The Hubbard model at high dimensions: self-consistent weak coupling theory

We examine the one-band Hubbard model on a simple cubic lattice in the limit of high spatial dimensions by means of standard weak coupling theory. Various infinite series of self-energy insertions are considered as approximations for the proper self-energy. Due to the diagram collapse in the limit of infinite dimensions it is feasible to solve the resulting equations for the one-particle propagator self-consistently. The resulting one-particle spectra show a narrow peak near the Fermi energy and pronounced band tails. In particular, the relevance of the considered approximations for mass enhancement is studied.Research performed within the program of the Sonderforschungsbereich 341 supported by the Deutsche Forschungsgemeinschaft     

Mixtures of correlated bosons and fermions: Dynamical mean‐field theory for normal and condensed phases

We derive a dynamical mean‐field theory for mixtures of interacting bosons and fermions on a lattice (BF‐DMFT). The BF‐DMFT is a comprehensive, thermodynamically consistent framework for the theoretical investigation of Bose‐Fermi mixtures and is applicable for arbitrary values of the coupling parameters and temperatures. It becomes exact in the limit of high spatial dimensions d or coordination number Z of the lattice. In particular, the BF‐DMFT treats normal and condensed bosons on equal footing and thus includes the effects caused by their dynamic coupling. Using the BF‐DMFT we investigate two different interaction models of correlated lattice bosons and fermions, one where all particles are spinless (model I) and one where fermions carry a spin one‐half (model II). In model I the local, repulsive interaction between bosons and fermions can give rise to an attractive effective interaction between the bosons. In model II it can also lead to an attraction between the fermions.     

Non-Abelian bosonization and topological aspects of BCS systems

It is shown that many features of the low energy behaviour of weak coupling BCS systems are topological in character. This is done by writing the BCS action as a Fermi surface sum of (1 + 1)-dimentional non-Abelian actions, each one of which can be bosonized à la Witten. This process leads to the correct current generating parts (as found by Cross) in the effective action for the gap function, and in addition a Wess-Zumino term. The mass and spin currents are calculated for the three-dimensional theory, and it is shown that upon averaging over all directions on the Fermi surface, the contribution from the Wess-Zumino term vanishes for a pure spin singlet or pure spin triplet gap, but not otherwise.     

The boundary condition integration technique: results for the Hubbard model in 1D and 2D

We study models of strongly correlated electrons in one-and two dimensions. We exactly diagonalize small clusters with general boundary conditions (BC) and integrate over all possible BC. This technique recovers the kinetic energy part of the (extended lattice) Hamiltonianexactly in a grand-canonical formulation. A continuous range of particle densities may be described with this technique and the momentum space can be probed for arbitrary momenta. For the Hubbard Hamiltonian we recover details of the Mott-insulating behaviour for the momentum distribution function at half filling, both in 1D and 2D. Off half-filling the shape of thecanonical Fermi surface is strongly distorted in 2D with respect to thegrand canonical Fermi surface. The shape of the grand canonical Fermi surface obtained by this finite-size technique reduces in the weak-coupling limit exactly to that of the infinite-lattice Fermi sea.email: UPH 301 at DDOHR Z11     

Progress in lattice gauge theory

Two topics of lattice gauge theory are reviewed. They include string tension and β-function calculations by strong coupling Hamiltonian methods for SU(3) gauge fields in 3 + 1 dimensions, and a 1/N-expansion for discrete gauge and spin systems in all dimensions. The SU(3) calculations give solid evidence for the coexistence of quark confinement and asymptotic freedom in the renormalized continuum limit of the lattice theory. The crossover between weak and strong coupling behavior in the theory is seen to be a weak coupling but non-perturbative effect. Quantitative relationships between perturbative and non-perturbative renormalization schemes are obtained for the O(N) nonlinear sigma models in 1 + 1 dimensions as well as the range theory in 3 + 1 dimensions. Analysis of the strong coupling expansion of the β-function for gauge fields suggests that it has cuts in the complex 1/g²-plane. A toy model of such a cut structure which naturally explains the abruptness of the theory's crossover from weak to strong coupling is presented. The relation of these cuts to other approaches to gauge field dynamics is discussed briefly.The dynamics underlying first order phase transitions in a wide class of lattice gauge theories is exposed by considering a class of models-P(N) gauge theories - which are soluble in the N → ∞ limit and have non-trivial phase diagrams. The first order character of the phase transitions in Potts spin systems for N #62; 4 in 1 + 1 dimensions is explained in simple terms which generalizes to P(N) gauge systems in higher dimensions. The phase diagram of Ising lattice gauge theory coupled to matter fields is obtained in a $\text{1}\text{N}$ expansion. A one-plaquette model (1 time-0 space dimensions) with a first-order phase transitions in the N → ∞ limit is discussed.     

Anomalous Nernst effect in type-II Weyl semimetals

Topological Weyl semimetals (WSM), a new state of quantum matter with gapless nodal bulk spectrum and open Fermi arc surface states, have recently sparked enormous interest in condensed matter physics. Based on the symmetry and fermiology, it has been proposed that WSMs can be broadly classified into two types, type-I and type-II Weyl semimetals. While the undoped, conventional, type-I WSMs have point like Fermi surface and vanishing density of states (DOS) at the Fermi energy, the type-II Weyl semimetals break Lorentz symmetry explicitly and have tilted conical spectra with electron and hole pockets producing finite DOS at the Fermi level. The tilted conical spectrum and finite DOS at Fermi level in type-II WSMs have recently been shown to produce interesting effects such as a chiral anomaly induced longitudinal magnetoresistance that is strongly anisotropic in direction and a novel anomalous Hall effect. In this work, we consider the anomalous Nernst effect in type-II WSMs in the absence of an external magnetic field using the framework of semi-classical Boltzmann theory. Based on both a linearized model of time-reversal breaking WSM with a higher energy cut-off and a more realistic lattice model, we show that the anomalous Nernst response in these systems is strongly anisotropic in space, and can serve as a reliable signature of type-II Weyl semimetals in a host of magnetic systems with spontaneously broken time reversal symmetry.     

Fractionalized fermi liquids

In spatial dimensions d>or=2, Kondo lattice models of conduction and local moment electrons can exhibit a fractionalized, nonmagnetic state (FL(*)) with a Fermi surface of sharp electronlike quasiparticles, enclosing a volume quantized by (rho(a)-1)(mod 2), with rho(a) the mean number of all electrons per unit cell of the ground state. Such states have fractionalized excitations linked to the deconfined phase of a gauge theory. Confinement leads to a conventional Fermi liquid state, with a Fermi volume quantized by rho(a)(mod 2), and an intermediate superconducting state for the Z2 gauge case. The FL(*) state permits a second order metamagnetic transition in an applied magnetic field.     

Soft fermi surfaces and breakdown of Fermi-liquid behavior

Electron-electron interactions can induce Fermi surface deformations which break the point-group symmetry of the lattice structure of the system. In the vicinity of such a "Pomeranchuk instability" the Fermi surface is easily deformed by anisotropic perturbations, and exhibits enhanced collective fluctuations. We show that critical Fermi surface fluctuations near a d-wave Pomeranchuk instability in two dimensions lead to large anisotropic decay rates for single-particle excitations, which destroy Fermi-liquid behavior over the whole surface except at the Brillouin zone diagonal.     

Layered superconductors in high magnetic fields

We investigate the possible occurrance of partially depaired states in superconducting intercalated layered systems. Those states are discussed as a possible explanation of the high critical fields found in some of these materials. It is shown that the Chandrasekhar-Clogston limit does not apply to those states mentioned above and that the maximum field compatible with superconductivity is a sensitive function of the shape of the Fermi surface. Mean free path and spin-orbit effects on the partially depaired state are investigated. An experiment is proposed to decide between the partially depaired state and a large spin-orbit scattering rate as possible explanations for the large critical fields.     

Antiferromagnetism in metals: from the cuprate superconductors to the heavy fermion materials

The critical theory of the onset of antiferromagnetism in metals, with concomitant Fermi surface reconstruction, has recently been shown to be strongly coupled in two spatial dimensions. The onset of unconventional superconductivity near this critical point is reviewed: it involves a subtle interplay between the breakdown of fermionic quasiparticle excitations on the Fermi surface and the strong pairing glue provided by the antiferromagnetic fluctuations. The net result is a logarithm-squared enhancement of the pairing vertex for generic Fermi surfaces, with a universal dimensionless coefficient independent of the strength of interactions, which is expected to lead to superconductivity at the scale of the Fermi energy. We also discuss the possibility that the antiferromagnetic critical point can be replaced by an intermediate 'fractionalized Fermi liquid' phase, in which there is Fermi surface reconstruction but no long-range antiferromagnetic order. We discuss the relevance of this phase to the underdoped cuprates and the heavy fermion materials.     

Universality of the continuum limit of lattice quantum theories

The lattice approximation of the naïve continuum action in quantum mechanics or in field theory is not uniquely determined. We investigate to what extent corrections to the lattice action, which vanish in the naïve continuum limit, affect the continuum limit when taking quantum fluctuations into account. In the quantum mechanical case, modifications of the lattice action may induce non-trivial corrections to the potential of the system and thereby change the structure of the theory completely. We verify this statement analytically as well as numerically by performing a Monte Carlo simulation. In the field theoretical case we argue that the lattice corrections considered do not affect the physics of the continuum limit, at least not for asymptotically free gauge field theories. In four dimensions, one might encounter finite renormalization of CP violating terms.     

Continuum QCD2 from a fixed-point lattice action

Gauge invariant expectation values for lattice gauge theory with a general local action in two dimensions may be expressed as functions of the single plaquette averages. The value of these averages at the fixed point of the renormalization group can be determined exactly, and the corresponding lattice theory is shown to reproduce the continuum results. The limit N_e = ∞ is investigated in detail, and fixed point values for all the averages are explicitly determined. Wilson's action results agree only to first order in weak coupling.     

Noncoplanar magnetic ordering driven by itinerant electrons on the pyrochlore lattice

Exchange interaction tends to favor collinear or coplanar magnetic orders in rotationally invariant spin systems. Indeed, such magnetic structures are usually selected by thermal or quantum fluctuations in highly frustrated magnets. Here we show that a complex noncoplanar magnetic order with a quadrupled unit cell is stabilized by itinerant electrons on the pyrochlore lattice. Specifically, we consider a Kondo-lattice model with classical localized moments at quarter filling. The electron Fermi "surface" at this filling factor is topologically equivalent to three intersecting Fermi circles. Perfect nesting of the Fermi lines leads to magnetic ordering with multiple wave vectors and a definite handedness. The chiral order might persist without magnetic order in a chiral spin liquid at finite temperatures.     

Berry curvature on the fermi surface: anomalous Hall effect as a topological fermi-liquid property

The intrinsic anomalous Hall effect in metallic ferromagnets is shown to be controlled by Berry phases accumulated by adiabatic motion of quasiparticles on the Fermi surface, and is purely a Fermi-liquid property, not a bulk Fermi sea property like Landau diamagnetism, as has been previously supposed. Berry phases are a new topological ingredient that must be added to Landau Fermi-liquid theory in the presence of broken inversion or time-reversal symmetry.     

Spin-incoherent behavior in the ground state of strongly correlated systems

It is commonly believed that strongly interacting one-dimensional Fermi systems with gapless excitations are effectively described by Luttinger liquid theory. However, when the temperature of the system is high compared to the spin energy, but small compared to the charge energy, the system becomes "spin incoherent." We present numerical evidence showing that the one-dimensional "t-J-Kondo" lattice, consisting of a t-J chain interacting with localized spins, displays all the characteristic signatures of spin-incoherent physics, but in the ground state. We argue that similar physics may be present in a wide range of strongly interacting systems.     

Electrical resistivity near Pomeranchuk instability in two dimensions

We analyze the dc charge transport in the quantum critical regime near a d-wave Pomeranchuk instability in two dimensions. The transport decay rate is linear in temperature everywhere on the Fermi surface except at cold spots on the Brillouin zone diagonal. For pure systems, this leads to a dc resistivity proportional to T(3/2) in the low-temperature limit. In the presence of impurities the residual impurity resistance at T=0 is approached linearly at low temperatures.     

Strong-coupling study of the Gribov ambiguity in lattice Landau gauge

We study the strong-coupling limit β=0 of lattice SU(2) Landau gauge Yang–Mills theory. In this limit the lattice spacing is infinite, and thus all momenta in physical units are infinitesimally small. Hence, the infrared behavior can be assessed at sufficiently large lattice momenta. Our results show that at the lattice volumes used here, the Gribov ambiguity has an enormous effect on the ghost propagator in all dimensions. This underlines the severity of the Gribov problem and calls for refined studies also at finite β. In turn, the gluon propagator only mildly depends on the Gribov ambiguity.     

Stability of Fermi surfaces and K theory

Nonrelativistic Fermi liquids in d+1 dimensions exhibit generalized Fermi surfaces: (d-p)-dimensional submanifolds in the (k,omega)-space supporting gapless excitations. We show that the universality classes of stable Fermi surfaces are classified by K theory, with the pattern of stability determined by Bott periodicity. The Atiyah-Bott-Shapiro construction implies that the low-energy modes near a Fermi surface exhibit relativistic invariance in the transverse p+1 dimensions. This suggests an intriguing parallel between nonrelativistic Fermi liquids and D-branes of string theory.     

Exact diagonalization study of 2D Hubbard model on honeycomb lattice: Semi-metal to insulator transition

Phase transition in a honeycomb lattice is studied by the means of the two-dimensional Hubbard model and the exact diagonalization dynamical mean field theory at zero temperature. At low energies, the dispersion relation is shown to be a linear function of the momentum. In the limit of weak interactions, the system is in the semi-metal phase. By increasing the on site interaction a semi-metal to insulator transition takes place in the paramagnetic phase. Calculation of double occupancy shows such a phase transition is of the second order. The respective phase transition point and critical on-site interaction are determined using renormalized Fermi velocity factor.