Suppressed Density of States in Self-Generated Coulomb Glasses

We investigate the structure of metastable states in self-generated Coulomb glasses. In dramatic contrast to disordered electron glasses, we find that these states lack marginal stability. Such absence of marginal stability is reflected by the suppression of the single-particle density of states into an exponentially soft gap of the form $g(\epsilon) \sim e^{-V / \xi |\epsilon|}$. To analytically explain this behavior, we extend the stability criterion of Efros and Shklovskii to incorporate local charge correlations, in qualitative agreement with our numerical findings. Our work suggests the existence of a new class of self-generated glasses dominated by strong geometric frustration.

Introduction -How and why glasses form is one the most intriguing open questions of modern science. A glass is a rigid material yet lacks crystalline order. It has been recognized thousands of years ago that glasses can be made by fast-cooling a liquid below its solidus temperature to avoid crystallization. The resulting supercooled liquid displays an exponential increase of its viscosity. As such, glasses are inherently non-equilibrium states of matter, even though on any reasonable experimental timescale the material appears to be static. [1][2][3][4] The natural question is why the crystal does not nucleate inside the supercooled liquid phase. One of the possible answers revolves around the existence of local short-range order in both the liquid, the supercooled liquid and the glass phase. If the short-ranged density correlations in the liquid are manifestly different than those characterizing the preferred crystalline state, a glass can be formed through fast-cooling. The system then freezes into a locally ordered frustrated configuration that requires a macroscopic number of rearrangements to lower the energy and realize the standard crystalline form.
An example is the icosahedral short-range order first proposed [5] and later observed in metallic glasses such as Ti-Zr-Ni. [6][7][8] A single icosahedron is much denser packed than any crystalline structure like fcc or hcp, which suggests that in the liquid phase the dominant density correlations are icosahedral. However, one cannot fill space with icosahedral order, leading to glassy behavior at low temperatures.
A more recent example is found in self-generated electron glasses. [9][10][11] In the organic layered materials of the θ-family, the electrons display distinct glassy features after fast-cooling to avoid a stripe ordering transition. Upon cooling, the local charge order present in the hightemperature liquid strengthens even further within the glassy regime.
In this Letter we propose, based on the assumption of glassiness due to locally frustrated order, that such self-generated glasses display a characteristic soft gap in the single-particle density of states. This implies that self-generated glasses are not marginally stable, [12] which would require saturation of the Efros-Shklovskii bound [13,14]. We observe that the presence of local charge order stabilizes the glass and suppresses the density of states to g( ) ∼ e −V /ξ| | where V is the interaction strength, and ξ is the finite correlation length of the local order. As can be seen in Fig. 1, this form is consistent with numerical simulations.
To derive this, we provide analytical arguments based on the Efros-Shklovskii stability criterion, which are then further constrained by the local charge order. We numerically verify our claim using a model of long-range interacting spinless electrons on a half-filled triangular lattice. This model was introduced earlier to describe the glassy behavior of electrons in the θ-organic compounds. [15,16] We find a remarkable agreement between our analytical form of the density of states and the numerical results. Finally, we address the limitations and implications of our model.
Definitions -We consider a model system of particles arXiv:1605.01822v2 [cond-mat.stat-mech] 11 Jun 2016 on an underlying lattice. On each lattice site the density is given by n i = 0, 1, and the total energy is given by the Coulomb repulsion between the particles, where n is the average number of particles, r ij the distance between site i and j and V some unit of energy. We define the on-site energies as i = j =i V | rij | (n j − n) such that the total energy is E = 1 2 i i (n i − n). It is important to note that we consider self-generated glasses only with a translationally invariant Hamiltonian, in contrast to the traditional electron glasses that require quenched disorder [17]. Details on the glassy behavior of this model are presented in Ref. [15], where we also showed that the effective classical description is still valid upon including a small quantum hopping term.
In the glass at low temperatures we assume the particles are frozen into a stable nonperiodic configuration. The stability requirement implies that if we move a particle from site i to site j, the total energy of the system should increase, [13] The ground state of the system naturally satisfies this single-particle stability criterion. Any configuration that satisfies the stability criterion is a metastable state [33].
Following the arguments mentioned in the introduction, we assume that these metastable states have some kind of local short-range order. This is characterized by the density correlation function where the average is over the ensemble Γ M S of metastable states. Generically the density correlation function has the structure where the sum runs over the short-range ordering wavevectors M and ξ is the correlation length. At large distances this implies that Numerical studies -We start by numerically studying the ensemble of metastable states for the model of particles on a half-filled (n = 1/2) triangular lattice of size N = L 2 with periodic boundary conditions, introduced indicates an exponential number of metastable states. Right: The distribution of the energy density of the unique metastable states, for L = 12, 18, 24, and 36. The distribution is fitted with a Gaussian around the same average energy E/N = −0.167 (which is higher than the striped ground state energy density [15]) and width σ(L). The standard deviation σ(L) decreases with increasing system size, as shown in the inset, suggesting that in the thermodynamic limit there are infinitely many metastable states with the same energy density.
in Ref. [15,16] and in Eqn. (1). The interaction potential contains both a Coulomb tail and nearest-neighbor repulsion, tunable by the parameter x, The long-range nature of the Coulomb interaction is taken care of using Ewald summation. [18] When only nearest neighbor interactions in the second term in Eqn. (6) are present, there is an exponentially large number of degenerate ground states that are not separated by barriers. The inclusion of a Coulomb tail lifts the degeneracy and creates barriers between different configurations.
To obtain these metastable states numerically, we start with a completely random configuration and lower the energy by random single-particle moves until the stability criterion is explicitly met. We find for x = 1 that the number of metastable states increases exponentially with the system size, N ms ∼ 2 αN [34], see Fig. 2. This scaling is consistent with typical models of glass formation that exhibit an exponential number of metastable states, such as the random Sherrington-Kirkpatrick model [1,2,2,20].
The ensemble of metastable states has a narrow distribution of energies. In the thermodynamic limit, all metastable states have the same energy density E/N , see extract from Π( k) by fitting to Eqn. (4) seems to be independent of the value of x.
We next proceed to compute the density of states. This is obtained by making a histogram of the on-site energies. In Fig. 1 we show the density of states for x = 1 averaged over 996 metastable states on a N = 48 × 48 lattice. The best fit of the density of states g( ) at low energies is given by the functional form g( ) = a| | −3/2 e −b/| | , rather than the powerlaw form expected from the Efros-Shklovskii bound.
The fitting parameter in the exponential (b) turns out to be proportional to xV /ξ. We compute the density of states for a range of values of x and fit it with the exponential shape. The fits themselves are shown in the top panel of Fig. 4. The parameter b as a function of x is shown in the lower panel, where we find that b is proportional to x. Upon dividing by x we see that the correlation length ξ extracted from the density of states is independent of x, consistent with the actual correlation length seen in Π( k) in Fig. 3. Note that we are not able to quantitatively equate the correlation length from Π( k) and the one obtained from g( ).
The Coulomb Gap -The spectrum of a long-ranged ordered state would have a hard gap in the spectrum. This gap is now smeared due to the spatial charge fluctuations reflecting the amorphous nature of the metastable states. We will present now analytical arguments describing these charge fluctuations, which quantitatively capture the form of the resulting soft gap.
For a given metastable state in the ensemble Γ M S , we define the origin as some empty site with onsite energy 0 asymptotically close to zero. The probability that a site at the position r is occupied equals Because of the local order, the density correlation func- tion Π( r) is the product of an oscillating function and an exponentially decaying function. Since the precise wavevector of the local order is irrelevant to further considerations, we will only consider distances r that are commensurate with the wavelength of the order, that is M · r is a multiple of 2π; as we will see, considerations for these distances will lead to strong constraints. We will denote the density correlation function at such commensurate sites by Π( r) ∼ 1 √ r e −r/ξ . Next we introduce the local distribution of on-site energies g r ( ) at distances r. Since an occupied site has negative onsite energy, the probability to find a particle at position r equals Note that the spatial average of this local density of states equals the total density of states, 1 N i g ri ( ) = g( ). Our single assumption is that the local density of states is only restricted by the stability criterion of Eqn. (2). This criterion requires that there cannot be particles at distance r with onsite energy in the range − V | r| < < 0. Consequently, g r ( ) must be zero in this range. Outside this excluded region (i.e., at lower energies), we assume that the local density of states is equal to the total density of states. Equating Eqn. (7) with Eqn. (8) at commen-surate sites, we thus find where the stability constraint of Eqn. (2) sets the upper bound on the energy integration. [35] Taking the derivative with respect to the distance r of both sides of Eqn. (9), we find at large distances This implies that at low energies, the functional shape of the density of states should be consistent with our numerical results. Note that this gap is stronger than the usual Coulomb gap in systems with quenched disorder, where the Efros-Shklovskii bound is saturated g( ) = C| | d−1 , with C a universal disorder-independent pre-factor [13,14,22]. For large disorder strength there are no charge correlations other than the correlation hole around k = 0 associated with the Coulomb tail. In this limit, the assumptions underlying Eqn (9) are invalid and the Efros-Shklovskii bound can be saturated. However, when disorder is weak compared to the Coulomb energy scale it is an open question how the gap changes from a power-law to an exponentially soft gap.
Outlook -We proposed that self-generated glasses with Coulomb interactions have an exponentially soft density of states, following Eqn. (11). This relation is satisfied in a simple model of particles on a triangular lattice that was shown earlier to form a glass [15,16].
In self-generated electron glasses, our result can be directly studied by performing tunneling experiments. The structure of the low-energy density of states can be measured indirectly via the DC conductivity. For structural glasses composed of atoms or molecules it is difficult to measure the density of states directly. In general, the existence of single-particle excitations of arbitrary low energy lead to characteristic dynamical properties such as crackling and avalanches. However, because our model does not saturate the Efros-Shklovskii bound, we do not expect a scale-invariant avalanche distribution. [12,[23][24][25] We verified our hypothesis using a simple model of Coulombicaly interacting particles on a triangular lattice. It would be interesting to see whether models that exhibit icosahedral local order also display the exponentially soft gap. [8] The arguments presented in this Letter generalize trivially to an arbitrary power-law interaction of the form V /r γ . However, interactions that decay faster than the dimensionality of the system, V ∼ 1/r γ with γ > d do not lead to glassy behavior. Indeed we were not able to reproduce a soft gap for the triangular lattice model with dipolar interactions. It remains an interesting open question whether long-range interactions are a sine qua non for glass formation. [12,26] In this work we focused only on self-generated systems, in the absence of any quenched disorder. The reason is that for disordered systems there are typically no density correlations. [36] We verified this by introducing onsite disorder in the triangular lattice model studied. Very weak disorder W V , however, only influences the states very close to the Fermi level. For low energies W the standard power-law Coulomb gap was recovered, while the density of states for intermediate energies W V was relatively unchanged. This suggests that the physics of the exponentially soft gap is stable against very weak disorder, though the precise relation between quenched disordered Coulomb glasses and self-generated ones is a matter for future research.
In this Letter, our considerations and analysis centered on the zero-temperature ensemble of metastable states. At finite temperature the gap will be filled, and earlier results are consistent with an exponentially weak scaling at finite temperature, g( = 0, T ) ∼ T −1/2 exp(−V /T ) [16]. Again, notice the relative stability compared to systems with quenched disorder where g( = 0, T ) ∼ T . [27,28] Finally, real glasses are obtained by a fast quench after which the gap needs time to develop. In fact, it has been shown that the soft gap forms extremely slowly, with power-law or even logarithmic time dependence g( = 0, t) ∼ (log t) ξ [29][30][31][32]. In our case, however, the absence of marginal stability opens up the possibility of a true thermodynamic phase transition into a glass phase. How the glass, with its soft gap and the concomitant local density correlations, may be dynamically generated in various systems following a quench to nonzero temperatures is an interesting question for future research.  In order to count the number of metastable states, we start out with a completely random half-filled initial configuration. We then propose moves between two randomly picked sites (hence including nonlocal hops) and accept the move if it lowers the energy. Once the acceptance rate of such moves becomes too small, we systematically check all possible single-particle moves whether they lower the energy. If it does, we accept the move and restart checking all possible moves. This is repeated until the stability criterion is explicitly met.
Once we have a metastable configuration, we check whether it is the same as an earlier configuration. We do this by explicitly matching configurations, including all 6L 2 possible rotations and translations. The result for the counting of the number of metastable states is shown in Table I. We have for a given system size L a collection of M random initial configurations, from which a stable configuration is constructed as described above. This way we find a set of N ms (M ) unique metastable states.
For L = 4 we can explicitly check all possible configurations, and we find N ms = 3 unique metastable configurations. For L = 6, we see that the number of unique metastable configurations saturates to a value of N ms = 93. The approach to this saturation value for smaller values of M gives us a function N ms,6 (M ). For larger systems L > 6 we approximate the expected total number of metastable states by scaling α −1 N ms,6 (αM ). This works for L = 8 and L = 10. However, for L = 12 this estimate is conservative and provides a very weak lower bound on the total number of metastable states.
Finally, we also computed the complexity, defined as the entropy associated with the number of metastable states, S = 1 L 2 log N ms . If there are less than exponentially many metastable states, the complexity should vanish. The results suggest that this is not the case, and that the complexity approaches a value 0.14 − 0.16 in the thermodynamic limit. There are therefore exponentially many metastable states in our model, consistent with glass models with quenched disorder [1,2].