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Ashley M. Cook

Publications and source records attributed to Ashley M. Cook.

At least 19 recordsLinked to original sources
arXiv2026

Bose-Einstein condensation and superfluidity on a fuzzy sphere

According to Hohenberg's theorem, Bose-Einstein condensation (BEC) in two dimensions is impossible for any temperature $T>0$. By contrast, superfluidity does occur in two dimensions at finite temperatures; it emerges due to the breaking of Galilei invariance. Here we consider BEC and superfluidity on a compact two-dimensional space taking the form of a non-commutative ("fuzzy") sphere, where the scalar bosonic fields are promoted to $N\times N$ matrices. The dimension $N$ is related to the non-commutativity parameter of space and introduces an additional scale into the system. We find that non-commutativity favors ordered phases and so enhances BEC and superfluidity. We analyze BEC in ideal and weakly interacting Bose gases on a fuzzy sphere, finding in each case that the critical temperature of BEC is greater compared to that found in the case of a commutative sphere $S^2$. Then we investigate the superfluid response of weakly interacting Bose systems. To account for vortices in a superfluid, we show that, even on an ordinary sphere, the collective coordinates of vortices induce non-commutativity. With this in mind, we extend the definition of vortex defects to an inherently non-commutative sphere studied here, where the notion of a point is untenable. The non-commutativity is expected to be experimentally relevant to BEC and superfluidity since the fuzzy sphere has a thermodynamic limit distinct from the one defined over a plane, unlike the $S^2$ case. The significance of this difference is illustrated by the superfluid density calculation indicating that, in the large sphere limit, the normal fluid fraction on the fuzzy sphere yields a linear in $T$ dependence, while on a commutative $S^2$ it exhibits the usual two-dimensional $\sim T^3$ behavior. This linear dependence, arising directly from non-commutativity, is reminiscent of Uemura's law in cuprate high-$T_c$ superconductors.

cond-mat.quant-gas↗
arXiv2026

Transition between one- and two-dimensional topology in a Chern insulator of finite width

Topology in quantum systems is typically considered in infinite crystals in one, two, or higher integer dimensions. Here, we show that one can continuously transform a system between a topological phase associated with one dimension and a topological phase associated with two dimensions without closing the energy gap. In this process, the dimension of the system itself changes. Concretely, we investigate a modified version of the Qi-Wu-Zhang model and develop a procedure to smoothly shrink the width of the system in one direction. By tracking gaps which remain open throughout the modulation, we establish a smooth transition from a two-dimensional to a one-dimensional topological insulator. In between the system exhibits both one- and two-dimensional topology, and the way the system accomplishes the transition is by making the one-dimensional topology more robust as the width decreases, while the two-dimensional topology becomes less robust. Finally, we show how the gaps arise from hybridization of edge states due to the finite width.

cond-mat.mes-hall↗
arXiv2025

Microscopic field theories of the quantum skyrmion Hall effect

We construct effective field theories of the quantum skyrmion Hall effect from matrix Chern-Simons theory for $N$ electrons, corresponding to matrix dimension $N$. We first consider a quantum Hall droplet within finite $N$ matrix Chern-Simons theory. Taking into account the differential geometry of the matrix Chern-Simons droplet for a partially-filled fuzzy two-sphere, we first generalize the quantization procedure by replacing the Poisson bracket, a classical Lie derivative, with a quantum counterpart, the Lie derivative for a deformed fuzzy sphere. This yields the topological invariant introduced in earlier works on the quantum skyrmion Hall effect and previously unidentified fusion rules. This is consistent with treatment of a spin $S$ of multiplicity $2S+1$ as a quantum Hall droplet within matrix Chern-Simons theory for $N=2S+1$ spinless electrons and a generalization of a Jain composite particle for a Laughlin state. We then construct $D$-dimensional arrays of coupled small $N$ matrix Chern-Simons droplets as effective field theories of the quantum skyrmion Hall effect. In higher-symmetry constructions, this yields what appears to be a D+1 dimensional $U(N)$ Yang-Mills theory, but actually contains $\delta$ extra fuzzy dimensions from the finite $N$ MCS theory as well as deformations from $U(N)$ due to partial filling of the fuzzy spheres. In this construction, the Chern-Simons level is $k+1$ for each small $N$ droplet, while the entire array can be interpreted as an unbounded matrix Chern-Simons theory at level $k$. Such constructions at $k=2$ are consistent with earlier results for the multiplicative Chern insulator. We also formulate the quantum skyrmion Hall effect in terms of a Lagrangian for an array of potentially distinct, small $N$ droplets within anisotropic fuzzification. We discuss the relevance of these results to spin lattice models and lattice gauge theories.

hep-th↗
arXiv2024

Effective field theory of the quantum skyrmion Hall effect

Motivated by phenomenology of myriad recently-identified topologically non-trivial phases of matter, we introduce effective field theories (EFTs) for the quantum skyrmion Hall effect (QSkHE). We employ a single, unifying generalisation for this purpose: in essence, a lowest Landau level projection defining a non-commutative, fuzzy sphere with position coordinates proportional to SU(2) generators of matrix representation size $N\times N$, may host an intrinsically 2+1 dimensional, topologically non-trivial many-body state for small $N$ as well as large $N$. That is, isospin degrees of freedom associated with a matrix Lie algebra with $N \times N$ generators potentially encode some finite number of spatial dimensions for $N\ge 2$, a regime in which isospin has previously been treated as a label. This statement extends to more general $p$-branes subjected to severe fuzzification as well as membranes. As a consequence of this generalisation, systems with $d$ Cartesian spatial coordinates and isospin degrees of freedom encoding an additional $\delta$ fuzzy coset space coordinates can realise topologically non-trivial states of intrinsic dimensionality up to $d$+$\delta$+1. We therefore identify gauge theories with extra fuzzy dimensions generalised to retain dependence upon gauge fields over fuzzy coset spaces even for severe fuzzification (small $N$), as EFTs for the QSkHE. We furthermore generalise these EFTs to space manifolds with local product structure exploiting the dimensional hierarchy of (fuzzy) spheres. For this purpose, we introduce methods of anisotropic fuzzification and propose formulating topological invariants on fuzzy coset spaces as artifacts of projecting matrix Lie algebras to occupied subspaces. Importantly, we focus on phenomenology indicating the 2+1 D SU(2) gauge theory should be generalised using this machinery, and serves as a minimal EFT of the QSkHE.

hep-th↗
arXiv2024

Signatures of the quantum skyrmion Hall effect in the Bernevig-Hughes-Zhang model

Given recent discovery of the quantum skyrmion Hall effect, we re-examine the related canonical Bernevig-Hughes-Zhang (BHZ) model for the quantum spin Hall insulator. Within the framework of the quantum skyrmion Hall effect, isospin degree(s) of freedom of the BHZ model encode additional spatial dimensions. Consistent with this framework, we observe phenomena similar to those of the four dimensional Chern insulator, revealed by weakly breaking time-reversal symmetry. Bulk-boundary correspondence of these states includes real-space boundary orbital angular momentum textures and gapless boundary modes that are robust against magnetic disorder, consistent with compactified three dimensional boundary Weyl nodes (WN$_F$s) of the quantum skyrmion Hall effect. These theoretical findings are furthermore consistent with past experimental work reporting unexpected edge conduction in HgTe quantum wells under applied Zeeman and orbital magnetic fields. This past work is therefore potentially the first known experimental observation of signatures of the quantum skyrmion Hall effect beyond the quantum Hall effect.

cond-mat.mes-hall↗
arXiv2024

Multiplicative Chern insulator

We study multiplicative Chern insulators (MCIs) as canonical examples of multiplicative topological phases of matter. Constructing the MCI Bloch Hamiltonian as a symmetry-protected tensor product of two topologically non-trivial parent Chern insulators (CIs), we study two-dimensional (2D) MCIs and introduce 3D mixed MCIs, constructed by requiring the two 2D parent Hamiltonians share only one momentum component. We study the 2D MCI response to time reversal symmetric flux insertion, observing a $4\pi$ Aharonov-Bohm effect, relating these topological states to fractional quantum Hall states via the effective field theory of the quantum skyrmion Hall effect. As part of this response, we observe evidence of quantisation of a proposed topological invariant for compactified many-body states, to a rational number, suggesting higher-dimensional topology may also be relevant. Finally, we study effects of bulk perturbations breaking the symmetry-protected tensor product structure of the child Hamiltonian, finding the MCI evolves adiabatically into a topological skyrmion phase.

cond-mat.mes-hall↗
arXiv2024

Finite-size topological phases from semimetals

Topological semimetals are some of the topological phases of matter most intensely-studied experimentally. The Weyl semimetal phase, in particular, has garned tremendous, sustained interest given fascinating signatures such as the Fermi arc surface states and the chiral anomaly, as well as the minimal requirements to protect this three-dimensional topological phase. Here, we show that thin films of Weyl semimetals (which we call quasi-(3-1)-dimensional, or q(3-1)d) generically realize finite-size topological phases distinct from 3d and 2d topological phases of established classification schemes: response signatures of the 3d bulk topology co-exist with topologically-protected, quasi-(3-2)d Fermi arc states or chiral boundary modes due to a second, previously-unidentified bulk-boundary correspondence. We show these finite-size topological semimetal phases are realized by Hamiltonians capturing the Fermiology of few-layer Van der Waals material MoTe2 in experiment. Given the broad experimental interest in few-layer Van der Waals materials and topological semimetals, our work paves the way for extensive future theoretical and experimental characterization of finite-size topological phases.

cond-mat.mes-hall↗
arXiv2023

Crystalline finite-size topology

Topological phases stabilized by crystalline point group symmetry protection are a large class of symmetry-protected topological phases subjected to considerable experimental scrutiny. Here, we show that the canonical three-dimensional (3D) crystalline topological insulator protected by time-reversal symmetry $\mathcal{T}$ and four-fold rotation symmetry $\mathcal{C}_4$ individually or the product symmetry $\mathcal{C}_4 \mathcal{T}$, generically realizes finite-size crystalline topological phases in thin film geometry (a quasi-(3-1)-dimensional, or q(3-1)D, geometry): response signatures of the 3D bulk topology co-exist with topologically-protected, quasi-(3-2)D and quasi-(3-3)D boundary modes within the energy gap resulting from strong hybridisation of the Dirac cone surface states of the underlying 3D crystalline topological phase. Importantly, we find qualitative distinctions between these gapless boundary modes and those of strictly 2D crystalline topological states with the same symmetry-protection, and develop a low-energy, analytical theory of the finite-size topological magnetoelectric response.

cond-mat.str-el↗
arXiv2023

Robust quantisation of circular photogalvanic effect in multiplicative topological semimetals

Nonlinear response signatures are increasingly recognized as useful probes of condensed matter systems, in particular for characterisation of topologically non-trivial states. The circular photogalvanic effect (CPGE) is particularly useful in study of topological semimetals, as the CPGE tensor quantises for well-isolated topological degeneracies in strictly linearly-dispersing band structures. Here, we study multiplicative Weyl semimetal band-structures, and find that the multiplicative structure robustly protects the quantization of the CPGE even in the case of non-linear dispersion. Computing phase diagrams as a function of Weyl node tilting, we find a variety of quantised values for the CPGE tensor, revealing that the CPGE is also a useful tool in detecting and characterising parent topology of multiplicative topological states.

cond-mat.mes-hall↗
arXiv2023

Topological quantum criticality from multiplicative topological phases

Symmetry-protected topological phases (SPTs) characterized by short-range entanglement include many states essential to understanding of topological condensed matter physics, and the extension to gapless SPTs provides essential understanding of their consequences. In this work, we identify a fundamental connection between gapless SPTs and recently-introduced multiplicative topological phases, demonstrating that multiplicative topological phases are an intuitive and general approach to realizing concrete models for gapless SPTs. In particular, they are naturally well-suited to realizing higher-dimensional, stable, and intrinsic gapless SPTs through combination of canonical topological insulator and semimetal models with critical gapless models in symmetry-protected tensor product constructions, opening avenues to far broader and deeper investigation of topology via short-range entanglement.

cond-mat.str-el↗
arXiv2023

Observation of the anomalous Hall effect in a layered polar semiconductor

Progress in magnetoelectric materials is hindered by apparently contradictory requirements for time-reversal symmetry broken and polar ferroelectric electronic structure in common ferromagnets and antiferromagnets. Alternative routes could be provided by recent discoveries of a time-reversal symmetry breaking anomalous Hall effect in noncollinear magnets and altermagnets, but hitherto reported bulk materials are not polar. Here, we report the observation of a spontaneous anomalous Hall effect in doped AgCrSe$_2$, a layered polar semiconductor with an antiferromagnetic coupling between Cr spins in adjacent layers. The anomalous Hall resistivity 3 $\mu\Omega$ cm is comparable to the largest observed in compensated magnetic systems to date, and is rapidly switched off when the angle of an applied magnetic field is rotated to $\sim 80^{\circ}$ from the crystalline $c$-axis. Our ionic gating experiments show that the anomalous Hall conductivity magnitude can be enhanced by modulating the $p$-type carrier density. We also present theoretical results that suggest the anomalous Hall effect is driven by Berry curvature due to noncollinear antiferromagnetic correlations among Cr spins, which are consistent with the previously suggested magnetic ordering in AgCrSe$_2$. Our results open the possibility to study the interplay of magnetic and ferroelectric-like responses in this fascinating class of materials.

cond-mat.mtrl-sci↗
arXiv2023

Time-reversal invariant topological skyrmion phases

Topological phases realized in time-reversal invariant (TRI) systems are foundational to experimental study of the broader canon of topological condensed matter as they do not require exotic magnetic orders for realization. We therefore introduce topological skyrmion phases of matter realized in TRI systems as a foundational step towards experimental realization of topological skyrmion phases. A novel bulk-boundary correspondence hidden from the ten-fold way classification scheme is revealed by the presence of a non-trivial value of a $\mathbb{Z}_2$ spin skyrmion invariant. This quantized topological invariant gives a finer description of the topology in 2D TRI systems as it indicates the presence or absence of robust helical edge states for open boundary conditions, in cases where the $\mathbb{Z}_2$ invariant computed with projectors onto occupied states takes a trivial value. Physically, we show this hidden bulk-boundary correspondence derives from additional spin-momentum-locking of the helical edge states associated with the topological skyrmion phase. ARPES techniques and transport measurements can detect these signatures of topological spin-momentum-locking and helical gapless modes. Our work therefore lays the foundation for experimental study of these phases of matter.

cond-mat.mes-hall↗
arXiv2023

Quantum skyrmion Hall effect

We consider the problem of magnetic charges in $(2+1)$ dimensions for a torus geometry in real-space, subjected to an inverted Lorentz force due to an external electric field applied normal to the surface of the torus. We compute the Hall conductivity associated with transport of these charges for the case of negligible gapless excitations and global $\mathrm{U}(1)$ charge conservation symmetry, and find it is proportional to an integer-valued topological invariant $\mathcal{Q}$, corresponding to a magnetic quantum Hall effect (MQHE). We identify a lattice model realizing this physics in the absence of an external electric field. Based on this, we identify a generalization of the MQHE to be quantized transport of magnetic skyrmions, the quantum skyrmion Hall effect (QSkHE), with a $\mathrm{U}(1)$ easy-plane anisotropy of magnetic skyrmions and effective conservation of charge associated with magnetic skyrmions yielding incompressibility, provided a hierarchy of energy scales is respected. As the lattice model may be characterized both by a total Chern number and the topological invariant $\mathcal{Q}$, we furthermore outline a possible field theory for electric charges, magnetic charges, and correlations between magnetic and electric charges approximated as composite particles, on a two-torus, to handle the scenario of intermediate-strength correlations between electric and magnetic charges modeled as composite particles. We map this problem to a generalized $(4+1)$D theory of the quantum Hall effect for the composite particles.

cond-mat.str-el↗
arXiv2023

Multiplicative Majorana zero-modes

Topological qubits composed of unpaired Majorana zero-modes are under intense experimental and theoretical scrutiny in efforts to realize practical quantum computation schemes. In this work, we show the minimum four \textit{unpaired} Majorana zero-modes required for a topological qubit according to braiding schemes and control of entanglement for gate operations are inherent to multiplicative topological phases, which realize symmetry-protected tensor products -- and maximally-entangled Bell states -- of unpaired Majorana zero-modes known as multiplicative Majorana zero-modes. We introduce multiplicative Majorana zero-modes as topologically-protected boundary states of both one and two-dimensional multiplicative topological phases, using methods reliant on multiplicative topology to construct relevant Hamiltonians from the Kitaev chain model. We furthermore characterize topology in the bulk and on the boundary with established methods while also introducing techniques to overcome challenges in characterizing multiplicative topology. In the process, we explore the potential of these multiplicative topological phases for an alternative to braiding-based topological quantum computation schemes, in which gate operations are performed through topological phase transitions.

quant-ph↗
arXiv2023

Multiplicative topological semimetals

Exhaustive study of topological semimetal phases of matter in equilibriated electonic systems and myriad extensions has built upon the foundations laid by earlier introduction and study of the Weyl semimetal, with broad applications in topologically-protected quantum computing, spintronics, and optical devices. We extend recent introduction of multiplicative topological phases to find previously-overlooked topological semimetal phases of electronic systems in equilibrium, with minimal symmetry-protection. We show these multiplicative topological semimetal phases exhibit rich and distinctive bulk-boundary correspondence and response signatures that greatly expand understanding of consequences of topology in condensed matter settings, such as the limits on Fermi arc connectivity and structure, and transport signatures such as the chiral anomaly. Our work therefore lays the foundation for extensive future study of multiplicative topological semimetal phases.

cond-mat.mes-hall↗
arXiv2023

Time-reversal invariant finite-size topology

We report finite-size topology in the quintessential time-reversal (TR) invariant systems, the quantum spin Hall insulator (QSHI) and the three-dimensional, strong topological insulator (STI): previously-identified helical or Dirac cone boundary states of these phases hybridize in wire or slab geometries with one open boundary condition for finite system size, and additional, topologically-protected, lower-dimensional boundary modes appear for open boundary conditions in two or more directions. For the quasi-one-dimensional (q(2-1)D) QSHI, we find topologically-protected, quasi-zero-dimensional (q(2-2)D) boundary states within the hybridization gap of the helical edge states, determined from q(2-1)D bulk topology characterized by topologically non-trivial Wilson loop spectra. We show this finite-size topology furthermore occurs in 1T'-WTe2 in ribbon geometries with sawtooth edges, based on analysis of a tight-binding model derived from density-functional theory calculations, motivating experimental investigation of our results. In addition, we find quasi-two-dimensional (q(3-1)D) finite-size topological phases occur for the STI, yielding helical boundary modes distinguished from those of the QSHI by a non-trivial magneto-electric polarizability linked to the original 3D bulk STI. Finite-size topological phases therefore exhibit signatures associated with the non-trivial topological invariant of a higher-dimensional bulk. Finally, we find the q(3-2)D STI also exhibits finite-size topological phases, finding the first signs of topologically-protected boundary modes of codimension greater than 1 due to finite-size topology. Finite-size topology of four or higher-dimensional systems is therefore possible in experimental settings without recourse to thermodynamically large synthetic dimensions.

cond-mat.mes-hall↗
arXiv2022

Finite-size Topology

We show that topological characterization and classification in $D$-dimensional systems, which are thermodynamically large in only $D-\delta$ dimensions and finite in size in $\delta$ dimensions, is fundamentally different from that of systems thermodynamically large in all $D$-dimensions: as $(D-\delta)$-dimensional topological boundary states permeate into a system's $D$ dimensional bulk with decreasing system size, they hybridize to create novel topological phases characterized by a set of $\delta+1$ topological invariants, ranging from the $D$-dimensional topological invariant to the $(D-\delta)$-dimensional topological invariant. The system exhibits topological response signatures and bulk-boundary correspondences governed by combinations of these topological invariants taking non-trivial values, with lower-dimensional topological invariants characterizing fragmentation of the underlying topological phase of the system thermodynamically large in all $D$-dimensions. We demonstrate this physics for the paradigmatic Chern insulator phase, but show its requirements for realization are satisfied by a much broader set of topological systems.

cond-mat.mes-hall↗
arXiv2022

Multiplicative topological phases

Symmetry-protected topological phases of matter have challenged our understanding of condensed matter systems and harbour exotic phenomena promising to address major technological challenges. Considerable understanding of these phases of matter has been gained recently by considering additional protecting symmetries, different types of quasiparticles, and systems out of equilibrium. Here, we show that symmetries could be enforced not just on full Hamiltonians, but also on their components. We construct a large class of previously unidentified multiplicative topological phases of matter characterized by tensor product Hilbert spaces similar to the Fock space of multiple particles. To demonstrate our methods, we introduce multiplicative topological phases of matter based on the foundational Hopf and Chern insulator phases, the multiplicative Hopf and Chern insulators (MHI and MCI), respectively. The MHI shows the distinctive properties of the parent phases as well as non-trivial topology of a child phase. We also comment on a similar structure in topological superconductors as these multiplicative phases are protected in part by particle-hole symmetry. The MCI phase realizes topologically-protected gapless states that do not extend from the valence bands to the conduction bands for open boundary conditions, which respect the symmetries protecting topological phase. The band connectivity discovered in the MCI could serve as a blueprint for potential multiplicative topology with exotic properties.

cond-mat.str-el↗