We discuss the boundary critical behaviors of two dimensional quantum phase transitions with fractionalized degrees of freedom in the bulk, motivated by the fact that usually it is the $1d$ boundary that is exposed and can be conveniently probed in many experimental platforms. In particular, we mainly discuss boundary criticality of two examples: i. the quantum phase transition between a $2d$ $Z_2$ topological order and an ordered phase with spontaneous symmetry breaking; ii. the continuous quantum phase transition between metal and a particular type of Mott insulator (U(1) spin liquid). This theoretical study could be relevant to many purely $2d$ systems, where recent experiments have found correlated insulator, superconductor, and metal in the same phase diagram.
Time reversal invariant (TRI) topological superfluids (TSFs) and topological superconductors (TSCs) are robust symmetry protected gapped topological states. In this article, we study the evolution of these topological states in the presence of time reversal symmetry breaking (TRB) fields and/or sufficiently large TRI fields. Physically, one of the realizations of TRB fields can be internal spin exchange fields due to background magnetic ordering. We find that the fully gapped TSFs and TSCs are generically separated from other nodal states by various zero temperature quantum critical points tha…
For D-dimensional weakly interacting topological insulators in certain symmetry classes, the topological invariant can be calculated from a D- or (D+1)-dimensional integration over a certain curvature function that is expressed in terms of single-particle Green's functions. Based on the divergence of curvature function at the topological phase transition, we demonstrate how a renormalization group approach circumvents these integrations and reduces the necessary calculation to that for the Green's function alone, rendering a numerically efficient tool to identify topological phase transitions …
Nambu-Eliashberg theory for multi-scale quantum criticality : Application to ferromagnetic quantum criticality in the surface of three dimensional topological insulators
Critical transition points between symmetry-broken phases are characterized as fixed points in the renormalization group (RG) theory. We show that, following the standard Wilsonian procedure that traces out the large momentum modes, this well known fact can break down in non-Hermitian systems. Based on non-Hermitian Su-Schrieffer-Hegger (SSH)-type models, we propose a real-space decimation scheme to study the criticality between the topological and trivial phase. We provide concrete examples and an analytic proof to show that the real-space scheme perfectly overcomes the insufficiency of the s…
Decoding Equilibrium and Dynamical Criticality in the 2D Topological Order
University of Science and Technology Beijing（北京科技大学）；Beijing Normal University（北京师范大学）；China University of Petroleum-Beijing（中国石油大学（北京））；Nanjing University of Aeronautics and Astronautics（南京航空航天大学）
An Attempt is made to study the non-Hermitian effect on the topological quantum criticality and also in the physics of Majroana zero mode (MZMs). In this work, the effects and modifications done by the non-Hermitian factor $γ$ on the topological phases, criticality and also in the MZMs is studied. We use the zero mode solutions (ZMS) to construct the phase diagram. We find a correspondence between Hermitian and non-Hermitian model Hamiltonian. The MZMs appear at criticality and has a stability dependence on the new passage created because of the introduction of non-Hermiticity. The multicritic…
Majorana zero modes are well studied in the gapped phases of topological systems. We investigate Majorana zero modes at the topological quantum criticality in one dimensional topological superconducting model with longer range interaction. We identify stable localized Majorana zero modes appearing at criticality under certain conditions. Topological invariant number for these non-trivial criticalities is obtained from zeros of a complex function associated with the Hamiltonian. Behavior of parametric curve at criticalities validate the invariant obtained and account for the appearance of Major…
The topological phase transitions in static and periodically driven Kitaev chains are investigated by means of a renormalization group (RG) approach. These transitions, across which the numbers of static or Floquet Majorana edge modes change, are accompanied by divergences of the relevant Berry connections. These divergences at certain high symmetry points in momentum space form the basis of the RG approach, through which topological phase boundaries are identified as a function of system parameters. We also introduce several aspects to characterize the quantum criticality of the topological p…
The investigation and characterization of topological quantum phase transition between gapless phases is one of the recent interest of research in topological states of matter. We consider transverse field Ising model with three spin interaction in one dimension and observe a topological transition between gapless phases on one of the critical lines of this model. We study the distinct nature of these gapless phases and show that they belong to different universality classes. The topological invariant number (winding number) characterize different topological phases for the different regime of…
Deconfined criticality as intrinsically gapless topological state in one dimension
Institute for Advanced Study in Physics and School of Physics, Zhejiang University（浙江大学物理学院和高等研究院）；Department of Physics, Nanjing University（南京大学物理系）；University of Colorado, Boulder（科罗拉多大学博尔德分校）；Harvard University（哈佛大学）；Department of Physics, University of California, San Diego（加州大学圣地亚哥分校物理系）；Fuzhou University（福州大学）；College of Physics and Information Engineering, Fuzhou University（福州大学物理与信息工程学院）
Quantum Ising transition on the Cayley tree: boundary criticality, universality, and an effective Bethe line
Kavli Institute of Nanoscience, Delft University of Technology（代尔夫特理工大学卡弗里纳米科学研究所）；The Rudolf Peierls Centre for Theoretical Physics, Oxford University（牛津大学鲁道夫·派尔斯理论物理中心）
We demonstrate that the prototypical two-dimensional Chern insulator hosts exotic quantum multi-criticality in the presence of an appropriate periodic driving: a linear Dirac-like transition coexists with a nodal loop-like transition caused by emerging symmetries. The existence of multiple universality classes and scaling laws can be unambiguously captured by a single renormalization group approach based on the stroboscopic Floquet Hamiltonian, regardless of whether the topological transition is associated with the anomalous edge modes or not.
It is often thought that emergent phenomena in topological phases of matter are destroyed when tuning to a critical point. In particular, topologically protected edge states supposedly delocalize when the bulk correlation length diverges. We show that this is not true in general. Edge states of topological insulators or superconductors remain exponentially localized---despite a vanishing band gap---if the transition increases the topological index. This applies to all classes where the topological classification is larger than $Z_2$, notably including Chern insulators. Moreover, these edge sta…
In an attempt to theoretically investigate the quantum phase transition and criticality in topological models, we study Kitaev chain with longer-range couplings (finite number of neighbors) as well as truly long-range couplings (infinite number of neighbors). We carry out an extensive topological characterization of the momentum space to explore the possibility of obtaining higher order winding numbers and analyze the nature of their stability in the model. The occurrences of phase transitions from even-to-even and odd-to-odd winding numbers are observed with decreasing longer-rangeness in the…
Long-range effects induce some interesting behavior and considered as a gateway to understand the non-local behavior in the quantum systems. Especially, the long-range topological models became a platform for the realization of new quasi-particles, which are believed to be potential candidates for the topological qubits. In this work, we consider non-Hermitian Su-Schriffer-Heeger (SSH) model and discuss the interplay of non-Hermiticity and long-range effects. We use the approach of momentum space characterization, critical exponents and curvature renormalization group (CRG) method to understan…
Boundary Criticality at the Nishimori Multicritical Point
Institute for Advanced Study in Physics and School of Physics, Zhejiang University（浙江大学物理学院与高等研究院）；Institute for Advanced Study, Tsinghua University（清华大学高等研究院）；Department of Physics and Engineering Physics, Tulane University（杜兰大学物理与工程物理系）
Electrons undergoing a Mott transition may shed their charge but persist as neutral excitations of a quantum spin liquid (QSL). We introduce concrete two-dimensional models exhibiting this exotic behavior as they transition from superconducting or topological phases into fully charge-localized insulators. We study these Mott transitions and the confinement of neutral fermions at a second transition into a symmetry-broken phase. In the process, we also derive coupled-wire parent Hamiltonians for a non-Abelian QSL and a $\mathbb{Z}_4$ QSL.
We establish an important duality correspondence between topological order in quantum many body systems and criticality in ferromagnetic classical spin systems. We show how such a correspondence leads to a classical and simple procedure for characterization of topological order in an important set of quantum entangled states, namely the Calderbank-Shor-Steane (CSS) states. To this end, we introduce a particular quantum Hamiltonian which allows us to consider the existence of a topological phase transition from quantum CSS states to a magnetized state. We study the ground state fidelity in orde…
The effects of downfolding a Brillouin zone can open gaps and quench the kinetic energy by flattening bands. Quasiperiodic systems are extreme examples of this process, which leads to new phases and critical eigenstates. We analytically and numerically investigate these effects in a two dimensional topological insulator with a quasiperiodic potential and discover a complex phase diagram. We study the nature of the resulting eigenstate quantum phase transitions; a quasiperiodic potential can make a trivial insulator topological and induce topological insulator-to-metal phase transitions through…
The deconfined quantum critical point, a prototype Landau-forbidden transition, could exist in principle in the phase transitions involving symmetry protected topological phase, however, examples of such kinds of transition in physical systems are rare beyond one-dimensional systems. Here, using density-matrix renormalization group calculation, we unveil a bosonic integer quantum Hall phase in two-dimensional correlated honeycomb lattice, by full identification of its internal structure from the topological $\mathbf{K}$ matrix. Moreover we demonstrate that imbalanced periodic chemical potentia…
We consider spontaneous symmetry breaking transitions of strongly interacting two-dimensional Dirac fermions minimally coupled to mass and emergent gauge field order parameters. Using a renormalization group analysis, we show that the presence of gauge fields leads to novel fermion induced quantum (multi-)criticality where the putative emergent Lorentz invariance is violated. We illustrate this with the example of translational symmetry breaking due to charge-density wave order on the honeycomb lattice. Finally, we identify that topological phase transitions are well described by this effectiv…
The nonequilibrium dynamics of two dimensional Su-Schrieffer-Heeger model, in the presence of staggered chemical potential, is investigated using the notion of dynamical quantum phase transition. We contribute to expanding the systematic understanding of the interrelation between the equilibrium quantum phase transition and the dynamical quantum phase transition (DQPT). Specifically, we find that dynamical quantum phase transition relies on the existence of massless {\it propagating quasiparticles} as signaled by their impact on the Loschmidt overlap. These massless excitations are a subset of…
Quantum fluctuating loops in 2+1 dimensions give gapless many-body states that are beyond current field theory techniques. Microscopically, these loops can be domain walls between up and down spins, or chains of flipped spins similar to those in the toric code. The key feature of their dynamics is that the reconnection of a pair of strands is forbidden. This happens at previously-studied multi-critical points between topologically nontrivial phases. We show that this topologically constrained dynamics leads to universality classes with unusual scaling properties. For example, scaling operators…
