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We present a manifestly gauge-invariant description of Chern numbers associated with the Berry connection defined on a discretized Brillouin zone. It provides an efficient method of computing (spin) Hall conductances without specifying gauge-fixing conditions. We demonstrate that it correctly reproduces quantized Hall conductances even on a coarsely discretized Brillouin zone. A gauge-dependent integer-valued field, which plays a key role in the formulation, is evaluated in several gauges. An extension to the non-Abelian Berry connection is also given.

A criterion to determine the existence of zero-energy edge states is discussed for a class of particlehole symmetric Hamiltonians. A "loop" in a parameter space is assigned for each one-dimensional bulk Hamiltonian, and its topological properties, combined with the chiral symmetry, play an essential role. It provides a unified framework to discuss zero-energy edge modes for several systems such as fully gapped superconductors, two-dimensional d-wave superconductors, and graphite ribbons. A variants of the Peierls instability caused by the presence of edges is also discussed.Depending on several parameters such as hopping integrals or chemical potentials, and also on underlying crystalline lattices, a large variety of electronic structures are realized in condensed matter physics. Electron correlations also give rise to a plenty of quantum phases, forming non-trivial quasi-particle band structures. An interesting consequence of a rich band structure is the existence of edge states that may appear when boundaries are present. In the quantum Hall effect (QHE), this issue was discussed in terms of the origin of the quantization of a Hall conductance. [1,2,3,4,5] Recently, the ideas developed in the QHE have also been extended for other gapped many-body systems, and become essential to describing topological nature of several quantum phases. [6,7,8,9,10,11] Apart from these examples for gapped systems, edge states in gapless systems have attracted much attention recently. Example of these are d-wave superconductor (SC) with edges [12,14], or graphite ribbons [15], where the existence of edge states strongly depends on the shape of edges. For d-wave SC with edges, the zero bias conductance peak (ZBCP) due to zero-energy edge states was observed via a tunneling spectroscopy. [16,17] The issue addressed in this Letter is how to infer the existence of zero-energy eigen states localized on the boundaries in terms of properties of the bulk, and the symmetry. We first consider one-dimensional (1D) systems with a particle-hole symmetry, and then apply the results to systems in higher dimensions. Especially, we will demonstrate applications to fully gapped SC in conjunction with the Chern number, 2D d-wave SC, and graphite ribbons. In addition to these examples, the present work is also applicable to zero-modes in the 1D molecule polyacetylene [18], and quantum spin systems.We start with the following single-particle Hamiltonian on a 1D lattice:

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Inspired by a recent discovery of a peculiar integer quantum Hall effect ͑QHE͒ in graphene, we study QHE on a honeycomb lattice in terms of the topological quantum number, with two interests. First, how the zero-mass Dirac QHE around the center of the tight-binding band crosses over to the ordinary finite-mass fermion QHE around the band edges. Second, how the bulk QHE is related with the edge QHE for the entire spectrum including Dirac and ordinary behaviors. We find the following. ͑i͒ The zero-mass Dirac QHE ͓with xy = ϯ ͑2N +1͒e 2 / h , N: integer͔ persists, surprisingly, up to the van Hove singularities, at which the ordinary fermion behavior abruptly takes over. Here a technique developed in the lattice gauge theory enabled us to calculate the behavior of the topological number over the entire spectrum. This result indicates a robustness of the topological quantum number, and should be observable if the chemical potential can be varied over a wide range in graphene. ͑ii͒ To see if the honeycomb lattice is singular in producing the anomalous QHE, we have systematically surveyed over square ↔ honeycomb ↔-flux lattices, which is scanned by introducing a diagonal transfer tЈ. We find that the massless Dirac QHE ͓ϰ͑2N +1͔͒ forms a critical line, that is, the presence of Dirac cones in the Brillouin zone is preserved by the inclusion of tЈ and the Dirac region sits side by side with ordinary one persists all through the transformation. ͑iii͒ We have compared the bulk QHE number obtained by an adiabatic continuity of the Chern number across the square ↔ honeycomb ↔-flux transformation and numerically obtained edge QHE number calculated from the whole energy spectra for sample with edges, which shows that the bulk QHE number coincides, as in ordinary lattices, with the edge QHE number throughout the lattice transformation.

The entanglement entropy ͑von Neumann entropy͒ has been used to characterize the complexity of manybody ground states in strongly correlated systems. In this paper, we try to establish a connection between the lower bound of the von Neumann entropy and the Berry phase defined for quantum ground states. As an example, a family of translational invariant lattice free fermion systems with two bands separated by a finite gap is investigated. We argue that, for one-dimensional ͑1D͒ cases, when the Berry phase ͑Zak's phase͒ of the occupied band is equal to ϫ ͑odd integer͒ and when the ground state respects a discrete unitary particle-hole symmetry ͑chiral symmetry͒, the entanglement entropy in the thermodynamic limit is at least larger than ln 2 ͑per boundary͒, i.e., the entanglement entropy that corresponds to a maximally entangled pair of two qubits. We also discuss how this lower bound is related to vanishing of the expectation value of a certain nonlocal operator which creates a kink in 1D systems.

We propose the use of quantized Berry phases as a local topological order parameter of a gapped quantum liquid in any dimension that is invariant under some antiunitary operation. The Berry connection is constructed by the response of the quantum manybody state to a local perturbation. Due to the anti-unitary invariance, the Berry phases are quantized as 0 or unless the energy gap closes by the local perturbation. Nontrivial characterizations are demonstrated for ground states of frustrated Heisenberg models and manybody ground states of half-filled random-hopping models. The local topological-order parameters in the models provide quantized texture patterns of local singlet pairs and fermionic local covalent bonds. The Haldane phase of the spin 1 chain is also characterized by the uniform Berry phases. One of the challenges in modern condensed matter physics is to have a better understanding of quantum liquids that do not have a classical analogue. States of matter in classical systems are mostly described by local order parameters based on a concept of symmetry breaking. However, recent studies in decades have revealed that many of interesting quantum phenomena are not well characterized by the classical local order parameters, such as quantum Hall effects, exotic superconductors 1) and frustrated or doped quantum magnets.2-5) Quantum spins with S ¼ 1=2 and fermions are objects in the quantum limits that do not have classical correspondents. When they get together, one may find some classical degree of freedom to describe the states approximately. However, this is not always the case. A pair of S ¼ 1=2 spins with an exchange interaction forms a singlet and a triplet. The latter has a classical analogue as a small magnet, however, the singlet has no classical analogue. When the total system is composed of such singlet pairs, the system is also in the quantum limit as a singlet spin liquid. The most famous singlet spin liquid is the resonating valence bond (RVB) state proposed by Anderson as a possible basic platform of the high-T C .2) Also the valence bond solid (VBS) state and the Haldane phases of integer Heisenberg spin chains are quantum spin liquids of this class. [4][5][6][7] The ground state of a half-filled Kondo lattice, which is a superposition of singlet pairs between the conduction electrons and the localized spins also belongs to this class. 8) Some of the dimer models and spins with a string-net condensation can be solvable limits of such quantum liquids.9-11) For a fermion pair, when the hybridization between the pair is assumed to be secondary, one may use a classical number basis for the description. However, this classical picture breaks down in the strong coupling limit, where the state is labeled as a bonding or an antibonding state. These bonding and antibonding states are purely quantum variables, as covalent bonds, that exist on the link between the fermions sites. The fermionic manybody state, composed of the superposition of local covalent bonds, can be also understood as a typical qua...

We derive an efficient formula for Z 2 topological invariants characterizing the quantum spin Hall effect. It is defined in a lattice Brillouin zone, which enables us to implement numerical calculations for realistic models even in three dimensions. Based on this, we study the quantum spin Hall effect in Bi and Sb in quasi-two and three dimensions using a tight-binding model. Quantum spin Hall (QSH) effect [1][2][3][4][5] has been attracting much current interest as a new device of spintronics. [6][7][8][9] It is a topological insulator [10][11][12] analogous to the quantum Hall (QH) effect, but it is realized in time-reversal (T ) invariant systems. While QH states are specified by Chern numbers, 13,14) QSH states are characterized by Z 2 topological numbers, 2) which suggests that Z 2 invariants would have deep relationship with the Z 2 anomaly of the Majorana fermions. 15,16) Graphene has been expected to be in the QSH phase. 1,2)However, recent calculations have suggested that the spinorbit coupling in graphene is too small to reveal the QSH effect experimentally. 17,18) Recently, it has been pointed out that Bi thin film is another plausible material for QSH effect.19) Also by the idea of adiabatic deformation of the diamond lattice, it has been conjectured that Bi in three dimensions (3D) is in a topological phase. 20)While systems in two dimensions (2D) are characterized by a single Z 2 topological invariant, four independent Z 2 invariants are needed in 3D. [20][21][22][23] This makes it difficult to investigate realistic models, in which complicated many-band structure is involved. Therefore, for the direct study of Bi in 3D as well as for the search for other materials, to establish a simple and efficient computational method of Z 2 invariants in 3D is an urgent issue to be resolved.In this paper, we present a method of computing Z 2 invariants based on the formula derived by Fu and Kane 24) together with the recent development of computing Chern numbers in a lattice Brillouin zone. [25][26][27] This method is based on recent developments in lattice gauge theories [30][31][32][33][34][35] but simple enough to compute Z 2 invariants even for realistic 3D systems. Based on this, we study a tight-binding model for Bi and Sb.First, we derive a lattice version of the Fu-Kane formula.24) To this end, we restrict our discussions, for simplicity, to systems in 2D, where a single Z 2 invariant is relevant. Let T be the time-reversal transformation T ¼ i 2 K, and assume that the Hamiltonian in the momentum space HðkÞ transforms under T as T HðkÞT À1 ¼ HðÀkÞ. Let ðkÞ ¼ ðj1ðkÞi; . . . ; j2MðkÞiÞ denote the 2M dimensional ground state multiplet of the Hamiltonian: HðkÞjnðkÞi ¼ E n ðkÞjnðkÞi. 11,12) Assuming that the many-body energy gap is finite, we focus on topological invariants under the Uð2MÞ transformation ðkÞ ! ðkÞUðkÞ; UðkÞ 2 Uð2MÞ:As discussed, 2,27) the pfaffian defined by pðkÞ ¼ pf É y ðT ÉÞ characterizes the topological phases of T invariant systems. To be precise, the systems belong to topological insu...