A Feynman-Kac formula for the quantum Heisenberg ferromagnet. I

The Hamiltonian of the (anisotropic) quantum Heisenberg (anti-) ferromagnet on an arbitrary finite lattice is lifted to a Hamiltonian acting on sections of the bundle obtained by twisting a certain line bundle over the classical spin configuration space (which is a Kähler manifold) with the Dolbeault complex. This procedure is extended fromSU(2) to arbitrary compact semi-simple Lie groups and arbitrary irreducible representations. The Bott-Borel-Weil theorem gives a heat kernel representation for the original partition function in an external magnetic field. TheU(1)-gauged local Hamiltonian is the sum of the free, supersymmetric, twisted Dolbeault Laplace operator (multiplied by the inverse of an arbitrary small mass parameter) plus the lifted Hamiltonian.The resulting (Euclidean) Lagrangian is nonlocal and describes bosons which do and fermions which do not propagate through the lattice. All fields couple to the external magnetic field. The Lagrangian contains Yukawa and Luttinger type interactions.     

Exact solutions of a one-dimensional mixture of spinor bosons and spinor fermions

The exact solutions of a one-dimensional mixture of spinor bosons and spinor fermions with δ-function interactions are studied. Some new sets of Bethe ansatz equations are obtained by using the graded nest quantum inverse scattering method. Many interesting features appear in the system. For example, the wave function has the SU(2|2) supersymmetry. It is also found that the ground state of the system is partial polarized, where the fermions form a spin singlet state and the bosons are totally polarized. From the solution of Bethe ansatz equations, it is shown that all the momentum, spin and isospin rapidities at the ground state are real if the interactions between the particles are repulsive; while the fermions form two-particle bounded states and the bosons form one large bound state, which means the bosons condensed at the zero momentum point, if the interactions are attractive. The charge, spin and isospin excitations are discussed in detail. The thermodynamic Bethe ansatz equations are also derived and their solutions at some special cases are obtained analytically.     

Chiral p-wave pairing of ultracold fermionic atoms due to a quadratic band touching

We study the superfuild ground state of ultracold fermions in optical lattices with a quadratic band touching. Examples are a checkerboard lattice around half filling and a kagome lattice above one third filling. Instead of pairing between spin states, here we focus on pairing interactions between different orbital states. We find that our systems have only odd-parity(orbital) pairing instability while the singlet(orbital) pairing instability vanishes thanks to the quadratic band touching. In the mean field level, the ground state is found to be a chiral p-wave pairing superfluid(mixed with finite f-wave pairing order-parameters) which supports Majorana fermions.     

I. The isotopic Foldy-Wouthuysen representation and chiral symmetry

The paper introduces the isotopic Foldy-Wouthuysen representation. This representation was used to derive equations for massive interacting fermion fields. When the interaction Hamiltonian commutes with the matrix γ⁵, these equations possess chiral invariance irrespective of whether fermions have mass or are massless. The isotopic Foldy-Wouthuysen representation preserves the vector and axial currents irrespective of the fermion mass value. In the Dirac representation, the axial current is preserved only for massless fermions. In the isotopic Foldy-Wouthuysen representation, the ground state of fermions (vacuum) turns out to be degenerate, and therefore there is the possibility of spontaneously breaking parity (P — symmetry). This study considers the example of constructing a chirally symmetric quantum electrodynamics framework in the isotopic Foldy-Wouthuysen representation. A number of physical processes are calculated in the lowest orders of the perturbation theory. Final results of the calculations agree with the results of the standard quantum electrodynamics.     

Excitonic metal–insulator phase transition of the Mott type in compressed calcium

It has been experimentally found that, under the static compression of a calcium crystal at room temperature, it undergoes a series of structural phase transitions: face-centered cubic lattice → body-centered cubic lattice → simple cubic lattice. It has been decided to investigate precisely the simple cubic lattice (because it is an alternative lattice) with the aim of elucidating the possibility of the existence of other (nonstructural) phase transitions in it by using for this purpose the Hubbard model for electrons with half-filled ns-bands and preliminarily transforming the initial electronic system into an electron–hole system by means of the known Shiba operators (applicable only to alternative lattices). This transformation leads to the fact that, in the new system of fermions, instead of the former repulsion, there is an attraction between electrons and holes. Elementary excitations of this new system are bound boson pairs—excitons. This system of fermions has been quantitatively analyzed by jointly using the equation-of-motion method and the direct algebraic method. The numerical integration of the analytically exact transcendental equations derived from the first principles for alternative (one-, two-, and three-dimensional) lattices has demonstrated that, in systems of two-species (electrons + hole) fermions, temperature-induced metal–insulator phase transitions of the Mott type are actually possible. Moreover, all these crystals are in fact excitonic insulators. This conclusion is in complete agreement with the analytically exact calculations of the ground state of a one-dimensional crystal (with half-filled bands), which were performed by Lieb and Wu with the aim to find out the Mott insulator–metal transition of another type.     

Determination of the Electromagnetic Lagrangian from a System of Poisson Brackets

The Lagrangian and Hamiltonian formulations of electromagnetism are reviewed and the Maxwell equations are obtained from the Hamiltonian for a system of many electric charges. It is shown that three of the equations which were obtained from the Hamiltonian, namely the Lorentz force law and two Maxwell equations, can be obtained as well from a set of postulated Poisson brackets. It is shown how the results derived from these brackets can be used to reconstruct the original Lagrangian for the theory aided by some reasoning based on physical concepts.     

Emergent fermions and anyons in the kitaev model

We study the gapped phase of the Kitaev model on the honeycomb lattice using perturbative continuous unitary transformations. The effective low-energy Hamiltonian is found to be an extended toric code with interacting anyons. High-energy excitations are emerging free fermions which are composed of hard-core bosons with an attached string of spin operators. The excitation spectrum is mapped onto that of a single particle hopping on a square lattice in a magnetic field. We also illustrate how to compute correlation functions in this framework. The present approach yields analytical perturbative results in the thermodynamical limit without using the Majorana or the Jordan-Wigner fermionization initially proposed to solve this problem.     

Interaction versus dimerization in one-dimensional Fermi systems

In order to study the effect of interaction and lattice distortion on quantum coherence in one-dimensional Fermi systems, we calculate the ground state energy and the phase sensitivity of a ring of interacting spinless fermions on a dimerized lattice. Our numerical DMRG studies, in which we keep up to 1000 states for systems of about 100 sites, are supplemented by analytical considerations using bosonization techniques. We find a delocalized phase for an attractive interaction, which differs from that obtained for random lattice distortions. The extension of this delocalized phase depends strongly on the dimerization induced modification of the interaction. Taking into account the harmonic lattice energy, we find a dimerized ground state for a repulsive interaction only. The dimerization is suppressed at half filling, when the correlation gap becomes large. Received: 11 February 1998 / Revised: 1st April 1998 / Accepted: 30 April 1998     

Analytical study of four-wave mixing with large atomic coherence

Four-wave mixing in resonant atomic vapors based on maximum coherence induced by Stark-chirped rapid adiabatic passage (SCRAP) is investigated theoretically. We show the advantages of a coupling scheme involving maximum coherence and demonstrate how a large atomic coherence between a ground and an highly excited state can be prepared by SCRAP. Full analytic solutions of the field propagation problem taking into account pump field depletion are derived. The solutions are obtained with the help of an Hamiltonian approach which in the adiabatic limit permits to reduce the full set of Maxwell-Bloch equations to simple canonical equations of Hamiltonian mechanics for the field variables. It is found that the conversion efficiency reached is largely enhanced if the phase mismatch induced by linear refraction is compensated. A detailed analysis of the phase matching conditions shows, however, that the phase mismatch contribution from the Kerr effect cannot be compensated simultaneously with linear refraction contribution. Therefore, the conversion efficiency in a coupling scheme involving maximum coherence prepared by SCRAP is high, but not equal to unity. Received 16 August 2002 Published online 4 February 2003 RID="a" ID="a"e-mail: korsunsky@physik.uni-kl.de     

Semiclassical ordering in the large-N pyrochlore antiferromagnet

We study the semiclassical limit of the Sp(N) generalization of the pyrochlore lattice Heisenberg antiferromagnet by expanding about the N --> infinity saddlepoint in powers of a generalized inverse spin. To leading order, we write down an effective Hamiltonian as a series in loops on the lattice. Using this as a formula for calculating the energy of any classical ground state, we perform Monte Carlo simulations and find a unique collinear ground state. This state is not a ground state of linear spin-wave theory, and can therefore not be a physical (N = 1) semiclassical ground state.     

THE VARIATIONAL CALCULATION OF MASS GAP IN 2+1 DIMENSIONAL SU(2) LGT

A variational calculation of the mass gap in 2+1 dimensional SU(2) lattice gauge theory by using a Hamiltonian which possesses exact ground state and correct continuum limit is made.In the range 1.3≤1/g²≤7,a good scaling behaviour am=2.28g² is obtained,which is in agreement with weak-coupling perturbation theory and the results obtained by another Hamiltonian which does not possess correct continuum limit.     

On the connetion between Euclidean and Hamiltonian lattice field theories of vortices and anyons

In the (2+1)-dimensional non-compact Abelian lattice Higgs model Euclidean correlation functions of vortices and, after adding the Chern-Simons term to the action, correlation functions of transmuted matter fields (anyons) are set up. These correlation functions satisfy Osterwalder-Schrader positivity. Via the transition to continuous Euclidean time, vortex and anyon operators within a Hamiltonian lattice formulation are obtained, respectively, and their respective dual algebras are displayed.     

Exact results for strongly correlated fermions in 2 + 1 dimensions

We derive exact results for a model of strongly interacting spinless fermions hopping on a two-dimensional lattice. By exploiting supersymmetry, we find the number and type of ground states exactly. Exploring various lattices and limits, we show how the ground states can be frustrated, quantum critical, or combine frustration with a Wigner crystal. We show that on generic lattices the model is in an exotic "superfrustrated" state characterized by an extensive ground-state entropy.     

Chaotic polaronic and bipolaronic states in the adiabatic Holstein model

A rigorous proof for the existence of bipolaronic states is given for the adiabatic Holstein model for any lattice at any dimension, periodic or not, and for an arbitrary band filling, provided that the electron-phonon coupling (in dimensionless units) is large enough. The existence of mixed polaronic-bipolaronic states is also proven, but for larger electron-phonon coupling. These states consist of arbitrary distributions of bipolarons (or of bipolarons and polarons) localized in real space which can be simply labeled by pseudospin configurations as for a lattice gas model. The theory not only applies to periodic crystals, but also to quasicrystals, amorphous structures, polymer network, etc.When these bipolaronic and mixed polaronic-bipolaronic states exist, it is proven that: (1) These bipolaronic (and mixed polaronic-bipolaronic) states exhibit a nonzero phonon gap with a nonvanishing lower bound and an electronic gap at the Fermi energy. (2) These structures are insulating. The perturbation generated by any local change in the bipolaronic or polaronic distribution or by any charged impurity or defect decays exponentially at long distance. (3) These bipolaronic (and mixed polaronic-bipolaronic) states persist for any uniform magnetic field. (4) For large enough electron-phonon coupling, the ground state of the extended adiabatic Holstein model is a bipolaronic state when there is no uniform magnetic field or when it is small enough. It becomes a mixed polaronic-bipolaronic state for large enough magnetic field (note that the mixed polaronic-bipolaronic states are magnetic).In one-dimensional models, the ground state is an incommensurate (or commensurate) charge density wave (CDW) as predicted by Peierls (this result is not rigorous, but has been confirmed numerically). It is proven that the ground state becomes a bipolaronic charge density wave (BCDW) at large enough electron-phonon coupling. The existence of a transition by breaking of analyticity (TBA), which was numerically observed as a function of the electron-phonon coupling, is then confirmed. In that case, the shape of the effective bipolaron can be numerically calculated. It is observed that its size diverges at the TBA. The physical properties of BCDWs are rather different from those predicted by standard charge density wave theory. Bipolaronic charge density waves can also exist in models which are not only low-dimensional, but purely two- or three-dimensional.The technique for proving these theorems is an application of the concept of anti-integrability initially developed for Hamiltonian dynamical systems. It consists in proving that the eigenstates of the (trivial) Hamiltonian (called antiintegrable) obtained by canceling all electronic and lattice kinetic terms survive as a uniformly continuous function of the electronic kinetic energy terms in the Hamiltonian up to a certain threshold.     

Bogoliubov transformations and fermion condensates in lattice field theories

We apply generalized Bogoliubov transformations to the transfer matrix of relativistic field theories regularized on a lattice. We derive the conditions these transformations must satisfy to factorize the transfer matrix into two terms which propagate fermions and antifermions separately, and we solve the relative equations under some conditions. We relate these equations to the saddle point approximation of a recent bosonization method and to the Foldy-Wouthuysen transformations which separate positive from negative energy states in the Dirac Hamiltonian.     

Mass generation with adjoint fermions

The QCD-like gauge theory with adjoint fermions is considered in the field correlator formalism and the total spectrum of mesons and glueballs is obtained in agreement with available lattice data. A new state of a white fermion appears, as a bound state of the adjoint fermion and gluon with the mass close to that of glueball. It is shown, that the main features of spectra and thermodynamics of adjoint fermions can be explained by this new bound state.     

The symmetric heavy-light ansatz

The symmetric heavy-light ansatz is a method for finding the ground state of any dilute unpolarized system of attractive two-component fermions. Operationally it can be viewed as a generalization of the Kohn-Sham equations in density functional theory applied to N -body density correlations. While the original Hamiltonian has an exact Z₂ symmetry, the heavy-light ansatz breaks this symmetry by skewing the mass ratio of the two components. In the limit where one component is infinitely heavy, the many-body problem can be solved in terms of single-particle orbitals. The original Z₂ symmetry is recovered by enforcing Z₂ symmetry as a constraint on N -body density correlations for the two components. For the 1D, 2D, and 3D attractive Hubbard models the method is in very good agreement with exact Lanczos calculations for few-body systems at arbitrary coupling. For the 3D attractive Hubbard model there is very good agreement with lattice Monte Carlo results for many-body systems in the limit of infinite scattering length.     

A measure of conductivity for lattice fermions at finite density

We study the linear response to an external electric field of a system of fermions in a lattice at zero temperature. This allows to measure numerically the Euclidean conductivity which turns out to be compatible with an analytical calculation for free fermions. The numerical method is generalizable to systems with dynamical interactions where no analytical approach is possible.     

Fermion Condensates in Lower Dimensional Gauge Theories

Combining a unitary transformation,the variational method and the exact ground state of pure gauge Hamiltonian.we investigate syst matically the vacumm structure and spintaneous chiral-symmetry breaking in (1+1) and (2+1) dimensional Hamiltonian lattice gauge theories with fermions,and obtain nice scaling behaviors for 〈ψψ〉 extending to the weak coupling regime.This paper not only reproduces well the exact value in the continuum Schwinger model,but also predicts the values for the fermion condensates in QCD₂,QED₃ and QCD₃.