Incompressible Quantum Liquid Research Articles

An Algebraic Approach to Electron Interactions in Quantum Hall Systems

Let m denote the number of quasielectrons (QEs) in a quantum Hall system containing N particles altogether. We show in several general cases that for systems containing m QEs in a single angular momentum shell above N − m Fermions in an incompressible quantum liquid (IQL) state having filling factor ν =1/3 that there always exists a configuration whose symmetric correlation function G is nonzero. This extends recent comparable results concerning the IQL state. As a consequence, one can obtain (explicitly) a configuration having a nonzero G for all N ≤ 8 particle systems containing any number of QEs. To establish our result, we construct a family of multi-graphs on N vertices satisfying certain restraints on the degrees of the vertices and possessing the property that whenever one computes the linear symmetrization of the graph monomial of any member of the family, the result is always nonzero. The nonzero linear symmetrization that is obtained in each case is in fact an example of what is called a relative semi-invariant of a (generic) binary form of degree N. Thus, in addition to providing new correlation functions for systems of interacting Fermions containing QEs, our construction is of potential interest from both the invariant and graph theoretic standpoints.

An Algebraic Approach to FQHE Variational Wave Functions

Consider a system of $N$ electrons projected onto the lowest Landau level (LLL) with filling factor of the form $n/(2pn\pm1)<1/2$ and $N$ a multiple of $n$. We show that there always exists a two-dimensional symmetric correlation factor (arising as a nonzero symmetrization) for such systems and hence one can always write a variational wave function. This extends an earlier observation of Laughlin for an incompressible quantum liquid (IQL) state with filling factor equal to the reciprocal of an odd integer $ \geq 3$. To do so, we construct a family of $d$-regular multi-graphs on $N$ vertices for any $N$ whose graph-monomials have nonzero linear symmetrization and obtain, as special cases, the aforementioned nonzero correlations for the IQL state. The nonzero linear symmetrization that is obtained is in fact an example of what is called a binary invariant of type $(N,d)$. Thus, in addition to supplying new variational wave functions for systems of interacting Fermions, our construction is of potential interest from both the graph and invariant theoretic viewpoints.

On the absence of higher generations of incompressible daughter states of composite Fermion quasiparticles

Jain [1] introduced a simple mean-field (MF) composite Fermion (CF) picture by attaching to each electron in a quantum Hall system a flux tube producing a Chern- Simons magnetic field b(r) = 2pϕ0 Σi δ(r - ri)ẑ. Here ϕ0 = hc/e is the quantum of flux, and the sum is over all electron coordinates ri. He then averaged the total flux and the total charge (electronic plus positive background) over the entire sample. This MF picture gave a system of noninteracting CFs in an effective magnetic field B*0 = νB0. It predicted incompressible quantum liquid (IQL) states at filling factors ν = n(1 + 2pn)-1 for integral values of n. Chen and Quinn [2] demonstrated that Jain’s MF CF picture predicted the total angular momentum values of the lowest energy band of states for any value of the applied magnetic field B0. Justification of when the MFCF picture was valid was given by Wojs and Quinn [3], who extended the CF hierarchy scheme of Sitko et al. [4]. The CF hierarchy gave the Jain states for integrally filled CF Landau levels (CF LLs), and the Haldane hierarchy of all odd denominator fractions when quasiparticles in the highest (partially filled) CF angular momentum shell had interactions sufficiently similar to the Coulomb interactions of electrons in the lowest Landau level. Sitko et al. showed that the predictions of the CF hierarchy scheme were not always correct. By using a simple pair angular momentum identity and the concept of fractional grandparentage, Wojs and Quinn showed that higher generations of CFs could result from the interactions of the original CF quasiparticles only if their interaction energy VQP(L2) as a function of their pair angular momentum L2 increased with increasing L2 faster than L2(L2 + 1). For Laughlin quasielectrons of the ν = 1/3 IQL state this condition was not satisfied. Therefore, no second generation of CFs could occur. The observed IQL at electron filling factor ν = 4/11 can not be attributed to a daughter IQL state at νQE = 1/3 in a totally spin polarized system. Rather, it is an IQL state of electron pairs with pair angular momentum ιp = 2ι – 1. A description of the IQL state of electron pairs and its low energy excitations will be presented, and the possibility of a spin flip quantum phase transition as a function of well width and applied magnetic field will be discussed.

Correlation diagrams: an intuitive approach to correlations in quantum Hall systems

A trial wave function Ψ(1, 2,..., N) of an N electron system can always be written as the product of an antisymmetric Fermion factor F{zij} = Πi<jzij , and a symmetric correlation factor G{zij}. F results from the Pauli principle, and G is caused by Coulomb interactions. One can represent G diagrammatically [1] by distributing N points on the circumference of a circle, and drawing appropriate lines representing correlation factors (cfs) zij between pairs. Here, of course, zij = zi – zj, where zi is the complex coordinate of the ith electron. Laughlin correlations for the ν = 1/3 filled incompressible quantum liquid (IQL) state contain two cfs connecting each pair (i,j). For the Moore-Read state of the half-filled excited Landau level (LL), with ν = 2 + 1/2, the even value of N for the half-filled LL is partitioned into two subsets A and B, each containing N/2 electrons [2]. For any one partition (A,B), the contribution to G is given by GAB = Πi<j∈Azij2 Πk<ι∈B zkι2. The full G is equal to the symmetric sum of contributions GAB over all possible partitions of N into two subsets of equal size. For Jain states at filling factor ν = p/q < 1/2, the value of the single particle angular momentum ι satisfies the equation 2ι = ν-1N - Cν, with Cν = q + 1 - p. The values of (2ι, N) define the function space of G{zij}, which must satisfy a number of conditions. For example, the highest power of any zi cannot exceed 2ι + 1 - N. In addition, the value of the total angular momentum L of the lowest correlated state must satisfy the equation L = (N/2)(2ι+1 - N) - KG, where KG is the degree of the homogeneous polynomial generated by G. Knowing the values of L for IQL states (and for states containing a few quasielectrons or a few quasiholes) from Jain’s mean field CF picture allows one to determine KG. The dependence of the pair pseudopotential V(L2) on pair angular momentum L2 suggests a small number of correlation diagrams for a given value of the total angular momentum L. Correlation diagrams and correlation functions for the Jain state at ν = 2/5 and for the Moore-Read states will be presented as examples. The generalizations of the method of selecting G from small to larger systems will be discussed.

Fractional Chern Insulators in Harper-Hofstadter Bands with Higher Chern Number.

The Harper-Hofstadter model provides a fractal spectrum containing topological bands of any integer Chern number C. We study the many-body physics that is realized by interacting particles occupying Harper-Hofstadter bands with |C|>1. We formulate the predictions of Chern-Simons or composite fermion theory in terms of the filling factor ν, defined as the ratio of particle density to the number of single-particle states per unit area. We show that this theory predicts a series of fractional quantum Hall states with filling factors ν=r/(r|C|+1) for bosons, or ν=r/(2r|C|+1) for fermions. This series includes a bosonic integer quantum Hall state in |C|=2 bands. We construct specific cases where a single band of the Harper-Hofstadter model is occupied. For these cases, we provide numerical evidence that several states in this series are realized as incompressible quantum liquids for bosons with contact interactions.

Two-dimensional heavy fermions on the strongly correlated boundaries of Kondo topological insulators

Samarium hexaboride (SmB$_6$), a representative Kondo insulator, has been characterized recently as a likely topological insulator. It is also a material with strong electron correlations, evident by the temperature dependence of its bandgap and the existence of a nearly flat collective mode whose energy lies within the bandgap. Similar strong correlations can affect or even destabilize the two-dimensional metallic state of topological origin at the crystal boundary. Here we construct the minimal lattice model of the correlated boundary of topological Kondo insulators, and make phenomenological predictions for its possible ground states. Depending on the microscopic properties of the interface between the topological Kondo material and a conventional insulator, the boundary metal can exhibit a varied degree of hybridization between the $d$ and $f$ orbitals of the rare earth element, yielding a rich two-dimensional heavy fermion phenomenology. A pronounced participation of the $f$ orbitals is expected to create a heavy fermion Dirac metal, possibly unstable to a spin density wave, electron localization or even superconductivity. The opposite limit of "localized magnetic moments" helped by the partial Kondo screening on the crystal boundary can bring about a non-Fermi liquid of $d$ electrons that exhibits two-dimensional quantum electrodynamics, or other unconventional states. In addition, ultra-thin films made from topological Kondo insulators could open the possibility of creating exotic incompressible quantum liquids with non-Abelian fractional excitations, whose dynamics shaped by the strong Rashba spin-orbit coupling resembles that of fractional quantum Hall systems.

Constructing trial wave functions for a many electron system confined to a quantum well in a strong magnetic field

Incompressible quantum liquid (IQL) states occur when the single particle angular momentum l and the number N of electrons in the partially filled Landau level satisfy the equation 2l = ν−1N − Cν. ...

The study of spatial–temporal correlations of giant optical fluctuations of 2D electrons

Giant optical fluctuations of 2D electrons under the quantum Hall effect conditions were studied by the method of correlation functions. The phenomenological model which describes the correlation functions of 2D photoluminescence signals registered by means of different fibers in this regime is suggested. The explanation is given for various phase conditions of a 2D electron system at different values of a magnetic field and the formation of incompressible quantum liquid. The formation of incompressible electronic liquid within a big photoexcitation spot can be accompanied by generation of “rings” and “spots” of extended states.

Effective theory of fractional topological insulators in two spatial dimensions

Electrons subjected to a strong spin-orbit coupling in two spatial dimensions could form fractional incompressible quantum liquids without violating the time-reversal symmetry. Here we construct a Lagrangian description of such fractional topological insulators by combining the available experimental information on potential host materials and the fundamental principles of quantum field theory. This Lagrangian is a Landau-Ginzburg theory of spinor fields, enhanced by a topological term that implements a state-dependent fractional statistics of excitations whenever both particles and vortices are incompressible. The spin-orbit coupling is captured by an external static SU(2) gauge field. The presence of spin conservation or emergent U(1) symmetries would reduce the topological term to the Chern-Simons effective theory tailored to the ensuing quantum Hall state. However, the Rashba spin-orbit coupling in solid-state materials does not conserve spin. We predict that it can nevertheless produce incompressible quantum liquids with topological order but without a quantized Hall conductivity. We discuss two examples of such liquids whose description requires a generalization of the Chern-Simons theory. One is an Abelian Laughlin-like state, while the other has a new kind of non-Abelian many-body entanglement. Their quasiparticles exhibit fractional spin-dependent exchange statistics, and have fractional quantum numbers derived from the electron's charge and spin according to their transformations under time-reversal. In addition to conventional phases of matter, the proposed topological Lagrangian can capture a broad class of hierarchical Abelian and non-Abelian topological states, involving particles with arbitrary spin or general emergent SU(N) charges.

Interaction proximity effect at the interface between a superconductor and a topological insulator quantum well

A material whose electrons are correlated can affect electron dynamics across the interface with another material. Such a ``proximity effect'' can have several manifestations, from order parameter leakage to generated effective interactions. The resulting combination of induced electron correlations and their intrinsic dynamics at the surface of the affected material can give rise to qualitatively new quantum states. For example, the leaking of a superconducting order parameter into certain Rashba spin-orbit-coupled materials has been recently identified as a path to creating ``topological superconductors'' that can host Majorana particles of use in quantum computing. Here we analyze the other aspects of the superconducting proximity effect. The proximity-induced interactions are a promising path to incompressible quantum liquids with non-Abelian fractional quasiparticles in topological insulator quantum wells, which could also find applications in topological quantum computing. We discuss the operational and design principles of a heterostructure device that could realize such states. We apply field-theoretical methods to characterize the properties of induced interactions via the electron-phonon coupling and Cooper pair tunneling across the interface. We argue that bound-state Cooper pairs can be stabilized by the interaction proximity effect inside a topological insulator quantum well at experimentally observable energy scales. The condensation of spinful triplet pairs, enabled by the Rashba spin-orbit coupling and tunable by gate voltage, would lead to novel superconducting states and fractional topological insulators.

Charge and spin fractionalization in strongly correlated topological insulators

We construct an effective topological Landau–Ginzburg theory that describes general SU(2) incompressible quantum liquids of strongly correlated particles in two spatial dimensions. This theory characterizes the fractionalization of quasiparticle quantum numbers and statistics in relation to the topological ground-state symmetries, and generalizes the Chern–Simons, BF (‘background field’) and hierarchical effective gauge theories to an arbitrary representation of the SU(2) symmetry group. We mainly focus on fractional topological insulators with time-reversal symmetry, which are treated as SU(2) generalizations of the quantum Hall effect.

Real-space imaging of fractional quantum Hall liquids

Electrons in semiconductors usually behave like a gas--as independent particles. However, when confined to two dimensions under a perpendicular magnetic field at low temperatures, they condense into an incompressible quantum liquid. This phenomenon, known as the fractional quantum Hall (FQH) effect, is a quantum-mechanical manifestation of the macroscopic behaviour of correlated electrons that arises when the Landau-level filling factor is a rational fraction. However, the diverse microscopic interactions responsible for its emergence have been hidden by its universality and macroscopic nature. Here, we report real-space imaging of FQH liquids, achieved with polarization-sensitive scanning optical microscopy using trions (charged excitons) as a local probe for electron spin polarization. When the FQH ground state is spin-polarized, the triplet/singlet intensity map exhibits a spatial pattern that mirrors the intrinsic disorder potential, which is interpreted as a mapping of compressible and incompressible electron liquids. In contrast, when FQH ground states with different spin polarization coexist, domain structures with spontaneous quasi-long-range order emerge, which can be reproduced remarkably well from the disorder patterns using a two-dimensional random-field Ising model. Our results constitute the first reported real-space observation of quantum liquids in a class of broken symmetry state known as the quantum Hall ferromagnet.

Fractional quantum Hall states in two-dimensional electron systems with anisotropic interactions

We study the anisotropic effect of the Coulomb interaction on a 1/3-filling fractional quantum Hall system by using an exact diagonalization method on small systems in torus geometry. For weak anisotropy the system remains to be an incompressible quantum liquid, although anisotropy manifests itself in density correlation functions and excitation spectra. When the strength of anisotropy increases, we find the system develops a Hall-smectic-like phase with a one-dimensional charge density wave order and is unstable towards the one-dimensional crystal in the strong anisotropy limit. In all three phases of the Laughlin liquid, Hall-smectic-like, and crystal phases the ground state of the anisotropic Coulomb system can be well described by a family of model wave functions generated by an anisotropic projection Hamiltonian. We discuss the relevance of the results to the geometrical description of fractional quantum Hall states proposed by Haldane [ Phys. Rev. Lett. 107 116801 (2011)].

Deconfined fractionally charged excitation in any dimensions

An exact incompressible quantum liquid is constructed at the filling factor 1/m2 in the square lattice. It supports deconfined fractionally charged excitation. At the filling factor 1/m2, the excitation has fractional charge e/m2, where e is the electric charge. This model can be easily generalized to the n-dimensional square lattice (integer lattice), where the charge of excitations becomes e/mn.

Electronic properties of graphene in a strong magnetic field

The basic aspects of electrons in graphene (two-dimensional graphite) exposed to a strong perpendicular magnetic field are reviewed. One of its most salient features is the relativistic quantum Hall effect, the observation of which has been the experimental breakthrough in identifying pseudorelativistic massless charge carriers as the low-energy excitations in graphene. The effect may be understood in terms of Landau quantization for massless Dirac fermions, which is also the theoretical basis for the understanding of more involved phenomena due to electronic interactions. The role of electron-electron interactions both in the weak-coupling limit, where the electron-hole excitations are determined by collective modes, and in the strong-coupling regime of partially filled relativistic Landau levels are presented. In the latter limit, exotic ferromagnetic phases and incompressible quantum liquids are expected to be at the origin of recently observed (fractional) quantum Hall states. Furthermore, the electron-phonon coupling in a strong magnetic field is discussed. Although the present review has a dominant theoretical character, a close connection with available experimental observation is intended.

Composite Fermion Description of the Excitations of the Paired Pfaffian Fractional Quantum Hall State

We review the recently developed bi-partite composite fermion model, in the context of so-called Pfaffian incompressible quantum liquid with fractional and non-Abelian quasiparticle statistics, a promising model for describing the correlated many-electron ground state responsible for fractional quantum Hall effect at the Landau level filling factor ? = 5/2. We use the concept of composite fermion partitions to demonstrate the emergence of an essential ingredient of the non-Abelian braid statistics – the topological degeneracy of spatially indistinguishable configurations of multiple widely separated (non-interacting) quasiparticles.

Transient three-pulse four-wave mixing spectra of magnetoexcitons coupled with an incompressible quantum liquid

Transient three-pulse four-wave mixing spectra of magnetoexcitons coupled with an incompressible quantum liquid

Featureless Mott insulators

A family of the pair hopping models exhibiting the incompressible quantum liquid at fractional filling $1/m^D$ is constructed in $D$ dimensional lattice. Except in one dimension, the lattice is the generalized edge-shared triangular lattice, for example the triangular lattice in two dimensions and tetrahedral lattice in three dimensions. They obey the new symmetry, conservation of the center-of-mass position proposed by Seidel et al..\cite{Seidel2005} The uniqueness of the ground state is proved rigorously in the open boundary condition. The finiteness of the excitation energy is calculated by the single mode approximation.

Fractional Quantum Hall States with Non-Abelian Statistics

Using exact numerical diagonalization we have studied correlated many-electron ground states in a partially filled second Landau level. We consider filling fractions ? = 1/2 and 2/5, for which incompressible quantum liquids with non-Abelian anion statistics have been proposed. Our calculations include finite layer width, Landau level mixing and arbitrary deformation of the interaction pseudopotential. Computed energies, gaps, and correlation functions support the non-Abelian ground states at both ? = 1/2 (“Pfaffian”) and ? = 2/5 (“parafermion” state).

Incompressible liquid, stripes, and bubbles in rapidly rotating Bose atoms atν=1

We numerically study the system of rapidly rotating Bose atoms at the filling factor (ratio of particle number to vortex number) $\nu=1$ with the dipolar interaction. A moderate dipolar interaction stabilizes the incompressible quantum liquid at $\nu=1$. Further addition induces a collapse of it. The state after the collapse is a compressible state which has phases with stripes and bubbles. There are two types of bubbles with a different array. We also investigate models constructed from truncated interactions and the models with the three-body contact interaction. They also have phases with stripes and bubbles.