arXiv subjects

Bartholomew Andrews

Publications and source records attributed to Bartholomew Andrews.

13 recordsLinked to original sources

arXiv2026

ffsim: Faster simulation of fermionic quantum circuits

We present ffsim, an open-source software library for fast simulation of fermionic quantum circuits. ffsim exploits conservation of particle number and the z component of spin, symmetries present in a wide range of fermionic systems, to dramatically reduce memory usage and simulation time compared to general-purpose quantum circuit simulators. Compared to FQE, a library with similar functionality, ffsim differs in software design and is faster on a representative set of simulation benchmarks. Beyond state vector evolution by basic fermionic gates, ffsim offers a number of additional features including variational ansatzes, Hamiltonian time evolution via Trotter-Suzuki product formulas, efficient sampling of Slater determinants, seamless integration with Qiskit and PySCF, and comprehensive documentation. We demonstrate ffsim's capabilities on scientific applications involving quantum circuits of up to 64 qubits.

quant-ph↗

arXiv2024

Tensor Network Python (TeNPy) version 1

TeNPy (short for 'Tensor Network Python') is a python library for the simulation of strongly correlated quantum systems with tensor networks. The philosophy of this library is to achieve a balance of readability and usability for new-comers, while at the same time providing powerful algorithms for experts. The focus is on MPS algorithms for 1D and 2D lattices, such as DMRG ground state search, as well as dynamics using TEBD, TDVP, or MPO evolution. This article is a companion to the recent version 1.0 release of TeNPy and gives a brief overview of the package.

cond-mat.str-el↗

arXiv2024

Stability of fractional Chern insulators with a non-Landau level continuum limit

The stability of fractional Chern insulators is widely believed to be predicted by the resemblance of their single-particle spectra to Landau levels. We investigate the scope of this geometric stability hypothesis by analyzing the stability of a set of fractional Chern insulators that explicitly do not have a Landau level continuum limit. By computing the many-body spectra of Laughlin states in a generalized Hofstadter model, we analyze the relationship between single-particle metrics, such as trace inequality saturation, and many-body metrics, such as the magnitude of the many-body and entanglement gaps. We show numerically that the geometric stability hypothesis holds for Chern bands that are not continuously connected to Landau levels, as well as conventional Chern bands, albeit often requiring larger system sizes to converge for these configurations.

cond-mat.str-el↗

arXiv2024

HofstadterTools: A Python package for analyzing the Hofstadter model

The Hofstadter model successfully describes the behavior of non-interacting quantum particles hopping on a lattice coupled to a gauge field, and hence is ubiquitous in many fields of research, including condensed matter, optical, and atomic physics. Motivated by this, we introduce HofstadterTools (https://hofstadter.tools), a Python package that can be used to analyze the energy spectrum of a generalized Hofstadter model, with any combination of hoppings on any regular Euclidean lattice. The package can be applied to compute key properties of the band structure, such as quantum geometry and topology, as well as plot Hofstadter butterflies and Wannier diagrams that are colored according to their Chern numbers.

cond-mat.mes-hall↗

arXiv2024

Localization renormalization and quantum Hall systems

The obstruction to constructing localized degrees of freedom is a signature of several interesting condensed matter phases. We introduce a localization renormalization procedure that harnesses this property, and apply our method to distinguish between topological and trivial phases in quantum Hall and Chern insulators. By iteratively removing a fraction of maximally-localized orthogonal basis states, we find that the localization length in the residual Hilbert space exhibits a power-law divergence as the fraction of remaining states approaches zero, with an exponent of $ν=0.5$. In sharp contrast, the localization length converges to a system-size-independent constant in the trivial phase. We verify this scaling using a variety of algorithms to truncate the Hilbert space, and show that it corresponds to a statistically self-similar expansion of the real-space projector. This result accords with a renormalization group picture and motivates the use of localization renormalization as a versatile numerical diagnostic for quantum Hall systems.

cond-mat.mes-hall↗

arXiv2023

Self-similarity of spectral response functions for fractional quantum Hall states

Spectral response functions are central quantities in the analysis of quantum many-body states, since they describe the response of many-body systems to external perturbations and hence directly correspond to observables in experiments. In this paper, we evaluate a momentum-averaged dynamical density structure factor for the fermionic $ν=1/3$ fractional quantum Hall state on a torus, using the continued fraction method to compute the dynamical correlation function. We establish the scaling behavior of the screened Coulomb structure factor with respect to interaction range, and expose an inherent self-similarity of structure factors in the frequency domain. These results highlight the statistical properties of spectral response functions for fractional quantum Hall states and show how they can be efficiently approximated in numerical models.

cond-mat.str-el↗

arXiv2022

Fractional Chern insulators with a non-Landau level continuum limit

Recent developments in fractional quantum Hall (FQH) physics highlight the importance of studying FQH phases of particles partially occupying energy bands that are not Landau levels. FQH phases in the regime of strong lattice effects, called fractional Chern insulators, provide one setting for such studies. As the strength of lattice effects vanishes, the bands of generic lattice models asymptotically approach Landau levels. In this article, we construct lattice models for single-particle bands that are distinct from Landau levels even in this continuum limit. We describe how the distinction between such bands and Landau levels is quantified by band geometry over the magnetic Brillouin zone and reflected in the electromagnetic response. We analyze the localization-delocalization transition in one such model and compute a localization length exponent of 2.57(3). Moreover, we study interactions projected to these bands and find signatures of bosonic and fermionic Laughlin states. Most pertinently, our models allow us to isolate conditions for optimal band geometry and gain further insight into the stability of FQH phases on lattices.

cond-mat.str-el↗

arXiv2021

Stability, phase transitions, and numerical breakdown of fractional Chern insulators in higher Chern bands of the Hofstadter model

The Hofstadter model is a popular choice for theorists investigating the fractional quantum Hall effect on lattices, due to its simplicity, infinite selection of topological flat bands, and increasing applicability to real materials. In particular, fractional Chern insulators in bands with Chern number $|C|>1$ can demonstrate richer physical properties than continuum Landau level states and have recently been detected in experiments. Motivated by this, we examine the stability of fractional Chern insulators with higher Chern number in the Hofstadter model, using large-scale infinite density matrix renormalization group simulations on a thin cylinder. We confirm the existence of fractional states in bands with Chern numbers $C=1,2,3,4,5$ at the filling fractions predicted by the generalized Jain series [Phys. Rev. Lett. 115, 126401 (2015)]. Moreover, we discuss their metal-to-insulator phase transitions, as well as the subtleties in distinguishing between physical and numerical stability. Finally, we comment on the relative suitability of fractional Chern insulators in higher Chern number bands for proposed modern applications.

cond-mat.str-el↗

arXiv2021

Abelian topological order of $ν=2/5$ and $3/7$ fractional quantum Hall states in lattice models

Determining the statistics of elementary excitations supported by fractional quantum Hall states is crucial to understanding their properties and potential applications. In this paper, we use the topological entanglement entropy as an indicator of Abelian statistics to investigate the single-component $ν=2/5$ and $3/7$ states for the Hofstadter model in the band mixing regime. We perform many-body simulations using the infinite cylinder density matrix renormalization group and present an efficient algorithm to construct the area law of entanglement, which accounts for both numerical and statistical errors. Using this algorithm, we show that the $ν=2/5$ and $3/7$ states exhibit Abelian topological order in the case of two-body nearest-neighbor interactions. Moreover, we discuss the sensitivity of the proposed method and fractional quantum Hall states with respect to interaction range and strength.

cond-mat.str-el↗

arXiv2020

Absence of diagonal force constants in cubic Coulomb crystals

The quasi-harmonic model proposes that a crystal can be modeled as atoms connected by springs. We demonstrate how this viewpoint can be misleading: a simple application of Gauss' law shows that the ion-ion potential for a cubic Coulomb system can have no diagonal harmonic contribution and so cannot necessarily be modeled by springs. We investigate the repercussions of this observation by examining three illustrative regimes: the bare ionic, density tight-binding, and density nearly-free electron models. For the bare ionic model, we demonstrate the zero elements in the force constants matrix and explain this phenomenon as a natural consequence of Poisson's law. In the density tight-binding model, we confirm that the inclusion of localized electrons stabilizes all major crystal structures at harmonic order and we construct a phase diagram of preferred structures with respect to core and valence electron radii. In the density nearly-free electron model, we verify that the inclusion of delocalized electrons, in the form of a background jellium, is enough to counterbalance the diagonal force constants matrix from the ion-ion potential in all cases and we show that a first-order perturbation to the jellium does not have a destabilizing effect. We discuss our results in connection to Wigner crystals in condensed matter, Yukawa crystals in plasma physics, as well as the elemental solids.

cond-mat.mtrl-sci↗

arXiv2020

Fractional quantum Hall states for moiré superstructures in the Hofstadter regime

We study the transition of $ν=1/3$ and $2/5$ fractional quantum Hall states of the honeycomb Hofstadter model as we tune to a two-orbital moiré superlattice Hamiltonian, motivated by the flat bands of twisted bilayer graphene in a perpendicular magnetic field. In doing so, we address the extent to which these states survive in moiré systems and analyze the nature of the transition. Through the use of a Peierls substitution, we determine the Landau-level splitting for the moiré Hamiltonian, and study the structure of the Chern bands for a range of magnetic flux per plaquette. We identify topological flat bands in the spectrum at low energies, with numerically tractable lattice geometries that can support the fractional quantum Hall effect. As we tune the model, we find that the orbital-polarized $ν=1/3$ and $2/5$ states corresponding to the honeycomb Hofstadter model survive up to $\approx 30\%$ of typical moiré superlattice parameters, beyond which they transition into an insulating phase. We present evidence for this through density matrix renormalization group calculations on an infinite cylinder, by verifying the charge pumping, spectral flow, entanglement scaling, and conformal field theory edge counting. We conclude that fractional quantum Hall states from the Hofstadter model can persist up to hopping amplitudes of the same order as those typical for moiré superlattice Hamiltonians, which implies generally that fractional states for moiré superstructures can be discerned simply by analyzing the dominant terms in their effective Hamiltonians.

cond-mat.str-el↗

arXiv2019

Direct evaluation of the force constant matrix in quantum Monte Carlo

We develop a formalism to directly evaluate the matrix of force constants within a Quantum Monte Carlo calculation. We utilize the matrix of force constants to accurately relax the positions of atoms in molecules and determine their vibrational modes, using a combination of Variational and Diffusion Monte Carlo. The computed bond lengths differ by less than 0.007Å from the experimental results for all four tested molecules. For hydrogen and hydrogen chloride, we obtain fundamental vibrational frequencies within 0.1% of experimental results and ~10 times more accurate than leading computational methods. For carbon dioxide and methane, the vibrational frequency obtained is on average within 1.1% of the experimental result, which is at least 3 times closer than results using Restricted Hartree-Fock and Density Functional Theory with a Perdew-Burke-Ernzerhof (PBE) functional and comparable or better than Density Functional Theory with a semi-empirical functional.

physics.comp-ph↗

arXiv2018

Stability of fractional Chern insulators in the effective continuum limit of Harper-Hofstadter bands with Chern number $|C|>1$

We study the stability of composite fermion fractional quantum Hall states in Harper-Hofstadter bands with Chern number $|C|>1$. We analyze the states of the composite fermion series for bosons with contact interactions and (spinless) fermions with nearest-neighbor interactions. We examine the scaling of the many-body gap as the bands are tuned to the effective continuum limit $n_ϕ\to 1/|C|$. Near these points, the Hofstadter model realises large magnetic unit cells that yield bands with perfectly flat dispersion and Berry curvature. We exploit the known scaling of energies in the effective continuum limit in order to maintain a fixed square aspect ratio in finite-size calculations. Based on exact diagonalization calculations of the band-projected Hamiltonian, we show that almost all finite-size spectra yield the ground-state degeneracy predicted by composite fermion theory. We confirm that states at low ranks in the composite fermion hierarchy are the most robust and yield a clear gap in the thermodynamic limit. For bosons in $|C|=2$ and $|C|=3$ bands, our data for the composite fermion states are compatible with a finite gap in the thermodynamic limit. We also report new evidence for gapped incompressible states of fermions in $|C|>1$ bands, which have large entanglement gaps. For cases with a clear spectral gap, we confirm that the thermodynamic limit commutes with the effective continuum limit. We analyze the nature of the correlation functions for the Abelian composite fermion states and find that they feature $|C|^2$ smooth sheets. We examine two cases associated with a bosonic integer quantum Hall effect (BIQHE): For $ν=2$ in $|C|=1$ bands, we find a strong competing state with a higher ground-state degeneracy, so no clear BIQHE is found in the band-projected Hofstadter model; for $ν=1$ in $|C|=2$ bands, we present additional data confirming the existence of a BIQHE state.

cond-mat.str-el↗