Anyons in Quantum Many-Body Systems

Unlike in the lowest Landau level, a unified description of the second Landau level fractional quantum Hall effect (FQHE) has remained elusive. In this work, we propose that the parton theory of the FQHE, which goes beyond the composite fermion theory, accomplishes this in a natural fashion for a subset of experimentally observed sequences of FQHE states in the second Landau level. This work is motivated by the unexpected observation of FQHE at filling factor 2+6/13 a few years ago. We construct a parton wave function for this state and demonstrate that it has lower energy than the standard composite fermion state, has significant overlap with the exact Coulomb ground state for finite systems, and is robust to small changes in the interaction. We show how this fits well naturally with the anti-Pfaffian state at 5/2, and make several experimentally verifiable predictions that can reveal the topological nature of this state.

Interfaces between topologically distinct phases of matter reveal a remarkably rich phenomenology. We study the experimentally relevant interface between a Laughlin phase at filling factor ν = 1/3 and a Halperin 332 phase at filling factor ν = 2/5. Based on our recent construction of chiral topological interfaces in [arXiv:1806.06858], we study a family of model wavefunctions that captures both the bulk and interface properties. These model wavefunctions are built within the matrix product state framework. The validity of our approach is substantiated through extensive comparisons with exact diagonalization studies. We probe previously unreachable features of the low energy physics of the transition. We provide, amongst other things, the characterization of the interface gapless mode and the identification of the spin and charge excitations in the many-body spectrum. The methods and tools presented are applicable to a broad range of topological interfaces.

We consider an exactly solvable model in (3+1)d whose ground state is described by a topological field theory that has a 2-form gauge theory interpretation. The input data of this model is a finite abelian group and a 4-cocycle in the fourth cohomology group of the Eilenberg-Maclane space K(G,2). Using the so-called W-construction of K(G,2), we show algebraically and geometrically how the 4-cocycle reduces to an F-symbol and an R-matrix that satisfy the pentagon and the hexagon relations. We further emphasize to which extent our model is related to the Walker-Wang model for the braided fusion category of G-graded vector spaces. Finally, we study the excitations of this model by considering a generalization of Ocneanu's tube algebra.

We propose a systematic approach to constructing microscopic models with fractional excitations in three-dimensional (3D) space. Building blocks are quantum wires described by the (1+1)-dimensional conformal field theory (CFT) associated with a current algebra $g$. Wires are coupled with each other to form a 3D network through the current-current interactions of $g_1$ and $g_2$ CFTs that are related to the g CFT by a nontrivial conformal embedding $g \supset g_1 \times g_2$. The resulting model can be viewed as a layer construction of a 3D topological ordered state, in which the conformal embedding in each wire implements the anyon condensation between adjacent layers. Local operators acting on the ground state create point-like or loop-like deconfined excitations depending on the branching rule. We demonstrate our construction for a simple solvable model based on the conformal embedding $SU(2)_1 \times SU(2)_1 \supset U(1)_4 \times U(1)_4$. We show that the model possesses extensively degenerate ground states on a torus with deconfined quasiparticles, and that appropriate local perturbations lift the degeneracy and yield a 3D $Z_2$ gauge theory with a fermionic $Z_2$ charge.

We propose an order parameter to detect the symmetry fractionalization class of a given topological phase in 2D. We focus on quantum double models enriched with an internal symmetry group using their projected entangled pair state representations. The order parameter we have developed is purely two dimensional; without a need for dimensional reduction to 1D and it is calculated on the bulk. Moreover it depends only on the virtual symmetry operators which allows to identify the quantum phase independently of its actual system representation. We will give enough examples to cover many interesting cases in which the symmetry fractionalization class can be distinguished.

Abstract: Inspired by the recent experimental advances in the study of ultracold atoms trapped in optical lattices, we consider models of fermions hopping in ladder geometries and subject to artificial magnetic fluxes. By applying the concept of resonances in chiral currents, we find a parameter (momentum component of the current in Fourier space), distinguishing between trivial and quantum Hall (QH) phases in non-interacting cases. We aim for evidence about fractional QH phases: In case of nearest-neighbor Hubbard interactions, we identify a gap in the spin sector of the corresponding Luttinger liquid leading to a resonant state at fractional filling factor $\nu=1/2$. We support our analytic results with matrix product sates (MPS) simulations.

The behavior of topological phase transitions is governed by anyon condensation. The formalism of tensor network states not only enables the construction of anyonic excitations, but it also provides a natural tool to investigate the critical behavior of phase transitions which are governed by the condensation of anyons. We analyze phase transitions between topologically distinct phases by doing a variational optimization over the manifold of tensor network states with a well defined virtual symmetry. This approach allows us to use anyon condensation, which is characterized by certain condensate fractions, as a probe to study topological phase transitions in analogy to the use of local order parameters in the context to conventional (symmetry breaking) phase transitions.

We propose a standard time-of-flight experiment as a method to observe the anyonic statistics of quasiholes in a fractional quantum Hall state of ultracold atoms. The quasihole states can be stably prepared by pinning the quasiholes with localized potentials and a measurement of the mean square radius of the freely expanding cloud, which is related to the average total angular momentum of the initial state, offers direct signatures of the statistical phase. Our proposed method is validated by Monte Carlo calculations for

Anyons in fractional quantum Hall systems exhibit fractional charge and fractional quantum statistics. Particularly non-Abelian anyons are important in topological quantum computation. In the fractional quantum Hall effect, in the continuum, anyonic states are derived as conformal blocks of the underlying rational Conformal Field Theory, and we retrace the similar path to construct the states on lattices. We insert Ising non-Abelian anyons, both quasiholes and quasielectrons, in the Moore-Read lattice states and investigate their density profile, charge, shape and braiding statistics by using Metropolis Monte Carlo simulations. We show that the quasielectrons can be constructed and investigated in a way similar to the one for quasiholes. We also derive parent Hamiltonians for the anyonic states.

Conformal field theory has turned out to be a powerful tool to derive interesting lattice models with analytical ground states. In 1-D, the models exhibit criticality whose ground states are related to the Laughlin states while in 2-D the ground states are lattice versions of fractional quantum Hall states. The exact lattice models involve interactions over long distances, which is difficult to realize in experiments. It seems, however, that such long-range interactions should not be necessary, as the correlations decay exponentially in the bulk. This poses the question, whether the Hamiltonians can be truncated to contain only local interactions without changing the physics of the ground state. Previous studies have in a couple of cases with particularly much symmetry obtained such local Hamiltonians by a combination of guesswork and numerical optimization. Here, we propose a different strategy to construct truncated Hamiltonians, which does not rely on optimization, and which can be applied independent of the choice of lattice. We test the approach on 2-D models with Bosonic Laughlin-like ground states and find that the overlaps per site between the states constructed from conformal field theory and the ground states of the truncated models are higher than 0.98 for all the studied lattices. In 1-D, we investigate local lattice models of fermions and hardcore bosons. Based on computations of wavefunction overlaps, entanglement entropies, and two-site correlation functions for systems of up to 32 sites, we find that the ground state is close to the ground state of the exact critical model.

In this work, we describe the characterization of fracton phases in terms of their gapped excitations. In particular, we describe fusion and statistical processes in Abelian fracton phases. As compared to more conventional states, there are two key new features. First is the restricted mobility of excitations, which implies that statistical processes need not always take the form of familiar braiding processes. The fusion theory we develop encodes the mobility of excitations, which allows us to use it as a starting point to describe statistical processes. Second, the number of distinct excitation types in fracton phases in infinite, in contrast to conventional phases with intrinsic topological ordered (iTO). Moreover, if one considers excitations supported in a region with linear size $L$, the number of excitation types supported in the region grows exponentially with $L$. This strongly suggests that, in order to get a manageable theory, we need to impose some structure beyond what is present in the theory of conventional iTO phases. To build a theory that incorporates these features, we consider lattice translation symmetry. If we ignore translation symmetry, the fusion of excitations in an Abelian fracton phase is described by an infinite Abelian group, whose elements correspond to distinct excitation types. Translation symmetry acts on this Abelian group, giving it more structure and making it into a more manageable object to work with. Moreover, this action directly allows us to describe the mobility of excitations at the level of the fusion theory, which then forms the basis for a description of statistical processes.

Anyonic fractional charges $e^*$ have been detected by autocorrelation shot noise at a quantum point contact (QPC) between two fractional quantum Hall edges. We suggest that the autocorrelation noise is useful also for detecting Abelian anyonic fractional statistics [1]. We predict the noise of electrical tunneling current $I$ at the QPC of the fractional-charge detection setup, when anyons are dilutely injected [2], from an additional edge biased by a voltage, to the setup in equilibrium. At large voltages, the nonequilibrium noise is reduced below the thermal equilibrium noise by the value $2 e^* I$. This negative excess noise is opposite to the positive excess noise $2 e^* I$ of the conventional fractional-charge detection and also to usual positive autocorrelation noises of electrical currents. This is a signature of the Abelian fractional statistics, resulting from the effective braiding of an anyon thermally excited at the QPC around another anyon injected from the additional edge. [1] Byeongmok Lee, Cheolhee Han, and H.-S. Sim, Negative Excess Shot Noise by Anyon Braiding, submitted. not yet posted on arXiv. [2] Cheolhee Han, Jinhong Park, Yuval Gefen, and H.-S. Sim, Topological vacuum bubbles by anyon braiding, Nature Communications 7, 11131(2016).

Using our newly developed graph-based Projected Entangled-Pair State (gPEPS) tensor network method, we explore the phase diagram of the Kitaev-Heisenberg model on all of the three-dimensional (3D) lattices, belonging to the family of 3D Kitaev models, in the full parameter space. Our findings reveal the existence of different phases ranging from ferromagnetic, Neel, Stripy to Kitaev quantum spin liquids in different regimes of the parameter space on different 3D lattices. We capture possible phase transitions and phase boundaries by measuring expectation values of local operators such as magnetization and two-point correlators and by calculating the local fidelity and two-site entropy from the ground state wave function obtained with gPEPS method in the thermodynamic limit.

Composites formed from charged particles and magnetic flux tubes, proposed by Wilczek, are one model for anyons—particles obeying fractional statistics. Here we propose a scheme for realizing charged flux tubes, in which a charged object with an intrinsic magnetic dipole moment is placed between two semi-infinite blocks of a high-permeability ($\mu_r$) material, and the images of the magnetic moment create an effective flux tube. We show that the scheme can lead to a realization of Wilczek’s anyons, when a two-dimensional electron system, which exhibits the integer quantum Hall effect, is sandwiched between two blocks of the high-$\mu_r$ material with a temporally fast response (in the cyclotron and Larmor frequency range). The signature of Wilczek’s anyons is a slight shift of the resistivity at the plateau of the IQHE. Thus, the quest for high-$\mu_r$ materials at high frequencies, which is underway in the field of metamaterials, and the quest for anyons, are here found to be on the same avenue.

There is a class of surprisingly simple models of various quantum Hall states which can be easily stated yet automatically capture interesting properties of complicated non-Abelian ground states (with the simplest family encapsulating states from Laughlin up to Blok-Wen). They allow some computations to be done analytically -- for example, a direct computation of the anyonic phase of Laughlin quasiholes from microscopic definitions -- and there is some hope that they may be able to supply an interesting, tractable model of many more states, including hierarchy states. I will describe these models.

The recent observation of a half-integer quantized thermal Hall effect in α−RuCl3 is interpreted as a unique signature of a chiral spin liquid with a Majorana edge mode. A similar quantized thermal Hall effect is expected in chiral topological superconductors. The unavoidable presence of gapless acoustic phonons, however, implies that, in contrast to the quantized electrical conductivity, the thermal Hall conductivity $\kappa_xy$ is never exactly quantized in real materials. In this talk I will examine how phonons affect the quantization of the thermal conductivity, focusing on the edge theory. As an example, I consider a Kitaev spin liquid gapped by an external magnetic field coupled to acoustic phonons. The coupling to phonons destroys the ballistic thermal transport of the edge mode completely, as energy can leak into the bulk, thus drastically modifying the edge picture of the thermal Hall effect. Nevertheless, the thermal Hall conductivity remains approximately quantized, and in fact I argue that the coupling to phonons to the edge mode is a necessary condition for the observation of the quantized thermal Hall effect. The strength of this edge coupling does, however, not affect the conductivity. For sufficiently clean systems the leading correction to the quantized thermal Hall effect, $\Delta\kappa_xy/T∼sign(B)T^2$, arises from an intrinsic anomalous Hall effect of the acoustic phonons due to Berry phases imprinted by the chiral (spin) liquid in the bulk. This correction depends on the sign but not the amplitude of the external magnetic field.

Haah's cubic code is an model realizing fracton topolical order and is a promising candidate for self correcting quantum memory. We analyse the robustness of the topological order in Haah's cubic code in a homogeneous magnetic field using perturbative continuous unitary transformations and a mean field approach. We argue that there is a first order phase transition from the topological phase to the polarized phase.

Anyons are exotic particles, which differ from bosons and fermions in their statistical phase under exchange of two particles. They arise as quasi-particle excitations in fractional quantum Hall systems, but their direct observation is still challenging. A promising approach for microscopic access to anyons is the quantum simulation of the fractional quantum Hall effect with ultracold atoms. In these system, the large magnetic field can be created via a rapid rotation of the harmonic trap and the strongly-correlated states can be probed via single-atom resolved imaging. The Laughlin state of small systems can be adiabatically prepared by reaching rotation frequencies close to the centrifugal limit. Different protocols to probe the fractional statistics of the quasihole excitations have been proposed, including a spectroscopic signature by removing an atom into a different internal state or an interferometric signature by manipulating two quasiholes with optical tweezers. An alternative approach is the realization of the anyon-Hubbard model via a mapping to bosons with density-dependent Peierls phases. Here, I will present these ideas and discuss the experimental challenges and requirements on the way to a direct observation of anyons.

The realization of topological quantum phases of matter remains a key challenge to condensed matter physics and quantum information science. In this work, we demonstrate that progress in this direction can be made by combining concepts of tensor network theory with Majorana device technology. Considering the topological double semion string-net phase as an example, we exploit the fact that the representation of topological phases by tensor networks can be significantly simpler than their description by lattice Hamiltonians. The building blocks defining the tensor network are tailored to realization via simple units of capacitively coupled Majorana bound states. In the case under consideration, this defines a remarkably simple blueprint of a synthetic double semion string-net, and one may be optimistic that the required device technology will be available soon. Our results indicate that the implementation of tensor network structures via mesoscopic quantum devices may define a powerful novel avenue to the realization of synthetic topological quantum matter in general.

We use the Hamiltonian theory developed by Shankar and Murthy to study a quantum Hall system in a tilted magnetic field. With a finite width of the system in the $z$ direction, the parallel component of the magnetic field introduces anisotropy into the effective two-dimensional interactions. The effects of such anisotropy can be effectively captured by the recently proposed generalized pseudo-potentials. We find that the off-diagonal components of the pseudo-potentials lead to mixing of composite fermions Landau levels, which is a perturbation to the picture of $p$ filled Landau levels in composite-fermion theory. By changing the internal geometry of the composite fermions, such a perturbation can be minimized and one can find the corresponding activation gaps for different tilting angles, and we calculate the associated optimal metric. Our results show that the activation gap is remarkably robust against the in-plane magnetic field in the lowest Landau level.