge-4 e orders or nematic orders; by analyzing the competition between the effective massof the charge-4 e order and the cubic potential of the nematic order, we find a sizable regime wherethe charge-4 e order is favored. These two analysis consistently show that nematic superconductorscan provide a promising route to realize charge-4 e phases, which may apply to candidate nematicsuperconductors such as PbTaSe and twisted bilayer graphene. Introduction.—
Featuring the condensation of quartetswith four times the fundamental electron charges, charge-4 e superconductivity [1–4] is intrinsically distinct fromthe conventional charge-2 e superconductivity discoveredmore than a century ago. One hallmark of charge-4esuperconductors is the magnetic flux quantization withperiod hc/ e , which is half that of the usual super-conducting flux quantum. Unlike the conventional su-perconductivity that is well described by the seminalBardeen–Cooper–Schrieffer (BCS) theory, many proper-ties of charge-4 e superconductivity remain less well un-derstood. The mystery of this phase is not only be-cause it defies any BCS-type analysis, but also due to thelack of experimental realization so far. Previous studiesin Ref. [3] suggested that the charge-4 e superconductiv-ity can arise as a vestigial long-range order above thetransition temperature of a pair-density-wave (PDW) su-perconductor [5–17], whose order parameter varies peri-odically in the real space. The additional breaking oftranslational symmetry in the PDW state is essential inrealizing the charge-4 e superconductivity as it providesnontrivial topological defects—dislocations. As a conse-quence, if the dislocation proliferates as the temperatureis raised, the equilibrium state will restore the transla-tional symmetry by doubling the original charge-2 e con-densate, and therefore lead to the charge-4 e phase.The crucial ingredient of the underlying condensatehaving more than one quantum number [18–20] providesa general guide to search for charge-4 e superconductors.It was proposed that in certain two-dimensional high-temperature superconductors (e.g. La − x Ba x CuO andLa . − x Nd . Sr x CuO [21–23]) exhibiting stripe super-conducting orders [24, 25], the charge-4 e superconduc-tivity may occur above the transition temperature [3].Nevertheless, the experimental signature of the charge- 4 e superconductivity in these high-temperature super-conductors has not been observed so far. Apart fromPDW superconductors, the nematic superconductor thatbreaks both charge conservation and lattice rotationalsymmetry also hosts an order parameter that carriesmultiple quantum numbers. The additional topolog-ical defect in nematic superconductors is created bythe intersection of domain walls separating differentground states. More importantly, many experimentalprogresses have been made in achieving the nematic su-perconductivity in various systems. For instance, re-cent experiments found that the spin susceptibility be-low the superconducting temperature breaks the three-fold lattice rotational symmetry in doped topological in-sulator Cu x Bi Se [26–29], where the breaking of rota-tional symmetry suggests that the order parameter inthe superconducting phase is a two-dimensional E u rep-resentation of point group D d [30–32]. Providing thegrowing experimental evidence of the nematic super-conductivity in various systems, including doped topo-logical insulators Cu x Bi Se [26–29], Sr x Bi Se [33, 34]and Nb x Bi Se [35, 36], superconducting topologicalsemimetals PbTaSe [37], and more recently twisted bi-layer graphene [38, 39], as well as increasing interest innovel charge-4 e ordering, it is thus interesting to askwhether the multiple-component superconducting orderparameters in nematic superconductors is able to realizecharge-4 e phases above their charge-2 e transition tem-perature.In this paper, we answer this question in the affirma-tive; namely, we provide two analysis, suitable for 2D and3D, respectively, to support the possibility of a vestigialcharge-4 e superconducting phase from nematic supercon-ductors. In 2D, because a vortex is a point-like object, weanalyze the fate of various topological defects to deter- a r X i v : . [ c ond - m a t . s t r- e l ] F e b mine the phase diagram. After identifying three distincttopological defects that are responsible for three differ-ent orders out of the nematic superconducting phase, wefind that the competition between the superfluid stiffnessand the elastic constant leads to a rich phase diagram, asshown in Fig. 1. In particular, if the elastic constant ofthe material is less than one third of the superfluid stiff-ness, the energetically favored nematic vortices are pro-liferated, resulting in the novel charge-4 e phase, whenthe temperature gets raised above the transition tem-perature. In 3D, we employ a Ginzburg-Landau theorynear the transition point where the nematic supercon-ducting order parameter is well described by a simplefield theory up to quartic terms. We are able to showanalytically that the effective mass of the charge-4 e or-der is less than that of the nematic order by treating thethree-fold anisotropy perturbatively, which indicates thecharge-4 e order is favored. We further confirm our re-sults by numerically solving the saddle-point equation.Our result in 3D suggest that materials whose dispersionalong the third direction is weaker than the in-plane dis-persion such as in Sr x Bi Se are promising in realizingthe charge-4 e phase. Charge-4 e phase from proliferating topological defect.— We start by analyzing the symmetries in the nematic su-perconducting phase providing an appropriate languagefor clarifying the topological defects, and then turn torenormalization group analysis of the defect theory. Asmentioned above, we consider the order parameter of thenematic superconducting phase being a two-componentcomplex boson ∆ = (∆ x , ∆ y ) T carrying E u represen-tation in D d group, and each complex component ad-ditionally hosts the U (1) quantum number correspond-ing to the charge conservation, i.e., it is the conven-tional charge-2 e condensate. In the phase respectingtime-reversal symmetry, the relative phase between twocomponents ∆ x , ∆ y is pined at 0 or π . As we focus onthe time-reversal-invariant nematic superconductor, wehereafter assume the relative phase between ∆ x and ∆ y is pined at 0 or π in the analysis of low-energy physics.In terms of the basis ∆ ± = ∆ x ± i ∆ y , one can bring thethe four-dimensional field configuration into three phasemodes and one amplitude mode. Moreover, the time re-versal invariance makes one of the phase modes heavier,thus we have two phase modes and one amplitude modein the low-energy sector,∆ + = | ∆ | e i ( θ + φ ) , ∆ − = | ∆ | e i ( θ − φ ) . (1)where θ and φ denote the two phase modes. The φ fielddescribes the U (1) rotation between amplitudes of twocomponents (i.e. the spatial rotation), whereas the θ fieldis the U (1) phase conjugated to the global charge. Thetwo U (1) phases can also be understood by secondary or-ders, i.e., the nematic order Q ∼ ∆ †− ∆ + and the charge-4 e order ∆ e ∼ ∆ + ∆ − , that transform as Q → Qe iφ , ∆ e → ∆ e e iθ . (2) FIG. 1. Schematic phase diagram of the defect theory. κ , ρ and T denote the nematic stiffness, the superfluid stiffnessand temperature, respectively. The solid lines refer to phaseboundaries. Blow the dashed line, the three-fold anisotropy isrelevant from the lowest order calculation. Nevertheless, thetrue transition to the charge-4 e phase belongs to the three-state Potts model universality class [40] where the three-foldanisotropy is relevant at the transition point. The single-valueness of nematic superconducting or-der parameter ∆ ± uniquely determines that the topolog-ical defects are given by ( δθ, δφ ) = (2 π, , (0 , π ) , ( π, π ),where δθ , δφ denote the winding of the phase around adefect. Physically, they correspond to the superconduct-ing vortex, the nematic double vortex and the supercon-ducting half-vortex binding with a single nematic vortex,respectively. In the nematic superconducting phase, theeffective Hamiltonian characterizing the phase modes is H = ρ ∂θ ) + κ ∂φ ) − g cos 6 πφ, (3)where ρ is the superfluid stiffness (superfluid density)and κ is the nematic stiffness (elastic constant). No-tice that in writing this Hamiltonian, we have changedthe compactification of the phase mode from 2 π to 1 forconvenience. Note that cos 6 πφ is allowed owing to thethree-fold anisotropy, i.e., the action is invariant under φ → φ +1 /
3. Thermal proliferation of topological defectscan be then described by the dual bosons ˜ φ and ˜ θ [3], H = T ρ ( ∂ ˜ θ ) + T κ ( ∂ ˜ φ ) − g cos 6 πφ − g , cos 2 π ˜ θ − g , cos 2 π ˜ φ − g , cos π ˜ θ cos π ˜ φ, (4)where g , , g , and g , are couplings characterizing thestrength of creating (annihilating) each kind of topolog-ical defects, respectively, and T is the temperature.The standard renormalization group flow for Eq. (4)to the lowest-order reads dg , dl = (cid:16) − πρT (cid:17) g , , (5) dg , dl = (cid:16) − πκT (cid:17) g , , (6) dg , dl = (cid:104) − π T ( ρ + κ ) (cid:105) g , , (7) dg dl = (cid:18) − πTκ (cid:19) g . (8)The coefficient in front of the coupling on the right-handside of the renormalization group equation determineswhether the corresponding process is relevant or not.For instance, if Tρ > π , the creation and annihilationprocess of the superconducting vortex is relevant, lead-ing to the proliferation of superconducting phase defectsand, consequently, destroying the superconducting phasecoherence driving the system out of superconducting or-ders. In this way, the phase diagram as shown in Fig. 1is mapped out by the renormalization group equations.Clearly, when the nematic stiffness is less than the su-perfluid stiffness, more specifically κ < ρ , it is easierto create nematic vortices than superconducting vorticessuch that raising temperature can more efficiently pro-liferate the nematic vortices, restoring the lattice rota-tional symmetry but not the U (1) charge symmetry. Asa result, the charge-4 e order emerges since any charge-2 e condensate from the nematic superconductivity breakslattice rotational symmetry whereas the charge-4 e orderis blind to the nematic vortex as shown by the symmetrytransformation law in Eq. (2).Besides the interesting charge-4 e phase, the competi-tion between the nematic stiffness and the superfluid stiff-ness results in a rich phase diagram in Fig. 1. Namely,when < κρ <
3, raising the temperature causes a di-rect transition from the nematic superconductivity tothe normal phase since proliferating a superconductinghalf-vortex bounded with a nematic vortex is favored,whereas, when κρ >
3, a vestigial of nematic phaseemerges since proliferating the normal superconductingphase vortex is favored [41].
Charge- e phase from superconducting fluctuations.— In 3D, the Ginzburg-Landau theory works better sincethe quantum fluctuation is generally suppressed as di-mension increases, so we expect a mean-field analysis to3D nematic superconductivity near the phase boundarycan describe the essential physics. Such a theory wasalso used in Ref. [32] to investigate the vestigial nematicorder, but the authors did not analyze the possibility ofcharge-4 e orders. The Ginzburg-Landau theory of the ne-matic superconducting order parameter ∆ = (∆ x , ∆ y ) T near the phase boundary reads [32] S = (cid:90) p ∆ † G − p ∆ + (cid:90) x ( u (∆ † ∆) + v (∆ † τ y ∆) ) , (9)where (cid:82) p = (cid:82) dp (2 π ) , (cid:82) x = (cid:82) d x , and G − p = m ( p ) τ + m ( p ) τ z + m ( p ) τ x , (10)is the inverse propagator of the nematic superconductingorder parameter with m ( p ) = d (cid:107) ( p x + p y ) + d z p z + r , m ( p ) = d (cid:48) ( p x − p y ) + ¯ dp y p z , and m ( p ) = d (cid:48) p x p y +¯ dp x p z . Notice that while m ( p ) enjoys a continuous ro-tational symmetry, m ( p ) and m ( p ) lower it down tothree-fold rotation. r , u , v are real parameters allowedby the symmetry and d (cid:107) , d z , d (cid:48) , ¯ d characterize the kinetic energy. In the following, we consider v > u >
0, as thecase for v < τ yαβ τ yγδ = 2 δ αδ δ βγ − δ αβ δ γδ − τ xαβ τ xγδ − τ zαβ τ zγδ , (11) τ yαβ τ yγδ = δ αβ δ γδ − δ αγ δ βδ . (12)This leads to either nematic channel for the first oneEq. (11) or charge-4 e channel for the second one Eq. (12).After the decoupling, one is able to integrate out thequadratic nematic superconducting order parameter. Weleave the details to the Supplemental Materials [42], andpresent the main results here. The Ginzburg-Landau the-ory for the nematic order and the charge-4 e order (assum-ing to be homogeneous in real space) are given by S nem = Tr log χ − p + (cid:90) x (cid:18) v ( Q x + Q y ) − u (cid:48) R (cid:19) , (13) S e = 12 Tr log D − p + (cid:90) x (cid:18) v | ∆ e | − u (cid:48) R (cid:19) , (14)where u (cid:48) = u + v , Q x , Q y are the nematic orders, ∆ e is the charge-4 e order, and R ∼ ∆ † ∆ is a scalar thatdecouples the first quartic interaction in Eq. (9). TheTr log term comes from integrating out the nematic su-perconducting order parameter, and χ − p = G − p + R + Q x τ z + Q y τ x , and D − p = G − p + R + ∆ e ρ + + ∆ ∗ e ρ − are the modified inverse propagators in the presence ofthe nematic order and the charge-4 e order, respectively. ρ ± = ( ρ x ± iρ y ) where ρ x and ρ y are Pauli matricesacting on the “Nambu” space, i.e., ∆ e and ∆ ∗ e , and theextra factor in front of the Tr log term in Eq. (14) isdue to the redundancy of the enlarged Nambu space.The Fierz identity and the bosonic nature of the ne-matic superconducting order parameter ∆ cause thesimilar Ginzburg-Landau theory for the nematic orderEq. (13) and the charge-4 e orders Eq. (14). The only dif-ference comes from the Tr log term as the nematic ordercarries a real space (nematic) quantum number while thecharge-4 e order carries a Nambu space quantum number.Unlike the fermionic field, the bosonic nematic supercon-ducting field is blind to the Nambu space as one can easilyobserve that the inverse propagator G − p (10) is an iden-tity in the Nambu space. Thus if one artificially turns off m and m such that the nematic superconducting fieldhas an accidental continuous real space rotational sym-metry at the leading order, the nematic order and thecharge-4 e order will be degenerate since now the nematicsuperconducting field is blind to both nematic space andNambu space (the factor of in Eq. (14) is perfectly com-pensated by the enlarged Nambu space). So even withoutany calculation, one can identify a specific albeit artifi-cial point where the charge-4 e order is degenerate with (a) (b) FIG. 2. The Feynman diagrams that contribute to the ef-fective mass of the charge-4 e order and the nematic order,respectively. The difference of (a) and (b) arises from thedifferent vertices denoted by ρ and τ . the nematic order. Now the question is how the non-vanishing m and m , equivelantly d (cid:48) and ¯ d , affect thetwo instabilities. In the following, we provide an analyti-cal result showing the charge-4 e order is favored over thenematic order when d (cid:48) (cid:29) ¯ d , which makes quasi-2D mate-rials whose out-of-plane dispersion is much weaker thanthe in-plane dispersion promising candidates for realizingcharge-4 e phases. In addition, we also numerically solvethe saddle-point equation for both nematic orders andcharge-4 e orders, which confirms the analytical result.Near the transition point, the effective Ginzburg-Landau theory can be obtained by assuming order pa-rameters small and expanding the action order by or-der. The effective mass that is crucial in determiningthe phase boundary is given by evaluating the Feynmandiagram in Fig. 2, and as we discussed above the differ-ence lies in the vertices of the two orders as shown inFig. 2(a) and Fig. 2(b). Since nonvanishing m and m are diagonal in the Nambu space the contributions fromthe τ z and τ x kinetic energy are additive to the charge-4 e order’s mass; however, since they are not diagonal inthe nematic space the contributions tend to cancel eachother. We calculate the effective mass perturbatively in¯ d , and the lowest-order result is m e = m nem − δm , (15) δm = 1 d (cid:107) (cid:112) d z ( R + r ) 2 γ − (1 − γ ) log (cid:16) γ − γ (cid:17) πγ (1 − γ ) , (16)where γ = d (cid:48) d (cid:107) . It is not hard to see that δm is positiveand monotonic in 0 < γ <
1, indicating the anisotropyinduced by nonvanishing d (cid:48) favors the charge-4 e order.However, this enhancement of the charge-4 e instabil-ity needs to compete with the cubic potential of nematicorders arised from the three-fold anisotropy. The cu-bic term appears at least at the quadratic order in ¯ d and the linear order in d (cid:48) (actually, the dependence of d (cid:48) at quadratic order of ¯ d can be obtained explicitly), andthrough explicit calculation, the cubic term is S nem (cid:51) w (cid:90) ( Q x − Q x Q y ) , (17) w ≈ π d z ( R + r )] / ¯ d d (cid:48) d (cid:107) . (18) (a) (b) FIG. 3. The mean field phase diagram of the charge-4 e order(a) and the nematic order (b). | ∆ e | , and Q = (cid:112) Q x + Q y are the amplitude of the charge-4 e and the nematic orderrespectively. δr = r − r ∗ ∝ T − T ∗ , where r ∗ ( T ∗ ) denotesthe transition point (transition temperature) without vestigialorders. The parameters are d (cid:107) = 1 , d (cid:48) = 0 . , d z = 0 . , ¯ d =0 . , v = 1 , u = 4. It is a simple matter of fact that the cubic term en-hances the instability at least in the quadratic orders,i.e., δm ∝ w ∝ ¯ d , so the lowest order is the quarticorder in ¯ d . On the other hand, the enhancement of thecharge-4 e superconductivity δm appears already in thezeroth order of ¯ d . For the systems described by Eq. (9),there exists a parameter regime, where ¯ d (cid:28) d (cid:48) , such thatthe charge-4 e order is favored over the nematic order.Notice that we have neglected the sixth order terms like (cid:0) ∆ † ( τ z + iτ x )∆ (cid:1) + h.c. in Eq. (9) that are allowed bythe three-fold anisotropy, and these terms can also rendercubic contributions to the nematic order effective actionEq. (17). These higher order terms are in general irrel-evant for the long wavelength physics near and slightlyabove the nematic superconducting transition tempera-ture at which the Ginzburg-Landau theory Eq. (9) canapply. (Whereas if the temperature is lower than thetransition temperature, this becomes dangerously irrele-vant [15] and selects one of the degenerate ground states).Unlike the universal physics that can be obtained withthe help of the Ginzburg-Landau action in Eq. (9), it isnot clear away from the applicability of such an actionwhat the fate of the competition between the charge-4 e order and the nematic order is. Nevertheless, the 2D cal-culation shown in Fig. 1 provides a complementary qual-itative understanding: the large anisotropy from higher-order potentials can increase the nematic stiffness andprohibit the charge-4 e phase [41].To further support our result, we carry out a numeri-cal calculation of the saddle-point equations in both thecharge-4 e channel and the nematic channel. We leave thesaddle-point equation in the Supplemental Materials [42].The phase diagram is shown in Fig. 3, where | ∆ e | , and Q = (cid:113) Q x + Q y are the amplitude of the charge-4 e andthe nematic order respectively. δr = r − r ∗ ∝ T − T ∗ ,where r ∗ ( T ∗ ) denotes the transition point (transitiontemperature) without vestigial orders. That the systemis in favor of the charge-4 e phases when ¯ d (cid:28) d (cid:48) is demon-strated explicitly by T nem < T e in Fig. 3. One shouldalso notice the first-order jump of the nematic transitiondue to the cubic anisotropy, on the contrary, the phasetransition to the nematic phase at two dimensions is con-tinuous due to the strong quantum fluctuations [40]. Conclusions.—
We have revealed the promosing pos-sibility of realizing a novel charge-4 e superconductingphase above the transition temperature of time-reversal-invariant nematic superconductors in 2D and quasi-2D.Since experimental evidences of rotational symmetrybreaking superconducting phases in various systems, in-cluding doped topological insulators Cu x Bi Se [26–29],Sr x Bi Se [33, 34], and Nb x Bi Se [35, 36], supercon-ducting topological semimetal PbTaSe [37], and morerecently twisted bilayer graphene [38, 39], have accu-mulated, we believe that our results pave an importantstep toward experimentally realizing charge-4 e phases inquantum materials. Acknowledgement.—
We thank Wen Huang for help-ful discussions. This work is supported in part by theNSFC under Grant No. 11825404 (HY), the MOSTC un-der Grant Nos. 2016YFA0301001 and 2018YFA0305604(HY), and the Strategic Priority Research Programof Chinese Academy of Sciences under Grant No.XDB28000000 (HY). SKJ is supported by the SimonsFoundation via the It From Qubit Collaboration.
Note added:
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There are two ways to decouple the quartic term in Eq. (9), given by the following two different Fierz identities τ yαβ τ yγδ = 2 δ αδ δ βγ − δ αβ δ γδ − τ xαβ τ xγδ − τ zαβ τ zγδ , (S1) τ yαβ τ yγδ = δ αβ δ γδ − δ αγ δ βδ , (S2)which lead to two exact rewritings of Eq. (9) S = (cid:90) p ∆ † G − p ∆ + (cid:90) x ( u (cid:48) (∆ † ∆) − v [(∆ † τ x ∆) + (∆ † τ z ∆) ]) , (S3) S = (cid:90) p ∆ † G − p ∆ + (cid:90) x ( u (cid:48) (∆ † ∆) − v | ∆ T ∆ | ) , (S4)where u (cid:48) = u + v .The first equation (S3) is readily decoupled by nematic orders, ( Q x , Q y ), S = (cid:90) x (cid:18) ∆ † ( G − + R + Q x τ z + Q y τ x )∆ + 14 v ( Q x + Q y ) − u (cid:48) R ) (cid:19) , (S5)where R is another boson field that decouple u (cid:48) term, and it does not break any symmetry. On the other hand, thesecond equation (S4) is readily decoupled by charge-4 e SC orders, (∆ e , ∆ ∗ e ), S = (cid:90) x (cid:18)