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 understand the aspects of interplay. The longer-range (finite neighbors) effect produces higher winding numbers, where we observe a staircase of transitions among the even-even and odd-odd winding numbers which depends on the number of interacting neighbors. Here we also highlight the effect of multi-criticality in the system and show that they belong to a different universality class. The interplay of long-range (infinite neighbors) effect and non-Hermiticity produces fractional topological invariants, and we analyze them from the behavior of pseudo-spin vectors. We also determine the long-range and short-range limit of the model through universality class of critical exponents. Our work mainly showcases that how the study of criticality in topological system is interesting in exploring the interplay of non-Hermiticity and long-range effects.
Extended-range models are the interesting systems, which has been widely used to understand the non-local properties of the fermions at quantum scale. We aim to study the interplay between criticality and extended range couplings under various symmetry constraints. Here, we consider a two orbital Bernevig-Hughes-Zhang model in one dimension with longer (finite neighbor) and long-range (infinite neighbor) couplings. We study the behavior of model using scaling laws and universality class for models with Hermitian, parity-time ([Formula: see text]) symmetric and broken time-reversal symmetries. We observe the interesting results on multi-criticalities, where the universality class of critical exponent is different than the normal criticalities. Also, the results can be generalized by considering the interplay between criticalities and different symmetry classes of Hamiltonian. Also, with the introduction of extended-range of coupling, there occurs different criticalities, and we provide the analogy to characterize their universality classes. We also show the violation of Lorentz invariance at multi-criticalities and evaluation of short-range limit in long-range models as the highlights of this work.
In an attempt to theoretically investigate the quantum phase transition and criticality in topological models, we study a 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 system. We derive topological quantum critical lines and study them to understand the behavior of criticality. A suppression of higher order winding numbers is observed with decreasing longer-rangeness in the model. We show that the mechanism behind such phenomena is due to the superposition and vanishing of the topological quantum critical lines associated with the higher winding number. Through the study of the Berry connection, we show the possible different behaviors of critical lines when they undergo superposition along with the corresponding critical exponents. We analyze the behavior of the long-range models through the momentum space characterization. We also provide the exact solution for the problem and discuss the experimental aspects of the work.
Abstract Disorders and long-range hoppings can induce exotic phenomena in condensed matter and artificial systems. We study the topological and dynamical properties of the quasiperiodic Su–Schrier–Heeger model with long-range hoppings. It is found that the interplay of quasiperiodic disorder and long-range hopping can induce topological Anderson insulator phases with non-zero winding numbers ω = 1 , 2 , and the phase boundaries can be consistently revealed by the divergence of zero-energy mode localization length. We also investigate the nonequilibrium dynamics by ramping the long-range hopping along two different paths. The critical exponents extracted from the dynamical behavior agree with the Kibble–Zurek mechanic prediction for the path with W = 0.90 . In particular, the dynamical exponent of the path crossing the multicritical point is numerical obtained as 1 / 6 ∼ 0.167 , which agrees with the unconventional finding in the previously studied XY spin model. Besides, we discuss the anomalous and non-universal scaling of the defect density dynamics of topological edge states in this disordered system under open boundary condictions.
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 parameter space. We observe the evidence of two multi-critical points, one is topologically trivial and the other one is topologically active. Topological quantum phase transition between the gapless phases on the critical line occurs through the non-trivial multi-critical point in the Lifshitz universality class. We calculate and analyze the behavior of Wannier state correlation function close to the multi-critical point and confirm the topological transition between gapless phases. We show the breakdown of Lorentz invariance at this multi-critical point through the energy dispersion analysis. We also show that the scaling theories and curvature function renormalization group can also be effectively used to understand the topological quantum phase transitions between gapless phases. The model Hamiltonian which we study is more applicable for the system with gapless excitations, where the conventional concept of topological quantum phase transition fails.
The long-range Su-Schrieffer-Heeger (SSH) model, in which the second nearest-neighbor hopping is taken into account, exhibits a topological phase diagram that contains winding numbers
w=0,1, and 2. In the clean system, the change in winding number stems from the band-touching phenomenon. In the presence of disorder, the renormalization of energy band and Fermi level results in the nonzero density of states in the energy gap. These midgap states cause the crossover phenomenon and the divergence of localization length at a critical disorder strength
Uc in the finite SSH system. In this study, we numerically computed the mean winding number and localization length for the disordered SSH system. We find that the disorder is able to drive phase transitions between different mean winding numbers: $w=0\ensuremath{\rightarrow}1$, $0\ensuremath{\rightarrow}2$, $1\ensuremath{\rightarrow}2$, and $2\ensuremath{\rightarrow}1$ in the weak disorder regime. By investigating the wave function distribution and the self-energy, the nonzero mean winding numbers correspond to the so-called topological Anderson insulating (TAI) phases. The finite size scaling for the mean winding number in the TAI phase is shown. For describing the phase transitions in the thermodynamic limit, we apply the criterion of band gap closure resulting from the broadening of energy band and Fermi level to determine the critical disorder strength. The critical disorder strength for self-consistent Born approximation (SCBA)
UcSCBA and that for first Born approximation (FBA)
UcFBA are numerically calculated.
UcSCBA is found to match with
Uc qualitatively. Nonetheless, SCBA indicates the different roles of band shifts and Fermi level broadening near the topological phase transitions. Band shift/Fermi level broadening is more dominant for the transitions from low-to-high/high-to-low winding number. Interestingly, for the transition from bulk insulator to TAI,
UcFBA is quantitatively closer to
Uc than
UcSCBA as long as the renormalized band gap is zero within FBA.
Abstract The dynamics of quantum phase transitions of topological systems has attracted interest. Here we consider the effect of long-range hopping terms on the dynamic behavior. The long-range hoppings preserve chiral symmetry and decay as a function of distance, with an exponent β. We also consider a situation where the hoppings are modulated by oscillations. We compare the results with those obtained for short-range models such as the Su–Schrieffer–Heeger model or its extension to second neighboring cells. While often the lowest critical times coincide with the first peak of the Loschmidt rate, several anomalous cases are identified. Also, the number of critical times often exceeds the absolute value of the change of the winding number in a quench. Various correlation functions are considered, and their time-dependence may often be used to identify the early-time dynamical phase transitions.