---
title: 'Mott Glass: Gapless, Incompressible Phase'
url: https://www.emergentmind.com/topics/mott-glass-mg
type: topic
---

# Mott Glass: Gapless, Incompressible Phase

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 [1506.08029][1712.08245][2407.03430].

## 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 [1506.08029][1511.07895].

Across bosonic and spin realizations, the phase distinctions used in the literature are as follows [1506.08029][2407.03430][1511.07895]:

| 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, $\chi_u\sim \exp(-\Delta/T)$ [2509.03604][1511.07895]. In the disordered two-dimensional Bose-Hubbard model at filling $\rho=1$, the compressibility in a narrow glassy regime was found to follow
$$
\kappa(T)\sim \exp(-b/T^\alpha)+c
$$
with $\alpha<1$ and $c$ vanishing or very small, leading to the interpretation of either a true MG or a nearly incompressible anomalous Bose glass [1110.3213].

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 [1712.08245][1110.3338].

## 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 [2509.03604]. 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 [1110.3338][1712.08245].

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 [1608.06862][1712.08245]. 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 $1/|x|^{1+\sigma}$, the localized phase is a Bose glass for $\sigma>\sigma_c=3-\pi^2/3\simeq -0.289868$, but becomes a Mott glass with vanishing compressibility and gapless conductivity for $\sigma<\sigma_c$ [2407.03430]. In a one-dimensional quantum fluid with long-range interactions ${\cal V}_\sigma(x)\sim 1/|x|^{1+\sigma}$ or ${\cal V}_\sigma(x)\sim -|x|^{-1-\sigma}$, any amount of disorder suppresses Wigner crystallization when $-3/2<\sigma\leq0$ and produces a Mott glass with vanishing compressibility and gapless optical conductivity; for $\sigma<-3/2$, the Wigner crystal remains stable [2007.02993].

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 $S=1/2$ 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 [2509.03604].

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 $O(3)$ universality class with $\nu\approx0.71$, $\beta\approx0.366$, and $z=1$. With random dilution, by contrast, increasing the interlayer coupling at fixed dilution produces a **two-step sequence**:
1. Néel-ordered $\rightarrow$ Mott glass,
2. Mott glass $\rightarrow$ quantum disordered phase [2509.03604].

The MG phase in this system is characterized by the stretched-exponential susceptibility
$$
\chi_u(T)\sim \exp(-b/T^\alpha),\qquad 0<\alpha<1.
$$
In log-linear plots of $\ln \chi_u$ versus $T^{-\alpha}$, the simulations produced straight lines in the MG regime. As the coupling increases beyond the MG regime, $\alpha$ tends toward $1$, marking crossover into a fully gapped regime [2509.03604].

At the Néel-to-MG transition, disorder modifies the critical exponents. For random dilution $P=0.2$, the reported values are $\nu=1.01(2)$, $\beta=0.504(2)$, and $z=1.236(1)$, in contrast to the clean-limit values $\nu\approx0.71$, $\beta\approx0.366$, and $z=1$. The critical point was located using Binder ratio, spin stiffness, and staggered magnetization finite-size scaling, and consistency of $z$ was checked באמצעות the critical susceptibility scaling $\chi_u\sim T^{2/z-1}$. This behavior was interpreted as consistent with the Harris criterion because the clean correlation-length exponent satisfies $\nu_{\rm clean}\approx0.71<2/d=1$ in two dimensions [2509.03604].

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 $\alpha=0.21(1)$ at $g=1.3$, $\alpha=0.58(1)$ at $g=1.5$, and $\alpha=0.71(2)$ at $g=1.7$ for $p=0.3$ [1511.07895]. However, not all disordered spin models exhibit disorder-modified criticality. In a different disordered two-dimensional $S=1/2$ Heisenberg model with random dimer patterns on coupled ladders, the intermediate phase was again identified as a gapless Mott glass with $\chi\sim\exp(-b/T^\alpha)$, but the Néel-to-glass transition retained standard $O(3)$ critical exponents, implying a violation of the Harris criterion in that system [1407.1113].

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 [1506.08029]. Within that field theory, the disorder variance enters through
$$
g=\overline{r^2}-(\overline r)^2,
$$
with RG variables
$$
I_0=\frac{1}{1+\overline r},\qquad \lambda=I_0^2 g.
$$
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 [1506.08029].

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,
$$
\kappa\sim L^{-y_\tau},\qquad \rho_s\sim L^{-y_s},
$$
with $y_\tau=0.53(5)$ and $y_s=1.47(6)$ at dilution $p=1/3$. These exponents satisfy the generalized Josephson relations
$$
y_\tau=d-z,\qquad y_s=d+z-2,
$$
using $d=2$ and the dynamical exponent $z=1.52(3)$ [1712.08245].

The same rotor model also exhibits a **superfluid-Mott-glass quantum multicritical point** on the percolation threshold. The reported multicritical exponents are $z=1.72(2)$, $\beta/\nu=0.41(2)$, and $\gamma/\nu=2.90(5)$ [1608.06862]. 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 $\nu\approx1.09\pm0.04$ and a fractal dimension $d_f\approx1.31\pm0.07$ [1110.3338].

Analytical scaling theory for the quantum transition between magnetically ordered and MG phases in disordered Bose systems and quantum antiferromagnets proposes
$$
z=d-\beta/\nu,\qquad \phi=z\nu,
$$
and predicts a superuniversal density of states for localized excitations (“fractons”) [1701.08624]. 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
$$
K_c(\sigma)=\frac{3}{2}-\frac{\sigma}{2},
$$
and the insulator is a Mott glass for
$$
\sigma<\sigma_c=3-\frac{\pi^2}{3}\simeq -0.289868.
$$
The MG is then characterized by vanishing compressibility and gapless conductivity, and the exactly solvable case $\sigma=-1$ in the semiclassical limit $K\to0^+$ gives
$$
\Re[\sigma(\omega)]\sim |\omega|^3\exp\!\left(-\frac{\omega^4}{C}\right)
$$
for some constant $C$ [2407.03430].

A related nonperturbative FRG analysis of one-dimensional quantum fluids with long-range interactions concluded that for $-3/2<\sigma\leq0$, disorder converts the Wigner crystal into a Mott glass with $\kappa=0$ and gapless optical conductivity $\Re[\sigma(\omega)]\sim\omega^2$; only for $\sigma<-3/2$ does the Wigner crystal remain stable [2007.02993]. 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 $\mathrm{Re}[\sigma(\omega)]\propto\omega^2$ [1803.04426]. 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 [2001.05682].

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 $T=0$, and exhibits non-exponential low-temperature specific heat described by the local-gap model
$$
C_V(T)\sim t^{-5/4}\exp\left(-2\sqrt{c x_0/t}\right),
$$
with $t=k_B T/J_c$ and $x_0=\log(2x)$ [1109.4403].

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 [1712.08245][1608.06862]. 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 [1506.08029].

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 [2509.03604][1712.08245][2407.03430][1109.4403]. 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 [2001.05682][1110.3213].

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.

Source: https://www.emergentmind.com/topics/mott-glass-mg