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PHYSICAL REVIEW RESEARCH 2, 013034 (2020)

Mass hierarchy in collective modes of pair-density-wave superconductors

Shao-Kai Jian ,1,2 Michael M. Scherer,3 and Hong Yao1,4,* 1Institute for Advanced Study, Tsinghua University, Beijing 100084, China 2Condensed Theory Center, Department of , University of Maryland, College Park, Maryland 20742, USA 3Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany 4State Key Laboratory of Low Dimensional Physics, Tsinghua University, Beijing 100084, China

(Received 16 October 2018; revised manuscript received 5 February 2019; published 10 January 2020)

We study collective modes near the quantum critical point of a pair-density-wave (PDW) superconductor in two-dimensional Dirac systems. The fate of gaps of various collective modes is investigated by functional renormalization. For incommensurate PDW superconductors, we show that the gapless Leggett mode, protected by the emergent U (1) symmetry, can induce an exponentially small Higgs mass compared to the superconducting gap. Further, for commensurate PDW superconductors, we find an emergent mass hierarchy in the collective modes, i.e., the masses of Leggett , , and the superconducting gap can differ by several magnitudes in the infrared. This may shed light on a mechanism underlying the hierarchy problem in the of physics.

DOI: 10.1103/PhysRevResearch.2.013034

I. INTRODUCTION where ± correspond to two superconducting condensates that are related by time reversal (or inversion) symmetry. Collective modes in superconductors are among the most Under a translation operation, the order parameters transform fascinating emergent phenomena in ±iQ as ± → e ±, where Q = Q · r. Therefore, the phases [1,2], and are further related to the famous Anderson-Higgs of the two complex order parameters manifest themselves as mechanism [3–6]. In the case of charge-neutral , order parameters of global charge conservation and translation spontaneous breaking of global U (1) symmetry provides mas- symmetry. It also manifests in secondary orders induced by sive amplitude and gapless phase fluctuations at low ener- the PDW, i.e., the 2Q charge-density wave (CDW) ρCDW ∼ gies. However, in the context of charged superconductors, ∗ +− and the charge-4e SC 4e ∼ +− [13]. Upon the i.e., interacting with dynamic , the gapless iθ± i(θ+−θ− ) ± → e ± ρ → e ρ Goldstone mode is “eaten” by gauge resulting in transformation , CDW CDW and → ei(θ++θ− ) massive transverse photons and the Meissner effect [7–9]; in 4e 4e. this case, the amplitude mode is also known as Higgs mode. It is clear that the induced CDW order is proportional to the Another collective mode—the Leggett mode [10]—appears in Josephson coupling between the two condensates described by + and −. For incommensurate momentum Q,any superconductors described by a superconducting (SC) order ∗ N − N parameter with multiple components. While the U (1) trans- local Josephson coupling, i.e., ( + ) for any integer , formation from charge conservation corresponds to a uniform is forbidden by the translational symmetry of the Landau phase shift in all components, the Leggett modes describe the theory and consequently, in addition to the global charge U relative phase fluctuations between different condensates. conservation, another (1) symmetry which is characterized One intriguing manifestation of a multicomponent super- by the phase difference between the two condensates emerges. conductor is the pair-density-wave (PDW) superconductor Consequently, the Leggett mode is gapless which is protected U whose order parameter transforms as a nontrivial representa- by the emergent (1) symmetry in the incommensurate PDW tion of both U (1) and lattice translation operations [11–24] phase. Surprisingly, we find that the fluctuations of the gapless (see Ref. [25] for a recent review). Recently, experimental Leggett mode can dramatically renormalize the mass of Higgs evidences of PDW ordering in cuprate high-temperature su- mode. Specifically, we show that the Higgs mass is exponen- perconductors were reported [26]. The superconducting order tially small compared to the superconducting gap—i.e., the parameter of a generic PDW reads gap in spectrum—at low energies, and thus opens up the possibility of a detectable Higgs mode [27–31]inPDW iQ·r −iQ·r (r) = +(r)e + −(r)e , (1) superconductors. On the other hand, for commensurate momentum Q with commensurability N, the minimum integer satisfying 2NQ = π× *[email protected] 2 integer, the emergent U (1) symmetry mentioned above is lowered to a discrete ZN symmetry and the Landau free Published by the American Physical Society under the terms of the energy can then allow the following Josephson coupling term ∗ N N Creative Commons Attribution 4.0 International license. Further at order N, h[(+−) + H.c.] = 2h|+−| cos N(θ+ − distribution of this work must maintain attribution to the author(s) θ−), where h is a constant. In this situation, the Leggett and the published article’s title, journal citation, and DOI. mode will obtain a mass proportional to the strength of the

2643-1564/2020/2(1)/013034(8) 013034-1 Published by the American Physical Society JIAN, SCHERER, AND YAO PHYSICAL REVIEW RESEARCH 2, 013034 (2020)

as valley degrees of freedom and are denoted by n =±.We y consider a finite intravalley pairing, i.e., n ∼ ψnσ ψn = 0, which breaks the translation symmetry of the underlying lattice. Under translation of the primitive lattice constant Dirac and the order parameters transform as ψ± → ±iK ±i2K e ψ± and ± → e ±, where K = 2π/3 and we set the lattice constant to unity. Thus the intravalley pairing state is a PDW superconducting state with commensurability N = 3. Such a state can, for example, be realized in the honeycomb model with nearest- and next-nearest-neighbor interactions [21]. In the charged PDW phase described above, its low-energy physics is described by the Abelian Higgs model    − 2  F 2 r 2 u 4 S = + |(∂μ − iAμ) | + | | + | | 4e2 n 2 n 4! n x n=±  u 2 2  3 ∗3 + |+| |−| + h (+− + H.c.) +··· , (2) 4 FIG. 1. (a) Schematic flow diagram near the   ≡ 3 between a (semi)metal and a PDW superconductor. The two axes where x d x, A and F are the vector potential and represent the tuning parameter and the Josephson coupling strength, the field strength, respectively, and e is the effective charge respectively. There are two fixed points, indicated by red points, of Cooper pairs. The particle-hole symmetry of supercon- corresponding to the superconducting (SC) transition point and the ductors rules out a linear kinetic term [43,44]. Note that, Nambu-Goldstone (NG) fixed point. The black and blue arrows in Eq. (2), the gapped fermions are ignored for simplic- indicate the flow for the incommensurate and commensurate PDW ity as their inclusion does not qualitatively change the states, respectively, distinguished by whether the Josephson coupling discussion of the Higgs mechanism. In terms of ampli- iθ± is vanishing or not. Panels (b), (c), and (d) are the flow diagrams of tude and phase modes, ± = ϕ±e , the kinetic energy 2 the potential coefficients in the incommensurate PDW phase. The is expressed as |(∂μ − iAμ)±| =|∂μϕ± + iϕ±(∂μθ¯ − Aμ ± horizontal axis represents the flow parameter t ≡ log  /. 2 0 ∂μθ )| , where θ¯ = (θ+ + θ−)/2 and θ = (θ+ − θ−)/2 corre- sponding to Goldstone and Leggett modes, respectively. In N unitary gauge, Aμ → Aμ + ∂μθ¯, the Goldstone mode is eaten order-N Josephson coupling JN ∝ 2h|+−| . As a result, the commensurability N provides a knob to tune the mass by the gauge field via the Higgs mechanism, while the gauge of the Leggett mode: for larger commensurability (larger N), field obtains a mass. The mass of the is set by | | the Josephson coupling is more irrelevant, and the Leggett the amplitude of the SC order parameter, i.e., ± , which is boson mass gets smaller at low energies. In 2 + 1 dimensions, comparable to the SC gap of the Dirac fermions. Thus, as far the Josephson coupling is dangerously irrelevant for N  3, as the physics below the SC gap is concerned, both the eaten- which will result in an interesting hierarchy [32–35]ofthe up Goldstone mode and the gauge boson can be neglected various masses of collective modes as we will show below. [45]. Note that it is reasonable to neglect the eaten-up Gold- In the following, we implement a functional renormaliza- stone mode and the gauge boson in both incommensurate and tion group (FRG) approach [36–38] to investigate collective commensurate PDW phases, although the discussion above is = modes in both incommensurate and commensurate PDW su- focused on the PDW phase with commensurability N 3. perconductors in Dirac systems. The FRG is a nonperturba- tive approach to evaluate the effective action—namely, the III. COLLECTIVE MODES IN INCOMMENSURATE PDW one-particle irreducible generating functional—at any energy scale below the cutoff. Importantly, it allows one to study An incommensurate PDW can occur, e.g., through intraval- generic potential functions irrespective of whether they are ley pairing in a nematic Dirac semimetal that breaks the perturbatively renormalizable or beyond [39–41]. Thus the C3 symmetry of the underlying honeycomb lattice. Without FRG method is a suitable approach for the investigation of C3 symmetry, the Dirac point is still locally stable, but the  bosonic collective modes, where the effective potential is momentum of the Dirac point is no longer locked at K or K . crucial for the determination of various gaps in the symmetry- A possible example is the twisted bilayer , where an broken phase. intermediate C3 nematic semimetal phase is proposed [46]. At a generic momentum, the PDW is incommensurate with the underlying lattice and an additional U (1) symmetry emerges II. PDW STATE IN HONEYCOMB DIRAC SEMIMETALS as discussed above. The three boson degrees of freedom, i.e., We consider the PDW state of spinless fermions on a two Higgs modes ϕ± and one Leggett mode θ, can be changed ±iθ honeycomb lattice close to a quantum phase transition [21,42] to three real bosons, φ± and φ, i.e., ± = ϕ±e → ± = as a primary example [see Fig. 1(a) for a schematic phase φ± ± iφ. diagram]. The half-filled honeycomb lattice hosts two Dirac Now, we are ready to write down the bare action for cones at K and K in the Brillouin zone, which are referred to the incommensurate PDW state with two Dirac fermions

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S = S + S with −1 = 0 1 where D is the space-time dimension and KD    D−1π D/2 / ∂φ 2  ∂φ 2 2 (D 2). The RG flow of U can be projected ( ) ( n ) † ρ S0 = + + Hn n , (3) to the flow of the minimum of the potential 0 and 2 2 n λ  x n=±  the interaction coefficients i. Their dimensionless  λ versions are given asρ ¯ ≡ Z 2−Dρ , λ¯ ≡ Z−2D−4λ , = λ ρ − ρ 2 + 2 ρ − ρ 2 0 b 0 2 b 2 S1 11( + −) ( n 0 ) 2 ≡ −2 −1D−4 2,... 2 g¯ Z f Zb g . The mass terms appearing in x n=±  Eq. (6) and the following flow equations are evaluated λ = φ ,φ + 3 ρ − ρ 3 + †σ y φ μx + φμy , at generic field configurations, m± m±( ± ) and ( n 0 ) g n ( n n ) n (4) 6 mL = mL (φ±,φ). At the minimum of the potential, two Higgs 2 2 modes are given by m+ = 2λ2ρ0 and m− = 2λ2ρ0 + 8λ11ρ0, ± where is the in Nambu space with Pauli while the Leggett mode remains massless, mL = 0, due to i i matrices μ and σ acting on Nambu space and Dirac space, the emergent U (1) symmetry. The superconducting gap is ρ 2 2 respectively. In Eq. (4), the terms concern the bosonic = 2g ρ0. The flow equation for the Yukawa coupling action, while the last term originates from Yukawa coupling reads x y z in Nambu space. H± =−iω ± kxσ + kyσ μ is the kinetic  η 4 D+1 1 − f Z2 term of the fermions and we have further introduced ρ± ≡ 2 g KD D+1 f 1 ∂g = 1 (φ2 + φ2 ). λ characterizes the boson potential and g is 22 + 2 22 + 2 22 + 2 2 ± i 2D Z f Z f Zb m+ the Yukawa coupling. We consider a time-reversal invariant η Z 1 − b PDW phase, so the minimum√ of the potential is chosen to be + 1 − 2 + b D+2 located at φ±,min = 2ρ0 and φmin = 0. In Eq. (4), there is 22 + 2 22 + 2 22 + 2 2 Zb m− Zb mL (Zb m+) no Josephson coupling because of the incommensurability of  η η − b − b the PDW phase under consideration and the emergent U (1) Zb 1 + 2Zb 1 + + D 2 − D 2 . (7) symmetry renders the Leggett mode massless. 2 2 2 2 2 (Z  + m−) 22 + 2 b Zb mL

IV. FUNCTIONAL RG ANALYSIS Note that in the symmetry-breaking phase, the renormaliza- tion of Yukawa coupling g2 [48,49] is not vanishing. Finally, We use the FRG approach to study the superconducting the anomalous dimensions are related to field renormalization −1 gap and the masses of the collective modes. The exact flow factors by ηi =−Z ∂Zi, 1 (2) −1 i equation [36] reads ∂ = Tr[∂R( + R) ], where  2 2 D+2 2 D 2  8K λ ρ Z  8KDg  Z denotes the flowing effective action with energy scale and η = D 2 0 b + f (2) b 2 is the second functional derivative of the effective action 2 + 2 2 2 + 2 ZbD D(Zb m+) Zb mL with respect to boson and fermion fields. Furthermore, R is a   η η + − 4 f 22 3 − f suitable cutoff function. We implement the extended local po- D 2 − Z f  × D 1 − 4 2 , tential approximation (LPA ) considering the following ansatz 3 2 (8) (D − 2) Z22 + 2 Z22 + 2 of effective action  = 0 + 1 with f f D+2 2    ηb 8KDZb g ∂φ 2  ∂φ 2 η f = 1 − ( ) ( n ) † + 22 + 2  = Z + Z + Z H , D 1 D Z f 0 b 2 b 2 f n n n   x n=±   1  1 1  λ × + . (9)  = λ ρ − ρ 2 + 2 ρ − ρ 2 2 2 1 11( + −) ( n 0 ) 2 Z 2 + m2 Z 2 + m2 2 n b n b L x n=±  λ In Fig. 1, we show the flow diagrams for the couplings 3 3 † y x y 2 + (ρ − ρ ) + g σ (φ μ + nφμ ) , (5) λ¯ 2,λ11 andg ¯ as a function of flow parameter t = log 0/. 6 n 0 n n n Here, 0 is the cutoff energy of the bare action. The initial , ∈{ , }, values of the RG flow are chosen in the PDW regime close to where Zi i b f are field renormalization factors. We λ¯ implicitly assume Higgs and Leggett modes have the same the transition point. The flow diagram of 2,Fig.1(b),shows two plateaus corresponding to the PDW transition point and field renormalization factor. Note that the fields φ±, ±, and φ in the effective action are expectation values and different the Nambu-Goldstone (NG) fixed point of the broken U (1) from the fields in the bare action. For notational convenience symmetry owing to the incommensurability. Note that the we use the same symbols. PDW transition point is a critical point, while the NG fixed = 2 − 2 θ 2 − 2 point is a stable fixed point characterizing the gapless Leggett With cutoff functions [47] Rb Zb( k ) ( k ) 2  modes. The flow diagram ofg ¯ ,Fig.1(c), only shows the and R f = Z H r ( ), where r (x) = (x − 1)θ(x2 − 1), the n f n f k f PDW transition plateau, because the fermions are gapped out flow equation for the bosonic potential U reads in the SC phase and decouple from the low energy sector at   η η g2 D+1  − b − b the NG fixed point. Thus, the flow of ¯ is set by its canonical KDZb 1 D+2 1 D+2 ∂U = + dimension at the NG fixed point. The flow diagram of the D Z 2 + m2 Z 2 + m2 n b n b L dimensionful λ11 shows its irrelevance at the critical point η 2D+1  − f [21]. KDZ 1 + − 2 f D 1 , (6) After identifying two fixed points, we can study in more D Z22 + 2g2ρ n f n details the flow of the SC gap and Higgs boson mass towards

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FIG. 2. Flow diagrams of the superconducting gap and mass of amplitude mode at the incommensurate PDW phase. We have scaled FIG. 3. Flow diagrams of the potential coefficient λ¯ ,SCgap, 2 2 2 the initial value to 1. Panel (a) shows the flow of and m+. and masses of various collective modes at the commensurate PDW Notice that the flow quantities have dimension mass squared. Panel phase. We have scaled the initial value to 1. Panel (a) shows the 2 2 (c) shows the flow of the ratio m+/ . 2 flow of dimensionless coefficient λ¯ 2 . Panel (b) shows the flow of superconducting gap, Higgs boson mass, and Leggett boson mass 2 2 2 low energies and we focus on one of the Higgs modes, i.e., denoted by , m+,andmL, respectively. Notice that the flow 2 m+, for simplicity. We show the flow of the dimensionful quantities have dimension mass squared. 2 2 squared masses and m+ in Fig. 2(a). At the energy scale controlled by the PDW transition point, the RG flows of SC Leggett boson mass is proportional to the strength of the gap and Higgs boson mass are almost identical. At lower Josephson coupling. energies, the physics is controlled by the NG fixed point: To study the flow of collective modes in the commensurate while 2 stops flowing because it decouples from the low 2 PDW, we set initial values in the PDW regime close to the energy sector, m+ continues to flow due to fluctuations of transition point. Figure 3(a), the flow diagram of λ¯ ,shows 2 2 the massless Leggett mode. Eventually, m+ flows to zero two fixed points corresponding to the PDW transition point at extremely low energies [50]. Figure 2(b) shows the flow and the NG fixed point similar to that in the incommensurate of the ratio between the SC gap and Higgs boson mass. PDW state. However, the NG fixed point is unstable in the After the system enters the energy scale controlled by the commensurate PDW state due to the runaway flow of λ¯ 2 after NG fixed point, the Higgs mass gets exponentially smaller the NG fixed point. This behavior originates in the Josephson compared to the SC gap at low energy. This provides a robust coupling which is dangerously irrelevant at the PDW transi- energy window where the Higgs modes are detectable in an tion point and triggers the runaway flow of λ¯ 2.InFig.3(b), incommensurate PDW superconductor. The response of the we show the flow diagrams of the SC gap and the masses of Higgs mode to external probe is similar to that in neutral the collective modes. In the energy range controlled by the / SC superfluid [50,51]. PDW transition point, the flows of SC gap 2 and Higgs mass 2 m+ are identical. More interestingly, the flow of the Leggett V. COLLECTIVE MODES IN COMMENSURATE PDW 2 boson mass mL is faster, because of the irrelevance of h at the We now study the case of commensurate PDW. Due to the PDW transition. Note that it is a robust feature insensitive to commensurability N = 3, we add a Josephson coupling term the initial values [50]. Thus it provides an energy window to S ∝ 3 ∗3 + H.c. to the action which couples the two SC detect the Leggett boson, i.e., the (dangerous) irrelevance of J + − the Josephson coupling makes the Leggett mode detectable in condensates. In terms of real bosons SJ reads   commensurate PDW.  In the energy range controlled by the NG fixed point = ρ3 − φ + φ 3 φ + φ 3 + . . . SJ h 8 n [( + i ) ( − i ) H c ] only the Higgs boson mass continues to flow, similar to the x n  incommensurate PDW state, while both SC gap and Leggett ρ3 We added n n such that SJ is nonnegative√ and the minimum boson mass stop running. Finally, when the system reaches of the potential is still at φ±,min = 2ρ0, φmin = 0. Including lower energies, all masses stop flowing and remain finite: a term corresponding to SJ in the truncation, the flow equa- unlike the incommensurate PDW state where the enhanced tions of the potential and fermion anomalous dimension are U (1) symmetry protects gapless Leggett modes, there is no the same as Eqs. (6) and (9), except the masses are different protected gapless mode in the commensurate PDW phase. due to the presence of the Josephson coupling. The flow The presence of two fixed points gives rise to an interesting equation of boson anomalous dimensions is emergent hierarchy of boson masses [33], which may shed light on a mechanism underlying the hierarchy problem in the D+2 2 2 D 2 8K Z  (λ + 288hρ ) ρ 8KDg  Z η = D b 2 0 0 + f standard model of [32]. b 2 2 + 2 2 2 + 2 DZb D(Zb m+) Zb mL   η η + − 4 f 22 3 f VI. CONCLUDING REMARKS D 2 − Z − × D 1 f − 4 2 . (10) (D − 2) Z22 + 2 3 Z22 + 2 2 By using the FRG method, we show that (a) in an incom- f f mensurate PDW superconductor, the Higgs mass is exponen- The masses of Higgs mode and Leggett mode are given by tially smaller than the superconducting gap near the super- 2 = λ ρ 2 = ρ2 m+ 2 2 0 and mL 288h 0 , respectively. Note that the conductor transition point, due to the gapless fluctuations of

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Leggett modes, and (b) in the commensurate PDW phase, the Leggett boson mass is finite but exponentially small compared to the Higgs boson mass and superconducting gap, i.e., a mass hierarchy of the collective modes emerges. While we studied the PDW state in Dirac semimetals as an explicit example, we note that the results are robust in nodeless PDW superconductors: in the incommensurate PDW state, the gapless Leggett modes—which are protected by the emergent U (1) symmetry—can strongly renormalize the Higgs mass. These findings are in general correct in nodeless incommensurate PDW superconductors. On the other hand, FIG. 4. Flow of the mass of the Leggett mode and the mass in commensurate PDW superconductors, the mass hierarchy ratio for different choices of the initial parameters, namely, at (ρ ¯0 − between Higgs boson and Leggett boson masses relies on the ρ¯ ∗ )/ρ¯ ∗ = 10%, 1%, and 0.1% away from the critical point in the fact that the Josephson coupling is dangerously irrelevant at symmetry-broken phase. the PDW transition point. This happens generically in PDW states with high commensurability.

ACKNOWLEDGMENTS nonperturbative functional RG approach, we find that the quantum fluctuations render the Josephson coupling irrele- We thank Shuai Yin for helpful discussions. This work is vant at the critical point, which is also in agreement with supported in part by the NSFC under Grant No. 11825404 simpler perturbative arguments. Here, we add another non- (S.-K.J. and H.Y.), the Simons Foundation via the It From perturbative reasoning, which further supports our finding Qubit Collaboration (S.-K.J.), the MOSTC under Grants No. for the PDW transition in the Dirac semimetal as consid- 2016YFA0301001 and No. 2018YFA0305604 (H.Y.), the ered in the manuscript: the PDW transition for this specific Strategic Priority Research Program of Chinese Academy system features emergent (SUSY) [21]. More of Sciences under Grant No. XDB28000000 (H.Y.), Beijing explicitly, it is described by two decoupled copies of N = Municipal Science & Technology Commission under Grant 2 Wess-Zumino supersymmetric theory leading to an exact No. Z181100004218001 (H.Y.), and Beijing Natural Science scaling dimension of complex boson, [±] = 2/3, and con- Foundation under Grant No. Z180010 (H.Y.). M.M.S. was N straining the quantum fluctuation such that [±] = N × [±] supported by the DFG, Projektnummer 277146847 – SFB [52]. Since two valleys are decoupled at criticality, we have 1238 (project C02, C04). ∗ N ∗ N N [(+−) ] = [(+) ] + [(−) ] = 4N/3 and [h¯N ] = 3 − 4N/3. As a result, the mass ratio is APPENDIX 1. Mechanism of the emergent mass hierarchy − / m2 h¯ ( )¯ρN−2()  3 4N 3 Here we would like to articulate more about the origin of L ∝ N 0 0 , 2 (A2) the mass hierarchy between Leggett mode and Higgs mode in m+ λ¯ 2() 0 the commensurate pair-density-wave (PDW) superconductor. In short, it is due to the irrelevance of Josephson coupling at criticality. More specifically, this mass hierarchy can be where 0 is the energy cutoff and  represents the energy quantified by the inverse ratio of the masses of the Higgs mode scale. Becauseρ ¯0() and λ¯ 2() stay roughly constant near 2 ∝ λ ρ 2 ∝ ρN−1 m+ 2 0 and the Leggett mode mL hN 0 , where hN is the transition point, we can see the reason of mass hierarchy a Josephson coupling with commensurability N, is due to the irrelevance of Josephson coupling. Moreover, the m2 h¯ ρ¯ N−2 mass hierarchy is more apparent for the larger N. L ∝ N 0 . (A1) We clearly exhibit this result by showing the renormaliza- 2 λ¯ m+ 2 tion group flow of the mass for different choices of tuning Near the critical point, the RG flows of the dimensionlessρ ¯0 parameters away from criticality; see Fig. 4. This confirms and λ¯ 2 show plateaus, i.e., their values stay nearly constant. that an initial value which is closer to the critical point induces 2 / 2 On the other hand, at the critical point the dimensionless h¯N a smaller Leggett mass and mass ratio mL m+ as argued will flow to zero asymptotically, because, here, the Josephson above. Note the small increase of the mass ratio in the figure is coupling is irrelevant for large enough N (N  3). The longer due to the Nambu-Goldstone (NG) fixed point, which further the RG flow is dominated by the critical point, the smaller the decreases the Higgs mass but not Leggett mass. 2 / 2 ratio mL m+ becomes, independent of the initial value of the The above results explain the mass hierarchy between Josephson coupling. This can be controlled by tuning closer the Leggett mode and the Higgs mode. Next, we provide a to criticality. Eventually, in the deep infrared, both masses physical reason for the mass hierarchy of the Higgs mode freeze-out and give a constant tiny but finite mass ratio—the and the superconducting (SC) gap. From Fig. 2 and Fig. 3 mass hierarchy. in the main text, we can see that the mass hierarchy emerges 2 /2 ∝ The canonical scaling dimension of h¯N is (2 − D)N + D. near the NG fixed point. Because the mass ratio is m+ 2 For D = 3 and N = 3, h¯N is marginal at tree level and λ¯ 2/g¯ , the hierarchy is due to the different IR behavior of λ¯ 2 quantum fluctuations have to be considered. Employing the andg ¯2. The FRG equations near the incommensurate PDW

013034-5 JIAN, SCHERER, AND YAO PHYSICAL REVIEW RESEARCH 2, 013034 (2020) transition are given by λ¯ 2 2 λ¯ 2 3 2 g¯ 2 2 ∂λ¯ =−λ¯ + − + , 2 2 2 3 2 3 2 π (1 + 2λ¯ρ¯0 ) (1 + 2¯g ρ0 ) 3π  (A3) 4 2 2 g¯ 1 ∂g¯ =−g¯ + − 1 2 2 2 6π (1 + 2¯g ρ¯ ) 1 + 2λ¯ 2ρ¯0  FIG. 5. One-loop diagrams containing singular polarization. The 1 dashed lines and the wavy lines represent Higgs mode propagator + (1 + 2¯g2ρ¯ ) − 1 . (A4) 2 and Leggett mode propagator, respectively. (1 + 2λ¯ 2ρ¯0 ) The above equation allows us to directly analyze the behavior of the system at the NG fixed point: since the NG fixed point set λ11 = 0 for simplicity, i.e.,   describes the symmetry broken phase, the relevant tuning  ρ0 parameter diverges, i.e.,ρ ¯ →∞, and the RG equations L = (∂φ )2 + (λ ρ + 2λ ρ )φ2 + λ φ3 + φ φ2 0 n 2 0 11 0 n 2 2 n n reduce to n λ¯ 2 λ 2 2 2 2 4 ∂λ¯ =−λ¯ + , (A5) + (∂φ) + φ . (A10) 2 2 3π 2 8 2 2 ∂g¯ =−g¯ . (A6) The polarization operator of the Higgs mode is given by 2  The flow ofg ¯ is set by its canonical dimension because 1 dDk 1 (q) = the quantum corrections vanish due to the decoupling of the 4 (2π )D k2(k + q)2 fermion from the low-energy sector. With the solutions of the √ 3−D π / −  − / RG equation at the NG fixed point, = 1 1 2 (D 2 1) (2 D 2). π D/2  / − / 4−D 1  4 (4 ) (D 2 1 2) q λ¯ () = , g¯2() = g¯2() 0 , 2 2 + 1 − 2   (A11) 2 2 3π λ¯ 2 (0 ) 3π 0 (A7) Since (q) = 1 √1 in 2 + 1D, the self-energy suffers from 32 q2 we can get the mass ratio an IR singularity, making the direct observation of longitudi- 2 2 nal fluctuations difficult. Indeed, the spectral function of the m+ 3π  ∝ . (A8) longitudinal susceptibility at one-loop order is given by 2 2   2¯g ( 0 ) 0 π  χ  =  δ ω − 2 + λ ρ Physically, the mass hierarchy is due to the fact that the φ φ ( q 2 0 ) n n 2 + λ ρ fermion decouples from the low-energy sector at the NG fixed 4 q 2 0 point, while the Higgs mode does not. λ2ρ 1 1 + 0 0  (ω2 − q2 ), 2 2 2 128 (ω − q − λ2ρ0 ) ω2 − q2 2. Observability of Higgs mode in incommensurate (A12) PDW superconductors In this section, we discuss the observability of Higgs modes where  is the step function. The first term is the quasiparticle in PDW superconductors (SC) in 2 + 1D. One necessary peak of the Higgs mode and the second term comes from the condition of a well-defined Higgs mode is Lorentz symme- decay to the Leggett mode. In the static limit, χ  ∼ ω−1, φnφn try. In condensed matter physics, which typically provides making the Higgs peak difficult to observe. Note that such a a nonrelativistic environment, the particle-hole symmetry in situation is ubiquitous for the amplitude mode in spontaneous superconductors plays an essential role similar to Lorentz continuous symmetry breaking in 2 + 1D; the gapless Gold- symmetry [44]. In the incommensurate PDW SC, it is not stone mode fluctuations inhibit probing the longitudinal sus- obvious whether the Higgs mode can be probed in experi- ceptibility [51,53]. In order to probe the signature of the Higgs ments due to the gapless fluctuations of the Leggett mode. modes, one can instead consider the scalar susceptibility. The φ2+φ2 The effective theory, i.e., Eqs. (3) and (4), in terms of Higgs δρ ≡ √1 1 | |2 − ρ = φ + √n scalar is defined as n ρ ( n 0 ) n ρ , and Leggett modes is 2 0 2 2 2 0   and the scalar susceptibility is  ρ0 L = (∂φ )2 + (λ ρ + 2λ ρ )φ2 + λ φ3 + φ φ2 n 2 0 11 0 n 2 2 n n 1 n χδρ δρ = χφ φ + √ χφ φ2 + χφ φ2 n n n n ρ n n n λ 2 0 2 2 4 + (∂φ) + φ − 4λ11ρ0φ+φ− 1 1 8 + χφ2φ2 + χφ2φ2 + χφ2φ2 . (A13) ρ n n ρ n λ  8 0 4 0 11 2 + (φ+ − φ−) (2 2ρ0 + φ+ + φ−). (A9) 4 We calculate the susceptibility by a weak coupling expansion Note that λ11 only couples the Higgs modes. To address the (loop expansion), and present the one-loop result. The singu- effect of the gapless fluctuation of the Leggett mode, one can lar loop diagrams shown in Fig. 5 lead to the singular part of

013034-6 MASS HIERARCHY IN COLLECTIVE MODES OF … PHYSICAL REVIEW RESEARCH 2, 013034 (2020) the susceptibility, the amplitude-angle decomposition, =||eθ ,itisthetwo 2 1 derivatives appearing in the coupling ||(∂θ) that lead χ singular = λ2ρ χ  χ − λ χ  +  δρ δρ (q) 2 0 0(q) (q) 0(q) 2 0(q) (q) (q) to a suppression. We emphasize that the Higgs mode in n n 4ρ 0 condensed-matter systems is not like the Bogoliubov quasi- q4 = (q), (A14) particle that can be detected by tunneling experiments. As 2 2 4ρ0(q + λ2ρ0 ) a neutral collective mode, the Higgs mode is not easy to χ = 1 1 excite and detect. The different self-energy from the different where 0(q) 2+λ ρ is the bare propagator. One can see 2 q 2 0 decomposition of order parameters by perturbative calculation 4 in the singular at low energy is suppressed by a factor q , i.e., will not lead to a direct experimental observation. If both   (ω2 − q2 )2 perturbative calculations remain qualitatively correct when χ singular ω, =  ω2 − 2 . δρ δρ ( q) ( q ) high-order corrections are included, the lesson above tells n n 128ρ (ω2 − q2 + λ ρ )2 0 2 0 us that the Higgs mode can be observed by a suitable ob- (A15) servable in experiments. For example, besides the contribu-  tion from the in a superconductor, The static scalar susceptibility is given by χδρ δρ (ω) = √ n n 3 the amplitude mode can lead to an excess contribution to √π δ ω − λ ρ + ω + λ ρ ( 2 0 ) ρ ω2− 2+λ ρ 2 regular terms, 4 2 0 128 0 ( q 2 0 ) the dynamical conductivity. The leading contribution from the where we can see clearly that the singularity is well amplitude mode is theoretically shown to have a hard gap suppressed. As a consequence, the peak of the Higgs modes at the frequency set by the mass of the Higgs mode [51]. can be observed experimentally in the scalar susceptibility The experiments in disordered NbN and InO film confirm without covering from the IR divergence. the extra contribution in dynamical conductivity near the The suppression in the above calculation is actually due superconductor- transition [30]. to the different decompositions of the order parameter. In

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