Lifetimes of local excitations in disordered dipolar quantum systems
aa r X i v : . [ c ond - m a t . d i s - nn ] F e b Lifetimes of local excitations in disordered dipolar quantum systems
Rahul Nandkishore and Sarang Gopalakrishnan Department of Physics and Center for Theory of Quantum Matter,University of Colorado at Boulder, Boulder CO 80309, USA Department of Physics, Pennsylvania State University, University Park PA 16802, USA
When a strongly disordered system of interacting quantum dipoles is locally excited, the exci-tation relaxes on some (potentially very long) timescale. We analyze this relaxation process, bothfor electron glasses with strong Coulomb interactions—in which particle-hole dipoles are emergentexcitations—and for systems (e.g., quantum magnets or ultracold dipolar molecules) made up ofmicroscopic dipoles. We consider both energy relaxation rates ( T times) and dephasing rates ( T times), and their dependence on frequency, temperature, and polarization. Systems in both twoand three dimensions are considered, along with the dimensional crossover in quasi-two dimensionalgeometries. A rich set of scaling laws is found. I. INTRODUCTION
The dynamical behavior of well isolated and stronglydisordered quantum many body systems has been arous-ing great interest, from both the condensed matter andquantum information communities. Under certain cir-cumstances, such systems can display the phenomenonof ‘many body localization’ (MBL) [1–4] (for reviews,see [5–7]), whereby they preserve a memory of their ini-tial conditions forever in local observables, hence serv-ing as good quantum memories. The existing theoryof MBL describes systems with interactions that are short range in real space, whereas physical systems fre-quently contain long range interactions—e.g., Coulombor dipolar interactions. For interactions that fall off as apower law with distance, it has been argued that MBL isalways ultimately destabilized by nonperturbative pro-cesses due to rare regions [8, 9]. However, such puta-tive non-perturbative effects only manifest on timescaleswhich diverge faster than a power law of the disorderstrength [10], and which as such are not relevant for mostexperiments in the strongly disordered regime (but seeRef. [11]). We will not consider such non-perturbativeeffects in this paper. However, for sufficiently slowly de-caying power laws, there is also a perturbative instabilityof the MBL phase [12–17] (although see [18] for excep-tions), which is expected to manifest on experimentallyrelevant timescales. Even when the MBL phase is pertur-batively unstable on the longest timescales, however, theframework of “nearly MBL” systems offers a useful per-spective on the dynamics of well isolated but strongly dis-ordered long range interacting systems—which remainsan important open problem. Dipolar systems present a particularly interesting sub-class of long range interacting systems. Strongly disor-dered systems of interacting dipoles arise microscopically in multiple experimentally relevant contexts, including:(i) dipolar molecules in optical lattices [19]; (ii) dense en-sembles of nitrogen-vacancy centers in diamond [20–22];(iii) dipolar quantum magnets, such as lithium holmiumfluoride [23]. In addition, emergent dipolar excitationsdominate the low energy physics of the electron glass [24]. Dipoles at non-zero temperature in two and three dimen-sions are known to exhibit a perturbative instability toMBL [13, 14]. At the same time, the presence of a longrange (and highly anisotropic) interaction makes dipolarsystems challenging for exact theoretical analysis, partic-ularly in an out-of-equilibrium setting. However, in thelimit of strong disorder, the dipoles that contribute toslow dynamics and relaxation interact via sparse, long-range resonances; in this limit, therefore, one can con-struct controlled theories of response and dynamics.The excitations of a system of interacting dipoles arein general complicated delocalized modes. However, inthe strongly disordered regime, the dynamics of eachdipolar excitation can be separated into “fast” local pre-cession and “slow” relaxation through coupling to otherdipoles. In this sense, each dipole can be regarded asa long-lived excitation of the system, and interactionsbetween dipoles matter because they cause individualdipoles to dephase and relax. The relaxational dynamicsof dipolar systems can be usefully described in terms ofthe population relaxation time (bit flip time) T and thedephasing time T , which in turn can be directly mea-sured experimentally by interferometric methods [25–27]or via ‘pulsed’ experiments such as 2D coherent spec-troscopy [28, 29]. This latter experimental techniquewas recently applied to study the relaxational dynam-ics of the emergent dipolar excitations in phosphorousdoped silicon [30], where the frequency, temperature anddoping dependence of the T and T times was experi-mentally measured and explained in terms of a dipolarhopping model. Ref. [30] operated entirely with a threedimensional system, at effectively zero temperature (i.e.temperatures lower than experimental probe frequency).In this paper, we analyze the relaxational dynamics ofdipolar systems across a wide range of settings. In par-ticular, we determine the frequency, temperature, andpolarization dependence of the T and T times for sys-tems in two and three dimensions, as well as the dimen-sional crossover in quasi-two dimensional systems (slabgeometries). We find that relaxation in dipolar systems,especially in two dimensions, is due to subtle collectiveeffects, which nevertheless give rise to simple scaling laws(Table I).We emphasize at the outset that frequency dependentrelaxation rates are not the same as ac conductivity. Theac conductivity describes the response of some referenceequilibrium state to a weak time-dependent perturbation;at finite frequencies, it is nonzero both for metals and An-derson (or many-body) localized insulators. Even thoughthere is no relaxational dynamics in insulators, the linear-response ac conductivity remains nonzero because it isdominated by sharp absorption resonances. Instead, re-laxational dynamics has to do with the lifetimes of cer-tain excited states that are locally out of equilibrium:i.e., it is the timescale on which such states, once cre-ated, return to equilibrium. Formally, relaxation ratesare a property of multiple-time correlation functions ofspatially local operators, such as the local charge den-sity operator or the local fermion creation operator. Themost basic observable that measures “relaxation” is thelinewidth of a local spectral function; however, this is of-ten dominated by inhomogeneous broadening [30], so oneneeds more delicate spatially local observables [26, 27]or pulsed techniques [30] to identify relaxation. The con-trast between these concepts is particularly clean if oneconsiders a non-interacting Anderson insulator. The acconductivity is given by the Mott formula [31] σ ( ω ) ∼ ω .However, if one prepares an initial state where some lowenergy localized orbitals are unoccupied and some higherenergy localized orbitals are occupied, then this initialstate is nevertheless an eigenstate of the dynamics, andas such the relaxation rates are zero i.e. the state livesforever. Relaxation rates are, however, related to thezero-frequency limit of the conductivity, as well as thefield-dependence of nonlinear response [32, 33]. Our goalin this paper is to determine relaxation rates in dipolarsystems.This paper is structured as follows. In Sec. II we char-acterize the various regimes of the electron glass and ofmicroscopic dipolar ensembles via a density-of-states ex-ponent γ . In the rest of the paper we present a unifiedtheory of dipolar relaxation; the origin of the dipoles en-ters our analysis purely through the exponent γ . Weaddress, in turn, three-dimensional systems (Sec. III),two-dimensional systems (Sec. IV), and slab geometries(Sec. V); in each case we discuss both zero-temperatureand finite-temperature behavior. Finally, we generalizeour results to the case where polarization rather thantemperature sets the entropy of the ensemble (Sec. VI).Throughout, we assume we are dealing with an isolated system, decoupled from phonons or any external envi-ronment. (For a discussion of imperfectly isolated nearlocalized systems, see, e.g., [34]). II. DENSITY OF STATES OF DIPOLES
As we discussed in the introduction, dipolar systemsarise in many physical settings. Excitation lifetimes in allof these systems are set by very similar mechanisms, but scale d T = 0
T > T − , ω ( γ = 1) e − c/ | ω | γ − ( γ > T γ T − T γ ( typical ) T γ ( resonant ) T − T γ ( T ∗ ) − T γ/ TABLE I: Summary of main results for scaling ofrelaxation rates in three and two dimensions withfrequency at zero temperature, and with temperature.For the density of states exponent γ see Table II. system regime γ Electron glass Mott (cid:16) Vξ | log ω | < ω (cid:17) (cid:16) Vξ | log ω | > ω (cid:17) TABLE II: Density of states of low-energy dipolarexcitations in the systems considered here.the resulting rates are sensitive to the density of statesof low-frequency dipolar excitations. In all the cases ofinterest, ρ ( ω ) ∼ ω γ − (up to logarithmic corrections).Therefore, at low temperatures, the fraction of dipolesthat are thermally active scales as T γ . In this sectionwe discuss the values of γ in various cases; in subsequentsections we will present unified expressions for rates (thatapply to all the dipolar systems of interest) in terms ofthis exponent γ . Our results in this section are not new,but are presented for completeness. A. Electron glass
The low energy excitations of the electron glass involvean electron moving from a low energy localized orbital toa nearby higher energy localized orbital. This localizedparticle-hole excitation can be thought of as an emergentelectric dipole.The reduction to dipoles is standard (see, e.g., thesupplement to [30]) but for completeness we outline themain steps here. To fix units, consider a general micro-scopic Hamiltonian describing spinful electrons subjectto Coulomb interactions and a random potential: H = X iσ ε i n i + X h ij i ,σ t ij c † iσ c jσ + X i = j,σ V | r i − r j | n iσ n jσ . (1)The on-site energies E i are drawn from some distributionof width W , and the hopping terms from some distribu-tion with characteristic scale t . The third energy scale inthe problem is set by the Coulomb potential; we denoteby V the scale of the Coulomb potential at the typicalnearest neighbor distance between electronic orbitals (inour units this distance is set to unity). In what follows wewill essentially ignore the spin degree of freedom, whichwas argued to be unimportant for THz experiments in[30]. Our focus is on relaxation of the electric dipoles.For a discussion of the spin sector, see e.g. [35].In the electron glass, the randomness comes microscop-ically from the positions r i . Thus, t ij = t e −| r i − r j | /r ,i.e., the wavefunction overlap between two orbitals. Sincethis is exponentially decaying we truncate it at thenearest-neighbor scale. Similarly, the on-site potential ateach site comes from treating Coulomb interactions at theHartree level. Since the hopping is exponentially sensi-tive to the inter-site spacing, while the Coulomb interac-tion falls off as a power law, one can increase the ratio ofhopping to interactions by increasing the density of car-riers. At some critical density, the single-particle statesdelocalize and the system undergoes a metal-insulatortransition. Our analysis is confined to the insulatingside of this transition. We have introduced the prob-lem schematically, as the details do not affect the scalingbehavior that is of primary interest here.We first consider the single-particle problem, with in-teraction effects included (e.g.) at the Hartree-Fock level.Single-particle eigenstates are localized with a localiza-tion length ξ . In three dimensions this in turn requires t < W . There is a corresponding energy scale δ ξ , whichis the level spacing between single-particle orbitals in thesame localization volume. (At large ξ it scales roughlyas t/ξ . Deep in the localized phase, ξ ≪ δ ξ ∼ t .) We will only con-sider temperatures and frequencies ω, T ≪ δ ξ . At theselow frequencies, there are no excitations in a typical win-dow of the sample, i.e., it looks locally “gapped.” Theexcitations that do exist in this frequency window arethose involving charge transfer over a scale that is largecompared with ξ . Such large-scale rearrangements canbe arbitrarily low-energy; however, a local operator hasvanishing matrix elements between spatially separatedstates. The physically relevant low-energy excitations—i.e., those that can be excited by local probes—insteadinvolve atypical orbitals that are centered around pairsof sites i and j that are much farther than ξ apart, butenergetically “resonant,” in the sense that ε i ≈ ε j . Thesingle-particle eigenstates on a resonant pair are (approx-imately) symmetric and antisymmetric linear combina-tions of orbitals centered at i and j . When one of theseeigenstates is occupied and the other is empty, the reso-nant pair is a two-level system with a dipole moment setby its radius r = | r i − r j | . A well-known result due toMott is that resonant pairs, or dipoles, with a transitionfrequency ω have a typical size r ω ∼ ξ log( W/ω ). Theeffective Hamiltonian for these emergent “active” dipolestakes the form H pair = X α E α τ zα + X α = β c ′ V p α p β r αβ ( τ + α τ − β + h . c . ) . (2)where c ′ is an prefactor of order unity that is not impor-tant for the argument, the τ operators are Pauli opera- tors acting on the emergent dipoles, α labels emergentdipoles, and p α ∼ ξ log( W/ E α ) is the size of emergentdipole α .It remains to establish the density of states of theseactive dipoles. There are two low-frequency regimes: Mott regime:
When the transition frequency ω is rela-tively large, i.e., ω > ∼ V / | r ω | , interactions do not af-fect the occupation numbers of the relevant single-particle eigenstates. For a resonant pair to be ac-tive, the lower eigenstate of the pair must be within ω of the Fermi energy. The density of resonant pairsat frequency ω then scales as ωnξr d − ω /W , where r ω ∼ ξ log( W/ω ) and n is the dipole density. Shklovskii-Efros regime:
When ω < ∼ V / | r ω | , inter-actions qualitatively rearrange the ground statethrough a mechanism similar to Coulomb blockade.As long as either site forming a resonant pair hasa single-particle energy within V / | r ω | of the Fermienergy, the doubly occupied state is energeticallyunfavorable, so the pair of sites is singly occupiedin the ground state. Therefore, the phase space forthe TLS goes as V nξ | r ω | d − /W , which is essen-tially constant at low frequencies [36].In addition to these “low-frequency” regimes where lo-calized dipolar excitations are well-defined, there is ahigh-frequency regime where the eigenstates of (2) aredelocalized and local dipolar excitations relax rapidly.We will not consider this high frequency regime. Notethat the Shklovskii-Efros regime always exists and de-scribes the low energy density of states of the electronglass. The ‘Mott’ regime exists IFF the crossover fre-quency set by the relation V /r ω = ω is below the uppercutoff δ ξ i.e. IFF Vξ log( W ξ/V ) < δ ξ .To summarize, in the electron glass we generically have γ = 1 at the lowest frequencies, but there is potentiallyalso an intermediate-frequency “Mott” regime for which γ = 2. B. Microscopic dipoles
We now turn to systems consisting of microscopicdipoles, such as quantum magnets with strong dipolarinteractions [37], lattices of ultracold dipolar molecules,and dense ensembles of nitrogen-vacancy centers in dia-mond. The T = 0 behavior of excitations in such randomspin systems can be classified according to whether theybreak a continuous symmetry or not [38]; in either case,the excitations are generically bosonic in character, andcan be thought of as spin waves in a random medium.If the ground state spontaneously breaks a contin-uous symmetry, the low energy excitations are Gold-stone modes. At nonzero frequency the Goldstone modeswould be localizable, absent the dipolar interaction; thelong-range tail of the dipolar interaction creates reso-nances among these localized orbitals. (In fact, evenshort-range interactions are expected to cause delocaliza-tion in this limit, since the nearly delocalized Goldstonemodes can act as a bath [39].) Thus, there are no well-defined localized excitations at low frequency in this case,and we will not consider it further. In the case where theground state does not spontaneously break a continuoussymmetry, there are no Goldstone modes, and all singleparticle localization lengths are bounded at strong disor-der. In this case it universally follows [38] from stabilityof the ground state that the low energy density of statesfor non-Goldstone bosonic excitations in random mediamust scale as ρ (˜ ω ) ∼ ˜ ω . Thus for generic systems ofmicroscopic dipoles that do not spontaneously break acontinuous symmetry, we have γ = 5. III. RELAXATION IN THREE DIMENSIONS
In the rest of this paper we will take the dipolardensity-of-states exponent γ as an input, and construct ageneral theory of dipolar relaxation. In three dimensions,the long-range tail of the dipolar interaction guaranteesthat all excitations will eventually delocalize [40], but thetimescales diverge at low frequencies and temperatures. A. Zero-temperature relaxation
We first consider zero-temperature relaxation. (Moregenerally this section concerns relaxation of excitationsabove zero-entropy states, which could also, e.g., be fullypolarized states that are far from the ground state.) Thisoccurs through the long-range hopping of a single dipolarexcitation of frequency ω . Within perturbation theory inthe dipolar interaction, an excitation can only hop to an-other of similar frequency—for example, an excitation offrequency ω can only hop to other states in the frequencywindow (0 , ω ), which occur with density ∼ ω γ . The dis-tance between neighboring dipolar excitations thereforescales as ω − γ/ , and the interaction between them scalesas ω γ . If γ = 1, the typical interaction between excita-tions at frequency ω scales the same way as the typicaldetuning (this case is therefore “marginal” [30]). If γ > γ >
1, one can estimate the characteristic lifetimeas follows. Starting from a typical spin the number ofresonances within a radius R scales as log R [40]. Thecriterion for finding a resonance is that the number ofpartners times the matrix element should equal the typ-ical detuning, i.e., ω γ log R ∼ ω , which giveslog R ∼ ω − γ ⇒ Γ ω ∼ exp( − c | ω | − γ ) . (3)Thus the excitation lifetimes diverge with an essentialsingularity at low frequencies. If γ = 1, on the other hand, the detuning and matrixelement scale the same way with distance. Thus a dipoleat low frequency ω finds a resonant partner at a distanceof order ω − / —i.e., it only depends on ω through theoverall density of the resonant network. Thus an excita-tion decays at a rate Γ ∼ ω : a scaling that was describedas the “marginal Fermi glass” [30]. In this marginal case,logarithmic corrections play an important part at low fre-quencies. In the specific case of the electron glass at lowtemperature, there are logarithmic enhancements to thedensity of states for dipoles as well as the dipole moments(and thus the interaction between dipoles), as discussedin the previous section. Putting these together we findthe scaling Γ( ω ) ∼ ω | log ω | . (4)We discuss the numerical prefactors in the Appendix.Note that Eq. (4) implies that at very low frequencies,individual particle-hole excitations are not well-definedbut instead become strongly coupled. Thus (within ouranalysis) the ultra-low-temperature state is apparently a“non-Fermi-glass” whose excitations are highly collectivein terms of the original fermions (though they are simpleif one regards them as delocalized dipoles). However, thisregime does not onset until exponentially low tempera-tures and frequencies, which may be difficult to accessexperimentally. B. Finite-temperature relaxation
At finite temperature, interactions between dipoles area crucial additional channel for relaxation. The effectiveHamiltonian for interacting dipoles takes the form [14] H pair = X α E α τ zα + X α = β Jr αβ ( τ + α τ − β + h . c . + λτ zα τ zβ ) , (5)which is the same up to relabellings as Eq.2 except for a ZZ interaction (which can be ignored at zero tempera-ture, when there is only a single excited dipole).At finite temperature, 3D dipolar systems host abath of delocalized thermal excitations, as identified byBurin [13]. We present this argument for general dimen-sions d , then specialize to d = 3. In general, the densityof dipolar excitations of size R in a d -dimensional samplewith dipolar interactions scales as R d − . At tempera-ture T , the density of thermally excited dipoles scales as T γ R d − . Since all of these excitations have similar de-tunings, they hybridize if they overlap. If we pick oneresonance, the number of thermally excited overlappingexcitations scales as T γ R d − . There is an associatedthermal length R ( T ) ∼ T − γ/ (2 d − at which thermallyexcited resonances form a percolating hopping networkwith characteristic timescale τ Burin ∼ T − γ/ (2 d − . (6)Specializing to three dimensions, we find that the char-acteristic timescale of the Burin resonances simply scalesas T − γ .In three dimensions, the density of the percolating net-work (as computed above) coincides with that of ther-mally activated spins. Thus, for scaling purposes, onecan regard all thermally active spins as being on the res-onant network. (We will see below that the situationin two dimensions is drastically different.) We now con-sider the relaxation of an inserted excitation at frequency ω . To relax, this excitation must find a partner that iswithin T γ of it in energy. The nearest such excitationis at R ∼ T − γ/ . A Golden Rule calculation yields theresult that Γ ∼ T γ from this mechanism.To understand the low-temperature limit, one shouldalso address how the thermal bath interferes with co-herent zero-temperature hopping. Experimentally [30],going to finite temperature seems to suppress coher-ent hopping, as one might expect [41]. Thus excita-tion lifetimes are non-monotonic. Although this suppres-sion can be quantitatively significant, it is not expectedto be parametric and we will not address it here. Atour level of analysis, the excitation hops to the nearestzero-temperature or bath-induced resonance, whicheveris closer. So we conclude thatΓ ∼ ( min [ ω, T ] γ = 1 , min (cid:2) exp( − c/ | ω | γ − ) , T γ (cid:3) γ > . (7)Thus in the low-temperature Shklovskii-Efros regime ofthe electron glass, the effects of finite temperature arerelatively obvious (viz. we just replace frequency by tem-perature), but away from this regime low temperaturescan dramatically enhance the energy relaxation rate.We now comment briefly on dephasing times. In thethree-dimensional case, the interaction of a typical degreeof freedom with the nearest thermal excitation scales as T γ . The fluctuations of thermal excitations cause de-phasing, so in the present case dephasing scales para-metrically the same way as energy relaxation, so that T ≃ T . (At zero temperature there is no separate de-phasing channel, so T saturates the bound T ≤ T . IV. RELAXATION IN TWO DIMENSIONS
The dipolar interaction in two dimensions falls off suf-ficiently fast that single-particle excitations can be local-ized. In the orthogonal symmetry class (relevant for mostsystems of dipoles), localization occurs even for weak dis-order. For the strongly disordered systems we are consid-ering here, single dipolar excitations can be taken to betightly localized, so that interactions between thermallyexcited dipoles are crucial in determining excitation life-times. Since zero-temperature relaxation is absent, weturn directly to the case of finite temperatures.In two dimensions, the Burin mechanism outlinedabove (6) leads to the conclusion that the spacing of the percolating cluster R ( T ) ∼ /T γ , so that the characteris-tic timescale (6) of the “bath” formed by the percolatingnetwork is τ ∼ T − γ . (8)Note a crucial difference from the three-dimensional case:while the density of thermal spins scales as T γ , the spinson the percolating network are parametrically sparser,with their density scaling as T γ . It is therefore impor-tant to distinguish between three types of excitations:(i) the inserted excitation at frequency ω whose lifetimewe are considering, (ii) generic thermal excitations, whichare not part of the percolating network, and (iii) excita-tions on the percolating network. While excitations onthe percolating network have a lifetime ∼ T − γ , “off-network” excitations relax much more slowly, as we willnow discuss.One can make a naive estimate of the lifetimes of off-resonant spins along the following lines. The interac-tion between a typical spin and the nearest network spinscales as T γ , and so does the maximum energy exchangeallowed by the bandwidth of the network. To find anallowed transition, the typical spin must find a partnerdetuned from it by ∼ T γ . The nearest such partner is atdistance R ∼ T − γ/ , giving the dipolar matrix element T γ/ and the Fermi’s Golden Rule rate Γ ∼ T γ . Whilethis is certainly a lower bound on the relaxation rate, weargue that relaxation in fact occurs much faster, becausethe estimate above neglects spectral diffusion .Spectral diffusion [16, 42, 43] is a phenomenon bywhich the frequency of a spectral transition shifts overtime because the relevant degree of freedom is interact-ing with other, slowly fluctuating, degrees of freedom. Inthe system we are considering, thermally active spins arepresent at density T γ , and interact with each other viastatic shifts of typical size T γ/ . Thus, a spectral lineaveraged over a very long time would have an apparentlinewidth of T γ/ . This apparent broadening, conven-tionally denoted 1 /T ∗ , can in principle be undone usingspin echo. It is therefore not a true decay timescale:the true decay timescales are 1 /T (the timescale onwhich the spectral line shifts its frequency) and 1 /T (thetimescale on which the excitation actually decays). When T ≫ T ∗ , the apparent width of the spectral line interpo-lates from 1 /T (for short averaging times) to 1 /T ∗ (forlong averaging times).We estimate T in this case using two distinct ap-proaches, which agree on the final answer. First, wenote that the level spacing at the Burin scale R ∼ /T γ goes as 1 /R ∼ T γ ; meanwhile, the broadening of eachlevel due to spectral diffusion scales as T γ/ , which isparametrically larger than the level spacing. Therefore,spectral diffusion randomly brings each individual dipoleinto and out of resonance. Assuming spectral diffusionis effective, each level is part of the “network” about T γ of the time. It relaxes at the rate T γ while it is onresonance, thus giving a total relaxation rate T γ .We can also arrive at this result using a more directself-consistent approach, as follows. We are concernedwith the problem of a two-level system (consisting of the“source” and “destination” sites of the dipolar hoppingpair of interest) coupled to a bath of thermal fluctuators(i.e., thermally active spins). Thus its Hamiltonian isof the form H = h ( t ) σ z + ∆ σ x , where h ( t ) is the time-dependent field due to thermal fluctuators. These fluctu-ators come in two flavors: (i) “network” spins that fluc-tuate on a timescale T γ and shift the energy of the levelby a corresponding amount T γ , and (ii) “off-networkspins” that shift the energy by a much larger amount T γ/ but also fluctuate much more slowly at some as-yet-unspecified rate 1 /T (which could potentially be as slowas T γ , from the naive estimate above). The power spec-trum of the noise coming from type (ii) spins has the form h h ( ω, t ) i ∼ f ( ω − ω ( t )), where f is a function peakedat ω ( t ) and of width ∼ /T , and t is a much longertimescale (associated with spectral diffusion), such that h ω ( t ) ω (0) i ∼ (1 /T ∗ ) exp( − t/T ). For a given powerspectrum of h ( t ) this two-level system has a decay rate1 /T . However the power spectrum itself is implicitlya function of 1 /T (which is also the rate at which thethermal fluctuators flip), while the off diagonal matrixelement ∆ is implicitly a function of the power spectrum(since the bandwidth of the noise governs how far in spacethe nearest accessible partner is).Since the decay rate and spectral lineshape dependnontrivially on each other, we solve for both of them self-consistently. In principle one can do this explicitly for thetwo-level system specified above, following Refs. [44, 45].To simplify our analysis we assume that the noise spec-tral function can be characterized by a time-dependentapparent bandwidth Γ( t ) that interpolates between 1 /T and 1 /T ∗ by “filling in” a new spectral strip of width1 /T at a rate 1 /T . ThusΓ( t ) = /T t ≃ T t/T T ≪ t ≪ T /T ∗ /T ∗ t ≫ T /T ∗ (9)To set up the rest of the self-consistency loop we firstestimate T as a function of Γ( t ), which (for consistency)must itself be evaluated over the timescale T on which adecay process happens. We estimate T using the GoldenRule: to find a spin at detuning Γ one must go a distancesuch that R ∼ / Γ, so the matrix element is Γ / andtherefore, using the Golden Rule,1 /T ∼ Γ( T ) (10)Finally we need to evaluate T . We make this estimateself consistently following [25]. At some distance R fromthe central spin, there are T γ R other thermally excitedspins, each flipping at the rate 1 /T , and T γ R spins onthe Burin network flipping at the rate T γ . The dipolarcoupling at distance R is V /R . We fix R self-consistentlyso that V /R = (1 /T ) T γ R + T γ T γ R : once we haveincluded the effects of thermal fluctuations to this dis-tance, the line has broadened by enough that it is no longer able to resolve more distant fluctuations. Thisyields the ultimate expression1 /T ∼ ( T γ /T + T γ ) / , (11)One can check that the set of equations (9,10,11) has thefollowing self-consistent scaling solution:1 /T ∼ T γ , /T ∼ T γ . (12)This seems to be the only self-consistent solution: if wesuppose Γ( T ) ∼ /T then the resulting decay rate is tooslow for Eq. (9) to be consistent, whereas if we assumeΓ( T ) ∼ /T ∗ the decay rate is too fast.An interesting aspect of these results is that the effectof the Burin network spins on the decay of typical spinsis marginal, scaling the same way in Eq. (11) as the self-generated contribution to T . Thus the 2D system is onthe cusp of being a self-sustaining metallic state throughspectral diffusion. (However, absent the Burin networkthere would also be a self-consistent solution with all de-cay rates equal to zero, which the present problem cannothave.) V. DIMENSIONAL CROSSOVERS
We now turn to the crossover between the 2D and 3Dbehaviors discussed above, in a slab that has finite thick-ness ℓ in the z direction, but infinite in the x − y plane.(This can be achieved either by making a finite-widthsample or by imposing a strong magnetic or electric fieldgradient along one direction (say the z axis) of a three-dimensional sample, thus detuning dipoles at different z .In the latter setup the physics discussed in [46, 47] wouldalso come into play.) A. Slabs at zero temperature
At asymptotically low frequencies, a slab behaves as atwo-dimensional system; in particular, in the orthogonalclass, all dipolar excitations are weakly localized. How-ever, at weak disorder, the localization length divergesexponentially in the slab width ℓ . Thus there is a qualita-tive distinction between two regimes of dipolar hopping:a regime in which each dipole finds a resonant partner us-ing the long-range three-dimensional dipolar interactionbefore the crossover set by ℓ , and which acts effectivelylike a metal; and a regime in which ℓ is shorter thanthe resonance scale R c ∼ exp( c/ | ω | γ − ) [generic case] or R c ∼ /ω / [marginal case], so that excitations remaintightly localized at zero temperature. (In the electronglass these crossovers all happen when the pair radius r ω ≪ ℓ .)The resonance scale R c increases monotonically as ω decreases; therefore a slab of thickness ℓ has a frequency ω ∗ ( ℓ ) below which dipoles become manifestly sharp. Forthe electron glass, at large ℓ this crossover always hap-pens in the Shklovskii-Efros regime, with ω ∗ ∼ ℓ − whereas at small ℓ the crossover can happen in the Mottregime, with ω ∗ ∼ / log( ℓ ). We emphasize that thechange in apparent dimension is a crossover rather than atrue localization transition, since technically the dipolesare always localized; however, in two dimensions the lo-calization length in the weakly localized regime is verylarge and systems in this regime can be regarded as ef-fectively metallic. We expect the crossover from weak tostrong localization to happen as follows: in the weakly lo-calized regime, the lineshape of a dipolar excitation con-sists of a large number of very finely spaced lines thatcannot be resolved in practice. As the localization lengthdecreases, the lines become less finely spaced and there-fore resolvable. Finally, in the strongly localized limit,there is essentially just one central line left with appre-ciable weight. In contrast, for dipolar systems in the sym-plectic class, there is the possibility of a true transition,with localized behavior at small ℓ (effectively, strongerdisorder) giving way to delocalized behavior at larger ℓ (effectively, weaker disorder). B. Slabs at finite temperature
At finite temperature, as we have discussed, Burin res-onances and spectral diffusion play an important role inrelaxation. Recall that the line broadening from spectraldiffusion comes mainly from the nearest thermally ac-tive fluctuator. Whether we use the two dimensional orthe three dimensional formula for spectral diffusion de-pends on whether the distance to the nearest fluctuator ismore or less than ℓ . Thus, the dimensional crossover forspectral diffusion happens when the density of thermalexcitations n ex ≈ /ℓ . In contrast the ‘Burin crossover’happens at n ex ≈ W/ ( Jnℓ ), which is a larger value of n ex at strong disorder (which we have assumed). Here W is the disorder scale, J the hopping scale, and n thedensity of dipoles on the lattice. Thus, we enter the ‘twodimensional’ regime for the Burin analysis at a highertemperature than that at which we enter the ‘two di-mensional’ regime for spectral diffusion.In this intermediate temperature range, the coupling tothe nearest thermally active neighbor scales as T γ (set-ting T ∗ ), but the nearest “network” spin is much fur-ther, since the percolating network density takes its two-dimensional value T γ . The spin-echo lifetime T crossesover at a timescale that is intermediate between theselimits. The two-dimensional self-consistent analysis wedeveloped above can be extended to this situation, butthere are many cases (depending on whether each scaleis three-dimensional or two-dimensional) and we will notconsider these in detail. VI. CROSSOVERS NEAR MAXIMUMPOLARIZATION
In experiments involving cold atoms, nitrogen-vacancycenters, etc. it is often more natural to tune the po-larization than the temperature [48]. Since the dipolarinteraction in these systems can generically be treatedin the secular approximation, it approximately conservesthe total number of spin-up dipoles. Thus one can re-gard high-polarization ensembles as effectively being atlarge chemical potential µ . The analysis above carriesover with minor changes to this setup, if we replace thetemperature with the density of minority-state spins, andset the density-of-states exponent γ = 1.There is one distinction worth noting: at low temper-ature, the thermal excitations are disproportionately inlow-energy, well localized states, whereas for high polar-ization they are in random states. Therefore at large po-larization there is a potential additional relaxation chan-nel due to delocalized high-energy excitations. This doesnot, however, matter in the strongly disordered regimeconsidered here, since all single-particle states are as-sumed to be well localized. VII. DISCUSSION
We have determined the frequency and temperaturedependence of relaxation rates for systems of quantumdipoles, whether microscopic or emergent (as in the elec-tron glass). At zero temperature, dipoles relax via long-range coherent hopping, which is possible in three dimen-sions (where long-range hopping resonances proliferate)but not in two dimensions. In three dimensions, there-fore, relaxation occurs at zero temperature. For micro-scopic dipoles (and the electron glass in its intermediate-temperature Mott regime), quasiparticle lifetimes growexponentially as their frequency goes to zero, so localdipoles are asymptotically sharp excitations. For theelectron glass at sufficiently low temperatures (i.e., inits Shklovskii-Efros regime), or for dipolar spin ensem-bles that are at maximum polarization rather than intheir ground state, the excitation lifetimes are marginal,scaling the same way as the excitation frequency up tolog corrections. For the electron glass specifically, theselog corrections enhance dipole hopping (see Appendix),and local dipolar excitations are no longer the right vari-ables to think about the problem at exponentially lowfrequencies and temperatures. (What the ‘right’ degreesof freedom are is not clear to us, but exotic physics couldconceivably be involved [49].) Finite-temperature effectsprovide an additional transport channel.Although energy transport is (apparently) present inthe zero-temperature limit, and local particle-hole exci-tations relax, the dc electrical conductivity of the elec-tron glass is zero in linear response, since moving chargeacross the system requires uphill transitions, which azero-temperature environment cannot facilitate. The dy-namics of realistic electron glasses at very low frequenciesinvolve subtleties that go beyond the approximations wemade here, and would be interesting to revisit for a morecomplete model that includes spin degrees of freedom,large-scale rearrangements, etc.In two dimensions, the situation is quite different:dipolar excitations are localized and therefore infinitelysharp at all frequencies in the zero-temperature limit,and delocalize at finite temperatures only through subtleinteraction effects. We found that the T and T timesboth scale as power laws of temperature, but with dis-tinct exponents. “Diagonal” interaction effects (governedby the timescale T ∗ ∼ T − γ/ are parametrically strongerthan relaxation effects T , > ∼ T − γ , making dipoles intwo dimensions at low temperatures a promising plat-form for exploring dynamical signatures associated withmany-body localization. In quasi-two dimensional sys-tems (slab geometries) there is a crossover from threedimensional behavior at high frequency/temperature totwo dimensional behavior at low frequency/temperature,with crossover scales that we have identified. Our resultsin all of these cases are collected in Table I.It would be interesting to go beyond the scaling the-ory outlined herein to also calculate (potentially loga-rithmic) prefactors. It would also be interesting to deter-mine the relation between relaxation times (calculatedherein) and transport coefficients. We have also iden-tified certain regimes where the ‘dipolar’ approach em-ployed herein breaks down - understanding the behaviorin these regimes is an important open problem. Finally,thermopower coefficients might also be interesting to ex-plore. We leave these problems to future work. ACKNOWLEDGMENTS
We acknowledge useful conversations with Peter Ar-mitage, Mikhail Lukin, and Markus M¨uller. This mate-rial is based in part (RN) upon work supported by theAir Force Office of Scientific Research under award num-ber FA9550-20-1-0222. S.G. acknowledges support fromNSF DMR-1653271.
Appendix A: Crossovers in the 3D electron glass
We now explore, in some more detail, the dynamicalcrossovers in the three-dimensional electron glass at zerotemperature. While the scaling of the relaxation rateswas addressed in the main text, here we also provide somediscussion of the prefactors, which play an important partin determining the crossover timescales. a. Mott regime
Let us start by considering the dynamics in the Mottregime (assuming it exists). In the language of the single- particle dipolar hopping model (2), we are concernedwith the dynamics of a dipolar excitation that is ini-tially localized on resonant pair α (which has frequency E α = ω ). We have postulated weak interaction i.e. thedisorder is strong enough that nearest-neighbor hops areoff shell. Instead, an excitation at frequency ω can onlyhybridize with a partner at similar (or lower) frequency.Excitations in the frequency window (0 , ω ) occur withdensity ∼ ω nξr ω /W , where n is the dipole density.Thus the spacing between two nearby such excitationsscales as ω − / , and the dipolar interaction matrix ele-ment between them scales as ∼ V r ω ω nξr ω /W . Gener-ically, however, the detuning between these excitationsis ∼ ω (by construction). Asymptotically, therefore, ex-citations at low frequencies in the Mott regime will beoff-shell with their near neighbors. and will be paramet-rically longer-lived. One can estimate the characteristicexcitation lifetime as follows. Starting from a typical spinthe total number of resonances nearer than a distance R scales as log R . An excitation at frequency ω finds a res-onant partner when log R ∼ W / ( V r ω ωnξ ). The char-acteristic interaction energy at this scale, and thereforethe excitation lifetime, also scale as Γ ω ∼ V exp( − ˜ c/ | ω | ),i.e., the excitation lifetime is nonperturbative in ω .Could there be a regime with ω < V r ω ω nξr ω /W ,such that typical excitations at ω were resonant, and therelaxation rate simply followed the typical matrix ele-ment between such excitations ( ∼ ω )? It is straightfor-ward to see this is not possible. The condition above canbe expressed (dropping logarithmic factors which are notlarge by postulate) as ω > WV nξ Wξ , but the first term onthe right hand side is larger than one (by the weak in-teraction postulate), and the second term is larger than δ ξ (since we need W > t to have single particle states lo-calized), so we conclude that the above condition is onlysatisfied for frequencies larger than δ ξ , but δ ξ was theupper cutoff for our theory. We thus conclude that thescaling Γ ω ∼ exp( − ˜ c/ | ω | ) holds throughout the regime V /r ω < ω < δ ξ . b. Shklovskii-Efros regime The Mott regime crosses over to the Shklovskii-Efrosregime at a frequency such that ω SE r ω SE ≃ V . In theShklovskii-Efros regime, Coulomb interactions rearrangethe ground state such that any resonant pair split by ω < ω SE with either of its partners within V /r ω ofthe Fermi energy is singly occupied in the ground state.One can estimate the density of states of pairs at fre-quency ω as follows. The first half of the pair must bewithin V /r ω of the Fermi energy, and the density of suchsites scales as V / [ W r ω ]. Its partner is in a shell of ra-dius r ω and thickness ξ ; multiplying these, we get thatthe density of dipoles of transition frequency ω scales as n ( ω ) ∼ V nξr ω /W .We now consider diagonalizing the Hamiltonian (2) inthe Shklovskii-Efros regime for states in the frequencywindow ( ω/c, cω ) where c is some arbitrarily chosen num-ber of order unity. These dipoles occupy a fraction ofsites ∼ ωn ( ω ), and have a characteristic dipole moment r ω . The interaction between adjacent dipoles in this fre-quency window is V r ω ωn ( ω ) = ωr ω V ξ/W ∼ ω | log ω | .For small enough ω , therefore, the interactions amongnearest-neighbor dipoles at frequency ω become strongenough that they are generically resonant. Below thiscrossover scale, set by ω c ∼ exp (cid:0) − ( W/V ξ ) / (cid:1) , dipolarexcitations are strongly coupled, relative to their naturalfrequency. 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