Thermal effects on collective modes in disordered s-wave superconductors
Abhisek Samanta, Anirban Das, Nandini Trivedi, Rajdeep Sensarma
TThermal effects on collective modes in disordered s -wave superconductors Abhisek Samanta, ∗ Anirban Das, Nandini Trivedi, and Rajdeep Sensarma † Physics Department, Technion, Haifa 32000, Israel School of Physical Sciences, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India Department of Physics, The Ohio State University, Columbus, Ohio, USA 43201 Department of Theoretical Physics, Tata Institute of Fundamental Research, Mumbai 400005, India. (Dated: March 1, 2021)We investigate the effect of thermal fluctuations on the two-particle spectral function for a disordered s -wavesuperconductor in two dimensions, focusing on the evolution of the collective amplitude and phase modes.We find three main effects of thermal fluctuations: (a) the phase mode is softened with increasing temperaturereflecting the decrease of superfluid stiffness; (b) remarkably, the non-dispersive collective amplitude modes atfinite energy near q = [0 , and q = [ π, π ] survive even in presence of thermal fluctuations in the disorderedsuperconductor; and (c) the scattering of the thermally excited fermionic quasiparticles leads to low energyincoherent spectral weight that forms a strongly momentum-dependent background halo around the phase andamplitude collective modes and broadens them. Due to momentum and energy conservation constraints, thishalo has a boundary which disperses linearly at low momenta and shows a strong dip near the [ π, π ] point in theBrillouin zone. I. INTRODUCTION
The quantum phase transition between superconductingand insulating phases of two dimensional films, driven by in-creasing disorder , provides a paradigm for the complex in-terplay of interaction and localization . The single parti-cle fermionic spectrum remains gapped throughout the transi-tion and the fluctuations of the local superconducting orderparameter, that describe the phase (Goldstone) and amplitude(Anderson-Higgs) collective modes, are the key low energyexcitations that drive this phase transition .The Higgs mode in superconductors has been an active areaof research for a long time. While the observation of theHiggs particle in particle colliders has been hailed as oneof the recent successes in that field, the corresponding modehas not been observed in a clean superconductor. This is dueto the fact that in a clean superconductor, this mode sits atthe two-particle continuum threshold and is damped . Earlypredictions that the mode can be seen as a subgap feature insystems with accompanying charge density order has recentlybeen experimentally verified . The area has also received alot of attention due to clean observation of the Higgs mode ina charge neutral ultracold atomic system near the superfluid-insulator transition. More recently, observation of low energyoptical spectral weight in disordered superconductors closeto a disorder driven superfluid-insulator transition has led toa conjecture that this weight is due to the Higgs mode, basedon earlier work on optical conductivity in clean systems .More recent theoretical work has looked at the questionof the contribution of collective modes to optical conductivityin disordered superconductors at zero temperature.In an earlier work , we had studied the evolution of two-particle pair spectral functions in a disordered superconductorand presented their full momentum and frequency dependenceas a function of disorder at zero temperature. We had foundthe expected (a) continuum of two-particle excitations, abovean energy threshold equal to twice the single-particle gap, and (b) linearly dispersing low energy collective modes. In addi-tion, surprisingly, we found additional spectral weight at finiteenergies below the two-particle continuum in the long wave-length limit. The weight in this non-dispersive feature, whichwas spectrally separated from the linearly dispersing collec-tive modes, increased with increasing disorder strength. Wewere able to correlate this non-dispersive spectral weight withthe Higgs mode and the low energy Higgs weight was con-centrated in this additional spectral feature in the two-particlepair spectral function. One obvious question is how does thatpicture change in the presence of temperature? It is crucialto understand the combined role of both thermal and quantumfluctuations in order to make connections with experimentaldata.We had also found that at arbitrarily weak disorder, the zeromomentum Anderson-Higgs mode that sits at the threshold ofthe two-particle continuum, shifts non-perturbatively withinthe two-particle gap. This subgap feature of the Anderson-Higgs mode is distinguishable from the low energy phase di-vergence at all disorder values, making it a possible candi-date to observe in energy resolved spectroscopies. There-fore, the natural question that arises is: what is the fate ofthe Anderson-Higgs mode as a function of temperature? Is itstill possible to separate this mode from the phase fluctuationsat finite temperatures? We address these important questionsin this work.Our theoretical approach involving functional integrals al-lows us to investigate the different contributions of the ampli-tude, the phase, and amplitude-phase mixing fluctuations tothe two-particle spectral function at finite temperature. Thekey features of our analysis are the following: i) We obtainthe evolution of the Anderson-Higgs and the Goldstone modeas a function of both temperature and disorder. ii) We find thatsmall thermal fluctuations induce additional low energy inco-herent spectral weight that forms lobes below the two-particlegap, which compete with the collective mode structure in theamplitude sector, but keep the phase sector mostly unaffected. a r X i v : . [ c ond - m a t . s up r- c on ] F e b iii) In presence of weak disorder, the subgap Anderson-Higgsmode can be observed separately from the low energy phasepile-up in an energy resolved way even in presence of moder-ately high temperatures, thereby making it a robust feature ofdisordered superconductors. We note that an alternative ap-proach based on an effective classical Monte Carlo has beenused to treat disordered superconductors at finite tempera-ture , but it does not provide momentum resolved informa-tion about the amplitude and phase fluctuations.The two-particle continuum at T = 0 is formed microscopi-cally by breaking up a Cooper pair into a pair of single-particleexcitations, as shown in Fig. 1 (A) and requires a threshold en-ergy of twice the single-particle gap. On the other hand, thecollective mode is better described in terms of the long wave-length fluctuations of the amplitude and phase of the conden-sate of the Cooper-pairs (i.e. the order parameter). As temper-ature is raised, changes in the pair spectral function occur bytwo processes: (a) The collective mode dispersion flattens asthe superfluid stiffness is reduced at finite temperatures due tothermally excited quasiparticles. (b) Additionally, another in-coherent continuum is formed due to scattering of these ther-mally populated quasiparticles, as shown in Fig. 1 (B). Thisleads to a low energy diffuse background weight and conse-quent broadening of the collective modes. In a clean system,the incoherent thermal excitations occur only below an up-per energy cut-off (cid:15) cut − off ( q ) determined by energy and mo-mentum conservation in the scattering process. (cid:15) cut − off ( q ) varies linearly at long wavelengths and shows a prominent diparound the commensurate wave-vector [ π, π ] .In a weakly disordered system, the behavior of the energy-cutoff and its momentum-dependence continues to hold withsmall corrections. As a result, the non-dispersive spectralweight observed at finite subgap energy at long wavelengthsremains sharp at finite temperatures for weakly disorderedsystems. For strongly disordered systems, the constraint dueto momentum conservation in a scattering process is no longerapplicable, and we find the incoherent spectral weight as adiffused halo without sharply defined boundaries. Since thelow energy weight in the diffuse halo comes from the scat-tering of thermally excited quasiparticles, it is exponentiallysuppressed at low temperatures, and significant weight devel-ops only when a fraction of the critical temperature T c is ap-proached.In disordered systems, the presence of a reasonably sharpthreshold of the incoherent weight at long wavelengths leadsto a clear visibility of the long wavelength finite energy weightin the Higgs spectrum. This spectral feature, which was seenclearly in the T = 0 calculations , thus survives thermal fluc-tuations in the system. This is a key insight that we obtainfrom these calculations.The rest of the paper is organized as follows: In Section IIwe discuss the model Hamiltonian for disordered supercon-ductors and the finite temperature mean-field theory. In Sec-tion III, we first discuss the technical details of the finite tem-perature gaussian fluctuation calculation, before turning ourattention to the pair spectral function in clean systems at finitetemperature in Section III A. In Section III B, we focus on the T=0
Collective Mode Scattering of QP T ≠ Creation of QP pairCollective Mode
T=0 T ≠ Thermally Excited
QPCreation of QP pair
𝛥 𝜔 > 2
𝛥 𝜔 ~ 0
A Bq q 𝜔𝜔 FIG. 1. (A) At T = 0 , the breaking of the Cooper pair into twoquasiparticles leads to a two-particle continuum in the pair spectralweight with a threshold of . The lower part of the figure depicts aschematic of the features of the pair spectral function at T = 0 . (B)At finite temperature, the scattering of thermally excited quasiparti-cles leads to additional low energy incoherent spectral weight. Thisleads to a halo behind the collective modes, as shown in the lowerpart of the figure. finite temperature evolution of the pair spectral function in thedisordered system. Finally, we conclude with a brief overviewof our calculation and key results. II. MEAN-FIELD THEORY AT FINITE TEMPERATURES
We study the attractive Hubbard model on a square lattice inthe presence of on-site non-magnetic impurities. The Hamil-tonian for the model is given by, H = − t (cid:88) (cid:104) rr (cid:48) (cid:105) σ ( c † rσ c r (cid:48) σ + h.c ) − U (cid:88) r n r ↑ n r ↓ + (cid:88) r ( v r − µ ) n r (1)where c † rσ ( c rσ ) is the creation (annihilation) operator for anelectron with spin σ on site r , and µ is the chemical potential.Nearest neighbour hopping between two electrons is governedby t , and U is the attractive interaction between two electronson the same site which leads to Cooper pairing. Here, v r isan on-site random potential, which is drawn independentlyon every site r from a uniform distribution of zero mean andwidth V , i.e. v r ∈ [ − V / , V / . Therefore, V correspondsto the strength of the disorder. This model has been studiedpreviously at zero temperature within a spatially inhomoge-neous Bogoliubov de-Gennes (BdG) mean-field theory. Morerecently the two-particle spectral functions in this model at T = 0 have been studied within a gaussian expansion aroundthe BdG solution . In this section we investigate the mean-field theory at finite temperatures, while later sections will bedevoted to considering the fluctuations around the mean-field . .
52 0 0 . . . . . O P T V /t = 00 . (b) . . . . . . . . ⇢ s T V /t = 00 . (c)(a) . . . . . . . . E g a p T V /t = 00 . T /t T /t T /t
FIG. 2. (a) Single-particle excitation gap E gap , (b) Superconducting order parameter ∆ OP , and (c) superfluid stiffness ρ s as a function oftemperature T , obtained from finite T BdG calculation. We present data for clean system ( V = 0 ) and three disorder values, V = 0 . t , t and t . E gap and ∆ OP vanish at T c = 1 . t in the clean system. All three parameters, E gap , ∆ OP and ρ s have reduced significantly at T = 0 ,while the decrease in mean-field T c is much smaller. The data have been obtained on a × square lattice and averaged over 15 disorderrealizations. theory at finite temperatures.Within a functional integral formalism, the partition func-tion for the model is given by, Z = (cid:90) D [ ¯ f σ , f σ ] e − S [ ¯ f σ ,f σ ] . (2)Here the imaginary time ( τ ) action S in terms of the fermionfields ( ¯ f σ ( r, τ ) , f σ ( r, τ ) ) is given by S = (cid:90) β dτ (cid:88) rr (cid:48) ,σ ¯ f σ ( r, τ ) (cid:2) ∂ τ δ rr (cid:48) + H rr (cid:48) (cid:3) f σ ( r (cid:48) , τ ) − U (cid:88) r ¯ f ↑ ( r, τ ) ¯ f ↓ ( r, τ ) f ↓ ( r, τ ) f ↑ ( r, τ ) , (3)where β = 1 /T and the single-particle Hamiltonian H rr (cid:48) = − tδ (cid:104) rr (cid:48) (cid:105) − ( µ − v r ) δ rr (cid:48) . We introduce twoHubbard-Stratonovich auxiliary fields, ∆( r, τ ) that couples tothe particle-particle channel ( ¯ f ↑ ( r, τ ) ¯ f ↓ ( r, τ ) ), and the field ξ ( r, τ ) that couples to the density channel ( ¯ f ( r, τ ) f ( r, τ ) ), inorder to construct a quadratic theory in the fermion fields.Integrating out the fermions, and considering a static butspatially varying saddle point profile of the auxiliary fields, ∆( r, τ ) = ∆ ( r ) and ξ ( r, τ ) = ξ ( r ) , lead to the BdG mean-field theory. The BdG self-consistency equations at finite T are given by, ∆ ( r ) = U (cid:88) n u n ( r ) v ∗ n ( r ) (1 − F n ( T )) , (4) ξ ( r ) = U (cid:88) n (1 − F n ( T )) | v n ( r ) | + F n ( T ) | u n ( r ) | , (5)and (cid:104) n (cid:105) = 2 N s (cid:88) r ξ ( r ) U , (6)where (cid:104) n (cid:105) is the average density of electrons in the sys-tem with N s number of sites. Here [ u n ( r ) , v n ( r )] are theeigenvector of the BdG matrix corresponding to the eigen-value E n and n runs over positive eigenvalues ( E n > ) only. The Fermi function at a temperature T is given by F n ( T ) = e En/T +1 . We solve the BdG self-consistency equa-tions (Eqn. 4-6) on a × square lattice with an inter-action strength U/t = 5 and at an average fermion density (cid:104) n (cid:105) = 0 . . We consider 15 disorder realizations for eachdisorder.Before we discuss the results of the mean-field theory at fi-nite temperatures, we note the main features of the mean-fieldtheory at zero temperature for disordered superconductors :(i) The distribution of the local pairing amplitude evolves froma sharp distribution around an average value for low disorderto a broad distribution with peaks around zero for large disor-der, which indicates the destruction of superconductivity. (ii)The distribution of local densities evolves from a sharp uni-modal distribution at low disorder to a broad bimodal distri-bution at large disorder. This indicates the formation of su-perconducting puddles or patches in the background of non-superconducting regions at large disorder. (iii) The formationof superconducting patches is further confirmed by the spatialdistribution of the pairing amplitude that shows cluster for-mation on the scale of the coherence length in typical disor-der configurations. (iv) The single-particle gap remains finiteand large at strong disorder, while the average order parame-ter and the superfluid density both decrease monotonically atlarge disorder. We will next compare and contrast these fea-tures to the behavior at finite temperatures. Temperature dependence of single-particle gap : Fig. 2 (a)shows the single-particle gap in the system as a function oftemperature. The gap for a clean superconductor ( V = 0 )vanishes around T c = 1 . t . Note that while the T = 0 gap has reduced by a factor of between the clean case and V = 3 t , the decrease in the mean-field T c is much smaller.A similar trend is seen in Fig. 2 (b) where we plot the averagepairing amplitude ∆ OP (averaged over sites and over disorderrealizations) as a function of temperature for different valuesof disorder. Once again we note that while ∆ OP ( T = 0) re-duces by a factor of four as we go from V = 0 to V = 3 t , T c . . . n P ( n ) V /t = 1 . T/t = 0 . . . . . . . . . P ( ¢ ) V /t = 0 . T/t = 0 . . . . . . . . . . . . . . P ( ¢ ) V /t = 1 . T/t = 0 . . . . . . . . . . . . . P ( ¢ ) V /t = 3 . T/t = 0 . . . . . (b)(a) (c)(e) . . . n P ( n ) V /t = 0 . T/t = 0 . . . . . (d) . . . n P ( n ) V /t = 3 . T/t = 0 . . . . . (f) FIG. 3. Distribution of (a)-(c) local superconducting pairing amplitude ∆( r ) and (d)-(f) local density n ( r ) obtained from finite T BdGcalculation for different values of disorder V and temperature T . With increasing disorder, P (∆) changes from a sharp distribution aroundthe average ∆ to a broad distribution with peaks around 0. For a fixed V , the shape of P (∆) does not change much with temperature, whilethe distribution shifts towards the lower value of ∆ . On the other hand, P ( n ) becomes bimodal around the average density (cid:104) n (cid:105) = 0 . inpresence of strong disorder, and with increasing the temperature, the distribution becomes narrower. only changes from . t to . t . These two trends taken to-gether show that within the mean-field theory disorder is muchmore effective at reducing/killing superconductivity at T = 0 compared to its effect on reducing T c of the system. Temperature dependence of superfluid stiffness : A similartrend is seen in the temperature variation of the superfluidstiffness ρ s (see Appendix A), which is plotted in Fig. 2 (c)with increasing disorder. While the T = 0 value of ρ s reducesby a factor of as the disorder is ramped up from the cleancase to V = 3 t , the transition temperature T c only changesfrom . t to . t . We note that in two dimensions, the finitetemperature transition will be a BKT type transition, with atransition temperature lower than the mean-field T c . Distributions : It is useful to look at how the distribution of thelocal order parameter ∆( r ) and the local density n ( r ) changeswith temperature and disorder strength. In Fig. 3 (a) and (b),we plot the distribution of ∆( r ) for V = 0 . t and V = t respectively. Each plot shows the distribution for a range oftemperatures. In each of these cases, we see that the shape ofthe distribution does not change much with temperature, al-though the distribution shifts to lower values of ∆ , consistentwith the decrease of ∆ with temperature. In Fig. 3 (c), wesee a similar trend with a pile-up around ∆ = 0 . Note thatfor V = t and V = 3 t , T = t is above T c and we simply get all the weight at ∆ = 0 . We plot the distribution of localdensities for V = 0 . t , V = t and V = 3 t in Fig. 3 (d),(e) and (f) respectively. The density distribution goes from aunimodal distribution at low disorder to a bimodal distributionat high disorder. At all values of disorder, the distribution nar-rows with increasing temperature, with the effect clearly visi-ble at large disorder strengths. At large disorder, the bimodaldistribution comes from the formation of superconducting andnon-superconducting patches. Increasing temperature leads tosmoother density profile between the patches and hence to anarrowing of the density distributions. III. GAUSSIAN FLUCTUATIONS AND PAIR SPECTRALFUNCTIONS
The primary motivation of this work is to understand howthe fluctuations around the mean-field theory that dominatethe two-particle pair spectral function at low energies, evolvewith temperature in a disordered superconductor. To this end,we include the spatio-temporal fluctuations of the ∆ fieldthrough ∆( r, τ ) = (∆ ( r ) + η ( r, τ )) e iθ ( r,τ ) , (7)where η ( r, τ ) and θ ( r, τ ) are the amplitude and the phase fluc-tuations respectively around the BdG saddle point solution ∆ ( r ) . We expand the action to second order in the fluctu- ations to obtain the gaussian action S G corresponding to thefluctuations of the order parameter at finite temperature T (for T = 0 , see Ref. 27) S G = (cid:88) rr (cid:48) (cid:88) ω m (cid:0) η ( r, ω m ) θ ( r, ω m ) (cid:1) (cid:18) D − ( r, r (cid:48) , ω m ) D − ( r, r (cid:48) , ω m ) D − ( r, r (cid:48) , ω m ) D − ( r, r (cid:48) , ω m ) (cid:19) (cid:18) η ( r (cid:48) , − ω m ) θ ( r (cid:48) , − ω m ) (cid:19) , (8)where ω m = (2 m ) π/β is the bosonic Matsubara frequency.We analytically continue from Matsubara to real frequencies to construct the real time inverse fluctuation propagators. Wenote that our formalism does not suffer from issues of numer-ical analytic continuation.The inverse fluctuation propagator corresponding to the amplitude fluctuation, D − is given by, D − ( r, r (cid:48) , ω ) = 1 U δ rr (cid:48) + 12 (cid:88) E n ,E n (cid:48) > f (1) nn (cid:48) ( r ) f (1) nn (cid:48) ( r (cid:48) ) χ nn (cid:48) ( ω ) + 12 (cid:88) E n ,E n (cid:48) > f (2) nn (cid:48) ( r ) f (2) nn (cid:48) ( r (cid:48) ) ζ nn (cid:48) ( ω ) , (9)where f nn (cid:48) ( r ) = [ u n ( r ) u n (cid:48) ( r ) − v n ( r ) v n (cid:48) ( r )] , and f nn (cid:48) ( r ) = [ u n ( r ) v n (cid:48) ( r ) + v n ( r ) u n (cid:48) ( r )] (10)are the matrix elements related to the BdG wave functions and the temperature dependent functions χ and ζ are given by, χ nn (cid:48) ( ω ) = (cid:32) ω + i + − E n − E n (cid:48) − ω + i + + E n + E n (cid:48) (cid:33) (1 − F n ( T ) − F n (cid:48) ( T )) , and ζ nn (cid:48) ( ω ) = (cid:32) ω + i + + E n − E n (cid:48) − ω + i + − E n + E n (cid:48) (cid:33) ( F n ( T ) − F n (cid:48) ( T )) . (11)It is useful to analyze the structure of χ and ζ , since they oc-cur in all the matrix elements of the inverse fluctuation prop-agators and provide insight about the microscopic processesthat control the temperature dependence of the pair spectralfunction. Here ζ represents (upto matrix elements, which donot change its singularity structure) the probability amplitudeof scattering a Bogoliubov quasiparticle from one state to theother. Note that F n ( T ) = 0 for all gapped states at T = 0 andhence this term does not contribute to the collective modesaround the ground state. A simple way to understand this isthat the quasiparticles need to be present in the first place tobe scattered, and at T = 0 , none of the gapped modes areexcited in the system. As temperature is raised, this ampli-tude becomes finite. It is important to note that the singular-ities of this function occur when ω = E n − E n (cid:48) , and henceat very low energies. Thus at finite temperatures, ζ is com-plex at low energies, with an amplitude that increases withtemperature. We will later see that these scattering processesplay a very important role in determining the low energy pairspectral function at finite temperatures. We now consider thestructure of χ , which represents (upto matrix elements, whichdo not change the singularity structure of these functions) the amplitude for creating a pair of Bogoliubov quasiparticles.This is reflected in the singularities at ω = ± ( E n + E n (cid:48) ) .Hence, for ω < E gap , where the fermionic single-particlegap E gap corresponds to the lowest positive eigenvalue of theBdG Hamiltonian, χ is purely real, while it takes complex val-ues for ω > E gap . If we consider the numerator of χ , it isevident that the numerator goes to at T = 0 . So the T = 0 spectral function is completely dominated by this term. Asthe temperature is raised, the numerator decreases; however χ remains real at low energies below the two-particle continuumas long as the single-particle gap remains finite.The inverse fluctuation propagator for the phase fluctuation, D − is given by, D − ( r, r (cid:48) , ω ) = ˜ D dia ( r, r (cid:48) ) + ω κ ( r, r (cid:48) , ω ) + Λ( r, r (cid:48) , ω ) , (12)where ˜ D dia is the diamagnetic response of the system, κ isthe frequency dependent compressibility and Λ( r, r (cid:48) , ω ) is re-lated to the paramagnetic current-current correlator on the lat-tice. The exact formulas for ˜ D dia , κ and Λ are given in Ap-pendix B.Finally, the inverse fluctuation propagator corresponding to amplitude-phase mixing, D − is given by D − ( r, r (cid:48) , ω ) = − iω (cid:88) E n,n (cid:48) > f (1) nn (cid:48) ( r ) f (2) nn (cid:48) ( r (cid:48) ) χ nn (cid:48) ( ω ) + f (1) nn (cid:48) ( r (cid:48) ) f (2) nn (cid:48) ( r ) ζ nn (cid:48) ( ω ) . (13)We invert the matrix D − αβ ( r, r (cid:48) , ω ) to obtain the fluctua-tion propagators D αβ ( r, r (cid:48) , ω ) and the corresponding spec-tral functions, P αβ ( r, r (cid:48) , ω ) = − π Im D αβ ( r, r (cid:48) , ω ) . Here P ( r, r (cid:48) , ω ) = − π Im (cid:104) η ( r, ω + i + ) η ( r (cid:48) , − ω + i + ) (cid:105) cor-responds to amplitude or Higgs fluctuations, P ( r, r (cid:48) , ω ) = − π Im (cid:104) θ ( r, ω + i + ) θ ( r (cid:48) , − ω + i + ) (cid:105) denotes the phase fluc-tuations while the amplitude-phase mixing is governed by P ( r, r (cid:48) , ω ) = − π Im (cid:104) η ( r, ω + i + ) θ ( r (cid:48) , − ω + i + ) (cid:105) . How-ever, the phase fluctuation propagators are not directly mea-sureable in experiments, where probes couple to the electrondensity or current. As shown in Ref. 27, the experimentallymeasureable pair spectral function P ( r, r (cid:48) , ω ) = (cid:88) αβ P αβ ( r, r (cid:48) , ω ) , (14)where P = P , P ( r, r (cid:48) , ω ) = ∆ ( r ) P ( r, r (cid:48) , ω ) , P ( r, r (cid:48) , ω ) = ∆ ( r (cid:48) ) P ( r, r (cid:48) , ω ) , and P ( r, r (cid:48) ω ) =∆ ( r )∆ ( r (cid:48) ) P ( r, r (cid:48) , ω ) .Note that in a translation invariant system, P and P are re-lated by simple scaling factors. However in a disordered sys- tem, where the pairing amplitude ∆ ( r ) is varying in space,the spatial correlations of P and P will be quite different andhence it is important to study the physically measureable cor-relations. A. Pair spectral function in a clean superconductor
In this paper, we are primarily interested in studying thetemperature dependence of the collective modes and the re-sultant two-particle spectral functions for a disordered super-conductor. We start with the behavior of the temperature de-pendence of the two-particle spectral function P ( q, ω ) in theclean limit ( V /t = 0 ). This allows us to interpret the lowenergy spectral functions in terms of a temperature broad-ened collective mode and a background spectral weight aris-ing from the scattering of thermally excited quasiparticles.This framework will then be used to investigate the pair spec-tral functions in the disordered case.For a clean system, the problem simplifies considerablysince the fluctuation propagators are diagonal in the momen-tum basis; e.g. D − ( q, ω ) = 1 U + 12 (cid:88) k (cid:104) f (1) k ( q ) (cid:105) χ k ( q, ω ) + (cid:104) f (2) k ( q ) (cid:105) ζ k ( q, ω ) , (15)where f (1) k ( q ) = u k u k + q − v k v k + q , f (2) k ( q ) = u k v k (cid:48) + v k u k (cid:48) , (16)with χ k ( q, ω ) = (cid:18) ω + i + − E k − E k (cid:48) ) − ω + i + + E k + E k (cid:48) ) (cid:19) (1 − F k ( T ) − F k (cid:48) ( T )) (17) ζ k ( q, ω ) = (cid:18) ω + i + + E k − E k (cid:48) ) − ω + i + − E k + E k (cid:48) ) (cid:19) ( F k ( T ) − F k (cid:48) ( T )) . (18)In the above formulae, we have used the standard BCS spec-trum E k = (cid:112) ξ k + ∆ with ξ k = − t (cos k x + cos k y ) − µ , ∆ the uniform pairing amplitude and u k = 1 / ξ k /E k ) = 1 − v k .Fig. 4 (a)-(d) shows the amplitude spectral function P ( q, ω ) , while Fig. 4 (e)-(h) shows the phase spectral func-tion P ( q, ω ) in the clean system with increasing tempera-ture. Here the attractive interaction U = 5 t and the density is set to 0.875. Let us first focus on the pair spectral func-tions at T = 0 (Fig. 4 (a) and (e)). There is diffuse continuumspectral weight above ω > , corresponding to propaga-tion of two Bogoliubov quasiparticles. Note that at T = 0 ,the ζ terms do not contribute, while χ is complex only for ω > . For ω < , there is a coherent dispersing peakat the collective mode frequencies determined by the vanish-ing of the determinant of the inverse fluctuation propagator. T/T C =0.18 (b) T/T C =0.54 (d) T/T C =0.36 (c) (h) T/T C =0.54 (g) T/T C =0.36 (f) T/T C =0.18 (e) T/T C =0 P T/T C =0 (a) ωω P FIG. 4. Spectral functions in clean superconductor: (a)-(d) amplitude spectral function P and (e)-(h) phase spectral function P of thepair spectral function P ( q, ω ) in a clean superconductor ( V /t = 0 ) shown as a density plot in q (along the principle axis of the 2D Brillouinzone of the square lattice) and ω for increasing temperatures: T = 0 , T = 0 . T c , T = 0 . T c and T = 0 . T c (corresponding to T = 0 , T = 0 . t, T = 0 . t and T = 0 . t respectively). At small temperatures, both the spectral functions P and P consist of thecollective modes below two-particle continuum. The temperature dependent background halo is clearly seen in P for T = 0 . T c and T = 0 . T c , while P is mostly dominated by strong collective modes. All data for the clean superconductors have been obtained on a × square lattice. The mode disperses linearly at low momenta. The collectivemode is a mixture of amplitude and phase fluctuations at fi-nite momenta, but reduces to a pure phase Goldstone mode as q → . As the temperature is raised to T = 0 . T c (Fig. 4(b)), T = 0 . T c (Fig. 4 (c)) and T = 0 . T c (Fig. 4 (d)), athermally broadened collective mode is clearly present ridingon a distinct background halo.The background halo, which is due to the scattering of thequasiparticles already present in the system (the ζ terms), in-creases in intensity with increasing temperature. This inco-herent spectral weight has some interesting characteristics. Ateach q , there is an upper bound of energy beyond which thereis no incoherent spectral weight, till one reaches ω = 2∆ .This limiting energy, which is the maximum of | E k − E k + q | for a fixed q , disperses linearly at small q and shows a sharpdip around the Γ and M ( [ π, π ] ) points. In Fig. 5 (a) we plotthe dispersion of the quasiparticle energy E k as a function of k in the Brillouin zone. We see that the wavevector q = [ π, π ] only connects points in the Brillouin zone where the valuesof E k do not differ much, leading to a dip in the temperaturedependent background halo around the M point. Although only one such connection is shown in the figure, one can eas-ily see that this is true in general. The same also holds for the Γ point. On the contrary, the wave-vector q = [ π, can con-nect points in the Brillouin zone where the values of E k canvary from a small to large value, and hence the lobe like struc-ture extends up to a large value of ω . As a result one can seethat the collective mode both at q = 0 and q = [ π, π ] remainsharp, while there is considerable broadening at intermediatemomenta. This is clearly seen in Fig. 4 (c) and (d) wherethe apparent width of the collective mode shrinks near the M point when the collective mode lies above the band of inco-herent spectral weight. This is also shown in Fig. 5 (b), wherewe plot the line cuts of the Higgs spectral function along theenergy axis (EDC or energy distribution curve) for fixed val-ues of momenta at the largest temperature T = 0 . T c . Nearthe zone center, the spectral weight lies above the two parti-cle continuum. As we move along the q x axis, the modes at [ π/ , and [ π, do not show up as sharp peaks due to thelarge background incoherent weight. However, at [ π, π ] , onecan clearly see two bumps in the spectral function, the lowerone coming from the incoherent scattering of quasiparticles (b) . . . . . . . P !q = [0 , ⇡/ , ⇡, ⇡, ⇡ ] T/T C =0.54 . . . . P ! q = [0 , ⇡/ , ⇡, ⇡, ⇡ ] (c) T/T C =0.54 (a) [ 𝜋 , 𝜋 ] [ 𝜋 , ] 𝜞 XM (d) ° X M ° q . . . . æ ! T/T c = 0 . . .
36 0 . . . . . . . . . T/T c . . . c s (e) FIG. 5. (a): Color plot of the dispersion of a clean superconductorin the square lattice Brillouin zone. The wave-vector [ π, π ] , shownwith white arrow, connects momenta where the dispersion is almostsame. This leads to a low threshold for incoherent spectral weightdue to quasiparticle scattering near [ π, π ] . On the other hand, thewave-vector [ π, connects momenta with large difference in dis-persion. Hence the energy threshold for incoherent weight is largenear [ π, . (b) and (c): The EDC curves for (b) P and (c) P are shown for specific q values at T = 0 . T c . (d): Width of thedispersive collective modes in ω , ( σ ω , obtained from the phase sec-tor P ) as a function of q values for different temperature values of T . We notice that that collective mode remains sharp at Γ and M points, while it broadens with temperature at intermediate momenta.(e): The sound velocity is plotted as a function of T /T c . It remainsalmost constant up to T /T c ∼ . , then rapidly decreases to zeronear T c . and the upper one corresponding to the coherent collectivemode in the system.We note that the background halo is more clearly seen inthe Higgs spectral functions, while the phase spectral func-tions (Fig. 4 (e)-(h)) are primarily dominated by the largecollective mode peak. The robust linear dispersion of thephase mode allows us to extract the speed of sound from thelong wavelength linear dispersion. This speed of sound c s is plotted as a function of T /T c in Fig. 5 (e). We see that atlow temperatures c s remains almost constant, whereas near T /T c ∼ . it starts decreasing and drops to zero at T c . Thebackground spectral weight in the phase sector is clearly seenonly around T = 0 . T c (Fig. 4 (h)), where the character-istics are similar to that of the Higgs spectral weight. TheEDC curves for the phase spectral function at fixed momentaare plotted in Fig. 5 (c). Here it is clear that the coherentspectral weight in the collective mode is much larger thanthe incoherent spectral weight. Hence, near the collectivemode frequency, one can expand the phase spectral function P ( q, ω ) ∼ Z ( q ) / ( ω − ω ( q ) + iσ ω ( q )) . The large coherentspectral weight in the phase channel allows us to extract anenergy width of the peak, σ ω from the line cuts in Fig. 5 (c).This width is plotted as a function of momenta for differenttemperatures in Fig. 5 (d). It is clear that the collective moderemains sharp at q = [0 , and [ π, π ] , while the broadening atintermediate momenta increases with increasing temperature. B. Pair spectral function in disordered superconductor
We now consider the key issue which we want to study inthis paper: how do the collective modes evolve with temper-ature in a disordered superconductor. In a disordered sys-tem, momentum is not a good quantum number for a singledisorder realization. For each such realization, we first con-struct the pair spectral function P αβ as a matrix in the realspace co-ordinates r and r (cid:48) . We then work with the cen-ter of mass co-ordinate R = ( r + r (cid:48) ) / and relative co-ordinate d = ( r − r (cid:48) ) , and average the spectral function P ( d, R, ω ) over several disorder realizations. The disorder av-eraging restores translation invariance, i.e. the averaged quan-tity is a function only of d and ω . Averaging over R , we get P ( d, ω ) = 1 /N s (cid:104) (cid:80) R P ( d, R, ω ) (cid:105) ( N s being the number oflattice sites and (cid:104)(cid:105) corresponds to disorder average). We thenFourier transform the spectral function in d to express it as afunction of q and ω , i.e. P ( q, ω ) . The variation of this disor-der averaged spectral function with momentum q and energy ω will be our key tool to study the behaviour of finite temper-ature collective modes.We first consider the spectral function of the disordered sys-tem at T = 0 (worked out in Ref. 27), so that we have a ref-erence to understand the finite temperature variations. Theamplitude and phase spectral functions P and P are plot-ted as a function of q and ω for a weakly disordered systemwith V = 0 . t in Fig. 6 (a) and (e) respectively. Whilethe phase spectral function (Fig. 6 (e)) is almost unchangedfrom the clean case, with a linearly dispersing collective modedominating at low energies, the amplitude spectral functionshows dramatic change (Fig. 6 (a)). In contrast to the cleancase, where at q = 0 , the Higgs mode sits at the thresholdof the two-particle continuum at an energy of , a non-dispersive mode appears at an energy below two-particle con-tinuum ( E gap ) in this case. At q = 0 this subgap mode isidentified as the disorder-induced Higgs mode in a supercon-ductor . The T = 0 spectral functions for a moderately dis- T/T C =0.18 (b)(f) T/T C =0.36 (c) T/T C =0.18 P T/T C =0.54 (d)(h) T/T C =0.54 (g) T/T C =0.36 (j) T/T C =0.54 P ! q = [0 , ⇡/ , ⇡, ⇡, ⇡ ] (i) T/T C =0.54 . . . . P !q = [0 , ⇡/ , ⇡, ⇡, ⇡ ] T/T C =0 (a) P P (e) T/T C =0 P ωω FIG. 6. Spectral functions in a weakly disordered superconductor: (a)-(d) amplitude spectral function P and (e)-(h) phase spectral function P of the pair spectral function P ( q, ω ) in the presence of weak disorder V /t = 0 . shown as a density plot in q and ω , for increasingtemperatures: T = 0 , T = 0 . T c , T = 0 . T c and T = 0 . T c (corresponding to T = 0 , T = 0 . t, T = 0 . t and T = 0 . t respectively). Note the appearance of the subgap Higgs peak in the amplitude sector P at low T . While this non-dispersive Higgs moderemains unaffected at momenta Γ = [0 , and M = [ π, π ] with increasing T , it gets overwhelmed by the temperature induced backgroundhalo at other momenta. The coherent collective modes in the phase sector remain mostly unaffected at low temperatures, but they are thermallybroadened at large temperatures. The EDC curves for (i) P and (j) P are shown, for specific q values and for a temperature T = 0 . T c .The EDC of P clearly shows the Higgs mode at the Γ point and two distinct modes (the lower broad peak is due to the temperature inducedquasiparticle scattering, and the higher energy peak is due to disorder) at the M point. The EDC curves for P show thermally broadeneddispersive collective mode peaks. All the disorder data have been obtained on a × lattice, and averaged over 15 disorder realizations. ordered system with V = t is shown in Fig. 7 (a) (amplitude)and (e) (phase) respectively. The non-dispersive mode in theamplitude spectral function gains more spectral weight and isconsiderably broadened, while the phase spectral function isrelatively unchanged with disorder.Next we study the effect of disorder on the amplitude and phase spectral functions at finite temperatures. Fig. 6 (b)-(d)shows the amplitude spectral function P ( q, ω ) , and Fig. 6(f)-(h) shows the phase spectral function in presence of a weakdisorder V /t = 0 . with increasing temperature. The mostvisible change in the amplitude spectral functions is the ap-pearance of the low energy continuum weight or the halo in0 T/T C =0.2 (b) T/T C =0.6 (d)(f) (g) (h) T/T C =0.4 (c) T/T C =0.2 T/T C =0.6T/T C =0.4 (j) T/T C =0.6 (i) T/T C =0.6 P !q = [0 , ⇡/ , ⇡, ⇡, ⇡ ] P ! q = [0 , ⇡/ , ⇡, ⇡, ⇡ ] T/T C =0 (a) P (e) T/T C =0 P ωω FIG. 7. Spectral functions in a moderately disordered superconductor: (a)-(d) amplitude spectral function P and (e)-(h) phase spectralfunction P of the pair spectral function P ( q, ω ) in the presence of moderate disorder V /t = 1 . shown as a density plot in q and ω , forincreasing temperatures: T = 0 , T = 0 . T c , T = 0 . T c and T = 0 . T c (with T c = 1 . ). The subgap Higgs mode gets broadened and itslow energy threshold comes down in energy. With increase in temperature, contrary to the weak disorder, the lobe structure of the backgroundhalo is not seen in amplitude sector, and the Higgs mode remains prominent at all temperatures. The EDC curves for (i) P and (j) P areshown for specific q values and for T = 0 . T c . The EDC curves for P also show that the non-dispersive Higgs mode dominates at alltemperatures. On the other hand, the phase sector is dominated by the dispersing collective modes, which only get thermally broadened atlarge temperatures. the background of the collective mode. As explained in thesection on clean superconductors, this weight represents thescattering of the Bogoliubov quasiparticles and increases withtemperature. However, for each q there is an upper boundof energy up to which the background weight exists. Thisbackground cut-off disperses linearly at small q and shows apronounced dip around the [ π, π ] point. Therefore the non-dispersing Higgs mode near q = 0 and q = [ π, π ] remainsunaffected at finite temperatures (Fig. 6 (b), (c) and (d)). We also note that the small Higgs component in the linearly dis-persing collective mode is overwhelmed by the backgroundincoherent weight as temperature increases (Fig. 6 (c) and(d), corresponding to T /T c = 0 . and T /T c = 0 . ), sothe only coherent weight in the amplitude spectral functionat these finite temperatures is related to the disorder-inducedHiggs mode. To see this feature clearly, we plot some energydistribution curves (EDCs) of the amplitude spectral functionat T = 0 . T c in Fig. 6 (i). These are line-cuts of the data1 . . . ! æ q T/T c = 0 . V /t = 0 . . .
00 0 . . . ! æ q T/T c = 0 . V /t = 0 . . . (c)(b) . . . ! æ q T /T c = 0 . V /t = 0 . . . (a) FIG. 8. (a)-(c): Width of the collective modes in q , ( σ q , obtained from the phase spectral function P ) as a function of ω below two particlecontinuum, for (a) T = 0 . T c , T = 0 . T c and T = 0 . T c respectively. We note that for a fixed T , σ q increases with increase in disorder.At small disorder the width does not change much with energy, while the larger disorder shows a broad peak. Since with increasing disorderthe collective mode structure comes down to lower energy (Fig. 6 and 7), the threshold ω at which σ q vanishes also decreases with disorder. in Fig. 6 (d) at fixed values of q . The non-dispersive mode at q = 0 is clearly seen as a peak. At q = [ π, π ] , there are twopeaks, with the lower broad peak corresponding to the inco-herent quasiparticle scattering, and the sharper higher energypeak (at energies similar to the q = 0 peak) corresponding tothe non-dispersive Higgs mode.On the other hand, the phase spectral function P remainsmostly unaffected in the presence of weak disorder even atfinite temperatures, as seen in Fig. 6 (f)-(h). While the collec-tive mode is thermally broadened, it still dominates the lowenergy phase spectral function function. It is interesting tonote that the mode near [ π, π ] remains sharp at finite tempera-tures as the incoherent spectral weight lies below the energy ofthis mode. As temperature is increased (Fig. 6 (g) and (h)), thebackground halo with the two lobe structure becomes moreprominent even in the phase spectral function. The variationof the phase spectral function with energy at fixed momenta at T = 0 . T c is plotted in Fig. 6 (j). The curves show thermallybroadened dispersive peaks of the collective modes.We now increase the disorder to a moderate value of V = t and study the spectral functions with increasing temperature.With increase in disorder, the non-dispersive Higgs mode getsbroadened and its lower end comes down towards the zero en-ergy. This mode also gains much more spectral weight. This isseen in Fig. 7 (a) where we plot the spectral function at T = 0 .As temperature is increased (Fig. 7 (b)-(d)), we once againsee a diffuse background halo, but at this moderate disorder,the sharp lobe structure of the halo, which was present in theclean and the weakly disordered system, is absent. Since thesharp boundaries resulted from simultaneous momentum andenergy conservation in quasiparticle scattering, and momen-tum conservation is strongly broken in each disorder realiza-tion at these moderate disorders, the background halo is morediffuse in this case. However, as in the weakly disorderedcase, the non-dispersive Higgs mode remains prominent at allfinite temperatures. This is clearly seen in Fig. 7 (i), wherewe plot the variation of the amplitude spectral function withenergy at fixed momenta. The curves at all the momenta show a broad peak at roughly the same energy corresponding to thenon-dispersive Higgs mode in the system. We note that atthis moderate disorder, the edge of the continuum perceptiblycomes down with increasing temperature, showing the soften-ing of the gap in the system.In contrast, the phase spectral function is dominated by thelinearly dispersing collective mode, which is robust to boththe presence of disorder and temperature (Fig. 7 (e)-(h)). Atthe largest temperature of T = 0 . T c , the collective modenear the [ π, or [ π, π ] point is broadened, but a distinct peakcan still be observed, as seen in the EDCs plotted in (Fig. 7(j)).We have already seen that the phase spectral function con-sists of a dominant dispersing collective mode. However in adisordered system, momentum is not a good quantum number,and one would expect the collective modes to be broadened inmomentum space due to elastic scattering from the impuri-ties. To estimate the effect of this scattering, we consider thehalf width of the spectral function peak in the phase channelat different fixed values of ω from the momentum distribu-tion curves (MDCs). We only take into account the collectivemode line between Γ and X points where the mode is dis-persing, and limit our study to energies well below two par-ticle continuum. In Fig. 8 (a)-(c) we plot this width σ q as afunction of ω for three different temperatures, T = 0 . T c , T = 0 . T c and T = 0 . T c respectively. Each plot containsthe width for three different disorder values, a weak disorderof V = 0 . t , a moderate disorder of V = t and a strong dis-order of V = 3 t respectively. As expected, we observe thatfor any fixed T , σ q increases with increasing disorder. Whilethe low disorder width does not change much with energy ofthe collective modes, the width at moderate and high disor-ders show a broad peak as a function of collective mode fre-quency. We also find that σ q vanishes at a threshold ω whichdecreases with disorder. This happens because the whole col-lective mode structure itself comes down when we increasedisorder (see Fig. 6 and 7).We have already seen that the non-dispersive mode in the2 (a) (b) (c) . . . . . T/T c . . . ¢ ! V /t = 3 . . . . . . !V /t = 1 T /T c = 0 . . . !V /t = 0 . T /T c = 0 . . . P P P P FIG. 9. (a)-(b): Amplitude ( P , shown as dotted lines) and phase contribution ( P , shown as solid lines) to the pair spectral function P at q = [0 , shown for two different disorder values, V = 0 . t and V = t , as a function of ω , and for increasing temperatures. (c): Theseparation between the Higgs and the phase peak, ∆ ω plotted as a function of ω . We notice that at small temperatures, they are separated formoderately large values of disorder, while it falls rapidly to zero at a critical temperature T sc with increase in disorder. amplitude channel produces finite subgap spectral weight at q = 0 , while the linearly dispersing collective mode has largeweight at zero energy in the phase channel (the Goldstonemode). The phase peak and the amplitude peak are spec-trally separated at T = 0 , and hence this mode should bespectroscopically observable. We now consider whether a fi-nite temperature will erase this spectral separation and renderthis mode invisible. We have plotted the amplitude and phasecontribution to the spectral function for a weak disorder of V /t = 0 . (Fig. 9 (a)) and a moderate disorder of V = t (Fig. 9 (b)). In both these cases, we find that as temperatureis increased, the peak positions remain unchanged while thebroadening increases, but the separate phase and amplitudefeatures are observable upto a reasonably high temperature.Thus this feature is also robust to turning on temperature inthe system. We note that inclusion of density fluctuations canalter the spectral separation of these features .To systematically track the separation between the subgapHiggs peak and the low energy phase peak, we define a pa-rameter ∆ ω which indicates to the separation between themin energy. In Fig. 9 (c), we plot ∆ ω as a function of temper-ature for disorder V /t = 0 . , and 3. Here we extend ouranalysis up to large temperature values, keeping in mind thatthe BdG theory does not work well close to T c . We notice thatat small temperature, the Higgs and the phase modes are sep-arated for moderately large value of disorder. However, withincrease in temperature, the separation decreases monotoni-cally and vanishes at a critical temperature T sc . T sc decreaseswith increase in disorder, which suggests that the sharp featureof the Higgs mode is more robust in presence of temperature atsmall disorder and the robustness goes away with increase indisorder. The momentum and energy resolved MEELS spec-troscopy should observe this Higgs mode separately fromthe phase pileup in an energy resolved way.In conclusion, in this work we have extended our previ-ous studies on two-particle spectral function for disordered s -wave superconductors to finite temperatures. Using a func-tional integral formalism and gaussian expansion around the inhomogeneous saddle point, we have studied the two-particlespectral function at small and moderately high temperatures,both in clean and disordered superconductors. We derive theanalytical formulas for inverse fluctuation propagators at fi-nite temperature, continued to real frequency. We present thefull q − ω dependence of the amplitude and phase sectors ofthe spectral function, and therefore study the evolution of the Higgs and the Goldstone mode with temperature and disor-der. We show that at finite temperatures, additional low en-ergy incoherent spectral weight appears in the form of lobes.In presence of disorder, these temperature dependent back-ground halo competes with the collective modes in the am-plitude sector. However, we find that if the disorder is nottoo strong, the non-dispersive Higgs mode which appears asa subgap feature at q = [0 , remains unaffected in presenceof moderately high temperatures. Therefore, the Higgs modecan be seen in an energy resolved way separately from thelow energy phase pile-up, even at experimentally accessibletemperatures. ACKNOWLEDGMENTS
A.S and R.S. acknowledge the computational facilitiesof the Department of Theoretical Physics, TIFR Mum-bai. N.T. acknowledges support from DOE grant DE-FG02-07ER46423.
Appendix A: Superfluid stiffness at finite temperature
We use Bogoliubov transformation in a disordered super-conductor, which diagonalizes the effective mean-field Hamil-tonian for the negative U Hubbard model, with energy E n andthe corresponding eigenfunction [ u n ( r ) , v n ( r )] . Here n runsover the positive eigenvalues i.e. E n > . The current opera-3tor is defined as j xr = it (cid:88) σ (cid:16) c † r +ˆ xσ c rσ − c † rσ c r +ˆ xσ (cid:17) , (A1)and the local kinetic energy associated with the x -directedhopping is given by K xr = − t (cid:88) σ (cid:16) c † r +ˆ xσ c rσ + c † rσ c r +ˆ xσ (cid:17) . (A2)Now the superfluid stiffness by the Kubo formula is given by, D s π = (cid:104)− K x (cid:105) − Λ xx ( q x = 0 , q y → , iω p = 0) , (A3)where iω p is the Bosonic Matsubara frequency. The first termrepresents the diamagnetic response to an external magnetic field which is given by, (cid:104)− K x (cid:105) = 4 tN (cid:88) n u n ( r +ˆ x ) u n ( r ) F n ( T )+ v n ( r +ˆ x ) v n ( r )(1 − F n ( T )) (A4)The second term is the paramagnetic response given by thedynamical transverse current-current correlation function, Λ xx ( q , iω p ) = 1 N (cid:90) /T dτ e iω p τ (cid:104) j x ( q , τ ) , j x ( − q , (cid:105) (A5)which is calculated to be, Λ xx ( q , iω p ) = 2 t N (cid:88) nm A nm ( q )( A nm ( q ) + B nm ( − q )) iω p + ( E n − E m ) ( F n ( T ) − F m ( T )) . (A6)In the above equation, n and m run over all eigenvalues (both E n < and E n > ), and A nm and B nm are given by, A nm ( q ) = (cid:88) i e − i q .r ( u n ( r + ˆ x ) u m ( r ) − u n ( r ) u m ( r + ˆ x )) D nm ( q ) = (cid:88) i e − i q .r ( v n ( r + ˆ x ) v m ( r ) − v n ( r ) v m ( r + ˆ x )) . Appendix B: Inverse fluctuation propagator for phasefluctuations
The inverse fluctuation propagator for the phase fluctuationis given by, D − ( r, r (cid:48) , ω ) = ˜ D dia ( r, r (cid:48) ) + ω κ ( r, r (cid:48) , ω ) + Λ( r, r (cid:48) , ω ) . The diamagnetic response ˜ D dia is related to the local kineticenergy K ( r, r (cid:48) ) = − t (cid:88) E n > v n ( r ) v n ( r (cid:48) )[1 − F n ( T )]+ u n ( r ) u n ( r (cid:48) ) F n ( T ) through the relation ˜ D dia ( r, r (cid:48) ) = − δ rr (cid:48) (cid:88) (cid:104) rr (cid:105) K ( r, r ) + 12 δ (cid:104) rr (cid:48) (cid:105) K ( r, r (cid:48) ) (B1)Here, the frequency dependent compressibility κ is given by density density correlator, κ ( r, r (cid:48) , ω ) = 18 (cid:88) E n,n (cid:48) > f (2) nn (cid:48) ( r ) f (2) nn (cid:48) ( r (cid:48) ) χ nn (cid:48) ( ω ) + f (1) nn (cid:48) ( r ) f (1) nn (cid:48) ( r (cid:48) ) ζ nn (cid:48) ( ω ) , (B2)while Λ( r, r (cid:48) , ω ) is related to the paramagnetic current-current correlator on the lattice Λ( r, r (cid:48) , ω ) = (cid:88) (cid:104) rr (cid:105)(cid:104) r (cid:48) r (cid:105) J ( r, r , r (cid:48) , r , ω ) − J ( r, r , r , r (cid:48) , ω ) − J ( r , r, r (cid:48) , r , ω ) + J ( r , r, r , r (cid:48) , ω ) (B3)where J ( r, r , r (cid:48) , r , ω ) = − t (cid:88) E nn (cid:48) > f (3) nn (cid:48) ( r, r ) f (3) nn (cid:48) ( r , r (cid:48) ) χ nn (cid:48) ( ω ) + f (4) nn (cid:48) ( r, r ) f (4) nn (cid:48) ( r , r (cid:48) ) ζ nn (cid:48) ( ω ) . (B4)4The new matrix elements f (3) and f (4) are given by f (3) nn (cid:48) ( r, r (cid:48) ) = [ u n ( r ) v n (cid:48) ( r (cid:48) ) − v n ( r ) u n (cid:48) ( r (cid:48) )] , and f (4) nn (cid:48) ( r, r (cid:48) ) = [ u n ( r ) u n (cid:48) ( r (cid:48) ) + v n ( r ) v n (cid:48) ( r (cid:48) )] . (B5) ∗ [email protected] † [email protected] Allen M Goldman and Nina Markovic. Superconductor-insulator transitions in the two-dimensional limit.
Physics Today ,51(11):39–44, 1998. Benjamin Sac´ep´e, Thomas Dubouchet, Claude Chapelier, MarcSanquer, Maoz Ovadia, Dan Shahar, Mikhail Feigel’Man, and LevIoffe. Localization of preformed cooper pairs in disordered super-conductors.
Nature Physics , 7(3):239–244, 2011. Vsevolod F Gantmakher and Valery T Dolgopolov.Superconductor–insulator quantum phase transition.
Physics-Uspekhi , 53(1):1, 2010. Yonatan Dubi, Yigal Meir, and Yshai Avishai. Nature of thesuperconductor–insulator transition in disordered superconduc-tors.
Nature , 449(7164):876–880, 2007. G Kopnov, O Cohen, M Ovadia, K Hong Lee, Chee CheongWong, and D Shahar. Little-parks oscillations in an insulator.
Physical review letters , 109(16):167002, 2012. Nandini Trivedi, Richard T Scalettar, and Mohit Randeria.Superconductor-insulator transition in a disordered electronic sys-tem.
Physical Review B , 54(6):R3756, 1996. MV Feigel’man and MA Skvortsov. Universal broadening ofthe bardeen-cooper-schrieffer coherence peak of disordered su-perconducting films.
Physical review letters , 109(14):147002,2012. IS Burmistrov, IV Gornyi, and AD Mirlin. Enhancement of thecritical temperature of superconductors by anderson localization.
Physical review letters , 108(1):017002, 2012. Amit Ghosal, Mohit Randeria, and Nandini Trivedi. Inhomoge-neous pairing in highly disordered s-wave superconductors.
Phys-ical Review B , 65(1):014501, 2001. Karim Bouadim, Yen Lee Loh, Mohit Randeria, and NandiniTrivedi. Single-and two-particle energy gaps across the disorder-driven superconductor–insulator transition.
Nature Physics ,7(11):884–889, 2011. ME Gershenson, VN Gubankov, and Yu E Zhuravlev. Interactionand localization effects in two-dimensional film of superconduc-tor at t¿ tc.
Solid State Communications , 45(2):87–90, 1983. Madhavi Chand, Garima Saraswat, Anand Kamlapure, MintuMondal, Sanjeev Kumar, John Jesudasan, Vivas Bagwe, LaraBenfatto, Vikram Tripathi, and Pratap Raychaudhuri. Phasediagram of the strongly disordered s-wave superconductor nbnclose to the metal-insulator transition.
Physical Review B ,85(1):014508, 2012. Manuel Endres, Takeshi Fukuhara, David Pekker, Marc Che-neau, Peter Schau β , Christian Gross, Eugene Demler, StefanKuhr, and Immanuel Bloch. The ‘higgs’ amplitude mode atthe two-dimensional superfluid/mott insulator transition. Nature ,487(7408):454–458, 2012. Ryusuke Matsunaga, Naoto Tsuji, Hiroyuki Fujita, Arata Sug-ioka, Kazumasa Makise, Yoshinori Uzawa, Hirotaka Terai, ZhenWang, Hideo Aoki, and Ryo Shimano. Light-induced collectivepseudospin precession resonating with higgs mode in a supercon- ductor.
Science , 345(6201):1145–1149, 2014. Daniel Sherman, Uwe S Pracht, Boris Gorshunov, Shachaf Poran,John Jesudasan, Madhavi Chand, Pratap Raychaudhuri, MasonSwanson, Nandini Trivedi, Assa Auerbach, et al. The higgs modein disordered superconductors close to a quantum phase transition.
Nature Physics , 11(2):188–192, 2015. Philip W Anderson. Coherent excited states in the theory of super-conductivity: Gauge invariance and the meissner effect.
Physicalreview , 110(4):827, 1958. Ryo Shimano and Naoto Tsuji. Higgs mode in superconductors.
Annual Review of Condensed Matter Physics , 11:103–124, 2020. David Pekker and CM Varma. Amplitude/higgs modes in con-densed matter physics.
Annu. Rev. Condens. Matter Phys. ,6(1):269–297, 2015. Georges Aad, Tatevik Abajyan, B Abbott, J Abdallah, S Ab-del Khalek, Ahmed Ali Abdelalim, R Aben, B Abi, M Abolins,OS AbouZeid, et al. Observation of a new particle in the searchfor the standard model higgs boson with the atlas detector at thelhc.
Physics Letters B , 716(1):1–29, 2012. Serguei Chatrchyan, Vardan Khachatryan, Albert M Sirunyan,Armen Tumasyan, Wolfgang Adam, Ernest Aguilo, ThomasBergauer, M Dragicevic, J Er¨o, C Fabjan, et al. Observation ofa new boson at a mass of 125 gev with the cms experiment at thelhc.
Physics Letters B , 716(1):30–61, 2012. PB Littlewood and CM Varma. Amplitude collective modes in su-perconductors and their coupling to charge-density waves.
Physi-cal Review B , 26(9):4883, 1982. M-A M´easson, Yann Gallais, Maximilien Cazayous, BertrandClair, Pierre Rodiere, Laurent Cario, and Alain Sacuto. Amplitudehiggs mode in the 2 h- nbse 2 superconductor.
Physical Review B ,89(6):060503, 2014. Daniel Podolsky, Assa Auerbach, and Daniel P. Arovas. Visibilityof the amplitude (higgs) mode in condensed matter.
Phys. Rev. B ,84:174522, Nov 2011. Snir Gazit, Daniel Podolsky, and Assa Auerbach. Fate of the higgsmode near quantum criticality.
Phys. Rev. Lett. , 110:140401, Apr2013. T Cea, Claudio Castellani, G¨otz Seibold, and Lara Benfatto. Non-relativistic dynamics of the amplitude (higgs) mode in supercon-ductors.
Physical review letters , 115(15):157002, 2015. Tommaso Cea and Lara Benfatto. Nature and raman signatures ofthe higgs amplitude mode in the coexisting superconducting andcharge-density-wave state.
Physical Review B , 90(22):224515,2014. Abhisek Samanta, Amulya Ratnakar, Nandini Trivedi, and Ra-jdeep Sensarma. Two-particle spectral function for disordered s-wave superconductors: Local maps and collective modes.
Physi-cal Review B , 101(2):024507, 2020. Sabyasachi Tarat and Pinaki Majumdar. Tunneling spectroscopyacross the superconductor-insulator thermal transition. arXivpreprint arXiv:1406.5423 , 2014. Abhisek Samanta, Prashant Gupta, Nandini Trivedi, and RajdeepSensarma.
Unpublished . Anshul Kogar, Melinda S Rak, Sean Vig, Ali A Husain, Fe-lix Flicker, Young Il Joe, Luc Venema, Greg J MacDougall, Tai C Chiang, Eduardo Fradkin, et al. Signatures of exci-ton condensation in a transition metal dichalcogenide.