In this work, we present a theory for topological phases for quantum systems on tree graphs. Conventionally, topological phases of matter have been studied in regular lattices, but also in quasicrystals and amorphous settings. We consider specific generalizations of regular tree graphs, and explore their topological properties. Unlike conventional systems, infinite quantum trees are not finite-dimensional, allowing for novel phenomena. We find a proliferation of topological zero modes present throughout the entire system, indicating that the bulk also acts as a boundary. We then go on to show that only three symmetry classes host stable topological phases in contrast to the usual five symmetry classes per dimension. Finally, we introduce what we call the Su-Schrieffer-Heeger tree which is topologically non-trivial even in the absence of inner degrees of freedom and does not possess any gapped trivial phases. We realize this system in an electronic circuit and show that our theory matches with experiments.
In this work, we present a theory for topological phases for quantum systems on tree graphs. Conventionally, topological phases of matter have been studied in regular lattices, but also in quasicrystals and amorphous settings. We consider specific generalizations of regular tree graphs, and explore their topological properties. Unlike conventional systems, infinite quantum trees are not finite-dimensional, allowing for novel phenomena. We find a proliferation of topological zero modes present throughout the entire system, indicating that the bulk also acts as a boundary. We then go on to show that only three symmetry classes host stable topological phases in contrast to the usual five symmetry classes per dimension. Finally, we introduce what we call the Su-Schrieffer-Heeger tree which is topologically non-trivial even in the absence of inner degrees of freedom and does not possess any gapped trivial phases. We realize this system in an electronic circuit and show that our theory matches with experiments.
Abstract The dynamical and topological properties of non-Hermitian systems have attracted great attention in recent years. In this work, we establish an intrinsic connection between two classes of intriguing phenomena—topological phases and dynamical quantum phase transitions (DQPTs)—in non-Hermitian systems. Focusing on one-dimensional models with chiral symmetry, we find DQPTs following the quench from a trivial to a non-Hermitian topological phase. Moreover, the critical momenta and critical time of the DQPTs are found to be directly related to the topological invariants of the non-Hermitian system. We further demonstrate our theory in three prototypical non-Hermitian lattice models, the lossy Kitaev chain (LKC), the LKC with next-nearest-neighbor hoppings, and the nonreciprocal Su–Schrieffer–Heeger model. Finally, we suggest a proposal to experimentally verify the found connection by a nitrogen-vacancy center in diamond.
In this paper, we study five-dimensional Dirac fermions of which the extra-dimension is compactified on quantum graphs. We find that there is a nontrivial correspondence between matrices specifying boundary conditions at the vertex of the quantum graphs and zero-dimensional Hamiltonians in gapped free-fermion systems. Based on the correspondence, we provide a complete topological classification of the boundary conditions in terms of noninteracting fermionic topological phases. The ten symmetry classes of topological phases are fully identified in the language of five-dimensional Dirac fermions, and topological numbers of the boundary conditions are given. In analogy with the bulk-boundary correspondence in noninteracting fermionic topological phases, the boundary condition topological numbers predict four-dimensional massless fermions localized at the vertex of the quantum graphs and thus govern the low energy physics in four dimensions.
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.
We study the interplay of spontaneous symmetry breaking and topological properties in interacting one-dimensional models. We solve these models using bozonization and identify topologically non-trivial phases by counting the additional degeneracy (affiliated with the edge modes) of a finite-size system relative to the infinite one. We find even if the mean-field solution is topological, this may not be true when it arises from spontaneous symmetry breaking, including in the Su-Schrieffer-Heeger (SSH) model. This implies that the original SSH model, as presented by Su, Schrieffer, and Heeger, is topologically trivial, as opposed to its mean-field version. A spinful version, on the other hand, does exhibit a topologically non-trivial phase. In that state, both mean-field solutions are topologically non-trivial and correspond to non-interacting SSH chains in the opposite phases with the winding number ν=1. We show that this phase is protected by a chiral symmetry, similar to the non-interacting phases.
We demonstrate the existence of topological phase transitions in interacting, symmetry-protected quantum matter at finite temperatures. Using a combined numerical and analytical approach, we study a one-dimensional Su-Schrieffer-Heeger model with added Hubbard interactions, where no thermodynamic phase transition occurs at finite temperatures. The transition is signalled by a quantized, nonlocal bulk topological order parameter. It is driven by defects, which are enabled by the combination of interaction and thermal activation, with no counterpart in the noninteracting limit. The defects localize topological zero modes, which, when sufficiently abundant, cause the order parameter to vanish. This phenomenon, interpreted via bulk-boundary correspondence, reflects the loss of a topological edge mode at a well-defined critical temperature in the thermodynamic limit. Unlike zero-temperature topological transitions, these finite-temperature transitions lack thermodynamic signatures but remain observable in controlled quantum systems, such as ultracold fermionic atoms in optical lattices.
The interplay of symmetry, topology, and many-body effects in the classification of phases of matter poses a formidable challenge in condensed-matter physics. Such many-body effects are typically induced by inter-particle interactions involving an action at a distance, such as the Coulomb interaction between electrons in a symmetry-protected topological (SPT) phase. In this work, we show that similar phenomena also occur in certain relativistic theories with interactions mediated by gauge bosons, and constrained by gauge symmetry. In particular, we introduce a variant of the Schwinger model or quantum electrodynamics (QED) in 1+1 dimensions on an interval, which displays dynamical edge states localized on the boundary. We show that the system hosts SPT phases with a dynamical contribution to the vacuum $\ensuremath{\theta}$-angle from edge states, leading to a new type of topological QED in 1+1 dimensions. The resulting system displays an SPT phase which can be viewed as a correlated version of the Su-Schrieffer-Heeger topological insulator for polyacetylene due to nonzero gauge couplings. We use bosonization and density-matrix renormalization group techniques to reveal the detailed phase diagram, which can further be explored in experiments of ultra-cold atoms in optical lattices.