Transport and spectral signatures of transient fluctuating superfluids in the absence of long-range order
TTransport and spectral signatures of transient fluctuating superfluids in the absence oflong-range order
Yonah Lemonik and Aditi Mitra
Center for Quantum Phenomena, Department of Physics,New York University, 726 Broadway, New York, New York, 10003, USA (Dated: August 22, 2019)Results are presented for the quench dynamics of a clean and interacting electron system, wherethe quench involves varying the strength of the attractive interaction along arbitrary quench trajec-tories. The initial state before the quench is assumed to be a normal electron gas, and the dynamicsis studied in a regime where long-range order is absent, but nonequilibrium superconducting fluc-tuations need to be accounted for. A quantum kinetic equation using a two-particle irreducibleformalism is derived. Conservation of energy, particle-number, and momentum emerge naturally,with the conserved currents depending on both the electron Green’s functions and the Green’s func-tions for the superconducting fluctuations. The quantum kinetic equation is employed to derivea kinetic equation for the current, and the transient optical conductivity relevant to pump-probespectroscopy is studied. The general structure of the kinetic equation for the current is also justifiedby a phenomenological approach that assumes model-F in the Halperin-Hohenberg classification,corresponding to a non-conserved order-parameter coupled to a conserved density. Linear responseconductivity and the diffusion coefficients in thermal equilibrium are also derived, and connectionswith Aslamazov-Larkin fluctuation corrections are highlighted. Results are also presented for thetime-evolution of the local density of states. It is shown that Andreev scattering processes result inan enhanced density of states at low frequencies. For a quench trajectory corresponding to a suddenquench to the critical point, the density of states is shown to grow in a manner where the time afterthe quench plays the role of the inverse detuning from the critical point.
I. INTRODUCTION
Recent years have seen impressive advances in ultrafastmeasurements of strongly correlated systems . In theseexperiments, a pump field strongly perturbs the sys-tem, while a weak probe field studies the eventual time-evolution over time scales that can range from femto-seconds to nano-seconds. Thus one can study the fullnonequilibrium dynamics of the system from short times,to its eventual thermalization at long times. Moreover,the access to probe fields ranging from x-rays to mid-infrared allows one to probe dynamics from short lengthand time scales to longer length and time scales associ-ated with collective modes.Among the various pump-probe studies, notable exam-ples are experiments that show the appearance of highlyconducting, superconducting-like states, that can persistfrom a few to hundreds of pico-seconds . The micro-scopic origin of this physics is not fully understood. Ex-planations range from destabilization by the pump laserof a competing order in favor of superconductivity , toselective pumping of phonon modes that can engineer theeffective electronic degrees of freedom so as to enhancethe attractive Hubbard-U, and hence T c . There arealso theoretical studies that argue that the metastablehighly conducting state observed in experiments may nothave a superconducting origin .The resulting superconducting-like states are tran-sient in nature, where traditional measurements of su-perconductivity, such as the Meissner effect , cannotbe performed. On the other hand, time-resolved trans-port and angle-resolved photo-emission spectroscopy (tr- ARPES) are more relevant probes for such transientstates. Thus, a microscopic treatment that makes pre-dictions for transient conductivity and spectral featuresof a highly nonequilibrium system, accounting for dy-namics of collective modes, is needed. This is the goal ofthe paper.In what follows, we model the pump field as a quan-tum quench in that it perturbs a microscopic param-eter of the system, such as the strength of the attractiveHubbard-U. We assume that, as in the experiments ,the electronic system is initially in the normal phase. Thequench involves tuning the magnitude of the attractiveHubbard-U along different trajectories. How supercon-ducting fluctuations develop in time, and how they affecttransient transport and spectral properties are studied.Denoting the superconducting order-parameter as ∆, weassume that all throughout the dynamics, the systemnever develops true long range order, (cid:104) ∆( t ) (cid:105) = 0. Thusour approach is complementary to other studies wherethe starting point is usually a fully ordered supercon-ducting state, and the mean-field dynamics of the order-parameter (cid:104) ∆( t ) (cid:105) is probed . In our case, the non-trivial dynamics appears in the fluctuations (cid:104) ∆( t )∆( t (cid:48) ) (cid:105) which we study using a two particle irreducible (2PI)formalism. We justify the selection of diagrams usinga 1 /N approach where N denotes an orbital degree offreedom. The resulting choice of diagrams is equivalentto a self-consistent random-phase-approximation (RPA)in the particle-particle channel. The physical meaningof the RPA is that the superconducting fluctuations areweakly interacting with each other.While in this paper we study a clean system, a comple- a r X i v : . [ c ond - m a t . s up r- c on ] A ug mentary study of the transient optical conductivity of adisordered system appears in Ref. 30, where a Kubo for-malism approach was employed, and the quench dynam-ics arising from fluctuation corrections of the Aslamazov-Larkin (AL) and Maki-Thompson (MT) typewas explored. We avoid a Kubo-formalism approach inthis paper because conserving approximations are harderto make, as compared to a quantum kinetic equation ap-proach. This is an observation well known from otherstudies for transport in thermal equilibrium performedfor example in the particle-hole channel . Since we arein a disorder-free system, our quantum kinetic equationsconserve momentum. Thus, to obtain any non-trivialcurrent dynamics, we have to break Galilean invariance.We do this through the underlying lattice dispersion,while neglecting Umpklapp processes.Complementary studies of transient conductivityinvolve phenomenological time-dependent Ginzburg-Landau (TDGL) theories , and microscopic ap-proaches based on a Bardeen-Cooper-Schrieffer (BCS)mean-field treatment of the order-parameter dynamics .A microscopic mean field approach has also been usedto study transient spectral densities , while nonequilib-rium dynamical mean field theory has been used to studytransient spectral properties of a BCS superconductor .The outline of the paper is as follows. In Section II wepresent the model. In this section we also outline a quali-tative analysis for the transient conductivity employing aclassical Model-F scenario for the fluctuation dynamics.In Section III we derive the quantum kinetic equationsusing a two particle irreducible (2PI) formalism, and out-line how the transient dynamics respects all the conser-vation laws (particle, energy, momentum) of the system.In Section IV we simplify the 2PI quantum kinetic equa-tion to that of an effective classical equation for the decayof the current employing a separation of time-scales. Inthe process we justify the model-F scenario discussed inSection II. In Section V we assume thermal equilibrium,and derive the conductivity and diffusion constants. Wediscuss these in the context of Aslamazov-Larkin (AL)and Maki-Thompson (MT) corrections . In Section VIwe present results for the transient conductivity for somerepresentative quench profiles. In section VII we presentresults for the transient local density of states, and fi-nally in Section VIII we present our conclusions. Manyintermediate steps of the derivations are relegated to theappendices. II. MODEL
The Hamiltonian describes fermions with spin and anadditional orbital degree on a regular lattice in d spa-tial dimensions. Fermions are created by the operator c † σn ( r i ), where σ = ± n = 1 . . . N labels theorbital index and r i gives the lattice site. These have theFourier transform ˜ c σn ( k ) = (cid:80) r i e i(cid:126)k · (cid:126)r i c σn ( r i ). The dy- namics of the fermions is governed by the Hamiltonian: H ( t ) ≡ H + H int ( t ) , (1) H ≡ (cid:90) BZ d d k (2 π ) d (cid:88) σ,n ( (cid:15) k − µ )˜ c † σn ( k )˜ c σn ( k ) , (2) H int ≡ U ( t ) N (cid:88) r i ,n,m,σ c † σ,n ( r i ) c †− σ,n ( r i ) c − σ,m ( r i ) c σ,m ( r i ) , (3)where (cid:15) k is the dispersion, and µ is the chemical poten-tial. We will assume for simplicity that at this chemicalpotential there is only a single Fermi surface. We explic-itly allow for the interaction U ( t ) to vary with time.In addition, to probe current dynamics, we introducean electric field (cid:126)E = − ∂ t (cid:126)A by minimal coupling (cid:126)k → (cid:126)k − q (cid:126)A . Accounting for the charge of the electron q = − e , this appears in the Hamiltonian via (cid:126)k → (cid:126)k + e (cid:126)A ,or equivalently, (cid:126) ∇ /i → (cid:126) ∇ /i + e (cid:126)A . In the microscopicderivation we will set (cid:126) = 1.In thermal equilibrium at a temperature T , and forspatial dimension d >
2, the above system becomes su-perconducting below a critical temperature
T < T C . At d = 2, there is a Berezenskii-Kosterlitz-Thouless (BKT)transition at T < T
BKT where the system shows onlyquasi-long range order. For
T > T
BKT , although the sys-tem has no long range order, yet superconducting fluctu-ations play a key role in transport. The lower the spatialdimensions, the more important the role of fluctuations.In particular for spatial dimension d = 2, well knownresults for the optical conductivity exist for a stronglydisordered system . While our derivation is valid in anyspatial dimension, we will present results for spatial di-mensions d = 2 where fluctuation effects are most pro-nounced. The experiments also study either a layeredsuperconducting system, or three dimensional systemswhere the pump field is effectively a surface perturbation.This makes the spatial dimension d = 2 also experimen-tally relevant.Before we go into details, we note that our microscopictreatment leads to the following expression (reported inRef. 44) for the dynamics of the current j i , generated byan applied electric field E i , ∂ t j i ( t ) − E i ( t ) = − τ r (cid:20) A ( t ) j i ( t ) − α (cid:90) t dt (cid:48) (cid:26) B ( t, t (cid:48) ) j i ( t (cid:48) ) + C ( t, t (cid:48) ) E i ( t (cid:48) ) (cid:27)(cid:21) . (4)Above τ r , A, B, C are kinetic coefficients. We have set e = 1 , ρ/ ¯ m = 1, where ρ is the electron density, and ¯ m isthe electron effective mass. When B = C = 0, the aboveequation simply implies a time-dependent Drude scat-tering rate τ − r A ( t ), which in our model arises becauseelectrons scatter off superconducting fluctuations whosedensity is changing with time due to the quench. Theappearance of memory terms B, C is non-trivial, and be-fore deriving them, we will justify the appearance of thesememory terms through a phenomenological approach inthe next subsection.
A. Model F dynamics
In a finite temperature phase transition the dynamicsof the fluctuations may be understood by treating themas classical stochastic fields. This will be seen by directcalculation later in the paper. However for now, we seekto write a Langevin equation for the (classical) field ∆.In the theory of dynamical critical phenomena, we mustaccount not only for the fluctuating mode, ∆, but alsofor any conserved quantities which are coupled to ∆. Thepairing field ∆ couples directly to the density ρ , as givenby the commutator [∆ , ρ ] = 2∆. This can be translatedinto classical terms by the usual prescription [ · , · ] /i (cid:126) →{· , ·} where {· , ·} is the classical Poisson bracket.In the framework of dynamical critical phenomena wespecify the static free energy (see Ref. 42, Eq. (6.3)) F [∆ , ρ ] ≡ (cid:90) d d r (cid:20) e Φ δρ + 12 C ( δρ ) + r ( t ) | ∆ | + 12 M | ( i(cid:126) ∇ − e (cid:126)A )∆ | + u | ∆ | + γ ( δρ ) | ∆ | (cid:21) . (5)Here δρ is density minus the equilibrium density, Φ is thescalar potential, C is the compressibility, r ( t ) is the timedependent detuning from the critical point, (cid:126)A ( t ) is anexternal vector potential, and u , γ are phenomenologi-cal parameters. Note that model E has γ = 0, whilemodel F has γ (cid:54) = 0.The field ∆ then has the equation of motion, (cid:18) ∂ ∆ ∂t − { ∆ , ρ } ∂ F ∂ρ (cid:19) = − Γ ∆ ∂ F ∂ ∆ ∗ + ξ ∆ . (6)Using Eq. (5), we obtain, (cid:18) ∂ ∆ ∂t + 2 i ∆ (cid:20) e Φ + δρC + γ | ∆ | (cid:21)(cid:19) = − Γ ∆ (cid:20) r ( t ) + γ δρ + 2 u | ∆ | + 12 M ( i ∇ i − eA i ) (cid:21) ∆ + ξ ∆ , (7)where ξ ∆ is a Gaussian white noise obeying (cid:104) ξ ∆ ( r, t ) ξ ∆ ∗ ( r (cid:48) , t (cid:48) ) (cid:105) = 2 T Γ ∆ δ ( r − r (cid:48) ) δ ( t − t (cid:48) ), andΓ ∆ is a dimensionless phenomenological damping.We will denote the total current by a dissipative com-ponent j dis , and a superfluid component j sc , j i = j i sc + j i dis . The equation of motion for the density ρ is givenby, ∂δρ∂t − { ρ, ∆ } ∂ F ∂ ∆ − { ρ, ∆ ∗ } ∂ F ∂ ∆ ∗ = (cid:126) ∇ · (cid:126)j dis , (8)where the right hand side (rhs) can be interpreted as adefinition of j dis . Under assumptions that the currentmay be written as a gradient of a scalar, and that it reaches a well defined steady state in the presence of adc electric field, (cid:126)j dis = − Γ j dis (cid:32) (cid:126) ∇ ∂ F ∂ρ + e ∂ (cid:126)A∂t (cid:33) + (cid:126)ξ j dis . (9)Above, (cid:126)ξ j dis is a Gaussian white noise, whose strengthis now controlled by T Γ j dis , where Γ j dis is another phe-nomenological dissipation. Using the fact that ∂ t (cid:126)A = − (cid:126)E , the steady-state dissipative current for a spatiallyhomogeneous system is (setting e = 1) j i dis = Γ j dis E i . (10)Now using, { ρ, ∆ } ∂ F ∂ ∆ + { ρ, ∆ ∗ } ∂ F ∂ ∆ ∗ = iM (cid:20) ∆( i ∇ i + 2 eA i ) ∆ ∗ − ∆ ∗ ( i ∇ i − eA i ) ∆ (cid:21) = (cid:126) ∇ · (cid:126)j sc , (11)we find that the definition of the superfluid current nat-urally emerges, (cid:126)j sc = 2 M Im (cid:104) ∆( (cid:126) ∇ − ie (cid:126)A )∆ ∗ (cid:105) . (12)Substituting Eq. (11) in Eq. (8), we have, ∂δρ∂t = (cid:126) ∇ · (cid:126)j sc + (cid:126) ∇ · (cid:126)j dis . (13)We now relate these equations to our equations of mo-tion Eq. (4). The expectation value for the supercurrentis (cid:126)j sc = 2 M Im (cid:104) ∆( x ) (cid:126) ∇ ∆ ∗ ( x ) (cid:105) = 2 M (cid:90) d d q (2 π ) d (cid:126)q (cid:104) ∆ q ∆ ∗ q (cid:105) = 2 (cid:90) d d q (2 π ) d (cid:126)qM F ( q, t ) , (14)where F ( q ) is the equal time ∆ q ∆ ∗ q correlator at momen-tum q . Now we show that the integral or memory termproportional to α in Eq. (4) comes from a term of theform ∝ (cid:90) d d q (cid:126)qM ∂ t F ( q, t ) , (15)and that this term is precisely ∂ t j sc . The remainder cantherefore be identified with the dissipative current givingan equation ∂ t j i dis − E i ( t ) = − τ − r A ( t ) j i dis . (16)This should be compared to the expectation value forthe dissipative current Eq. (10). (Note that the dampingcoefficient A is not to be confused with the vector poten-tial A i ). Thus in the dc limit of the kinetic equation, weidentify Γ j dis = τ r A − . Note that the model F dynam-ics are less general in the sense that they assume this dclimit, and therefore that all perturbations are slow com-pared with Γ j dis . The kinetic equation does not makethis assumption.Now let us evaluate ∂ t j sc . We simplify the general-ized time-dependent Ginzburg-Landau theory in Eq. (7)by going into Fourier space, and dropping all non-linearterms, ∂ ∆ q ∂t = − Γ ∆ (cid:20) r ( t ) + 12 M ( q i + 2 eA i ( t )) + γ δρ q ( t ) (cid:21) ∆ q + ξ ∆ . (17)Above we have encoded the effect of the electric field intwo places, one in the minimal coupling (cid:126)q → (cid:126)q + 2 e (cid:126)A ,and second in the change in the electron density at mo-mentum q , where by symmetry, δρ q ( t ) = cj i ( t ) q i , (18)where c is a phenomenological constant. Going forward,we absorb this constant into a redefinition of γ appearingin the combination γ δρ q ( t ).Defining, λ q ( t ) = Γ ∆ (cid:20) r ( t ) + q M (cid:21) = Γ ∆ (cid:15) q ( t ) , (19)in the absence of an electric field, the solution is∆ q ( t ) = (cid:90) t dt (cid:48) ξ ∆ ( t (cid:48) ) e − (cid:82) tt (cid:48) dt (cid:48)(cid:48) λ q ( t (cid:48)(cid:48) ) . (20)Above we have adopted boundary conditions where thesuperconducting order-parameter and fluctuations arezero at t = 0. This is because we are interested inquenches where there are initially t ≤ t > F ( q, E = 0 , t ) = (cid:104) ∆ q ∆ ∗ q (cid:105) = 2Γ ∆ T (cid:90) t dt (cid:48) e − (cid:82) tt (cid:48) dt (cid:48)(cid:48) λ q ( t (cid:48)(cid:48) ) . (21)In the presence of an electric field, and to leading orderin it, λ q changes to λ q + δλ q , with δλ q ( s ) = Γ ∆ M (2 e ) (cid:126)q · (cid:90) ts dt (cid:48) (cid:126)E ( t (cid:48) ) + Γ ∆ γ q i j i ( s ) , (22)where the first term is due to the order-parameter be-ing charged, and the second term is due to the cou-pling of the order-parameter to the normal electronswhose density is perturbed by the electric field. We havealso used that the electric field is (cid:126)E = − ∂ t (cid:126)A , so that (cid:126)A ( t ) = − (cid:82) ts dt (cid:48) (cid:126)E ( t (cid:48) ). Expanding Eq. (21) in δλ q , which is equivalent to ex-panding in the electric field, F ( q, t ) (cid:39) ∆ T (cid:90) t dt (cid:48) e − (cid:82) tt (cid:48) dt (cid:48)(cid:48) λ q ( t (cid:48)(cid:48) ) (cid:20) − (cid:90) tt (cid:48) duδλ q ( u ) (cid:21) . Splitting the integral in the exponent, (cid:82) tt (cid:48) dt (cid:48)(cid:48) = (cid:82) tu dt (cid:48)(cid:48) + (cid:82) ut (cid:48) dt (cid:48)(cid:48) , and using Eq. (21), we arrive at the following ex-pression for the superconducting fluctuations at momen-tum q , to leading non-zero order in the applied electricfield, for an arbitrary time-dependent detuning r ( t ), F ( q, t ) = F ( q, E = 0 , t ) + δF ( q, t ) ,δF ( q, t ) = − (cid:90) t duδλ q ( u ) e − (cid:82) tu dt (cid:48)(cid:48) λ q ( t (cid:48)(cid:48) ) F ( q, E = 0 , u ) . (23)
1. Aslamazov-Larkin (AL)
We now briefly show how one may recover the familiarAL conductivity in thermal equilibrium. For this it suf-fices to revert to model E by setting γ = 0. For a staticelectric field, and a system in thermal equilibrium, allcouplings are time-independent. Thus Eq. (23) becomes, δF ( q, t ) = − F ( q ) (cid:90) t duδλ q ( u ) e − t − u ) λ q . (24)Using Eq. (22) for γ = 0, δλ q ( u ) = Γ ∆ M (2 e )( t − u ) (cid:126)q · (cid:126)E . Taking the long time t → ∞ limit, and employingthe equilibrium expression for F ( q ) = T /(cid:15) q , where (cid:15) q = r + q / M , the change in the density of superconductingfluctuations due to the applied electric field is, δF q = Γ − T(cid:15) q (cid:126)qM · e (cid:126)E. (25)The current is j i = 2 e (cid:88) q q i M δF ( q ) = Γ − e (cid:88) q T(cid:15) q q i q j M E j , (26)leading to the AL conductivity in d dimensions, σ AL = 2 e Γ − T (cid:90) d d q (2 π ) d q dM r + q / M ) . (27)In d = 2, this reduces to σ d =2AL = e π Γ ∆ Tr . (28)A microscopic treatment involving electrons in a disor-dered potential shows that Γ − = π/
8, giving the wellknown expression, σ d =2AL = e Tr . In our disorder-freemodel, Γ ∆ takes a different value.
2. Kinetic equation for the current
We now return to the derivation of the dynamics of thecurrent in model F. For the current dynamics we need todifferentiate Eq. (23) with time, ∂ t F ( q, E, t ) = ∂ t F ( q, E = 0 , t ) − δλ q ( t ) F ( q, E = 0 , t )+ 4 λ q ( t ) (cid:90) t duδλ q ( u ) e − (cid:82) tu dt (cid:48)(cid:48) λ q ( t (cid:48)(cid:48) ) F ( q, E = 0 , u ) . (29)The first term on the rhs, being symmetric in momentumspace, does not contribute to the current. The secondterm on the rhs simply provides a time-dependent Drudescattering rate. It is the last term, namely the memoryterm proportional to the electric field, that is unique tohaving long lived superconducting fluctuations. We focusonly on this term, and using Eq. (22), write it as, ∂ t j i sc ( t ) = 2 (cid:90) d d q (2 π ) d q i M ∂ t F ( q, t )= 8Γ ∆ M (cid:90) d d q (2 π ) d q i q j λ q ( t ) (cid:90) t du (cid:20) γ j j ( u )+ 2 eM (cid:90) tu dt (cid:48) E j ( t (cid:48) ) (cid:21) e − (cid:82) tu dt (cid:48)(cid:48) λ q ( t (cid:48)(cid:48) ) F ( q, u ) . (30)Above, we have dropped the E = 0 label in F . In thesecond term, we will find it convenient to replace the timeintegrals as follows (cid:82) t du (cid:82) tu dt (cid:48) → (cid:82) t dt (cid:48) (cid:82) t (cid:48) du .Comparing equations (4) and (30), we identify, τ − r αB ( t, t (cid:48) ) = 8Γ ∆ γ dM (cid:90) d d q (2 π ) d q λ q ( t ) × e − (cid:82) tt (cid:48) dt (cid:48)(cid:48) λ q ( t (cid:48)(cid:48) ) F ( q, t (cid:48) ) . (31)The second memory term may be identified as, τ − r αC ( t, t (cid:48) ) = 8Γ ∆ dM (2 e ) (cid:90) d d q (2 π ) d q λ q ( t ) × (cid:90) t (cid:48) due − (cid:82) tu dt (cid:48)(cid:48) λ q ( t (cid:48)(cid:48) ) F ( q, u ) . (32)Comparing Eq. (31) and Eq. (32), we obtain, C ( t, t (cid:48) ) = 2 eM γ (cid:90) t (cid:48) dsB ( t, s ) . (33)The microscopic treatment that follows not only justifiesmodel F, it also provides an explicit calculation for theparameters τ − r , A ( t ) , α, Γ ∆ , Γ j dis , M, r, γ . III. PI EQUATIONS OF MOTIONA. Properties of PI formalism
We briefly recapitulate the 2PI formalism here. A de-tailed explanation can be found for example in Refs 45–48. The 2PI formalism begins with the Keldysh action S for the Hamiltonian H . This is written in terms ofGrassmann fields ψ σ,n, ± ( r, t ), ¯ ψ σ,n, ± ( r, t ) where σ labelsthe spin, n labels the orbital quantum number, and ± la-bel the two branches of the Keldysh contour . In termsof these, the Keldysh action is given as S K [ ψ, ¯ ψ ] ≡ (cid:90) dt (cid:26) i (cid:88) σ = ↑ , ↓ ; n,r ¯ ψ σn + ( r, t ) ∂∂t ψ σn + ( r, t ) − H (cid:2) ¯ ψ + ( t ) , ψ + ( t ) (cid:3) − i (cid:88) σ = ↑ , ↓ ; n,r ¯ ψ σn − ( r, t ) ∂∂t ψ σn − ( r, t )+ H (cid:2) ¯ ψ − ( t ) , ψ − ( t ) (cid:3) (cid:27) , (34)where H (cid:2) ¯ ψ ± ( t ) , ψ ± ( t ) (cid:3) indicates the substitution of ψ ± ( t ) for c ( t ) in the Hamiltonian, Eq. (3), in the ob-vious way. Now we consider a classical field h that cou-ples to a general bilinear of the Grassmann field, and istherefore given by h σ (cid:48) n (cid:48) p (cid:48) σ n p ( r, t, r (cid:48) , t (cid:48) ), where p , p (cid:48) run over ± . In order to simplify notation, we combine the fiveindices, spin, orbital, Keldysh, position and time into asingle vector index, so that the Grassman field becomesa vector ψ i and the source field a matrix h ij .From the action and the source field, we form a gener-ating functional I [ h ]: I [ h ] = (cid:90) D (cid:2) ¯ ψ, ψ (cid:3) exp (cid:20) iS K [ ¯ ψ, ψ ] + i (cid:88) ij ¯ ψ i h ij ψ j (cid:21) . (35)The first derivative of I is precisely the Green’s functionin the presence of the source field hG ij [ h ] = δ I δh ji = i (cid:90) D (cid:2) ¯ ψ, ψ (cid:3) (cid:0) ¯ ψ j ψ i (cid:1) e iS K + i ¯ ψ · h · ψ . (36)In order to work with the physical Green’s function,rather than the unphysical source field h , we perform aLagrange inversion. First we invert Eq. (36) which gives G as a function of h , to implicitly define h as a function G .Then we construct the Lagrange transformed functional W , W [ G ] = I [ h ( G )] − h ( G ) G. (37)This has the property that, δ W δG = − h ( G ) . (38)In particular in the physical situation when h = 0 we havethat δ W /δG = 0. Thus given the functional W , we havereduced the problem of calculating G to a minimizationproblem.The functional W can be calculated in terms of a per-turbation series in U , giving W [ G ] = i G − ) + i (cid:2) g − G (cid:3) + Γ [ G ] , (39)here the Tr operates on the entire combined index space,and g − is the bare electron Green’s function. The func-tional Γ [ G ] is constructed as follows. First the Feynmanrules for the Hamiltonian H are written down. In thepresent case, this is a single fermionic line and a four-fermion interaction. Then all two-particle irreduciblebubble diagrams are drawn. A diagram is two particleirreducible (2PI) if it cannot be disconnected by deletingup to two fermionic lines. The functional Γ [ G ] is givenby the sum of all the diagrams. Lastly the diagrams areinterpreted as in usual diagrammatic perturbation the-ory except that the fermionic line is not the bare Green’sfunction g , but instead the full Green’s function G .The end result is the minimization condition δ W δG = − i G − + i g − + δ Γ δG = 0 , (40) → G − = g − − Σ , Σ [ G ] = 2 iδ Γ /δG. (41)This is the Dyson equation for the Green’s function G where the self energy Σ[ G ] ≡ iδ Γ /δG is a self-consistentfunction of G . As no approximations have been madeEq. (41) is a restatement of the original many-body prob-lem, and clearly cannot be solved. The advantage of theformalism is that we may replace Γ with an approxi-mate functional Γ (cid:48) and still be guaranteed to preserveconservation laws and causality, which is not the case ifone directly approximates Σ.We assume that all symmetries are maintained by thesolution to Eq. (41). Therefore the Green’s function canbe written in terms of the original spin, orbital, Keldysh,space, and time indices as, G σ ,n ,p ; σ ,n ,p ( r , t ; r , t ) = δ σ σ δ n ,n × G p p ( r − r ; t , t ) , (42)so that only the Keldysh indices p , p will be explicitlynoted. + + + ........ FIG. 1: 2PI Ladder diagrams for Γ (cid:48) . Solid lines areelectron Green’s functions, while the dashed lines arethe interaction U. B. RPA approximation to the 2PI generatingfunctional
The approximation we make is the random phase ap-proximation (RPA) in the particle-particle channel, or,equivalently the N → ∞ where N is the number of or-bital degrees of freedom. That is we approximate Γ byΓ (cid:48) where the latter include the set of all closed ladder diagrams,Γ (cid:48) ≡ − i · D D ≡ (cid:0) U − − Π (cid:1) − , (43)Π p p ( r , r ; t , t ) ≡ i G p p ( r − r ; t , t )] , (44)shown diagrammatically in Fig. 1, which gives the equa-tion, Σ p p ( r , t ; r , t ) = iD p p ( r , t ; r , t ) × G p p ( r , t ; r , t ) . (45)Above we have used that Σ(1 ,
2) = 2 iδ Γ /δG (2 , G and D given by W (cid:48) [ G, D ] = i (cid:2) ln( G − ) + g − G + ln( D − ) + U − D (cid:3) + Γ (cid:48) [ G, D ] , (46)Γ (cid:48) [ G, D ] ≡ − i · D, (47) δ W (cid:48) δD = − i D − + i U − − i , (48) δ W (cid:48) δG = − i G − + i g − − i . (49)This may be interpreted as the functional for electronsinteracting with a fluctuating pairing field whose Green’sfunction is D . The Γ (cid:48) is the minimal diagram whichhas interaction between the fermions and the fluctuatingpairing field. C. RPA equations of motion
We now proceed to analyze the RPA equations of mo-tion. As is customary, we perform a unitary rotation ofthe Keldysh space, defining G K,A,R by G K ≡ ( G ++ + G + − + G − + + G −− ) / , (50) G R ≡ ( G ++ − G + − + G − + − G −− ) / , (51) G A ≡ ( G ++ + G + − − G − + − G −− ) / , (52)0 = G ++ − G + − − G − + + G −− , (53)the last equality holding by causality. We likewise defineΣ K,A,R , D
K,A,R , Π K,A,R . In this basis we may write theequations of motion as (setting e = 1) (cid:20) i∂ t − (cid:15) (cid:18) i(cid:126) ∇ − (cid:126)A ( r , t ) (cid:19)(cid:21) g R ( r , t ; r , t )= δ ( t − t ) , (54a) G − R = g − R − Σ R , (54b) G K = G R · Σ K · G A , (54c) D − R = U − − Π R , (54d) D K = D R · Π K · D A , (54e)and where (cid:15) denotes the single particle dispersion, thesymbol · convolution, and Σ , Π are given byΣ R ( x, y ) = i (cid:20) D R ( x, y ) G K ( y, x ) + D K ( x, y ) G A ( y, x ) (cid:21) , (54f)Σ K ( x, y ) = i (cid:20) D K ( x, y ) G K ( y, x ) + D R ( x, y ) G A ( y, x )+ D A ( x, y ) G R ( y, x ) (cid:21) , (54g)Π R ( x, y ) = i G K ( x, y ) G R ( x, y ) , (54h)Π K ( x, y ) = i (cid:2) G K ( x, y ) + G R ( x, y ) + G A ( x, y ) (cid:3) , (54i)and where x and y stand for combined space, and timeindices. The function G A is given by G A = G † R , where O ( x, y ) † = O ( y, x ) ∗ , and likewise for D A , Σ A , Π A . Wenote that we also have the relationship G † K = − G K We now convert the definition of G K into a more usefulform by evaluating G − R · G K − G K · G − A . On the onehand, substituting in the definition of G − R,A this is, G − R · G K − G K · G − A = g − R · G K − G K · g − A − Σ R · G K + G K · Σ A , (55)on the other hand substituting in G K = G R · Σ K · G A weobtain G − R · G K − G K · G − A = Σ K · G A − G R · Σ K . (56)Therefore we obtain the fundamental equation g − R · G K − G K · g − A = Σ K · G A − G R · Σ K +Σ R · G K − G K · Σ A . (57)Setting the two times equal and using the form of g R weobtain, ∂ t iG K ( r , t ; r , t ) − (cid:20) (cid:15) (cid:16) i(cid:126) ∇ r − (cid:126)A ( r , t ) (cid:17) − (cid:15) (cid:16) − i(cid:126) ∇ r − (cid:126)A ( r , t ) (cid:17)(cid:21) G K ( r , t ; r , t ) = S ( r t, r t ) ,S ( r t, r t (cid:48) ) ≡ (cid:90) dy (cid:8) Σ K ( r , t ; y ) G A ( y, r , t (cid:48) )+ Σ R ( r , t ; y ) G K ( y ; r , t (cid:48) ) − G R ( r , t ; y )Σ K ( y ; r , t (cid:48) ) − G K ( r , t ; y )Σ A ( y ; r , t (cid:48) ) (cid:9) . (58)Above y includes both space and time coordinates, andwe have given the collision term S on the rhs for unequaltimes as we will need it to prove different conservationlaws in the next section. D. Conservation laws
We emphasize that the true Green’s function are foundby minimizing with respect to the full 2PI potential Γ ,whereas the functions G , D defined above are only ap-proximations, found by minimizing with respect to theapproximate potential Γ (cid:48) . Therefore it is not aprioriclear that there are any conserved quantities that cor-respond to the conserved quantities of the true Hamilto-nian. However, the existence of such quantities is guaran-teed by the 2PI formalism. Here we show that the naiveexpression for the conserved density n ( r, t ) ≡ iG K ( r, t ; r, t ) , (59)is correct. Setting r = r in Eq. (58), ∂ t iG K ( r, t ; r, t ) − (cid:20) (cid:15) (cid:16) i(cid:126) ∇ r − (cid:126)A ( r , t ) (cid:17) − (cid:15) (cid:16) − i(cid:126) ∇ r − (cid:126)A ( r , t ) (cid:17)(cid:21) G K ( r , t ; r , t ) (cid:12)(cid:12)(cid:12)(cid:12) r = r = r = S ( rt, rt ) . (60)Inspecting the second term on the left hand side (lhs) wesee that it is a total derivative and the equation can bewritten as ∂ t n ( r, t ) − (cid:126) ∇ · (cid:126)j ( r, t ) = S ( rt, rt ) , (61) (cid:126)j = (cid:126)v (cid:18) i(cid:126) ∇ − (cid:126)A ( r , t ) , i(cid:126) ∇ + (cid:126)A ( r , t ) (cid:19) iG K ( r , t ; r , t ) (cid:12)(cid:12)(cid:12)(cid:12) r = r = r , where the velocity (cid:126)v generalized for an arbitrary disper-sion (cid:15) , is defined as( (cid:126)x + (cid:126)y ) · (cid:126)v ( (cid:126)x, (cid:126)y ) = (cid:15) ( (cid:126)x ) − (cid:15) ( − (cid:126)y ) . (62)This has the form of a continuity equation for the con-served quantity n because S ( rt, rt ) = 0 . (63)We demonstrate this in Appendix A 1. Thus the particlenumber n ( r, t ) = iG K ( rt ; rt ) is a conserved quantity.Conservation of momentum follows similarly by takingthe definition of the momentum p α ( r, t ) = (cid:20) i (cid:0) ∇ αr − ∇ αr (cid:1) − A α ( r, t ) (cid:21) iG K ( r , t ; r , t ) (cid:12)(cid:12)(cid:12)(cid:12) r = r = r . (64)Evaluating the time-derivative of p α , and using Eq. (58),one obtains, ∂ t p α ( r, t ) − n ( r, t ) E α ( r, t ) + (cid:15) αβγ j β ( r, t ) B γ ( r, t ) − ∇ βr T αβ ( r, t ) = 0 , (65) T αβ = (cid:20) i (cid:0) ∇ αr − ∇ αr (cid:1) − A α ( r, t ) (cid:21) v β ( i ∇ − A ( r , t ) , i ∇ + A ( r , t )) iG K ( r t, r t ) (cid:12)(cid:12)(cid:12)(cid:12) r = r = r + δ αβ U − ( t ) iD K ( rt, rt ) . (66)Above, the electric field is E α = − ∂ t A α and the magneticfield is B α = (cid:15) αβγ (cid:0) ∇ β A γ − ∇ γ A β (cid:1) , and T αβ is themomentum tensor. Above, we have used the relation i (cid:20) ∇ αr − ∇ αr (cid:21) S ( r t, r t ) (cid:12)(cid:12)(cid:12)(cid:12) r = r = r == iU − ( t ) ∇ αr D K ( rt, rt ) . (67) which is proved in Appendix A 2. We also note that since S ( rt, rt ) = 0, A α ( r, t ) S ( rt, rt ) = 0. Thus, the momentumtensor has a contribution from the full electron Green’sfunction G , and also from the full superconducting fluc-tuations D .Finally conservation of energy is obtained by operatingon the kinetic equation i ( ∂ t − ∂ t ) and setting, r = r , t = t . This yields, ∂ t E ( r, t ) − (cid:126) ∇ · (cid:126)j E = (cid:126)E · (cid:126)j + Q ext ( r, t ) , (68) E ( r, t ) = i ∂ t − ∂ t ) iG K ( r, t ; r, t ) | t = t = t − iU − ( t ) D K ( r, t ) , (69) j β E = i ∂ t − ∂ t ) v β (cid:18) i ∇ − A ( r , t ) , i ∇ + A ( r , t ) (cid:19) iG K ( r , t , r , t ) (cid:12)(cid:12)(cid:12) r = r = rt = t = t , (70) Q ext = − iD K ( r, t ) ∂ t U − ( t ) . (71)The necessary relation for S needed to derive the aboveis i ∂ t − ∂ t ) S ( rt , rt ) | t = t = t = i∂ t (cid:0) U − D K ( r, t ; r, t ) (cid:1) − iD K ( r, t ; r, t ) ∂ t U − ( t ) . (72)The proof of the above is in Appendix A 3. The aboveshows that the energy is not conserved due to Joule heat-ing (cid:126)E · (cid:126)j , from an applied electric field, and also whenthe parameters of the Hamiltonian are explicitly time-dependent ∂ t U ( t ) (cid:54) = 0. IV. REDUCTION TO A CLASSICAL KINETICEQUATION
In this section we project the full quantum kineticequation onto the dynamics of the current. This canbe done by noting some important separation of energy-scales associated with critical slowing down of the super-conducting fluctuations.
A. Critical slowing down
Although the RPA approximation produces a closedset of equations of motion, Eq. (54), these are a set of non-linear coupled integral-differential equations, andtherefore cannot be solved without further reduction.The key to reducing the equations further is the phe-nomenon of critical slowing down . This may bedemonstrated by considering the particle-particle polar-ization in equilibrium. As the equilibrium system is timeand space-translation invariant, we consider the FouriertransformΠ R ( q, ω ) = (cid:90) d d rdt Π R ( r, t, r (cid:48) , t (cid:48) ) e − i(cid:126)q · ( (cid:126)r − (cid:126)r (cid:48) )+ iω ( t − t (cid:48) ) = i (cid:90) d d kdη G R (cid:16) k + q , η + ω (cid:17) G K (cid:16) − k + q , − η + ω (cid:17) . (73)Assuming for the moment that G R has a Fermi-liquidform G R ( k, ω ) = Zω − (cid:15) k + i/ τ + G inc , (74)where Z is the quasiparticle residue, τ − is the de-cay rate and is ∝ max( ω , T ), while G inc is a smoothincoherent part whose effect is negligible at low ener-gies. We also assume quasi-equilibrium where G K isrelated to G R by the fluctuation dissipation theorem G K ( k, ω ) = 2Im [ G R ( k, ω )] tanh( ω/ T ). We expect thatΠ R ( r, t ; r, t (cid:48) ) decays exponentially on the scale T − . Thusfrom the perspective of any dynamics that is slower than T − , the polarization is effectively short ranged in time.Thus, in terms of the Fourier expansion, we only needthe leading behavior in ω/T . Note that the short timedynamics showing the onset of quasi-equilibrium of theelectron distribution function to a temperature T can alsobe studied by the numerical time-evolution of the RPADyson equations .Now the Green’s function D R is precisely the linearresponse to a superconducting fluctuation. Therefore,if U < U c , where U c denotes the critical value of thecoupling, the system is in the disordered phase and D R should decay exponentially with time. If U > U c thesystem is super-critical and superconducting fluctuationsshould grow exponentially in time. In terms of theFourier transform D R ( q, ω ), this means that there is apole in the complex ω plane which crosses the real axisat precisely the phase transition, U c ( T ). Therefore wemay write D − R ( q, ω ) = U − − Π R ( q, ω )= Z ( q ) (cid:20) λ ( q, U ) + iω/T (cid:21) + O (cid:18) ω T (cid:19) , (75)where the function λ ( q = 0 , U c ( T )) = 0. There is acritical regime in q and U − U c where λ ( q, U ) (cid:28) T . Wefurther define a thermal wavelength q T by the expression λ ( q T , U c ) = T , which we estimate as q T ∼ T /v F , where v F is the Fermi velocity.On the assumption, to be explicitly shown later, thatthe transport is dominated by processes that are muchslower than T − , we therefore replaceΠ R ( q ; t, t (cid:48) ) = U − ( t ) − Z ( q ) δ ( t − t (cid:48) ) (cid:20) λ ( q, U ( t )) + ∂ t (cid:21) , (76)where we have explicitly included the possibility that U ( t ) is time dependent. Therefore the equations of mo-tion for D are,( ∂ t + λ q ( t )) D R ( q, t, t (cid:48) ) = T ν − δ ( t − t (cid:48) ); (77) λ q ( t ) = r ( t ) + q / M, (78)where r ( t ) is an effective detuning arising from the thetime-dependence of the interaction U ( t ). Once D R isknown then, D K follows from Eq. (54e), and the factthat Π K ( t, t (cid:48) ) = 4 iνδ ( t − t (cid:48) ).We consider two cases below. One where λ q ( t ) = θ ( t ) λ q + θ ( − t ) r i , with r i /T = O (1). This represents arapid quench, where the detuning changes from an initialvalue to a final value at a rate that is faster than or ofthe order of the temperature. The second situation weconsider is an arbitrary trajectory for the detuning r ( t ).The solution for bosonic propagators for the former, namely the rapid quench, at times t, t (cid:48) ≥ ⇒ D R ( q, t, t (cid:48) ) = θ ( t − t (cid:48) ) T ν − e − λ q ( t − t (cid:48) ) , (79) iD K ( q, t, t (cid:48) ) = 2 T ν − Tλ q (cid:16) e − λ q | t − t (cid:48) | − e − λ q ( t + t (cid:48) ) (cid:17) , (80)=2 [ D R ( q, t, t (cid:48) ) F ( q, t (cid:48) )+ F ( q, t ) D A ( q, t, t (cid:48) )] , (81) F ( q, t ) ≡ Tλ q (cid:0) − e − λ q t (cid:1) . (82)Note the function F ( q, t ) changes with rate λ q which forsmall q is (cid:28) T . Also note that at t = t (cid:48) = 0, the densityof superconducting fluctuations as measured by D K ( t = t (cid:48) = 0) or F ( t = 0) is zero, consistent with an initialcondition where the initial detuning is large and positive,and hence far from the critical point.For an arbitrary trajectory r ( t ) and hence λ q ( t ), thesolutions for the bosonic propagators, generalize as fol-lows, D R ( q, t, t (cid:48) ) = θ ( t − t (cid:48) ) T ν − e − (cid:82) tt (cid:48) dt (cid:48)(cid:48) λ q ( t (cid:48)(cid:48) ) , (83) iD K ( q, t, t (cid:48) ) =2 [ D R ( q, t, t (cid:48) ) F ( q, t (cid:48) )+ F ( q, t ) D A ( q, t, t (cid:48) )] , (84) F ( q, t ) ≡ T (cid:90) t dxe − (cid:82) tx dyλ q ( y ) . (85)We show in Appendix B that the above equations for D R,A,K imply an effective model for a bosonic field obey-ing Langevin dynamics, where the Langevin noise isdelta-correlated with a strength proportional to the tem-perature T . B. Linear response to electric field
The previous derivation of the kinetic equationsEq. (58) is correct for arbitrary electric fields strengths,and therefore includes non-linear effects. We now ex-pand the Dyson equation to linear order in E ( t ) in or-der to evaluate the linear response. Note that since g R ( k, t, t (cid:48) ) = e − i (cid:82) tt (cid:48) ds(cid:15) ( k i + A i ( s )) , it is the combination, g R ( (cid:126)k − (cid:126)A ( t )) = e − i (cid:82) tt (cid:48) ds(cid:15) ( k i + A i ( s ) − A i ( t )) which is gauge-invariant. The leading change in the electron Green’sfunctions at the gauge invariant momentum is, g R ( t, t (cid:48) ) = − iθ ( t − t (cid:48) ) e − i(cid:15) k ( t − t (cid:48) ) × (cid:18) − i (cid:90) tt (cid:48) ds(cid:126)v k · (cid:104) (cid:126)A ( s ) − (cid:126)A ( t ) (cid:105)(cid:19) , (86) n k = n eq k + δn k ( t ) , (87)where n k = (cid:82) d d ( r − r ) e − ik ( r − r ) iG K ( r , r ), and δn is assumed to be of order E .We now write the kinetic equation (58) after Fouriertransforming with respect to the spatial coordinates0 r − r . The change in density at the gauge-invariantmomentum is ddt n k − A ( t ) = ∂ t n k ( t ) − ∂ t (cid:126)A · (cid:126) ∇ k n eq k = I C + (cid:126)E · (cid:126) ∇ k n eq k , (88)where I C is the rhs of the kinetic equation (58) evaluatedat the gauge-invariant momentum (cid:126)k − (cid:126)A ( t ). Substitutingfor I C , and shifting the internal momentum q by q − A ( t ), we obtain ∂ t n k ( t ) − (cid:126)E · (cid:126) ∇ k n eq k = 4 (cid:88) q (cid:90) dt (cid:48) Im (cid:20) iD K ( (cid:126)q − (cid:126)A ( t ) , t, t (cid:48) ) δ Π (cid:48) A ( (cid:126)k − (cid:126)A ( t ) , (cid:126)q − (cid:126)A ( t ) , t (cid:48) , t )+ iD R ( (cid:126)q − (cid:126)A ( t ) , t, t (cid:48) ) δ Π (cid:48) K ( (cid:126)k − (cid:126)A ( t ) , (cid:126)q − (cid:126)A ( t ) , t (cid:48) , t )+ δiD K ( (cid:126)q − (cid:126)A ( t ) , t, t (cid:48) )Π (cid:48) A ( (cid:126)k − (cid:126)A ( t ) , (cid:126)q − (cid:126)A ( t ) , t (cid:48) , t )+ iδD R ( (cid:126)q − (cid:126)A ( t ) , t, t (cid:48) )Π (cid:48) K ( (cid:126)k − (cid:126)A ( t ) , (cid:126)q − (cid:126)A ( t ) , t, t (cid:48) ) (cid:21) , (89)where the δ in Eq. (89) indicates the first order variationwith respect to the electric field. As expected the con-servation laws continue to hold after this approximation.Above, the quantity Π (cid:48) ( k, q, t, t (cid:48) ) is defined as,Π (cid:48) R,A ( (cid:126)k − (cid:126)A ( t ) , (cid:126)q − (cid:126)A ( t ) , t, t (cid:48) ) = i G R,A ( (cid:126)k − (cid:126)A ( t ) , t, t (cid:48) ) G K ( − (cid:126)k + (cid:126)q − (cid:126)A ( t ) , t, t (cid:48) ) , (90)Π (cid:48) K ( (cid:126)k − (cid:126)A ( t ) , (cid:126)q − (cid:126)A ( t ) , t, t (cid:48) ) = i (cid:20) G K ( (cid:126)k − (cid:126)A ( t ) , t, t (cid:48) ) G K ( − (cid:126)k + (cid:126)q − (cid:126)A ( t ) , t, t (cid:48) )+ G R ( (cid:126)k − (cid:126)A ( t ) , t, t (cid:48) ) G R ( − (cid:126)k + (cid:126)q − (cid:126)A ( t ) , t, t (cid:48) )+ G A ( (cid:126)k − (cid:126)A ( t ) , t, t (cid:48) ) G A ( − (cid:126)k + (cid:126)q − (cid:126)A ( t ) , t, t (cid:48) ) (cid:21) , (91)Π R,A,K ( q, t, t (cid:48) ) = (cid:88) k Π (cid:48) R,A,K ( k, q, t, t (cid:48) ) . (92)Note that all the momenta appear in the gauge invariantcombination (cid:126)k − (cid:126)A ( t ). Above, the last line highlights therelation between Π (cid:48) , and the polarization Π.The fluctuations are modified by the electric fieldthrough their dependence on Π R : δD R = D R · δ Π R · D R , (93) δD K = δD R · Π K · D A + D R · δ Π K · D A + D R · Π K · δD A = D R · δ Π R · D K + D R · δ Π K · D A + D K · δ Π A · δD A . (94)In what follows, since we are interested in the responseof the current, we multiply the linearized kinetic equa-tion (89) by v k and integrate over k . C. Large fluctuations limit
We make one further simplification, which is based onthe fact that in the critical regime D K (cid:29) D R by a factor T /λ q . Thus we will only keep the terms that are highestorder in D K . D. Projections of equation to finite number ofmodes
The kinetic equation is an integral-differential equationin which the unknowns δn k ( t ) appear linearly. Concep-tually therefore it may be solved by standard methods.However, even though it is linear, it is non-local in timeand not time translation invariant. This makes directanalytical and numerical solution difficult.We therefore make an additional simplification that in-stead of considering the full space of solutions, we insteadproject entirely onto the current mode. This means thatwe would like to fix, δn k ( t ) ? = ¯ mρ J i ( t ) ∇ ik n eq k , (95) m − ij = ∂ (cid:15) ( k ) ∂k i ∂k j = ∂ k j v ik ; ¯ m − δ ij = − ρ (cid:88) k m − ij n eq k , (96)where ρ = − (cid:80) k n eq k is the density of fermions, ¯ m is theeffective mass and J i ( t ) is some function to be deter-mined. The ? in Eq. (95) is to question the correctnessof this equation. This is because conservation of momen-tum P i = (cid:80) k k i n k causes Eq. (95) to fail qualitatively.As ∂ t P i = ρE i , an electric field pulse, say for simplicitytaken to be a delta-function in time, will generate a netmomentum. Moreover, by conservation of momentum,the initial perturbation δn k = V i k i ∂n eq /∂(cid:15), (97)with V i arbitrary, will never decay. Therefore instead ofEq. (95) we decompose the occupation number as δn k ( t ) = ¯ mρ (cid:2) J ir ( t )( v ik − γk i / ¯ m ) + γk i P i ( t ) / ¯ m (cid:3) ∂ (cid:15) n eq k , (98) γ ≡ ¯ m (cid:80) k (cid:126)v k · (cid:126)k∂ (cid:15) n (cid:80) k (cid:126)k · (cid:126)k∂ (cid:15) n , (99) J i tot ≡ (cid:88) k v ik δn k = (1 − γ ) J ir + γP i / ¯ m. (100)The parameter γ gives the amount of current that is car-ried by the momentum mode, which does not relax. Inthe limit γ → γ →
1, as in a Galilean1invariant system, the current and momentum are propor-tional and there is no relaxing current.Multiplying the kinetic equation by v ik − γk i / ¯ m and k i , and summing over k we obtain, ∂ t J ir ( t ) − ρ ¯ m E i = (cid:88) k,q v ik − γk i /m − γ (cid:90) dt (cid:48) Im (cid:20) × iD K ( q, t, t (cid:48) ) δ Π (cid:48) A ( k, q, t (cid:48) , t )+ iD R ( q, t, t (cid:48) ) δ Π (cid:48) K ( k, q, t (cid:48) , t )+ δiD K ( q, t, t (cid:48) )Π (cid:48) A ( k, q, t (cid:48) , t )+ iδD R ( q, t, t (cid:48) )Π (cid:48) K ( k, q, t, t (cid:48) ) (cid:21) , (101) ∂ t P i ( t ) − ρE i ( t ) = 0 . (102)Above, in the first equation we have used that (cid:80) k ( v k − γk/ ¯ m ) δn k = (1 − γ ) J ir as follows from Eq. (100). Thesecond equation above just follows from conservation ofmomentum. We also do not write the explicit depen-dence of the momentum labels on the vector potentialas it is understood that it is always the gauge-invariantcombination that appears.As the momentum mode has trivial dependence due tomomentum conservation, we will simply drop it and set γ = 0 in Eq. (101). In this limit, δn k ( t ) = ¯ mρ J ir ( t ) ∇ ik n eq k . (103)These are now a closed set of equations in terms of thesingle unknown function J r ( t ) which we will denote sim-ply as J ( t ). The remaining step is to evaluate the variousterms. We note that D K ( t, t (cid:48) ) is generally larger than D R by the factor T /λ q . Thus in what follows we will onlyretain the first and third terms on the right hand side ofEq. (101). E. Time dependent kinetic coefficients
We begin by evaluating the first term, proportional to δ Π iR ( q, t, t (cid:48) ) in Eq. (101). There are two contributions,one from varying n k , and we denote it by K J . The secondis from varying g R , and we denote it by K E (cid:90) dt (cid:48) (cid:88) k,q v ik iD K ( q, t, t (cid:48) ) δ Π (cid:48) A ( k, q, t (cid:48) , t )= K iJ ( t ) + K iE ( t ) . (104)In Appendix C 1 we show that the term coming fromvarying n k is K iJ ( t ) (cid:39) (cid:88) q F q ( t ) J j ( t ) ˜ Q ijJ ( t ) , ˜ Q ij ( t ) ≈ ¯ mπq r q l ρ (cid:104) m − jr m − il (cid:105) FS . (105)To estimate the magnitude of the above term, we neglectfactors of order one, write ρ ∼ νk F /m , and obtain thatthis term is ∼ ( q/k F ) /ν . For the term coming from varying g R , we show in Ap-pendix C 2 that, K iE ( t ) (cid:39) (cid:88) q F q ( t ) ρE j ( t )¯ m ˜ Q ijE ( t ) , ˜ Q ijE ( t ) ≈ π ¯ mq r q l T ρ ζ (cid:48) ( − (cid:104) m − jr m − il (cid:105) FS . (106)The resulting term has the form ∝ ( q/k F ) F q ( t ) ρE ( t ) /νT ¯ m , and is thus a parametri-cally small correction to the drift term ρE/ ¯ m . Thereforewe neglect it for the remainder.We now turn to the third term ∼ δD K Π (cid:48) A in Eq. (101).We begin by considering the change in δD K . This in turndepends on δD R , which from Eq. (77) is given by[ ∂ t + λ q ] δD R = − δλ q D R . (107)There are two contributions to δλ q . One is from vary-ing δn via λ q = U − − Re (cid:20) Π R ( q ) (cid:21) = U − + 14 (cid:88) k n k + q/ + n − k + q/ (cid:15) k + q/ + (cid:15) − k + q/ . (108)Writing the above as a small q expansion, we define M such that, λ q = r + q M , (109)where we find that, on expanding, (cid:88) k n k + q/ + n − k + q/ (cid:15) k = (cid:88) k n k (cid:15) k + q i q j (cid:88) k ∇ ik ∇ jk n k (cid:15) k = (cid:88) k n k (cid:15) k + q d (cid:88) k ∇ k n k (cid:15) k , ⇒ M = ∂ q λ q = 14 d (cid:88) k ∇ k n k (cid:15) k . (110)where d is the spatial dimension. The above equationalso shows, M ∼ T /v F Thus, the change in λ q coming from the change in theelectron distribution δn k , may be evaluated as follows,(below we also use δn k = − δn − k ), δλ q = 14 (cid:88) k δn k + q/ + δn − k + q/ (cid:15) k + q/ + (cid:15) − k + q/ = 14 (cid:88) k δn k + q/ − δn k − q/ (cid:15) k + q/ + (cid:15) − k + q/ (cid:39) q i (cid:88) k ∇ ik δn k (cid:15) k ( q → . (111)Using the relation between δn k and the current inEq. (103), and the expression for M in Eq. (110), we2find, δλ q = q i ¯ m ρ J j (cid:88) k ∇ ik ∇ jk n eq k (cid:15) k = q i ¯ m ρ J i d (cid:88) k ∇ k n eq k (cid:15) k = 2 ¯ mρM q i J i . (112)The second reason for the change in λ q is due tothe direct coupling to the electric field, where defining δA i ( s ) = A i ( s ) − A i ( t ), λ q ( s ) = (cid:20) q i + 2 δA i ( s ) (cid:21) / M, ⇒ δλ q ( s ) = 2 q i δA i ( s ) M = 2 q i M (cid:2) A i ( s ) − A i ( t ) (cid:3) = − q i M (cid:90) ts dt (cid:48) ∂ t (cid:48) A i ( t (cid:48) ) , ⇒ δλ q ( s ) = 2 q i M (cid:90) ts dt (cid:48) E i ( t (cid:48) ) . (113)Thus the total change in λ q is δλ q ( s ) = 2 ¯ mq i J i ( s ) M ρ + 2 q i M (cid:90) ts dt (cid:48) E i ( t (cid:48) )= 2 ¯ mq i ρM (cid:20) J i ( s ) + (cid:90) ts dt (cid:48) ρE i ( t (cid:48) )¯ m (cid:21) . (114) δλ q changes both D R and D K , where the change in theformer is δD R ( t, t (cid:48) ) = e − λ q ( t − t (cid:48) ) (cid:20) − (cid:90) tt (cid:48) dsδλ q ( s ) (cid:21) . (115)Note that δD K = ( δD R )Π K D A + D R ( δ Π K ) D A + D R Π K ( δD A ). Since the variation δ Π K produces a termthat is smaller by a factor of F q , we neglect it. Using thezeroth order calculation that i Π K ∝ νδ ( t − t (cid:48) ), we have, iδD K ( t, t (cid:48) ) = ν (cid:90) dsδD R ( t, s ) D A ( s, t (cid:48) )+ ν (cid:90) dsD R ( t, s ) δD A ( s, t (cid:48) ) . (116)Substituting in for the result for δD R , using, δD R ( t, t (cid:48) ) = δD A ( t (cid:48) , t ) ∗ , we obtain (see Appendix C 3 for details),4 (cid:90) t dt (cid:48) (cid:88) k,q Im (cid:20) v ik iδD K ( q, t, t (cid:48) )Π (cid:48) A ( k, q, t (cid:48) , t ) (cid:21) = (cid:88) q αq i q j λ q (cid:90) t due − t − u ) λ q F q ( u ) × (cid:20) J j ( u ) + (cid:90) tu dt (cid:48) ρE j ( t (cid:48) )¯ m (cid:21) . (117) The coefficient α is given by, α ≡ T ¯ mM νρ (cid:88) k m − ii n eq ( (cid:15) k )4 (cid:15) k + λ q = T ¯ mM νρ (cid:90) d(cid:15) κ ( (cid:15) ) n eq (cid:15) + λ ,κ ( (cid:15) ) ≡ (cid:88) k m − ii ( k ) δ ( (cid:15) − (cid:15) k ) . (118)The above shows that αq ∼ ( q/k F ) /ν , which is com-parable to the local terms. Moreover the sign of α forgeneric band structures is such as to oppose the local intime term coming from iD K δ Π A .The function κ ( (cid:15) ) may be Taylor expanded to give κ ( (cid:15) ) ≈ ν ¯ m (1 + b(cid:15)/E F + · · · ), where the parameter b isa material dependent parameter of order 1. The sign of b is not fixed in general. However for a ’simple’ bandstructure, the parameter b is negative. Thus, α ≈ TM ρ (cid:90) d(cid:15) (cid:18) b (cid:15)E F (cid:19) n eq ( (cid:15) )4 (cid:15) + λ = bTM ρE F (cid:90) d(cid:15) (cid:15)n eq ( (cid:15) )4 (cid:15) ≈ bT log( E F /T ) M ρE F ≈ bv F T log( E F /T ) T νE F ≈ k F ν b log( E F /T ) . (119)For generic quench profiles, the only change that isneeded, is the replacement e − λ q t → e − (cid:82) t dsλ q ( s ) . (120)In what follows we set ρ/ ¯ m = 1. In these units, theratio J/E gives a Drude scattering time, and also equalsthe conductivity. The kinetic equation for the currentfor general quench trajectories becomes Eq. (4), and forconvenience we rewrite it below, ∂ t J i ( t ) − E ( t ) = − τ r (cid:20) A ( t ) J i ( t ) − α (cid:90) t dt (cid:48) (cid:26) B ( t, t (cid:48) ) J i ( t (cid:48) ) + C ( t, t (cid:48) ) E i ( t (cid:48) ) (cid:27)(cid:21) . (121)We define the dimensionless quantity, A ( t ) ∝ (cid:88) q q F ( q, t ) . (122)The memory terms are, B ( t, s ) ∝ (cid:88) q (cid:110) q λ q e − (cid:82) ts du λ q ( u ) F q ( s ) (cid:111) , (123) C ( t, s ) = (cid:90) s ds (cid:48) B ( t, s (cid:48) ) . (124)3We now simplify the expressions for A, B, C usingthe derived expressions for the time-dependent supercon-ducting fluctuations. A ( t ) = 1 T d/ (cid:90) t ds e − (cid:82) ts du r ( u ) ( t − s ) d/ . (125)Similarly the memory term is, B ( t, s ) ∝ (cid:88) q (cid:110) q λ q e − (cid:82) ts du λ q ( u ) F q ( s ) (cid:111) (126)= 1 T d/ (cid:90) s dt (cid:48) e − (cid:82) tt (cid:48) du r ( u ) × (cid:18) (1 + d/
2) + 2 r ( t )( t − t (cid:48) )( t − t (cid:48) ) d/ (cid:19) . (127)For the case of the sudden quench, since r ( t ) = θ ( t ) r + r i , the above expressions for A, B, C simplify consider-ably. For r i /T = O (1) so that at t = 0 , F ( q, t = 0) = 0,and for a quench to the critical point r = 0, equations(125), (127) and (124) give, A cr ( t ) = (cid:20) − /d (1 + T t ) d/ (cid:21) , (128) B cr ( t, s ) = T (cid:26) T ( t − s ) + 1] d/ − T t + 1] d/ (cid:27) , (129) C cr ( t, s ) = 2 d (cid:26) T ( t − s ) + 1] d/ − d/ T s ( T t +1) [ T t + 1] d/ (cid:27) . (130)Above we have regularized the integrals such that a shorttime cutoff of T − has been introduced. It is also helpfulto study the system at non-zero detuning. For a suddenquench to a distance r from the critical point, equations(125), (127) and (124) give, A ( r, t ) = 1 − (cid:18) rT (cid:19) d/ Γ (cid:18) − d , rt (cid:19) , (131) B ( r, t, s ) = T e − r ( t − s ) ( T ( t − s )) d/ − T e − rt ( T t ) d/ , (132) C ( r, t, s ) = (cid:18) rT (cid:19) d/ (cid:20) Γ (cid:18) − d , r ( t − s ) (cid:19) − Γ (cid:18) − d , rt (cid:19)(cid:21) − T se − rt ( T t ) d/ . (133) F. Charge Diffusion
We now adapt the kinetic equation to the case wherethe current is being driven by a density gradient. FromEq. (58), Fourier transforming with respect to the dif-ference in position coordinates r − r , and performing a gradient expansion with respect to the center of masscoordinate r = r + r , ∂ t n k ( r, t ) + (cid:126)v k · (cid:126) ∇ n k ( r, t ) = S ( r, k, t ) , (134)where S ( k ) = (cid:82) d d ( r − r ) e − ik ( r − r ) S ( r t, r t ). A spa-tial density gradient leads to a current J i = (cid:80) k v ik n k .Multiplying Eq. (134) with v k , taking a sum on k , andusing the fact that v k S ( k ) in the presence of the cur-rent can be written as a contribution coming from smallchanges to the superconducting propagator, and smallchanges to the polarization bubble as summarized on therhs of Eq. (89). The rhs can be simplified as before, lead-ing to the rhs of Eq. (121). This leads to, ∂ t J i + (cid:88) k v ik v jk ∇ j n k ( r, t ) = − τ r (cid:20) A ( t ) J i ( t ) − α (cid:90) t dt (cid:48) (cid:26) B ( t, t (cid:48) ) J i ( t (cid:48) )+ (cid:90) dt (cid:48) C ( t, t (cid:48) ) E i ( t (cid:48) ) (cid:27)(cid:21) . (135)The coefficients A, B are given in equations (125), (127)for a general quench profile, and in equations (128), (129)for a rapid quench to the critical point. Although thereis no external electric field, the density gradient inducesa current, and the electric field on the rhs is a linearresponse to this current.
V. CONDUCTIVITY AND DIFFUSIONCOEFFICIENT IN THERMAL EQUILIBRIUM
In thermal equilibrium, one may simply take the longtime limit of the expressions in equations (128), (129),and (130). Then, at zero detuning r = 0, A eq ( r = 0) = 1 ,B eq ( r = 0 , t − s ) = T [ T ( t − s ) + 1] d/ ,C eq ( r = 0 , t − s ) = (2 /d )[ T ( t − s ) + 1] d/ . (136)It is also useful to analyze these coefficients when thesystem has equilibrated at a non-zero detuning r awayfrom the critical point. In this case, for d = 2 we have, A eq ( r ) = A eq ( r = 0) (cid:20) rT ln(2 r/T ) (cid:21) ,B eq ( r, t − s ) = T e − r ( t − s ) [1 + T ( t − s )] ,C eq ( r, t − s ) (cid:12)(cid:12)(cid:12)(cid:12) r ( t − s ) (cid:29) = T r e − r ( t − s ) [1 + T ( t − s )] . (137)Above due to the separation of time-scales discussed inthe previous section, | r | /T (cid:28)
1. We note that in Fourier4space,˜ B eq ( r, ω = 0) = 1 + 2 rT (cid:20) γ + ln(2 r/T ) (cid:21) , (138)˜ C eq ( r, ω = 0) = 12 r (cid:20) rT (cid:26) γ + ln(2 r/T ) (cid:27)(cid:21) . (139)Above γ is Euler Gamma.We now discuss the linear response conductivity inthermal equilibrium. In this case, all the coefficients A, B, C are time-translation invariant. Fourier trans-forming Eq. (121), we obtain, J ( ω ) = 1 + τ − r α ˜ C eq ( ω ) iω + τ − r (cid:18) − α ˜ B eq ( ω ) (cid:19) E ( ω ) . (140)Writing, J = J diss + J fl , (141)where the first is a dissipative current arising due toDrude scattering, while the second is a current arisingdue to the superconducting fluctuations. Then, takingthe dc limit of Eq. (140), J diss = τ r E = σ diss E, (142)while the current from the superconducting fluctuations(neglecting B which gives small correction to the Drudescattering rate) is, J fl = α ˜ C eq E = σ fl E. (143)The above implies that the fluctuation conductivity σ fl gives a correction to the dissipative conductivity τ r byan amount = τ − r α ˜ C eq ( r, ω = 0). Noting that τ r ∼ T − ,and using Eq. (139), this implies a fluctuation conductiv-ity correction that goes as ∝ αT /r . While this is qualita-tively the same as the fluctuation AL conductivity ,we note that for the ultra clean case considered here, ma-terial dependent parameters such as α which determinethe strength of Galilean invariance breaking lattice ef-fects, are unavoidable, and give a non-universal materialdependent prefactor.The MT correction is also discussed in the context ofa disordered system , and is well defined only aslong as the mean free path is shorter than the inelasticscattering time . Thus the MT conductivity does notemerge naturally in the clean limit we consider here.Now we discuss the diffusion constant D defined as J i = − D ∇ i n . Due to time-translation invariance inequilibrium, we write Eq. (135) in Fourier space. Wealso note that the electric field generated as a responseto the current, is given by J = τ r E , up to fluctuationcorrections. This gives the diffusion constant, D ( ω ) = v F / iω + τ − r (cid:26) − α (cid:18) ˜ B eq ( ω ) + ˜ C eq ( ω ) τ r (cid:19)(cid:27) . (144) Close to the critical point, C dominates over B . More-over, Taylor expanding in α , the dc limit of the diffusionconstant is, D ( ω = 0) ≈ v F τ r (cid:20) α τ r r (cid:21) . (145)Thus the fluctuation correction for the diffusion constanthas qualitatively the same form as that for the conductiv-ity in being ∝ αT /r , where we have used that τ r ∼ T − . VI. SOLVING THE KINETIC EQUATION FORTHE CONDUCTIVITY
The solution of the kinetic equation for the current waspresented in much detail in Ref. 44. For completeness wesummarize some of the findings. Two kinds of quenchtrajectories were studied. One was a rapid quench fromdeep in the disordered phase to a distance r ≥ r = 0, and returned back tothe disordered phase. Regardless of the details of thetrajectory, the current showed slow dynamics because ofslowly relaxing superconducting fluctuations. In the ki-netic equation Eq. (121), this physics is encoded in thememory terms ( B, C ).In addition, the dynamics of the current for a criticalquench to r = 0, was found to show universal behav-ior, with a power-law aging in the conductivity σ ( t, t (cid:48) ),a result also found for a disordered system . Moreover,the dynamics at non-zero detuning r was shown to obeyscaling collapse. However it should be noted that, forthe clean system studied here, the exponents entering thescaling behavior were non-universal in that they dependon T τ r .For a smooth quench it was shown that transientlyenhanced superconducting fluctuations create transienta low resistance current carrying channel, whose signa-ture is a suppression of the Drude scattering rate at lowfrequencies. Note that the Drude scattering rate in theabsence of time-translation invariance is defined as, τ Dr ( t, ω ) = − Im [ σ ( ω, t )] ω Re [ σ ( ω, t )] , (146)with σ ( ω, t ) = (cid:90) dτ e iωτ σ ( t + τ, t ) . (147)In this section we revisit the smooth quench, definedby the trajectory r ( t ) /T = 1 − θ ( t ) (1 + (cid:15) ) ( tT / e − tT/ . (148)Above, r/T starts out being 1, smoothly approaches − (cid:15) at a time T t ∗ = 30, and then smoothly returnsback to its initial value of r/T = 1. We consider two5cases, one where (cid:15) = 0 (see Fig. 2), and one where thequench is super-critical in that (cid:15) = 0 .
05. For the lattercase, for a finite time, the parameters of the Hamilto-nian correspond to that of an ordered phase since r < r ≥
0. In particular, critical slowing down prevents truelong range order to develop over time scales over whichthe microscopic parameters are varying.Note that in the numerical simulation, we apply anelectric field pulse which is a delta-function in time. Inparticular, for an electric field pulse of unit strength cen-tered at t (cid:48) , the conductivity equals the current, σ ( t, t (cid:48) ) = J ( t ). Fourier transforming with respect to t − t (cid:48) , we canobtain the conductivity for arbitrary ω . However, in anactual experiment, the finite temporal width of the elec-tric field pulse places a limit on the lowest frequency ac-cessible. Nevertheless, the physics of a suppressed Drudescattering at low frequencies is visible over a sufficientlybroad range of frequencies, as shown in figures 2 and 3,for this to be a feasible experimental observation.Top panels of Fig. 2 and Fig. 3 show how the super-conducting fluctuations, at several different wavelengths,evolve for a critical and a super-critical quench respec-tively. The lower panels of the same figures show howthe corresponding Drude scattering rate, for different fre-quencies, evolve in time. When the density of supercon-ducting fluctuations peak, the Drude scattering rate atlow frequencies ω < T dips, with the effect being more en-hanced for the super-critical quench. Moreover the Drudescattering rate is strongly dispersive in that different fre-quency components of the Drude scattering rate peak atdifferent times.Recall that when the system returns to the normalphase, since we are in the clean limit, the steady-stateDrude scattering rate is zero. A slow power-law approachto thermal equilibrium is also visible in the long time tailsof the Drude scattering rate that are found to persist longafter the detuning has returned back to its initial value inthe normal phase. The slow dynamics highlighted aboveprovide signatures in time-resolved optical conductivitywhere, although the system lives for too short a time fortrue long range order to develop, the transient dynamicscan still show clear signatures of superconducting fluctu-ations. T t
T/rF (0 , t ) F ( T/ v, t ) F ( T/ v, t ) (a) Fluctuations F ( q, t ) for three different q and for a criticalquench where the detuning varies smoothly as r ( t ) /T = 1 . − θ ( t )( T t/ e − Tt/ . In thermal equilibrium F ( q = 0) = T /r .As a reference,
T /r ( t ) is plotted. −
20 0 20 40 60 80 100 120
T t . . . . . . . . . τ − D r ω = 0 . ω = 0 . ω = 0 . ω = 0 . ω = 0 . (b) Drude scattering rate for the critical quench r ( t ) /T = 1 . − θ ( t )( T t/ e − Tt/ . The high frequency Drude scattering ratefollows the profile of the fluctuations (top figure) by smoothlyincreasing and decreasing in time. On the other hand, thelow frequency Drude scattering rate first increases, followed bybeing suppressed at approximately when the superconductingfluctuations peak. FIG. 2: Critical quench6
T t r/TF (0 , t ) F ( T/ v, t ) F ( T/ v, t ) (a) Fluctuations F ( q, t ) for three different q and for a super-critical quench where the detuning varies smoothly as r ( t ) /T =1 . − θ ( t )(1 + 0 . T t/ e − Tt/ , becoming negative for acertain length of time. In thermal equilibrium F ( q = 0) = T /r .As a reference, 100 ∗ r ( t ) /T is plotted. −
20 0 20 40 60 80 100 120
T t . . . . . . τ − D r ω = 0 . ω = 0 . ω = 0 . ω = 0 . ω = 0 . (b) Drude conductivity for a super-critical quench where r ( t ) /T = 1 . − θ ( t ) ∗ (1 + 0 . T t/ e − Tt/ . Qualitativelysimilar behavior as in Fig. 2, but all the effects are more am-plified. As the density of superconducting fluctuations F ( q, t )(top figure) grow in time, the high frequency Drude scatter-ing rate increases, while the low frequency Drude scatteringrate is suppressed at approximately when the superconductingfluctuations peak. The behavior in frequency and time is alsomore dispersive for the super-critical quench than the criticalquench. FIG. 3: Super-critical quench
VII. SPECTRAL PROPERTIES
We now present results for the time-evolution of theelectron spectral properties. Results for the electron life-time obtained from the imaginary part of the electronself-energy, appear Ref. 53, where it was shown that thebehavior is non-Fermi liquid like due to enhanced An-dreev scattering of the electrons at the Fermi energy. Inthis section we discuss how this scattering affects the lo-cal density of states.For simplicity we consider a rapid quench from deepin the normal phase where F ( q ) = 0 initially, to a finalarbitrarily small detuning r ≥ (cid:20) T ∂ t + 2 (cid:18) v T q + r (cid:19)(cid:21) F ( q , t ) = 2 , (149)where the solution of the above is identical to Eq. (82)with λ q /T = r + v q /T . We have assumed that F depends only isotropically on q , and follow a conventionwhere r is dimensionless. We define the dimensionlessvariable x = v q /T . The electron self energy usingEq. (54f) and in the limit of D K (cid:29) D R , and in twospatial dimensions is,Σ R,A ( k, t , t ) = i (cid:90) d q (2 π ) D K ( q, t , t ) × G A,R ( − k + q, t , t ) . (150)Going into Wigner coordinates which involves Fouriertransforming with respect to relative coordinates t − t ,while keeping the mean time t = ( t + t ) / D K , and performing an angular integral, we obtain,Σ R ( ω, ε ) T = + Gi4 π (cid:90) ∞ dxF ( x, t ) × (cid:34)(cid:18) ω + ε + Σ A ( − ω, ε ) T (cid:19) − x (cid:35) − / , (151)Σ A ( − ω, ε ) T = + Gi4 π (cid:90) ∞ dxF ( x, t ) × (cid:34)(cid:18) − ω + ε + Σ R ( ω, ε ) T (cid:19) − x (cid:35) − / , (152)where Gi ≡ Tπνv . (153)We start in the perturbative limit by setting Σ A = i + ,Σ R = − i + . First let us study the equilibrium case. Here7 F = 1 / ( x + r ). Thus we have,Σ R ( ω, ε ) T = Gi8 π (cid:90) ∞ dxx + r × (cid:34)(cid:18) ω + ε + Σ A ( − ω, ε ) T (cid:19) − x (cid:35) − / . (154)Define ¯ x ≡ xr ; z ≡ ω + ε + i + T (cid:114) r , (155)Σ R ( ω, ε ) T = Gi8 π (cid:114) r (cid:90) ∞ d ¯ x x (cid:104)(cid:0) z (cid:1) − ¯ x (cid:105) − / = Gi8 π √ r f ( z +0 ) , (156)where Re (cid:20) f ( z ) (cid:21) ≡ √ z + 1 log z + √ z − z + √ z . (157)The density of states is, ν ( ω ) = − ν π (cid:90) dε (cid:61) (cid:2) G R ( ε, ω ) (cid:3) . (158)To evaluate this integral, we need to locate the pole in ε complex plane: ω − ε ∗ ( ω ) − Σ R ( ε ∗ ( ω ) , ω ) = 0. Then thevalue of the integral follows from the residue ν ( ω ) = ν π (cid:61) (cid:20) πi ∂ Σ R ∂ε | ε = ε ∗ (cid:21) . (159)Thus, the perturbative change in the density of statesis obtained from Taylor expanding, δν ( ω ) ν = − Re (cid:20) ∂ ε Σ R ( ω, ε ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ε = ω = − Re (cid:114) r Gi8 π (cid:114) r ∂ z f ( z ) | z = ωT √ r = Gi8 πr g (cid:18) ωT √ r (cid:19) , (160)where g ( z ) = − Re (cid:20) ∂ z f ( z ) (cid:21) . (161)Using Eq. (157), one obtains the following asymptoticbehavior, δν ( ω ) ν = Gi4 πr (cid:40) − ω (cid:28) T √ r rT ω log (cid:16) ω rT (cid:17) ω (cid:29) T √ r (162)Note that this shift is perturbative for all ω , that is δν/ν (cid:28)
1, when Gi / πr (cid:28) r is O (1)for t < r = 0 for t >
0. In this case, the solutionof Eq. (149) gives, F ( x, t ) = 1 − e − xT t x . (163) The above implies,Σ R ( ω, ε ) T = Gi8 π (cid:90) ∞ dx − e − xT t x × (cid:34)(cid:18) ω + ε + i + T (cid:19) − x (cid:35) − / , (164)define ¯ x = 2 xT t ; z ≡ ω + ε + i + T √ tT , (165)Σ R ( ω, ε ) T = Gi4 π (cid:114) T t (cid:90) ∞ d ¯ x − e − ¯ x ¯ x (cid:104)(cid:0) z (cid:1) − ¯ x (cid:105) − / = Gi8 π (cid:16) √ T t (cid:17) h ( z +0 ) . (166)In Appendix D we show that h has the following asymp-totic form,Re [ h ( z )] = z | z | (cid:28) z − (cid:18) log (cid:0) z (cid:1) + Γ (cid:0) (cid:1)(cid:19) | z | (cid:29) δν ( ω ) ν = − Gi4 π √ √ T t Re (cid:2) T ∂ ε h (cid:0) z (cid:1)(cid:3) | ε = ω = − Gi8 π (2 T t ) Re [ ∂ z h ( z )] | z =2 √ T t ωT (168)= Gi4 π (cid:40) − T t ω (cid:28) (cid:112)
T /t T ω log (cid:0) ω t/T (cid:1) ω (cid:29) (cid:112) T /t (169)Note that the asymptotic behaviors of the critical andequilibrium cases agree if we set r = T t . This followsfrom the fact that the fluctuations F ( x, t ) behave in acomparable manner in these two cases. A. Self-Consistent Regime (Results for ω = 0 ) We now turn to the full solution that does not require δν (cid:28) ν . We will first work out the equilibrium case. Thecritical quench case will then follow from the substitutionof r = T t .By adding ± ω + ε , the self-consistent equation for theself-energy in equations (151), (152) may be written as, z + = z + γf ( z − ) , (170) z − = z − + γf ( z + ) , (171)where z + ≡ ω + ε + Σ A ( − ω, ε ) T (cid:114) r , (172) z − ≡ − ω + ε + Σ R ( ω, ε ) T (cid:114) r , (173) γ = Gi8 πr . (174)8The above self-consistent equations can be solved in cer-tain limits (see Appendix E) giving the following resultsfor the density of states at zero frequency, ν ( ω = 0) ∝ ν log (Gi / πr ) . (175)The critical quench case then follows by the substitution r = T t , ν ( ω = 0 , t ) ∝ ν log (Gi T t/ π ) . (176)Thus we find that there is no true steady state of thezero frequency density of states due to the solution con-tinuing to evolve logarithmically in time. This time-evolution, although slow, happens because there is nonon-zero temperature critical point in d = 2. To ob-tain the correct thermalized steady-state, vortices needto be accounted for in our treatment. However, for thetransient regime relevant for experiments, where no su-perconducting gap has developed yet, and the dynamicsis governed by weakly interacting superconducting fluc-tuations, our results are still valid. VIII. CONCLUSIONS
We have studied quench dynamics of an interactingelectron system along general quench trajectories usinga 2PI approach. To leading non-trivial order in 1 /N ,our treatment reduced to RPA in the particle-particlechannel. While the quantum kinetic equations are exact,leading to proper definitions of the conserved densitiesand currents, progress was made in solving them by ex-ploiting a separation of time-scales. This could be seenby noting that thermalization of the electron distributionfunction sets in on a time-scale of T − , while the collec-tive modes relax at a rate given by the detuning to thesuperconducting critical point, which can be arbitrarilyslow. This is the phenomena of critical slowing down,and we derived an effective classical equation for the de-cay of the current in this regime. The dynamics of thecurrent was also justified using a phenomenological map-ping to model F in the Halperin-Hohenberg classificationscheme. Going forward, we envision that such mappingsmay be helpful in other contexts when understandingtime-resolved experiments that aim to probe collectivemodes.Results were also presented for the local density ofstates which was found to be enhanced at low frequen-cies due to Andreev scattering processes. For a criticalquench, the self-consistent equations showed an equiva-lence between the time after the quench and the inversedetuning.Future directions involve studying the coarseningregime, in particular how true long range order develops.The reverse quench where the initial state is ordered,also needs to be explored, going beyond mean-field. Ex-periments involving ultra-fast manipulation of spin and charge order also require generalizing our study to quenchdynamics in the particle-hole channel.Acknowledgments: This work was supported by theUS National Science Foundation Grant nsf-dmr 1607059. Appendix A: Proving conservation laws1. Proof of S ( rt, rt ) = 0 Using Eq. (54), S ( rt, rt ) = i (cid:90) dy (cid:26) G A ( y ; r, t ) G A ( y ; r, t ) D R ( r, t ; y )+ G K ( y ; r, t ) G A ( y ; r, t ) D K ( r, t ; y ) (cid:27) + i (cid:90) dy (cid:26) G K ( y ; r, t ) G K ( y ; r, t ) D R ( r, t ; y )+ G A ( y ; r, t ) G K ( y ; r, t ) D K ( r, t ; y ) (cid:27) − i (cid:90) dy (cid:26) G R ( r, t ; y ) G R ( r, t ; y ) D A ( y ; r, t )+ G R ( r, t ; y ) G K ( r, t ; y ) D K ( y ; r, t ) (cid:27) − i (cid:90) dy (cid:26) G K ( r, t ; y ) G R ( r, t ; y ) D K ( y ; r, t )+ G K ( r, t ; y ) G K ( r, t ; y ) D A ( y ; r, t ) (cid:27) . (A1)The above terms can be rearranged to give, S ( rt, rt ) = 2 (cid:82) dy (cid:26) Π K ( y ; r, t ) D R ( r, t ; y )+ Π A ( y ; r, t ) D K ( r, t ; y ) (cid:27) − (cid:82) dy (cid:26) Π K ( r, t ; y ) D A ( y ; r, t )+ Π R ( r, t ; y ) D K ( y ; r, t ) (cid:27) . (A2)Using Eq. (54e), the second and fourth terms above arerewritten to give, S ( rt, rt ) = 2 (cid:90) dy (cid:26) D R ( r, t ; y )Π K ( y ; r, t )+ (cid:18) D R Π K D A (cid:19) r,t ; y Π A ( y ; r, t ) (cid:27) − (cid:90) dy (cid:26) Π K ( r, t ; y ) D A ( y ; r, t )+Π R ( r, t ; y ) (cid:18) D R Π K D A (cid:19) y ; r,t (cid:27) . (A3)9Using Eq. (54d) and writing, Π A,R D A,R = U − D A,R − D A,R Π A,R = D A,R U − − S ( rt, rt ) = 2 U − ( t ) (cid:20) D R Π K D A (cid:21) r,t ; r,t − U − ( t )Tr (cid:20) D R Π K D A (cid:21) r,t ; r,t = 0 . (A4)
2. Proof of Eq. (67)
It is convenient to write S for unequal positions andtimes, and express the self-energy Σ in terms of the G, D propagators. Doing this we obtain, S ( r t , r t ) = 12 (cid:90) y (cid:20) i (cid:26) D K ( r t , y ) G K ( y, r t )+ D R ( r t , y ) G A ( y, r t )+ D A ( r t , y ) G R ( y, r t ) (cid:27) G A ( y, r t )+ i (cid:26) D R ( r t , y ) G K ( y, r t )+ D K ( r t , y ) G A ( y, r t ) (cid:27) G K ( y, r t ) (cid:21) − (cid:90) y (cid:20) iG R ( r t , y ) (cid:26) D K ( y, r t ) G K ( r t , y )+ D R ( y, r t ) G A ( r t , y ) + D A ( y, r t ) G R ( r t , y ) (cid:27) + iG K ( r t , y ) (cid:26) D A ( y, r t ) G K ( r t , y )+ D K ( y, r t ) G R ( r t , y ) (cid:27)(cid:21) . (A5)In order to prove momentum conservation we need tostudy the action of the spatial gradients on S , i (cid:20) ∇ αr − ∇ αr (cid:21) S ( r t , r t ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:90) r (cid:48) t (cid:48) (cid:20)(cid:26) i ∇ αr D K ( r t , r (cid:48) t (cid:48) ) (cid:27) Π A ( r (cid:48) t (cid:48) , r t )+ (cid:26) i ∇ αr D R ( r t , r (cid:48) t (cid:48) ) (cid:27) Π K ( r (cid:48) t (cid:48) , r t )+ (cid:90) r (cid:48) t (cid:48) (cid:20) Π R ( r t , r (cid:48) t (cid:48) ) (cid:26) i ∇ αr D K ( r (cid:48) t (cid:48) , r t ) (cid:27) + Π K ( r t , r (cid:48) t (cid:48) ) (cid:26) i ∇ αr D A ( r (cid:48) t (cid:48) , r t ) (cid:27)(cid:21) . (A6)where we have used that( ∂ r − ∂ r ) [ G a (1 (cid:48) , G b (1 (cid:48) ,
2) + G b (1 (cid:48) , G a (1 (cid:48) , (cid:12)(cid:12)(cid:12)(cid:12) = 0. Now using that (cid:90) r (cid:48) t (cid:48) (cid:26) i ∇ αr D K ( r t , r (cid:48) t (cid:48) ) (cid:27) Π A ( r (cid:48) t (cid:48) , r t ) = (cid:90) r (cid:48) t (cid:48) ,.,.. i ∇ αr D R ( r t , . )Π K ( ., .. ) D A ( .., r (cid:48) t (cid:48) )Π A ( r (cid:48) t (cid:48) , r t ) (cid:12)(cid:12)(cid:12)(cid:12) r = r = (cid:90) r (cid:48) t (cid:48) ,.,.. i ∇ αr D R ( r t , . )Π K ( ., .. ) (cid:20) D A ( .., r t ) U − ( t ) (cid:12)(cid:12)(cid:12)(cid:12) r = r − δ .., (cid:21) = i ∇ αr D K ( r t , r t ) U − ( t ) (cid:12)(cid:12)(cid:12)(cid:12) r = r − (cid:90) r (cid:48) t (cid:48) (cid:26) i ∇ αr D R ( r t , r (cid:48) t (cid:48) ) (cid:27) Π K ( r (cid:48) t (cid:48) , r t ) , and performing a similar analysis for second last term inEq. (A6), we obtain, i (cid:20) ∇ αr − ∇ αr (cid:21) S ( r t, r t ) (cid:12)(cid:12)(cid:12)(cid:12) r = r = r == iU − ( t ) ∇ αr D K ( r t, r t ) (cid:12)(cid:12)(cid:12)(cid:12) r = r = r + iU − ( t ) ∇ αr D K ( r t, r t ) (cid:12)(cid:12)(cid:12)(cid:12) r = r = r = iU − ( t ) ∇ αr D K ( rt, rt ) . (A7)
3. Proof of Eq. (72)
The manipulations involved are, i ∂ t − ∂ t ) S ( rt , rt ) | t = t = t = (cid:90) r (cid:48) t (cid:48) (cid:20)(cid:26) i∂ t D K ( r t , r (cid:48) t (cid:48) ) (cid:27) Π A ( r (cid:48) t (cid:48) , r t )+ (cid:26) i∂ t D R ( r t , r (cid:48) t (cid:48) ) (cid:27) Π K ( r (cid:48) t (cid:48) , r t )+ (cid:90) r (cid:48) t (cid:48) (cid:20) Π R ( r t , r (cid:48) t (cid:48) ) (cid:26) i∂ t D K ( r (cid:48) t (cid:48) , r t ) (cid:27) + Π K ( r t , r (cid:48) t (cid:48) ) (cid:26) i∂ t D A ( r (cid:48) t (cid:48) , r t ) (cid:27)(cid:21) = i∂ t D K ( r, t ; r, t ) U − ( t ) | t = t = t + i∂ t U − ( t ) D K ( r, t ; r, t ) | t = t = t = iU − ( t ) [ ∂ t D K ( r, t ; r, t ) + ∂ t D K ( r, t ; r, t )] | t = t = t = iU − ∂ t D K ( r, t ; r, t )= i∂ t (cid:0) U − D K ( r, t ; r, t ) (cid:1) − iD K ( r, t ; r, t ) ∂ t U − ( t ) . (A8) Appendix B: Equivalence to Langevin dynamics
Using the notation of Ref. 49, the bosonic quantum(∆ q ) and classical (∆ c ) fields defined on the Keldysh con-0tour, have the following correlators, D R (1 ,
2) = − i (cid:104) ∆ c (1)∆ ∗ q (2) (cid:105) ,D A (1 ,
2) = − i (cid:104) ∆ q (1)∆ ∗ c (2) (cid:105) ,D K (1 ,
2) = − i (cid:104) ∆ c (1)∆ ∗ c (2) (cid:105) , (B1)above 1 , D K = D R · Π K · D A , then the above correlators canequivalently be written as a path integral over the bosonicfields as follows, Z K = (cid:90) D (cid:2) ∆ c,q , ∆ ∗ c,q (cid:3) e iS K , (B2) S K = (cid:90) d (cid:90) d (cid:0) ∆ ∗ c ∆ ∗ q (cid:1) (cid:18) D − A D − R Π K (cid:19) , (cid:18) ∆ c ∆ q (cid:19) . (B3)According to Eq. (77), and the line below, D − R ( q, t, t (cid:48) ) = νT ( ∂ t + λ q ( t )) δ ( t − t (cid:48) );Π K ( t, t (cid:48) ) = 4 iνδ ( t − t (cid:48) ) . (B4)One may introduce an auxiliary field ξ that decouplesthe | ∆ q | term in the action, (for notational conveniencewe only highlight the time coordinate) e − ν (cid:82) dt ∆ q ∆ ∗ q = (cid:90) D [ ξ, ξ ∗ ] e − i (cid:82) dt [ ξ ∆ ∗ q + ξ ∗ ∆ q ] e − ν (cid:82) dtξξ ∗ . (B5)Writing the action in terms of this auxiliary field, S K = (cid:90) d (cid:90) d (cid:20) ∆ ∗ q D − R ∆ c + ∆ ∗ c D − A ∆ q − ξ ∆ ∗ q − ξ ∗ ∆ q + i ν ξξ ∗ (cid:21) . (B6)Requiring δS/δ ∆ ∗ q = 0 gives, D − R ∆ c = ξ, (B7)where (cid:104) ξ ∗ ( t ) ξ ( t (cid:48) ) (cid:105) = 4 νδ ( t − t (cid:48) ). Giving the explicit ex-pression for D R , and restoring the momentum label, νT ( ∂ t + λ q ( t )) ∆ c ( q, t ) = ξ q ( t ) , (B8)It is convenient to rescale ∆ c → (cid:112) νT ∆ c , then, theLangevin equation is,( ∂ t + λ q ( t )) ∆ c ( q, t ) = (cid:114) Tν ξ q ( t ) , (B9)implying that ∆ c obeys Langevin dynamics where thenoise is delta-correlated with a strength equal to the tem-perature T . Appendix C: Derivation of kinetic coefficients
In this section we derive the kinetic coefficients assum-ing a rapid quench profile λ q ( t ) = θ ( t ) λ q + θ ( − t ) r i , where r i /T = O (1). This condition simply ensures that thedensity of superconducting fluctuations is zero at t = 0.The generalization to general quench profiles is formallystraightforward, and summarized in the main text.
1. Derivation of Eq. (105)
The term obtained from varying n k is, K iJ ( t ) ≡
12 Im (cid:90) dt (cid:48) (cid:88) k,q iD K ( q, t, t (cid:48) ) × i (cid:20) δn k + q/ ( t (cid:48) ) + δn − k + q/ ( t (cid:48) ) (cid:21)(cid:20) v ik + q/ + v i − k + q/ (cid:21) × e i ( (cid:15) k + q/ + (cid:15) − k + q/ ) ( t − t (cid:48) ) . (C1)Substituting Eq. (81) for D K , and Eq. (103) for δn k , wesee that the above can be rewritten as K iJ ( t ) = (cid:90) t dt (cid:48) (cid:88) q Q ijJ ( t, t (cid:48) ) F q ( t (cid:48) ) J j ( t (cid:48) ) ,Q ijJ ( t, t (cid:48) ) ≡ ¯ mTρν Im (cid:88) k (cid:20) ∇ jk n eq k + q/ + ∇ jk n eq − k + q/ (cid:21) × i (cid:16) v ik + q/ + v i − k + q/ (cid:17) e i ( (cid:15) k + q/ + (cid:15) − k + q/ + iλ q ) ( t − t (cid:48) ) . (C2)Following the discussion in the main text, the term Q ijJ decays exponentially with rate T , whereas J and F change at a slower rate. Therefore Q J appears like adelta function when integrated against F ( t ) J ( t ), and wemay write K iJ ( t ) ≈ (cid:88) q F q ( t ) J j ( t ) ˜ Q ijJ ( t ) , ˜ Q ijJ ( t ) = (cid:90) t dt (cid:48) Q ijJ ( t, t (cid:48) ) ≈ − ¯ mTρν Im (cid:88) k × (cid:16) ∇ jk n eq k + q/ + ∇ jk n eq − k + q/ (cid:17) (cid:16) v ik + q/ + v i − k + q/ (cid:17) (cid:15) k + q/ + (cid:15) − k + q/ + iλ q ≈ v F q (cid:28) T (cid:18) − ¯ mTρν (cid:19) Im (cid:88) k q r q l m − il ∇ rk ∇ jk n eq k (cid:15) k + iλ q ≈ λ q (cid:28) T π ¯ mTρν (cid:88) k (cid:16) q r q l m − il ∇ rk ∇ jk n eq k (cid:17) δ ( (cid:15) k + (cid:15) − k )= π ¯ mT ρν (cid:88) k (cid:32) q r q l m − il ∂ (cid:15) k ∂ rk ∂ jk ∂ (cid:15) k n eq k (cid:33) δ ( (cid:15) k ) ≈ ¯ mπq r q l ρ (cid:104) m − jr m − il (cid:105) FS . (C3)1In the second line above we assume t − (cid:29) T so we arenot too close to the quench. In the third line we take q (cid:28) T /v F , and we also assume λ q /T (cid:28)
2. Derivation of Eq. (106)
Using Eq. (81) for D K , and accounting for the changein g R due to the electric field in Eq. (86), we find, K iE ( t ) = (cid:90) t dt (cid:48) (cid:88) q Q ijE ( t, t (cid:48) ) F q ( t (cid:48) ) ρ ¯ m E j ( t (cid:48) ) ,Q ijE ( t, t (cid:48) ) E j ( t (cid:48) ) ≡ − Im ¯ mTρν (cid:88) k (cid:18) n eq k + q/ + n eq − k + q/ (cid:19) × (cid:18) v ik + q/ + v i − k + q/ (cid:19) × (cid:90) tt (cid:48) ds v jk (cid:0) A j ( t ) − A j ( s ) (cid:1) e i ( (cid:15) k + q/ + (cid:15) − k + q/ + iλ q ) ( t − t (cid:48) ) . (C4)Assume that the frequency of the electric field ω (cid:28) T - in this case we may approximate the integral (cid:90) tt (cid:48) ds ( A j ( t ) − A j ( s )) ≈ − E j ( t )( t − t (cid:48) ) / . (C5)Following the discussion in the main text, the term Q ijE decays exponentially with rate T , whereas J and F change at a slower rate. Therefore Q E appears like adelta function when integrated against F ( t ) J ( t ), and we may write, as we did for Q ijJ , K iE ( t ) ≈ (cid:88) q F q ( t ) ρE j ( t )¯ m ˜ Q ijE ( t ) , ˜ Q ijE ( t ) = (cid:90) t dt (cid:48) Q ijE ( t, t (cid:48) ) ≈ ¯ mT ρν Im (cid:88) k (cid:16) n eq k + q/ + n eq − k + q/ (cid:17) × i (cid:16) v ik + q/ + v i − k + q/ (cid:17) (cid:16) v jk + q/ + v j − k + q/ (cid:17) ( (cid:15) k + q/ + (cid:15) − k + q/ + iλ q ) ≈ ¯ mTρν Im (cid:88) k i q l q r m − il m − jr n eq k (2 (cid:15) k + iλ q ) ≈ π ¯ mq r q l ρ (cid:104) m − jr m − il (cid:105) FS (cid:90) dxx − d dx tanh( x/ ≈ π ¯ mq r q l ρT ζ (cid:48) ( − (cid:104) m − jr m − il (cid:105) FS . (C6)
3. Derivation of Eq (117)
Substituting Eq. (114) into Eq. (115), and assuming, t > t (cid:48) , we obtain, iδD K ( t, t (cid:48) ) = − ν (cid:90) t (cid:48) dsD R ( t, s ) D A ( s, t (cid:48) ) (cid:20) (cid:90) ts duδλ q ( u )+ (cid:90) t (cid:48) s duδλ q ( u ) (cid:21) (C7)= − ν (cid:90) t (cid:48) duδλ q ( u ) (cid:90) u dsD R ( t, s ) D A ( s, t (cid:48) ) − ν (cid:90) tt (cid:48) duδλ q ( u ) (cid:90) t (cid:48) dsD R ( t, s ) D A ( s, t (cid:48) ) . (C8)Using the definition of D K = D R · Π K · D A withΠ K ( t, t (cid:48) ) = 4 νδ ( t − t (cid:48) ), the above becomes, iδD K ( t, t (cid:48) ) = − (cid:90) t (cid:48) due − λ q ( t (cid:48) − u ) δλ q ( u ) iD K ( t, u ) − (cid:90) tt (cid:48) duδλ q ( u ) iD K ( t, t (cid:48) ) . (C9)Then we use Eq. (81) to write,2 iδD K ( t > t (cid:48) ) = − T ν e − λ q ( t − t (cid:48) ) (cid:90) t (cid:48) du e − λ q ( t (cid:48) − u ) δλ q ( u ) F q ( u ) − T ν e − λ q ( t − t (cid:48) ) (cid:90) tt (cid:48) duδλ q ( u ) F q ( t (cid:48) )= − T ν e − λ q ( t − t (cid:48) ) (cid:90) t du e − λ q ( t (cid:48) − u ) δλ q ( u ) F q ( u ) − T ν e − λ q ( t − t (cid:48) ) (cid:90) tt (cid:48) duδλ q ( u ) (cid:16) F q ( t (cid:48) ) − e − λ q ( t (cid:48) − u ) F q ( u ) (cid:17) = − T ν e λ q ( t − t (cid:48) ) (cid:90) t du e − λ q ( t − u ) δλ q ( u ) F q ( u ) − T ν e − λ q ( t − t (cid:48) ) (cid:90) tt (cid:48) duδλ q ( u ) (cid:16) F q ( t (cid:48) ) − e − λ q ( t (cid:48) − u ) F q ( u ) (cid:17) ≈ − T ν e λ q ( t − t (cid:48) ) (cid:90) t du e − λ q ( t − u ) δλ q ( u ) F q ( u ) + T ν e − λ q ( t − t (cid:48) ) ( t − t (cid:48) ) δλ q ( t ) F q ( t ) , (C10)where in the last line we used the fact that t − t (cid:48) is of order T − as iD K appears along with Π A . We discuss thislast term further below and show it to be parametrically smaller than the local terms calculated earlier.We insert the second part into the collision integralgiving,4 (cid:90) t dt (cid:48) ImΠ (cid:48) A ( t (cid:48) , t ) (cid:18) e − λ q ( t − t (cid:48) ) Tν ( t − t (cid:48) ) δλ q ( t ) F q ( t ) (cid:19) ∝ δλ q ( t ) F q ( t )Im (cid:88) k i n eq k + q/ + n eq − k + q/ ( (cid:15) k + q/ + (cid:15) − k + q/ − iλ q ) ≈ q → δλ q ( t ) F q ( t )Im (cid:88) k i n eq ( (cid:15) k )(2 (cid:15) k − iλ q ) ≈ δλ q ( t ) F q ( t )Im i (cid:90) d(cid:15) ν n eq ( (cid:15) )(2 (cid:15) − iλ q ) ≈ λ q → δλ q ( t ) F q ( t )Re (cid:90) d(cid:15) ν P ∂ (cid:15) n eq ( (cid:15) ) (cid:15) = 0 . (C11)Thus this term is parametrically smaller than the local term already calculated.Now we consider the full term4 (cid:90) t dt (cid:48) Im (cid:20) v ik iδD K ( t, t (cid:48) )Π (cid:48) A ( t (cid:48) , t ) (cid:21) = 12 Im (cid:90) t dt (cid:48) (cid:88) k,q i (cid:16) v ik + q/ + v i − k + q/ (cid:17) iδD K ( t (cid:48) , t ) (cid:16) n eq k + q/ + n eq − k + q/ (cid:17) e i ( (cid:15) k + q/ + (cid:15) − k + q/ )( t − t (cid:48) ) (cid:21) ≈ Im (cid:20) T ν (cid:88) k,q i (cid:90) t dt (cid:48) (cid:16) v ik + q/ + v i − k + q/ (cid:17) (cid:16) n eq k + q/ + n eq − k + q/ (cid:17) e i ( (cid:15) k + q/ + (cid:15) − k + q/ − iλ q )( t − t (cid:48) ) × (cid:90) t du (cid:16) − e − t − u ) λ q δλ q ( u ) F q ( u ) (cid:17)(cid:21) ≈ Im (cid:20) T ν (cid:88) k,q q j m − ij n eq k + q/ + n eq − k + q/ (cid:15) k + q/ + (cid:15) − k + q/ − iλ q (cid:90) t du e − t − u ) λ q δλ q ( u ) F q ( u ) (cid:21) ≈ (cid:88) q q l T ν (cid:88) k m − il n eq ( (cid:15) k ) λ q (cid:15) k + λ q (cid:90) t due − t − u ) λ q δλ q ( u ) F q ( u )= (cid:88) q T q l ν (cid:32)(cid:88) k m − il n eq ( (cid:15) k ) λ q (cid:15) k + λ q (cid:33) (cid:90) t due − t − u ) λ q F q ( u ) 2 q j ¯ mρM (cid:20) J j ( u ) + (cid:90) tu dt (cid:48) ρE j ( t (cid:48) )¯ m (cid:21) = (cid:88) q αq i q j λ q (cid:90) t due − t − u ) λ q F q ( u ) (cid:20) J j ( u ) + (cid:90) tu dt (cid:48) ρE j ( t (cid:48) )¯ m (cid:21) . (C12)3Above, in the second last line, we used the expression for δλ q in Eq. (114). Appendix D: Derivation of Eq. (167)
Let us now discuss the limiting behaviors of h ( z ). If z (cid:28)
1, the integral is restricted to the region ¯ x (cid:28) − e − ¯ x ) / ¯ x ≈
1. Therefore in thislimit we have Re [ h ] ≈ z . If | z | (cid:29) x = 1. The lower part (cid:90) d ¯ x − e − ¯ x ¯ x (cid:104)(cid:0) z (cid:1) − ¯ x (cid:105) − / ∼ (cid:90) d ¯ x − e − ¯ x ¯ x z × (cid:32) x (cid:0) z (cid:1) + · · · (cid:33) ∝ z + O (cid:16)(cid:0) z (cid:1) − (cid:17) . (D1)Whereas in the part from 1 to ∞ the exponential maybe neglected leaving the leading part (cid:90) ∞ d ¯ x x (cid:104)(cid:0) z (cid:1) − ¯ x (cid:105) − / ∼ log (cid:0) z (cid:1) z . (D2)Thus we have Eq. (167). Appendix E: Derivation of Eq. (175)
We record here some facts about f ( z ) as a function inthe complex plane. It may be written as f ( z ) = 1 √ z + 1 (cid:34) log (cid:32) √ z + 1 + z √ z + 1 − z (cid:33) − πi sgn (cid:61) z (cid:35) . (E1)It is analytic in the upper and lower half planes separatelywith a branch cut along the real axis. The points z = ± i which appear singular are in fact smooth. One may alsoconvince one self that | f ( z ) | ≤ π for all z , reaching thislimit only when z = 0 ± iδ . It is also an injective functionof z . In addition, we have that as z → ∞ , f ( z ) ∼ zz , (E2)and that f is pure imaginary on the imaginary axis.Causality requires that z ± have no zeros or branch cutsin the upper half complex ω plane, but there will be an-alytic structure as a function of ε .Let us first test the validity of the perturbative approx-imation. To zeroth order we set z ± = z ± . To first orderwe substitute this into the rhs of equation (170), (171),and obtain, z ± = z ± + γf (cid:0) z ∓ (cid:1) , (E3) and to second order z ± = z ± + γf (cid:0) z ∓ + γf (cid:0) z ± (cid:1)(cid:1) ≈ z ± + γf (cid:0) z ∓ (cid:1) + γ f (cid:48) (cid:0) z ∓ (cid:1) f (cid:0) z ± (cid:1) . (E4)Comparing the first and second correction we see thatthe latter is smaller when γ (cid:28) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f (cid:0) z ∓ (cid:1) f (cid:48) (cid:0) z ∓ (cid:1) f (cid:0) z ± (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (E5)Considering the various limits, we see that the rhs is al-ways at least O (1) so the perturbative treatment is al-ways valid if γ (cid:28)
1. Making the very coarse approxima-tion | f | ∼ min(1 , | z − | ), we get roughly γ (cid:28) min(1 , | z | ) · min(1 , | z − | ) . (E6)This may still be satisfied even if γ ≥
1. If ω/T (cid:29) / Gi, the perturbative limit will hold for all ε . With theweaker condition ω /T (cid:29) Gi, the perturbative limitholds except in the narrow region | ε ± ω | /T (cid:28) √ (cid:15) , whichgives only a small contribution to δν ( ω ).Now let us consider the behavior in the simplest casewhen z ± = 0. In this case we have a self-consistent solu-tion when z ± = ∓ iy , y real, y >
0, since this is equivalentto finding a solution to − iy ( γ ) = γf ( iy ( γ )) , (E7)which is possible since f is imaginary on the imaginaryaxis. In the limiting case of γ (cid:29)
1, we get y ∼ γ log y which gives to logarithmic accuracy, y ∼ (cid:112) γ log γ. (E8)In the limit γ (cid:28)
1, using the expansion of f ( z ) for smallargument, we obtain z ± ≈ ∓ iγπ + O ( γ ).Now let us advance to the case z ± = c ∈ R . Recall ω = 0. This has a self-consistent solution when z + = z = z ∗− , z − c = γf ( z ∗ ) , (E9)since f ( z ) ∗ = f ( z ∗ ). The pertubative condition in thiscase is γ (cid:28) / | f (cid:48) ( c ) | ∼ c / log | c | . (E10)Thus the perturbative regime begins when c (cid:29) y ( γ ).In this limit, we obtain the usual perturbative result bysetting z ≈ c .In the opposite limit, let us assume | z | (cid:29) , | c | . Onthese assumptions, writing z = − i | z | e − iφ , and comparingreal and imaginary parts of( z − c ) z ∗ = γ log( − ( z ∗ ) ) ⇒ | z | − ic | z | e iφ = 2 γ [log | z | + iφ ] , | z | + c | z | sin φ = γ log | z | , (E11)& − c | z | cos φ = 2 γφ. (E12)On the assumption | z | (cid:29) | c | the first line above gives | z | = y ( γ ). The second line can be rearranged to give, φ cos φ = − c y ( γ )2 γ . (E13)Thus φ goes from π/ − π/ c goes from −∞ to ∞ .We may split this into two limits. When c (cid:28) γ/y ( γ ) = y ( γ ) / log y ( γ ), we have that φ (cid:28) φ = − cy ( γ ) / γ , and thus z = − i | z | (1 − iφ ) = − iy ( γ ) + c log( γ ) / . (E14)In the other limit, cy ( γ ) /γ (cid:29) φ is close to ± π/
2. Let us take c >
0, then φ = − π πγcy ( γ ) . (E15)so that z is given by, z = − ie iπ/ | z | (cid:18) − i πγcy ( γ ) (cid:19) = y ( γ ) − i πγc . (E16)This is almost identical to the perturbative calculationwith a slight correction to the real part.We can collect all of these limits, z = − iy ( γ ) + c log( γ ) / c (cid:28) y ( γ ) / log( γ ) y ( γ ) − i πγc y ( γ ) / log( γ ) (cid:28) c (cid:28) y ( γ ) c + γc (cid:2) log c − πi (cid:3) y ( γ ) (cid:28) c (E17)where in the last case, we have used the perturbativeexpansion in Eq. (E4).We are now in a position to estimate the density ofstates for ω = 0, γ (cid:29)
1, and c = ε/T √ r . Starting from ν ( ω ) ν = 1 π (cid:61) (cid:90) ∞−∞ dε √ rT z ( ω = 0 , ε ) . (E18)Combining the positive and negative integrals over ε or c , ν ( ω ) ν = 1 π (cid:61) (cid:90) ∞ dc (cid:20) z ( c ) + 1 z ( − c ) (cid:21) . (E19)Next use the fact that at ω = 0, z ( c ) = − z ( − c ) ∗ , whichfollows from the above calculation. This can also beseen from noting that f ( − z ) = − f ( z ). Then, using z + ( c ) = c + iδ + γf ( z − ( c )) = c + iδ + γf ( z ∗ + ( c )). Tak-ing c → − c and conjugating, this becomes, z ∗ + ( − c ) = − c − iδ + γf ( z + ( − c )) = − c − iδ − γf ( − z + ( − c )). Thus z + ( − c ) ∗ = − z + ( c ) is a solution of the above equationas this simultaneously requires f ( − z + ( − c )) = f ( z ∗ + ( c )).Using this gives, ν ( ω = 0) ν = 2 π (cid:90) ∞ dc (cid:20) (cid:61) z ( c )[ (cid:60) z ( c )] + [ (cid:61) z ( c )] (cid:21) . (E20)Now we split this integral into three regions, accordingto the asymptotics above, (cid:90) ∞ dc (cid:20) (cid:61) z ( c )[ (cid:60) z ( c )] + [ (cid:61) z ( c )] (cid:21) = I + I + I , (E21)where, I ≡ (cid:90) y ( γ ) / log γ dc (cid:20) (cid:61) z ( c )[ (cid:60) z ( c )] + [ (cid:61) z ( c )] (cid:21) (E22)= (cid:90) y ( γ ) / log γ dc (cid:20) y ( γ )[ c log( γ ) / + [ y ( γ )] (cid:21) , define u ≡ c log γy ( γ ) ,I = 1log γ (cid:90) du (cid:20) u/ + 1 (cid:21) ∝ γ . (E23) I ≡ (cid:90) y ( γ ) y ( γ ) / log y ( γ ) dc (cid:20) (cid:61) z ( c )[ (cid:60) z ( c )] + [ (cid:61) z ( c )] (cid:21) (E24)= (cid:90) y ( γ ) y ( γ ) / log γ dc (cid:20) πγ/cy ( γ ) + [ πγ/c ] (cid:21) , define u = c · y ( γ ) /γ ≈ c ˙log( γ ) /y ( γ ) ,I = γy ( γ ) (cid:90) log y ( γ )1 du (cid:20) π/u π/u ] (cid:21) ≈ γy ( γ ) (cid:90) log y ( γ )1 du πu ∝ γ log log y ( γ ) ∝ γ . (E25)Here we are treating log log γ as an O (1) constant, whichhas been the order of our precision throughout. I ≡ (cid:90) ∞ y ( γ ) dc (cid:20) (cid:61) z ( c )[ (cid:60) z ( c )] + [ (cid:61) z ( c )] (cid:21) (E26)= (cid:90) ∞ y ( γ ) dc (cid:20) γπ/cc + [ γπ/c ] (cid:21) ≈ (cid:90) ∞ y ( γ ) dc γπc = 2 π γy ( γ ) = 2 π log y ( γ ) ≈ π log γ . (E27)Thus we see that all terms contribute ∝ / log γ andtherefore we have that ν ( ω = 0) ∝ ν log (Gi / πr ) . (E28)5 C. L. Smallwood, J. P. Hinton, C. Jozwiak, W. Zhang,J. D. Koralek, H. Eisaki, D.-H. Lee, J. Orenstein, andA. Lanzara, Science , 1137 (2012). J. Zhang and R. Averitt, Annual Review of Materials Re-search , 19 (2014). C. L. Smallwood, W. Zhang, T. L. Miller, C. Jozwiak,H. Eisaki, D.-H. Lee, and A. Lanzara, Phys. Rev. B ,115126 (2014). S. Wall, S. Yang, L. Vidas, M. Chollet, J. M. Glow-nia, M. Kozina, T. Katayama, T. Henighan, M. Jiang,T. A. Miller, D. A. Reis, L. A. Boatner, O. Delaire, andM. Trigo, Science , 572 (2018). J. W. Harter, D. M. Kennes, H. Chu, A. de la Torre,Z. Y. Zhao, J.-Q. Yan, D. G. Mandrus, A. J. Millis, andD. Hsieh, Phys. Rev. Lett. , 047601 (2018). J. McIver, B. Schulte, F.-U. Stein, T. Matsuyama,G. Jotzu, G. Meier, and A. Cavalleri, arXiv:1811.03522(unpublished). M. Mitrano, S. Lee, A. A. Husain, L. Delacretaz, M. Zhu,G. de la Pena Munoz, S. Sun, Y. I. Joe, A. H. Reid,S. F. Wandel, G. Coslovich, W. Schlotter, T. van Driel,J. Schneeloch, G. D. Gu, S. Hartnoll, N. Goldenfeld, andP. Abbamonte, arXiv:1808.04847 (2018). P. Tengdin, W. You, C. Chen, X. Shi, D. Zusin, Y. Zhang,C. Gentry, A. Blonsky, M. Keller, P. M. Oppeneer, H. C.Kapteyn, Z. Tao, and M. M. Murnane, Science Advances (2018). D. Fausti, R. I. Tobey, N. Dean, S. Kaiser, A. Dienst,M. C. Hoffmann, S. Pyon, T. Takayama, H. Takagi, andA. Cavalleri, Science , 189 (2011). M. Mitrano, A. Cantaluppi, D. Nicoletti, S. Kaiser, A. Pe-rucchi, S. Lupi, P. D. Pietro, D. Pontiroli, M. Ricc´o, S. R.Clark, D. Jaksch, and A. Cavalleri, Nature , 461(2016). E. Pomarico, M. Mitrano, H. Bromberger, M. A. Sentef,A. Al-Temimy, C. Coletti, A. St¨ohr, S. Link, U. Starke,C. Cacho, R. Chapman, E. Springate, A. Cavalleri, andI. Gierz, Phys. Rev. B , 024304 (2017). K. A. Cremin, J. Zhang, C. C. Homes, G. D. Gu, Z. Sun,M. M. Fogler, A. J. Millis, D. N. Basov, and R. D. Averitt,arXiv:1901.10037 (unpublished). J.-i. Okamoto, A. Cavalleri, and L. Mathey, Phys. Rev.Lett. , 227001 (2016). S. Rajasekaran, J. Okamoto, L. Mathey, M. Fechner,V. Thampy, G. D. Gu, and A. Cavalleri, Science ,575 (2018). M. Knap, M. Babadi, G. Refael, I. Martin, and E. Demler,Phys. Rev. B , 214504 (2016). M. A. Sentef, A. F. Kemper, A. Georges, and C. Kollath,Phys. Rev. B , 144506 (2016). M. A. Sentef, Phys. Rev. B , 205111 (2017). D. M. Kennes, E. Y. Wilner, D. R. Reichman, and A. J.Millis, Nature Physics , 479 (2017). Y. Murakami, N. Tsuji, M. Eckstein, and P. Werner, Phys.Rev. B , 045125 (2017). G. Chiriac`o, A. J. Millis, and I. L. Aleiner, Phys. Rev. B , 220510 (2018). M. Tinkham,
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