Charge glass from supercooling topological-ordered liquid

TL;DR

The paper investigates glassy crystallization dynamics emerging from a topologically ordered Coulomb-like liquid on a triangular lattice, using a minimal charge-Ising model with a flux-conserving topological sector and triplet excitations. Through kinetic Monte Carlo simulations of single-charge transfers and a zero-triplet quench, the authors show that crystallization proceeds by diffusion of triplets across macroscopic distances, enabling flux-sector changes and yielding a diffusion-limited, slow ordering process. The resulting time-temperature-transformation diagram features a nose temperature $T_{\rm nose}$; below it, ordering scales with the maximal triplet density and exhibits an Arrhenius-like dependence, while above it a nucleation barrier dominates and the onset is delayed, with an effective Avrami exponent near $n\approx 0.5$, reflecting diffusion. These findings reproduce key features of charge glass observed in $\theta$-(BEDT-TTF)$_2$X(SCN)$_4$, link triplet excitations to the slow dynamics, and introduce the concept of crystallization by diffusion in a topologically ordered liquid.

Abstract

Topological order characterizes a class of quantum and classical many-body liquid states that escape the conventional classification by spontaneous symmetry breaking. Many properties of the topological-ordered states still await a clear understanding, and nature of phase transition dynamics is one of them. Normally, when a liquid freezes into a solid, crystallization starts with nucleation and a solid domain quickly grows on the surface of the expanding nucleus, and the domains evolve into macroscopic size. In this work, we reveal that the crystallization of the topological-ordered liquid proceeds in a fundamentally different way. The topological-ordered phase is characterized by a global conserved quantity and its conjugate fractional charge, which we call a flux and a triplet in our working system of the charge Ising model on a triangular lattice. In contrast to the normal crystallization process, the phase transition is driven by the diffusive motion of triplets, which is required to change the value of conserved fluxes to exit the topological-ordered phase. In order to complete crystallization, triplets must spend a divergently long time to diffuse over a macroscopic distance across the system, which results in glassy behavior. Reflecting the diffusive motion of triplets, the initial crystallization process shows slowing down with unusually small Avrami exponent $\sim0.5$. These anomalous dynamics are specific to the crystallization from topological-ordered liquid, and well account for the main features of charge glass behavior exhibited by the organic conductors, $θ$-(BEDT-TTF)$_2$X(SCN)$_4$.

Figures (7)

• Figure 1: (Color online) (a) The numerically obtained Time-Temperature-Transformation (TTT) diagram showing the time evolution of the ordered fraction, $\Delta_{\rm CO}$, after the system is quenched to the temperature, $T_{\rm end}$, in the zero-triplet quench protocol. (b) Schematic illustration of crystallization by diffusion: triplets, represented by upward and downward triangles, diffuse through the system and induce charge ordering. (c) Schematic illustration of conventional crystallization, where small ordered domains grow at the surfaces of existing ones and expand into macroscopic regions.
• Figure 2: (Color online) Model and topological constraint of the charge Ising model on a triangular lattice. (a) Geometry of the lattice and definition of the nearest, second, and third neighbor interactions $V_{1}$, $V_{2}$, and $V_{3}$. (b) Schematic phase diagram of Hamiltonian, Eq. (\ref{['eq:Hamiltonian']}). At zero temperature, diagonal and horizontal charge-ordered phases become ground states for $V_3<\frac{V_2}{2}$ and $V_3>\frac{V_2}{2}$, respectively. At higher temperatures, equilibrium states are described mostly by the states satisfying the Coulomb rule, provided $V_2$ and $V_3$ are sufficiently smaller than $V_1$. (c) Typical charge configurations. Dashed blue lines show frustrated bonds. For the isotropic configuration, the horizontal frustrated bonds are highlighted with green ovals. Each row supports the same number of frustrated bonds: $L_1=2$. (d) Mapping to the dual honeycomb lattice, showing (top) a Coulomb-phase configuration and (bottom) a pair of triplets created by shifting a single charge along a bond indicated by a green oval. Orange triangles show triplet positions. (e) Illustration of triplet propagation: a pair of triplets are created, diffuse along orange arrows, 1, 2, 3, and annihilate after circulating the system, thereby changing the global flux sector. The top (bottom) panel shows the propagation of upward (downward) triangle, which has negative (positive) charge in the arrow representation. Accordingly, the flux value $L_1$ increases (decreases) by 2, as indicated by blue arrows.
• Figure 3: (Color online) Two distinct kinetic regimes of crystallization from a topological ordered liquid. (a,b) Time evolution of the ordered fraction $\Delta_{\rm CO}$ and the triplet density $n_{\rm tpt}$ after quenches to temperatures $T_{\rm end}$ below $T_{\rm nose}$, showing that ordering begins only after $n_{\rm tpt}$ reaches its maximum $n_{\rm tpt}^{\rm (max)}$. (c,d) Corresponding results above $T_{\rm nose}$, where ordering is delayed by a long plateau in $n_{\rm tpt}$, indicating a nucleation bottleneck. (e) Ordering time $t_{\rm CO}$ plotted against inverse temperature $1/T$. At low $T$, $t_{\rm CO}$ is proportional to the inverse of maximal triplet density: $t_{\rm CO}\sim1.85/n_{\rm tpt}^{\rm (max)}$. $n_{\rm tpt}^{\rm (max)}$ follows Arrhenius law, $n_{\rm tpt}^{\rm (max)}\propto\exp(-\frac{0.57V_1}{T})$ as shown with the green solid line. (f) $\log(t_{\rm CO}/t_0)^{-1}$ is plotted against the temperature, $T$. The solid line shows a linear fitting predicted by classical nucleation theory: $\log(t_{\rm CO}/t_0)^{-1}\sim A(T_{\rm c} - T)$ with $t_0=1650$, $A=201.01$, and $T_{\rm c}=0.0789$.
• Figure 4: (Color online) Slow ordering process in the initial stage. (a,b) Time evolution of the ordered fraction $\Delta_{\rm CO}$ and the triplet density $n_{\rm tpt}$ after quenches to $T_{\rm end}=0.06$ from different initial temperatures $T_{\rm init}$ together with the result of the zero-triplet protocol. The growth of $\Delta_{\rm CO}$ accelerates for higher $T_{\rm init}$, reflecting the larger initial triplet population. (c) Log-log plot of (a) with power-law fit $Kt^n$ where $K=2.2\times10^{-4}$ and $n=0.604$ (dashed line), evidencing sublinear growth. (d) Scaling of $\Delta_{\rm CO}$ with the rescaled time $t(L_{0}/L)^{2}$ with $L_0=144$ for the three system sizes, $L=144, 168$, and $192$.The data collapse demonstrates that the ordering time diverges as $L^{2}$. The inset shows $\Delta_{\rm CO}$ in the initial time range, plotted against unscaled time, $t$.
• Figure S1: (Color online) Time evolution of the order fraction $\Delta_{\rm CO}$ obtained by KMC method for the zero-triplet quench to $T_{\rm end}=0.06$. Time series for 6 different samples, and the average over 1000 samples are shown.