Quasiparticle Vacuum Research Articles

Robust formulation of Wick's theorem for computing matrix elements between Hartree-Fock-Bogoliubov wavefunctions.

Numerical difficulties associated with computing matrix elements of operators between Hartree-Fock-Bogoliubov (HFB) wavefunctions have plagued the development of HFB-based many-body theories for decades. The problem arises from divisions by zero in the standard formulation of the nonorthogonal Wick's theorem in the limit of vanishing HFB overlap. In this Communication, we present a robust formulation of Wick's theorem that stays well-behaved regardless of whether the HFB states are orthogonal or not. This new formulation ensures cancellation between the zeros of the overlap and the poles of the Pfaffian, which appears naturally in fermionic systems. Our formula explicitly eliminates self-interaction, which otherwise causes additional numerical challenges. A computationally efficient version of our formalism enables robust symmetry-projected HFB calculations with the same computational cost as mean-field theories. Moreover, we avoid potentially diverging normalization factors by introducing a robust normalization procedure. The resulting formalism treats even and odd number of particles on equal footing and reduces to Hartree-Fock as a natural limit. As proof of concept, we present a numerically stable and accurate solution to a Jordan-Wigner-transformed Hamiltonian, whose singularities motivated the present work. Our robust formulation of Wick's theorem is a most promising development for methods using quasiparticle vacuum states.

Two-orbital quantum discord in fermion systems

A simple expression to compute the quantum discord between two orbitals in fermion systems is derived using the parity superselection rule. As the correlation between orbitals depends on the basis chosen, we discuss a special orbital basis, the natural one. We show that quantum correlations between natural orbital pairs disappear when the pairing tensor is zero, i.e. the particle number symmetry is preserved. The Hartree-Fock orbitals within a Slater determinant state, a Hartree-Fock-Bogoliubov quasiparticle orbitals in a quasiparticle vacuum, or the ground state of a Hamiltonian with particle symmetry and their corresponding natural orbitals are some relevant examples of natural basis and their corresponding states. Since natural orbitals have that special property, we seek for the quantum discord in non-natural orbital basis. We analyze our findings in the context of the Lipkin-Meshkov-Glick and Agassi models.

Torque equilibrium spin wave theory of Raman scattering in an anisotropic triangular lattice antiferromagnet with Dzyaloshinskii-Moriya interaction

We apply torque equilibrium spin wave theory (TESWT) to investigate an anisotropic XXZ antiferromagnetic model with Dzyaloshinskii-Moriya interaction in a triangular lattice. Considering the quasiparticle vacuum as our reference, we provide an accurate analysis of the noncollinear ground state of a frustrated triangular lattice magnet using the TESWT formalism. We elucidate the effects of quantum fluctuations on the ordering wave vector based on model system parameters. We study the single-magnon dispersion, the two-magnon continuum using the spectral function, and the Raman spectrum of bimagnon and trimagnon excitations. We present our results for the $HH,\phantom{\rule{0.28em}{0ex}}VV$, and the $HV$ polarization Raman geometry dependence of the bimagnon and trimagnon excitation spectra where $H\phantom{\rule{0.28em}{0ex}}(V)$ represents horizontal (vertical) polarization. Our calculations show that both the $HH$ and the $HV$ polarization spectra can be used to determine the degree of anisotropy of our system. We calculate the Raman spectra of ${\mathrm{Ba}}_{3}{\mathrm{CoSb}}_{2}{\mathrm{O}}_{9}$ and ${\mathrm{Cs}}_{2}{\mathrm{CuCl}}_{4}$.

Symmetry restoration in the mean-field description of proton-neutron pairing

We show that the symmetry-restored paired mean-field states (quasiparticle vacua) properly account for isoscalar versus isovector nuclear pairing properties. Full particle-number, spin, and isospin symmetries are restored in a simple SO(8) proton-neutron pairing model, and prospects to implement a similar approach in a realistic setting are delineated. Our results show that, provided all symmetries are restored, the pictures based on pair-condensate and quartet-condensate wave functions represent equivalent ways of looking at the physics of nuclear proton-neutron pairing.

Condensation of quasiparticles and density modulation beyond the superfluid critical velocity

We extend our earlier study of the ground state of a bosonic quasiparticle Hamiltonian by investigating the effect of a constant external velocity field. Below a critical velocity the ground state is a quasiparticle vacuum, corresponding to a pure superfluid phase at zero temperature. Beyond the critical velocity energy minimization leads to a macroscopic condensation of quasiparticles at a nonzero wave vector ${\mathbf{k}}_{\mathbf{v}}$ parallel to the velocity $\mathbf{v}$. Simultaneously, physical particles also undergo a condensation at ${\mathbf{k}}_{\mathbf{v}}$ and, to a smaller extent, at $\ensuremath{-}{\mathbf{k}}_{\mathbf{v}}$. Together with the Bose-Einstein condensation at $\mathbf{k}=\mathbf{0}$, the three coexisting condensates give rise to density modulations of wave vectors ${\mathbf{k}}_{\mathbf{v}}$ and $2{\mathbf{k}}_{\mathbf{v}}$. For larger $|\mathbf{v}|$ our model predicts a bifurcation of ${\mathbf{k}}_{\mathbf{v}}$ with corresponding two pure condensates and no density modulation.

Pair-transfer probability in open- and closed-shell Sn isotopes

Approximations made to estimate two-nucleon transfer probabilities in ground-state to ground-state transitions and physical interpretation of these probabilities are discussed. Probabilities are often calculated by approximating both ground states, of the initial nucleus A and of the final nucleus A\pm 2 by the same quasiparticle vacuum. We analyze two improvements of this approach. First, the effect of using two different ground states with average numbers of particles A and A\pm2 is quantified. Second, by using projection techniques, the role of particle number restoration is analyzed. Our analysis shows that the improved treatment plays a role close to magicity, leading to an enhancement of the pair-transfer probability. In mid-shell regions, part of the error made by approximating the initial and final ground states by a single vacuum is compensated by projecting onto good particle number. Surface effects are analyzed by using pairing interactions with a different volume-to-surface mixing. Finally, a simple expression of the pair-transfer probability is given in terms of occupation probabilities in the canonical basis. We show that, in the canonical basis formulation, surface effects which are visible in the transfer probability are related to the fragmentation of single-particle occupancies close to the Fermi energy. This provides a complementary interpretation with respect to the standard quasiparticle representation where surface effects are generated by the integrated radial profiles of the contributing wave functions.

Probing the pairing interaction through two-neutron transfer reactions

The treatment of the pairing interaction in mean-field-based models is addressed. In particular, the possibility to use pair transfers as a tool to better constrain this interaction is discussed. First, pairing inter- actions with various density dependencies (surface/volume mixing) are used in the microscopic Hartree-Fock- Bogoliubov + quasiparticle random-phase approximation model to generate the form factors to be used in reac- tion calculations. Cross sections for (p,t) two-neutron transfer reactions are calculated in the one-step zero-range distorted-wave Born approximation for some Tin isotopes and for incident proton energies from 15 to 35 MeV. Three different surface/volume mixings of a zero-range density-dependent pairing interaction are employed in the microscopic calculations and the sensitivity of the cross sections to the different mixings is analyzed. Differences among the three different theoretical predictions are found espacially for the nucleus 136 Sn and they are more important at the incident proton energy of 15 MeV. We thus indicate (p,t) two-neutron transfer reactions with very neutron-rich Sn isotopes and at proton energies around 15 MeV as good experimental cases where the sur- face/volume mixing of the pairing interaction may be probed. In the second part of the manuscript, ground-state to ground-state transitions are investigated. Approximations made to estimate two-nucleon transfer probabilities in ground-state to ground-state transitions and the physical interpretation of these probabilities are discussed. Probabilities are often calculated by approximating both ground states of the initial nucleus A and of the final nucleus A ±2 by the same quasiparticle vacuum. We analyze two improvements of this approach. First, the effect of using two different ground states with average numbers of particles A and A ±2 is quantified. Second, by using projection techniques, the role of particle number restoration is analyzed. Our analysis shows that the improved treatment plays a role close to magicity, leading to an enhancement of the pair-transfer probability. In midshell regions, part of the error made by approximating the initial and final ground states by a single vacuum is com- pensated by projecting onto a good particle number. Surface effects are analyzed by using pairing interactions with a different volume/surface mixing. Finally, a simple expression of the pair-transfer probability is given in terms of occupation probabilities in the canonical basis. We show that, in the canonical basis formulation, surface effects that are visible in the transfer probability are related to the fragmentation of single-particle occupancies close to the Fermi energy. This provides a complementary interpretation with respect to the standard quasiparti- cle representation where surface effects are generated by the integrated radial profiles of the contributing wave functions.

REGULARIZATION OF MULTI-REFERENCE ENERGY DENSITY FUNCTIONAL CALCULATIONS

We report the first application of a recently proposed regularization procedure for multi-reference energy density functionals, which removes spurious divergent or non-continuous contributions to the binding energy, to a general configuration mixing. As an example, we present a calculation that corresponds to the particle-number and angular momentum projection of axially symmetric time-reversal invariant quasiparticle vacua of different quadrupole deformation for the nucleus 18 O . The SIII parameterization of the Skyrme energy functional is used.

Configuration mixing within the energy density functional formalism: Removing spurious contributions from nondiagonal energy kernels

Multi reference calculations along the lines of the generator coordinate method or the restoration of broken symmetries within the nuclear energy density functional (EDF) framework are becoming a standard tool in nuclear structure physics. These calculations rely on the extension of a single-reference energy functional, of the Gogny or the Skyrme types, to nondiagonal energy kernels. There is no rigorous constructive framework for this extension so far. The commonly accepted way proceeds by formal analogy with the expressions obtained when applying the generalized Wick theorem to the nondiagonal matrix element of a Hamilton operator between two product states. It is pointed out that this procedure is ill defined when extended to EDF calculations as the generalized Wick theorem is taken outside of its range of applicability. In particular, such a procedure is responsible for the appearance of spurious divergences and steps in multi reference EDF energies, as was recently observed in calculations restoring particle number or angular momentum. In the present work, we give a formal analysis of the origin of this problem for calculations with and without pairing, i.e., constructing the density matrices from either Slater determinants or quasiparticle vacua. We propose a method to regularize nondiagonal energy kernels such that divergences and steps are removed from multi reference EDF energies. Such a removal is a priori quasiparticle-basis dependent. A special feature of the method we use to proceed to the actual regularization is that it singles out one basis among all possible ones. The regularization method is applicable to calculations based on any symmetry restoration or generator coordinate but is limited to EDFs depending only on integer powers of the normal and anomalous density matrices. Eventually, the method is formally illustrated for particle-number restoration and is specified to configuration mixing calculations based on Slater determinants.

Hartree-Fock-Bogoliubov theory of polarized Fermi systems

Condensed Fermi systems with an odd number of particles can be described by means of polarizing external fields having a time-odd character. We illustrate how this works for Fermi gases and atomic nuclei treated by density-functional theory or Hartree-Fock-Bogoliubov theory. We discuss the method based on introducing two chemical potentials for different superfluid components, whereby one may change the particle-number parity of the underlying quasiparticle vacuum. Formally, this method is a variant of noncollective cranking, and the procedure is equivalent to the so-called blocking. We present and exemplify relations between the two-chemical-potential method and the cranking approximation for Fermi gases and nuclei.

Signature of magnetic monopole and Dirac string dynamics in spin ice

Magnetic monopoles have eluded experimental detection since their prediction nearly a century ago by Dirac. Recently it has been shown that classical analogues of these enigmatic particles occur as excitations out of the topological ground state of a model magnetic system, dipolar spin ice. These quasi-particle excitations do not require a modification of Maxwell's equations, but they do interact via Coulombs law and are of magnetic origin. In this paper we present an experimentally measurable signature of monopole dynamics and show that magnetic relaxation measurements in the spin ice material $Dy_{2}Ti_{2}O_{7}$ can be interpreted entirely in terms of the diffusive motion of monopoles in the grand canonical ensemble, constrained by a network of "Dirac strings" filling the quasi-particle vacuum. In a magnetic field the topology of the network prevents charge flow in the steady state, but there is a monopole density gradient near the surface of an open system.

Gravitational dynamics in Bose-Einstein condensates

Analogue models for gravity intend to provide a framework where matter and gravity, as well as their intertwined dynamics, emerge from degrees of freedom that have a priori nothing to do with what we call gravity or matter. Bose-Einstein condensates (BEC) are a natural example of an analogue model since one can identify matter propagating on a (pseudo-Riemannian) metric with collective excitations above the condensate of atoms. However, until now, a description of the ``analogue gravitational dynamics'' for such model was missing. We show here that in a BEC system with massive quasiparticles, the gravitational dynamics can be encoded in a modified (semiclassical) Poisson equation. In particular, gravity is of extreme short range (characterized by the healing length) and the cosmological constant appears from the noncondensed fraction of atoms in the quasiparticle vacuum. While some of these features make the analogue gravitational dynamics of our BEC system quite different from standard Newtonian gravity, we nonetheless show that it can be used to draw some interesting lessons about ``emergent gravity'' scenarios.

Utilizing group property of Bogoliubov transformation

We revisit the Bogoliubov transformation as a representation of the group of unitary operators of Balian and Brezin. We show that the group property is best utilized when we treat successive transformations of quasiparticles and their vacuum at the same time. In particular, we establish a one-to-one correspondence between sets of quasiparticle operators and their vacua using the group property. The correspondence determines the quasiparticle vacuum uniquely including the phase, which is inevitable in treating probability amplitudes in a consistent fashion.

Variational wave functions for homogenous Bose systems

We study variational wave functions of the product form, factorizing according to the wave vectors k, for the ground state of a system of bosons interacting via positive pair interactions with a positive Fourier transform. Our trial functions are members of different orthonormal bases in Fock space. Each basis contains a quasiparticle vacuum state and states with an arbitrary finite number of quasiparticles. One of the bases is that of Valatin and Butler (VB), introduced fifty years ago and parametrized by an infinite set of variables determining Bogoliubov's canonical transformation for each k. In another case, inspired by Nozieres and Saint James the canonical transformation for k=0 is replaced by a shift in the creation/annihilation operators. For the VB basis we prove that the lowest energy is obtained in a state with {approx}{radical}(volume) quasiparticles in the zero mode. The number of k=0 physical particles is of the order of the volume and its fluctuation is anomalously large, resulting in an excess energy. The same fluctuation is normal in the second type of optimized bases, the minimum energy is smaller and is attained in a vacuum state. Associated quasiparticle theories and questions about the gap in their spectrum are also discussed.

Strong coupling Kondo lattice model as a Fermi gas

The strong coupling half-filled Kondo lattice model is an important example of a strongly interacting dense Fermi system for which conventional Fermi gas analysis has thus far failed. We remedy this by deriving an exact transformation that maps the model to a dilute gas of weakly interacting electron and hole quasiparticles that can then be analyzed by conventional dilute Fermi gas methods. The quasiparticle vacuum is a singlet Mott insulator for which the quasiparticle dynamics are simple. Since the transformation is exact, the electron spectral weight sum rules are obeyed exactly. Subtleties in understanding the behavior of electrons in the singlet Mott insulator can be reduced to a fairly complicated but precise relation between quasiparticles and bare electrons. The theory of free quasiparticles can be interpreted as an exactly solvable model for a singlet Mott insulator, providing an exact model in which to explore the strong coupling regime of a singlet Kondo insulator.

Natural orbits of atomic Cooper pairs in a nonuniform Fermi gas

We examine the basic mode structure of atomic Cooper pairs in an inhomogeneous Fermi gas. Based on the properties of Bogoliubov quasiparticle vacuum, the single particle density matrix and the anomalous density matrix share the same set of eigenfunctions. These eigenfunctions correspond to natural pairing orbits associated with the BCS ground state. We investigate these orbits for a Fermi gas in a spherical harmonic trap and construct the wave function of a Cooper pair in the form of Schmidt decomposition. The issue of spatial quantum entanglement between constituent atoms in a pair is addressed.

Multidimensional cut-off technique, odd-dimensional Epstein zeta functions and Casimir energy of massless scalar fields

Quantum fluctuations of massless scalar fields represented by quantum fluctuations of the quasiparticle vacuum in a zero-temperature dilute Bose-Einstein condensate may well provide the first experimental arena for measuring the Casimir force of a field other than the electromagnetic field. This would constitute a real Casimir force measurement - due to quantum fluctuations - in contrast to thermal fluctuation effects. We develop a multidimensional cut-off technique for calculating the Casimir energy of massless scalar fields in $d$-dimensional rectangular spaces with $q$ large dimensions and $d-q$ dimensions of length $L$ and generalize the technique to arbitrary lengths. We explicitly evaluate the multidimensional remainder and express it in a form that converges exponentially fast. Together with the compact analytical formulas we derive, the numerical results are exact and easy to obtain. Most importantly, we show that the division between analytical and remainder is not arbitrary but has a natural physical interpretation. The analytical part can be viewed as the sum of individual parallel plate energies and the remainder as an interaction energy. In a separate procedure, via results from number theory, we express some odd-dimensional homogeneous Epstein zeta functions as products of one-dimensional sums plus a tiny remainder and calculate from them the Casimir energy via zeta function regularization.

Stability diagram and growth rate of parametric resonances in Bose-Einstein condensates in one-dimensional optical lattices

A Bose-Einstein condensate in an optical lattice exhibits parametric resonances when the intensity of the lattice is periodically modulated in time. These resonances correspond to an exponential growth of the population of counter-propagating Bogoliubov excitations. A suitable linearization of the Gross-Pitaevskii (GP) equation is used to calculate the stability diagram and the growth rates of the unstable modes. The results agree with the ones extracted from time-dependent GP simulations, supporting our previous claim (M. Kraemer et al., Phys. Rev. A (2005) in press) concerning the key role of parametric resonances in the response observed by Stoeferle et al. (Phys. Rev. Lett. 92, 130403 (2004)) in the superfluid regime. The role of the seed excitations required to trigger the parametric amplification is discussed. The possible amplification of the quantum fluctuations present in the quasiparticle vacuum, beyond GP theory, is also addressed, finding interesting analogies with similar processes in nonlinear quantum optics and with the dynamic Casimir effect. Our results can be used in exploiting parametric instabilities for the purpose of spectroscopy, selective amplification of a particular excitation mode and for establishing a new type of thermometry.

Quasi-particle entanglement: redefinition of the vacuum and reduced density matrix approach

A scattering approach to entanglement in mesoscopic conductors with independent fermionic quasi-particles is discussed. We focus on conductors in the tunnelling limit, where a redefinition of the quasi-particle vacuum transforms the wavefunction from a many-body product state of non-interacting particles to a state describing entangled two-particle excitations out of the new vacuum (Samuelsson, Sukhorukov and Büttiker 2003 Phys. Rev. Lett.91 157002). The approach is illustrated with two examples: (i) a normal–superconducting system, where the transformation is made between Bogoliubov–de Gennes quasi-particles and Cooper pairs, and (ii) a normal system, where the transformation is made between electron quasi-particles and electron–hole pairs. This is compared to a scheme where an effective two-particle state is derived from the manybody scattering state by a reduced density matrix approach.

On the linear term in the nuclear symmetry energy

The nuclear symmetry energy calculated in the RPA from the pairing plus symmetry force Hamiltonian with equidistant single-nucleon levels is for mass number A=48 approximately proportional to T(T+1.025), where T is the isospin quantum number. An isovector character of the pair field assumed to produce the observed odd–even mass staggering is essential for this result. The RPA contribution to the symmetry energy cannot be simulated by adding to the Hartree–Fock–Bogolyubov energy a term proportional to the isospin variance in the Bogolyubov quasiparticle vacuum, and there are significant corrections to the approximation which consist in adding half the isocranking angular velocity. The present calculation employs a smaller single-nucleon level spacing than used in a previous investigation of the model.