We analyze the entanglement spectrum of Laughlin states on the torus and show that it is arranged in towers, each of which is generated by modes of two spatially separated chiral edges. This structure is present for all torus circumferences, which allows for a microscopic identification of the prominent features of the spectrum by perturbing around the thin-torus limit.
We analyze the entanglement spectrum of Laughlin states on the torus and show that it is arranged in towers, each of which is generated by modes of two spatially separated chiral edges. This structure is present for all torus circumferences, which allows for a microscopic identification of the prominent features of the spectrum by perturbing around the thin-torus limit.
We present a detailed analysis of bipartite entanglement in the non-Abelian Moore-Read fractional quantum Hall state of bosons and fermions on the torus. In particular, we show that the entanglement spectra can be decomposed into intricate combinations of different sectors of the conformal field theory describing the edge physics, and that the edge level counting and tower structure can be microscopically understood by considering the vicinity of the thin-torus limit. We also find that the boundary entropy density of the Moore-Read state is markedly higher than in the Laughlin states investigated so far. Despite the torus geometry being somewhat more involved than in the sphere geometry, our analysis and insights may prove useful when adopting entanglement probes to other systems that are more easily studied with periodic boundary conditions, such as fractional Chern insulators and lattice problems in general.
We analytically and numerically analyze the one-dimensional "thin-torus" limit of Fractional Topological Insulators in a series of simple models exhibiting exactly flat bands with local hopping. These models are the one-dimensional limit of two dimensional Chern Insulators, and the Hubbard-type interactions projected into their lowest band take particularly simple forms. By exactly solving the many-body interacting spectrum of these models, we show that, just like in the Fractional Quantum Hall effect, the zero modes of the thin-torus limit are CDW states of occupation numbers satisfying generalized Pauli principles. As opposed to the FQH where the thin-torus CDW appear in orbital space, in the thin-torus FCI states, the CDW states are in real-space. We show the counting of the quasihole excitations in the energy spectrum cannot distinguish between a CDW state and a FQH state. However, by exactly computing the entanglement spectrum for the thin-torus states, we show that it can qualitatively and quantitatively distinguish between a CDW and a fractional topological state such as the FCI. We then discover a previously unknown separation of energy scales of the full FQH energy spectrum in the thin torus limit and find that Chern insulator models exhibiting strong isotropic FCI states have a similar structure in their thin-torus limit spectrum. We close by numerically computing the evolution of energy and entanglement spectra from the thin-torus to the isotropic limit. Our results can also be interpreted as an analysis of one-body, 1-dimensional topological insulators stabilized by inversion symmetry in the presence of interactions.
The fractional quantum Hall effect (FQHE), now entering it's fourth decade, continues to draw attention from the condensed matter community. New experiments in recent years are raising hopes that it will be possible to observe quasi-particles with non-abelian anyonic statistics. These particles could form the building blocks of a quantum computer.The quantum Hall states have topologically protected energy gaps to the low-lying set of excitations. This topological order is not a locally measurable quantity but rather a non-local object, and it is one of the keys to it's stability. From an early stage understanding of the FQHE has been facilitate by constructing trial wave functions. The topological classification of these wave functions have given further insight to the nature of the FQHE.An early, and successful, wave function construction for filling fractions ν=p/(2p+1) was that of composite fermions on planar and spherical geometries. Recently, new developments using conformal field theory have made it possible to also construct the full Haldane-Halperin hierarchy wave functions on planar and spherical geometries. In this thesis we extend this construction to a toroidal geometry, i.e. a flat surface with periodic boundary conditions.One of the defining features of topological states of matter in two dimensions is that the ground state is not unique on surfaces with non trivial topology, such as a torus. The archetypical example is the fractional quantum Hall effect, where a state at filling fraction ν=p/q, has at least a q-fold degeneracy on a torus. This has been shown explicitly for a few cases, such as the Laughlin states and the the Moore-Read states, by explicit construction of candidate electron wave functions with good overlap with numerically found states. In this thesis, we construct explicit torus wave functions for a large class of experimentally important quantum liquids, namely the chiral hierarchy states in the lowest Landau level. These states, which includes the prominently observed positive Jain sequence at filling fractions ν=p/(2p+1), are characterized by having boundary modes with only one chirality.Our construction relies heavily on previous work that expressed the hierarchy wave functions on a plane or a sphere in terms of correlation functions in a conformal field theory. This construction can be taken over to the torus when care is taken to ensure correct behaviour under the modular transformations that leave the geometry of the torus unchanged. Our construction solves the long standing problem of engineering torus wave functions for multi-component many-body states. Since the resulting expressions are rather complicated, we have carefully compared the simplest example, that of ν=2/5, with numerically found wave functions. We have found an extremely good overlap for arbitrary values of the modular parameter τ, that describes the geometry of the torus.Having explicit torus wave functions allows us to use the methods developed by Read and Read \& Rezayi to numerically compute the quantum Hall viscosity. Hall viscosity is conjectured to be a topologically protected macroscopic transport coefficient characterizing the quantum Hall state. It is related to the shift of the same QH-fluid when it is put on a sphere. The good agreement with the theoretical prediction for the 2/5 state strongly suggests that our wave functions encodes all relevant topologically information.We also consider the Hall viscosity in the limit of a very thin torus. There we find that the viscosity changes as we approach the thin torus limit. Because of this we study the Laughlin state in that limit and see how the change in viscosity arises from a change in the Hamiltonian hopping elements. Finally we conclude that there are both qualitative and quantitative difference between the thin and the square torus. Thus, one has to be careful when interpreting results in the thin torus limit.
The edge of the electronic fractional quantum Hall (FQH) system obeys the law of the chiral Luttinger liquid theory due to its intrinsic topological properties and the relation of bulk-edge correspondence. However, in a realistic experimental system, such as the usual Hall bar setup, the softening of the background confinement potential can induce the reconstruction of the edge spectrum, which breaks the chirality and universality of the FQH edge. The entanglement spectrum (ES) of the FQH ground state has the same counting structure as that of the energy spectrum indicating the topological characters of the quantum state. In this work, we report that the ES can also have an edge reconstruction while sweeping the area of the subsystem in real space cut. Moreover, we found that the critical area of the subsystem matches accurately with the intrinsic building block of the FQH liquids, namely the correlation hole of the FQH liquids. The above results seem to be universal after our study of a series of typical FQH states, such as two Laughlin states at $\ensuremath{\nu}=1/3$ and 1/5, and the Moore-Read state for $\ensuremath{\nu}=5/2$.
"Topological ordered" phases such as gapped quantum spin-liquids and fractional quantum Hall states possess ground state degeneracy on a torus. We show that the topological nature of this degeneracy has interesting consequences for the entanglement structure of the degenerate ground states. This leads to a simple method for detecting topological order, which is ready-made for numerical schemes such as density matrix renormalization group. The method is valid for phases that do or do not possess edge states alike. We demonstrate it by calculating the entanglement spectrum and the topological entanglement entropy for a chiral spin-liquid state on a 18 site system.
The fractional quantum Hall effect (FQHE), now entering it's fourth decade, continues to draw attention from the condensed matter community. New experiments in recent years are raising hopes that it will be possible to observe quasi-particles with non-abelian anyonic statistics. These particles could form the building blocks of a quantum computer. The quantum Hall states have topologically protected energy gaps to the low-lying set of excitations. This topological order is not a locally measurable quantity but rather a non-local object, and it is one of the keys to it's stability. From an early stage understanding of the FQHE has been facilitate by constructing trial wave functions. The topological classification of these wave functions have given further insight to the nature of the FQHE. An early, and successful, wave function construction for filling fractions ν=p/(2p+1) was that of composite fermions on planar and spherical geometries. Recently, new developments using conformal field theory have made it possible to also construct the full Haldane-Halperin hierarchy wave functions on planar and spherical geometries. In this thesis we extend this construction to a toroidal geometry, i.e. a flat surface with periodic boundary conditions. One of the defining features of topological states of matter in two dimensions is that the ground state is not unique on surfaces with non trivial topology, such as a torus. The archetypical example is the fractional quantum Hall effect, where a state at filling fraction ν=p/q, has at least a q-fold degeneracy on a torus. This has been shown explicitly for a few cases, such as the Laughlin states and the the Moore-Read states, by explicit construction of candidate electron wave functions with good overlap with numerically found states. In this thesis, we construct explicit torus wave functions for a large class of experimentally important quantum liquids, namely the chiral hierarchy states in the lowest Landau level. These states, which includes the prominently observed positive Jain sequence at filling fractions ν=p/(2p+1), are characterized by having boundary modes with only one chirality. Our construction relies heavily on previous work that expressed the hierarchy wave functions on a plane or a sphere in terms of correlation functions in a conformal field theory. This construction can be taken over to the torus when care is taken to ensure correct behaviour under the modular transformations that leave the geometry of the torus unchanged. Our construction solves the long standing problem of engineering torus wave functions for multi-component many-body states. Since the resulting expressions are rather complicated, we have carefully compared the simplest example, that of ν=2/5, with numerically found wave functions. We have found an extremely good overlap for arbitrary values of the modular parameter τ, that describes the geometry of the torus. Having explicit torus wave functions allows us to use the methods developed by Read and Read \\& Rezayi to numerically compute the quantum Hall viscosity. Hall viscosity is conjectured to be a topologically protected macroscopic transport coefficient characterizing the quantum Hall state. It is related to the shift of the same QH-fluid when it is put on a sphere. The good agreement with the theoretical prediction for the 2/5 state strongly suggests that our wave functions encodes all relevant topologically information. We also consider the Hall viscosity in the limit of a very thin torus. There we find that the viscosity changes as we approach the thin torus limit. Because of this we study the Laughlin state in that limit and see how the change in viscosity arises from a change in the Hamiltonian hopping elements. Finally we conclude that there are both qualitative and quantitative difference between the thin and the square torus. Thus, one has to be careful when interpreting results in the thin torus limit.