We consider two-dimensional topological insulators as prototypes of symmetry protected topological phases where spin-orbit coupling gives rise to the topological insulator phase. The appearance of additional non-topological terms might cause quantum phase transitions into trivial phases when this topologically trivial term overweights the spin-orbit coupling. Usually such a phase transition is associated with closing of the bulk gap. In contrast, time-reversal symmetry breaking terms immediately destroy the topological phase without closing of the bulk gap unless the axial spin symmetry remains preserved. These findings are discussed in the context of topological insulators in the presence of interactions.
We consider two-dimensional topological insulators as prototypes of symmetry protected topological phases where spin-orbit coupling gives rise to the topological insulator phase. The appearance of additional non-topological terms might cause quantum phase transitions into trivial phases when this topologically trivial term overweights the spin-orbit coupling. Usually such a phase transition is associated with closing of the bulk gap. In contrast, time-reversal symmetry breaking terms immediately destroy the topological phase without closing of the bulk gap unless the axial spin symmetry remains preserved. These findings are discussed in the context of topological insulators in the presence of interactions.
We consider two-dimensional Chern insulators and time-reversal invariant topological insulators and discuss the effect of perturbations breaking either particle-number conservation or time-reversal symmetry. The appearance of trivial mass terms is expected to cause quantum phase transitions into trivial phases when such a perturbation overweighs the topological term. These phase transitions are usually associated with a bulk-gap closing. In contrast, the chiral Chern insulator is unaffected by particle-number breaking perturbations. Moreover, the [Formula: see text] topological insulator undergoes phase transitions into topologically trivial phases without bulk-gap closing in the presence of any of such perturbations. In certain cases, these phase transitions can be circumvented and the protection restored by another U(1) symmetry, e.g. due to spin conservation. These findings are discussed in the context of interacting topological insulators.
When time-reversal symmetry is present there is a sharp distinction between topological and trivial band insulators which ensures that, as parameters are varied, these phases are separated by a phase transition at which the bulk gap closes. Surprisingly we find that even in the absence of time-reversal symmetry, gapless regions originating from the phase boundaries persist. Moreover, the critical line generically opens up to enclose Chern insulating phases that are thin but of finite extent in the phase diagram. We explain the topological origin of this effect in terms of quantized charge pumping, showing in particular that it is robust to the effect of disorder and interactions.
Topological quantum phase transitions are characterized by changes in global topological invariants. These invariants classify many-body systems beyond the conventional paradigm of local order parameters describing spontaneous symmetry breaking. For noninteracting electrons, it is well understood that such transitions are continuous and always accompanied by a gap closing in the energy spectrum, given that the symmetries protecting the topological phase are maintained. Here, we demonstrate that a sufficiently strong electron-electron interaction can fundamentally change the situation: we discover a topological quantum phase transition of first-order character in the genuine thermodynamic sense that occurs without a gap closing. Our theoretical study reveals the existence of a quantum critical endpoint associated with an orbital instability on the transition line between a 2D topological insulator and a trivial band insulator. Remarkably, this phenomenon entails unambiguous signatures related to the orbital occupations that can be detected experimentally.
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.
Topological phases of matter are generally characterized by topological properties of energy bands of a system. Their transitions under preserved symmetries occur through closing a gap of energy bands, leading to topologically protected edge states in energy spectra in topological phases. Here we predict a new topological phase that emerges through closing a gap of bands constructed by energy bands, instead of through closing an energy gap with preserved symmetries. From this perspective, topological phases may arise from topological properties of the "bands of bands" associated with their gap closure and corresponding edge states. We demonstrate this idea by studying a tight-binding model. We find that the Wannier bands constructed by energy bands exhibit a gap closure associated with a change of a winding number, while the energy bands remain gapped and trivial without any zero energy modes. In addition, the topological Wannier bands give rise to quantized edge polarizations. Since the emergence of this topological phase does not involve any energy gap closure, we expect its appearance under unitary time evolution. Indeed, this phase appears as we perform a quench dynamics. Our study opens a new direction for exploring topological phases beyond conventional energy band characterization.
We consider a one-dimensional, time-reversal-invariant system with attractive interactions and spin-orbit coupling. Such a system is gapless due to the strong quantum fluctuations of the superconducting order parameter. However, we show that a sharply defined topological phase with protected, exponentially localized edge states exists. If one of the spin components is conserved, the protection of the edge modes can be understood as a consequence of the presence of a spin gap. In the more general case, the localization of the edge states arises from a gap to single-particle excitations in the bulk. We consider specific microscopic models and demonstrate both analytically and numerically (using density matrix renormalization group calculations) that they can support the topologically nontrivial phase.