Auxiliary fields and the sign problem

The auxiliary-field quantum Monte Carlo method is reviewed. The Hubbard-Stratonovich transformation converts an interacting Hamiltonian into a non-interacting Hamiltonian in a time-dependent stochastic field, allowing calculation of the resulting functional integral by Monte Carlo methods. The method is presented in a sufficiently general form to be applicable to any Hamiltonian with oneand two-body terms, with special reference to the Heisenberg model and one- and many-band Hubbard models. Many physical correlation functions can be related to correlation functions of the auxiliary field; general results are given here. Issues relating to the choice of auxiliary fields are addressed; operator product identities change the relative dimensionalities of the attractive and repulsive parts of the interaction. Frequently the integrand is not positive-definite, rendering numerical evaluation unstable. If the auxiliary field violates time-reversal invariance, the integrand is complex and this sign problem becomes a phase problem. The origin of this sign or phase is examined from a number of geometrical and other viewpoints and illustrated by simple examples: the phase problem by the spin 1/2 Heisenberg model, and the sign problem by the attractive SU(N) Hubbard model on a triangular molecule with negative hopping integrals. In the latter case, widely studied in the Jahn Teller literature, the sign is due neither to fermions nor spin, but to frustration. This system is used to illustrate a number of suggested interpretations of the sign problem

Magnetic correlations in paramagnetic iron

A method is described for calculating the extent of short-range order (SRO) in the paramagnetic state of magnetic transition metals. An energy V is calculated for a set of configurations of the exchange field. An entropy S is attached to each configuration; this is the logarithm of the number of configurations with the same degree of long- and short-range magnetic order. The SRO is then found by minimizing the resulting free energy V-TS with respect to a near-neighbour correlation. The resulting Curie temperature Tc and the magnetic entropy are in good agreement with experiment. The corresponding nearest neighbour correlation is small (approximately cos 74°) at Tc

Symmetry reduction of Fourier kernels

Symmetry reduction of Fourier kernel

Strictly localised triplet dimers on one- and two-dimensional lattices

Electrons may form inter-site pairs (dimers) by a number of mechanisms. For example, long-range (Fröhlich) electron-phonon interactions and strong on-site Hubbard U allow formation of small light bipolarons in some lattices. We identify circumstances under which triplet dimers are strictly localised by interference in certain one- and two-dimensional lattices. We assume a U-V Hamiltonian with nearest- and next-nearest-neighbour hopping integrals t and t', large positive U and attractive nearest- and next-nearest-neighbour interactions V and V'. In the square ladder and some two-dimensional bilayers, if the dimer Hilbert space is restricted to nearest- and next-nearest-neighbour dimers, triplet dimers become strictly localised for certain values of these parameters. For example, in a square ladder with t' = t and V' = V, all triplet bands become flat due to exact cancellation of hopping paths. We identify the localised eigenstates for all flat bands in each lattice. We show that many of the flat bands persist for arbitrary t/t' so long as other restrictions still apply

High-temperature superconductivity from realistic Coulomb and Frohlich interactions

In recent years ample experimental evidence has shown that charge carriers in high-temperature superconductors are strongly correlated but also coupled with lattice vibrations (phonons), signalling that the true origin of high-Tc superconductivity can only be found in a proper combination of Coulomb and electron-phonon interactions. On this basis, we propose and study a model for high-Tc superconductivity which accounts for realistic Coulomb repulsion, strong electron-phonon (Frohlich) interaction and residual on-site (Hubbard ~U ) correlations without any ad-hoc assumptions on their relative strength and interaction range. In the framework of this model, which exhibits a phase transition to a superconducting state with a critical temperature Tc well in excess of 100K, we emphasize the role of ~U as the driving parameter for a BEC/BCS crossover. Our model lays a microscopic foundation for the polaron-bipolaron theory of superconductivity. We argue that the high-Tc phenomenon originates in competing Coulomb and Frohlich interactions beyond the conventional BCS description

Erratum: effects of lattice geometry and interaction range on polaron dynamics

Erratum: effects of lattice geometry and interaction range on polaron dynamic

Effects of lattice geometry and interaction range on polaron dynamics

We study the effects of lattice type on polaron dynamics using a continuous-time quantum Monte-Carlo approach. Holstein and screened Froehlich polarons are simulated on a number of different Bravais lattices. The effective mass, isotope coefficients, ground state energy and energy spectra, phonon numbers, and density of states are calculated. In addition, the results are compared with weak and strong coupling perturbation theory. For the Holstein polaron, it is found that the crossover between weak and strong coupling results becomes sharper as the coordination number is increased. In higher dimensions, polarons are much less mobile at strong coupling, with more phonons contributing to the polaron. The total energy decreases monotonically with coupling. Spectral properties of the polaron depend on the lattice type considered, with the dimensionality contributing to the shape and the coordination number to the bandwidth. As the range of the electron-phonon interaction is increased, the coordination number becomes less important, with the dimensionality taking the leading role

Singlet and triplet bipolarons on the triangular lattice

We study the Coulomb–Fröhlich model on a triangular lattice, looking in particular at states with angular momentum. We examine a simplified model of crab bipolarons with angular momentum by projecting onto the low energy subspace of the Coulomb–Fröhlich model with large phonon frequency. Such a projection is consistent with large long-range electron–phonon coupling and large repulsive Hubbard U. Significant differences are found between the band structure of singlet and triplet states: The triplet state (which has a flat band) is found to be significantly heavier than the singlet state (which has mass similar to the polaron). We test whether the heavier triplet states persist to lower electron–phonon coupling using continuous time quantum Monte Carlo (QMC) simulation. The triplet state is both heavier and larger, demonstrating that the heavier mass is due to quantum interference effects on the motion. We also find that retardation effects reduce the differences between singlet and triplet states, since they reintroduce second order terms in the hopping into the inverse effective mass

Superlight small bipolarons

Recent angle-resolved photoemission spectroscopy (ARPES) has identified that a finite-range Fröhlich electron-phonon interaction (EPI) with c-axis polarized optical phonons is important in cuprate superconductors, in agreement with an earlier proposal by Alexandrov and Kornilovitch. The estimated unscreened EPI is so strong that it could easily transform doped holes into mobile lattice bipolarons in narrow-band Mott insulators such as cuprates. Applying a continuous-time quantum Monte-Carlo algorithm (CTQMC) we compute the total energy, effective mass, pair radius, number of phonons and isotope exponent of lattice bipolarons in the region of parameters where any approximation might fail taking into account the Coulomb repulsion and the finite-range EPI. The effects of modifying the interaction range and different lattice geometries are discussed with regards to analytical strong-coupling/non-adiabatic results. We demonstrate that bipolarons can be simultaneously small and light, provided suitable conditions on the electron-phonon and electron-electron interaction are satisfied. Such light small bipolarons are a necessary precursor to high-temperature Bose-Einstein condensation in solids. The light bipolaron mass is shown to be universal in systems made of triangular plaquettes, due to a novel crab-like motion. Another surprising result is that the triplet-singlet exchange energy is of the first order in the hopping integral and triplet bipolarons are heavier than singlets in certain lattice structures at variance with intuitive expectations. Finally, we identify a range of lattices where superlight small bipolarons may be formed, and give estimates for their masses in the anti-adiabatic approximation

Superlight small bipolarons in the presence of strong Coulomb repulsion

We study a lattice bipolaron on a staggered triangular ladder and triangular and hexagonal lattices with both long-range electron-phonon interaction and strong Coulomb repulsion using a novel continuous-time quantum Monte-Carlo (CTQMC) algorithm extended to the Coulomb-Frohlich model with two particles. The algorithm is preceded by an exact integration over phonon degrees of freedom, and as such is extremely efficient. The bipolaron effective mass and bipolaron radius are computed. Lattice bipolarons on such lattices have a novel crablike motion, and are small but very light in a wide range of parameters, which leads to a high Bose-Einstein condensation temperature. We discuss the relevance of our results with current experiments on cuprate high-temperature superconductors and propose a route to room temperature superconductivity