Laughlin State Research Articles

Precursor of the Laughlin state of hard-core bosons on a two-leg ladder

We study hard core bosons on a two leg ladder lattice under the orbital effect of a uniform magnetic field. At densities which are incommensurate with flux, the ground state is a Meissner state, or a vortex state, depending on the strength of the flux. When the density is commensurate with the flux, analytical arguments predict the existence of a ground state of central charge $c = 1$, which displays signatures compatible with the expected Laughlin state at $\nu=1/2$. This differs from the coupled wire construction of the Laughlin state in that there exists a nonzero backscattering term in the edge Hamiltonian. We construct a phase diagram versus density and flux in order to delimit the region where this precursor to the Laughlin state is the ground state, by using a combination of bosonization and numerics based on the density matrix renormalization group (DMRG) and exact diagonalization. We obtain the phase diagram from local observables and central charge. We use bipartite charge fluctuations to deduce the Luttinger parameter for the edge Luttinger liquid corresponding to the Laughlin state. The properties studied with local observables are confirmed by the long distance behavior of correlation functions. Our findings are consistent with a calculation of the many body ground state transverse conductivity in a thin torus geometry for parameters corresponding to the Laughlin state. The model considered is simple enough such that the precursor to the Laughlin state could be realized in current ultracold atom, Josephson junction array, and quantum circuit experiments.

Supersymmetry in the fractional quantum Hall regime

Supersymmetry (SUSY) is a symmetry transforming bosons to fermions and vice versa. Indications of its existence have been extensively sought after in high-energy experiments. However, signatures of SUSY have yet to be detected. In this manuscript we propose a condensed matter realization of SUSY on the edge of a Read-Rezayi quantum Hall state, given by filling factors of the form $\nu =\frac{k}{k+2}$, where $k$ is an integer. As we show, this strongly interacting state exhibits an $\mathcal{N}=2$ SUSY. This allows us to use a topological invariant - the Witten index - defined specifically for supersymmetric theories, to count the difference between the number of bosonic and fermionic zero-modes in a circular edge. In our system, we argue that the edge hosts $k+1$ protected zero-modes. We further discuss the stability of SUSY with respect to generic perturbations, and find that much of the above results remain unchanged. In particular, these results directly apply to the well-established $\nu=1/3$ Laughlin state, in which case SUSY is a highly robust property of the edge theory. These results unveil a hidden topological structure on the long-studied Read-Rezayi states.

Non-AbelianSU(N−1)-singlet fractional quantum Hall states from coupled wires

The construction of fractional quantum Hall (FQH) states from the two-dimensional array of quantum wires provides a useful way to control strong interactions in microscopic models and has been successfully applied to the Laughlin, Moore-Read, and Read-Rezayi states. We extend this construction to the Abelian and non-Abelian $SU(N-1)$-singlet FQH states at filling fraction $\nu=k(N-1)/[N+k(N-1)m]$ labeled by integers $k$ and $m$, which are potentially realized in multi-component quantum Hall systems or $SU(N)$ spin systems. Utilizing the bosonization approach and conformal field theory (CFT), we show that their bulk quasiparticles and gapless edge excitations are both described by an $(N-1)$-component free-boson CFT and the $SU(N)_k/[U(1)]^{N-1}$ CFT known as the Gepner parafermion. Their generalization to different filling fractions is also proposed. In addition, we argue possible applications of these results to two kinds of lattice systems: bosons interacting via occupation-dependent correlated hoppings and an $SU(N)$ Heisenberg model.

Thermometry for Laughlin States of Ultracold Atoms.

Cooling atomic gases into strongly correlated quantum phases requires estimates of the entropy to perform thermometry and establish viability. We construct an ansatz partition function for models of Laughlin states of atomic gases by combining high temperature series expansions with exact diagonalization. Using the ansatz we find that entropies required to observe Laughlin correlations with bosonic gases are within reach of current cooling capabilities.

Fractional quantum Hall effect in strained graphene: Stability of Laughlin states in disordered pseudomagnetic fields

We address the question of the stability of the fractional quantum Hall effect in the presence of pseudomagnetic disorder generated by mechanical deformations of a graphene sheet. Neglecting the potential disorder and taking into account only strain-induced random pseudomagnetic fields, it is possible to write down a Laughlin-like trial ground-state wave function explicitly. Exploiting the Laughlin plasma analogy, we demonstrate that in the case of fluctuating pseudomagnetic fluxes of a relatively small amplitude, the fractional quantum Hall effect is always stable upon the deformations. By contrast, in the case of bubble-induced pseudomagnetic fields in graphene on a substrate (a small number of large fluxes) the disorder can be strong enough to cause a glass transition in the corresponding classical Coulomb plasma, resulting in the destruction of the fractional quantum Hall regime and in a quantum phase transition to a nonergodic state of the lowest Landau level.

Microwave spectroscopic observation of a Wigner solid within the ν=1/2 fractional quantum Hall effect

The fractional quantum Hall effect (FQHE) states at half integer Landau fillings ($\nu$) have long been of great interest, since they have correlations that differ from those of the fundamental Laughlin states found at odd denominators. At $\nu=1/2$ the FQHE has been observed in wide or double quantum wells, and is ascribed to the two-component Halperin-Laughlin $\Psi_{331}$ state. $\Psi_{331}$ excitations carry charge $\pm e/4$, like the carriers of $\nu=5/2$ states which are of interest in quantum computation. Further, such an excitation (quasiparticle or -hole) of $\Psi_{331}$ has unequal, opposite charge in the top and bottom layers, and hence an up or down dipole moment. Here we report evidence for a Wigner solid (WS) of such dipolar quasiholes from a quantitative study of the microwave spectra of a wide quantum well (WQW) at $\nu$ close to 1/2.

Lattice effects on Laughlin wave functions and parent Hamiltonians

We investigate lattice effects on wave functions that are lattice analogues of bosonic and fermionic Laughlin wave functions with number of particles per flux $\nu=1/q$ in the Landau levels. These wave functions are defined analytically on lattices with $\mu$ particles per lattice site, where $\mu$ may be different than $\nu$. We give numerical evidence that these states have the same topological properties as the corresponding continuum Laughlin states for different values of $q$ and for different fillings $\mu$. These states define, in particular, particle-hole symmetric lattice Fractional Quantum Hall states when the lattice is half-filled. On the square lattice it is observed that for $q\leq 4$ this particle-hole symmetric state displays the topological properties of the continuum Laughlin state at filling fraction $\nu=1/q$, while for larger $q$ there is a transition towards long-range ordered anti-ferromagnets. This effect does not persist if the lattice is deformed from a square to a triangular lattice, or on the Kagome lattice, in which case the topological properties of the state are recovered. We then show that changing the number of particles while keeping the expression of these wave functions identical gives rise to edge states that have the same correlations in the bulk as the reference lattice Laughlin states but a different density at the edge. We derive an exact parent Hamiltonian for which all these edge states are ground states with different number of particles. In addition this Hamiltonian admits the reference lattice Laughlin state as its unique ground state of filling factor $1/q$. Parent Hamiltonians are also derived for the lattice Laughlin states at other fillings of the lattice, when $\mu\leq 1/q$ or $\mu\geq 1-1/q$ and when $q=4$ also at half-filling.

Success and failure of the plasma analogy for Laughlin states on a torus

We investigate the nature of the plasma analogy for the Laughlin wave function on a torus describing the quantum Hall plateau at . We first establish, as expected, that the plasma is screening if there are no short nontrivial paths around the torus. We also find that when one of the handles has a short circumference—i.e. the thin-torus limit—the plasma no longer screens. To quantify this we compute the normalization of the Laughlin state, both numerically and analytically. In the thin torus limit, the analytical form of the normalization simplify and we can reconstruct the normalization and analytically extend it back into the 2D regime. We find that there are geometry dependent corrections to the normalization, and this in turn implies that the plasma in the plasma analogy is not screening when in the thin torus limit. Despite the breaking of the plasma analogy in this limit, the analytical approximation is still a good description of the normalization for all tori, and also allows us to compute hall viscosity at intermediate thickness.

Fractional quantum Hall states of dipolar fermions in a strained optical lattice

We study strongly correlated ground states of dipolar fermions in a honeycomb optical lattice with spatial variations in hopping amplitudes. Similar to a strained graphene, such nonuniform hopping amplitudes produce valley-dependent pseudomagnetic fields for fermions near the two Dirac points, resulting in the formation of Landau levels. The dipole moments polarized perpendicular to the honeycomb plane yield a long-range repulsive interaction. By exact diagonalization in the zeroth-Landau-level basis, we show that this repulsive interaction stabilizes a variety of valley-polarized fractional quantum Hall states such as Laughlin and composite-fermion states. The present system thus offers an intriguing platform for emulating fractional quantum Hall physics in a static optical lattice. We calculate the energy gaps above these incompressible states, and discuss the temperature scales required for their experimental realization.

Creating fractional quantum Hall states with atomic clusters using light-assisted insertion of angular momentum

We describe a protocol to prepare clusters of ultracold bosonic atoms in strongly-interacting states reminiscent of fractional quantum Hall states. Our scheme consists in injecting a controlled amount of angular momentum to an atomic gas using Raman transitions carrying orbital angular momentum. By injecting one unit of angular momentum per atom, one realizes a single-vortex state, which is well described by mean field theory for large enough particle numbers. We also present schemes to realize fractional quantum Hall states, namely the bosonic Laughlin and Moore-Read states. We investigate the requirements for adiabatic nucleation of a such topological states, in particular comparing linear Landau-Zener ramps and arbitrary ramps obtained from optimized control methods. We also show that this protocol requires excellent control over the isotropic character of the trapping potential.

Geometric defects in quantum Hall states

We describe a geometric (or gravitational) analogue of the Laughlin quasiholes in the fractional quantum Hall states. Analogously to the quasiholes these defects can be constructed by an insertion of an appropriate vertex operator into the conformal block representation of a trial wavefunction, however, unlike the quasiholes these defects are extrinsic and do not correspond to true excitations of the quantum fluid. We construct a wavefunction in the presence of such defects and explain how to assign an electric charge and a spin to each defect, and calculate the adiabatic, non-abelian statistics of the defects. The defects turn out to be equivalent to the genons in that their adiabatic exchange statistics can be described in terms of representations of the mapping class group of an appropriate higher genus Riemann surface. We present a general construction that, in principle, allows to calculate the statistics of $\mathbb Z_n$ genons for any "parent" topological phase. We illustrate the construction on the example of the Laughlin state and perform an explicit calculation of the braiding matrices. In addition to non-abelian statistics geometric defects possess a universal abelian overall phase, determined by the gravitational anomaly.

Interaction-driven fractional quantum Hall state of hard-core bosons on kagome lattice at one-third filling

There has been a growing interest in realizing topologically nontrivial states of matter in band insulators, where a quantum Hall effect can appear as an intrinsic property of the band structure. While the on-going progress is under way with a number of directions, the possibility of realizing novel interaction-generated topological phases, without the requirement of a nontrivial invariant encoded in single-particle wavefunction or band structure, can significantly extend the class of topological materials and is thus of great importance. Here, we show an interaction-driven topological phase emerging in an extended Bose-Hubbard model on kagome lattice, where the non-interacting band structure is topological trivial with zero Berry curvature in the Brillouin zone. By means of an unbiased state-of-the-art density-matrix renormalization group technique, we identify that the groundstate in a broad parameter region is equivalent to a bosonic fractional quantum Hall Laughlin state, based on the characterization of universal properties including groundstate degeneracy, edge excitations and anyonic quasiparticle statistics. Our work paves a way of finding interaction induced topological phase at the phase boundary of conventionally ordered solid phases.

Laughlin states and their quasiparticle excitations on the torus

We provide a full derivation of Laughlin's Jastrow-type wave functions for quantized Hall states subject to periodic boundary conditions using an operator formalism. The construction includes the quasi-hole and the technically more challenging quasi-electron excitation, which was left as an open problem in the classic paper by Haldane and Rezayi [Phys. Rev. B 31, 2529 (1985)].

Energy projection and modified Laughlin states

We develop a method to efficiently calculate trial wave functions for quantum Hall systems which involve projection onto the lowest Landau level. The method essentially replaces the lowest Landau level projection by projection onto the $M$ lowest eigenstates of a suitably chosen Hamiltonian acting within the lowest Landau level. The resulting ``energy projection'' is a controlled approximation to the exact lowest Landau level projection which improves with increasing $M$. It allows us to study the projected trial wave functions for system sizes close to the maximal sizes that can be reached by exact diagonalization and can be straightforwardly applied in any geometry. As a first application and test case, we study a class of trial wave functions first proposed by Girvin and Jach [Girvin and Jach, Phys. Rev. B 29, 5617 (1984)], which are modifications of the Laughlin states involving a single real parameter. While these modified Laughlin states probably represent the same universality class exemplified by the Laughlin wave functions, we show by extensive numerical work for systems on the sphere and torus that they provide a significant improvement of the variational energy, overlap with the exact wave function and properties of the entanglement spectrum.

Formulation of the relativistic quantum Hall effect and parity anomaly

We present a relativistic formulation of the quantum Hall effect on Haldane sphere. An explicit form of the pseudopotential is derived for the relativistic quantum Hall effect with/without mass term. We clarify particular features of the relativistic quantum Hall states with the use of the exact diagonalization study of the pseudopotential Hamiltonian. Physical effects of the mass term to the relativistic quantum Hall states are investigated in detail. The mass term acts as an interpolating parameter between the relativistic and non-relativistic quantum Hall effects. It is pointed out that the mass term unevenly affects the many-body physics of the positive and negative Landau levels as a manifestation of the "parity anomaly". In particular, we explicitly demonstrate the instability of the Laughlin state of the positive first relativistic Landau level with the reduction of the charge gap.

SU(2)1chiral edge modes of a critical spin liquid

Protected chiral edge modes are a well-known signature of topologically ordered phases like the Fractional Quantum Hall States. Recently, using the framework of projected entangled pair states (PEPS) on the square lattice, we constructed a family of chiral Resonating Valence Bond states with $\mathbb{Z}_2$ gauge symmetry. Here we revisit and analyze in full details the properties of the edge modes as given by their Entanglement Spectra on a cylinder. Surprisingly, we show that the latter can be well described by a chiral SU(2)$_1$ Conformal Field Theory (CFT), as for the $\nu=1/2$ (bosonic) gapped Laughlin state, although our numerical data suggest a critical bulk compatible with an emergent $U(1)$ gauge symmetry. We propose that our family of PEPS may physically describe a boundary between a chiral topological phase and a trivial phase.

Emergence of Fractional Statistics for Tracer Particles in a Laughlin Liquid.

We consider a thought experiment where two distinct species of 2D particles in a perpendicular magnetic field interact via repulsive potentials. If the magnetic field and the interactions are strong enough, one type of particles forms a Laughlin state and the other type couples to Laughlin quasiholes. We show that, in this situation, the motion of the second type of particles is described by an effective Hamiltonian, corresponding to the magnetic gauge picture for noninteracting anyons. The argument is in accord with, but distinct from, the Berry phase calculation of Arovas, Schrieffer, and Wilczek. It suggests possibilities to observe the influence of effective anyon statistics in fractional quantum Hall systems.

Classification and properties of symmetry-enriched topological phases: Chern-Simons approach with applications toZ2spin liquids

We study 2+1 dimensional phases with topological order, such as fractional quantum Hall states and gapped spin liquids, in the presence of global symmetries. Phases that share the same topological order can then differ depending on the action of symmetry, leading to symmetry enriched topological (SET) phases. Here we present a K-matrix Chern-Simons approach to identify all distinct phases with Abelian topological order, in the presence of unitary or anti-unitary global symmetries . A key step is the identification of an edge sewing condition that is used to check if two putative phases are indeed distinct. We illustrate this method for the case of $Z_2$ topological order ($Z_2$ spin liquids), in the presence of an internal Z$_2$ global symmetry. We find 6 distinct phases. The well known quantum number fractionalization patterns account for half of these states. Phases also differ due to the addition of a symmetry protected topological (SPT) phase. Also, we allow for the unconventional possibility that anyons are exchanged by the symmetry. This leads to 2 additional phases with symmetry protected Majorana edge modes. Other routes to realizing protected edge states in SET phases are identified. Symmetry enriched Laughlin states and double semion theories are also discussed. Two surprising lessons that emerge are: (i) gauging the global symmetry of distinct SET phases always lead to different topological orders with the same total quantum dimension, (ii) gauge theories with distinct Dijkgraaf-Witten topological terms may have the same topological order.

Multipole expansion in the quantum hall effect

The effective action for low-energy excitations of Laughlin's states is obtained by systematic expansion in inverse powers of the magnetic field. It is based on the W-infinity symmetry of quantum incompressible fluids and the associated higher-spin fields. Besides reproducing the Wen and Wen-Zee actions and the Hall viscosity, this approach further indicates that the low-energy excitations are extended objects with dipolar and multipolar moments.

Fractional quantum Hall effect in HgTe quantum wells

We study the possibility of fractional quantum Hall effects in HgTe quantum wells using exact diagonalization. Our results show that Laughlin states, the Moore-Read state, and the Read-Rezayi $Z_3$ state can all be supported. However, near the level crossing point (of the single-particle spectrum), the gap can be destroyed by Landau level mixing, and the Moore-Read state and the Read-Rezayi state dominate over their respective competing states only for wide wells. For smaller well widths the Moore-Read state crosses over to the composite fermion Fermi sea, while the Read-Rezayi state loses its dominance over the hierarchy state.