Mott Glass: Gapless, Incompressible Phase

• Mott Glass is a disorder-induced quantum phase characterized by vanishing compressibility yet gapless excitations and is identified via rare-region and Griffiths phenomena.
• It is diagnosed in bosonic systems through zero compressibility and in magnetic systems by a vanishing uniform susceptibility as temperature approaches zero.
• Experimental and numerical studies in disordered quantum magnets and ultracold atoms highlight its unique critical behavior and relevance in quantum phase transitions.

Mott glass (MG) denotes a disorder-induced quantum phase that is gapless yet incompressible. In bosonic formulations this means vanishing compressibility, $\kappa=0$, while in quantum magnets mapped to bosons the corresponding signature is a vanishing uniform magnetic susceptibility as $T\to0$. Across the literature, MG is distinguished from the Bose glass, which is gapless and compressible, and from the Mott insulator, which is incompressible and gapped; its phenomenology is typically tied to rare-region physics, Griffiths effects, and disorder-driven quantum criticality (Thomson et al., 2015, Lerch et al., 2017, Dupuis et al., 2024).

1. Definition and phase diagnostics

The defining feature of the Mott glass is the coexistence of incompressibility with gapless low-energy excitations. In disordered bosonic systems, incompressibility is expressed directly by $\kappa=0$. In disordered quantum magnets, the same distinction is usually made through the uniform susceptibility $\chi_u$, because magnetic susceptibility corresponds to compressibility in the boson language (Thomson et al., 2015, Ma et al., 2015).

Across bosonic and spin realizations, the phase distinctions used in the literature are as follows (Thomson et al., 2015, Dupuis et al., 2024, Ma et al., 2015):

Phase	Compressibility / $\chi_u(T\to0)$	Low-energy response
Mott insulator	zero	gapped
Bose glass	finite	gapless
Mott glass	zero	gapless

A frequently used thermodynamic diagnostic of MG is a stretched-exponential low-temperature response. In disordered bilayer Heisenberg magnets, the uniform susceptibility in the MG regime follows

$\chi_u(T)\sim \exp(-b/T^\alpha),\qquad 0<\alpha<1,$

whereas the Néel phase has finite $\chi_u$ as $T\to0$ and the gapped quantum disordered phase shows ordinary activated behavior, $\chi_u\sim \exp(-\Delta/T)$ (Li et al., 3 Sep 2025, Ma et al., 2015). In the disordered two-dimensional Bose-Hubbard model at filling $\rho=1$, the compressibility in a narrow glassy regime was found to follow

$T\to0$0

with $T\to0$1 and $T\to0$2 vanishing or very small, leading to the interpretation of either a true MG or a nearly incompressible anomalous Bose glass (Wang et al., 2011).

The literature therefore treats gaplessness alone as insufficient for phase identification. The operational distinction between MG and BG is the vanishing or finite zero-temperature compressibility, respectively, even when both phases are glassy and supported by rare regions (Lerch et al., 2017, Iyer et al., 2011).

2. Rare regions, Griffiths physics, and microscopic origin

The standard physical picture of MG is based on rare regions embedded in an otherwise insulating background. In the interlayer-bond-diluted bilayer Heisenberg model, the MG emerges from isolated, gapless antiferromagnetic clusters inside a gapped paramagnetic background; the cluster-size distribution near the percolation threshold is crucial and was described as an archetype of Griffiths phases triggered by disorder (Li et al., 3 Sep 2025). In disordered bosonic systems, closely related language is used: rare, locally superfluid regions generate gapless excitations, yet their statistical weight can remain too small to produce finite compressibility (Iyer et al., 2011, Lerch et al., 2017).

Particle-hole symmetry is a recurrent structural condition in many bosonic realizations of MG. In the site-diluted quantum rotor model, the insulating glass phase is MG specifically because particle-hole symmetry is preserved; breaking that symmetry turns the intermediate phase into a Bose glass (Puschmann et al., 2016, Lerch et al., 2017). This suggests one robust route to incompressibility in a gapless glass: rare regions can generate arbitrarily low excitation energies without yielding a macroscopic density response.

A second route arises from long-range correlated disorder or long-range interactions. In a one-dimensional Bose gas with disorder correlations decaying as $T\to0$3, the localized phase is a Bose glass for $T\to0$4, but becomes a Mott glass with vanishing compressibility and gapless conductivity for $T\to0$5 (Dupuis et al., 2024). In a one-dimensional quantum fluid with long-range interactions $T\to0$6 or $T\to0$7, any amount of disorder suppresses Wigner crystallization when $T\to0$8 and produces a Mott glass with vanishing compressibility and gapless optical conductivity; for $T\to0$9, the Wigner crystal remains stable (Daviet et al., 2020).

These constructions give a common interpretation: MG combines a mechanism that suppresses compressibility with a disorder landscape that still supports arbitrarily low-energy neutral or cluster excitations.

3. Diluted bilayer Heisenberg magnets

A particularly explicit realization of MG was reported for the $\kappa=0$0 antiferromagnetic Heisenberg model on a bilayer square lattice with interlayer bond dilution, studied by stochastic series expansion quantum Monte Carlo. In this model, dilution acts only on the vertical interlayer bonds, preserving all spins and only suppressing or removing the vertical couplings; this setup was motivated by physical situations such as oxygen vacancies in nickelates (Li et al., 3 Sep 2025).

The study distinguished regular and random dilution. With regular dilution, tuning the interlayer coupling drives a direct transition from a Néel-ordered phase to a quantum disordered phase, consistent with the $\kappa=0$1 universality class with $\kappa=0$2, $\kappa=0$3, and $\kappa=0$4. With random dilution, by contrast, increasing the interlayer coupling at fixed dilution produces a two-step sequence:

1. Néel-ordered $\kappa=0$5 Mott glass,
2. Mott glass $\kappa=0$6 quantum disordered phase (Li et al., 3 Sep 2025).

The MG phase in this system is characterized by the stretched-exponential susceptibility

$\kappa=0$7

In log-linear plots of $\kappa=0$8 versus $\kappa=0$9, the simulations produced straight lines in the MG regime. As the coupling increases beyond the MG regime, $\chi_u$0 tends toward $\chi_u$1, marking crossover into a fully gapped regime (Li et al., 3 Sep 2025).

At the Néel-to-MG transition, disorder modifies the critical exponents. For random dilution $\chi_u$2, the reported values are $\chi_u$3, $\chi_u$4, and $\chi_u$5, in contrast to the clean-limit values $\chi_u$6, $\chi_u$7, and $\chi_u$8. The critical point was located using Binder ratio, spin stiffness, and staggered magnetization finite-size scaling, and consistency of $\chi_u$9 was checked באמצעות the critical susceptibility scaling $\chi_u(T\to0)$0. This behavior was interpreted as consistent with the Harris criterion because the clean correlation-length exponent satisfies $\chi_u(T\to0)$1 in two dimensions (Li et al., 3 Sep 2025).

The earlier dimer-diluted bilayer Heisenberg model exhibited the same three-phase structure—Néel order, MG, and gapped quantum paramagnet—and the same stretched-exponential susceptibility, with example fit values $\chi_u(T\to0)$2 at $\chi_u(T\to0)$3, $\chi_u(T\to0)$4 at $\chi_u(T\to0)$5, and $\chi_u(T\to0)$6 at $\chi_u(T\to0)$7 for $\chi_u(T\to0)$8 (Ma et al., 2015). However, not all disordered spin models exhibit disorder-modified criticality. In a different disordered two-dimensional $\chi_u(T\to0)$9 Heisenberg model with random dimer patterns on coupled ladders, the intermediate phase was again identified as a gapless Mott glass with $\chi_u(T)\sim \exp(-b/T^\alpha),\qquad 0<\alpha<1,$0, but the Néel-to-glass transition retained standard $\chi_u(T)\sim \exp(-b/T^\alpha),\qquad 0<\alpha<1,$1 critical exponents, implying a violation of the Harris criterion in that system (Ma et al., 2014).

This comparison shows that MG phenomenology can be robust even when the associated critical universality class is not.

4. Bosonic field theories and critical scaling

In dimerized quantum antiferromagnets, MG is often analyzed through an exact mapping to hard-core bosons: singlets become the vacuum and triplet excitations become bosons. Using bond-operator techniques, strong-coupling expansion, the replica method, and one-loop renormalization group, the disordered dimer system was found to support a Bose glass away from the tips of the Mott lobes and a Mott glass precisely at the particle-hole symmetric lobe tips (Thomson et al., 2015). Within that field theory, the disorder variance enters through

$\chi_u(T)\sim \exp(-b/T^\alpha),\qquad 0<\alpha<1,$2

with RG variables

$\chi_u(T)\sim \exp(-b/T^\alpha),\qquad 0<\alpha<1,$3

A central conclusion of this approach is that replica symmetry breaking (RSB) is necessary to reproduce the physical compressibility of the glass phases: one-step Parisi RSB yields finite compressibility in the Bose glass, while the RSB contribution vanishes at the lobe tip, where the incompressible Mott glass appears (Thomson et al., 2015).

In particle-hole symmetric rotor models, the transition between superfluid and MG exhibits conventional finite-size scaling with disorder-modified exponents. Large-scale Monte Carlo simulations of the two-dimensional site-diluted quantum rotor model found power-law critical behavior of both compressibility and superfluid density,

$\chi_u(T)\sim \exp(-b/T^\alpha),\qquad 0<\alpha<1,$4

with $\chi_u(T)\sim \exp(-b/T^\alpha),\qquad 0<\alpha<1,$5 and $\chi_u(T)\sim \exp(-b/T^\alpha),\qquad 0<\alpha<1,$6 at dilution $\chi_u(T)\sim \exp(-b/T^\alpha),\qquad 0<\alpha<1,$7. These exponents satisfy the generalized Josephson relations

$\chi_u(T)\sim \exp(-b/T^\alpha),\qquad 0<\alpha<1,$8

using $\chi_u(T)\sim \exp(-b/T^\alpha),\qquad 0<\alpha<1,$9 and the dynamical exponent $\chi_u$0 (Lerch et al., 2017).

The same rotor model also exhibits a superfluid-Mott-glass quantum multicritical point on the percolation threshold. The reported multicritical exponents are $\chi_u$1, $\chi_u$2, and $\chi_u$3 (Puschmann et al., 2016). A complementary strong-disorder RG study of a two-dimensional random rotor model described the MG-to-superfluid transition as a percolation-type process controlled by an unstable finite-disorder fixed point and estimated $\chi_u$4 and a fractal dimension $\chi_u$5 (Iyer et al., 2011).

Analytical scaling theory for the quantum transition between magnetically ordered and MG phases in disordered Bose systems and quantum antiferromagnets proposes

$\chi_u$6

and predicts a superuniversal density of states for localized excitations (“fractons”) (Syromyatnikov, 2017). Together, these results place MG criticality within the broader theory of dirty-boson quantum phase transitions, but without collapsing it onto the compressible Bose-glass case.

5. One-dimensional long-range problems and the question of existence

One-dimensional systems have been central to the question of whether MG is a sharply defined phase or only a phenomenological crossover label. In the one-dimensional Bose gas with long-range correlated disorder, the superfluid-insulator transition occurs at

$\chi_u$7

and the insulator is a Mott glass for

$\chi_u$8

The MG is then characterized by vanishing compressibility and gapless conductivity, and the exactly solvable case $\chi_u$9 in the semiclassical limit $T\to0$0 gives

$T\to0$1

for some constant $T\to0$2 (Dupuis et al., 2024).

A related nonperturbative FRG analysis of one-dimensional quantum fluids with long-range interactions concluded that for $T\to0$3, disorder converts the Wigner crystal into a Mott glass with $T\to0$4 and gapless optical conductivity $T\to0$5; only for $T\to0$6 does the Wigner crystal remain stable (Daviet et al., 2020). In this construction, MG inherits incompressibility from the long-range interaction sector and gapless optical response from disorder-induced localized excitations.

At the same time, the linearly confining disordered Schwinger model became the subject of a direct controversy. A Gaussian variational treatment argued for a Mott glass produced by the interplay of localization and confinement, with vanishing compressibility but gapless optical conductivity $T\to0$7 (Chou et al., 2018). A subsequent nonperturbative FRG study rejected that conclusion, finding instead that the stable low-energy phase is an incompressible state with gapped optical conductivity, similar to a conventional Mott insulator, and explicitly stating that a Mott-glass phase does not exist in that model (Dupuis, 2020).

This disagreement is not merely terminological. It shows that the phrase “Mott glass” cannot be inferred from incompressibility alone; the low-frequency dynamical response remains a decisive criterion.

6. Experimental realizations and broader significance

The clearest experimental realization discussed in the cited literature is Br-doped DTN. In this doped quantum magnet, the finite-field low-temperature phase is a Bose glass of field-induced magnetic quasiparticles, while at zero field the state becomes an incompressible Mott glass. The reported zero-field MG is gapless, has vanishing magnetic susceptibility at $T\to0$8, and exhibits non-exponential low-temperature specific heat described by the local-gap model

$T\to0$9

with $\chi_u\sim \exp(-\Delta/T)$0 and $\chi_u\sim \exp(-\Delta/T)$1 (Yu et al., 2011).

Other proposed platforms include granular superconductors, ultracold atoms in disordered optical lattices, and doped quantum magnets that map onto particle-hole symmetric rotor or boson models (Lerch et al., 2017, Puschmann et al., 2016). In cold-atom language, the MG is significant because compressibility-based diagnostics that cleanly distinguish Bose glass from Mott insulator can fail at particle-hole symmetric points where the glass is incompressible (Thomson et al., 2015).

A recurrent misconception in the broader disordered-boson literature is that the only generic glassy insulator is the Bose glass. Multiple numerical, analytical, and experimental works contradict that simplification. The disordered bilayer Heisenberg studies, rotor-model simulations, long-range FRG analyses, and the DTN experiment all identify a phase that is gapless yet incompressible (Li et al., 3 Sep 2025, Lerch et al., 2017, Dupuis et al., 2024, Yu et al., 2011). At the same time, the Schwinger-model controversy and the “Mott glass or anomalous Bose glass” interpretation of the disordered two-dimensional Bose-Hubbard model show that the phenomenology can be subtle and model-dependent (Dupuis, 2020, Wang et al., 2011).

Within this body of work, MG is best understood not as a single universal mechanism but as a class of disorder-induced gapless incompressible states. Its realizations differ in microscopic origin—particle-hole symmetry, long-range correlated disorder, long-range interactions, or cluster-based Griffiths physics—but they share the same diagnostic core: vanishing compressibility or uniform susceptibility together with gapless excitations and glassy rare-region dynamics.