footnote

##  Chapter 0 Quantum Glasses

###  1 Introduction

The interplay of disorder, interactions and quantum mechanics leads to a host
of interesting phenomena. A particularly intriguing aspect is the breakdown of
ergodicity in a way that differs significantly from that of standard
spontaneous symmetry breaking. This can arise as a consequence of quenched
disorder, as in spin glasses, or structural disorder, think of amorphous
density order. Both quenched and self-generated disorder entail a clustering
in phase space associated with glass attributes like memory effects and
extremely slow relaxations [1, 2, 3]. The replica method allows one to access
the phase space structure, and replica symmetry breaking (RSB) signals the
existence of a large number of metastable states separated by substantial free
energy barriers. Even though barriers may be finite in low dimensions,
preventing a genuine thermodynamic glass transition, the replica method has
proven useful for these cases as well.

As a result of frustration and the competition between many nearly degenerate
ordering patterns, the glassy order is often rather soft and gives rise to an
unusually high density of low energy excitations. In or close to a quantum
glass phase those may have a strong influence on electronic transport and have
often been invoked as candidates to explain the non-Fermi liquid behavior of
the resistivity of complex materials, such as high temperature superconductors
(cuprates), heavy fermion systems, but also systems close to a Mott-Anderson
metal-insulator transition.

Apart from this “glassy” route, very slow dynamics and non-ergodicity in
quantum systems can be due to other mechanisms too, especially in low
dimensions where the effects of disorder are very strong. On the one hand,
under a real space renormalization group treatment, disordered quantum spin
chains typically flow to fixed points characterized by infinite randomness
[4], with extended regions of parameter space where Griffiths–McCoy
singularities are very important [5, 6]. Thereby the low frequency response is
dominated by nearly isolated degrees of freedom only weakly coupled to others
further away.

On the other hand, non-interacting quantum particles in one dimension subject
to a disordered potential Anderson localize and stop diffusing on the scale of
the elastic mean free path [7, 8]. Its interacting counterpart, many-body
localization (MBL) [9, 10, 11, 12], relies on the discreteness of non-
interacting excitations and the rareness of resonant interactions, which then
fail to induce diffusion of energy. Thus, MBL can only occur in isolation,
with no coupling to the continuum of a bath which would reinstate transport
and ergodicity. In contrast, the above discussed glass phases are stable
against a bath, as they rely on the rugged structure of the free-energy
landscape. A further difference between quantum glasses and MBL systems is
that the latter are highly susceptible to local inclusions where disorder is
weak. Those act as sources of dynamical chaos, and it is believed that only in
low-dimensional, discrete lattices such nucleation centers of local baths do
not destroy MBL [13, 14, 15] (although the existence of MBL even in 1d has
been questioned recently [16, 17, 18]). In contrast, glasses, like standard
symmetry broken phases, are stabilized in high dimensions (and RSB is certain
to occur above a $d_{c}$ [19]). In summary, while both quantum glassiness and
MBL entail non-ergodicity and impede full thermalization, they do so for
fundamentally different reasons.

Other obstructions to full ergodicity exist, e.g., integrable systems with an
extensive number of conserved quantities [20], systems hosting many-body scars
(a set of eigenstates that violate the eigenstate thermalization hypothesis)
[21] or cases with fragmented Hilbert spaces [22]. However, none of these is
robust against generic perturbations of the Hamiltonian, as even modest ones
reinstate equilibrating dynamics rather quickly. We do not consider them as
quantum glassy and we will not discuss them here. A further way to kill
ergodicity is the sufficiently strong coupling to an Ohmic bath of a single
quantum degree of freedom, which can get dressed with many bath modes, so as
to acquire an infinite mass, and its dynamics freeze out [23, 24]. We shall
not discuss this effect of non-glassy “local ergodicity” breaking either,
except for noting that the coupling of quantum systems to a bath usually
results in the reduction of fluctuations, and thus extends the regime of
stability of the glass [25, 26].

In this review we focus on quantum glasses that arise in high dimensions,
where disorder is so relevant as to produce amorphous order and phase space
clusterization, but does not flow to infinite randomness. This structure
withstands both thermal and quantum fluctuations, which soften the energy
landscape and eventually melt the glass. For certain types of glasses the
quantum melting differs drastically from the thermal route.

Since this chapter is part of a book devoted to replica symmetry breaking, we
focus the discussion on equilibrium aspects and we structure it as follows. In
Sec. 2 we overview some general aspects of quantum glassy dynamics. Section 3
reminds the gist of the replica treatment of mean field glasses. In Sec. 4 we
classify different universality classes of (mean field) glasses according to
the type of interactions and environments, and discuss their phenomenology, in
particular the nature of their low energy collective modes which we obtain
both from Landau theory and an effective potential approach in Sec. 5. In Sec.
6 we discuss whether an isolated quantum glass can explore its clusterized
phase space by tunneling or stays localized. The interplay between the glass
and localization routes to non-equilibrium is a multi-faceted topic briefly
reviewed in Sec. 7. Section 8 presents some experimental glassy materials.
Finally, in Sec. 9 we survey fermionic models closely connected to the
paramagnetic phase of Heisenberg spin glasses, a.k.a. the Sachdev-Ye-Kitaev
(SYK) model, that may fertilize the study of spin glasses via interesting
dynamical analogies.

###  2 Dynamics in quantum glassy phases

A particularly interesting aspect of quantum glasses is their dynamics, both
in and out of equilibrium. We broadly survey them here and discuss selected
questions in more detail in later sections.

####  1 Short-time dynamics

Distinct quantum aspects of glasses appear on microscopic time scales. The
high-frequency dynamics depends crucially on the symmetries of the order
parameter and the nature of the quantum fluctuations. In the low-frequency
limit, instead, the excitations feel the softness of the glassy landscape,
which, in the mean-field limit, exhibits surprisingly universal spectral
features whose frequency scaling merely depends on the glass being metallic or
insulating. We will review the salient results in Secs. 4, 5 and 9.

####  2 Moderately long-time dynamics

The intermediate to long-time dynamics of quantum glassy systems are
essentially identical to those of their classical counterparts. Indeed, one
expects that on time scales $t\gg\hbar/T$ all degrees of freedom decohere and
any quantum effects are washed out. After a quench from the disordered phase,
the glass hovers over saddles in the free energy landscape and slowly relaxes
its energy without ever getting deeply trapped into rare deep minima, in a
manner dubbed weak ergodicity breaking [27, 28, 29, 30, 31]. The response
slows down increasingly with time (the glass “ages”). In solvable mean field
models, the fluctuation-dissipation relation evaluated on those long time
scales takes the classical form, albeit with an effective temperature $T_{\rm
eff}$ that exceeds the bath temperature (even at $T=0$). The latter is
formally related to Parisi’s order parameter $Q(x)$ as in the classical limit
[1]. Moreover, the equations governing the slow evolution of the response and
correlation functions are invariant under time reparametrizations, $t\to f(t)$
[32]. Such an invariance also occurs in the large $M$ limit of the $SU(M)$
Heisenberg and the SYK models. Recent progress in this direction will be
reviewed in Sec. 9. Finally, as in the classical limit, one can tune the
initial state to be within one of the valleys in the free-energy landscape and
show that the subsequent dynamics will remain confined to it (at least for
times exponentially large in system size).

A short summary of the real-time dynamics of quantum disordered systems on
these time scales can be found in Ref. [33]. We will not extend this
discussion further here, our aim being to focus mainly on equilibrium
properties.

####  3 Extremely long-time dynamics

On extremely long time scales, crossing over or tunneling under barriers
between metastable valleys should be possible. In mean-field models barriers
scale with system size and thermal activation over them is exponentially slow.
Since the latter is the root cause for the failure of simulated annealing in
finding the ground state of NP-hard problems, there was hope that quantum
annealing could achieve a significant speed-up [34]. The idea was to
initialize the system under strong quantum fluctuations (e.g. a transverse
field) which induce a simple ground state. Then, by invoking the adiabatic
theorem, one should reach the classical ground state upon turning off the
fluctuations sufficiently slowly. The caveat is that exponentially slow
switch-off rates are required to follow the ground state while it delicately
hybridizes distant low-energy valleys in phase space. Still, one may ask which
of the exponentially long times (barrier crossing or adiabatic ground state
preparation) is more efficient to explore complex landscapes [35], an idea
that was explored experimentally in quantum Ising spin glasses [36], cf. Sec.
1. However, an even bigger obstacle to quantum annealing [37, 38] is posed by
the fact that NP-hard problems have quantum first order transitions [39, 40]
(see Sec. 1).

A related dynamical question arises for an isolated quantum spin glass – the
setting considered in MBL: after preparation in a deep valley, can it tunnel
to others or does it remain localized? We discuss it in Sec. 6.

###  3 Replica treatment of quantum mean-field glasses

Most theoretical insights on quantum glasses have been obtained from mean-
field $SU(M)$ models, where all $i=1,\dots,N$ spins interact with all others
via random couplings with variance proportional to $J^{2}$ and conveniently
scaled with $N$ and $M$. While this looks remote from any real physical glass,
long range couplings can be emulated via light-matter coupling in multi-mode
cavities, see Sec. 3. Their equilibrium properties were derived with a
replicated imaginary-time path integral formulation of the partition function
[41]. A saddle-point evaluation allows one to define the glass order
parameter111The normalization factor $M^{2}$ captures the leading dependence
for $M\to\infty$. In general one should naturally divide by the number of spin
components ($M^{2}-1$).

$Q_{ab}(\tau)={\frac{1}{NM^{2}}}\sum_{i}[\langle{\rm
T}\,{\boldsymbol{\sigma}}_{i}^{a}(\tau)\cdot{\boldsymbol{\sigma}}_{i}^{b}(0)\rangle]$
(1)

where $a,b=1,\dots,n$ are the replica indices, $\tau$ the imaginary time and
${\rm T}$ the time-ordering operator. Depending on the complexity of the
model, the free-energy density can either be just a functional of the saddle-
point $Q_{ab}(\tau)$, or the partition function can be transformed into one
for a single spin with $Q_{ab}(\tau)$ self-consistently given by the
expectation value of its imaginary-time correlations (see, e.g., [42]). In all
cases, $Q_{a\neq b}$ are $\tau$-independent, vanish in the disordered phase,
and adopt an RSB structure in the glass phase. The diagonal elements
$Q_{aa}(\tau)\equiv q_{d}(\tau)$ are independent of the replica index but
depend on time, being $\beta\hbar$ periodic, in the whole phase diagram. In
certain cases (typically models in which also the single-spin partition
function can be calculated exactly), $Q_{ab}$ satisfies a generic differential
equation of the form

${G_{0}^{-1}(\tau)}_{ac}Q_{cb}(\tau)=\delta_{ab}\delta(\tau)+\int_{0}^{\beta\hbar}d\tau^{\prime}\;\Sigma_{ac}(\tau-\tau^{\prime})Q_{cb}(\tau^{\prime})\;,$
(2)

with a sum over repeated indices. ${G_{0}^{-1}}$ is a differential operator,
typically diagonal in replica indices, and its order (first or second time
derivative, further time dependence) depends on the kind of degrees of freedom
and whether the system is coupled to a bath (microscopic dynamics). $\Sigma$
is a self-energy which typically depends on time only via $Q$. Note though
that this equation is not causal, and is simpler to solve in frequency space.
We will specify $G_{0}^{-1}$ and $\Sigma$ in several concrete cases below. The
physical properties follow from $Q_{ab}$. Of special importance is the out-of-
phase local susceptibility

$\chi_{\rm loc}^{\prime\prime}(\omega)={\rm Im}\,\tilde{q}_{d}(\omega+{\rm
i}0^{+})\;,$ (3)

with $\tilde{q}_{d}$ the Fourier transform of $q_{d}$. The right-hand-side is
the analytic continuation to real $\omega$ of $\tilde{q}_{d}(\omega_{n})$,
with $\omega_{n}$ the Matsubara frequencies. $\chi^{\prime\prime}_{\rm loc}$
contains essential information on the spectrum of collective excitations since

$\displaystyle\chi_{\rm
loc}^{\prime\prime}(\omega)=\frac{\pi}{N}\sum_{im}|\langle\psi_{m}|\sigma_{i}^{1}|\psi_{0}\rangle|^{2}\left[\delta(\omega-
E_{m}+E_{0})-\delta(\omega+E_{m}-E_{0})\right],$

(at $T=0$) where $(\psi_{m},E_{m})$ are the many-body eigenstates and
energies, with $0$ referring to the ground state. A continuous glass
transition is identified by the condition $J\chi_{\rm loc}=1$, with $\chi_{\rm
loc}$ the zero frequency limit of the local susceptibility, $\chi_{\rm
loc}=\int_{0}^{\beta\hbar}d\tau\,q_{d}(\tau)$.

###  4 Classification of mean-field quantum glasses

The local degrees of freedom, their interactions and the environment they
couple to define different mean-field classes, distinguished by their phase
diagrams and properties. We list three distinguishing aspects of spin models
and discuss them in turn below.

  1. –

The number of interacting spins decides on the organization of phase space as
elucidated, for instance, by the replica analysis and its symmetry breaking
scheme. This distinction drastically affects the nature and the order of the
quantum glass transition.

  2. –

The symmetry group in spin space is important. It matters whether the various
spin-spin interaction terms mutually commute (as in Ising and rotor models) or
not (as in Heisenberg systems). In the latter case quantum fluctuations are
substantially stronger, entailing a weaker glassy order and softer
excitations, both at low and high $T$.

  3. –

The low-frequency dynamics depend on whether the quantum glass is embedded in
an insulating, gapped host, or couples to a gapless bath with Ohmic spectrum,
as in metallic glasses. For a given environment – insulating or metallic – the
dynamically relevant states of all mean-field models, in spite of quantitative
differences, display a striking universality in their low-frequency response
within the glass phase (Sec. 5).

Note that these spin glass models do not cover all possible glassy quantum
systems. Some other cases are pinned elastic systems (including vortex
lattices in superconductors, Wigner crystals, charge and spin-density waves or
disordered liquid crystals) in which frustrating disorder originates, e.g.,
from substrate impurities [43]. The interested reader may consult Ref. [44].
Nonetheless, once coarse-grained at the collective pinning scale, these
systems resemble short-range coupled spin systems in random fields.

####  1 Pair vs multi-spin interactions

##### Pair interactions ($p=2$) - continuous glass transition

Usually one considers Ising-like interactions when discussing the distinction
between pair and multi-spin interactions. Ising-type glasses arise in numerous
contexts, as they describe generic interacting two level systems (TLS). The
simplest representative is the transverse field Ising model (TFIM)

$\hat{\mathcal{H}}=-\sum_{ij}J_{ij}\,\hat{\sigma}_{i}^{z}\hat{\sigma}_{j}^{z}+\sum_{i}\,{\mathbf{H}}_{i}\cdot\hat{\boldsymbol{\sigma}}_{i}$
(4)

where
$\hat{\boldsymbol{\sigma}}_{i}=(\hat{\sigma}^{x}_{i},\hat{\sigma}^{y}_{i},\hat{\sigma}^{z}_{i})$,
with the usual Pauli matrices acting in the Hilbert space of the $i$’th TLS.
In finite dimensions the first sum extends over nearest neighbours of a
lattice. The interaction strengths $J_{ij}$ are drawn from a probability
distribution, $P(J_{ij})$, typically chosen to be Gaussian with zero mean
$[J_{ij}]=0$ and variance $[J_{ij}^{2}]=J^{2}/(2c)$. $J$ is $O(1)$ and $c$ is
the connectivity of the lattice. Mean values over $P$ are indicated with
square brackets. The ${\mathbf{H}}_{i}$ are local fields. Most often one
restricts to a homogeneous transverse field,
${\mathbf{H}}_{i}=\Gamma{\mathbf{e}}_{x}$. Since the last term does not
commute with the interactions, it induces non-trivial quantum dynamics. In the
absence of longitudinal fields $H_{i}^{z}$, the Hamiltonian possesses an Ising
symmetry which is spontaneously broken at sufficiently low $T$ and weak
transverse fields $\Gamma<\Gamma_{c}$. For $H_{i}^{z}\neq 0$ (as in many
experimental systems of interest) the only potential phase transition is an
RSB one, which however occurs only in high enough dimensions.

Allowing each spin to interact with all others, $c\to N-1$, places the spins
on a complete graph. The scaling of the variance, $[J_{ij}^{2}]\approx
J^{2}/(2N)$, ensures a non-trivial thermodynamic limit, $N\to\infty$. The
Hamiltonian becomes the quantum extension [41] of the Sherrington-Kirkpatrick
(SK) spin-glass (TFSK), which is tractable with the replica trick. Again, only
the replica diagonal, $q_{d}(\tau)\equiv Q_{aa}(\tau)=[\langle{\rm
T}\sigma^{z}_{a}(\tau)\sigma^{z}_{a}(0)\rangle]$ depends on the imaginary time
$\tau$, while the RSB structure of the off-diagonal elements remains basically
the same as in the classical limit. Glassy freezing is signalled by a finite
Edwards-Anderson parameter $q_{\rm EA}\equiv\lim_{\tau\to\infty}q_{d}(\tau)$
(at $T=0$). The resulting single-site problem, with the coupling to the rest
encapsulated self-consistently in $Q_{ab}$, is of similar difficulty as the
impurity problems of Dynamical Mean Field Theory [45].

From the stability criterion $1=J\chi_{\rm loc}$, a $(T/J,\,\Gamma/J)$ phase
diagram with a second order phase transition between a paramagnetic and a
spin-glass phase is predicted [41]. Quantum fluctuations depress the
transition temperature but do not destroy the transition. For $\Gamma\to 0$
the classical SK $T_{c}=J$ is recovered and for $T\to 0$ a quantum critical
point at $\Gamma_{c}=1.52J$ is found [46]. Order sets in when the entropy loss
and/or the loss in transverse field energy are compensated by the gain in
interaction energy. Close to the phase transition both scale like $O(m^{2})$,
where $m\ll 1$ is a typical ordered moment of a single spin. The glass
transition, whether classical or quantum, can thus be viewed as a condensation
into the first magnetization mode to become unstable. The glass transition
differs crucially from standard ordering phenomena, however, because a large
number of modes turn soft almost simultaneously. This entails many local
minima with different ordering patterns. Moreover, throughout the entire glass
phase there are permanently some gapless, critical modes at the verge of
condensing.

Interestingly enough, a “static approximation” in which $q_{d}(\tau)$ is
replaced by a $\tau$-independent variational parameter captures the essence of
the phase transition. A more detailed dynamical analysis for the Ising case
[47] and related rotor models with $M\gg 1$ components [48] shows that the
spectral gap closes as
$\Delta\sim[(\Gamma-\Gamma_{c})/\ln(\Gamma-\Gamma_{c})]^{1/2}$ at
$\Gamma_{c}$, assuming a gapless spectral function $\chi^{\prime\prime}_{\rm
loc}(\omega)\sim\omega$, associated with algebraic decay in imaginary time,
$q_{d}(\tau)\sim 1/\tau^{2}$.

Determining $Q_{ab}$ and $q_{d}(\tau)$ deeper in the glass phase requires the
solution of a quantum impurity problem in a frozen field whose distribution is
controlled by full RSB in the off-diagonal $Q_{ab}$ [49]. The latter ensures
marginal stability, which in turn implies a gapless spectrum everywhere in the
glass phase. Moreover, the limit of small $\Gamma$ admits a scaling form for
the mean field solution. From this it follows that the low-frequency spectral
function becomes independent of $\Gamma$,

$\displaystyle\chi_{\rm loc}^{\prime\prime}(\omega;\Gamma\ll
J)=0.59\,\omega/J^{2},$ (5)

which we will interpret physically in Sec. 5.

#####  $p>2$ \- discontinuous onset of glassy order

Models in which $p>2$ classical spins interact simultaneously are particularly
interesting since at the mean field level their dynamics are described by
equations identical to those arising in the Mode-Coupling Theory of structural
glasses. It is therefore believed that they capture the physics of the
structural glass transition and the glassy phase [1, 2]. Moreover, such multi-
spin Hamiltonians are ubiquitous in combinatorial optimization problems when
translated into questions about ground states of statistical mechanics
systems. Defined on diluted graphs, the $p$-spin models are, e.g., closely
connected to the $K$-satisfiability problem [50].

A typical multi-spin Hamiltonian reads [51, 52, 40, 53, 25, 26]

$\hat{\mathcal{H}}=-\sum_{i_{1}\neq\dots\neq i_{p}}J_{i_{1}\dots
i_{p}}\hat{\sigma}^{z}_{i_{1}}\dots\,\hat{\sigma}^{z}_{i_{p}}+\sum_{i}\Gamma_{i}\hat{\sigma}_{i}^{x}\;,$
(6)

with the parameter $p$ taking any integer value $p\geq 3$. The model is
defined on a hypergraph, the sum running over all $p$-uplets. The couplings
are random independent variables with variance $p!{J}^{2}/(2N^{p-1})$.

The important difference w.r.t. pairwise interacting models lies in the
scaling of the loss of entropy or transverse field energy as compared to the
interaction energy gain with a putative small emerging magnetization pattern
$m_{i}=\langle\sigma_{i}^{z}\rangle$. The former still scale as $O(m^{2})$,
but the energy gain is only $O(m^{p})$ and cannot compensate for this loss if
$m$ is small. The only possibility for a phase transition is that the order
parameter, $q_{\rm EA}=1/N\sum m_{i}^{2}$ jumps discontinuously to a finite
value at a critical point. Technically, this is reflected by a one-step RSB.
Part of the liquid entropy is transferred to configurational entropy without a
thermodynamic phase transition (the free energy is smooth). The Gibbs weight
distributes over an exponential number of metastable states. When quantum
fluctuations are switched on, and temperature is sufficiently low, the
transition necessarily has to change nature since the configurational entropy
cannot contribute to the free energy. The phase transition thus becomes truly
first order [29] and the ground states on the two sides of the transition are
essentially unrelated. For this reason quantum annealing is not expected to be
useful for $(p>2)$-spin systems and other systems with a 1-step RSB.

In quantum and classical glasses of this type alike, the proliferation of
metastable states affects the dynamics, but does not necessarily change the
static observables. Over a finite range of free-energy densities one finds
exponentially many local minima, the fewer the lower the free-energy. As
temperature decreases the Gibbs weight concentrates on lower energy states. A
thermodynamic freeze-out transition occurs once the lowest $O(1)$ states
dominate. Consequently, the dynamical glassy phase extends beyond the
thermodynamic one [54, 27, 28].

The $p$-spin potential was also used to model a quantum particle in a random
potential, in which the interactions are between $p$-uplets of coordinate
positions on an $N$-dimensional sphere. Conventional kinetic energy is given
to the particle. In these cases an equation of the kind of Eq. (2) applies
with ${G_{0}^{-1}}_{\\!\\!ab}=(d_{\tau}^{2}+\mu)\delta_{ab}$ with $\mu$ a
Lagrange multiplier enforcing the spherical constraint and
$\Sigma_{ab}=J^{2}Q^{\bullet(p-1)}_{ab}$ where $\bullet$ denotes a normal
power. Self-energies of this very same kind appear in the SYK model, Sec. 9.

####  2 Commuting vs non-commuting interactions

##### Commuting interactions

The disordered quantum rotor model is defined by [48]

$\hat{\mathcal{H}}=\frac{g}{2M}\sum_{i}\hat{\mathbb{L}}_{i}^{2}+\frac{M}{\sqrt{N}}\sum_{i<j}J_{ij}\,\hat{\mathbf{n}}_{i}\cdot\hat{\mathbf{n}}_{j}\qquad\;\;\hat{\mathbf{n}}_{i}^{2}=1\;\;\forall
i\;.$ (7)

The $M$ components $\hat{n}_{i}^{\mu}$ of the $i$th unit-length rotor
$\hat{\mathbf{n}}_{i}$ commute with each other, unlike the components of
quantum spins. As a consequence, all interaction terms mutually commute.
$\hat{L}_{i}^{\mu\nu}$ (with $\mu<\nu$, $\mu,\nu=1,\dots,M$) are the
$M(M-1)/2$ components of the angular-momentum generator $\hat{\mathbb{L}}_{i}$
in rotor space, and
$[L_{i}^{\mu\nu},n_{j}^{\sigma}]=i\delta_{ij}(\delta_{\mu\sigma}n_{j}^{\nu}-\delta_{\nu\sigma}n_{j}^{\mu})$.
The $J_{ij}$ are $O(1)$ randomly distributed uncorrelated exchange constants.
As $g\to 0$ the model reduces to the classical, infinite-range, $M$-component
spin glass, the limit $M\to 1$ being very similar to the Ising model.

There is no Berry phase in the real action of the path integral representation
of the partition function, and the $O(M)$ symmetric saddle-point [48] in the
$M\to\infty$ limit yields a self-consistency equation, Eq. (2), with
${G_{0}^{-1}}_{\\!\\!ab}=g^{-1}(d^{2}/d\tau^{2}+\mu)\delta_{ab}$, where $\mu$
enforces the constraint $\hat{\mathbf{n}}_{i}^{2}=1$, and one has
$Q_{ab}=q_{d}\delta_{ab}$ and $\Sigma_{ab}=Q_{ab}$. In the glass phase the
same equation is obeyed by $q_{d}(\tau)-q_{\rm EA}$, while $Q_{a\neq b}=q_{\rm
EA}$ becomes non-zero, without however breaking the replica symmetry. The
integro-differential equation is easy to solve in frequency space and yields
spectral functions that we will review in Sec. 5. The quantum critical
behavior was found to be in the TFSK universality class, which was
rationalized by showing that $1/M$ corrections do not modify the critical
exponents and the low frequency spectrum.

##### Non-commuting interactions - $SU(M)$ spins

The most relevant model in this class is the quantum Heisenberg spin glass,

$\hat{\mathcal{H}}=-\sum_{i\neq
j}J_{ij}\,\hat{\boldsymbol{\sigma}}_{i}\cdot\hat{\boldsymbol{\sigma}}_{j}\;.$
(8)

A static approximation used to evaluate the instability condition for $S=1/2$
[41] yields a surprisingly good estimate for $T_{g}\approx\sqrt{3}/12\approx
0.14\,J$, obtained with Quantum Monte Carlo calculations [55]. The substantial
reduction with respect to the classical $T_{g}=J/4$ is due to quantum
fluctuations.

With the static approximation, though, one cannot access the spin dynamics. A
crucial step forward was taken in [56] where $SU(2)$ was promoted to $SU(M)$
and a Schwinger boson representation of the spins, with the number of bosons
constrained to be $n_{b}=SM$, was adopted. A first analysis in the $(M,S)$
parameter space was performed in the large $M$ limit. Soon after, a complete
solution with $M\to\infty$ and fixed $S$ was derived, and global aspects, some
of them also valid for $SU(2)$, discussed [57, 58]. The solution is formulated
in terms of the Green functions of the bosons,
$G^{ab}_{B}(\tau)\equiv-M^{-1}\sum_{\mu}[\langle{\rm
T}\hat{b}_{\mu}^{a}(\tau)\hat{b}_{\mu}^{b{\dagger}}(0)\rangle]$ and their
self-consistency equations are again of the form of Eq. (2) with

$\begin{split}{G_{0}^{-1}}_{\\!\\!ab}=(d_{\tau}+\mu^{a})\delta_{ab}\;,\qquad\Sigma_{B}^{ab}(\tau)&=J^{2}G^{ab}_{B}(\tau)G^{ab}_{B}(\tau)G^{ab}_{B}(-\tau)\;.\end{split}$
(9)

$\mu^{a}$ is a chemical potential which fixes $G_{B}^{aa}(\tau=0)=-S$. The
disorder-averaged spin-spin correlator is obtained from $G_{B}$ by

$\displaystyle{q_{d}}(\tau)\equiv\frac{1}{M^{2}}[\langle\hat{\boldsymbol{\sigma}}^{a}(\tau)\cdot\hat{\boldsymbol{\sigma}}^{a}(0)\rangle]=G_{B}^{aa}(\tau)G_{B}^{aa}(-\tau)\;.$
(10)

For large $S$ one finds an essentially classical glass transition at
$T_{g}\sim S^{2}J$, where the bosonic spinons condense. The 4-spinon
interaction term resembles the $p=4$-spin problem, and for similar reasons one
finds a 1-step RSB transition. This leaves open the choice of the size of
replica blocks, which selects the energy of the targeted metastable states.
Usually one uses the condition of marginal stability (vanishing of the
“replicon” mode), as those are reached dynamically [31] and exhibit a gapless
spectrum. Extremization of the free-energy density instead describes
dynamically inaccessible states with a gap in $\chi_{\rm
loc}^{\prime\prime}(\omega)$.

At large $S$, quantum fluctuations introduce a frequency scale $\sim SJ$,
which sets the crossover temperature where collective quantum dynamics
emerges. Consistent with the Landau analysis of Sec. 5 marginal states are
found to have $\chi^{\prime\prime}_{\rm loc}(\omega)\sim\omega$ and a low-$T$
specific heat that scales as $T^{3}$ [59, 60] (contrary to the linear $T$
dependence originally claimed in [57, 58]).

The case of small $S\ll 1$ exhibits much stronger quantum fluctuations and is
thus more interesting. Here, the spin-fluid, paramagnetic regime survives down
to much lower temperatures $T\ll J$ and is radically different from the
classical paramagnet of the high $T$ and large $S$ limits. Below $T\sim J$,
the system enters a gapless, quantum critical regime in which the Green
function assumes long-time tails, $G^{aa}_{B}(\tau)\sim 1/(J\tau)^{1/2}$,
while the melon-bubble self-energy is unusually large,
$\Sigma(\omega)\sim(\omega J)^{1/2}\gg\omega$, reflecting the strong
scattering among spinons. From the spin-spin correlation $q_{d}(\tau)\sim
1/(J\tau)$ one extracts a susceptibility very similar to the one of marginal
Fermi liquids: $J\chi^{\prime\prime}_{\rm
loc}(\omega)\simeq\tanh(\hbar\omega/2T)$ implying $\chi^{\prime}_{\rm
loc}\simeq\ln(J/T)$. One further finds linear in $T$ specific heat and a
residual low-temperature entropy. This holds in the paramagnetic regime
$T_{g}<T<J$, for an extended range of $S$ all the way up to $S\simeq 1$. These
properties are very similar to those of black holes to which this model and
SYK is related.

From Eq. (10) one sees that the onset of glassy order modifies the Green
functions beyond the time scale $\tau^{*}=(\omega^{*})^{-1}=(q_{\rm
EA}J)^{-1}$ where $G_{B}(\tau)G_{B}(-\tau)\sim q_{\rm EA}$. At that scale the
spinons get confined, and $Q(\tau)$ crosses over to $Q(\tau)-q_{\rm
EA}\sim\tau^{*}/(J\tau^{2})$. This corresponds to a linear spectral function
$J\chi_{\rm loc}^{\prime\prime}(\omega)\sim\omega/(q_{\rm EA}J)$, as is found
in all insulating mean-field glasses, cf. Sec. 5. Note, however, that weak
glass order implies a potentially large prefactor $\sim 1/q_{\rm EA}$,
reflecting very slow collective modes.

The bosonic representation is a priori best suited for large $S$. Fermionic
$SU(M)$ representations are instead believed to capture better the physically
most relevant case $S=1/2$. Such an approach yields essentially identical
results for the paramagnetic phase [61], though no glass transition is found
at $M=\infty$ in this case [56], since quantum fluctuations are much stronger.
Yet, a $1/M$ expansion [62] yields an instability to a glass phase where a
fermionic bilinear condenses at $T_{g}\sim J\exp(-\sqrt{\pi M})$ (now with
continuous RSB – as the effective action is quadratic in the emerging order
parameter, like in $p=2$ models). One expects a similar suppression of the
order parameter, $q_{\rm EA}\sim T_{g}/J$ (which ensures that at $T\sim T_{g}$
the low-$T$ linear spectral function matches that of the paramagnetic regime).
Exact diagonalization for all-to-all coupled $SU(2)$ spins [63, 64] indeed
yields $q_{\rm EA}\approx 0.02$. However, currently accessible system sizes of
$\sim$ 20 spins do not allow to disentangle the regime $\omega<\omega^{*}$
from the broadened peak $\sim q_{\rm EA}\delta(\omega)$ reflecting static spin
glass order. Nonetheless, they do exhibit spectral features as predicted by
the fractionalized large $M$ approach for $\omega>\omega^{*}$.

##### Heisenberg glasses and Many-Body Localization

We close this section by remarking that Heisenberg spin chains with coupling
$J$ under strong, but random i.i.d. local fields
${\mathbf{H}}_{i}=H^{x}_{i}{\mathbf{e}}_{x}$ drawn from a distribution of
width $W\gg J$, has become one of the standard models for MBL [10]. Indeed,
the typically large mismatch between the local gaps $H_{i}^{x}$ cannot be
bridged resonantly by the weak couplings $J$, and thus coherent, bath-free
dynamics is expected to remain stuck close to an arbitrary initial state. It
is worth pointing out again that MBL is favored by completely opposite
ingredients than a spin glass phase: it requires the non-commuting
interactions to be weak, and the dimensionality should be low. In contrast, in
the parameter range where a glass with non-trivial energy landscape exists,
one expects rapid and full thermalization within any local minimum of the
energy landscape, independently of the coupling to a bath.

####  3 Insulating vs metallic environment

Remarkably, all insulating mean-field quantum glasses in their marginally
stable glassy states exhibit the same linear scaling $\chi^{\prime\prime}_{\rm
loc}(\omega)\sim\omega$, albeit with quantitatively different prefactors, and
a low-$T$ specific heat $c_{V}\sim T^{3}$. This contrasts with a spin glass
coupled to a gapless bath, e.g. via Kondo coupling to itinerant conduction
band electrons, as in heavy fermion materials [65]. The dissipation due to the
low-energy bath degrees of freedom slows down the collective modes and thereby
substantially enhances the low-frequency spin spectral function and the spin
contribution to the specific heat, as we will review via a Landau approach in
the next section.

###  5 Landau theory & the low frequency spectrum

A Landau expansion was reviewed in [66] and we only sketch it here. The aim is
to construct an effective action for the glass order parameter $Q_{ab}$ by
integrating out all other degrees of freedom. This approach is expected to
succeed for phase transitions in which the spins undergo standard ordering, as
e.g. in the TFIM and rotor models. However, in Heisenberg-type models, where
at least for $M\gg 1$ the spins fractionalize into deconfined spinons in the
paramagnetic phase, it will likely fail to capture the entire dynamical
crossover functions correctly. Methods as the ones for deconfined quantum
critical points [67] may instead be required. Still, even in these cases the
Landau approach may capture well the glassy phase in its low-frequency,
spinon-confined regime.

Considering symmetry restrictions and constraints on the replica structure,
and assuming the absence of time-reversal symmetry-breaking fields, the
resulting mean-field action close to the quantum glass transition takes the
form [68]

$\displaystyle n\beta{\cal F}[\delta Q]=\sum_{a}\int d\tau\left.{\cal
L}[\delta Q_{aa}]\right.+u\sum_{a}\int
d\tau\left[{\delta}Q_{aa}(\tau,\tau)\right]^{2}$
$\displaystyle\;\;\;-c_{3}\sum_{abc}\int d\tau_{1}d\tau_{2}d\tau_{3}\delta
Q_{ab}(\tau_{1},\tau_{2})\delta Q_{bc}(\tau_{2},\tau_{3})\delta
Q_{ca}(\tau_{3},\tau_{1})+O(\delta Q^{4})\;,$

where $\delta
Q(\tau,\tau^{\prime})=Q_{ab}(\tau,\tau^{\prime})-c\delta_{ab}\delta(\tau-\tau^{\prime})$,
with the constant $c$ fixed to remove the uninteresting and non-universal
short-time dynamics. The coefficients are bare correlations of the degrees of
freedom coupling to $Q$, but only in the linear term their time-dependence is
relevant. The leading low-frequency dependencies, distinguish whether the
degrees of freedom are gapped (insulating) or gapless like in a Fermi sea
(metallic), and are

$\displaystyle{\cal L}_{\rm ins}[\delta Q_{aa}]$
$\displaystyle=(r+\partial_{\tau_{1}}\partial_{\tau_{2}})\delta
Q_{aa}(\tau_{1},\tau_{2})\;,$ (11) $\displaystyle{\cal L}_{\rm met}[\delta
Q_{aa}]$ $\displaystyle=\left[r^{\prime}-(\tau_{1}-\tau_{2})^{-2}\right]\delta
Q_{aa}(\tau_{1},\tau_{2})\;,$

where ${\tau_{1}=\tau_{2}=\tau}$. In the metallic case we dropped the
subleading time derivatives, obtaining a leading $|\omega_{n}|$ dependence in
Matsubara space, while the coefficient in insulating glasses displays a faster
$\omega_{n}^{2}$ dependence.

####  1 Insulating glasses

Minimizing ${\cal L}_{\rm ins}$ with respect to $\delta Q$ close to the
quantum transition at $T=0$ yields [68]
$\delta\tilde{Q}_{aa}(\omega_{n})=-\sqrt{\omega_{n}^{2}+\Delta^{2}}$ (dropping
numerical coefficients), which implies the spectral function

$\displaystyle\chi^{\prime\prime}_{\rm
loc}(\omega)=\textrm{sgn}(\omega)\sqrt{\omega^{2}-\Delta}\,\Theta(|\omega|-\sqrt{\Delta})\;.$
(12)

The parameter-dependent energy $\Delta$ plays the role of a spectral gap,
which closes, $\Delta\rightarrow 0$, upon approaching the transition from the
paramagnetic side. However, remarkably, within the glassy phase the gap
remains pinned to zero, $\Delta=0$, which entails a linear low frequency
spectral function:

$\displaystyle\chi_{\rm loc}^{\prime\prime}(\omega)=\omega\;.$ (13)

This dynamical crossover agrees with the exact solution of large $M$ quantum
rotors and the critical behavior of the TFIM. Allowing for RSB, one finds
marginally stable solutions with a vanishing replicon and gapless spectrum.

Note that, as anticipated above, the high-frequency spectrum predicted by the
Landau approach differs qualitatively from the spin susceptibility
$\chi^{\prime\prime}_{\rm loc}(\omega)\sim{\rm sgn}(\omega)$ of the large $M$
limit of Heisenberg models, which hinges on spinon deconfinement.

####  2 Metallic glasses

Minimizing ${\cal L}_{\rm met}$ close to the transition one finds [69, 65],
$\delta Q_{aa}(\omega_{n})=-\sqrt{|\omega_{n}|+\Upsilon}$, whereby the
crossover energy $\Upsilon$ vanishes again at the quantum transition,
$\Upsilon\rightarrow 0$, and stays pinned to $\Upsilon=0$ in the glassy phase.
Here the spectral function is everywhere gapless and approaches

$\displaystyle\chi^{\prime\prime}_{\rm
loc}(\omega)=\frac{1}{\sqrt{2}}\frac{\omega}{\sqrt{\Upsilon+\sqrt{\omega^{2}+\Upsilon^{2}}}}\overset{\Upsilon\to
0}{\longrightarrow}{\rm sgn}(\omega)\frac{|\omega|^{1/2}}{\sqrt{2}}$ (14)

in the glass phase, which, at low frequencies, is substantially stronger than
the one of insulating glasses, Eq. (13). Again, the Landau approach does not
capture the high-frequency features associated with spinon deconfinement [64].

####  3 Physical interpretation of glassy spectral functions

The low-frequency spectral functions in the glass phase can be interpreted as
arising from of set of collective random spin-density modes that behave as
either underdamped or overdamped harmonic oscillators.

In the glass phase a local minimum of the free-energy landscape (expressed as
a functional of local magnetizations) has a Hessian with positive eigenvalues
$\lambda_{k}$ which may be interpreted as generalized spring constants. The
marginal stability of the relevant glass states assures that their
distribution is gapless, and random matrix theory suggests that at small
$\lambda$, it behaves as $\rho(\lambda)\equiv
N^{-1}\sum_{k}\delta(\lambda-\lambda_{k})=c\sqrt{\lambda}$.

In the insulating case one expects the Hessian normal modes to behave as
independent harmonic oscillators with an effective mass $M$, set by the
typical microscopic time scale of the spin dynamics. The associated oscillator
frequencies then scale as $\omega_{k}=\sqrt{\lambda_{k}/M}$, with a spectral
density $\rho(\omega)\sim cM^{3/2}\omega^{2}$. This correctly predicts the
specific heat to scale as $c_{V}(T)\sim T^{3}$ at low $T$, rationalizing the
result of [59] (with the notable exception of quantum particles at jamming
where $\rho(\lambda)\sim\lambda^{-1/2}$ [70]). The typical mean-square
displacement of oscillator mode $k$ scales as $\left\langle
x_{k}^{2}\right\rangle\sim\hbar/M\omega_{k}\sim\hbar/\sqrt{M\lambda_{k}}$, and
from the Lehmann representation, Eq. (3), one expects the spectral function to
scale as

$\displaystyle\chi^{\prime\prime}_{\rm loc}(\omega)$ $\displaystyle\sim$
$\displaystyle\rho(\omega)\left\langle x^{2}\right\rangle_{\omega}\sim
c\sqrt{M}\omega\;,$ (15)

which rationalizes the linear scaling in Eq. (13).

The random oscillator picture is further supported by the scaling of the ratio
$\chi^{\prime\prime}_{\rm loc}(\omega)/\omega$. Exact calculations for rotors
[68] or a particle in a random environment [29] indeed obtained a ratio
$\sim\sqrt{M}$. In the TFSK, an effective potential construction yields
$c\sim\sqrt{\Gamma}/J^{2}$ for $\Gamma\ll J$, while one expects $M\sim
1/\Gamma$. This correctly predicts the $\Gamma$-independent ratio in Eq. (5).

For metallic environments, the oscillator dynamics is instead expected to be
overdamped with a friction coefficient $\eta$ proportional to the prefactor of
$|\omega_{n}|\tilde{Q}_{aa}$. In this case the characteristic frequency of
mode $k$ is given by the relaxation rate, $\omega_{k}\sim\lambda_{k}/\eta$,
implying a mode density $\rho(\omega)\sim c\eta^{3/2}\omega^{1/2}$ with larger
weight at low frequencies. Accordingly, one expects corrections $\delta
c_{V}(T)\sim T^{3/2}$ to the specific heat, as indeed predicted in [65]. The
typical oscillator displacement follows from the scaling
$\hbar\omega_{k}\sim\lambda_{k}x_{k}^{2}$, that is, $\langle
x_{k}^{2}\rangle\sim\hbar/\eta$, independent of the mode. The spectral
function should then scale as

$\displaystyle\chi^{\prime\prime}_{\rm loc}(\omega)$ $\displaystyle\sim$
$\displaystyle\left\langle x^{2}\right\rangle_{\omega}\rho(\omega)\sim
c\sqrt{\eta\omega}\;,$ (16)

which again correctly reproduces the result of the Landau theory given in Eq.
(14), as well as the scaling with $\eta$.

Unfortunately, little is known about the collective modes in short-range
glasses. Those may play an important role in electronic processes such as
hopping transport in insulators, as potential glue for strong-coupling
superconductors, or as inelastic scatterers affecting the electrical
resistance in metallic glasses.

###  6 Tunneling and eigenstate localization

It is interesting to ask whether and how quantum tunneling allows one to
explore the clusterized landscape of quantum models with a 1-step RSB pattern.
Having initialized a quantum Ising glass in a deep energy valley, tunnelling
might not take place, even in the thermodynamic limit, since the remaining
discreteness of the eigenstates (within one valley) may prevent resonant
intervalley coupling.

This problem was first analyzed for $p>2$ models [71]. Computing the tunneling
amplitude between valleys of equal internal energy $E$ and equating it to the
intra-valley level spacing determines a minimal transverse field
$\Gamma_{c}(E)$, below which wavefunctions and thus quantum dynamics, remain
valley-localized. This situation is globally non-ergodic, yet delocalized
within one valley - an unusual situation, believed to be impossible in finite
dimensional models of MBL. The delocalization threshold $\Gamma_{c}(E)$ was
found to lie consistently below the threshold $\Gamma_{d}(E)$ where glassy
dynamics sets in. Indeed, $\Gamma_{c}$ only senses the energy barriers in
phase space, while dynamics can be exponentially trapped by entropic (free
energy) barriers, in spite of percolation in the energy landscape.

At finite $p$, once quantum tunneling resonantly connects different valleys of
energy $E$, the eigenstates hybridize all classical configurations having
energy close to $E$, since the wavefunctions behave ergodically within a
valley. This is not so in the strict limit $p\to\infty$ [72, 73, 74] where the
random energy model results, and each valley contains only one single
configuration. In this limit, only states in a very narrow energy shell
hybridize, its width being the exponentially small tunneling between those
states.

It was claimed that tunneling under barriers can be exploited for a quantum
algorithm to efficiently find other valleys of equal internal energy [35].

###  7 Interplay of glassy order with localization phenomena

The interplay of glassy freezing with other quantum phenomena, especially with
the localization of fermions or bosons, is a rich subject. Below, we discuss
how (fermionic) electron glasses become metals, and how glasses of bosons may
become superfluid.

####  1 Quantum electron glasses

The classical Coulomb glass is a disordered insulator with unscreened Coulomb
interactions between localized electrons, realized in e.g. doped
semiconductors [75, 76, 77]. A standard model is

${\mathcal{H}}_{\rm
Cb}=\sum_{i}\epsilon_{i}n_{i}+\sum_{i<j}\frac{n_{i}n_{j}}{r_{ij}}$ (17)

with random energy $\epsilon_{i}$ on site $i$ drawn from a box distribution
$[-W/2,W/2]$. The local occupation number is $n_{i}=0,1$ and $r_{ij}$ is the
distance between sites $i$ and $j$. The number of electrons is a fixed
fraction of the sites, which are arranged on a lattice or at random.

The unscreened, long range $1/r$ interactions between localized electrons
enforce the Efros-Shklovskii (ES) Coulomb gap in the single-particle density
of states, $\rho(E)$. It has long been conjectured that the stability
constraint $\rho(E)\leq{\rm const.}\times E^{d-1}/e^{2d}$ (which assures that
particle-hole recombinations do not lower the energy) is only marginally
satisfied. A mean-field treatment with marginal full RSB predicts this
critical property [78, 79, 80]. This phenomenon is directly analogous to the
appearance of a linear density of local fields in the classical SK model,
which is known to saturate similar stability bounds [81].

Upon increasing the density of dopants the hopping $t_{ij}$ between the sites
$i$ becomes important and a term $\sum_{i<j}(t_{ij}c_{j}^{\dagger}c_{i}+h.c.)$
must be added to ${\mathcal{H}}_{\rm Cb}$ in Eq. (17). Single-particle
eigenfunctions $\psi_{\alpha}$ now spread over several sites. When written in
the basis of $\psi_{\alpha}$, the interactions take the generic form
$\sum_{\alpha,\beta,\gamma,\delta}U_{\alpha\beta\gamma\delta}c^{\dagger}_{\alpha}c^{\dagger}_{\beta}c_{\gamma}c_{\delta}$,
which couples all wavefunctions $\psi_{\alpha}$ that overlap in real space.
Close to the metal-insulator transition this term resembles the fermionic
models reviewed in Sec. 9.

Eventually localization breaks down and the wavefunctions become extended
[82]. At the same time screening sets in, the effective Coulomb interactions
become short range and the density of states at the Fermi level becomes
finite. Close to criticality the growing Coulomb gap suppresses the occurrence
of strong resonances for low-energy electrons, so that their wavefunctions
concentrate less on rare paths. This results in a larger fractal dimension as
compared to the ones of non-interacting critical wavefunctions [83, 84].

Already in the proximity of the insulating state one finds multiple solutions
to Hartree-Fock equations. This may signal the emergence of metastability [85,
86], which then extends all the way into the insulating electron glass. Slow
relaxation and aging in gated 2d electron gases in silicon [87, 88] at
metallic densities was taken as an indication for electron glassy behavior.
However, the existence of a genuine glass transition in $d=3$ remains debated
even in the classical limit, as the random potential breaks a potential Ising
symmetry so that only the replica symmetry is left to be broken. Numerical
simulations have not fully clarified the situation [89, 90].

In Sec. 3 we will discuss the related case of fermionic atoms with photon-
mediated long-range interactions. Interestingly, the latter allow for a
genuine glass transition (with RSB) already in the metallic phase.

####  2 Superfluidity and glassy density order

Experiments that had suggested superflow in solid, but possibly amorphous
Helium [91] had raised the interesting question whether and how (glassy)
density order of bosons could coexist with the a priori competing superfluid
order. For models with mean field-like, frustrated density interactions (such
as e.g. realized in multimode cavities) and locally hopping bosons, it was
indeed found that the two orders can coexist [92], while they try to avoid
each other locally: if the local superfluid order parameter is high, the
glassy density order is weak, and vice versa [93]. This phase is the bosonic
analogue of the metallic glass of fermionic atoms discussed in Sec. 3. For a
specific 3d model whose quantum Hamiltonian maps onto the Fokker-Planck
equation of Brownian hard spheres, the existence of such a superglass phase
could be inferred explicitly, drawing on the knowledge on the classical glass
phase of hard spheres [94].

###  8 Experimental realizations of quantum glasses

Let us now review a few selected examples of quantum glasses in correlated
materials and possible realizations in artificial structures involving light-
matter coupling.

####  1 Quantum Ising spin glasses

For long, LiY1-xHoxF4 was considered to be “the” quantum spin-glass
realization [95]. The rare earth magnetic ion Holmium (Ho) has an Ising
doublet with a large magnetic moment of $5.4\mu_{B}$ as its crystal field
ground state. As the content of Ho is decreased below $x\approx 44\%$, the
material turns from a dipolar ferromagnet to a dipolar-interacting Ising spin
glass. Quantum fluctuations are introduced by a transverse field $H_{t}$,
which at second order in perturbation theory hybridizes the two states of the
Ho ground-doublet, resulting in an effective transverse field for the Ising
doublet $\Gamma\sim H_{t}^{2}/\Delta$, where $\Delta$ is the gap in the
crystal field spectrum.

Initially, the variation of the susceptibility with $\Gamma$ for $x=0.167$ was
interpreted in terms of a first order transition at low temperatures [96].
With decreasing $\Gamma$ the linear susceptibility reflects a rather sudden
slowdown of dynamics and the onset of low frequency $1/f$ noise. However, the
non-linear susceptibility does not diverge, ruling out a continuous glass
transition. A first order transition, instead, is at odds with the rule of
thumb that two-body interactions usually entail continuous transitions. It was
later argued [97] that no equilibrium glass transition can occur at large
$H_{t}$, since the latter induces finite transverse moments whose internal
longitudinal fields break the Ising symmetry explicitly. In the absence of a
de Almeida-Thouless line, one could thus at best expect a regime of correlated
spin clusters, whose size diverges as $\Gamma\to 0$. This might explain the
flat maximum in the non-linear susceptibility as a function of $\Gamma$, and
its decrease with $T$. However, to fully understand the dynamic response of
LiY1-xHoxF4, it will be crucial to include the hyper-fine coupling to the Ho
nuclear spin, especially in the quantum glassy low $T$ regime [98].

Interestingly, the susceptibility at a given point ($\Gamma,T$) in the glassy
regime depends strongly on the annealing protocol [36]. Quantum annealing
results in significantly faster typical relaxation rates and larger
$\chi^{\prime}(\omega)$ than thermal annealing. Moreover, it results in a
nearly $\Gamma$-independent dissipation $\chi^{\prime\prime}(\omega)$ at low
$\omega$, which was interpreted to signal a critical state. The authors
suggested that faster response in quantum annealed samples implies more
complete relaxation than under thermal annealing. However, the opposite
conclusion seems equally possible: quantum annealing remains stuck closer to
the marginal surface of the rugged energy landscape, leaving more fluctuating
regions with small barriers and faster relaxation times. Instead the thermally
annealed state, if the system managed to cross barriers, looks more “aged”,
being entrenched in deeper and more stable valleys, and thus displaying slower
response.

####  2 Quantum Heisenberg spin glasses

Heisenberg quantum spin glasses attract a lot of interest because of the
intriguing SYK-like spectral features they might display in their paramagnetic
phase, as well as above a moderate frequency scale – provided the mean-field
and large-$M$ predictions survive for $SU(2)$ spins in 3d. A prominent example
is the cuprate La2-xSrxCuO4 [99] which was recently shown to host spin glass
order at low temperatures, all the way up to optimal doping, albeit with
strong short-range antiferromagnetic correlations. If critical spectral
features as predicted for fully frustrated mean-field Heisenberg glasses also
survive in such real materials, they might possibly be at the root of the
linear temperature dependence of the resistivity often observed in this type
of materials [100].

In heavy fermion compounds, such as e.g. Y1-xUxPd3, the dilute, randomly
positioned local moments of U couple to conduction electrons that mediate RKKY
interactions. Such materials are promising candidates to find genuine metallic
quantum glass phases and quantum glass transitions to a paramagnet, where
Kondo screening of the moments dominates. A quantum spin glass phase and
interesting non-Fermi liquid behavior at higher temperatures was indeed
reported for $0.2<x<0.4$ of the above compound [101]. For a more thorough
review of quantum glass candidates and their magnetic properties we refer to
[102].

####  3 Mean-field quantum glasses in optical cavities

Random and long-range interactions among cold atoms can be generated via
light-matter coupling in multi-mode cavities with sufficiently random mode
functions [103, 104]. We distinguish realizations with non-intinerant and
itinerant atoms, respectively.

##### Non-itinerant fermionic atoms

Laser-trapped, immobile atoms act as local TLS (Ising spins). They are coupled
by an effective long range Ising interaction $J_{ij}$ obtained from
integrating out the photons of the enclosing optical cavity. In cavities with
a large number of modes, the long-ranged $J_{ij}$ are nearly random, owing to
the complex structure of the mode amplitudes. Such systems effectively realize
a TFSK model, whereby the interaction may assume a non-vanishing mean
$[J_{ij}]=J_{0}$ [105], and can be tuned via the driving parameters of the
optical cavity. Upon dialling up the average coupling $J_{0}$ a ferromagnetic
component of the spin freezing may arise as in classical analogues. Here it
essentially realizes Dicke’s superradiant phase. Under off-equilibrium
conditions, as introduced by the driving of the laser cavity and the leakage
of photons, the (steady state) phase diagram remains robust, but the critical
behavior is modified [106].

##### Itinerant fermionic atoms

If the atoms are able to move, an interesting interplay between amorphous
charge order and atomic delocalization occurs. In contrast to the case of
Coulomb interactions (Sec. 1), the interactions remain long-ranged across the
glass and the localization transitions. This allows for a glassy density order
to develop upon decreasing the kinetic energy of the fermions while they are
still in a metallic phase. Glassy density order in turn generates increased
local disorder which eventually localizes the fermions.

The instability of the ergodic metal to glassy order and the single particle
localization threshold are well separated in a lattice model with one
fermionic level per site and close to half-filling. Thus, a metallic glass
intervenes between the insulator and the ergodic metal [107]. In contrast, for
more dilute filling the transition between ergodic metal and glassy insulator
becomes first order, and in an extended parameter regime the two phases can
coexist. How one phase nucleates out of the less favorable one remains an
interesting open question.

###  9 Fermion models and the Sachdev-Ye-Kitaev model

Kitaev [108] proposed to use the Majorana fermion system

$\hat{\mathcal{H}}_{SYK}=\frac{1}{i^{q/2}}\sum_{i_{i}<i_{2}<\dots<i_{q}}^{N}J_{i_{i}i_{2}\dots
i_{q}}\hat{\psi}_{i_{1}}\hat{\psi}_{i_{2}}\dots\hat{\psi}_{i_{q}}\;,$ (18)

with $\hat{\psi}_{i}=\hat{\psi}^{\dagger}_{i}$,
$\\{\hat{\psi}_{i},\hat{\psi}_{j}\\}=2\delta_{ij}$, quenched random couplings
with zero mean and $[J^{2}_{i_{1}\dots i_{q}}]=(q-1)!J^{2}/N^{q-1}$, and four
fermion ($q=4$) interactions, as a toy model for near-extremal black holes.
Being very similar to the fermionic representation of the $SU(M\to\infty)$
Heisenberg model, it has no glassy phase. The main reason to propose the
connection with gravity is that it becomes approximately conformal in the
infrared (dropping time-derivatives, low-frequency limit), with a time-
reparametrization symmetry which is broken to SL(2,R), as expected in
blackhole theories which develop a nearly AdS2 background [100]. A “Schwarzian
action” describes the cost of reparametrizations [109]. Moreover, there is
non-zero entropy at $T\to 0$ (taken after $N\to\infty$) and the specific heat
is linear in $T$ – thermodynamic properties that are expected in the black
hole context as well. Finally, this system is a “maximally chaotic/perfect
scrambler” meaning that the bound $\lambda_{T}\leq 2\pi T/\hbar$ on the
Lyapunov exponent (a consequence of the fluctuation-dissipation theorem
[110]), defined from the exponential growth of out-of-time-order correlations
[111], is saturated.

It was soon realized that Kitaev’s proposal is very close to the Sachdev-Ye
model, introduced and studied with the aim of describing non-Fermi liquid
behavior in condensed matter systems hosting quantum spins (see Sec. 2). The
above quenched disordered fermion system (with no glassy phase), subsequently
called SYK, thus became a popular model in high-energy theory. Potential
experimental realizations were proposed in disordered graphene flakes under
strong magnetic fields [112, 113].

In order to make a connection with the formalism we described in the rest of
the chapter, the relevant correlator in the SYK model is
$Q(\tau)=-[\langle{\rm T}\hat{\psi}_{i}(\tau)\hat{\psi}_{j}(0)\rangle]$ (the
fermionic analogue of $G_{B}$ of Sec. 2), which satisfies an equation like Eq.
(2) in its replica-diagonal form (no need of off-diagonal terms here) with

$G_{0}^{-1}=d_{\tau}\;,\qquad\qquad\Sigma(\tau)=J^{2}Q^{3}(\tau)\;.$ (19)

Let us now comment on some interesting properties of the SYK model. Time-
reparametrization symmetry also emerges in mean-field glass models, where
sigma models for the reparametrizations were phenomenologically proposed (but
not derived) [32]. Coupling two (or more) SYKs, popular in high-energy studies
to mimic wormholes, is similar to coupling real replicas in disordered
systems, a procedure that has long been used in the glass literature to access
properties of the free-energy landscape [114]. Moreover, the relation of SYK
with tensor models without quenched disorder [115], parallels the ideas of
self-generated disorder put forward in the 90s to link $p$-spin models to the
Mode-Coupling Theory for structural glasses [1]. In the SYK context, the trick
is to use tensor models such that in a perturbative expansion only melon
diagrams for the two-time function, and ladder diagrams for the four-time
function, survive. With quenched disorder this structure occurs thanks to the
average over randomness.

These results motivated numerous studies of other quantum spin glasses [116,
117], in particular those discussed in Sec. 1. Let us just mention here the
analysis of chaos in the paramagnetic and marginal glassy phases of the
spherical $p$-spin models [117]. Quantum fluctuations were found to make the
paramagnetic phase less and the spin glass phase more chaotic. In the
classical limit $\hbar\to 0$, a crossover from strong to weak chaos, as marked
by a maximum in $\lambda_{T}$, arises well above $T_{d}$, concomitant with the
onset of two-step slow relaxation.

###  10 Conclusion and outlook

This short review on quantum glassy systems focuses on the discussion of their
equilibrium properties. We have tried to tie connections between the different
systems and models presented. On the analytic side, we mostly discussed
results obtained in different mean-field limits, where the replica method and
replica symmetry breaking schemes can be safely applied. The phenomenology of
these disordered and glassy mean-field models is very rich and has a wide
scope. Indeed, many applications beyond physics have been exploited, notably
to optimization problems, but more recently fruitful connections with
unexpected fields, like gravity, have been uncovered, too. As usual, finite-
dimensional physically relevant problems are not amenable to an exact
treatment and much of what is known about them is either phenomenological or
numerical.

Putting this summary together we came across a number of open issues which
would be interesting to study. We just mention three of them. Revisiting the
quantum glass transitions with deconfining spinons, especially in insulating
Heisenberg systems, should be within analytic reach. We have only
superficially discussed the interplay between glassiness and localization.
There are certainly many interesting open questions there: how and when does
glassiness set in? How does a localized quantum glass melt into a metal?
Finally, it would be welcome to establish deeper connections between SYK
insights (such as the treatment of the reparametrization invariance) and glass
physics.

## References

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