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F. D. M. Haldane

Publications and source records attributed to F. D. M. Haldane.

At least 37 records · Page 2Linked to original sources
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

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↗
arXiv2016

Ground State Properties of Antiferromagnetic Chains with Unrestricted Spin: Integer Spin Chains as Realisations of the O(3) Non-Linear Sigma Model

A continuum limit treatment of planar spin chains with arbitrary S is presented. The difference between integer and half-integer spins is emphasised. While isotropic half-integer spin chains are gapless,and have power-law decay of correlations at T = 0 with exponent eta = 1, integer spin systems have a singlet ground state with a gap for S=1 excitations and exponential decay of correlations. The easy-plane to easy-axis transition is described. Note: this is a verbatim transcription of ILL preprint SP-81/95, which is cited in a number of places, but was rejected for publication in 1981, and remained unpublished in this original form, which gives somewhat different arguments for the central result as compared to the later paper finally published in Phys. Lett. 93A, 464 (1983); many thanks to Jeno Solyom for preserving this historical document.

cond-mat.other↗
arXiv2016

Fractional Quantum Hall Bilayers at Half-Filling: Tunneling-driven Non-Abelian Phase

Multicomponent quantum Hall systems with internal degrees of freedom provide a fertile ground for the emergence of exotic quantum liquids. Here we investigate the possibility of non-Abelian topological order in the half-filled fractional quantum Hall (FQH) bilayer system driven by the tunneling effect between two layers. By means of the state-of-the-art density-matrix renormalization group, we unveil "finger print" evidence of the non-Abelian Moore-Read Pfaffian state emerging in the intermediate-tunneling regime, including the ground-state degeneracy on the torus geometry and the topological entanglement spectroscopy (entanglement spectrum and topological entanglement entropy) on the spherical geometry, respectively. Remarkably, the phase transition from the previously identified Abelian $(331)$ Halperin state to the non-Abelian Moore-Read Pfaffian state is determined to be continuous, which is signaled by the continuous evolution of the universal part of the entanglement spectrum, and discontinuities in the excitation gap and the derivative of the ground-state energy. Our results not only provide a "proof-of-principle" demonstration of realizing a non-Abelian state through coupling different degrees of freedom, but also open up a possibility in FQH bilayer systems for detecting different chiral $p-$wave pairing states.

cond-mat.str-el↗
arXiv2016

Geometry of Landau orbits in the absence of rotational symmetry

The integer quantum Hall effect (IQHE) is usually modeled using Galilean-invariant or rotationally-invariant Landau levels. However, these are not generic symmetries of electrons moving in a crystalline background, even in the low-density continuum limit. We present a treatment of the IQHE which abandons the Galilean dispersion relation while keeping inversion symmetry. We define an emergent metric $g^n_{ab}$ for each Landau level with a reformulation of the Hall viscosity. The metric is then used to define a guiding-center coherent state and the wavefunctions are holomorphic functions of $z^*$ times a Gaussian. By numerically studying cases with quartic dispersion, we show that the number of the zeroes of the wavefunction encircled by the semiclassical orbit, denoted by $n$, defines a "topological spin" $s_n$ by $s_n=n+\frac{1}{2}$, with its original definition as an "intrinsic angular momentum" no longer valid without rotational symmetry. At the end of the paper we show our results for the density and current responses which differentiate between diagonal and Landau-level-mixing terms. In conclusion, this treatment extracts topological information without resort to Galilean or rotational symmetry, and also reveals more generic geometric structures.

cond-mat.mes-hall↗
arXiv2015

Probing the geometry of the Laughlin state

It has recently been pointed out that phases of matter with intrinsic topological order, like the fractional quantum Hall states, have an extra dynamical degree of freedom that corresponds to quantum geometry. Here we perform extensive numerical studies of the geometric degree of freedom for the simplest example of fractional quantum Hall states -- the filling $ν=1/3$ Laughlin state. We perturb the system by a smooth, spatially dependent metric deformation and measure the response of the Hall fluid, finding it to be proportional to the Gaussian curvature of the metric. Further, we generalize the concept of coherent states to formulate the bulk off-diagonal long range order for the Laughlin state, and compute the deformations of the metric in the vicinity of the edge of the system. We introduce a "pair amplitude" operator and show that it can be used to numerically determine the intrinsic metric of the Laughlin state. These various probes are applied to several experimentally relevant settings that can expose the quantum geometry of the Laughlin state, in particular to systems with mass anisotropy and in the presence of an electric field gradient.

cond-mat.str-el↗
arXiv2015

Topological nodal Cooper pairing in doped Weyl metals

We generalize the concept of Berry connection of the single-electron band structure to the two-particle Cooper pair states between two Fermi surfaces with opposite Chern numbers. Because of underlying Fermi surface topology, the pairing Berry phase acquires non-trivial monopole structure. Consequently, pairing gap functions have the topologically-protected nodal structure as vortices in the momentum space with the total vorticity solely determined by the monopole charge $q_p$. The pairing nodes behave as the Weyl-Majorana points of the Bogoliubov-de Gennes pairing Hamiltonian. Their relation with the connection patterns of the surface modes from the Weyl band structure and the Majorana surface modes inside the pairing gap is also discussed. Under the approximation of spherical Fermi surfaces, the pairing symmetry are represented by monopole harmonic functions. The lowest possible pairing channel carries angular momentum number $j=|q_p|$, and the corresponding gap functions are holomorphic or anti-holomorphic functions on Fermi surfaces.

cond-mat.str-el↗
arXiv2015

Topological Characterization of Non-Abelian Moore-Read State using Density-Matrix Renormailzation Group

The non-Abelian topological order has attracted a lot of attention for its fundamental importance and exciting prospect of topological quantum computation. However, explicit demonstration or identification of the non-Abelian states and the associated statistics in a microscopic model is very challenging. Here, based on density-matrix renormalization group calculation, we provide a complete characterization of the universal properties of bosonic Moore-Read state on Haldane honeycomb lattice model at filling number $ν=1$ for larger systems, including both the edge spectrum and the bulk anyonic quasiparticle (QP) statistics. We first demonstrate that there are three degenerating ground states, for each of which there is a definite anyonic flux threading through the cylinder. We identify the nontrivial countings for the entanglement spectrum in accordance with the corresponding conformal field theory. Through inserting the $U(1)$ charge flux, it is found that two of the ground states can be adiabatically connected through a fermionic charge-$\textit{e}$ QP being pumped from one edge to the other, while the ground state in Ising anyon sector evolves back to itself. Furthermore, we calculate the modular matrices $\mathcal{S}$ and $\mathcal{U}$, which contain all the information for the anyonic QPs. In particular, the extracted quantum dimensions, fusion rule and topological spins from modular matrices positively identify the emergence of non-Abelian statistics following the $SU(2)_2$ Chern-Simons theory.

cond-mat.str-el↗
arXiv2015

The Fractional Quantum Hall States at $ν=13/5$ and $12/5$ and their Non-Abelian Nature

We investigate the nature of the fractional quantum Hall (FQH) state at filling factor $ν=13/5$, and its particle-hole conjugate state at $12/5$, with the Coulomb interaction, and address the issue of possible competing states. Based on a large-scale density-matrix renormalization group (DMRG) calculation in spherical geometry, we present evidence that the physics of the Coulomb ground state (GS) at $ν=13/5$ and $12/5$ is captured by the $k=3$ parafermion Read-Rezayi RR state, $\text{RR}_3$. We first establish that the state at $ν=13/5$ is an incompressible FQH state, with a GS protected by a finite excitation gap, with the shift in accordance with the RR state. Then, by performing a finite-size scaling analysis of the GS energies for $ν=12/5$ with different shifts, we find that the $\text{RR}_3$ state has the lowest energy among different competing states in the thermodynamic limit. We find the fingerprint of $\text{RR}_3$ topological order in the FQH $13/5$ and $12/5$ states, based on their entanglement spectrum and topological entanglement entropy, both of which strongly support their identification with the $\text{RR}_3$ state. Furthermore, by considering the shift-free infinite-cylinder geometry, we expose two topologically-distinct GS sectors, one identity sector and a second one matching the non-Abelian sector of the Fibonacci anyonic quasiparticle, which serves as additional evidence for the $\text{RR}_3$ state at $13/5$ and $12/5$.

cond-mat.str-el↗
arXiv2014

Guiding-center Hall viscosity and intrinsic dipole moment along edges of incompressible fractional quantum Hall fluids

The discontinuity of guiding-center Hall viscosity (a bulk property) at edges of incompressible quantum Hall fluids is associated with the presence of an intrinsic electric dipole moment on the edge. If there is a gradient of drift velocity due to a non-uniform electric field, the discontinuity in the induced stress is exactly balanced by the electric force on the dipole. The total Hall viscosity has two distinct contributions: a "trivial" contribution associated with the geometry of the Landau orbits, and a non-trivial contribution associated with guiding-center correlations. We describe a relation between the guiding-center edge-dipole moment and "momentum polarization", which relates the guiding-center part of the bulk Hall viscosity to the "orbital entanglement spectrum(OES)". We observe that using the computationally-more-onerous "real-space entanglement spectrum (RES)" just adds the trivial Landau-orbit contribution to the guiding-center part. This shows that all the non-trivial information is completely contained in the OES, which also exposes a fundamental topological quantity $γ$ = $\tilde c-ν$, the difference between the "chiral stress-energy anomaly" (or signed conformal anomaly) and the chiral charge anomaly. This quantity characterizes correlated fractional quantum Hall fluids, and vanishes in uncorrelated integer quantum Hall fluids.

cond-mat.str-el↗
arXiv2014

Entanglement entropy of the $ν=1/2$ composite fermion non-Fermi liquid state

The so-called ``non-Fermi liquid'' behavior is very common in strongly correlated systems. However, its operational definition in terms of ``what it is not'' is a major obstacle against theoretical understanding of this fascinating correlated state. Recently there has been much interest in entanglement entropy as a theoretical tool to study non-Fermi liquids. So far explicit calculations have been limited to models without direct experimental realizations. Here we focus on a two dimensional electron fluid under magnetic field and filling fraction $ν=1/2$, which is believed to be a non-Fermi liquid state. Using the composite fermion (CF) wave-function which captures the $ν=1/2$ state very accurately, we compute the second Rényi entropy using variational Monte-Carlo technique and an efficient parallel algorithm. We find the entanglement entropy scales as $L\log L$ with the length of the boundary $L$ as it does for free fermions, albeit with a pre-factor twice that of the free fermion. We contrast the results against theoretical conjectures and discuss the implications of the results.

cond-mat.str-el↗
arXiv2014

Nature of Quasielectrons and the Continuum of Neutral Bulk Excitations in the Laughlin Quantum Hall Fluids

We construct model wavefunctions for a family of single-quasielectron states supported by the $ν=1/3$ fractional quantum Hall (FQH) fluid. The charge $e^*$ = $e/3$ quasielectron state is identified as a composite of a charge-$2e^*$ quasiparticle and a $-e^*$ quasihole, orbiting around their common center of charge with relative angular momentum $n\hbar > 0$, and corresponds precisely to the "composite fermion" construction based on a filled $n=0$ Landau level plus an extra particle in level $n > 0$. An effective three-body model (one $2e^*$ quasiparticle and two $-e^*$ quasiholes) is introduced to capture the essential physics of the neutral bulk excitations.

cond-mat.str-el↗
arXiv2014

Attachment of Surface "Fermi Arcs" to the Bulk Fermi Surface: "Fermi-Level Plumbing" in Topological Metals

The role of "Fermi arc" surface-quasiparticle states in "topological metals" (where some Fermi surface sheets have non-zero Chern number) is examined. They act as "Fermi-level plumbing" conduits that transfer quasiparticles among groups of apparently-disconnected Fermi sheets with non-zero Chern numbers to maintain equality of their chemical potentials, which is required by gauge invariance. Fermi arcs have a chiral tangential attachment to the surface projections of sheets of the bulk Fermi Surface: the total Chern number of each projection equals the net chirality of arc-attachments to it. Information from the Fermi arcs is needed to unambiguously determine the quantized part of the anomalous Hall effect that is not determined at the bulk Fermi surface.

cond-mat.str-el↗
arXiv2013

Identifying Non-Abelian Topological Order through Minimal Entangled States

The topological order is equivalent to the pattern of long-range quantum entanglements, which cannot be measured by any local observable. Here we perform an exact diagonalization study to establish the non-Abelian topological order through entanglement entropy measurement. We focus on the quasiparticle statistics of the non-Abelian Moore-Read and Read-Rezayi states on the lattice boson models. We identify multiple independent minimal entangled states (MESs) in the groundstate manifold on a torus. The extracted modular $\mathcal{S}$ matrix from MESs faithfully demonstrates the Majorana quasiparticle or Fibonacci quasiparticle statistics, including the quasiparticle quantum dimensions and the fusion rules for such systems. These findings support that MESs manifest the eigenstates of quasiparticles for the non-Abelian topological states and encode the full information of the topological order.

cond-mat.str-el↗
arXiv2013

Minimal Entangled States and Modular Matrix for Fractional Quantum Hall Effect in Topological Flat Bands

We perform an exact diagonalization study of the topological order in topological flat band models through calculating entanglement entropy and spectra of low energy states. We identify multiple independent minimal entangled states, which form a set of orthogonal basis states for the ground-state manifold. We extract the modular transformation matrices S (U) which contains the information of mutual (self) statistics, quantum dimensions and fusion rule of quasi-particles. Moreover, we demonstrate that these matrices are robust and universal in the whole topological phase against different perturbations until the quantum phase transition takes place.

cond-mat.str-el↗
arXiv2013

Quantum Phase Transitions and the ν=5/2 Fractional Hall State in Wide Quantum Wells

We study the nature of the ν=5/2 quantum Hall state in wide quantum wells under the mixing of electronic subbands and Landau levels. We introduce a general method to analyze the Moore-Read Pfaffian state and its particle-hole conjugate, the anti-Pfaffian, under periodic boundary conditions in a "quartered" Brillouin zone scheme containing both even and odd numbers of electrons. We examine the rotational quantum numbers on the torus, and show spontaneous breaking of the particle-hole symmetry can be observed in finite-size systems. In the presence of electronic-subband and Landau-level mixing the particle-hole symmetry is broken in such a way that the anti-Pfaffian is unambiguously favored, and becomes more robust in the vicinity of a transition to the compressible phase, in agreement with recent experiments.

cond-mat.str-el↗
arXiv2012

Model Wavefunctions for the Collective Modes and the Magneto-roton Theory of the Fractional Quantum Hall Effect

We construct model wavefunctions for the collective modes of fractional quantum Hall systems. The wavefunctions are expressed in terms of symmetric polynomials characterized by a root partition and a "squeezed" basis, and show excellent agreement with exact diagonalization results for finite systems. In the long wavelength limit, the model wavefunctions reduce to those predicted by the single-mode approximation, and remain accurate at energies above the continuum of roton pairs.

cond-mat.str-el↗