Effective Hamiltonian Research Articles

Necessary Condition of Self-Organisation in Nonextensive Open Systems

In this paper, we focus on evolution from an equilibrium state in a power law form by means of q-exponentials to an arbitrary one. Introducing new q-Gibbsian equalities as the necessary condition of self-organization in nonextensive open systems, we theoretically show how to derive the connections between q-renormalized entropies (ΔS˜q) and q-relative entropies (KLq) in both Bregman and Csiszar forms after we clearly explain the connection between renormalized entropy by Klimantovich and relative entropy by Kullback-Leibler without using any predefined effective Hamiltonian. This function, in our treatment, spontaneously comes directly from the calculations. We also explain the difference between using ordinary and normalized q-expectations in mean energy calculations of the states. To verify the results numerically, we use a toy model of complexity, namely the logistic map defined as Xt+1=1-aXt2, where a∈[0,2] is the map parameter. We measure the level of self-organization using two distinct forms of the q-renormalized entropy through period doublings and chaotic band mergings of the map as the number of periods/chaotic-bands increase/decrease. We associate the behaviour of the q-renormalized entropies with the emergence/disappearance of complex structures in the phase space as the control parameter of the map changes. Similar to Shiner-Davison-Landsberg (SDL) complexity, we categorize the tendencies of the q-renormalized entropies for the evaluation of the map for the whole control parameter space. Moreover, we show that any evolution between two states possesses a unique q=q* value (not a range for q values) for which the q-Gibbsian equalities hold and the values are the same for the Bregmann and Csiszar forms. Interestingly, if the evolution is from a=0 to a=ac≃1.4011, this unique q* value is found to be q*≃0.2445, which is the same value of qsensitivity given in the literature.

Weak radiative decay Lambda _{c}^{+}rightarrow Sigma ^{+}gamma using light-cone sum rules

We calculate the decay width of the Lambda _{c}^{+}rightarrow Sigma ^{+}gamma using light-cone sum rules. For the initial quark radiation an effective Hamiltonian is constructed, where the internal quark line shrinks to a point. The final quark radiation is studied within the full theory. The leading twist light-cone distribution amplitudes of the Sigma ^+ serve as the non-perturbative input for the sum rules calculation, and the perturbative kernel is calculated at leading order. The branching fraction we obtain is mathcal{B}(Lambda _{c}^{+}rightarrow Sigma ^{+}gamma )=1.03pm 0.36 times 10^{-4}, which is below the recent upper limits <2.6 times 10^{-4} and <4.4 times 10^{-4} given by the Belle collaboration and the BESIII collaboration, respectively.

Possible restoration of particle-hole symmetry in the 5/2-quantized Hall state at small magnetic field

Motivated by the experimental observation of a quantized 5/2 thermal conductance at filling $\nu=5/2$, a result incompatible with both the Pfaffian and the Antipfaffian states, we have pushed the expansion of the effective Hamiltonian of the $5/2$ quantized Hall state to third-order in the parameter $\kappa=E_c/\hbar \omega_c \propto 1/\sqrt{B}$ controlling the Landau level mixing , where $E_c$ is the Coulomb energy and $\omega_c$ the cyclotron frequency. Exact diagonalizations of this effective Hamiltonian show that the difference in overlap with the Pfaffian and the AntiPfaffian induced at second-order is reduced by third-order corrections and disappears around $\kappa=0.4$, suggesting that these states are much closer in energy at smaller magnetic field than previously anticipated. Furthermore, we show that in this range of $\kappa$ the finite-size spectrum is typical of a quantum phase transition, with a strong reduction of the energy gap and with level crossings between excited states. These results point to the possibility of a quantum phase transition at smaller magnetic field into a phase with an emergent particle-hole symmetry that would explain the measured $5/2$ thermal conductance of the $5/2$ quantized Hall state.

Density Matrix Renormalization Group for Transcorrelated Hamiltonians: Ground and Excited States in Molecules.

We present the theory of a density matrix renormalization group (DMRG) algorithm which can solve for both the ground and excited states of non-Hermitian transcorrelated Hamiltonians and show applications in molecular systems. Transcorrelation (TC) accelerates the basis set convergence rate by including known physics (such as, but not limited to, the electron-electron cusp) in the Jastrow factor used for the similarity transformation. It also improves the accuracy of approximate methods such as coupled cluster singles and doubles (CCSD) as shown by recent studies. However, the non-Hermiticity of the TC Hamiltonians poses challenges for variational methods like DMRG. Imaginary-time evolution on the matrix product state (MPS) in the DMRG framework has been proposed to circumvent this problem, but this is currently limited to treating the ground state and has lower efficiency than the time-independent DMRG (TI-DMRG) due to the need to eliminate Trotter errors. In this work, we show that with minimal changes to the existing TI-DMRG algorithm, namely, replacing the original Davidson solver with the general Davidson solver to solve the non-Hermitian effective Hamiltonians at each site for a few low-lying right eigenstates, and following the rest of the original DMRG recipe, one can find the ground and excited states with improved efficiency compared to the original DMRG when extrapolating to the infinite bond dimension limit in the same basis set. An accelerated basis set convergence rate is also observed, as expected, within the TC framework.

Normal ordering of the angular momentum cylindrical ladder operators and their products with Wigner D0,ε 1 functions.

The operator canonical perturbation theory (CPT) is an efficient tool for solving the molecular vibration-rotation Schrödinger equation. The corresponding Watson Hamiltonian can be written using angular momentum cylindrical ladder operators (Jz, J± = Jx ∓ iJy) possessing the Lie algebra su(2) commutation relations [J+, J-] = 2Jz, [Jz, J±] = ±J±. The reduced effective Hamiltonians suitable for fitting to observed spectra are traditionally based on Hermitian basis sets, e.g., (J2)lJz n,(J+ m+J- m)+. It is beneficial to re-express such Hamiltonians using sums of normal ordered products of ladder operators Jz aJ+ bJ- c. For instance, in the CPT, the unitary transformations reduce to the normal ordering problem. Similarly, the dipole moment transition probabilities can be evaluated using Wigner functions, D0,ε 1(ε=0,±1), possessing complex commutation relations with Jα-operators. The related line strengths are proportional to the squared matrix elements of the unitary transformed dipole moment operator given by a polynomial in normal ordered products D0,ε 1Jz aJ+ bJ- c. We have applied the technique of combinatorial calculations associated with the classical representation theory and obtained compact formulas for normal ordering of the typical raw products Jz aJ+ bJ- cJz dJ+ eJ- f and Jz aJ+ bJ- cD0,ε 1. The theory of universal enveloping Lie algebras and representation theory show that the resulting formulas cannot be further improved from the mathematical point of view. The results have a wide scope of applications in other fields of physics, including other operators with the same commutation properties. The resulting expressions for normal orderings and routines for evaluation of matrix elements, as well as the systematic tests of working formulas, were coded in Fortran and the software is available through the GitHub hosting service.

Time‐average stochastic control based on a singular local Lévy model for environmental project planning under habit formation

This study applies the theory of stochastic control to an environmental project planning to counteract against the sediment starvation problem in river environments. This can be considered as a time‐average inventory problem to time‐discretely control a continuous‐time system driven by a non‐smooth jump process under habit formation disturbing project implementation. The system is modeled such that the sediment storage dynamics are physically consistent with certain experimental results. Further, the habit formation is modeled as simple linear dynamics and serves as a constraint related to the replenishment amount of the sediment. We show that the time‐average control problem is not necessarily ergodic. Consequently, the effective Hamiltonian may become a non‐constant. Thereafter, ratcheting cases as extreme cases of the irreversible habit formation are considered, owing to them being unique exactly solvable non‐ergodic control problems. The optimality equation associated with a regularized and hence well‐defined control problem is verified. Furthermore, a finite difference scheme is examined against the exactly solvable case and then applied to more complicated cases.

Effect of local Coulomb interaction on Majorana corner modes: Weak and strong correlation limits

Here we present an analysis of the evolution of Majorana corner modes realizing in a higher-order topological superconductor (HOTSC) on a square lattice under the influence of local Coulomb repulsion. The HOTSC spectral properties were considered in two regimes: when the intensities of many-body interactions are either weak or strong. The weak regime was studied using the mean-field approximation with self-consistent solutions carried out both in the uniform case and taking into account of the boundary of the finite square-shaped system. It is shown that in the uniform case the topologically nontrivial phase on the phase diagram is widened by the Coulomb repulsion. The boundary effect, resulting in an inhomogeneous spatial distribution of the correlators, leads to the appearance of the crossover from the symmetric spin-independent solution to the spin-dependent one characterized by a spontaneously broken symmetry. In the former the corner states have energies that are determined by the overlap of the excitation wave functions localized at the different corners. In the latter the corner excitation energy is defined by the Coulomb repulsion intensity with a quadratic law. The crossover is a finite size effect, i.e. the larger the system the lesser the critical value of the Coulomb repulsion. In the strong repulsion regime we derive the effective HOTSC Hamiltonian in the atomic representation and found a rich variety of interactions induced by virtual processes between the lower and upper Hubbard subbands. It is shown that Majorana corner modes still can be realized in the limit of the infinite repulsion. Although the boundaries of the topologically nontrivial phase are strongly renormalized by Hubbard corrections.

Disorder-induced anomalous Hall effect in type-I Weyl metals: Connection between the Kubo-Streda formula in the spin and chiral basis

We study the anomalous Hall effect (AHE) in tilted Weyl metals with weak Gaussian disorder under the Kubo-Streda formalism in this work. To separate the three different contributions, namely the intrinsic, side jump and skew scattering contribution, it is usually considered necessary to go to the eigenstate (chiral) basis of the Kubo-Streda formula. However, it is more straight-forward to compute the total Hall current in the spin basis. For the reason, we develop a systematic and transparent scheme to separate the three different contributions in the spin basis for relativistic systems by building a one-to-one correspondence between the Feynman diagrams of the different mechanisms in the chiral basis and the products of the symmetric and anti-symmetric part of the polarization operator in the spin basis. We obtain the three contributions of the AHE in tilted Weyl metals by this scheme and found that the side jump contribution exceeds both the intrinsic and skew scattering contribution for the low-energy effective Hamiltonian. We compared the anomalous Hall current obtained from our scheme with the results from the semi-classical Boltzmann equation approach under the relaxation time approximation and found that the results from the two approaches agree with each other in the leading order of the tilting velocity.

Halo-independent bounds on the non-relativistic effective theory of WIMP-nucleon scattering from direct detection and neutrino observations

We combine experimental constraints from direct detection searches and from neutrino telescopes looking for WIMP annihilations in the Sun to derive halo-independent bounds on each of the 28 WIMP-proton and WIMP-neutron couplings of the effective non-relativistic Hamiltonian that drives the scattering process off nuclei of a WIMP of spin 1/2. The method assumes that the velocity distribution is normalized to one and homogeneous at the the solar system scale, as well as equilibrium between WIMP capture and annihilation in the Sun, and requires to fix the WIMP annihilation channels (we assume bb̅). We consider a single non-vanishing coupling at a time, and find that for most of the couplings the degree of relaxation of the halo-independent bounds compared to those obtained by assuming the Standard Halo Model is with few exceptions relatively moderate in the low and high WIMP mass regimes, where it can be as small as a factor of ≃ 2, while in the intermediate mass range between 10 GeV and 200 GeV it can be as large as ∼ 103. An exception to this general pattern, with more moderate values of the bound relaxation, is observed in the case of spin-dependent WIMP-proton couplings with no or a comparatively small momentum suppression, for which WIMP capture is strongly enhanced because it is driven by scattering events off 1H , which is the most abundant target in the Sun. Within this class of operators the relaxation is particularly small for interactions that are driven by only the velocity-dependent term, for which the solar capture signal is enhanced compared to the direct detection one, thanks to the highest speed of scattering WIMPs within the Sun due to the larger gravitational acceleration.

Hadronic structure on the light front. II. QCD strings, Wilson lines, and potentials

This is the second paper on hadronic wave functions in the light-front formulation. We first consider only confinement effects, using the classical Nambu-Gotto string with massive end points in the light-cone gauge. We derive the light-front Hamiltonian and show how to solve it, using an expansion in suitable basis functions. We next discuss the correlators of Wilson lines on the light front, leading to an effective Hamiltonian that includes spin effects. At the end, we consider a separate problem of instanton-induced interactions, at the origin of the pion as a massless Goldstone mode on the light front.

Thermoelectricity in bilayer graphene superlattices

Low-dimensional thermoelectricity is based on the redistribution-accumulation of the electron density of states by reducing the dimension of thermoelectric structures. Superlattices are the archetype of these structures due to the formation of energy minibands and minigaps. Here, we study for the first time the thermoelectric response of gated bilayer graphene superlattices (GBGSLs). The study is based on the four-band effective Dirac Hamiltonian, the hybrid matrix method and the Landauer-Büttiker formalism. We analyze the Seebeck coefficient, the power factor, figure of merit, output power and efficiency for different temperatures and different superlattice structural parameters. We pay special attention to the impact of not only minibands and minigaps on the thermoelectric properties, but also to intrinsic resonances in bilayer graphene structures such as Breit-Wigner, Fano and hybrid resonances. In particular, we analyze the interplay between minibands and Fano resonances as a possible mechanism to improve the thermoelectric response of GBGSLs. We also compute the density of states to know if the redistribution-accumulation of electron states is implicated in the thermoelectric response of GBGSLs.

Competing chiral d -wave superconductivity and magnetic phases in the strong-coupling Hubbard model on the honeycomb lattice

We address the competing superconducting and magnetic phases on the honeycomb lattice, in a field-theoretic approach suitable to yield a low-energy perturbative theory for the strong-coupling limit of the Hubbard model, both at half filling and in the low and high hole-doped regimes. The effective low-lying Hamiltonian is presented in terms of charge (Grassmann fields) and spin [SU(2) gauge fields] degrees of freedom. We analyze the competing phases by calculating the ground-state energy, electronic spectrum, and other observables associated with the $s$- and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}+i{d}_{xy}$-wave superconducting phases, doped antiferromagnetic, and doped ferromagnetic states. We find that, while the antiferromagnetic order has the lowest ground-state energy for low hole doping near half filling, a dominant superconducting state with chiral ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}+i{d}_{xy}$-wave symmetry emerges in the vicinity of the Van Hove singularity in the high hole-doped regime, with the presence of a quantum first-order transition accompanied by spatial phase separation. We highlight that advances in the understanding of chiral superconducting states on the honeycomb lattice are relevant to a number of doped compounds including the graphene monolayer system.

Separation of quadrupole, spin, and charge across the magnetic phases of a one-dimensional interacting spin-1 gas

We study the low-energy collective properties of a one-dimensional spin-1 Bose gas using bosonization. After giving an overview of the technique, emphasizing the physical aspects, we apply it to the $S=1$ Bose-Hubbard Hamiltonian and find a separation of the quadrupole-spin-charge sectors, confirmed by time--matrix-product states (time-MPS) numerical simulations. Additionally, through the single-particle spectrum, we show the existence of the superfluid--Mott-insulator transition and the point at which the physics are described by a Heisenberg-like Hamiltonian. The magnetic phase diagrams are found for both the superfluid and insulating regimes; the latter is determined by decomposing the complete Heisenberg bilinear-biquadratic Hamiltonian, which describes the Mott insulator, into simpler, effective Hamiltonians. This allows us to keep our methods flexible and transferable to other interesting interacting condensed matter systems.

Continuous similarity transformation for critical phenomena: Easy-axis antiferromagnetic XXZ model

We apply continuous similarity transformations (CSTs) to the easy-axis antiferromagnetic XXZ-model on the square lattice. The CST flow equations are truncated in momentum space by the scaling dimension $d$ so that all contributions with $d\le 2$ are taken into account. The resulting quartic magnon-conserving effective Hamiltonian is analyzed in the zero-, one-, and two-magnon sector. In this way, a quantitative description of the ground-state energy, the one-magnon dispersion and its gap as well as of two-magnon bound states is gained for anisotropies ranging from the gapped Ising model to the gapless Heisenberg model. We discuss the critical properties of the gap closing as well as the evolution of the one-magnon roton mininum. The excitation energies of two-magnon bound states are calculated and their decay into the two-magnon continuum is determined via the inverse participation ratio.

Quantum chaos and Hénon–Heiles model: Dirac’s variational approach with Jackiw–Kerman function

A simple semiclassical Hénon–Heiles model is constructed based on Dirac’s time-dependent variational principle. We obtain an effective semiclassical Hamiltonian using a Hartree-type two-body trial wave function in the Jackiw–Kerman form. Numerical results show that quantum effects can in fact induce chaos in the nonchaotic regions of the classical Hénon–Heiles model.

Quantum speed limit of a single atom in a squeezed optical cavity mode

We theoretically study the quantum speed limit of a single atom trapped in a Fabry–Perot microresonator. The cavity mode will be squeezed when a driving laser is applied to the second-order nonlinear medium, and the effective Hamiltonian can be obtained under the Bogoliubov squeezing transformation. The analytical expression of the evolved atom state can be obtained by using the non-Hermitian Schrödinger equation for the initial excited state, and the quantum speed limit time coincides very well for both the analytical expression and the master equation method. From the perspective of quantum speed limit, it is more conducive to accelerate the evolution of the quantum state for the large detuning, strong driving, and coupling strength. For the case of the initial superposition state, the form of the initial state has more influence on the evolution speed. The quantum speed limit time is not only dependent on the system parameters but also determined by the initial state.

R2022: A DFT/MRCI Ansatz with Improved Performance for Double Excitations

A reformulation of the combined density functional theory and multireference configuration interaction method (DFT/MRCI) is presented. Expressions for ab initio matrix elements are used to derive correction terms for a new effective Hamiltonian. On the example of diatomic carbon, the correction terms are derived, focusing on the doubly excited 1Δg state, which was problematic in previous formulations of the method, as were double excitations in general. The derivation shows that a splitting of the parameters for intra- and interorbital interactions is necessary for a concise description of the underlying physics. Results for 1La and 1Lb states in polyacenes and 1Au and 1Ag states in mini-β-carotenoids suggest that the presented formulation is superior to former effective Hamiltonians. Furthermore, statistical analysis reveals that all the benefits of the previous DFT/MRCI Hamiltonians are retained. Consequently, the here presented formulation should be considered as the new standard for DFT/MRCI calculations.

Pontryagin-optimal control of a non-Hermitian qubit

Open-system quantum dynamics described by non-Hermitian effective Hamiltonians have become a subject of considerable interest. Studies of non-Hermitian physics have revealed general principles, including relationships between the topology of the complex eigenvalue space and the breakdown of adiabatic control strategies. We study here the control of a single non-Hermitian qubit, similar to recently realized experimental systems in which the non-Hermiticity arises from an open spontaneous emission channel. We review the topological features of the resulting non-Hermitian Hamiltonian and then present two distinct results. First, we illustrate how to realize any continuous and differentiable pure-state trajectory in the dynamics of a qubit that are conditioned on no emission. This result implicitly provides a workaround for the breakdown of standard adiabatic following in such non-Hermitian systems. Second, we use Pontryagin's maximum principle to derive optimal trajectories connecting boundary states on the Bloch sphere, using a cost function which balances the desired dynamics against the controller energy used to realize them. We demonstrate that the latter approach can effectively find trajectories which maintain high state purity even in the case of inefficient detection.

Solving Anderson Impurity Model by the Effective Hamiltonian Theory

We apply the Self-Consistent Effective Hamiltonian Theory (SCEHT), which uses a general variational Fermionic many-body wave function to generate an effective Hamiltonian in a quadratic form, to the Anderson impurity model. The chiral symmetry-breaking quadratic effective Hamiltonian is solved exactly for the single Fermion excitation spectrum. We validate the theory by numerically solving a model problem. The solution shows the correct Kondo resonance in the quasi-particle density of states.

Complex Berry phase and imperfect non-Hermitian phase transitions

In many classical and quantum systems described by an effective non-Hermitian Hamiltonian, spectral phase transitions, from an entirely real-energy spectrum to a complex spectrum, can be observed as a non-Hermitian parameter in the system is increased above a critical value. A paradigmatic example is provided by systems possessing parity-time ($\mathcal{PT}$) symmetry, where the energy spectrum remains entirely real in the unbroken $\mathcal{PT}$ phase while a transition to complex energies is observed in the broken $\mathcal{PT}$ phase. Such spectral phase transitions are universally sharp. However, when the system is slowly and periodically cycled, the phase transition can become smooth, i.e., imperfect, owing to the complex Berry phase associated to the cyclic adiabatic evolution of the system. This remarkable phenomenon is illustrated by considering the spectral phase transition of the Wannier-Stark ladders in a $\mathcal{PT}$-symmetric class of two-band non-Hermitian lattices subjected to an external dc field, revealing that a nonvanishing imaginary part of the Zak phase---the Berry phase picked up by a Bloch eigenstate evolving across the entire Brillouin zone---is responsible for imperfect spectral phase transitions.