arXiv subjects

F. D. M. Haldane

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

At least 19 recordsLinked to original sources

arXiv2026

Fermi Surface Geometry from Charge Fluctuations in Three-Dimensional Metals

For three-dimensional non-interacting multi-band metals, we show that important information about the shape and the quantum geometry of Fermi surfaces is encoded in the subleading logarithmic term of bipartite charge fluctuations. This logarithmic term is related to the dimensionless $|\mathbf{q}|^3$-coefficient of the structure factor in momentum space, and both quantities can be expressed as Fermi surface integrals of the Fermi surface curvature tensor and the quantum metric tensor. When the real-space partition surface is a quadric (i.e., sphere or ellipsoid), the logarithmic coefficient satisfies a topological bound depending only on the Euler characteristic and the Chern number of the Fermi surface, illustrating a non-trivial interplay between topology and quantum topology in multi-band metals.

cond-mat.mes-hall↗

arXiv2025

Global Phase Diagram of D-wave Superconductivity in the Square-Lattice $t\text{-}J$ Model

The Hubbard and closely related $t\text-J$ models are exciting platforms for unconventional superconductivity (SC). Through state-of-the-art density matrix renormalization group calculations using the grand canonical ensemble, we address open issues regarding the ground-state phase diagram of the extended $t\text{-}J$ model on a square lattice in the parameter regime relevant to cuprate superconductors. On large 8-leg cylinders, we demonstrate that the pure $t\text{-}J$ model with only nearest-neighbor hoppings and superexchange interactions, for a wide range of doping ($δ=0.1-0.2$), hosts robust d-wave superconductivity possibly coexisting with weak unidirectional pair density wave. Furthermore, a small next nearest neighbor hopping $t_2$ suppresses pair and charge density waves, resulting in a uniform d-wave SC phase in both electron- and hole-doped cuprate model systems. Our work validates the $t\text-J$ model as a proper minimum model for the emergence of superconductivity in cuprate superconductors.

cond-mat.supr-con↗

arXiv2024

A numerical study of bounds in the correlations of fractional quantum Hall states

We numerically compute the guiding center static structure factor $\bar S(\bf k)$ of various fractional quantum Hall (FQH) states to $\mathcal{O}\left((k\ell)^6\right)$ where $k$ is the wavenumber and $\ell$ is the magnetic length. Employing density matrix renormalization group on an infinite cylinder of circumference $L_y$, we study the two-dimensional limit using $L_y/ξ\gg 1$, where $ξ$ is the correlation length. The main findings of our work are: 1) the ground states that deviate away from the ideal conformal block wavefunctions, do not saturate the Haldane bound, and 2) the coefficient of $O\left((k\ell)^6\right)$ term appears to be bounded above by a value predicted by field theories proposed in the literature. The first finding implies that the graviton mode is not maximally chiral for experimentally relevant FQH states.

cond-mat.str-el↗

arXiv2023

Incompressible Quantum Hall fluids as Electric Quadrupole fluids

A new picture of both integer and fractional incompressible quantum Hall fluids as fluids carrying a electric quadrupole is introduced. This clarifies their geometric properties, provides a generic expression for Hall viscosity, and allows removal of the ingredient of continuous rotational symmetry (absent when electrons move in a crystalline background) from their description.

cond-mat.str-el↗

arXiv2023

Two-dimensional inversion symmetry as the fundamental symmetry of incompressible quantum Hall fluids

Two dimensional inversion symmetry ($180^{\circ}$ rotations in the ``Hall plane'' that hosts the incompressible electron fluid that exhibits the quantized Hall effect) is identified as its fundamental unbroken symmetry. A consequence is that the integers $p$ and $q$ which define both the Landau level filling factor $ν$ = $p/q$ and the elementary fractional charge $\pm e/q$ of topological excitations, cannot have a common divisor greater than 2.

cond-mat.mes-hall↗

arXiv2022

Neutral Excitations of Quantum Hall States: a Density Matrix Renormalization Group Study

We use the dynamical structure factors of the quantum Hall states at $ν=1/3$ and $ν=1/2$ in the lowest Landau level to study their excitation spectrum. Using the density matrix renormalization group in combination with the time-dependent variational principle on an infinite cylinder geometry, we extract the low energy properties. At $ν=1/3$, a sharp magnetoroton mode and the two-roton continuum are present and the finite-size effects can be understood using the fractional charge of the quasi-particle. At $ν=1/2$, we find low energy modes with linear dispersion and the static structure factor $\bar s(q) \sim (q\ell)^3$ in the limit $q\ell \rightarrow 0$. The properties of these modes agree quantitatively with the predictions of the composite-fermion theory placed on the infinite cylinder.

cond-mat.str-el↗

arXiv2021

Multiple Magnetorotons and Spectral Sum Rules in Fractional Quantum Hall Systems

We study numerically the the charge neutral excitations (magnetorotons) in fractional quantum Hall systems, concentrating on the two Jain states near quarter filling, $ν=2/7$ and $ν=2/9$, and the $ν=1/4$ Fermi-liquid state itself. In contrast to the $ν=1/3$ states and the Jain states near half filling, on each of the two Jain states $ν=2/7$ and $ν=2/9$ the graviton spectral densities show two, instead of one, magnetoroton peaks. The magnetorotons have spin 2 and have opposite chiralities in the $ν=2/7$ state and the same chirality in the $ν=2/9$ state. We also provide a numerical verification of a sum rule relating the guiding center spin $\bar s$ with the spectral densities of the stress tensor

cond-mat.str-el↗

arXiv2021

Gauge-invariant perturbation expansion in powers of electric charge for the density-of-states of a network model for charged-particle motion in a uniform background magnetic flux density

An explicitly-gauge-invariant expansion in powers of $e/\hbar$ times the magnetic flux density is formally obtained for the density of states (as characterized by the trace of the resolvent $\widehat G$ = $(ω- \hat h)^{-1}$) of a charged particle moving on a Hermitian quantum network that is embedded in a Euclidean background that supports a uniform magnetic flux density. The explicit expressions, given here up to third order in the flux density, are also valid for the ``local trace'' (the trace of $\widehat P_i \widehat G$, where $\widehat P_i$ is the projector on a network node), and do not appear to have been previously given.

cond-mat.other↗

arXiv2021

Energetics of the PH-Pfaffian state and the 5/2-fractional quantum Hall effect

We present a method for the exact construction of the fully particle-hole symmetric PH-Pfaffian ground state and its charged excitations on a sphere. We adopt the Moore-Read state, but with a nonholomorphic pairing component as in previous studies, and project it to the lowest Landau level. We study the energetics as well as other properties of these states and find that in a pure system interacting with the Coulomb forces the PH-Pfaffian cannot compete with either the Moore-Read state or its particle-hole conjugate, the anti-Pfaffian state, as an explanation for the 5/2-effect.

cond-mat.mes-hall↗

arXiv2021

Graviton Chirality and Topological Order in the Half-filled Landau Level

The fractional quantum Hall state at Landau level (LL) filling factor 5/2 is extremely interesting because it is likely the first non-Abelian state, but its precise nature remains unclear after decades of study. We demonstrate this can be resolved by studying the chirality of its graviton excitations, using circularly polarized Raman scattering. We discuss the advantage of this bulk probe over the existing edge probes.

cond-mat.mes-hall↗

arXiv2020

Microscopic Diagnosis of Universal Geometric Responses in Fractional Quantum Hall Liquids

Topological quantum liquids contain internal degrees of freedom that are coupled to geometric response. Yet, an explicit and microscopic identification of geometric response remains difficult. Here, taking notable fractional quantum Hall (FQH) states as typical examples, we systematically investigate a promising protocol -- the Dehn twist deformation on the torus geometry, to probe the geometric response of correlated topological states and establish the relation between such response and the universal properties of pertinent states. Based on analytical derivations and numerical simulations, we find that the geometry-induced Berry phase encodes novel features for a broad class of FQH states at the Laughlin, hierarchy, Halperin and non-Abelian Moore-Read fillings. Our findings conclusively demonstrate that the adiabatic Dehn twist deformation can faithfully capture the geometry of elementary FQH droplets and intrinsic modular information including topological spin and chiral central charge. Our approach provides a powerful way to reveal topological orders of generic FQH states and allows us to address previously open questions.

cond-mat.str-el↗

arXiv2019

Non-Abelian Statistics in one dimension: topological momentum spacings and SU(2) level $k$ fusion rules

We use a family of critical spin chain models discovered recently by one of us [M. Greiter, Mapping of Parent Hamiltonians, Springer, Berlin/Heidelberg 2011] to propose and elaborate that non-Abelian, SU(2) level $k=2S$ anyon statistics manifests itself in one dimension through topological selection rules for fractional shifts in the spacings of linear momenta, which yield an internal Hilbert space of, in the thermodynamic limit degenerate states. These shifts constitute the equivalent to the fractional shifts in the relative angular momenta of anyons in two dimensions. We derive the rules first for Ising anyons, and then generalize them to SU(2) level $k$ anyons. We establish a one-to-one correspondence between the topological choices for the momentum spacings and the fusion rules of spin \half spinons in the SU(2) level $k$ Wess--Zumino--Witten model, where the internal Hilbert space is spanned by the manifold of allowed fusion trees in the Bratelli diagrams. Finally, we show that the choices in the fusion trees may be interpreted as the choices between different domain walls between the $2S+1$ possible, degenerate dimer configurations of the spin $S$ chains at the multicritical point.

cond-mat.str-el↗

arXiv2019

Chiral Gravitons in Fractional Quantum Hall Liquids

We elucidate the nature of neutral collective excitations of fractional quantum Hall liquids in the long-wavelength limit. We demonstrate that they are chiral gravitons carrying angular momentum - 2, which are quanta of quantum motion of an internal metric, and show up as resonance peaks in the systems response to what is the fractional Hall analog of gravitational waves. Relation with existing and possible future experimental work that can detect these fractional quantum Hall gravitons and reveal their chirality are discussed.

cond-mat.str-el↗

arXiv2018

Fermi-surface geometry and "Planckian dissipation"

A purely Fermi-surface formula is proposed for the Ohmic "minimum metallic conductivity" tensor of clean metals with "Planckian limit" inelastic dissipation. This revises a recent proposal by Legros et al.

cond-mat.str-el↗

arXiv2018

Pomeranchuk Instability of Composite Fermi Liquids

Nematicity in quantum Hall systems has been experimentally well established at excited Landau levels. The mechanism of the symmetry breaking, however, is still unknown. Pomeranchuk instability of Fermi liquid parameter $F_{\ell} \le -1$ in the angular momentum $\ell=2$ channel has been argued to be the relevant mechanism, yet there are no definitive theoretical proofs. Here we calculate, using the variational Monte Carlo technique, Fermi liquid parameters $F_\ell$ of the composite fermion Fermi liquid with a finite layer width. We consider $F_{\ell}$ in different Landau levels $n=0,1,2$ as a function of layer width parameter $η$. We find that unlike the lowest Landau level, which shows no sign of Pomeranchuk instability, higher Landau levels show nematic instability below critical values of $η$. Furthermore, the critical value $η_c$ is higher for the $n=2$ Landau level, which is consistent with observation of nematic order in ambient conditions only in the $n=2$ Landau levels. The picture emerging from our work is that approaching the true 2D limit brings half-filled higher Landau-level systems to the brink of nematic Pomeranchuk instability.

cond-mat.str-el↗

arXiv2018

The origin of holomorphic states in Landau levels from non-commutative geometry, and a new formula for their overlaps on the torus

Holomorphic functions that characterize states in a two-dimensional Landau level been central to key developments such as the Laughlin state. Their origin has historically been attributed to a special property of "Schrödinger wavefunctions" of states in the "lowest Landau level". It is shown here that they instead arise in any Landau level as a generic mathematical property of the Heisenberg description of the non-commutative geometry of guiding centers. When quasiperiodic boundary conditions are applied to compactify the system on a torus, a new formula for the overlap between holomorphic states, in the form of a discrete sum rather than an integral, is obtained. The new formula is unexpected from the previous "lowest-Landau level Schrödinger wavefunction" interpretation.

cond-mat.str-el↗

arXiv2018

A modular-invariant modified Weierstrass sigma-function as a building block for lowest-Landau-level wavefunctions on the torus

A "modified" variant of the Weierstrass sigma, zeta, and elliptic functions is proposed whereby the zeta function is redefined by $ζ(z)$ $\mapsto$ $\tilde ζ(z)$ $\equiv$ $ζ(z) - γ_2z$, where $γ_2$ is a lattice invariant related to the almost-holomorphic modular invariant of the quasi-modular-invariant weight-2 Eisenstein series. If $ω_i$ is a primitive half-period, $\tildeζ(ω_i)$ = $πω_i^*/A$, where $A$ is the area of the primitive cell of the lattice. The quasiperiodicity of the modified sigma function is much simpler than that of the original, and it becomes the building block for the modular-invariant formulation of lowest-Landau-level wavefunctions on the torus. It is suggested that the "modified" sigma function is more natural than the original Weierstrass form, which was formulated before quasi-modular forms were understood. For the high-symmetry (square and hexagonal) lattices, the modified and original sigma functions coincide.

math-ph↗

arXiv2018

Geometry of flux attachment in anisotropic fractional quantum Hall states

Fractional quantum Hall (FQH) states are known to possess an internal metric degree of freedom that allows them to minimize their energy when contrasting geometries are present in the problem (e.g., electron band mass and dielectric tensor). We investigate the internal metric of several incompressible FQH states by probing its response to band mass anisotropy using infinite DMRG simulations on a cylinder geometry. We test and apply a method to extract the internal metric of a FQH state from its guiding center structure factor. We find that the response to band mass anisotropy is approximately the same for states in the same Jain sequence, but changes substantially between different sequences. We provide a theoretical explanation of the observed behavior of primary states at filling $ν= 1/m$ in terms of a minimal microscopic model of flux attachment.

cond-mat.str-el↗