Self-consistent Mean-field Calculations Research Articles

Decomposing Imaginary-Time Feynman Diagrams Using Separable Basis Functions: Anderson Impurity Model Strong-Coupling Expansion

We present a deterministic algorithm for the efficient evaluation of imaginary-time diagrams based on the recently introduced discrete Lehmann representation (DLR) of imaginary-time Green’s functions. In addition to the efficient discretization of diagrammatic integrals afforded by its approximation properties, the DLR basis is separable in imaginary-time, allowing us to decompose diagrams into linear combinations of nested sequences of one-dimensional products and convolutions. Focusing on the strong-coupling bold-line expansion of generalized Anderson impurity models, we show that our strategy reduces the computational complexity of evaluating an Mth-order diagram at inverse temperature β and spectral width ωmax from O((βωmax)2M−1) for a direct quadrature to O(M(log(βωmax))M+1), with controllable high-order accuracy. We benchmark our algorithm using third-order expansions for multiband impurity problems with off-diagonal hybridization and spin-orbit coupling, presenting comparisons with exact diagonalization and quantum Monte Carlo approaches. In particular, we perform a self-consistent dynamical mean-field theory calculation for a three-band Hubbard model with strong spin-orbit coupling representing a minimal model of Ca2RuO4, demonstrating the promise of the method for modeling realistic strongly correlated multiband materials. For both strong and weak coupling expansions of low and intermediate order, in which diagrams can be enumerated, our method provides an efficient, straightforward, and robust blackbox evaluation procedure. In this sense, it fills a gap between diagrammatic approximations of the lowest order, which are simple and inexpensive but inaccurate, and those based on Monte Carlo sampling of high-order diagrams. Published by the American Physical Society 2024

Pseudo-Goldstone Modes and Dynamical Gap Generation from Order by Thermal Disorder.

Accidental ground state degeneracies-those not a consequence of global symmetries of the Hamiltonian-are inevitably lifted by fluctuations, often leading to long-range order, a phenomenon known as "order-by-disorder" (ObD). The detection and characterization of ObD in real materials currently lacks clear, qualitative signatures that distinguish ObD from conventional energetic selection. We show that for order by thermal disorder (ObTD) such a signature exists: a characteristic temperature dependence of the fluctuation-induced pseudo-Goldstone gap. We demonstrate this in a minimal two-dimensional model that exhibits ObTD, the ferromagnetic Heisenberg-compass model on a square lattice. Using spin-dynamics simulations and self-consistent mean-field calculations, we determine the pseudo-Goldstone gap, Δ, and show that at low temperatures it scales as the square root of temperature, sqrt[T]. We establish that a power-law temperature dependence of the gap is a general consequence of ObTD, showing that all key features of this physics can be captured in a simple model of a particle moving in an effective potential generated by the fluctuation-induced free energy.

Nonlinear Transverse Magnetic Susceptibility under Electric Toroidal Dipole Ordering

An electric toroidal dipole (ETD) moment is one of the fundamental dipole moments as well as electric and magnetic ones. Although it directly couples to neither an electric nor magnetic field due to its spatial inversion and time-reversal parities, its ordered state leads to unconventional transverse responses of the conjugate physical quantities. We here theoretically investigate nonlinear transverse magnetic susceptibility under the ETD ordering. By performing a self-consistent mean-field calculation for a five $d$-orbital model under a tetragonal crystalline electric field and using the nonlinear Kubo formula, we show that a third-order transverse magnetic susceptibility corresponding to a uniform magnetization perpendicular to the external magnetic field becomes nonzero once the ETD moment is ordered under tetragonal crystalline electric field. Moreover, we find that spin-orbital entanglement and a low-lying first excited crystal-field level are important for realizing large transverse responses.

Time-dependent Dirac equation applied to one-proton radioactive emission

Relativistic energy-density functional (REDF) theory has been developed and utilized for self-consistent mean-field calculations of atomic nuclei. The proton-emitting radioactivity can provide a suitable reference to improve the predicting ability of REDF especially on the proton-drip line. One needs to consider the quantum tunneling effect, which plays an essential role in nucleon-emitting radioactive processes. However, the relativistic quantum tunneling has been less investigated compared with the nonrelativistic case. This work is devoted to a theoretical evaluation of one-proton ($1p$) radioactivity based on the relativistic Dirac formalism. For this purpose, I develop the time-dependent (TD) Dirac-spinor calculation to simulate the $1p$ emission. By utilizing the relativistic Hartree-Bogoliubov (RHB) calculation with the DD-PCX parameters, single-proton potentials for the time-dependent Dirac spinor are determined. The TD-Dirac calculation is applied to the $1p$ emissions from the $^{37}\mathrm{Sc}$ and $^{39}\mathrm{Sc}$ nuclei, which can be well approximated as the valence proton and the proton-close-shell cores. The sensitivity of $1p$-emission energy and decaying width to the mass number is demonstrated. Remarkable sensitivity exists due to the size of the system, which affects the nuclear part of potentials and energy levels, whereas the Coulomb barrier is common with the same atomic number. The calculated $1p$ energy and decaying lifetime are roughly consistent to the experimental limitation. The present TD-Dirac calculation is expected as applicable widely to proton-rich nuclides in order to improve the REDF by utilizing the $1p$-emission data.

Investigation of γ softness: Lifetime measurements in Ru104,106

Lifetimes of the ${2}_{1}^{+}, {4}_{1}^{+}, {6}_{1}^{+}, {2}_{\ensuremath{\gamma}}^{+}$, and ${3}_{\ensuremath{\gamma}}^{+}$ states in $^{104,106}\mathrm{Ru}$ were measured by using the recoil-distance Doppler-shift technique and the Cologne Plunger device. Low-lying excited states in both nuclei were populated in a $^{104}\mathrm{Ru}(^{18}\mathrm{O},^{18}\mathrm{O})^{104}\mathrm{Ru}$* inelastic scattering and in a $^{104}\mathrm{Ru}(^{18}\mathrm{O},^{16}\mathrm{O})^{106}\mathrm{Ru}$ two-neutron transfer reaction using the Cologne FN Tandem accelerator. The experimental energy levels and deduced electromagnetic transition probabilities are compared in the context of $\ensuremath{\gamma}$ softness and the mapped interacting boson model with input from the microscopic self-consistent mean-field calculation using a Gogny interaction. The newly obtained results for the $\ensuremath{\gamma}$ band give a more detailed insight about the triaxial behavior of $^{104,106}\mathrm{Ru}$. The results will be discussed in the context of $\ensuremath{\gamma}$ soft and rigid triaxial behavior which is present in the neutron-rich Ru isotopes.

Collective-model description of shape coexistence and intruder states in cadmium isotopes based on a relativistic energy density functional

Low-energy structure of even-even $^{108-116}$Cd isotopes is analyzed using a collective model that is based on the nuclear density functional theory. Spectroscopic properties are computed by solving the triaxial quadrupole collective Hamiltonian, with parameters determined by the constrained self-consistent mean-field calculations within the relativistic Hartree-Bogoliubov method employing a universal energy density functional and a pairing force. The collective Hamiltonian reproduces the observed quadrupole phonon states of vibrational character, which are based on the moderately deformed equilibrium minimum in the mean-field potential energy surface. In addition, the calculation yields a low-lying excited $0^+$ band and a $\gamma$-vibrational band that are associated with a deformed local minimum close in energy to the ground state, consistently with the empirical interpretation of these bands as intruder bands. Observed energy spectra, $B(E2)$, and $\rho^2(E0)$ values are, in general, reproduced reasonably well.

Skyrmion and vortex crystals in the Hubbard model

A mutual interplay between the charge and spin degrees of freedom in itinerant magnets leads to a plethora of topological spin textures, such as magnetic skyrmion and vortex crystals, in both centrosymmetric and noncentrosymmetric hosts. Meanwhile, their stabilization has been extensively studied in the system including the classical localized spins. We here study a realization of the skyrmion crystal in the centrosymmetric triangular-lattice Hubbard model, where the itinerant nature of electrons plays a more significant role. By performing the self-consistent mean-field calculations, we find that two types of skyrmion crystals with spatially nonuniform charge modulations appear in the ground state at a zero magnetic field. Moreover, we obtain another noncoplanar vortex crystal phase without a net scalar chirality in the vicinity of the skyrmion crystal phase. We show that the latter vortex crystal exhibits the topological phase transition to a different skyrmion phase in an applied magnetic field. Our results provide a possibility of the skyrmion and vortex crystals in itinerant magnets without the localized moments.

Mean-Field Study of Antiferromagnetic and Antiferroquadrupolar Orderings in Tetragonal CeCoSi

We investigate the stability of the multipolar orderings in f-electron material CeCoSi based on a self-consistent mean-field calculation for the effective localized model. This material has two ordered phases in the temperature–pressure phase diagram: the antiferromagnetic phase and the nonmagnetic phase, the latter of which has been suggested to be an antiferroquadrupolar phase. Meanwhile, the origin of the antiferroquadrupolar phase has been unclear, since a quadrupole degree of freedom is present only between the ground-state level and highly separated excited-state level under a tetragonal crystalline electric field (CEF), whose energy scale is much larger than the transition temperature. To understand the sequence of the phase transition from the paramagnetic phase, antiferroquadrupolar phase, and antiferromagnetic phase when decreasing the temperature, we examine the important interaction parameters in the effective localized model. We clarify that the 3z2 − r2-type of the antiferroquadrupolar interactions can renormalize the CEF level splitting, which might assist a quadrupolar ordering even in a tetragonal system without orbital degeneracy. Moreover, the stability of the antiferroquadrupolar and antiferromagnetic states in a magnetic field and the behavior of the magnetic/quadrupolar susceptibility are also presented for the information to identify the unknown order parameter in the nonmagnetic ordered phase.

Effect of configuration mixing on quadrupole and octupole collective states of transitional nuclei

A model is presented that simultaneously describes shape coexistence and quadrupole and octupole collective excitations within a theoretical framework based on the nuclear density functional theory and the interacting boson model. An optimal interacting-boson Hamiltonian that incorporates the configuration mixing between normal and intruder states, as well as the octupole degrees of freedom, is identified by means of self-consistent mean-field calculations using a universal energy density functional and a pairing interaction, with constraints on the triaxial quadrupole and the axially-symmetric quadrupole and octupole shape degrees of freedom. An illustrative application to the transitional nuclei $^{72}$Ge, $^{74}$Se, $^{74}$Kr, and $^{76}$Kr shows that the inclusion of the intruder states and the configuration mixing significantly lower the energy levels of the excited $0^+$ states, and that the predicted low-lying positive-parity states are characterized by the strong admixture of nearly spherical, weakly deformed oblate, and strongly deformed prolate shapes. The low-lying negative-parity states are shown to be dominated by the deformed intruder configurations.

2D gapless topological superfluids generated by pairing phases

We systematically investigate the ground state phase diagram and the finite temperature phase transitions for a Rydberg-dressed Fermi gas loaded in a bilayer optical lattice. When an effective finite-ranged attraction is induced, our self-consistent mean-field calculation shows that the gapped topological (p-wave) superfluids in each layer are coupled together by the s-wave pairing in an intermediate inter-layer distance with a spontaneously modulated phases between these two order parameters. The obtained ground state is a gapless topological superfluid with quantized topological charges characterizing the gapless points, leading to a zero energy flat band at the edges. Finally, we calculate the finite temperature phase diagrams of this two-dimensional gapless superfluid and observe two distinct critical temperatures, demonstrating the fruitful many-body effects on a paired topological superfluids.

Signatures of shape phase transitions in krypton isotopes based on relativistic energy density functionals

Spectroscopic properties that characterize the shape phase transitions in krypton isotopes with the mass $A\approx80$ region are investigated within the framework of the nuclear density functional theory. Triaxial quadrupole constrained self-consistent mean-field calculations that employ relativistic energy density functionals and a pairing interaction are carried out for the even-even nuclei $^{76-86}$Kr. The spectroscopic properties are computed by solving the triaxial quadrupole collective Hamiltonian, with the ingredients, i.e., the deformation-dependent moments of inertia and mass parameters, and the collective potential, determined by using the SCMF solutions as microscopic inputs. Systematic behaviors of the SCMF potential energy surfaces, the corresponding low-energy spectra, electric quadrupole and monopole transition probabilities, and the fluctuations in the triaxial quadrupole deformations indicate evolution of the underlying nuclear structure as functions of the neutron number, that is characterized by a considerable degree of shape mixing. A special attention is paid to the transitional nucleus $^{82}$Kr, which has been recently identified experimentally as an empirical realization of the E(5) critical-point symmetry.

Octupole correlations in collective excitations of neutron-rich N≈56 nuclei

Octupole correlations in the low-energy collective states of neutron-rich nuclei with the neutron number $N\approx56$ are studied within the interacting boson model (IBM) that is based on the nuclear density functional theory. The constrained self-consistent mean-field (SCMF) calculations using a universal energy density functional and a pairing interaction provide the potential energy surfaces in terms of the axially-symmetric quadrupole and octupole deformations for the even-even nuclei $^{86-94}$Se, $^{88-96}$Kr, $^{90-98}$Sr, $^{92-100}$Zr, and $^{94-102}$Mo. The SCMF energy surface is then mapped onto the energy expectation value of a version of the IBM in the boson condensate state, which consists of the neutron and proton monopole $s$, quadrupole $d$, and octupole $f$ bosons. This procedure determines the strength parameters of the IBM Hamiltonian, which is used to compute relevant spectroscopic properties. At the SCMF level, no octupole deformed ground state is obtained, while the energy surface is generally soft in the octupole deformation at $N\approx56$. The predicted negative-parity yrast bands with the bandhead state $3^-_1$ weakly depend on $N$ and become lowest in energy at $N=56$ in each of the considered isotopic chains. The model further predicts finite electric octupole transition rates between the lowest negative- and the positive-parity ground-state bands.

β decay and evolution of low-lying structure in Ge and As nuclei

A simultaneous calculation for the shape evolution and the related spectroscopic properties of the low-lying states, and the $\beta$-decay properties in the even- and odd-mass Ge and As nuclei in the mass $A\approx70-80$ region, within the framework of the nuclear density functional theory and the particle-core coupling scheme, is presented. The constrained self-consistent mean-field calculations using a universal energy density functional (EDF) and a pairing interaction determines the interacting-boson Hamiltonian for the even-even core nuclei, and the essential ingredients of the particle-boson interactions for the odd-nucleon systems, and of the Gamow-Teller and Fermi transition operators. A rapid structural evolution from $\gamma$-soft oblate to prolate shapes, as well as the spherical-oblate shape coexistence around the neutron sub-shell closure $N=40$, is suggested to occur in the even-even Ge nuclei. The predicted low-energy spectra, electromagnetic transition rates, and $\beta$-decay $\log{ft}$ values are in a reasonable agreement with experiment. The predicted $\log{ft}$ values reflect the structures of the wave functions for the initial and final nuclei of $\beta$ decay, which are, to a large extent, determined by the microscopic input provided by the underlying EDF calculation.

Two-neutrino double- β decay in the mapped interacting boson model

A calculation of two-neutrino double-$\beta$ ($2\nu\beta\beta$) decay matrix elements within the interacting boson model (IBM) that is based on the nuclear density functional theory is presented. The constrained self-consistent mean-field (SCMF) calculation using a universal energy density functional (EDF) and a pairing interaction provides potential energy surfaces with triaxial quadrupole degrees of freedom for even-even nuclei corresponding to the initial and final states of the $2\nu\beta\beta$ decays of interest. The SCMF energy surface is then mapped onto the bosonic one, and this procedure determines the IBM Hamiltonian for the even-even nuclei. The same SCMF calculation provides the essential ingredients of the interacting boson fermion-fermion model (IBFFM) for the intermediate odd-odd nuclei and the Gamow-Teller and Fermi transition operators. The EDF-based IBM and IBFFM provide a simultaneous description of excitation spectra and electromagnetic transition rates for each nucleus, and single-$\beta$ and $\beta\beta$ decay properties. The calculated $2\nu\beta\beta$-decay nuclear matrix elements are compared with experiment and with those from earlier theoretical calculations.

Competition between weak and α -decay modes in superheavy nuclei

The competition between $\ensuremath{\alpha}$- and ${\ensuremath{\beta}}^{+}/\text{EC}$-decay modes is studied systematically in nuclei along the $\ensuremath{\alpha}$-decay chains following the synthesis of superheavy nuclei with $Z=119$ and $Z=120$. A microscopic approach based on deformed self-consistent Hartree-Fock mean-field calculations with Skyrme forces and pairing correlations is used to describe the ${\ensuremath{\beta}}^{+}$ and electron capture weak decays, whereas the $\ensuremath{\alpha}$ decay is estimated from existing phenomenological expressions. It is shown that $\ensuremath{\alpha}$ decay is in most cases the dominant decay mode, but interesting instances are identified where the half-lives are comparable, opening the possibility of new pathways towards more neutron-rich nuclei in this region.

Metastable spin-phase diagrams in antiferromagnetic Bose-Einstein condensates

Spinor Bose-Einstein condensates under external magnetic fields exhibit well-characterized spin domains of its ground state due to spin-dependent interactions. At low temperatures, collision-induced spin-mixing instabilities may promote the condensate to dwell into metastable states occurring near the phase boundaries. In this work, we study theoretically the metastable spin-phase diagram of a spin-1 antiferromagnetic Bose-Einstein condensate at zero and finite temperatures. The approach makes use of Hartree-Fock theory and exploits the symmetry of the Hamiltonian and of the order parameters yielding a closed system of transcendental equations for the free energy, fully avoiding the use of selfconsistency. Our results are consistent with recent experiments and allow us to explain qualitatively the different types of observed quench dynamics. In addition, we found that similar phenomena should occur in antiferromagnetic spinor condensates with a sudden change in the temperature. It is shown also that the increase of temperature induces a traceable shift of the Ferromagnetic-Polar transition boundary, behavior previously not noticed by selfconsistent mean-field calculations.

Temperature and Density Conditions for Alpha Clustering in Excited Self-Conjugate Nuclei

Starting from experimental studies on alpha-clustering in excited self-conjugate nuclei (from 16O to 28Si), temperature and density conditions for such a clustering are determined. Measured temperatures have been found in the range of 5.5–6.0 MeV, whereas density values of 0.3–0.4 times the saturation density are deduced, i.e., 0.046 to 0.062 fm−3. Such a density domain is also predicted by constrained self-consistent mean field calculations. These results constitute a benchmark for alpha clustering from self-conjugate nuclei in relation to descriptions of stellar evolution and supernovae.

Shape evolution of Hg isotopes within the covariant density functional theory

The shape evolution in the neutron-deficient Hg region is investigated within the covariant density functional framework. We study in detail the chain of even–even mercury isotopes 190−200Hg using the relativistic point coupling model. The low-energy excitation spectrum and the B(E2) transitions rates of even–even nuclei are obtained as solutions of a five-dimensional collective Hamiltonian (5DCH) model, with parameters determined by constrained self-consistent mean-field calculations based on the relativistic energy density functional DD-PC1, and a finite-range pairing interaction. The calculations suggest a very interesting structure evolution with coexisting configurations for 190Hg, increased collectivity for the isotopes 192−198Hg and a more spherical structure for 200Hg.

Coupling of pairing and triaxial shape vibrations in collective states of γ -soft nuclei

In addition to shape oscillations, low-energy excitation spectra of deformed nuclei are also influenced by pairing vibrations. The simultaneous description of these collective modes and their coupling has been a long-standing problem in nuclear structure theory. Here we address the problem in terms of self-consistent mean-field calculations of collective deformation energy surfaces, and the framework of the interacting boson approximation. In addition to quadrupole shape vibrations and rotations, the explicit coupling to pairing vibrations is taken into account by a boson-number non-conserving Hamiltonian, specified by a choice of a universal density functional and pairing interaction. An illustrative calculation for $^{128}$Xe and $^{130}$Xe shows the importance of dynamical pairing degrees of freedom, especially for structures built on low-energy $0^+$ excited states, in $\gamma$-soft and triaxial nuclei.

Microscopic description of octupole collective excitations near N=56 and N=88

Octupole deformations and related collective excitations are analyzed using the framework of nuclear density functional theory. Axially-symmetric quadrupole-octupole constrained self-consistent mean-field (SCMF) calculations with a choice of universal energy density functional and a pairing interaction are performed for Xe, Ba, and Ce isotopes from proton-rich to neutron-rich regions, and neutron-rich Se, Kr, and Sr isotopes, in which enhanced octupole correlations are expected to occur. Low-energy positive- and negative-parity spectra and transition strengths are computed by solving the quadrupole-octupole collective Hamiltonian, with the inertia parameters and collective potential determined by the constrained SCMF calculations. Octupole-deformed equilibrium states are found in the potential energy surfaces of the Ba and Ce isotopes with $N\approx 56$ and 88. The evolution of spectroscopic properties indicates enhanced octupole correlations in the regions corresponding to $N\approx Z\approx 56$, $Z\approx 88$ and $Z\approx 56$, and $N\approx 56$ and $Z\approx 34$. The average $\beta_{30}$ deformation parameter and its fluctuation exhibit signatures of octupole shape phase transition around $N=56$ and 88.