The physics underlying the magnetization process of quantum antiferromagnets is revisited from the viewpoint of geometric phases. A continuum variant of the Lieb-Schultz-Mattis-type approach to the problem is put forth, where the commensurability condition of Oshikawa et al. derives from a Berry connection formulation of the system's crystal momentum. We then go on to formulate an effective field theory which can deal with higher dimensional cases as well. We find that a topological term, whose principle function is to assign Berry phase factors to space-time vortex objects, ultimately controls the magnetic behavior of the system. We further show how our effective action maps into a ${\mathbf{Z}}_{2}$ gauge theory under certain conditions, which in turn allows for the occurrence of a fractionalized phase with topological order.
The physics underlying the magnetization process of quantum antiferromagnets is revisited from the viewpoint of geometric phases. A continuum variant of the Lieb-Schultz-Mattis-type approach to the problem is put forth, where the commensurability condition of Oshikawa et al. derives from a Berry connection formulation of the system's crystal momentum. We then go on to formulate an effective field theory which can deal with higher dimensional cases as well. We find that a topological term, whose principle function is to assign Berry phase factors to space-time vortex objects, ultimately controls the magnetic behavior of the system. We further show how our effective action maps into a Z2 gauge theory under certain conditions, which in turn allows for the occurrence of a fractionalized phase with topological order.
On the basis of the principle that topological quantum phases arise from the scattering around space-time defects in higher dimensional unification, a geometric model is presented that associates with each quantum phase an element of a transformation group. 1 Introduction Within a physical theory there are often effects or phenomena studied in idealized form in order to gain insight into the peculiarities of the theory. In Quantum Mechanics one such effect is the appearance of topological quantum phases in the wave functions of particles moving freely in multiply connected space-times the prototype of this effect being the Aharonov-Bohm (AB) effect [1], the appearance of a phase factor in the wave function of an electron which moves around a magnetic flux line. A similar effect is the Aharonov-Casher (AC) effect [2] which is obtained from the AB effect by replacing the flux line and the electron by a charged line and a neutral particle with magnetic moment, respectively. There have al...
Topological phases of matter are purely quantum mechanical and have no classical analogue. Most phases in nature can be classified and studied classically through the concept of symmetry breaking and its theoretical description, Landau-Ginzburg field theory. In contrast to the general wisdom of Landau-Ginzburg field theory, the topological phases share the same symmetry as a trivial insulator and still are different phases. Having gapped spectrum in bulk, they support a metallic edge excition robust against symmetry-respecting perturbation or an emergent fractional excitation in bulk. Following after fractional quantum Hall fluids, many topologically orderd phases, such as spin liquids and topological insulators, have been found and studied. Spin liquids are disordered phases of frustrated antiferromagnets and do not freeze and order even at the lowest temperature. They support an electrically neutral spin-1/2 excitation, which does not exist in a microscopic scale, with emergent dynamical gauge field in bulk and do not have an adiabatic path to a trivial paramagnet phase. The topological insulators are time-reversal symmetric band insulators which cannot evolve smoothly into a trivial insulator with the symmetry, and they have been intensely studied theoretically and experimetally for the last decade. Though the topological insulators are inherently non-interacting systems, they have exotic gapless edge and surface states which demonstrate many interesting quantum phenomena such as fractionalization, axionic electromagnetism, and half quantum Hall effect. In this thesis, we study various quantum phenomena of the topological phases, mainly of the topological insulator and its close relatives in which the physics of spin liquids has been merged into. As they are intrinsically quantum many-body states, the quantum field theory is an invaluable tool to explore the venue of the phases and will be used thorougly in this work. First, we present a topological field theoretic description of the topological insulators and translational symmetric Z2 spin liquid. We demostrate that abelian BF theory is an effective field theory for the two and three dimensional topological insulators. We show that the phenomenologies of BF theory are consistent with those of the topological insulators such as gapless edge and surface excitations and response functions to external gauge field. The same form of the BF theory is also applicable to study the certain classes of Z2 spin liquids with the gapless edge states. We closely examine the possible gapless edge theory of the BF theory and the translational symmetric spin liquids, and we show that the effective BF theories capture the bulk properties as well as the gapless edge spectrum of the spin liquid. Also we discuss the zero-temperature phase diagram of a thin film three dimensional topological insulators with two competing mass terms: time-reversal symmetric exciton condensation and time-reversal symmetry breaking Zeeman effect. There are two phases, quantum spin Hall phase and quantum anomalous Hall phase, and both phases support fractional excitations. We derive an effective topological field theory for the fractional excitations and examine the origin of the fractional excitations. Next we consider the relatives of the topological insulators. First we discuss the proximate symmetry broken phases of topological Mott insulator, a U(1) spin liquid with fermionic spinons in the topological insulator state. The topologically non-trivial band structure of the spinon generates the axion term for the gauge field, and we show that the axion term changes the nature of the confined phase of the spin liquid. Contrast to the conventional confined phase which has only a bond order, the confined phase of the spin liquid is in general a coexisting phase of the two different orders : a current order and a bond order. We then consider another relative of the topological insulator: a bosonic symmetry protected topological phase in two dimensional space, with both PSU(N) and time reversal symmetry. We develop an effective field theory for the phase, which is a SU(N) principal chiral model with a theta-term, and reveal the physics of the topological phase in detail.Finally we consider a topological semimetal, namely a Weyl semimetal. We first demonstrate that the Weyl semimetal can be realized in the magnetically doped topological band insulators. We explicitly derive the low energy theory of Weyl points from the general continuum Hamiltonian and tight-binding model of the topological insulators. Then we study superconducting instabilities of doped Weyl semimetals. We consider a minimal model for a Weyl semimetal and study the superconducting instabilities induced by a short-ranged attractive interaction. With the interaction, we find two competing states: a fully gapped finite-momentum pairing state and a nodal even-parity pairing state. We show that, in a mean field approximateion, the finite-momentum pairing state wins energetically against the usual even-parity paired state. We also show that exotic modes are localized at the full and half quantum vortices of the finite-momentum pairing state.
T HE BERRY PHASE, A CENTRAL CONCEPT IN QUANTUM physics, is rooted in the state space structure of quantum theory and has a profound observable impact within a wide range of phenomena. The remarkably elegant interpretation of this geometric phase as the holonomy of a fiber bundle provides it with a mathematical framework capable of accounting for a broad spectrum of classical and quantum phenomena, such as, for example, the precession of a Foucault pendulum, the parallel transport of a vector on a sphere, the Aharonov-Bohm effect, the gauge theory of molecular physics, the electric polarization and orbital magnetization of solids, and the integer and fractional quantum Hall effects. The Berry phase is the central concept in topological band theory and provides the mechanism responsible for anyonic statistics in correlated many-body systems, a key element that could be used to achieve fault-tolerant quantum computation. In other words, the Berry phase is not just some fashionable idea useful for explaining a few exotic phenomena, but a generic feature of quantum mechanics and an essential ingredient necessary for a coherent understanding of basic quantum phenomena, such as, for example, the electronic properties of materials. It is therefore rather surprising that the full significance of geometric phases was practically ignored for more than fifty years prior to the publication of Berry’s work in 1984 [50]. In this chapter we discuss the basic physics that gives rise to these geometric phases, point out the elements that allow us to distinguish different types of holonomies, summarize the basic concepts relevant to understanding the mathematical structure of geometric phases, and mention a few relevant applications. The concepts and the mathematical framework introduced here will be used in subsequent chapters as the basis for describing the topological properties of condensed matter systems.
The triangular lattice antiferromagnet with S=1/2 spins and nearest-neighbor interactions is known to have long-range antiferromagnetic order, with nearest-neighbor spins at an angle of $120{}^{\ensuremath{\circ}}$. Numerical studies of quantum phases proximate to this state have been limited to small systems because of the sign problem in Monte Carlo simulations in imaginary time. We propose an effective lattice model for quantum fluctuations of the antiferromagnetic order and a sign-problem-free Monte Carlo algorithm enabling studies in large systems sizes. The model is a Z2 gauge theory coupled to gauge-charged scalars which have a relativistic dispersion in the continuum limit. Crucially, the gauge theory is odd, i.e., there is a static, background Z2 gauge charge on each site, accounting for the Berry phases of the half-odd-integer spins on each site. We present results of simulations on lattices of sizes up to $36\ifmmode\times\else\texttimes\fi{}36\ifmmode\times\else\texttimes\fi{}36$. Along with the antiferromagnetically ordered phase, our phase diagram has a valence bond solid state with a $\sqrt{12}\ifmmode\times\else\texttimes\fi{}\sqrt{12}$ unit cell and a gapped Z2 spin liquid. Deconfined critical points or phases in intermediate regions are not ruled out by our present simulations.
We develop a new theoretical framework for describing and analyzing exotic phases of strongly correlated electrons which support excitations with fractional quantum numbers. Starting with a class of microscopic models believed to capture much of the essential physics of the cuprate superconductors, we derive a new gauge theory---based upon a discrete Ising or Z2 symmetry---which interpolates naturally between an antiferromagnetic Mott insulator and a conventional d-wave superconductor. We explore the intervening regime, and demonstrate the possible existence of an exotic fractionalized insulator, the nodal liquid, as well as various more conventional insulating phases exhibiting broken lattice symmetries. A crucial role is played by vortex configurations in the Z2 gauge field. Fractionalization is obtained if they are uncondensed. Within the insulating phases, the dynamics of these Z2 vortices in two dimensions is described, after a duality transformation, by an Ising model in a transverse field, the Ising spins representing the Z2 vortices. The presence of an unusual Berry's phase term in the gauge theory leads to a doping-dependent ``frustration'' in the dual Ising model, being fully frustrated at half filling. The Z2 gauge theory is readily generalized to a variety of different situations, in particular, it can also describe three-dimensional insulators with fractional quantum numbers. We point out that the mechanism of fractionalization for d>1 is distinct from the well-known one-dimensional spin--charge separation. Other interesting results include a description of an exotic fractionalized superconductor in two or higher dimensions.