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Fractionalized conductivity at topological phase transitions

Yan-Cheng Wang, Meng Cheng, William Witczak‐Krempa, Zi Yang Meng

Published May 15, 2020• 1 citation
OpenAlex

Abstract

The experimental discovery of the fractionalized Hall conductivity revealed new types of quantum particles beyond bosons and fermions. These anyons are usually studied deep inside a topological phase. But can such fractionalization be detected at the phase transition point to a conventional phase? To answer this question, we study a quantum phase transition between a topological state called a $\mathbb{Z}_2$ spin liquid and a conventional superfluid using large-scale quantum Monte Carlo simulations. Our results show that the conductivity at the quantum critical point becomes a simple fraction of its value at the conventional insulator-to-superfluid transition. This opens the door for the experimental detection of anyons in a much broader regime, and has ramifications in the study of quantum materials and ultra-cold atomic gases.

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Abstract The experimental discovery of the fractional Hall conductivity in two-dimensional electron gases revealed new types of quantum particles, called anyons, which are beyond bosons and fermions as they possess fractionalized exchange statistics. These anyons are usually studied deep inside an insulating topological phase. It is natural to ask whether such fractionalization can be detected more broadly, say near a phase transition from a conventional to a topological phase. To answer this question, we study a strongly correlated quantum phase transition between a topological state, called a $${{\mathbb{Z}}}_{2}$$ Z 2 quantum spin liquid, and a conventional superfluid using large-scale quantum Monte Carlo simulations. Our results show that the universal conductivity at the quantum critical point becomes a simple fraction of its value at the conventional insulator-to-superfluid transition. Moreover, a dynamically self-dual optical conductivity emerges at low temperatures above the transition point, indicating the presence of the elusive vison particles. Our study opens the door for the experimental detection of anyons in a broader regime, and has ramifications in the study of quantum materials, programmable quantum simulators, and ultra-cold atomic gases. In the latter case, we discuss the feasibility of measurements in optical lattices using current techniques.

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The experimental discovery of the fractionalized Hall conductivity revealed new types of quantum particles beyond bosons and fermions. These anyons are usually studied deep inside a topological phase. But can such fractionalization be detected at the phase transition point to a conventional phase? To answer this question, we study a quantum phase transition between a topological state called a Z2\mathbb{Z}_2Z2​ spin liquid and a conventional superfluid using large-scale quantum Monte Carlo simulations. Our results show that the conductivity at the quantum critical point becomes a simple fraction of its value at the conventional insulator-to-superfluid transition. This opens the door for the experimental detection of anyons in a much broader regime, and has ramifications in the study of quantum materials and ultra-cold atomic gases.
Abstract The experimental discovery of the fractional Hall conductivity in two-dimensional electron gases revealed new types of quantum particles, called anyons, which are beyond bosons and fermions as they possess fractionalized exchange statistics. These anyons are usually studied deep inside an insulating topological phase. It is natural to ask whether such fractionalization can be detected more broadly, say near a phase transition from a conventional to a topological phase. To answer this question, we study a strongly correlated quantum phase transition between a topological state, called a Z2{{\mathbb{Z}}}_{2}Z2​ Z 2 quantum spin liquid, and a conventional superfluid using large-scale quantum Monte Carlo simulations. Our results show that the universal conductivity at the quantum critical point becomes a simple fraction of its value at the conventional insulator-to-superfluid transition. Moreover, a dynamically self-dual optical conductivity emerges at low temperatures above the transition point, indicating the presence of the elusive vison particles. Our study opens the door for the experimental detection of anyons in a broader regime, and has ramifications in the study of quantum materials, programmable quantum simulators, and ultra-cold atomic gases. In the latter case, we discuss the feasibility of measurements in optical lattices using current techniques.
The experimental discovery of the fractional Hall conductivity in two-dimensional electron gases revealed new types of quantum particles, called anyons, which are beyond bosons and fermions as they possess fractionalized exchange statistics. These anyons are usually studied deep inside an insulating topological phase. It is natural to ask whether such fractionalization can be detected more broadly, say near a phase transition from a conventional to a topological phase. To answer this question, we study a strongly correlated quantum phase transition between a topological state, called a [Formula: see text] quantum spin liquid, and a conventional superfluid using large-scale quantum Monte Carlo simulations. Our results show that the universal conductivity at the quantum critical point becomes a simple fraction of its value at the conventional insulator-to-superfluid transition. Moreover, a dynamically self-dual optical conductivity emerges at low temperatures above the transition point, indicating the presence of the elusive vison particles. Our study opens the door for the experimental detection of anyons in a broader regime, and has ramifications in the study of quantum materials, programmable quantum simulators, and ultra-cold atomic gases. In the latter case, we discuss the feasibility of measurements in optical lattices using current techniques.
Realizing strongly-correlated topological phases of ultracold gases is a central goal for ongoing experiments. And while fractional quantum Hall states could soon be implemented in small atomic ensembles, detecting their signatures in few-particle settings remains a fundamental challenge. In this work, we numerically analyze the center-of-mass Hall drift of a small ensemble of hardcore bosons, initially prepared in the ground state of the Harper-Hofstadter-Hubbard model. By extracting the Hall conductivity in a wide range of the magnetic flux, we identify an emergent Hall plateau compatible with a fractional Chern insulator state: the width of the plateau agrees with the spectral and topological properties of the prepared ground state, while the Hall conductivity approaches a fractional value determined by the many-body Chern number. A comparison with a direct application of Streda's formula is also discussed. Our calculations suggest that fractional Chern insulators can be detected in cold-atom experiments, using available detection methods.
We study a statistical mechanics model of two species of bosons with mutual statistics $\ensuremath{\theta}=2\ensuremath{\pi}/n$ in (2+1) dimensions. This model realizes a fractionalized topological phase of bosons, which is a fractionalized version of the boson integer quantum Hall effect. The model can be studied with sign-free Monte Carlo simulations. We study the phase transitions between the fractionalized topological phase and a trivial insulator, and between different topological phases. We find that these transitions are continuous, and we measure their critical exponents.
We investigate the disorder-driven phase transitions in bosonic fractional quantum Hall liquids at filling factors f=1/2f=1/2f=1/2 and f=1f=1f=1 in the lowest Landau level. We use the evolution of ground-state entanglement entropy, fidelity susceptibility, and Hall conductance with increasing disorder strength to identify the underlying phase transitions. The critical disorder strengths obtained from these different quantities are consistent with each other, validating the reliability of our numerical calculations based on exact diagonalization. At f=1/2f=1/2f=1/2, we observe a clear transition from the bosonic Laughlin state to a trivial insulating phase. At f=1f=1f=1, we identify a direct phase transition from the non-Abelian bosonic Moore-Read state to a trivial insulating phase, although some signs of a disorder-induced intermediate fractional quantum Hall phase were recently reported for the f=5/2f=5/2f=5/2 fermionic cousin.
Anyons in a topologically ordered phase can carry fractional quantum numbers with respect to the symmetry group of the considered system, one example being the fractional charge of the quasiparticles and quasiholes in the fractional quantum Hall effect. When such symmetry-fractionalized anyons condense, the resulting phase must spontaneously break the symmetry and display a local order parameter. In this paper, we study the phase diagram and anyon condensation transitions of a \mathbb{Z}_2 ℤ 2 topological order perturbed by Ising interactions in the Toric Code. The interplay between the global (``onsite’’) Ising ( \mathbb{Z}_2 ℤ 2 ) symmetry and the lattice space group symmetries results in a non-trivial symmetry fractionalization class for the anyons, and is shown to lead to two characteristically different confined, symmetry-broken phases. To understand the anyon condensation transitions, we use the recently introduced critical torus energy spectrum technique to identify a line of emergent 2+1D XY* transitions ending at a fine-tuned (Ising ^2 2 )* critical point. We provide numerical evidence for the occurrence of two symmetry breaking patterns predicted by the specific symmetry fractionalization class of the condensed anyons in the explored phase diagram. In combination with large-scale quantum Monte Carlo simulations we measure unusually large critical exponents \eta η for the scaling of the correlation function at the continuous anyon condensation transitions, and we further identify lines of (weakly) first order transitions in the phase diagram. As an important additional result, we discuss the phase diagram of a resulting 2+1D Ashkin-Teller model, where we demonstrate that torus spectroscopy is capable of identifying emergent XY/O(2) critical behaviour, thereby solving some longstanding open questions in the domain of the 3D Ashkin-Teller models. To establish the generality of our results, we propose a field theoretical description capturing the transition from a \mathbb{Z}_2 ℤ 2 topological order to either \mathbb{Z}_2 ℤ 2 symmetry broken phase, which is in excellent agreement with the numerical results.