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arXiv:2307.09690 · Physical Review Letters

Fast Fourier-Chebyshev Approach to Real-Space Simulations of the Kubo Formula

Santiago Giménez de Castro, João M. Viana Parente Lopes, Aires Ferreira, Dario Andres Bahamon

Published Feb 16, 2024• 16 citations
arXivHTMLPDF

Abstract

The Kubo formula is a cornerstone in our understanding of near-equilibrium transport phenomena. While conceptually elegant, the application of Kubo's linear-response theory to interesting problems is hindered by the need for algorithms that are accurate and scalable to large lattice sizes beyond one spatial dimension. Here, we propose a general framework to numerically study large systems, which combines the spectral accuracy of Chebyshev expansions with the efficiency of divide-and-conquer methods. We use the hybrid algorithm to calculate the two-terminal conductance and the bulk conductivity tensor of 2D lattice models with over 10^{7} sites. By efficiently sampling the microscopic information contained in billions of Chebyshev moments, the algorithm is able to accurately resolve the linear-response properties of complex systems in the presence of quenched disorder. Our results lay the groundwork for future studies of transport phenomena in previously inaccessible regimes.

DOI
10.1103/physrevlett.132.076302

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The Kubo formula is a cornerstone in our understanding of near-equilibrium transport phenomena. While conceptually elegant, the application of Kubo's linear-response theory to interesting problems is hindered by the need for algorithms that are accurate and scalable to large lattice sizes beyond one spatial dimension. Here, we propose a general framework to numerically study large systems, which combines the spectral accuracy of Chebyshev expansions with the efficiency of divide-and-conquer methods. We use the hybrid algorithm to calculate the two-terminal conductance and the bulk conductivity tensor of 2D lattice models with over 10^{7} sites. By efficiently sampling the microscopic information contained in billions of Chebyshev moments, the algorithm is able to accurately resolve the linear-response properties of complex systems in the presence of quenched disorder. Our results lay the groundwork for future studies of transport phenomena in previously inaccessible regimes.
We describe an efficient numerical approach to calculate the longitudinal and transverse Kubo conductivities of large systems using Bastin's formulation. We expand the Green's functions in terms of Chebyshev polynomials and compute the conductivity tensor for any temperature and chemical potential in a single step. To illustrate the power and generality of the approach, we calculate the conductivity tensor for the quantum Hall effect in disordered graphene and analyze the effect of the disorder in a Chern insulator in Haldane's model on a honeycomb lattice.
This thesis develops a rigorous large-scale response theory for gapless lattice Fermi systems in low dimensions, focusing on the mathematical validity of linear response -- the Kubo formula -- for transport in gapless systems, where the absence of a spectral gap prevents the direct use of standard adiabatic theorems. After reviewing fundamental results on many-body lattice fermions, we first analyse the transport properties of non-interacting, gapless, one-dimensional quantum systems and of the edge modes of two-dimensional topological insulators. We prove the validity of the Kubo formula, in the zero-temperature and infinite-volume limit, for perturbations that are weak and slowly varying in space and time. The proof relies on expressing the real-time Duhamel series in imaginary time, which ensures its convergence uniformly in the scaling parameter and system size, at low temperatures. This representation also reveals a key cancellation in the scaling limit, linked to the emergent anomalous chiral symmetry of relativistic one-dimensional fermions. As a consequence, in the combined low-temperature and scaling limit, the linear response provides the full physical response. In particular, the method yields a dynamical proof of the quantisation of edge conductance in two-dimensional quantum Hall systems. The analysis is then extended to weakly interacting, gapless one-dimensional fermionic systems, for which we obtain explicit expressions for the response functions in terms of renormalised Fermi velocities. Here, sharp estimates for interacting Euclidean correlation functions are established via rigorous renormalisation-group methods. The asymptotic exactness of linear response persists due to a cancellation in the scaling limit of the correlations -- reminiscent of bosonisation -- derived rigorously from emergent chiral Ward identities.
Accurate calculations of the spectral density in a strongly correlated quantum many-body system are of fundamental importance to study its dynamics in the linear response regime. Typical examples are the calculation of inclusive and semiexclusive scattering cross sections in atomic nuclei and transport properties of nuclear and neutron star matter. Integral transform techniques play an important role in accessing the spectral density in a variety of nuclear systems. However, their accuracy is in practice limited by the need to perform a numerical inversion which is often ill-conditioned. In the present work we extend a recently proposed quantum algorithm which circumvents this problem. We show how to perform controllable reconstructions of the spectral density over a finite energy resolution with rigorous error estimates. An appropriate expansion in Chebyshev polynomials allows for efficient simulations also on classical computers. We apply our idea to obtain the local density of states for graphene in a magnetic field as a proof of principle. This paves the way for future applications in nuclear and condensed matter physics.
Non-Hermitian systems host unconventional topological phenomena, while real-space spectral calculations in three dimensions remain computationally challenging. Here, we develop a tensor-network framework that combines a quantics-based matrix-product-operator representation of large real-space tight-binding Hamiltonians with a non-Hermitian kernel polynomial method. The Hermitized spectral problem is retained within the compact tensor-network representation, enabling real-space spectral function calculations for systems exceeding one billion lattice sites. Using this approach, we investigate a three-dimensional non-Hermitian higher-order topological insulator and resolve its corner-mode spectral response on a billion-site lattice, demonstrating the scalability and computational reach of the method. By tuning the gain--loss strength, we identify distinct in-gap corner-mode features in the weak- and strong-gain--loss regimes. Our results establish a scalable tensor-network framework for real-space spectral calculations in exceptionally large non-Hermitian lattice systems.
A computational method, in which a system is mapped to the time-dependent Schr\"odinger equation driven by a periodic external force, is formulated for computing linear response functions of quantum systems. This method, which scales linearly with the system size N, includes computing the Kubo-Greenwood formula for the dynamic conductivities of systems described by large-scale Hamiltonian matrices. In addition, a scaling approach, derived from this algorithm, is presented to determine the exponent of the conductivity $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{\delta}}$ near the metal-insulator quantum transition with high speed and accuracy.
Computing excitonic spectra in quasicrystal and super-moiré systems constitutes a formidable challenge due to the exceptional size of the excitonic Hilbert space. Here, we demonstrate a tensor-network method for the real-space Bethe-Salpeter Hamiltonian, allowing us to access the spectra of an excitonic 101810^{18}1018-dimensional Hamiltonian, and enabling the direct computation of bound-exciton spectral functions for systems exceeding one billion lattice sites, several orders of magnitude beyond the capabilities of conventional approaches. Our method combines a tensor-network encoding of the real-space Bethe-Salpeter Hamiltonian with a Chebyshev tensor network algorithm. This strategy bypasses explicit storage of the Hamiltonian while preserving full real-space resolution across widely different length scales. We demonstrate our methodology for one- and two-dimensional super-moiré systems, achieving the simultaneous resolution of atomistic and mesoscopic structures in the excitonic spectra in billion-size systems, showing exciton miniband formation and moiré-induced spatial confinement. Our results establish a real-space methodology enabling the simulation of excitonic physics in large-scale quasicrystal and super-moiré quantum matter.