Energy bands: Chern numbers and symmetry

Energy bands formed by rotation–vibrational states of molecules in the presence of symmetry and their qualitative modifications under variation of some control parameters are studied within the semi-quantum model. Rotational variables are treated as classical whereas a finite set of vibrational states is considered as quantum. In the two-state approximation the system is described in terms of a fiber bundle with the base space being a two-dimensional sphere, the classical phase space for rotational variables. Generically this rank 2 complex vector bundle can be decomposed into two complex line bundles characterized by a topological invariant, the first Chern class. A general method of explicit calculation of Chern classes and of their possible modifications under variation of control parameters in the presence of symmetry is suggested. The construction of iso-Chern diagrams which split the space of control parameters into connected domains with fixed Chern numbers is suggested. A detailed analysis of the rovibrational model Hamiltonian for a D₃ invariant molecule possessing two vibrational states transforming according to the two-dimensional irreducible representation is done to illustrate non-trivial restrictions imposed by symmetry on possible values of Chern classes.     

Spin Chern numbers and time-reversal-symmetry-broken quantum spin Hall effect

The quantum spin Hall （QSH） effect is considered to be unstable to perturbations violating the time-reversal （TR） symmetry. We review some recent developments in the search of the QSH effect in the absence of the TR symmetry. The possibility to realize a robust QSH effect by artificial removal of the TR symmetry of the edge states is explored. As a useful tool to characterize topological phases without the TR symmetry, the spin-Chern number theory is introduced.     

Application of green functions to finite Fermi systems with fixed numbers of particles and strong pairing

The single-particle Green function formalism (Gor’kov’s method) is applied to calculate the energies of finite superconducting Fermi systems with a finite number of particles. It is shown that strong pairing leads to equations similar to those from the Bardeen-Cooper-Schrieffer theory.     

Fragmentation of bands and Fermi surfaces in cuprate stripe phases

The band structure and evolution of the Fermi surfaces of stripe phases were studied using the t-t′-U Hubbard model in the mean field approximation. The appearance of quasi-one-dimensional “impurity” subbands caused by the localization of particles on domain walls inside the Hubbard gap is confirmed. Among vertical stripe phases parallel to y bonds, the Y₈ and Y₄ structures with distances l = 8a and 4a between domain walls were found to be stable. Fermi surface segments in antinodal or nodal directions were shown to correspond to an “ impurity” band or the main band related to the entire antiferromagnetic domain region. This is a probable explanation of the difference in the properties of ARPES spectra at different Fermi surface regions observed for La_2?xSr_xCuO₄. It was shown for the Y₈ structure that the topology of the Fermi surface changed and an isotropic pseudogap opened at the point corresponding to a p = 1/8 doping level. Attempts at relating this property to the anomalous suppression of T _c in LSCO at p = 1/8 encountered difficulties. The low dispersion of the impurity band and the wide gap separating it from the lower Hubbard band in diagonal stripe phases formed at p < 0.05 create prerequisites for the existence of the insulating state at nonzero doping.     

Weyl transformation in Fermi systems

In the path integral representation, the Hamiltonian in a quantum system is associated with the Hamiltonian in a classical system through the Weyl transformation. From this, it is possible to describe the time evolution in a quantum system by the Hamiltonian in a classical system. In a Bose system, the Weyl transformation is defined by the eigenstates of the canonical operators, since the Hamiltonian is given by a function of the canonical operators. On the other hand, in a Fermi system, the Hamiltonian is usually described by a function of the creation and annihilation operators, and hence the Weyl transformation is defined by the coherent states which are the eigenstate of an annihilation operator. Here, we formulate the Weyl transformation in Fermi systems in terms of the eigenstates of the canonical operators so as to clarify the correspondence between both systems. Using this, we can derive the path integral representation in Fermi systems.     

Collective coordinates in Fermi systems

The problem of developing a consistent perturbation theory for a Fermi system in the case in which the unperturbed system exhibits dynamical symmetry breaking is discussed, by using collective coordinate methods. By adapting to this problem the methods used in the quantization of gauge theories, it is shown how to deal with composite zero-frequency excitations in such a way that the resulting perturbation theory is free of infrared divergencies. Explicit calculations are carried out in the case of a simple quantum mechanical model representing a superfluid Fermi system.     

Weak currents in Fermi systems

In the framework of a relativistic field theory for the many-body system we use the hypothesis of a partially-conserved axial-vector current to relate weak-interaction coupling constants to strong-interaction parameters. The density-dependent renormalizations of weak-interaction coupling constants are investigated using the Ward-Takahashi identities of the σ-model. We compare the renormalization procedures for the vector current and for the axial-vector current and generally discuss differences between renormalizations at zero density and in the medium.     

The Fermi surface reconstruction in stripe phases of cuprates

Mean-field study of the stripe structures is conducted for a hole-doped Hubbard model. For bond-directed stripes, the Fermi surface consists of segments of an open surface and the boundaries of the hole pockets which appear in the diagonal region of momenta under certain conditions. Segments of the first type are due to one-dimensional bands of states localized on the domain walls. The relation of bands to the doping and temperature dependences of the Hall constant is discussed. In connection with the observation of quantum magnetic oscillations, a systematic search for the electron pockets has been carried out. It is shown that the formation of such pockets in bilayer models is quite possible.

     

The Fermi surface reconstruction in stripe phases of cuprates

Mean-field study of the stripe structures is conducted for a hole-doped Hubbard model. For bond-directed stripes, the Fermi surface consists of segments of an open surface and the boundaries of the hole pockets which appear in the diagonal region of momenta under certain conditions. Segments of the first type are due to one-dimensional bands of states localized on the domain walls. The relation of bands to the doping and temperature dependences of the Hall constant is discussed. In connection with the observation of quantum magnetic oscillations, a systematic search for the electron pockets has been carried out. It is shown that the formation of such pockets in bilayer models is quite possible. The article is published in the original.     

Gamma radiation in non-Markovian Fermi systems

The gamma-quanta emission is considered within the framework of the non-Markovian kinetic theory. It is shown that the memory effects have a strong influence on the spectral distribution of gamma quanta in the case of long-time relaxation regime. It is shown that the gamma radiation can be used as a probe for both the time-reversible hindrance force and the dissipative friction caused by the memory integral.     

Low-temperature collapse of the Fermi surface and phase transitions in correlated Fermi systems

A topological crossover, associated with the collapse of the Fermi surface in strongly correlated Fermi systems, is examined. It is demonstrated that in these systems, the temperature domain where standard Ferrai liquid results hold dramatically narrows, because the Landau regime is replaced by a classical one. The impact of the collapse of the Fermi surface on pairing correlations is analyzed. In the domain of the Lifshitz phase diagram where the Fermi surface collapses, splitting of the BCS superconducting phase transition into two different ones of the same symmetry is shown to occur.     

Luttinger theorem and imbalanced Fermi systems

The proof of the Luttinger theorem, which was originally given for a normal Fermi liquid with equal spin populations formally described by the exact many-body theory at zero temperature, is here extended to an approximate theory given in terms of a “conserving” approximation also with spin imbalanced populations. The need for this extended proof, whose underlying assumptions are here spelled out in detail, stems from the recent interest in superfluid trapped Fermi atoms with attractive inter-particle interaction, for which the difference between two spin populations can be made large enough that superfluidity is destroyed and the system remains normal even at zero temperature. In this context, we will demonstrate the validity of the Luttinger theorem separately for the two spin populations for any “Φ-derivable” approximation, and illustrate it in particular for the self-consistent t-matrix approximation.     

Coupling-matrix approach to the Chern number calculation in disordered systems

The Chern number is often used to distinguish different topological phases of matter in two-dimensional electron systems. A fast and efficient coupling-matrix method is designed to calculate the Chern number in finite crystalline and disordered systems. To show its effectiveness, we apply the approach to the Haldane model and the lattice Hofstadter model, and obtain the correct quantized Chern numbers. The disorder-induced topological phase transition is well reproduced, when the disorder strength is increased beyond the critical value. We expect the method to be widely applicable to the study of topological quantum numbers.