In clean and weakly disordered systems, topological and trivial phases having a finite bulk energy gap can transit to each other via a quantum critical point. In presence of strong disorder, both the nature of the phases and the associated criticality can fundamentally change. Here we investigate topological properties of a strongly disordered fermionic chain where the bond couplings are drawn from normal probability distributions which are defined by characteristic standard deviations. Using numerical strong disorder renormalization group methods along with analytical techniques, we show that the competition between fluctuation scales renders both the trivial and topological phases gapless with Griffiths like rare regions. Moreover, the transition between these phases is solely governed by the fluctuation scales, rather than the means, rendering the critical behavior to be determined by an infinite randomness fixed point with an irrational central charge. Our work points to a host of novel topological phases and atypical topological phase transitions which can be realized in systems under strong disorder.
In clean and weakly disordered systems, topological and trivial phases having a finite bulk energy gap can transit to each other via a quantum critical point. In presence of strong disorder, both the nature of the phases and the associated criticality can fundamentally change. Here we investigate topological properties of a strongly disordered fermionic chain where the bond couplings are drawn from normal probability distributions which are defined by characteristic standard deviations. Using numerical strong disorder renormalization group methods along with analytical techniques, we show that the competition between fluctuation scales renders both the trivial and topological phases gapless with Griffiths like rare regions. Moreover, the transition between these phases is solely governed by the fluctuation scales, rather than the means, rendering the critical behavior to be determined by an infinite randomness fixed point with an irrational central charge. Our work points to a host of novel topological phases and atypical topological phase transitions which can be realized in systems under strong disorder.
We study the effects of quenched disorder in a class of quantum chains with (p+1)-multispin interactions exhibiting a free fermionic spectrum, paying special attention to the case p=2. Depending if disorder couples to (i) all the couplings or just to (ii) some of them, we have two distinct physical scenarios. In case (i), we find that the transitions of the model are governed by a universal infinite-randomness critical point surrounded by quantum Griffiths phases similarly as happens to the random transverse-field Ising chain. In case (ii), we find that quenched disorder becomes an irrelevant perturbation: the clean critical behavior is stable and Griffiths phases are absent. Beyond the perturbative regime, disorder stabilizes a line of finite-randomness critical points (with nonuniversal critical exponents), that ends in a multicritical point of infinite-randomness type. In that case, quantum Griffiths phases also appear surrounding the finite-disorder transition point. We have characterized the correlation functions and the low-temperature thermodynamics of these chains. Our results are derived from a strong-disorder renormalization-group technique and from finite-size scaling analysis of the spectral gap computed exactly (up to sizes $\ensuremath{\sim}{10}^{7}$) via an efficient new numerical method recently introduced in the literature [Phys. Rev. B 104, 174206 (2021)].
We investigate the fate and robustness of topological strong zero modes (SZMs) in random Ising-Majorana chains using the SZM fidelity, FSZM, as a many-body diagnostic that quantifies how accurately SZM operators map the {\it entire} spectrum between opposite parity sectors. In clean systems, FSZM=1 in the topological phase, vanishes in the trivial regime, and takes the universal value 8/π at the (1+1)D Ising critical point. Here we study how quenched disorder modifies this picture across the infinite-randomness fixed point (IRFP) governing the criticality of the random chain. In both microcanonical and canonical ensembles, SZMs persist throughout the topological phase, including the gapless Griffiths regime, with fidelities converging exponentially to unity. At the IRFP, however, the fidelity distributions become ensemble dependent: the microcanonical ensemble displays bimodal peaks at {0.5,1}, while the canonical ensemble develops a triple-peak structure at {0,0.5,1} with power-law singularities. Our results establish FSZM as a robust probe of localization-protected topological order and uncover distinctive topological features of infinite-randomness criticality. Unlike the clean Ising CFT, where the finite critical value arises from a cancellation of power laws, the IRFP seems to exhibit an intrinsically stronger topological character. The edge-selective structure of the critical distributions may suggest a boundary manifestation of the average Kramers-Wannier duality symmetry at the IRFP.
Transition out of a topological phase is typically characterized by discontinuous changes in topological invariants along with bulk gap closings. However, as a clean system is geometrically punctured, it is natural to ask the fate of an underlying topological phase. To understand this physics we introduce and study both short- and long-ranged toy models where a one-dimensional topological phase is subjected to bond percolation protocols. We find that non-trivial boundary phenomena follow competing energy scales even while the global topological response is governed via geometrical properties of the percolated lattice. Using numerical, analytical, and appropriate mean-field studies we uncover the rich phenomenology and the various crossover regimes of these systems. In particular, we discuss the emergence of the ``fractured topological region'' where an overall trivial system contains a macroscopic number of topological clusters. Our study shows the interesting physics that can arise from an interplay of geometrical disorder within a topological phase.
This thesis focuses on the effects of strong electron-electron interactions on topological phase transitions.In very general terms, topological phase transitions are typically associated with the formation of gapless states at the boundary between two gapped phases.The opening of a gap is usually associated with a spontaneous symmetry breaking that can obliterate the topological nature of the system.In this work we highlight a completely different destiny for a topological transition in the presence of interaction and we develop a theoretical framework to address this physics identifying the physical origin for the unconventional scenario.In particular, we show that strong electronic correlation can lead to a gap opening at the topological transition without any symmetry breaking.This turns the topological transition discontinuous, thereby giving rise to a quantum critical point connecting the continuous region with the discontinuous one.This unconventional scenario establishes a surprising connection between a topological transition and quantum criticality.By solving a Bernevig-Hughes-Zhang model supplemented with a local symmetric interaction, we demonstrate that the crucial role of orbital polarization naturally emerges, already at the mean-field level.This local parameter effectively captures the essence of the topological transition, complementing the established evaluation of topological invariants and enabling a Landau-like description.Within this framework, we analyze the interplay between band topology and the dynamical effects of electronic correlations, achieved by developing a non-perturbative method for calculating the effective free energy, the Variational Gaussian Approximation, which is compared with both a simple static mean-field, which is not able to capture any critical behavior for the topological transition and with the accurate Dynamical Mean-Field Theory which provides an important benchmark for the accuracy of the approach.We show indeed that dynamical quantum fluctuations can lead to gap opening without any symmetry breaking.The growth of these quantum effects as the interaction is increased turns indeed the topological transition from continuous to discontinuous.This change of behavior is associated with the presence of massive Dirac fermion at the transition point showing a Gross-Neveu critical behaviour near the quantum critical endpoint.We identify the gap opening as a condensed matter analog of the Coleman-Weinberg mechanism of mass generation.The theory presented offers a broad scope for extension in various directions, ranging from changing the topological model and modifying the interactions, to incorporating non-local effects, or even applying it to entirely trivial strongly correlated models.
The quantum critical properties of interacting fermions in the presence of disorder are still not fully understood. While it is well known that for Dirac fermions, interactions are irrelevant to the noninteracting infinite randomness fixed point (IRFP), the problem remains largely open in the case of Majorana fermions which further display a much richer disorder-free phase diagram. Here, pushing the limits of density matrix renormalization group simulations, we carefully examine the ground state of a Majorana chain with both disorder and interactions. Building on appropriate boundary conditions and key observables such as entanglement, energy gap, and correlations, we strikingly find that the noninteracting Majorana IRFP is very stable against finite interactions, in contrast with previous claims.
We investigate the effects of quenched randomness on topological quantum phase transitions in strongly interacting two-dimensional systems. We focus first on transitions driven by the condensation of a subset of fractionalized quasiparticles (``anyons'') identified with ``electric charge'' excitations of a phase with intrinsic topological order. All other anyons have nontrivial mutual statistics with the condensed subset and hence become confined at the anyon condensation transition. Using a combination of microscopically exact duality transformations and asymptotically exact real-space renormalization group techniques applied to these two-dimensional disordered gauge theories, we argue that the resulting critical scaling behavior is ``superuniversal'' across a wide range of such condensation transitions and is controlled by the same infinite-randomness fixed point as that of the 2D random transverse-field Ising model. We validate this claim using large-scale quantum Monte Carlo simulations that allow us to extract zero-temperature critical exponents and correlation functions in (2+1)D disordered interacting systems. We discuss generalizations of these results to a large class of ground-state and excited-state topological transitions in systems with intrinsic topological order as well as those where topological order is either protected or enriched by global symmetries. When the underlying topological order and the symmetry group are Abelian, our results provide prototypes for topological phase transitions between distinct many-body localized phases.