# Lifetimes of local excitations in disordered dipolar quantum systems

Rahul Nandkishore  Department of Physics and Center for Theory of Quantum
Matter, University of Colorado at Boulder, Boulder CO 80309, USA  Sarang
Gopalakrishnan  Department of Physics, Pennsylvania State University,
University Park PA 16802, USA

###### Abstract

When a strongly disordered system of interacting quantum dipoles is locally
excited, the excitation relaxes on some (potentially very long) timescale. We
analyze this relaxation process, both for electron glasses with strong Coulomb
interactions—in which particle-hole dipoles are emergent excitations—and for
systems (e.g., quantum magnets or ultracold dipolar molecules) made up of
microscopic dipoles. We consider both energy relaxation rates ($T_{1}$ times)
and dephasing rates ($T_{2}$ times), and their dependence on frequency,
temperature, and polarization. Systems in both two and three dimensions are
considered, along with the dimensional crossover in quasi-two dimensional
geometries. A rich set of scaling laws is found.

##  I Introduction

The dynamical behavior of well isolated and strongly disordered quantum many
body systems has been arousing great interest, from both the condensed matter
and quantum information communities. Under certain circumstances, such systems
can display the phenomenon of ‘many body localization’ (MBL) Anderson (1958);
Fleishman and Anderson (1980); Gornyi  _et al._ (2005); Basko  _et al._ (2006)
(for reviews, see Nandkishore and Huse (2015); Abanin  _et al._ (2019);
Gopalakrishnan and Parameswaran (2020)), whereby they preserve a memory of
their initial conditions forever in local observables, hence serving as good
quantum memories. The existing theory of MBL describes systems with
interactions that are short range in real space, whereas physical systems
frequently contain long range interactions—e.g., Coulomb or dipolar
interactions. For interactions that fall off as a power law with distance, it
has been argued that MBL is always ultimately destabilized by nonperturbative
processes due to rare regions De Roeck and Huveneers (2017); Luitz  _et al._
(2017). However, such putative non-perturbative effects only manifest on
timescales which diverge faster than a power law of the disorder strength
Gopalakrishnan and Huse (2019), and which as such are not relevant for most
experiments in the strongly disordered regime (but see Ref. Léonard  _et al._
(2020)). We will not consider such non-perturbative effects in this paper.
However, for sufficiently slowly decaying power laws, there is also a
perturbative instability of the MBL phase Burin  _et al._ (1998); Burin
(2006); Yao  _et al._ (2014); Burin (2015); Tikhonov and Mirlin (2018);
Maksymov and Burin (2020) (although see Nandkishore and Sondhi (2017) for
exceptions), which is expected to manifest on experimentally relevant
timescales. Even when the MBL phase is perturbatively unstable on the longest
timescales, however, the framework of “nearly MBL” systems offers a useful
perspective on the dynamics of well isolated but strongly disordered long
range interacting systems—which remains an important open problem.

Dipolar systems present a particularly interesting subclass of long range
interacting systems. Strongly disordered systems of interacting dipoles arise
microscopically in multiple experimentally relevant contexts, including: (i)
dipolar molecules in optical lattices Yan  _et al._ (2013); (ii) dense
ensembles of nitrogen-vacancy centers in diamond Choi  _et al._ (2017a);
Kucsko  _et al._ (2018); Choi  _et al._ (2017b); (iii) dipolar quantum
magnets, such as lithium holmium fluoride Ghosh  _et al._ (2002). In addition,
emergent dipolar excitations dominate the low energy physics of the electron
glass Amir  _et al._ (2011). Dipoles at non-zero temperature in two and three
dimensions are known to exhibit a perturbative instability to MBL Burin
(2006); Yao  _et al._ (2014). At the same time, the presence of a long range
(and highly anisotropic) interaction makes dipolar systems challenging for
exact theoretical analysis, particularly in an out-of-equilibrium setting.
However, in the limit of strong disorder, the dipoles that contribute to slow
dynamics and relaxation interact via sparse, long-range resonances; in this
limit, therefore, one can construct controlled theories of response and
dynamics.

The excitations of a system of interacting dipoles are in general complicated
delocalized modes. However, in the strongly disordered regime, the dynamics of
each dipolar excitation can be separated into “fast” local precession and
“slow” relaxation through coupling to other dipoles. In this sense, each
dipole can be regarded as a long-lived excitation of the system, and
interactions between dipoles matter because they cause individual dipoles to
dephase and relax. The relaxational dynamics of dipolar systems can be
usefully described in terms of the population relaxation time (bit flip time)
$T_{1}$ and the dephasing time $T_{2}$, which in turn can be directly measured
experimentally by interferometric methods Nandkishore  _et al._ (2014); Serbyn
_et al._ (2014); Chiaro  _et al._ (2019) or via ‘pulsed’ experiments such as
2D coherent spectroscopy Hamm and Zanni (2011); Mukamel (1995). This latter
experimental technique was recently applied to study the relaxational dynamics
of the emergent dipolar excitations in phosphorous doped silicon Mahmood  _et
al._ (2021), where the frequency, temperature and doping dependence of the
$T_{1}$ and $T_{2}$ times was experimentally measured and explained in terms
of a dipolar hopping model. Ref. Mahmood  _et al._ (2021) operated entirely
with a three dimensional system, at effectively zero temperature (i.e.
temperatures lower than experimental probe frequency).

In this paper, we analyze the relaxational dynamics of dipolar systems across
a wide range of settings. In particular, we determine the frequency,
temperature, and polarization dependence of the $T_{1}$ and $T_{2}$ times for
systems in two and three dimensions, as well as the dimensional crossover in
quasi-two dimensional systems (slab geometries). We find that relaxation in
dipolar systems, especially in two dimensions, is due to subtle collective
effects, which nevertheless give rise to simple scaling laws (Table 1).

We emphasize at the outset that frequency dependent relaxation rates are not
the same as ac conductivity. The ac conductivity describes the response of
some reference equilibrium state to a weak time-dependent perturbation; at
finite frequencies, it is nonzero both for metals and Anderson (or many-body)
localized insulators. Even though there is no relaxational dynamics in
insulators, the linear-response ac conductivity remains nonzero because it is
dominated by sharp absorption resonances. Instead, relaxational dynamics has
to do with the lifetimes of certain _excited_ states that are locally out of
equilibrium: i.e., it is the timescale on which such states, once created,
return to equilibrium. Formally, relaxation rates are a property of multiple-
time correlation functions of spatially local operators, such as the local
charge density operator or the local fermion creation operator. The most basic
observable that measures “relaxation” is the linewidth of a local spectral
function; however, this is often dominated by inhomogeneous broadening Mahmood
_et al._ (2021), so one needs more delicate spatially local observables Serbyn
_et al._ (2014); Chiaro  _et al._ (2019) or pulsed techniques Mahmood  _et
al._ (2021) to identify relaxation. The contrast between these concepts is
particularly clean if one considers a non-interacting Anderson insulator. The
ac conductivity is given by the Mott formula Mott (1968)
$\sigma(\omega)\sim\omega^{2}$. However, if one prepares an initial state
where some low energy localized orbitals are unoccupied and some higher energy
localized orbitals are occupied, then this initial state is nevertheless an
eigenstate of the dynamics, and as such the relaxation rates are zero i.e. the
state lives forever. Relaxation rates are, however, related to the zero-
frequency limit of the conductivity, as well as the field-dependence of
nonlinear response Khemani  _et al._ (2015); Gopalakrishnan  _et al._ (2016).
Our goal in this paper is to determine relaxation rates in dipolar systems.

This paper is structured as follows. In Sec. II we characterize the various
regimes of the electron glass and of microscopic dipolar ensembles via a
density-of-states exponent $\gamma$. In the rest of the paper we present a
unified theory of dipolar relaxation; the origin of the dipoles enters our
analysis purely through the exponent $\gamma$. We address, in turn, three-
dimensional systems (Sec. III), two-dimensional systems (Sec. IV), and slab
geometries (Sec. V); in each case we discuss both zero-temperature and finite-
temperature behavior. Finally, we generalize our results to the case where
polarization rather than temperature sets the entropy of the ensemble (Sec.
VI). Throughout, we assume we are dealing with an isolated system, decoupled
from phonons or any external environment. (For a discussion of imperfectly
isolated near localized systems, see, e.g., Nandkishore and Gopalakrishnan
(2017)).

scale | $d$ | $T=0$ | $T>0$
---|---|---|---
$T^{-1}_{1,2}$ | $3$ | $\begin{array}[]{lr}\omega&(\gamma=1)\\\ e^{-c/|\omega|^{\gamma-1}}&(\gamma>1)\end{array}$ | $T^{\gamma}$
$T^{-1}_{1}$ | $2$ | 0 | $\begin{array}[]{lr}T^{4\gamma}&(typical)\\\ T^{3\gamma}&(resonant)\end{array}$
$T^{-1}_{2}$ | $2$ | 0 | $T^{3\gamma}$
$(T^{*}_{2})^{-1}$ | $2$ | 0 | $T^{3\gamma/2}$
Table 1: Summary of main results for scaling of relaxation rates in three and two dimensions with frequency at zero temperature, and with temperature. For the density of states exponent $\gamma$ see Table 2. system | regime | $\gamma$
---|---|---
Electron glass | Mott $\left(\frac{V}{\xi|\log\omega|}<\omega\right)$ | 2
| Shklovskii-Efros $\left(\frac{V}{\xi|\log\omega|}>\omega\right)$ | 1
Dipolar spins | No continuous SSB | 5
Table 2: Density of states of low-energy dipolar excitations in the systems
considered here.

##  II Density of states of dipoles

As we discussed in the introduction, dipolar systems arise in many physical
settings. Excitation lifetimes in all of these systems are set by very similar
mechanisms, but the resulting rates are sensitive to the density of states of
low-frequency dipolar excitations. In all the cases of interest,
$\rho(\omega)\sim\omega^{\gamma-1}$ (up to logarithmic corrections).
Therefore, at low temperatures, the fraction of dipoles that are thermally
active scales as $T^{\gamma}$. In this section we discuss the values of
$\gamma$ in various cases; in subsequent sections we will present unified
expressions for rates (that apply to all the dipolar systems of interest) in
terms of this exponent $\gamma$. Our results in this section are not new, but
are presented for completeness.

###  II.1 Electron glass

The low energy excitations of the electron glass involve an electron moving
from a low energy localized orbital to a nearby higher energy localized
orbital. This localized particle-hole excitation can be thought of as an
emergent electric dipole.

The reduction to dipoles is standard (see, e.g., the supplement to Mahmood
_et al._ (2021)) but for completeness we outline the main steps here. To fix
units, consider a general microscopic Hamiltonian describing spinful electrons
subject to Coulomb interactions and a random potential:

$H=\sum_{i\sigma}\varepsilon_{i}n_{i}+\sum_{\langle
ij\rangle,\sigma}t_{ij}c^{\dagger}_{i\sigma}c_{j\sigma}+\sum_{i\neq
j,\sigma}\frac{V}{|\mathbf{r}_{i}-\mathbf{r}_{j}|}n_{i\sigma}n_{j\sigma}.$ (1)

The on-site energies $E_{i}$ are drawn from some distribution of width $W$,
and the hopping terms from some distribution with characteristic scale $t$.
The third energy scale in the problem is set by the Coulomb potential; we
denote by $V$ the scale of the Coulomb potential at the typical nearest
neighbor distance between electronic orbitals (in our units this distance is
set to unity). In what follows we will essentially ignore the spin degree of
freedom, which was argued to be unimportant for THz experiments in Mahmood
_et al._ (2021). Our focus is on relaxation of the electric dipoles. For a
discussion of the spin sector, see e.g. Bhatt and Lee (1982).

In the electron glass, the randomness comes microscopically from the positions
$\mathbf{r}_{i}$. Thus,
$t_{ij}=t_{0}e^{-|\mathbf{r}_{i}-\mathbf{r}_{j}|/r_{0}}$, i.e., the
wavefunction overlap between two orbitals. Since this is exponentially
decaying we truncate it at the nearest-neighbor scale. Similarly, the on-site
potential at each site comes from treating Coulomb interactions at the Hartree
level. Since the hopping is exponentially sensitive to the inter-site spacing,
while the Coulomb interaction falls off as a power law, one can increase the
ratio of hopping to interactions by increasing the density of carriers. At
some critical density, the single-particle states delocalize and the system
undergoes a metal-insulator transition. Our analysis is confined to the
insulating side of this transition. We have introduced the problem
schematically, as the details do not affect the scaling behavior that is of
primary interest here.

We first consider the single-particle problem, with interaction effects
included (e.g.) at the Hartree-Fock level. Single-particle eigenstates are
localized with a localization length $\xi$. In three dimensions this in turn
requires $t<W$. There is a corresponding energy scale $\delta_{\xi}$, which is
the level spacing between single-particle orbitals in the same localization
volume. (At large $\xi$ it scales roughly as $t/\xi^{3}$. Deep in the
localized phase, $\xi\ll 1$ so this characteristic scale saturates to
$\delta_{\xi}\sim t$.) We will only consider temperatures and frequencies
$\omega,T\ll\delta_{\xi}$. At these low frequencies, there are no excitations
in a typical window of the sample, i.e., it looks locally “gapped.” The
excitations that do exist in this frequency window are those involving charge
transfer over a scale that is large compared with $\xi$. Such large-scale
rearrangements can be arbitrarily low-energy; however, a local operator has
vanishing matrix elements between spatially separated states. The physically
relevant low-energy excitations—i.e., those that can be excited by local
probes—instead involve atypical orbitals that are centered around pairs of
sites $i$ and $j$ that are much farther than $\xi$ apart, but energetically
“resonant,” in the sense that $\varepsilon_{i}\approx\varepsilon_{j}$. The
single-particle eigenstates on a resonant pair are (approximately) symmetric
and antisymmetric linear combinations of orbitals centered at $i$ and $j$.
When one of these eigenstates is occupied and the other is empty, the resonant
pair is a two-level system with a dipole moment set by its radius
$r=|\mathbf{r}_{i}-\mathbf{r}_{j}|$. A well-known result due to Mott is that
resonant pairs, or dipoles, with a transition frequency $\omega$ have a
typical size $r_{\omega}\sim\xi\log(W/\omega)$. The effective Hamiltonian for
these emergent “active” dipoles takes the form

$H_{\mathrm{pair}}=\sum_{\alpha}\mathcal{E}_{\alpha}\tau^{z}_{\alpha}+\sum_{\alpha\neq\beta}\frac{c^{\prime}Vp_{\alpha}p_{\beta}}{r_{\alpha\beta}^{3}}(\tau^{+}_{\alpha}\tau^{-}_{\beta}+\mathrm{h.c.}).$
(2)

where $c^{\prime}$ is an prefactor of order unity that is not important for
the argument, the $\tau$ operators are Pauli operators acting on the emergent
dipoles, $\alpha$ labels emergent dipoles, and
$p_{\alpha}\sim\xi\log(W/\mathcal{E}_{\alpha})$ is the size of emergent dipole
$\alpha$.

It remains to establish the density of states of these active dipoles. There
are two low-frequency regimes:

Mott regime

When the transition frequency $\omega$ is relatively large, i.e.,
$\omega\gtrsim V/|r_{\omega}|$, interactions do not affect the occupation
numbers of the relevant single-particle eigenstates. For a resonant pair to be
active, the lower eigenstate of the pair must be within $\omega$ of the Fermi
energy. The density of resonant pairs at frequency $\omega$ then scales as
$\omega n\xi r_{\omega}^{d-1}/W^{2}$, where $r_{\omega}\sim\xi\log(W/\omega)$
and $n$ is the dipole density.

Shklovskii-Efros regime

When $\omega\lesssim V/|r_{\omega}|$, interactions qualitatively rearrange the
ground state through a mechanism similar to Coulomb blockade. As long as
_either_ site forming a resonant pair has a single-particle energy within
$V/|r_{\omega}|$ of the Fermi energy, the doubly occupied state is
energetically unfavorable, so the pair of sites is singly occupied in the
ground state. Therefore, the phase space for the TLS goes as
$Vn\xi|r_{\omega}|^{d-2}/W^{2}$, which is essentially constant at low
frequencies Shklovskii and Efros (1981).

In addition to these “low-frequency” regimes where localized dipolar
excitations are well-defined, there is a high-frequency regime where the
eigenstates of (2) are delocalized and local dipolar excitations relax
rapidly. We will not consider this high frequency regime. Note that the
Shklovskii-Efros regime always exists and describes the low energy density of
states of the electron glass. The ‘Mott’ regime exists IFF the crossover
frequency set by the relation $V/r_{\omega}=\omega$ is below the upper cutoff
$\delta_{\xi}$ i.e. IFF $\frac{V}{\xi\log(W\xi/V)}<\delta_{\xi}$.

To summarize, in the electron glass we generically have $\gamma=1$ at the
lowest frequencies, but there is potentially also an intermediate-frequency
“Mott” regime for which $\gamma=2$.

###  II.2 Microscopic dipoles

We now turn to systems consisting of microscopic dipoles, such as quantum
magnets with strong dipolar interactions Silevitch  _et al._ (2007), lattices
of ultracold dipolar molecules, and dense ensembles of nitrogen-vacancy
centers in diamond. The $T=0$ behavior of excitations in such random spin
systems can be classified according to whether they break a continuous
symmetry or not Gurarie and Chalker (2003); in either case, the excitations
are generically bosonic in character, and can be thought of as spin waves in a
random medium.

If the ground state spontaneously breaks a continuous symmetry, the low energy
excitations are Goldstone modes. At nonzero frequency the Goldstone modes
would be localizable, absent the dipolar interaction; the long-range tail of
the dipolar interaction creates resonances among these localized orbitals. (In
fact, even short-range interactions are expected to cause delocalization in
this limit, since the nearly delocalized Goldstone modes can act as a bath
Nandkishore and Potter (2014).) 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 the ground state does not spontaneously break a
continuous symmetry, there are no Goldstone modes, and all single particle
localization lengths are bounded at strong disorder. In this case it
universally follows Gurarie and Chalker (2003) from stability of the ground
state that the low energy density of states for non-Goldstone bosonic
excitations in random media must scale as
$\rho(\tilde{\omega})\sim\tilde{\omega}^{4}$. Thus for generic systems of
microscopic dipoles that do not spontaneously break a continuous symmetry, we
have $\gamma=5$.

##  III Relaxation in three dimensions

In the rest of this paper we will take the dipolar density-of-states exponent
$\gamma$ as an input, and construct a general theory of dipolar relaxation. In
three dimensions, the long-range tail of the dipolar interaction guarantees
that all excitations will eventually delocalize Levitov (1990), but the
timescales diverge at low frequencies and temperatures.

###  III.1 Zero-temperature relaxation

We first consider zero-temperature relaxation. (More generally this section
concerns relaxation of excitations above _zero-entropy states_ , which could
also, e.g., be fully polarized states that are far from the ground state.)
This occurs through the long-range hopping of a single dipolar excitation of
frequency $\omega$. Within perturbation theory in the dipolar interaction, an
excitation can only hop to another of similar frequency—for example, an
excitation of frequency $\omega$ can only hop to other states in the frequency
window $(0,2\omega)$, which occur with density $\sim\omega^{\gamma}$. The
distance between neighboring dipolar excitations therefore scales as
$\omega^{-\gamma/3}$, and the interaction between them scales as
$\omega^{\gamma}$. If $\gamma=1$, the typical interaction between excitations
at frequency $\omega$ scales the same way as the typical detuning (this case
is therefore “marginal” Mahmood  _et al._ (2021)). If $\gamma>1$, on the other
hand, neighboring low-frequency excitations are detuned by much more than the
interaction between them. The dynamics in these two cases is therefore very
different, and we consider them in turn.

If $\gamma>1$, one can estimate the characteristic lifetime as follows.
Starting from a typical spin the number of resonances within a radius $R$
scales as $\log R$ Levitov (1990). The criterion for finding a resonance is
that the number of partners times the matrix element should equal the typical
detuning, i.e., $\omega^{\gamma}\log R\sim\omega$, which gives

$\log
R\sim\omega^{1-\gamma}\Rightarrow\Gamma_{\omega}\sim\exp(-c|\omega|^{1-\gamma}).$
(3)

Thus the excitation lifetimes diverge with an essential singularity at low
frequencies.

If $\gamma=1$, on the other hand, the detuning and matrix element scale the
same way with distance. Thus a dipole at low frequency $\omega$ finds a
resonant partner at a distance of order $\omega^{-1/3}$—i.e., it only depends
on $\omega$ through the overall density of the resonant network. Thus an
excitation decays at a rate $\Gamma\sim\omega$: a scaling that was described
as the “marginal Fermi glass” Mahmood  _et al._ (2021). In this marginal case,
logarithmic corrections play an important part at low frequencies. In the
specific case of the electron glass at low temperature, there are logarithmic
enhancements to the density of states for dipoles as well as the dipole
moments (and thus the interaction between dipoles), as discussed in the
previous section. Putting these together we find the scaling

$\Gamma(\omega)\sim\omega|\log^{3}\omega|.$ (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-defined but instead become strongly coupled. Thus (within our analysis)
the ultra-low-temperature state is apparently a “non-Fermi-glass” whose
excitations are highly collective in terms of the original fermions (though
they are simple if one regards them as delocalized dipoles). However, this
regime does not onset until exponentially low temperatures and frequencies,
which may be difficult to access experimentally.

###  III.2 Finite-temperature relaxation

At finite temperature, interactions between dipoles are a crucial additional
channel for relaxation. The effective Hamiltonian for interacting dipoles
takes the form Yao  _et al._ (2014)

$H_{\mathrm{pair}}=\sum_{\alpha}\mathcal{E}_{\alpha}\tau^{z}_{\alpha}+\sum_{\alpha\neq\beta}\frac{J}{r_{\alpha\beta}^{3}}(\tau^{+}_{\alpha}\tau^{-}_{\beta}+\mathrm{h.c.}+\lambda\tau^{z}_{\alpha}\tau^{z}_{\beta}),$
(5)

which is the same up to relabellings as Eq.2 except for a $ZZ$ interaction
(which can be ignored at zero temperature, when there is only a single excited
dipole).

At finite temperature, 3D dipolar systems host a bath of delocalized thermal
excitations, as identified by Burin Burin (2006). We present this argument for
general dimensions $d$, then specialize to $d=3$. In general, the density of
dipolar excitations of size $R$ in a $d$-dimensional sample with dipolar
interactions scales as $R^{d-3}$. At temperature $T$, the density of thermally
excited dipoles scales as $T^{\gamma}R^{d-3}$. Since all of these excitations
have similar detunings, they hybridize if they overlap. If we pick one
resonance, the number of thermally excited overlapping excitations scales as
$T^{\gamma}R^{2d-3}$. There is an associated thermal length $R(T)\sim
T^{-\gamma/(2d-3)}$ at which thermally excited resonances form a percolating
hopping network with characteristic timescale

$\tau_{\mathrm{Burin}}\sim T^{-3\gamma/(2d-3)}.$ (6)

Specializing to three dimensions, we find that the characteristic timescale of
the Burin resonances simply scales as $T^{-\gamma}$.

In three dimensions, the density of the percolating network (as computed
above) coincides with that of thermally activated spins. Thus, for scaling
purposes, one can regard all thermally active spins as being on the resonant
network. (We will see below that the situation in two dimensions is
drastically different.) We now consider the relaxation of an inserted
excitation at frequency $\omega$. To relax, this excitation must find a
partner that is within $T^{\gamma}$ of it in energy. The nearest such
excitation is at $R\sim T^{-\gamma/3}$. A Golden Rule calculation yields the
result that $\Gamma\sim T^{\gamma}$ from this mechanism.

To understand the low-temperature limit, one should also address how the
thermal bath interferes with coherent zero-temperature hopping. Experimentally
Mahmood  _et al._ (2021), going to finite temperature seems to suppress
coherent hopping, as one might expect Leggett  _et al._ (1987). Thus
excitation lifetimes are non-monotonic. Although this suppression can be
quantitatively significant, it is not expected to be _parametric_ and we will
not address it here. At our level of analysis, the excitation hops to the
nearest zero-temperature or bath-induced resonance, whichever is closer. So we
conclude that

$\Gamma\sim\left\\{\begin{array}[]{lr}\min\left[\omega,T\right]&\gamma=1,\\\
\min\left[\exp(-c/|\omega|^{\gamma-1}),T^{\gamma}\right]&\gamma>1.\end{array}\right.$
(7)

Thus in the low-temperature Shklovskii-Efros regime of the electron glass, the
effects of finite temperature are relatively obvious (viz. we just replace
frequency by temperature), but away from this regime low temperatures can
dramatically enhance the energy relaxation rate.

We now comment briefly on dephasing times. In the three-dimensional case, the
interaction of a typical degree of freedom with the nearest thermal excitation
scales as $T^{\gamma}$. The fluctuations of thermal excitations cause
dephasing, so in the present case dephasing scales parametrically the same way
as energy relaxation, so that $T_{2}\simeq T_{1}$. (At zero temperature there
is no separate dephasing channel, so $T_{2}$ saturates the bound $T_{2}\leq
2T_{1}$.

##  IV Relaxation in two dimensions

The dipolar interaction in two dimensions falls off sufficiently fast that
single-particle excitations can be localized. In the orthogonal symmetry class
(relevant for most systems of dipoles), localization occurs even for weak
disorder. For the strongly disordered systems we are considering here, single
dipolar excitations can be taken to be tightly localized, so that interactions
between thermally excited dipoles are crucial in determining excitation
lifetimes. Since zero-temperature relaxation is absent, we turn directly to
the case of finite temperatures.

In two dimensions, the Burin mechanism outlined above (6) leads to the
conclusion that the spacing of the percolating cluster $R(T)\sim
1/T^{\gamma}$, so that the characteristic timescale (6) of the “bath” formed
by the percolating network is

$\tau^{\mathrm{2D}}_{\mathrm{Burin}}\sim T^{-3\gamma}.$ (8)

Note a crucial difference from the three-dimensional case: while the density
of thermal spins scales as $T^{\gamma}$, the spins on the percolating network
are parametrically sparser, with their density scaling as $T^{2\gamma}$. It is
therefore important to distinguish between three types of excitations: (i) the
inserted excitation at frequency $\omega$ whose lifetime we are considering,
(ii) generic thermal excitations, which are not part of the percolating
network, and (iii) excitations on the percolating network. While excitations
on the percolating network have a lifetime $\sim T^{-3\gamma}$, “off-network”
excitations relax much more slowly, as we will now discuss.

One can make a naive estimate of the lifetimes of off-resonant spins along the
following lines. The interaction between a typical spin and the nearest
network spin scales as $T^{3\gamma}$, and so does the maximum energy exchange
allowed by the bandwidth of the network. To find an allowed transition, the
typical spin must find a partner detuned from it by $\sim T^{3\gamma}$. The
nearest such partner is at distance $R\sim T^{-3\gamma/2}$, giving the dipolar
matrix element $T^{9\gamma/2}$ and the Fermi’s Golden Rule rate $\Gamma\sim
T^{6\gamma}$. While this is certainly a _lower bound_ on the relaxation rate,
we argue that relaxation in fact occurs much faster, because the estimate
above neglects spectral diffusion.

Spectral diffusion GALPERIN  _et al._ (1991); Gornyi  _et al._ (2017);
Tikhonov and Mirlin (2018) is a phenomenon by which the frequency of a
spectral transition shifts over time because the relevant degree of freedom is
interacting with other, slowly fluctuating, degrees of freedom. In the system
we are considering, thermally active spins are present at density
$T^{\gamma}$, and interact with each other via static shifts of typical size
$T^{3\gamma/2}$. Thus, a spectral line averaged over a very long time would
have an apparent linewidth of $T^{3\gamma/2}$. This apparent broadening,
conventionally denoted $1/T_{2}^{*}$, can in principle be undone using spin
echo. It is therefore not a true decay timescale: the true decay timescales
are $1/T_{2}$ (the timescale on which the spectral line shifts its frequency)
and $1/T_{1}$ (the timescale on which the excitation actually decays). When
$T_{2}\gg T_{2}^{*}$, the apparent width of the spectral line interpolates
from $1/T_{2}$ (for short averaging times) to $1/T_{2}^{*}$ (for long
averaging times).

We estimate $T_{1}$ in this case using two distinct approaches, which agree on
the final answer. First, we note that the level spacing at the Burin scale
$R\sim 1/T^{\gamma}$ goes as $1/R^{2}\sim T^{2\gamma}$; meanwhile, the
broadening of each level due to spectral diffusion scales as $T^{3\gamma/2}$,
which is parametrically larger than the level spacing. Therefore, spectral
diffusion randomly brings each individual dipole into and out of resonance.
Assuming spectral diffusion is effective, each level is part of the “network”
about $T^{\gamma}$ of the time. It relaxes at the rate $T^{3\gamma}$ while it
is on resonance, thus giving a total relaxation rate $T^{4\gamma}$.

We can also arrive at this result using a more direct self-consistent
approach, as follows. We are concerned with the problem of a two-level system
(consisting of the “source” and “destination” sites of the dipolar hopping
pair of interest) coupled to a bath of thermal fluctuators (i.e., thermally
active spins). Thus its Hamiltonian is of the form
$H=h(t)\sigma^{z}+\Delta\sigma^{x}$, where $h(t)$ is the time-dependent field
due to thermal fluctuators. These fluctuators come in two flavors: (i)
“network” spins that fluctuate on a timescale $T^{3\gamma}$ and shift the
energy of the level by a corresponding amount $T^{3\gamma}$, and (ii) “off-
network spins” that shift the energy by a much larger amount $T^{3\gamma/2}$
but also fluctuate much more slowly at some as-yet-unspecified rate $1/T_{2}$
(which could potentially be as slow as $T^{6\gamma}$, from the naive estimate
above). The power spectrum of the noise coming from type (ii) spins has the
form $\langle h(\omega,t)^{2}\rangle\sim f(\omega-\omega_{0}(t))$, where $f$
is a function peaked at $\omega_{0}(t)$ and of width $\sim 1/T_{2}$, and $t$
is a much longer timescale (associated with spectral diffusion), such that
$\langle\omega_{0}(t)\omega_{0}(0)\rangle\sim(1/T_{2}^{*})\exp(-t/T_{2})$. For
a given power spectrum of $h(t)$ this two-level system has a decay rate
$1/T_{1}$. However the power spectrum itself is implicitly a function of
$1/T_{1}$ (which is also the rate at which the thermal fluctuators flip),
while the off diagonal matrix element $\Delta$ is implicitly a function of the
power spectrum (since the bandwidth of the noise governs how far in space the
nearest accessible partner is).

Since the decay rate and spectral lineshape depend nontrivially on each other,
we solve for both of them self-consistently. In principle one can do this
explicitly for the two-level system specified above, following Refs. Amir  _et
al._ (2009); Gopalakrishnan  _et al._ (2017). To simplify our analysis we
assume that the noise spectral function can be characterized by a time-
dependent apparent bandwidth $\Gamma(t)$ that interpolates between $1/T_{2}$
and $1/T_{2}^{*}$ by “filling in” a new spectral strip of width $1/T_{2}$ at a
rate $1/T_{2}$. Thus

$\Gamma(t)=\left\\{\begin{array}[]{lr}1/T_{2}&t\simeq T_{2}\\\
t/T_{2}^{2}&T_{2}\ll t\ll T_{2}^{2}/T_{2}^{*}\\\ 1/T_{2}^{*}&t\gg
T_{2}^{2}/T_{2}^{*}\end{array}\right.$ (9)

To set up the rest of the self-consistency loop we first estimate $T_{1}$ as a
function of $\Gamma(t)$, which (for consistency) must itself be evaluated over
the timescale $T_{1}$ on which a decay process happens. We estimate $T_{1}$
using the Golden Rule: to find a spin at detuning $\Gamma$ one must go a
distance such that $R^{2}\sim 1/\Gamma$, so the matrix element is
$\Gamma^{3/2}$ and therefore, using the Golden Rule,

$1/T_{1}\sim\Gamma(T_{1})^{2}$ (10)

Finally we need to evaluate $T_{2}$. We make this estimate self consistently
following Nandkishore  _et al._ (2014). At some distance $R$ from the central
spin, there are $T^{\gamma}R^{2}$ other thermally excited spins, each flipping
at the rate $1/T_{1}$, and $T^{2\gamma}R^{2}$ spins on the Burin network
flipping at the rate $T^{3\gamma}$. The dipolar coupling at distance $R$ is
$V/R^{3}$. We fix $R$ self-consistently so that
$V/R^{3}=(1/T_{1})T^{\gamma}R^{2}+T^{3\gamma}T^{2\gamma}R^{2}$: once we have
included the effects of thermal fluctuations to this distance, the line has
broadened by enough that it is no longer able to resolve more distant
fluctuations. This yields the ultimate expression

$1/T_{2}\sim(T^{\gamma}/T_{1}+T^{5\gamma})^{3/5},$ (11)

One can check that the set of equations (9,10,11) has the following self-
consistent scaling solution:

$1/T_{1}\sim T^{4\gamma},\quad 1/T_{2}\sim T^{3\gamma}.$ (12)

This seems to be the _only_ self-consistent solution: if we suppose
$\Gamma(T_{1})\sim 1/T_{2}$ then the resulting decay rate is too slow for Eq.
(9) to be consistent, whereas if we assume $\Gamma(T_{1})\sim 1/T_{2}^{*}$ the
decay rate is too fast.

An interesting aspect of these results is that the effect of the Burin network
spins on the decay of typical spins is _marginal_ , scaling the same way in
Eq. (11) as the self-generated contribution to $T_{2}$. Thus the 2D system is
on the cusp of being a self-sustaining metallic state through spectral
diffusion. (However, absent the Burin network there would also be a self-
consistent solution with all decay rates equal to zero, which the present
problem cannot have.)

##  V Dimensional crossovers

We now turn to the crossover between the 2D and 3D behaviors discussed above,
in a slab that has finite thickness $\ell$ in the $z$ direction, but infinite
in the $x-y$ plane. (This can be achieved either by making a finite-width
sample or by imposing a strong magnetic or electric field gradient 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 Khemani
_et al._ (2020); Gromov  _et al._ (2020) would also come into play.)

###  V.1 Slabs at zero temperature

At asymptotically low frequencies, a slab behaves as a two-dimensional system;
in particular, in the orthogonal class, all dipolar excitations are weakly
localized. However, at weak disorder, the localization length diverges
exponentially in the slab width $\ell$. Thus there is a qualitative
distinction between two regimes of dipolar hopping: a regime in which each
dipole finds a resonant partner using the long-range three-dimensional dipolar
interaction _before_ the crossover set by $\ell$, and which acts effectively
like a metal; and a regime in which $\ell$ is shorter than the resonance scale
$R_{c}\sim\exp(c/|\omega|^{\gamma-1})$ [generic case] or $R_{c}\sim
1/\omega^{1/3}$ [marginal case], so that excitations remain tightly localized
at zero temperature. (In the electron glass these crossovers all happen when
the pair radius $r_{\omega}\ll\ell$.)

The resonance scale $R_{c}$ increases monotonically as $\omega$ decreases;
therefore a slab of thickness $\ell$ has a frequency $\omega^{*}(\ell)$ below
which dipoles become manifestly sharp. For the electron glass, at large $\ell$
this crossover always happens in the Shklovskii-Efros regime, with
$\omega^{*}\sim\ell^{-3}$ whereas at small $\ell$ the crossover can happen in
the Mott regime, with $\omega^{*}\sim 1/\log(\ell)$. We emphasize that the
change in apparent dimension is a crossover rather than a true localization
transition, since technically the dipoles are always localized; however, in
two dimensions the localization length in the weakly localized regime is very
large and systems in this regime can be regarded as effectively metallic. We
expect the crossover from weak to strong localization to happen as follows: in
the weakly localized regime, the lineshape of a dipolar excitation consists of
a large number of very finely spaced lines that cannot be resolved in
practice. As the localization length decreases, the lines become less finely
spaced and therefore resolvable. Finally, in the strongly localized limit,
there is essentially just one central line left with appreciable weight. In
contrast, for dipolar systems in the symplectic class, there is the
possibility of a true transition, with localized behavior at small $\ell$
(effectively, stronger disorder) giving way to delocalized behavior at larger
$\ell$ (effectively, weaker disorder).

###  V.2 Slabs at finite temperature

At finite temperature, as we have discussed, Burin resonances and spectral
diffusion play an important role in relaxation. Recall that the line
broadening from spectral diffusion comes mainly from the nearest thermally
active fluctuator. Whether we use the two dimensional or the three dimensional
formula for spectral diffusion depends on whether the distance to the nearest
fluctuator is more or less than $\ell$. Thus, the dimensional crossover for
spectral diffusion happens when the density of thermal excitations
$n_{ex}\approx 1/\ell^{3}$. In contrast the ‘Burin crossover’ happens at
$n_{ex}\approx W/(Jn\ell^{3})$, 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 $n$ the density of dipoles on the lattice. Thus, we enter
the ‘two dimensional’ regime for the Burin analysis at a higher temperature
than that at which we enter the ‘two dimensional’ regime for spectral
diffusion.

In this intermediate temperature range, the coupling to the nearest thermally
active neighbor scales as $T^{\gamma}$ (setting $T_{2}^{*}$), but the nearest
“network” spin is much further, since the percolating network density takes
its two-dimensional value $T^{3\gamma}$. The spin-echo lifetime $T_{2}$
crosses over at a timescale that is intermediate between these limits. The
two-dimensional self-consistent analysis we developed above can be extended to
this situation, but there are many cases (depending on whether each scale is
three-dimensional or two-dimensional) and we will not consider these in
detail.

##  VI Crossovers near maximum polarization

In experiments involving cold atoms, nitrogen-vacancy centers, etc. it is
often more natural to tune the polarization than the temperature Ho  _et al._
(2017). Since the dipolar interaction in these systems can generically be
treated in the secular approximation, it approximately conserves the total
number of spin-up dipoles. Thus one can regard high-polarization ensembles as
effectively being at large chemical potential $\mu$. The analysis above
carries over with minor changes to this setup, if we replace the temperature
with the density of minority-state spins, and set the density-of-states
exponent $\gamma=1$.

There is one distinction worth noting: at low temperature, the thermal
excitations are disproportionately in low-energy, well localized states,
whereas for high polarization they are in random states. Therefore at large
polarization there is a potential additional relaxation channel due to
delocalized high-energy excitations. This does not, however, matter in the
strongly disordered regime considered here, since all single-particle states
are assumed to be well localized.

##  VII Discussion

We have determined the frequency and temperature dependence of relaxation
rates for systems of quantum dipoles, whether microscopic or emergent (as in
the electron glass). At zero temperature, dipoles relax via long-range
coherent hopping, which is possible in three dimensions (where long-range
hopping resonances proliferate) but not in two dimensions. In three
dimensions, therefore, relaxation occurs at zero temperature. For microscopic
dipoles (and the electron glass in its intermediate-temperature Mott regime),
quasiparticle lifetimes grow exponentially as their frequency goes to zero, so
local dipoles are asymptotically sharp excitations. For the electron glass at
sufficiently low temperatures (i.e., in its Shklovskii-Efros regime), or for
dipolar spin ensembles that are at maximum polarization rather than in their
ground state, the excitation lifetimes are marginal, scaling the same way as
the excitation frequency up to log corrections. For the electron glass
specifically, these log corrections enhance dipole hopping (see Appendix), and
local dipolar excitations are no longer the right variables to think about the
problem at exponentially low frequencies and temperatures. (What the ‘right’
degrees of freedom are is not clear to us, but exotic physics could
conceivably be involved Potter  _et al._ (2012).) Finite-temperature effects
provide an additional transport channel.

Although energy transport is (apparently) present in the zero-temperature
limit, and local particle-hole excitations relax, the dc electrical
conductivity of the electron glass is zero in linear response, since moving
charge across the system requires _uphill_ transitions, which a zero-
temperature environment cannot facilitate. The dynamics of realistic electron
glasses at very low frequencies involve subtleties that go beyond the
approximations we made here, and would be interesting to revisit for a more
complete 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 infinitely sharp at all frequencies in the zero-
temperature limit, and delocalize at finite temperatures only through subtle
interaction effects. We found that the $T_{1}$ and $T_{2}$ times both scale as
power laws of temperature, but with distinct exponents. “Diagonal” interaction
effects (governed by the timescale $T_{2}^{*}\sim T^{-3\gamma/2}$ are
parametrically stronger than relaxation effects $T_{1,2}\gtrsim T^{-3\gamma}$,
making dipoles in two dimensions at low temperatures a promising platform for
exploring dynamical signatures associated with many-body localization. In
quasi-two dimensional systems (slab geometries) there is a crossover from
three dimensional behavior at high frequency/temperature to two dimensional
behavior at low frequency/temperature, with crossover scales that we have
identified. Our results in all of these cases are collected in Table 1.

It would be interesting to go beyond the scaling theory outlined herein to
also calculate (potentially logarithmic) prefactors. It would also be
interesting to determine the relation between relaxation times (calculated
herein) and transport coefficients. We have also identified certain regimes
where the ‘dipolar’ approach employed herein breaks down - understanding the
behavior in these regimes is an important open problem. Finally, thermopower
coefficients might also be interesting to explore. We leave these problems to
future work.

###### Acknowledgements.

We acknowledge useful conversations with Peter Armitage, Mikhail Lukin, and
Markus Müller. This material is based in part (RN) upon work supported by the
Air Force Office of Scientific Research under award number FA9550-20-1-0222.
S.G. acknowledges support from NSF DMR-1653271.

##  Appendix A Crossovers in the 3D electron glass

We now explore, in some more detail, the dynamical crossovers in the three-
dimensional electron glass at zero temperature. While the scaling of the
relaxation rates was addressed in the main text, here we also provide some
discussion of the prefactors, which play an important part in determining the
crossover timescales.

####  A.0.1 Mott regime

Let us start by considering the dynamics in the Mott regime (assuming it
exists). In the language of the single-particle dipolar hopping model (2), we
are concerned with the dynamics of a dipolar excitation that is initially
localized on resonant pair $\alpha$ (which has frequency
$\mathcal{E}_{\alpha}=\omega$). We have postulated weak interaction i.e. the
disorder is strong enough that nearest-neighbor hops are off shell. Instead,
an excitation at frequency $\omega$ can only hybridize with a partner at
similar (or lower) frequency. Excitations in the frequency window
$(0,2\omega)$ occur with density $\sim\omega^{2}n\xi r_{\omega}^{2}/W^{2}$,
where $n$ is the dipole density. Thus the spacing between two nearby such
excitations scales as $\omega^{-2/3}$, and the dipolar interaction matrix
element between them scales as $\sim Vr_{\omega}^{2}\omega^{2}n\xi
r_{\omega}^{2}/W^{2}$. Generically, however, the detuning between these
excitations is $\sim\omega$ (by construction). Asymptotically, therefore,
excitations at low frequencies in the Mott regime will be off-shell with their
near neighbors. and will be parametrically longer-lived. One can estimate the
characteristic excitation lifetime as follows. Starting from a typical spin
the total number of resonances nearer than a distance $R$ scales as $\log R$.
An excitation at frequency $\omega$ finds a resonant partner when $\log R\sim
W^{2}/(Vr_{\omega}^{2}\omega n\xi)$. The characteristic interaction energy at
this scale, and therefore the excitation lifetime, also scale as
$\Gamma_{\omega}\sim V\exp(-\tilde{c}/|\omega|)$, i.e., the excitation
lifetime is nonperturbative in $\omega$.

Could there be a regime with $\omega<Vr_{\omega}^{2}\omega^{2}n\xi
r_{\omega}^{2}/W^{2}$, such that typical excitations at $\omega$ were
resonant, and the relaxation rate simply followed the typical matrix element
between such excitations ($\sim\omega^{2}$)? It is straightforward to see this
is not possible. The condition above can be expressed (dropping logarithmic
factors which are not large by postulate) as
$\omega>\frac{W}{Vn\xi^{2}}\frac{W}{\xi^{3}}$, but the first term on the right
hand side is larger than one (by the weak interaction postulate), and the
second term is larger than $\delta_{\xi}$ (since we need $W>t$ to have single
particle states localized), so we conclude that the above condition is only
satisfied for frequencies larger than $\delta_{\xi}$, but $\delta_{\xi}$ was
the upper cutoff for our theory. We thus conclude that the scaling
$\Gamma_{\omega}\sim\exp(-\tilde{c}/|\omega|)$ holds throughout the regime
$V/r_{\omega}<\omega<\delta_{\xi}.$

####  A.0.2 Shklovskii-Efros regime

The Mott regime crosses over to the Shklovskii-Efros regime at a frequency
such that $\omega_{SE}r_{\omega_{SE}}\simeq V$. In the Shklovskii-Efros
regime, Coulomb interactions rearrange the ground state such that any resonant
pair split by $\omega<\omega_{SE}$ with either of its partners within
$V/r_{\omega}$ of the Fermi energy is singly occupied in the ground state. One
can estimate the density of states of pairs at frequency $\omega$ as follows.
The first half of the pair must be within $V/r_{\omega}$ of the Fermi energy,
and the density of such sites scales as $V/[Wr_{\omega}]$. Its partner is in a
shell of radius $r_{\omega}$ and thickness $\xi$; multiplying these, we get
that the density of dipoles of transition frequency $\omega$ scales as
$n(\omega)\sim Vn\xi r_{\omega}/W^{2}$.

We now consider diagonalizing the Hamiltonian (2) in the Shklovskii-Efros
regime for states in the frequency window $(\omega/c,c\omega)$ where $c$ is
some arbitrarily chosen number of order unity. These dipoles occupy a fraction
of sites $\sim\omega n(\omega)$, and have a characteristic dipole moment
$r_{\omega}$. The interaction between adjacent dipoles in this frequency
window is $Vr_{\omega}^{2}\omega n(\omega)=\omega
r_{\omega}^{3}V^{2}\xi/W^{2}\sim\omega|\log^{3}\omega|$. For small enough
$\omega$, therefore, the interactions among nearest-neighbor dipoles at
frequency $\omega$ become strong enough that they are _generically_ resonant.
Below this crossover scale, set by
$\omega_{c}\sim\exp\big{(}-(W/V\xi^{2})^{2/3}\big{)}$, dipolar excitations are
strongly coupled, relative to their natural frequency. Perturbation theory in
the dipole-dipole interaction therefore ceases to apply. Since the dipoles are
no longer the true excitations of the system, we expect them to decay on a
timescale determined by the nearest neighbor dipole-dipole interaction.
However, the nature of the true ground state, and its thermal transport
properties, are delicate questions that are outside the scope of the present
work. It is also worth bearing in mind that this lowest frequency regime
occurs only at exponentially low frequencies (thus, exponentially long
timescales) and may be hard to see in practice because of e.g. dephasing due
to coupling to phonons or an external environment.

Before this crossover at very low frequencies, we expect to see an extensive
parameter range in which the ratio of interactions to bandwidth is essentially
constant. This is the regime that was recently identified as the “marginal
Fermi glass” Mahmood  _et al._ (2021) and seen in THz spectroscopy. In this
marginal regime, a dipole finds a resonant partner on a scale $\log R\sim
W^{2}/(V^{2}r_{\omega}^{3}\xi)$ that (up to the overall density of the
network) does not depend on $\omega$. In systems without a well-developed Mott
regime, $R$ is instead just the spacing between adjacent dipoles. Putting
these two cases together and neglecting logarithmic corrections, one gets

$\Gamma(\omega)\simeq\left\\{\begin{array}[]{lr}\omega\xi^{4}(V/W)^{2}&W/V\xi^{2}=O(1)\\\
\omega\exp(-c/\omega_{SE})&W/V\xi^{2}\gg 1.\end{array}\right.$ (13)

This is the characteristic “marginal” scaling of the relaxation rate
identified in Mahmood  _et al._ (2021).

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