arXiv subjects

Scott D. Geraedts

Publications and source records attributed to Scott D. Geraedts.

At least 19 recordsLinked to original sources

arXiv2017

Investigating anisotropic quantum Hall states with bi-metric geometry

We construct a low energy effective theory of anisotropic fractional quantum Hall (FQH) states. We develop a formalism similar to that used in the bi-metric approach to massive gravity, and apply it to describe abelian anisotropic FQH states in the presence of external electromagnetic and geometric backgrounds. We derive a relationship between the shift, the Hall viscosity, and a new quantized coupling to anisotropy, which we term "anisospin". We verify this relationship by numerically computing the Hall viscosity for a variety of anisotropic quantum Hall states using the density matrix renormalization group (DMRG). Finally, we apply these techniques to the problem of nematic order and clarify certain disagreements that exist in the literature about the meaning of the coefficient of the Berry phase term in the nematic effective action.

cond-mat.str-el↗

arXiv2017

Berry phase and model wavefunction in the half-filled Landau Level

We construct model wavefunctions for the half-filled Landau level parameterized by "composite fermion occupation-number configurations" in a two-dimensional momentum space which correspond to a Fermi sea with particle-hole excitations. When these correspond to a weakly-excited Fermi sea, they have large overlap with wavefunctions obtained by exact diagonalization of lowest-Landau-level electrons interacting with a Coulomb interaction, allowing exact states to be identified with quasiparticle configurations. We then formulate a many-body version of the single-particle Berry phase for adiabatic transport of a single quasiparticle around a path in momentum space, and evaluate it using a sequence of exact eigenstates in which a single quasiparticle moves incrementally. In this formulation the standard free-particle construction in terms of the overlap between "periodic parts of successive Bloch wavefunctions" is reinterpreted as the matrix element of a "momentum boost" operator between the full Bloch states, which becomes the matrix elements of a Girvin-MacDonald-Platzman density operator in the many-body context. This allows computation of the Berry phase for transport of a single composite fermion around the Fermi surface. In addition to a phase contributed by the density operator, we find a phase of exactly $π$ for this process.

cond-mat.str-el↗

arXiv2017

Integer quantum Hall transition in a $\textit{fraction}$ of a Landau level

We investigate the quantum Hall problem in the lowest Landau level in two dimensions, in the presence of an arbitrary number of $δ$-function potentials arranged in different geometric configurations. When the number of delta functions $N_δ$ is smaller than the number of flux quanta through the system ($N_ϕ$), there is a manifold of $(N_ϕ-N_δ)$ degenerate states at the original Landau level energy. We prove that the total Chern number of this set of states is +1 regardless of the number or position of the $δ$ functions. Furthermore, we find numerically that, upon the addition of disorder, this subspace includes a quantum Hall transition which is (in a well-defined sense) $\textit{quantitatively}$ the same as that for the lowest Landau level without $δ$-function impurities, but with a reduced number $N_ϕ' \equiv N_ϕ-N_δ$ of magnetic flux quanta. We discuss the implications of these results for studies of the integer plateau transitions, as well as for the many-body problem in the presence of electron-electron interactions.

cond-mat.dis-nn↗

arXiv2017

Lattice Monte Carlo for Quantum Hall States on a Torus

Monte Carlo is one of the most useful methods to study the quantum Hall problems. In this paper, we introduce a fast lattice Monte Carlo method based on a mathematically exact reformulation of the torus quantum Hall problems from continuum to lattice. We first apply this new technique to study the Berry phase of transporting composite fermions along different closed paths enclosing or not enclosing the Fermi surface center in the half filled Landau level problem. The Monte Carlo result agrees with the phase structure we found on small systems and confirms it on much larger sizes. Several other quantities including the Coulomb energy in different Landau levels, structure factor, particle-hole symmetry are computed and discussed for various model states. In the end, based on certain knowledge of structure factor, we introduce a algorithm by which the lattice Monte Carlo efficiency is further boosted by several orders.

cond-mat.str-el↗

arXiv2017

Lattice realization of a bosonic integer quantum Hall state - trivial insulator transition and relation to the self-dual line in the easy-plane NCCP1 model

We provide an explicit lattice model of bosons with two separately conserved boson species [$U(1)\times U(1)$ global symmetry] realizing a direct transition between an integer quantum Hall effect of bosons and a trivial phase, where any intermediate phase is avoided by an additional symmetry interchanging the two species. If the latter symmetry is absent, we find intermediate superfluid phases where one or the other boson species condenses. We know the precise location of the transition since at this point our model has an exact non-local anti-unitary particle-hole-like symmetry that resembles particle-hole symmetry in the lowest Landau level of electrons. We exactly map the direct transition to our earlier study of the self-dual line of the easy-plane NCCP1 model, in the mathematically equivalent reformulation in terms of two (new) particles with $π$-statistics and identical energetics. While the transition in our model is first-order, we hope that our mappings and recent renewed interest in such self-dual models will stimulate more searches for models with a continuous transition.

cond-mat.str-el↗

arXiv2017

Connection between Fermi contours of zero-field electrons and $ν=\frac12$ composite fermions in two-dimensional systems

We investigate the relation between the Fermi sea (FS) of zero-field carriers in two-dimensional systems and the FS of the corresponding composite fermions which emerge in a high magnetic field at filling $ν= \frac{1}{2}$, as the kinetic energy dispersion is varied. We study cases both with and without rotational symmetry, and find that there is generally no straightforward relation between the geometric shapes and topologies of the two FSs. In particular, we show analytically that the composite Fermi liquid (CFL) is completely insensitive to a wide range of changes to the zero-field dispersion which preserve rotational symmetry, including ones that break the zero-field FS into multiple disconnected pieces. In the absence of rotational symmetry, we show that the notion of `valley pseudospin' in many-valley systems is generically not transferred to the CFL, in agreement with experimental observations. We also discuss how a rotationally symmetric band structure can induce a reordering of the Landau levels, opening interesting possibilities of observing higher-Landau-level physics in the high-field regime.

cond-mat.str-el↗

arXiv2017

Composite fermions in bands with N-fold rotational symmetry

We study the effect of band anisotropy with discrete rotational symmetry $C_N$ (where $N\ge 2$) in the quantum Hall regime of two-dimensional electron systems. We focus on the composite Fermi liquid (CFL) at half filling of the lowest Landau level. We find that the magnitude of anisotropy transferred to the composite fermions decreases very rapidly with $N$. We demonstrate this by performing density matrix normalization group calculations on the CFL, and comparing the anisotropy of the composite fermion Fermi contour with that of the (non-interacting) electron Fermi contour at zero magnetic field. We also show that the effective interaction between the electrons after projecting into a single Landau level is much less anisotropic than the band, a fact which does not depend on filling and thus has implications for other quantum Hall states as well. Our results confirm experimental observations on anisotropic bands with warped Fermi contours, where the only detectable effect on the composite Fermi contour is an elliptical distortion ($N = 2$).

cond-mat.str-el↗

arXiv2017

Numerical study of anisotropy in a composite Fermi liquid

We perform density-matrix renormalization group studies of a two-dimensional electron gas in a high magnetic field and with an anisotropic band mass. At half-filling in the lowest Landau level, such a system is a Fermi liquid of composite fermions. By measuring the Fermi surface of these composite fermions, we determine a relationship between the anisotropy of composite fermion dispersion, $α_{CF}$, and the original anisotropy $α_F$ of the fermion dispersion at zero magnetic field. For systems where the electrons interact via a Coulomb interaction, we find $α_{CF}=\sqrt{α_F}$ within our numerical accuracy. The same result has been found concurrently in recent experiments. We also show results with other forms of the electron-electron interaction; this allows us (a) to benchmark our procedure against known exact results and (b) to show that the relationship between the anisotropies is dependent on the form of the interaction.

cond-mat.str-el↗

arXiv2017

Characterizing the many-body localization transition through the entanglement spectrum

We numerically explore the many body localization (MBL) transition through the lens of the {\it entanglement spectrum}. While a direct transition from localization to thermalization is believed to obtain in the thermodynamic limit (the exact details of which remain an open problem), in finite system sizes there exists an intermediate `quantum critical' regime. Previous numerical investigations have explored the crossover from thermalization to criticality, and have used this to place a numerical {\it lower} bound on the critical disorder strength for MBL. A careful analysis of the {\it high energy} part of the entanglement spectrum (which contains universal information about the critical point) allows us to make the first ever observation in exact numerics of the crossover from criticality to MBL and hence to place a numerical {\it upper bound} on the critical disorder strength for MBL.

cond-mat.dis-nn↗

arXiv2017

Emergent particle-hole symmetry in spinful bosonic quantum Hall systems

When a fermionic quantum Hall system is projected into the lowest Landau level, there is an exact particle-hole symmetry between filling fractions $ν$ and $1-ν$. We investigate whether a similar symmetry can emerge in bosonic quantum Hall states, where it would connect states at filling fractions $ν$ and $2-ν$. We begin by showing that the particle-hole conjugate to a composite fermion `Jain state' is another Jain state, obtained by reverse flux attachment. We show how information such as the shift and the edge theory can be obtained for states which are particle-hole conjugates. Using the techniques of exact diagonalization and infinite density matrix renormalization group, we study a system of two-component (i.e., spinful) bosons, interacting via a $δ$-function potential. We first obtain real-space entanglement spectra for the bosonic integer quantum Hall effect at $ν=2$, which plays the role of a filled Landau level for the bosonic system. We then show that at $ν=4/3$ the system is described by a Jain state which is the particle-hole conjugate of the Halperin (221) state at $ν=2/3$. We show a similar relationship between non-singlet states at $ν=1/2$ and $ν=3/2$. We also study the case of $ν=1$, providing unambiguous evidence that the ground state is a composite Fermi liquid. Taken together our results demonstrate that there is indeed an emergent particle-hole symmetry in bosonic quantum Hall systems.

cond-mat.str-el↗

arXiv2016

Absence of Many-Body Localization in a Single Landau Level

The dynamics of the highly excited states of a system projected into a single Landau level are analyzed. An analysis of level spacing ratios for finite size systems shows a clear crossover from extend (GUE) to localized (Poisson) statistics, indicating a many body localization transition. However, the location of this transition depends very strongly on system size, and appears to scale to infinite disorder in the thermodynamic limit. This result does not depend on the properties of the ground state (such as whether the ground state exhibits topological order), as expected for a transition of highly-excited eigenstates. We therefore conclude that many body localization does not exist in these systems. Our results demonstrate that a sub-thermodynamic number of single particle effectively extended states is sufficient to cause all many body states to become extended.

cond-mat.dis-nn↗

arXiv2016

Emergent local integrals of motion without a complete set of localized eigenstates

Systems where all energy eigenstates are localized are known to display an emergent local integrability, in the sense that one can construct an extensive number of operators that commute with the Hamiltonian and are localized in real space. Here we show that emergent local integrability does not require a complete set of localized eigenstates. Given a set of localized eigenstates comprising a nonzero fraction $(1-f)$ of the full many body spectrum, one can construct an extensive number of integrals of motion which are local in the sense that they have {\it nonzero weight} in a compact region of real space, in the thermodynamic limit. However, these modified integrals of motion have a `global dressing' whose weight vanishes as $\sim f$ as $f \rightarrow 0$. In this sense, the existence of a {\it non-zero fraction} of localized eigenstates is sufficient for emergent local integrability. We discuss the implications of our findings for systems where the spectrum contains delocalized states, for systems with projected Hilbert spaces, and for the robustness of quantum integrability.

cond-mat.stat-mech↗

arXiv2016

Many body localization and thermalization: insights from the entanglement spectrum

We study the entanglement spectrum in the many body localizing and thermalizing phases of one and two dimensional Hamiltonian systems, and periodically driven `Floquet' systems. We focus on the level statistics of the entanglement spectrum as obtained through numerical diagonalization, finding structure beyond that revealed by more limited measures such as entanglement entropy. In the thermalizing phase the entanglement spectrum obeys level statistics governed by an appropriate random matrix ensemble. For Hamiltonian systems this can be viewed as evidence in favor of a strong version of the eigenstate thermalization hypothesis (ETH). Similar results are also obtained for Floquet systems, where they constitute a result `beyond ETH', and show that the corrections to ETH governing the Floquet entanglement spectrum have statistical properties governed by a random matrix ensemble. The particular random matrix ensemble governing the Floquet entanglement spectrum depends on the symmetries of the Floquet drive, and therefore can depend on the choice of origin of time. In the many body localized phase the entanglement spectrum is also found to show level repulsion, following a semi-Poisson distribution (in contrast to the energy spectrum, which follows a Poisson distribution). This semi-Poisson distribution is found to come mainly from states at high entanglement energies. The observed level repulsion only occurs for interacting localized phases. We also demonstrate that equivalent results can be obtained by calculating with a single typical eigenstate, or by averaging over a microcanonical energy window - a surprising result in the localized phase. This discovery of new structure in the pattern of entanglement of localized and thermalizing phases may open up new lines of attack on many body localization, thermalization, and the localization transition.

cond-mat.stat-mech↗

arXiv2015

Monte Carlo study of phase transitions out of Symmetry-Enriched Topological phases of bosons in two dimensions

We study a statistical mechanics model of two species of bosons with mutual statistics $θ=2π/n$ 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.

cond-mat.stat-mech↗

arXiv2015

The half-filled Landau level: the case for Dirac composite fermions

One of the most spectacular experimental findings in the fractional quantum Hall effect is evidence for an emergent Fermi surface when the electron density is nearly half the density of magnetic flux quanta ($ν= 1/2$). The seminal work of Halperin, Lee, and Read (HLR) first predicted that at $ν= 1/2$ composite fermions--bound states of an electron and a pair of vortices--experience zero net magnetic field and can form a "composite Fermi liquid" with an emergent Fermi surface. In this paper we use infinite cylinder DMRG to provide compelling numerical evidence for the existence of a Fermi sea of composite fermions for realistic interactions between electrons at $ν= 1/2$. Moreover, we show that the state is particle-hole symmetric, in contrast to the construction of HLR. Instead, our findings are consistent if the composite fermions are massless Dirac particles, at finite density, similar to the surface state of a 3D topological insulator. Exploiting this analogy we devise a numerical test and successfully observe the suppression of $2k_F$ backscattering characteristic of Dirac particles.

cond-mat.str-el↗

arXiv2014

Exact Models for Symmetry-Protected Topological Phases in One Dimension

We present an exactly solvable model for one-dimensional symmetry-protected topological phases with $\mathbb{Z}_N\times\mathbb{Z}_N$ symmetry. The model works by binding point topological defects (domain walls) of one symmetry to charges of the other and condensing these bound states. Binding single topological defects to charges leads to symmetry-protected topological phases, while binding multiple topological defects to charges leads to phases with a combination of symmetry-breaking and topological properties.

cond-mat.stat-mech↗

arXiv2014

Model of Fractionalization of Faraday Lines in Compact Electrodynamics

Motivated by ideas of fractionalization and intrinsic topological order in bosonic models with short-range interactions, we consider similar phenomena in formal lattice gauge theory models. Specifically, we show that a compact quantum electrodynamics (CQED) can have, besides the familiar Coulomb and confined phases, additional unusual confined phases where excitations are quantum lines carrying fractions of the elementary unit of electric field strength. We construct a model that has $N$-tupled monopole condensation and realizes $1/N$ fractionalization of the quantum Faraday lines. This phase has another excitation which is a $Z_N$ quantum surface in spatial dimensions five and higher, but can be viewed as a quantum line or a quantum particle in four or three spatial dimensions respectively. These excitation have statistical interactions with the fractionalized Faraday lines; for example, in three spatial dimensions, the particle excitation picks up a Berry phase of $e^{i2π/N}$ when going around the fractionalized Faraday line excitation. We demonstrate the existence of this phase by Monte Carlo simulations in (3+1) space-time dimensions.

cond-mat.stat-mech↗

arXiv2013

Exact realization of Integer and Fractional Quantum Hall Phases in U(1)xU(1) models in (2+1)d

In this work we present a set of microscopic U(1)xU(1) models which realize insulating phases with a quantized Hall conductivity σ_{xy}. The models are defined in terms of physical degrees of freedom, and can be realized by local Hamiltonians. For one set of these models, we find that σ_{xy} is quantized to be an even integer. The origin of this effect is a condensation of objects made up of bosons of one species bound to a single vortex of the other species. For other models, the Hall conductivity can be quantized as a rational number times two. For these systems, the condensed objects contain bosons of one species bound to multiple vortices of the other species. These systems have excitations carrying fractional charges and non-trivial mutual statistics. We present sign-free reformulations of these models which can be studied in Monte Carlo, and we use such reformulations to numerically detect a gapless boundary between the quantum Hall and trivial insulator states. We also present the broader phase diagrams of the models.

cond-mat.str-el↗