Widom's Formula Research Articles

Exact formulas of the end-to-end Green's functions in non-Hermitian systems

Green's function in non-Hermitian systems has recently been revealed to be capable of directional amplification in some cases. The exact formulas for end-to-end Green's functions are significantly important for studies of both non-Hermitian systems and their applications. In this work, based on the Widom's formula, we derive exact formulas for the end-to-end Green's functions of single-band systems which depend on the roots of a simple algebraic equation. These exact formulas allow direct and accurate comparisons between theoretical results and experimentally measured quantities. In addition, we verify the prior established integral formula in the bulk region to agree with the result in our framework. We also find that the speed at which the Green's functions in the bulk region approach the prior established integral formula is not slower than an exponential decay as the system size increases. The correspondence between the signal amplification and the non-Hermitian skin effect is confirmed.

Nodal-line entanglement entropy: Generalized Widom formula from entanglement Hamiltonians

A system of fermions forming a Fermi surface exhibits a large degree of quantum entanglement, even in the absence of interactions. In particular, the usual case of a codimension one Fermi surface leads to a logarithmic violation of the area law for entanglement entropy, as dictated by the Widom formula. We here generalize this formula to the case of arbitrary codimension, which is of particular interest for nodal lines in three dimensions. We first rederive the standard Widom formula by calculating an entanglement Hamiltonian for Fermi surface systems, obtained by repurposing a trick commonly applied to relativistic theories. The entanglement Hamiltonian will take a local form in terms of a low-energy patch theory for the Fermi surface, though it is nonlocal with respect to the microscopic fermions. This entanglement Hamiltonian can then be used to derive the entanglement entropy, yielding a result in agreement with the Widom formula. The method is then generalized to arbitrary codimension. For nodal lines, the area law is obeyed, and the magnitude of the coefficient for a particular partition is non-universal. However, the coefficient has a universal dependence on the shape and orientation of the nodal line relative to the partitioning surface. By comparing the relative magnitude of the area law for different partitioning cuts, entanglement entropy can be used as a tool for diagnosing the presence and shape of a nodal line in a ground state wavefunction.

Entanglement entropy of composite Fermi liquid states on the lattice: In support of the Widom formula

Quantum phases characterized by surfaces of gapless excitations are known to violate the otherwise ubiquitous boundary law of entanglement entropy in the form of a multiplicative log correction: $S\sim L^{d-1} \log L$. Using variational Monte Carlo, we calculate the second R\'enyi entropy for a model wavefunction of the $\nu=1/2$ composite Fermi liquid (CFL) state defined on the two-dimensional triangular lattice. By carefully studying the scaling of the total R\'enyi entropy and, crucially, its contributions from the modulus and sign of the wavefunction on various finite-size geometries, we argue that the prefactor of the leading $L \log L$ term is equivalent to that in the analogous free fermion wavefunction. In contrast to the recent results of Shao et al. [PRL 114, 206402 (2015)], we thus conclude that the "Widom formula" holds even in this non-Fermi liquid CFL state. More generally, our results further elucidate---and place on a more quantitative footing---the relationship between nontrivial wavefunction sign structure and $S\sim L \log L$ entanglement scaling in such highly entangled gapless phases.

Matrix models, Toeplitz determinants and recurrence times for powers of random unitary matrices

The purpose of this paper is to study the eigenvalues [Formula: see text] of Ut where U is a large N×N random unitary matrix and t > 0. In particular we are interested in the typical times t for which all the eigenvalues are simultaneously close to 1 in different ways thus corresponding to recurrence times in the issue of quantum measurements. Our strategy consists in rewriting the problem as a random matrix integral and use loop equations techniques to compute the first-orders of the large N asymptotic. We also connect the problem to the computation of a large Toeplitz determinant whose symbol is the characteristic function of several arc segments of the unit circle. In particular in the case of a single arc segment we recover Widom's formula. Eventually we explain why the first return time is expected to converge toward an exponential distribution when N is large. Numerical simulations are provided along the paper to illustrate the results.

A Widom like formula for some Toeplitz plus Hankel determinants

A well known Widom formula expresses the determinant of a Toeplitz matrix Tn with Laurant polynomial symbol f in terms of the zeros of f. We give similar formulae for some even Toeplitz plus Hankel matrices. The formulae are based on an analytic representation of the determinant of such matrices in terms of Chebyshev polynomials.

Schur Polynomials, Banded Toeplitz Matrices and Widom's Formula

We prove that for arbitrary partitions $\mathbf{\lambda} \subseteq \mathbf{\kappa},$ and integers $0\leq c<r\leq n,$ the sequence of Schur polynomials $S_{(\mathbf{\kappa} + k\cdot\mathbf{1}^c)/(\mathbf{\lambda} + k\cdot\mathbf{1}^r)}(x_1,\dots,x_n)$ for $k$ sufficiently large, satisfy a linear recurrence. The roots of the characteristic equation are given explicitly. These recurrences are also valid for certain sequences of minors of banded Toeplitz matrices.In addition, we show that Widom's determinant formula from 1958 is a special case of a well-known identity for Schur polynomials.

198 Revision of professional roles: is this safe and effective?

A well known Widom formula expresses the determinant of a Toeplitz matrix Tn with Laurant polynomial symbol f in terms of the zeros of f. We give similar formulae for some even Toeplitz plus Hankel matrices. The formulae are based on an analytic representation of the determinant of such matrices in terms of Chebyshev polynomials.

Ion-ion correlations in electric double layers from Monte Carlo simulations and integral equation calculations

The anisotropic ion-ion correlation function g(R 1, R 2), where R 1 and R 2 are the coordinates of ions 1 and 2, is calculated for double layer systems composed of counterions between two charged, planar walls. The primitive model of the electrolyte is used. The Widom formula for the excess chemical potential is applied in the neighbourhood of a confined particle in the double layer to obtain g(R 1, R 2) from Monte Carlo (MC) simulations. The pair correlations are also calculated in the anisotropic RHNC approximation, where the reference hypernetted chain (RHNC) closure is applied at the pair (ion-ion) level and where the anisotropic bridge function is taken from an inhomogeneous reference system composed of a hard sphere fluid with the same density profile as the double-layer system. The results of the MC simulations and RHNC calculations are generally in very good agreement with each other. This demonstrates the fact that the anisotropic RHNC approximation is a very accurate method for calculating the v...

Ion-ion correlations in electric double layers from Monte Carlo simulations and integral equation calculations

The anisotropic ion-ion correlation function g(R 1, R 2), where R 1 and R 2 are the coordinates of ions 1 and 2, is calculated for double layer systems composed of counterions between two charged, planar walls. The primitive model of the electrolyte is used. The Widom formula for the excess chemical potential is applied in the neighbourhood of a confined particle in the double layer to obtain g(R 1, R 2) from Monte Carlo (MC) simulations. The pair correlations are also calculated in the anisotropic RHNC approximation, where the reference hypernetted chain (RHNC) closure is applied at the pair (ion-ion) level and where the anisotropic bridge function is taken from an inhomogeneous reference system composed of a hard sphere fluid with the same density profile as the double-layer system. The results of the MC simulations and RHNC calculations are generally in very good agreement with each other. This demonstrates the fact that the anisotropic RHNC approximation is a very accurate method for calculating the various double-layer properties. It also follows that the Widom technique is a very versatile tool in simulation studies of double-layer systems. To obtain 15 points for the pair correlation function by simulation takes about 60 times longer than to obtain 200 000 points by anisotropic RHNC calculations (the latter takes a few minutes on an IBM RS-6000/590 workstation).

Monte Carlo calculations of chemical potentials in ionic fluids by application of Widom's formula: Correction for finite-system effects

A method for correcting (single) ion chemical potentials obtained by Monte Carlo calculations using Widom's test particle formula is proposed. The method is shown to remove most of the considerable system-size dependence, which is found for these quantities.

Monte Carlo simulations of single ion chemical potentials. Results for the unrestricted primitive model

Widom's formula has been used to calculate single-ion activity coefficients for the primitive model of electrolyte systems, with unequal ionic radii, in Monte Carlo simulations based on the Metropolis sampling procedure. The results are found to be in reasonable agreement with values obtained by the MSA theory.

Monte Carlo simulations of single-ion chemical potentials. Preliminary results for the restricted primitive model

Widom's formula has been used to calculate activity coefficients for the restricted primitive model of electrolyte systems in Monte Carlo simulations based on the Metropolis sampling procedure. The agreement with literature data based on other approaches is found to be very satisfactory. The method can be used to evaluate single-ion chemical potentials.

Chemical potential by gradual insertion of a particle in Monte Carlo simulation.

The chemical potential of a fluid can be determined in a Monte Carlo simulation by gradually turning on or off the interaction of one particle with the other particles. This method, which can be employed at densities where a direct use of the Widom formula is impractical, has been tested on a two-dimensional fluid of particles with a Lennard-Jones pair interaction.

Comments on the widom Hall effect current formula

Recently Widom has derived a formula for the Hall current. The aim of the following comments is to clarify the relation between Widom's formula and the usual one.