Edge Exponents Research Articles

Electrical properties of the unit square plate

Numerical values of electrostatic properties of the cube are well investigated whereas numerical ones of the square plate are not. In this paper, we update the numerical values of electrostatic properties of the unit square plate, such as the capacitance, and the edge and corner singularity exponents. For the capacitance, we use the refined Brownian dynamics algorithm with “walk-on-planes” (WOP) method (a kind of first-passage algorithm) and for the edge singularity and corner exponents, the sphere last-passage algorithm.

On singularities of dynamic response functions in the massless regime of the XXZ spin-1/2 chain

This work extracts, by means of an exact analysis, the singular behavior of the dynamical response functions—the Fourier transforms of dynamical two-point functions—in the vicinity of the various excitation thresholds in the massless regime of the XXZ spin-1/2 chain. The analysis yields the edge exponents and associated amplitudes that describe the local behavior of the response function near a threshold. The singular behavior is derived starting from first principles considerations: the method of analysis does not rely, at any stage, on some hypothetical correspondence with a field theory or other phenomenological approaches. The analysis builds on the massless form factor expansion for the response functions of the XXZ chain obtained recently by the author. It confirms the non-linear Luttinger based predictions relative to the power-law behavior and of the associated edge exponents that arise in the vicinity of the dispersion relation of one massive excitation (hole, particle, or bound state). In addition, the present analysis shows that due to the lack of strict convexity of the particle dispersion relation and due to the presence of slow velocity branches of the bound states, there exist excitation thresholds with a different structure of edge exponents. These originate from multi-particle/hole/bound state excitations maximizing the energy at fixed momentum.

Classification of the quantum chaos in colored Sachdev-Ye-Kitaev models

The random matrix theory (RMT) can be used to classify both topological phases of matter and quantum chaos. We develop a systematic and transformative RMT to classify the quantum chaos in the colored Sachdev-Ye-Kitaev (SYK) model first introduced by Gross and Rosenhaus. Here we focus on the 2-colored case and 4-colored case with balanced number of Majorana fermion $N$. By identifying the maximal symmetries, the independent parity conservation sectors, the minimum (irreducible) Hilbert space, and especially the relevant anti-unitary and unitary operators, we show that the color degree of freedoms lead to novel quantum chaotic behaviours. When $N$ is odd, different symmetry operators need to be constructed to make the classifications complete. The 2-colored case only show 3-fold Wigner-Dyson way, and the 4-colored case show 10-fold generalized Wigner-Dyson way which may also have non-trivial edge exponents. We also study 2- and 4-colored hybrid SYK models which display many salient quantum chaotic features hidden in the corresponding pure SYK models. These features motivate us to develop a systematic RMT to study the energy level statistics of 2 or 4 un-correlated random matrix ensembles. The exact diagonalizations are performed to study both the bulk energy level statistics and the edge exponents and find excellent agreements with our exact maximal symmetry classifications. Our complete and systematic methods can be easily extended to study the generic imbalanced cases. They may be transferred to the classifications of colored tensor models, quantum chromodynamics with pairings across different colors, quantum black holes and interacting symmetry protected (or enriched) topological phases.

Quantum work of an optical lattice

A classic example of a quantum quench concerns the release of a interacting Bose gas from an optical lattice. The local properties of quenches such as this have been extensively studied however the global properties of these non-equilibrium quantum systems have received far less attention. Here we study several aspects of global non-equilibrium behavior by calculating the amount of work done by the quench as measured through the work distribution function. Using Bethe Ansatz techniques we determine the Loschmidt amplitude and work distribution function of the Lieb-Liniger gas after it is released from an optical lattice. We find the average work and its universal edge exponents from which we determine the long time decay of the Loshcmidt echo and highlight striking differences caused by the the interactions as well as changes in the geometry of the system. We extend our calculation to the attractive regime of the model and show that the system exhibits properties similar to the super Tonks-Girardaeu gas. Finally we examine the prominent role played by bound states in the work distribution and show that, with low probability, they allow for work to be extracted from the quench.

Particle-hole pairs and density–density correlations in the Lieb–Liniger model

We review the recently introduced thermodynamic form factors for pairs of particle-hole excitations on finite-entropy states in the Lieb–Liniger model. We focus on the density operator and we show how the form factors can be used for analytic computations of dynamical correlation functions. We derive a new representation for the form factors and we discuss some aspects of their structure. We rigorously show that in the small momentum limit (or equivalently, on hydrodynamic scales) a single particle-hole excitation fully saturates the spectral sum and we also discuss the contribution from two particle-hole pairs. Finally we show that thermodynamic form factors can be also used to study the ground state correlations and to derive the edge exponents.

Edge exponents in work statistics out of equilibrium and dynamical phase transitions from scattering theory in one-dimensional gapped systems

I discuss the relationship between edge exponents in the statistics of work done, dynamical phase transitions, and the role of different kinds of excitations appearing when a nonequilibrium protocol is performed on a closed, gapped, one-dimensional system. I show that the edge exponent in the probability density function of the work is insensitive to the presence of interactions and can take only one of three values: $+1/2,\phantom{\rule{0.16em}{0ex}}\ensuremath{-}1/2$, and $\ensuremath{-}3/2$. It also turns out that there is an interesting interplay between spontaneous symmetry breaking or the presence of bound states and the exponents. For instantaneous global protocols, I find that the presence of the one-particle channel creates dynamical phase transitions in the time evolution.

Universality lost: Relation between quantizations of the Hall conductance and the edge exponents in fractional quantum Hall effect

We note an implication of chiral Luttinger liquid based edge state description of the fractional quantum Hall effect. By considering several examples that involve backward moving neutral modes, arising from either composite fermions with reverse flux attached or edge reconstruction, we show that non-universality of the edge exponent implies non-universality of the Hall conductance, as measured in the two-terminal conductance.

Constraints on Conformal Field Theories in Diverse Dimensions from the Bootstrap Mechanism

Recently an efficient numerical method has been developed to implement the constraints of crossing symmetry and unitarity on the operator dimensions and operator product expansion coefficients of conformal field theories in diverse space-time dimensions. It appears that the calculations can be done only for theories lying at the boundary of the allowed parameter space. Here it is pointed out that a similar method can be applied to a larger class of conformal field theories, whether unitary or not, and no free parameter remains, provided we know the fusion algebra of the low lying primary operators. As an example we calculate using first principles, with no phenomenological input, the lowest scaling dimensions of the local operators associated with the Yang-Lee edge singularity in three and four space dimensions. The edge exponents compare favorably with the latest numerical estimates. A consistency check of this approach on the 3D critical Ising model is also made.

Form factor approach to dynamical correlation functions in critical models

We develop a form factor approach to the study of dynamical correlation functions of quantum integrable models in the critical regime. As an example, we consider the quantum non-linear Schrödinger model. We derive the long-distance/long-time asymptotic behavior of various two-point functions of this model. We also compute edge exponents and amplitudes characterizing the power-law behavior of dynamical response functions on the particle–hole excitation thresholds. These last results confirm predictions based on the non-linear Luttinger liquid method. Our results rely on a first principles derivation, based on a microscopic analysis of the model, without invoking, at any stage, any correspondence with a continuous field theory. Furthermore, our approach only makes use of certain general properties of the model, so that it should be applicable, possibly with minor modifications, to a wide class of (not necessarily integrable) gapless one-dimensional Hamiltonians.

Off-critical parafermions and the winding angle distribution of the O(n) model

Using an off-critical deformation of the identity of Duminil-Copin and Smirnov, we prove a relationship between half-plane surface critical exponents γ1 and γ11 as well as wedge critical exponents γ2(α) and γ21(α) and the exponent characterizing the winding angle distribution of the O(n) model in the half-plane, or more generally in a wedge of wedge-angle α. We assume only the existence of these exponents and, for some values of n, the conjectured value of the critical point. If we assume their values as predicted by conformal field theory, one gets complete agreement with the conjectured winding angle distribution, as obtained by CFT and Coulomb gas arguments. We also prove the exponent inequality γ1 − γ11 ⩾ 1, and its extension γ2(α) − γ21(α) ⩾ 1 for the edge exponents. We provide conjectured values for all exponents for n ∈ [ − 2, 2).

Dynamic and static density-density correlations in the one-dimensional Bose gas: Exact results and approximations

We discuss approximate formulas for the dynamic structure factor of the one-dimensional Bose gas in the Lieb-Liniger model that appear to be applicable over a wide range of the relevant parameters such as the interaction strength, frequency, and wave number. The suggested approximations are consistent with the exact results known in limiting cases. In particular, we encompass exact edge exponents as well as the Luttinger-liquid and perturbation theoretic results. We further discuss derived approximations for the static structure factor and the pair distribution function $g(x)$. The approximate expressions show excellent agreement with numerical results based on the algebraic Bethe ansatz.

Phase transitions at surfaces, edges, and corners

Results of large-scale Monte Carlo simulations of three-dimensional Ising models with edges and corners are reviewed. At the ordinary transition, angle dependent critical exponents are observed, whereas at the surface transition edge and corner critical exponents are non-universal and depend on the details of the model. The results obtained at the surface transition are compared to exact findings on critical two-dimensional Ising models with different types of defect lines.

The Inner-Shell Hole of Alkali Metals Studied by the DV-XαMethod

The discrete variational (DV) Xα method is applied to investigate the outermost inner-shell hole of all alkali metals. Calculations are performed for the model clusters M 9 and M 59 of alkali metals. The hole is assumed to be at the center of the cluster in the bcc structure. The hole potential of Rb and Cs is found to be extended to the cluster size. The result is applied to investigate the anomalous edge of X-ray absorption spectrum and the asymmetric shape of X-ray photoemission spectrum, but we could not find any peculiar characteristics expected for Cs. The obtained singularity indices α 's and the edge exponents α l 's of all the elements are very similar to those obtained by predecessors, failing to obtain a small value of α 0 for Cs which is claimed by experiment.

Numerical studies of critical percolation in three dimensions

Presents results of high-statistics simulations of the spreading of 3D percolation close to the critical point. The main results are: (i) a more precise estimate of the spreading (resp. 'chemical distance') dimension; (ii) a novel heuristic scaling theory for percolation near surfaces; and (iii) more precise values for critical surface and edge exponents. In addition, the author verifies the very precise results of Ziff and Stell (1988) for the percolation thresholds, and studies in detail corrections to scaling. The latter is made possible by simulating bond and site percolation, in a number of different geometries. In addition, in most cases one could measure more than one observable during each simulation, which provided quite severe internal consistency constraints. In some cases, the author found quite important deviations from scaling which are not easily described by a single power or logarithmic correction term.

Surface and edge exponents for the spreading of 3D percolation

Monte Carlo results are presented for the spreading of 3D percolation, where the seed consists of an (infinite) straight line. This line can either be in the bulk of the material into which spreading is possible, on a planar surface of the material, or on a rectangular edge. In the last two cases, spreading occurs only into angular regions with 180 degrees (resp 90 degrees ). While the mean distance grows at p=pc in all three cases with the same exponent, the average number of growth sites grows (resp vanishes) in all three cases with different exponents. In particular, this implies that at p=pc, the sites on the surface of a big cube belonging to the maximal cluster within that cube have fractal dimension <1.

Diffusion on a disordered Cayley tree

The model proposed recently by Brezini (1983) to calculate the average probability and the average size of the localisation domain for an electron being localised at a given site in a disordered Cayley tree is extended to the case of a uniform distribution for site energies. Thus, numerical results are presented in the limit of weak disorder. Particular attention is paid to the states near the mobility edge and critical exponents are deduced for the localisation length.

Spin-density calculations for core-electron photoemission and Auger electron line shapes, x-ray-edge exponents, and solid-state shifts

Spin-polarized density-functional conduction-electron screening calculations for core ionization and Auger transition for elements Li, Be, Na, Mg, Al, and Ca are performed. The screening is described by conduction-electron scattering phase shifts obtained from self-consistent-field ab initio calculations. Both spins are treated separately in order to study the spin-polarization effects. The core-electron states are resolved by a Hartree-Fock-Slater method, which makes it possible to apply the same local-density exchange and correlation approximation to all electrons, including the conduction electrons. X-ray-edge exponents and x-ray photoelectron (XPS) line singularity indices are extracted from the phase shifts. Auger line singularity indices are also evaluated. In the case of Na, Mg, and Al metals the results agree with those of the previous spin-symmetric calculations and experimental values. The screening of Li and Be is, however, strongly spin-polarized, which both increases the singularity index and decreases the edge exponents. This partly explains the broad $K$ edge of Li. The spin-polarized method seems to be necessary to describe the peculiar conduction-electron response to the core hole in Ca. The screening in this case is localized in energy, totally spin polarized, and of $d$-electron type. This should cause very broad x-ray edges and the most asymmetric XPS and Auger lines. By performing the corresponding free-atom calculations with the same method, the solid-state shifts (extra-atomic relaxation energies) for the photoelectron energies (binding energies) and Auger electron energies are obtained. The density-functional results agree better with the experiments than the results of the excitonic model or excited-atom model, for example. In the case of Ca the shift values are exceptionally large due to the localized $d$-electron screening.

Yang-Lee Edge Singularity from a Real-Space Renormalization-Group Method

The quantum one-dimensional transverse Ising model with an imaginary longitudinal field is studied at T=0 using a real-space renormalization-group method. Evaluation is made of the real part of the ground-state longitudinal magnetization which is related to the Yang-Lee distribution of zeros of the partition function for the two-dimensional classical Ising model. The edge exponents are found as sigma=-0.25 +- 0.05 and eta=-1.0 +- 0.02, in agreement with series expansion estimates.