Exact Density Of States Research Articles

Exact density of states and Yang–Lee edge singularity of the triangular-lattice ising model in an external magnetic field

Exact density of states and Yang–Lee edge singularity of the triangular-lattice ising model in an external magnetic field

Study of the frustrated Ising model on a square lattice based on the exact density of states

Study of the frustrated Ising model on a square lattice based on the exact density of states

Convergence estimation of flat-histogram algorithms based on simulation results

The present work considers the question of convergence of Wang–Landau, multi-canonical and stochastic approximation Monte Carlo algorithms, which was previously investigated on very different levels. In the case of the multi-canonical algorithms, a convergence behavior similar to either the original Wang–Landau or the stochastic approximation Monte Carlo algorithms is shown, depending on details of the algorithm. I suggest a method for error estimation based only on the results of simulations, not involving an information about the exact density of states, which is unknown for most practically interesting problems.

Charge- and spin-polarized currents in mesoscopic rings with Rashba spin-orbit interactions coupled to an electron reservoir

The electronic states of a mesoscopic ring are assessed in the presence of Rashba spin-orbit (SO) coupling and a $\text{U}(1)$ gauge field. Spin-symmetric coupling to an ideal lead is implemented following B\"uttiker's voltage probe. The exact density of states is derived using the reservoir uncoupled eigenstates as basis functions mixed by the reservoir coupling. The decay time of uncoupled electron eigenstates is derived by fitting the broadening profiles. Spin and charge persistent currents are computed in the presence of the SO interaction and the reservoir coupling for two distinct scenarios of the electron filling fraction. The degradation of the persistent currents depends uniformly on the reservoir coupling but nonuniformly in temperature, the latter due to the fact that currents emerge from different depths of the Fermi sea, and thus for some regimes of flux, they are provided with a protective gap. Such flux regimes can be tailored by the SO coupling for both charge and spin currents.

HP-sequence design for lattice proteins—An exact enumeration study on diamond as well as square lattice

We present an exact enumeration algorithm for identifying the native configuration--a maximally compact self-avoiding walk configuration that is also the minimum energy configuration for a given set of contact-energy schemes; the process is implicitly sequence-dependent. In particular, we show that the 25-step native configuration on a diamond lattice consists of two sheet-like structures and is the same for all the contact-energy schemes, {(-1, 0, 0); (-7, -3, 0); (-7, -3, -1); (-7, -3, 1)}; on a square lattice also, the 24-step native configuration is independent of the energy schemes considered. However, the designing sequence for the diamond lattice walk depends on the energy schemes used whereas that for the square lattice walk does not. We have calculated the temperature-dependent specific heat for these designed sequences and the four energy schemes using the exact density of states. These data show that the energy scheme (-7, -3, -1) is preferable to the other three for both diamond and square lattice because the associated sequences give rise to a sharp low-temperature peak. We have also presented data for shorter (23-, 21-, and 17-step) walks on a diamond lattice to show that this algorithm helps identify a unique minimum energy configuration by suitably taking care of the ground-state degeneracy. Interestingly, all these shorter target configurations also show sheet-like secondary structures.

Evaluation of an Efficient Monte Carlo Algorithm to Calculate the Density of States

Phase transitions and critical phenomena are the most universal phenomena in nature. To understand the phase transitions and critical phenomena of a given system as a continuous function of temperature and to obtain the partition function zeros in the complex temperature plane indicating most effectively phase transitions and critical phenomena, we need to calculate the density of states. Currently, Wang-Landau Monte Carlo algorithm is one of the most efficient Monte Carlo methods to calculate the approximate density of states. Using Wang-Landau Monte Carlo algorithm, the density of states for the Ising model on L x L square lattices (L = 4 ~ 32) with periodic boundary conditions is obtained, and the partition function zeros of the Ising model are evaluated in the complex temperature plane. By examining the behavior of the first partition function zero (partition function zero closest to the positive real axis), phase transitions and critical phenomena can be much more accurately analyzed. The approximate first zeros of the Ising ferromagnet, obtained from Wang-Landau algorithm, are quite close to the exact ones, indicating that it is a reliable method for calculating the density of states and the first partition function zeros. function zeros indicating most effectively phase transitions and critical phenomena, and to perform microcanonical analysis of phase transitions and critical phenomena, we need to calculate the density of states. Currently, Wang-Landau Monte Carlo algorithm (3) is one of the most efficient Monte Carlo methods to calculate the approximate density of states. In Wang-Landau Monte Carlo algorithm, the inverse of the density of states is employed as the sampling probability function, and the real values for the density of states can be obtained quickly due to its modification factor. Phase transitions and critical phenomena can also be understood based on the concept of partition function zeros. Fisher introduced the partition function zeros in the complex temperature plane utilizing the Onsager solution of the square-lattice Ising model in the absence of an external magnetic field (4). Fisher also showed that the partition function zeros in the complex temperature plane of the square-lattice Ising model determine its ferromagnetic and antiferromagnetic critical temperatures at the same time in the absence an external magnetic field. By calculating the partition function zeros and examining the behavior of the first partition function zero (partition function zero closest to the positive real axis) in the thermodynamic limit, phase transitions and critical phenomena can be much more accurately analyzed than examining the behavior of the specific heat per volume for real values of the temperature, which is plagued by the noise due to the subleading terms containing zeros other than the first ones (5−20). In the next section, the density of states and the partition function of the square-lattice Ising model are defined. In Section III, Wang-Landau Monte Carlo algorithm to calculate the approximate density of states is briefly explained. In Section IV, the concept of the partition function zeros in the complex temperature plane is introduced, the partition function zeros of the square-lattice Ising model are evaluated in the complex temperature plane using the exact density of states and the approximate density of states, and both results are compared.

Infrared catastrophe and tunneling into strongly correlated electron systems: Beyond the x-ray edge limit

We develop a nonperturbative method to calculate the electron propagator in low-dimensional and strongly correlated electron systems. The method builds on our earlier work using a Hubbard-Stratonovich transformation to map the tunneling problem to the x-ray edge problem, which accounts for the infrared catastrophe caused by the sudden introduction of a new electron into a conductor during a tunneling event. Here we use a cumulant expansion to include fluctuations about this x-ray edge limit. We find that the dominant effect of electron-electron interaction at low energies is to correct the noninteracting Green's function by a factor ${e}^{\ensuremath{-}S∕\ensuremath{\hbar}}$, where $S$ can be interpreted as the Euclidean action for a density field describing the time-dependent charge distribution of the newly added electron. Initially localized, this charge distribution spreads in time as the electron is accommodated by the host conductor, and during this relaxation process action is accumulated according to classical electrostatics with a screened interaction. The theory applies to lattice or continuum models of any dimensionality, with or without translational invariance. In one dimension the method correctly predicts a power-law density of states for electrons with short-range interaction and no disorder, and when applied to the solvable Tomonaga-Luttinger model, the exact density of states is obtained.

Fermionic Sherrington-Kirkpatrick models with Hubbard interaction: Magnetism and electronic structure

Models with range-free frustrated Ising spin- and Hubbard interaction are treated exactly by means of the discrete time slicing method. Critical and tricritical points, correlations, and the fermion propagator, are derived as a function of temperature T, chemical potential \mu, Hubbard coupling U, and spin glass energy J. The phase diagram is obtained. Replica symmetry breaking (RSB)-effects are evaluated up to four-step order (4RSB). The use of exact relations together with the 4RSB-solutions allow to model exact solutions by interpolation. For T=0, our numerical results provide strong evidence that the exact density of states in the spin glass pseudogap regime obeys \rho(E)=const |E-E_F| for energies close to the Fermi level. Rapid convergence of \rho'(E_F) under increasing order of RSB is observed. The leading term resembles the Efros-Shklovskii Coulomb pseudogap of localized disordered fermionic systems in 2D. Beyond half filling we obtain a quadratic dependence of the fermion filling factor on the chemical potential. We find a half filling transition between a phase for U>\mu, where the Fermi level lies inside the Hubbard gap, into a phase where \mu(>U) is located at the center of the upper spin glass pseudogap (SG-gap). For \mu>U the Hubbard gap combines with the lower one of two SG-gaps (phase I), while for \mu<U it joins the sole SG-gap of the half-filling regime (phase II). We predict scaling behaviour at the continuous half filling transition. Implications of the half-filling transition between the deeper insulating phase II and phase I for delocalization due to hopping processes in itinerant model extensions are discussed and metal-insulator transition scenarios described.

Determination of the density of states in quantum wells and quantum dot arrays by the capacitance-voltage method

The possibility of determining the electronic density of states in quantum wells and quantum dot arrays in heterostructures from the capacitance-voltage curve is investigated. In heterostructures fluctuations of the composition and geometrical dimensions play an important role. It is shown that to reconstruct the exact density of states from the measured capacitance-voltage curve is impossible, because this problem is ill-posed from the mathematical point of view. An approximate method is proposed for solving the problem, involving the determination of a “reduced” density of states. It is shown that the reduced density of states is close to the true density if the characteristic energy scale governing the variation of the latter is much greater than the thermal energy kT. The proposed method is used to find the density of states in the conduction band of a quantum well in an In0.22Ga0.78As/GaAs heterostructure.

Exact density of states of the lowest Landau level in a spin- system with uncorrelated disorders

Using the path-integral approach developed by Br\'ezin et al. [ Nucl.Phys.B235, 24 (1984) ], we obtain an analytical expression for the density of states of a spin-1/2 disordered two-dimensional electron gas in a strong, perpendicular magnetic field. The density of states of this system illustrates the interplay between the Zeeman splitting of Landau levels and the disorder-induced broadening. We find that the broadening and the band splitting of the Landau bands are enhanced due to the level repulsion from the mixing of two spin orientations caused by the random scatterings. The comparison between the spin-1/2 model and the double-layer system is also discussed.

SMOOTHED DENSITY OF STATES OF ELECTRONS FOR A MESOSCOPIC SYSTEM

For finite systems of a sphere, a disc, and a linear line, the smoothed electron density of states (DOS) is derived directly from the distribution of eigenvalues on one-particle model. The DOS for a sphere is expressed as the sum of three terms: volume, surface, and circumference. The change in the DOS from a microsystem to a macrosystem is clarified. Comparison with the results by Balian and Bloch is carried out and it is shown that the present results give better agreement with the exact DOS. The effect of boundary condition on the DOS is investigated.

Exact density of states and its critical behavior

We propose a fast algorithm for the exact computation of the density of states for arbitrary discrete systems on intermediate-size lattices. Results for the Ising model on lattices up to 13×13 are presented. We also discuss how signals for phase transitions may be observed by inspecting the density of states directly, and verify this with the numerical data for the Ising model. This procedure is based on the classical view of the density of states as the exponential of the entropy. Phase transition are then characterized by “straight sections” in the entropy, considered as a function of the energy. It is found that second- as well asfirst-order phase transitions may be so described, the difference between the two being in the way the length of the straight section goes to infinity with the size of the system. This viewpoint leads then to a short and physically transparent derivation of the Josephson inequality for critical exponents.

Fractional quantum Hall effect on the 2-sphere: a state-counting analysis

The authors consider a system of electrons confined to the surface of a (large) 2-sphere around a (strong) magnetic monopole, and interacting through a general rotational invariant potential. The Hilbert space for this system can be decomposed into subspaces with definite quantum numbers Jz, J(J+1), and N for respectively the rotation (Jz, J2) and particle number N operators. They study the systematics of the dimensions of these subspaces, which reveals many interesting patterns at low N. However, as N increases an overall smooth behaviour is approached, and they calculate this asymptotic behaviour as N to infinity . As a by-product of their analysis they find the exact density of states for the case when the electrons interact via a pair interaction V(ri,rj)=-v(1-cos theta ij), where theta ij is the spherical angle between the pair.

Defect chemistry and intrinsic carrier concentration for Hg1−x Cdx Te(s) for x = 0.20, 0.40, and 1.0

The theoretical equations governing the crystal-vapor equilibrium for Hg1−x Cdx Te(s) are summarized for a model with doubly-ionized native defects and applied to the data for x = 0.20, 0.40, and 1. The basic equations (Eq. (1) or (19)) are shown to contain those used elsewhere as special cases. Allowing the intrinsic carrier concentration, n1, to vary, but under constraints, to obtain an optimum fit and assuming a non-degenerate semiconductor, the hole con-centration-mercury pressure isotherms can be fitted satis-factorily and better than before with standard deviations between 22 and 24% for x = 0.20 and 0.40. A number of sets of model parameters give these fits, some of which give values as large as 10/cm at 500δC for the square root of the Schottky constant for ionized vacancies. In agreement with experiment, each of these parameter sets for x = 0.20 and 0.40 predicts a negligible dependence of the 77 and 192 K Hall mobilities upon equilibration temperature and predicts that metal saturation between 250 and 300δC with foreign donors in the 1015/cm3 range will result in elec-tron concentrations in the same range and dependent on saturation temperature to less than 26%. The enthalpy to create a “Hg-vacancy” and two holes is an invariant of the fits and is 2.00 ± 0.04 eV for x = 0.40 and 1.94 ± 0.02 eV for x = 0.20. Then ni is calculated independently assuming a parabolic valence band and a Kane conduction band. With an exact density of states for the latter obtained here, the momentum matrix element, P = 8.5 × 10−8 eV-cm, the spin-orbit splitting, Δ = 1 eV, and E (x, T) from Hansen et al., the experimental values below 300 K for 0 ≤x = 0.3 and at high temperature for x = 1 are fitted well with m h ∗ /m = 0.70. With these independently calculated values for ni the electron concentration-cadmium pressure isotherms for CdTe can be fitted to about 7% for either of two incon-sistent data sets assuming 2 × 1016 /cm3 foreign donors. However data for x = 0.40 can be fitted to only 42% and that for x = 0.20, which is degenerate, to only 28%.

The d-dimensional Landsberg gas

Expressions based on a recently proposed approximate density of states function are derived for thermodynamic functions of an ideal relativistic Bose gas in d dimensions. The results obtained are compared with those found using the exact density of states and agreement is found to be good in the three-dimensional case but less good in higher dimensions. It is noted that, in the extreme relativistic and non-relativistic limits, results agree exactly and, in the vicinity of the state of zero kinetic energy, the two densities of states exhibit smaller behaviour. Mention is made also of the application of this approximate density of states function to a consideration of the ideal relativistic Fermi gas.

Temperature dependence of the nuclear spin-magnon relaxation time in the two-dimensional antiferromagnetsK2MnF4andK2NiF4

The relaxation time ${T}_{1}$ of the out-of-layer $^{19}\mathrm{F}$ nuclei against decay to the magnons by mediation of hyperfine coupling has been studied as a function of temperature in the nearly ideal two-dimensional (2-D) quadratic-layer (QL) Heisenberg antiferromagnets ${\mathrm{K}}_{2}$Mn${\mathrm{F}}_{4}$ ($S=\frac{5}{2}$) and ${\mathrm{K}}_{2}$Ni${\mathrm{F}}_{4}$ ($S=1$). By virtue of the tetragonal symmetry, the lowest-order relaxation process involves three magnons; energy is conserved, but $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ is not. Theoretical expressions applicable to the QL structures are derived for ${T}_{1}$ due to the three-magnon process, including enhancement by four-magnon exchange scattering. In the numerical evaluation, the multiple summations over the Brillouin zone are carried out rigorously with the exact density of states, dispersion relation, and Bogoliubov coefficients, rather than the commonly used small-$\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ approximation. Exchange scattering turns out to play a significant role. The enhancement of the direct three-magnon relaxation rate is dependent on temperature, and amounts to up to a factor of 15, about four times larger than estimated previously. Then, generally quantitative agreement, ranging over three decades, is obtained with experiment, with the spin-wave and hyperfine parameters taken from other sources. At very low temperatures, the relaxation is impurity dominated. Residual effects due to multiple exchange scattering are estimated to add by some tens of percent. Another possible process, relaxation induced by electronic dipolar interaction, is worked out in detail for the 2-D case, only to be found negligible. Finally, breaking the symmetry by a magnetic field transverse to the tetragonal axis invokes a two-magnon process, starting off with ${T}_{1}^{\ensuremath{-}1}$ quadratically dependent on field, and dominating the three-magnon processes by two orders of magnitude at \ensuremath{\sim}35 kG. Again, the rigorous spin-wave result is found to be in excellent agreement with experiment.

Pseudofermion approach to binary disordered systems. II. Goldstone systems with off-diagonal disorder

A simple approximation is developed for the frequency spectrum of binary disordered Goldstone systems such as phonons in mixed crystals with both mass and force-constant disorder. This localized pseudofermion approximation is a simplification of the one-pseudofermion approximation previously discussed, which gives the coherent-potential approximation for systems with diagonal disorder. The approximation is shown to give properly analytic Green's functions, and to have the correct virtual-crystal, atomic, and dilute limits. Numerical results are presented for linear chains of atoms for comparison with the exact density of states, and results are given in the random-hopping electronic model for comparison with the theory of Blackman et al. It is found that the theory produces results very similar to those produced by the coherent-potential approximation for diagonal disorder. However, the neglect of pseudofermion propagation and certain overlap effects leads to somewhat incorrect locations of the impurity band edges. This effect vanishes when the disordered force constants superimpose linearly.

Calculation of electronic density of states using a self-consistent cluster-Bethe-lattice method

The electronic density of states for a periodic crystal is calculated by embedding a cluster of atoms in a Bethe-lattice medium. The structure of the Bethe-lattice Hamiltonian is determined by self-consistency requirements. The results for the density of states using this self-consistent cluster-Bethe-lattice method show excellent agreement with the exact density of states and significant improvement over previous calculations using the cluster-Bethe-lattice approach.

The density of states of a one-dimensional binary alloy by continued fractions

The density of states of a one-dimensional disordered alloy is calculated by a continued fraction technique and the results are found to agree well with those of the selfconsistent theory of Butler (1973) and with the exact density of states obtained from the Schmidt integral equation technique.

Influence of local fluctuations of concentration on the electronic density of states in disordered binary alloys

The authors present a method which improves the results of the coherent potential approximation (CPA) generally used to study the electronic properties of disordered alloys. They replace the disordered medium by an effective one characterized by a local coherent potential which varies from place to place and depends on the particular alloy configuration. The local electronic density of states is determined by equations similar to that found in the CPA, and then conveniently averaged over all the possible configurations in order to obtain the alloy density of states. As opposed to the CPA, this method yields tails in the allowed energy bands. As an illustration for the method, the authors calculate the electronic density of states of a simple one-dimensional model and compare it to the exact density of states and to the results of CPA calculation.