Semiclassical Formula Research Articles

SU(2)-Equivariant Quantum Channels: Semiclassical Analysis

We study completely positive and trace-preserving equivariant maps between operators on irreducible representations of SU(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{SU}\\,}}(2)$$\\end{document}. We find asymptotic approximations of channels in the limit of large output representation and we compute traces of functions of channel outputs. Our main tool is quantization using coherent states. We provide quantitative error bounds for various semiclassical formulas satisfied by quantizations of functions on the sphere.

CFT duals of Kerr-Taub-NUT and beyond

The duality relating the four-dimensional Kerr-Taub-NUT black hole to a thermal two-dimensional CFT with central charges cL = cR = 12J0 is analyzed in detail, generalizing an argument given recently for Kerr within the soft-hair approach. The hidden conformal symmetry is realized in the form of V irL × V irR diffeomorphisms which act non-trivially on the black hole horizon. Semiclassical formulae are derived for the temperature and central charges of the dual CFT. Assuming the applicability of the Cardy formula, these CFT quantities precisely reproduce the macroscopic Bekenstein-Hawking area law. Various further generalizations including the complete family of black holes in four dimensions are discussed.

DMRG study of the theta-dependent mass spectrum in the 2-flavor Schwinger model

We study the θ-dependent mass spectrum of the massive 2-flavor Schwinger model in the Hamiltonian formalism using the density-matrix renormalization group (DMRG). The masses of the composite particles, the pion and sigma meson, are computed by two independent methods. One is the improved one-point-function scheme, where we measure the local meson operator coupled to the boundary state and extract the mass from its exponential decay. Since the θ term causes a nontrivial operator mixing, we unravel it by diagonalizing the correlation matrix to define the meson operator. The other is the dispersion-relation scheme, a heuristic approach specific to Hamiltonian formalism. We obtain the dispersion relation directly by measuring the energy and momentum of the excited states. The sign problem is circumvented in these methods, and their results agree with each other even for large θ. We reveal that the θ-dependence of the pion mass at m/g = 0.1 is consistent with the prediction by the bosonized model. We also find that the mass of the sigma meson satisfies the semi-classical formula, Mσ/Mπ = 3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{3} $$\\end{document}, for almost all region of θ. While the sigma meson is a stable particle thanks to this relation, the eta meson is no longer protected by the G-parity and becomes unstable for θ ≠ 0.

Semiclassical perspective on Landau levels and Hall conductivity in an anisotropic cubic Dirac semimetal and the peculiar case of star-shaped classical orbits

We study an anisotropic cubic Dirac semimetal subjected to a constant magnetic field. In the case of an isotropic dispersion in the x−y plane, with parameters vx=vy, it is possible to find exact Landau levels, indexed by the quantum number n, using the typical ladder operator approach. Interestingly, we find that the lowest energy level (the zero-energy state in the case of kz=0) has a degeneracy that is 3 times that of other states. This degeneracy manifests in the Hall conductivity as a step at a zero chemical potential 3/2 the size of other steps. Moreover, as n→∞, we find energies En∝n3/2, which means the nth step as a function of the chemical potential roughly occurs at a value μ∝n3/2. We propose that these exciting features could be used to experimentally identify cubic Dirac semimetals. Subsequently, we analyze the anisotropic case vy=λvx, with λ≠1. First, we consider a perturbative treatment around λ≈1 and find that energies En∝n3/2 still hold as n→∞. To gain further insight into the Landau level structure for a maximum anisotropy, we turn to a semiclassical treatment that reveals interesting star-shaped orbits in phase space that close at infinity. This property is a manifestation of weakly localized states. Despite being infinite in length, these orbits enclose a finite phase space volume and permit finding a simple semiclassical formula for the energy, which has the same form as above. Our findings suggest that both isotropic and anisotropic cubic Dirac semimetals should leave similar experimental imprints. Published by the American Physical Society 2024

Generalized nonequilibrium Fermi's golden rule and its semiclassical approximations for electronic transitions between multiple states.

The nonequilibrium Fermi's golden rule (NE-FGR) approach is developed to simulate the electronic transitions between multiple excited states in complex condensed-phase systems described by the recently proposed multi-state harmonic (MSH) model Hamiltonian. The MSH models were constructed to faithfully capture the photoinduced charge transfer dynamics in a prototypical organic photovoltaic carotenoid-porphyrin-C60 molecular triad dissolved in tetrahydrofuran. A general expression of the fully quantum-mechanical NE-FGR rate coefficients for transitions between all pairs of states in the MSH model is obtained. Besides, the linearized semiclassical NE-FGR formula and a series of semiclassical approximations featuring Wigner and classical nuclear sampling choices and different dynamics during the quantum coherence period for the MSH model are derived. The current approach enables all the possible population transfer pathways between the excited states of the triad, in contrast to the previous applications that only addressed the donor-to-acceptor transition. Our simulations for two triad conformations serve as a demonstration for benchmarking different NE-FGR approximations and show that the difference between all levels of approximation is small for the current system, especially at room temperature. By comparing with nonadiabatic semiclassical dynamics, we observe similar timescales for the electronic population transfer predicted by NE-FGR. It is believed that the general formulation of NE-FGR for the MSH Hamiltonian enables a variety of applications in realistic systems.

Page curves and entanglement islands for the step-function Vaidya model of evaporating black holes

It was proposed recently that the fine-grained entropy of the Hawking radiation can be expressed by the semiclassical island formula, which reproduces the unitary Page curve. In this paper, we choose the “in” vacuum state and apply the quantum extremal surface construction to study the Page curve for the step-function Vaidya model of evaporating black holes in four dimensions, which is produced by the spherical null shells. Metrics of the three regions of this spacetimes are obtained. In addition, the entanglement islands for the step-function Vaidya model of evaporating black holes at very late times are studied. When cutoff surface A is located in Minkowski region III with uA< uH at very late times, we find that the location of the boundary of island ∂I depends on the value of 8M − vA + vI. Specifically, ∂I is inside, at or outside the horizon when 8M − vA + vI is less than, equal to or larger than zero respectively. Moreover, when cutoff surface A is located in Minkowski region III with uA> uH after the black hole evaporates completely, we find that entanglement island still exists and ∂I is located on an equal-time Cauchy surface of the observer A when {r}_{(A)}^2 ≥ 64GNκc.

Periodic orbit theory of Bethe-integrable quantum systems: an N-particle Berry–Tabor trace formula

One of the fundamental results of semiclassical theory is the existence of trace formulae showing how spectra of quantum mechanical systems emerge from massive interference among amplitudes related with time-periodic structures of the corresponding classical limit. If it displays the properties of Hamiltonian integrability, this connection is given by the celebrated Berry–Tabor trace formula, and the periodic structures it is built on are tori supporting closed trajectories in phase space. Here we show how to extend this connection into the domain of quantum many-body systems displaying integrability in the sense of the Bethe ansatz, where a classical limit cannot be rigorously defined due to the presence of singular potentials. Formally following the original derivation of Berry and Tabor (1976 Proc. R. Soc. A 349 101), but applied to the Bethe equations without underlying classical structure, we obtain a many-particle trace formula for the density of states of N interacting bosons on a ring, the Lieb–Liniger model. Our semiclassical expressions are in excellent agreement with quantum mechanical results for and 4 particles. For N = 2 we relate our results to the quantization of billiards with mixed boundary conditions. Our work paves the way towards the treatment of the important class of integrable many-body systems by means of semiclassical trace formulae pioneered by Michael Berry in the single-particle context.

Constructive Implementation of Semiclassical Asymptotic Formulas in a Neighborhood of a Generic Caustic Cusp

Constructive Implementation of Semiclassical Asymptotic Formulas in a Neighborhood of a Generic Caustic Cusp

Microcanonical action and the entropy of Hawking radiation

The island formula---an extremization prescription for generalized entropy---is known to result in a unitary Page curve for the entropy of Hawking radiation. This semiclassical entropy formula has been derived for Jackiw-Teitelboim (JT) gravity coupled to conformal matter using the ``replica trick'' to evaluate the Euclidean path integral. Alternatively, for eternal Anti--de Sitter black holes, we derive the extremization of generalized entropy from minimizing the microcanonical action of an entanglement wedge. The on-shell action is minus the entropy and arises in the saddle-point approximation of the (nonreplicated) microcanonical path integral. When the black hole is coupled to a bath, islands emerge from maximizing the entropy at fixed energy, consistent with the island formula. Our method applies to JT gravity as well as other two-dimensional dilaton gravity theories.

Interpretation of the X-Ray Emission Spectra of Liquid Water through Temperature and Isotope Dependence.

The interpretation of x-ray emission spectroscopy (XES) spectra in terms of their sensitivity to the hydrogen bonding and the consequent microheterogeneity in liquid water has been debated over a decade. To shed a light on this problem, we report the theoretical reproduction of the debated 1b_{1} peaks observed in the XES spectra of liquid water using semiclassical Kramers-Heisenberg formula. The essence of the temperature and isotope dependence of the 1b_{1} double peaks is explained by molecular dynamics simulations including full vibrational (O─H stretching, bending, and) modes, rotational combined with the density functional theory and core-hole induced dynamics. Some inconsistencies exist with the experimental XES profile, which illustrates the need to employ a more precise theoretical calculations for both geometry sampling and electronic structure using a more sophisticated procedure.

Trace formula for the magnetic Laplacian at zero energy level

The paper is devoted to the trace formula for the magnetic Laplacian associated with a magnetic system on a compact manifold. This formula is a natural generalization of Gutzwiller's semiclassical trace formula and reduces to it in the case when the magnetic field form is exact. It differs slightly from the Guillemin-Uribe trace formula considered in a previous paper of the author and Taimanov. Moreover, in contrast to that paper, the focus is on the trace formula at the zero energy level, which is a critical energy level. An overview of the main notions and results related to the trace formula at the zero energy level is presented, various approaches to its proof are described, and concrete examples of its computation are given. In addition, a brief review of Gutzwiller's trace formula for regular and critical energy levels is presented. Bibliography: 88 titles.

Hyperbolicity, irrationality exponents and the eta invariant

We consider the remainder term in the semiclassical limit formula of [2] for the eta invariant on a metric contact manifold, proving in general that it is controlled by volumes of recurrence sets of the Reeb flow. This particularly gives a logarithmic improvement of the remainder for Anosov Reeb flows, while for certain elliptic flows the improvement is in terms of irrationality measures of corresponding Floquet exponents.

Polaron Models with Regular Interactions at Strong Coupling

We study a class of polaron-type Hamiltonians with sufficiently regular form factor in the interaction term. We investigate the strong-coupling limit of the model, and prove suitable bounds on the ground state energy as a function of the total momentum of the system. These bounds agree with the semiclassical approximation to leading order. The latter corresponds here to the situation when the particle undergoes harmonic motion in a potential well whose frequency is determined by the corresponding Pekar functional. We show that for all such models the effective mass diverges in the strong coupling limit, in all spatial dimensions. Moreover, for the case when the phonon dispersion relation grows at least linearly with momentum, the bounds result in an asymptotic formula for the effective mass quotient, a quantity generalizing the usual notion of the effective mass. This asymptotic form agrees with the semiclassical Landau–Pekar formula and can be regarded as the first rigorous confirmation, in a slightly weaker sense than usually considered, of the validity of the semiclassical formula for the effective mass.

Complex collisions of ultracold molecules: A toy model

We introduce a model to study the collisions of two ultracold diatomic molecules in one dimension interacting via pairwise potentials. We present results for this system and argue that it offers lessons for real molecular collisions in three dimensions. We analyze the distribution of the adiabatic potentials in the hyperspherical coordinate representation as well as the distribution of near-threshold four-body bound states, systematically studying the effects of molecular properties, such as interaction strength, interaction range, and atomic mass. It is found that the adiabatic potential's nearest-neighbor energy level distribution transitions from significant level repulsion characteristic of chaos (Brody distribution) to nonchaotic (Poisson distribution) as the two molecules are separated. For the near-threshold four-atom bound states, the case where all atoms have equal masses shows a Poissonian spacing distribution, while the unequal-mass system exhibits significant level repulsion characterized by a nonzero Brody parameter. We derive a semiclassical formula for the density of states and extract from it simple scaling laws with potential depth and range. We find good agreement between the semiclassical predictions for the density of states and the full quantum mechanical calculations.

Entanglement dynamics of spins using a few complex trajectories

In this paper, we consider two spins initially prepared in a product of coherent states and study their entanglement dynamics due to a general interacting Hamiltonian. We adopt an approach that allowed the derivation of a semiclassical formula for the linear entropy of the reduced density operator, assumed as an entanglement quantifier. The resulting expression depends on sets of four trajectories, originated from the underlying classical description, and having mutually connected final phase-space points. Such classical elements, which are capable to reproduce the quantum entanglement even for long values of propagation time, arise when we assume a proper analytical continuation of the classical phase space onto a complex domain. We apply this theory to a particular physical system, showing that taking into account only a few sets of complex trajectories is enough to get an excellent agreement between the semiclassical linear entropy of the reduced density operator and its quantum counterpart.

A mechanical analog of Bohr's atom based on de Broglie's double-solution approach.

Motivated by recent developments of hydrodynamical quantum mechanical analogs [J. W. M. Bush, Annu. Rev. Fluid Mech. 47, 269-292 (2015)], we provide a relativistic model for a classical particle coupled to a scalar wave field through a holonomic constraint. In the presence of an external Coulomb field, we define a regime where the particle is guided by the wave in a way similar to the old de Broglie phase-wave proposal. Moreover, this dualistic mechanical analog of the quantum theory is reminiscent of the double-solution approach suggested by de Broglie in 1927 and is able to reproduce the Bohr-Sommerfeld semiclassical quantization formula for an electron moving in an atom.

Bohr's stopping-power formula derived for a classical free-electron gas

Bohr's centenary stopping-power formula is rederived for a free electron gas (FEG) system within the framework of nonrelativistic classical mechanics. A simple and more concise expression for the stopping power of charged particles in FEG is demonstrated on classical grounds. Using semiclassical arguments and the Euler-Maclaurin well-known mathematical formula, Bloch's correction that links Bethe's quantum theory to Bohr's classical model is also recovered. The proposed semiclassical stopping-power formula contains the main physical ingredients for a general stopping formula applicable for different systems and energies and facilitates computational calculations.

Geometric quantization of symplectic maps and Witten's asymptotic conjecture

We use the theory of Berezin-Toeplitz operators of Ma and Marinescu to study the spaces of holomorphic sections of a prequantizing line bundle over compact Kähler manifolds under deformations of the complex structure. We show that the parallel transport in the induced vector bundle over the deformation space behaves like a Toeplitz operator, and compute its first coefficient. We then use this result to establish a semi-classical trace formula for the induced quantization of symplectic maps, and give an application to Witten's asymptotic expansion conjecture for the quantum representations of the mapping class group.

Preheating from target space curvature and unitarity violation: Analysis in field space

We study particle production and unitarity violation caused by a curved target space right after inflation. We use the inflaton field value instead of cosmic time as the time variable, and derive a semiclassical formula for the spectrum of produced particles. We then derive a simple condition for unitarity violation during preheating, which we confirm by our semiclassical method and numerical solution. This condition depends not only on the target space curvature but also on the height of the inflaton potential at the end of inflation. This condition tells us, for instance, that unitarity is violated in running kinetic inflation and Higgs inflation, while unitarity is conserved in $\alpha$-attractor inflation and Higgs-Palatini inflation.

Semiclassical level density parameter with collective enhancements in nuclei

In this work, we calculate the level density parameter for several nuclei by deploying the semiclassical trace formula for spherically symmetric harmonic oscillator potential with spin-orbit interactions, which describes the quantum shell effects in terms of the classical periodic orbits. With the usage of recently proposed Laplace like formula of level density parameter by Canbula et al., the effects of collective rotations and vibrations as well as deformations in nuclei are incorporated. The comparison of our results with the experimental ones show a good agreement. Later on, the study is also extended to the calculation of nuclear level density and survival probability in several superheavy nuclei (SHN).