Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2312.02656 · Physical review. B./Physical review. B

Diffusion constants from the recursion method

Jiaozi Wang, Mats H. Lamann, Robin Steinigeweg, Jochen Gemmer B

Published Sep 6, 2024• 22 citations
arXivHTMLPDF

Abstract

Understanding the transport behavior of quantum many-body systems constitutes an important physical endeavor, both experimentally and theoretically. While a reliable classification into normal and anomalous dynamics is known to be notoriously difficult for a given microscopic model, even the seemingly simpler evaluation of transport coefficients in diffusive systems continues to be a hard task in practice. This fact has motivated the development and application of various sophisticated methods and is also the main issue of this paper. We particularly take a barely used strategy, which is based on the recursion method, and demonstrate that this strategy allows for accurate calculation of diffusion constants for different paradigmatic examples, including magnetization transport in nonintegrable spin-$1/2$ chains and ladders as well as energy transport in the mixed-field Ising model in one dimension.

DOI
10.1103/physrevb.110.104413

Related papers

Computational and Theoretical Insights into Multi-Body Quantum Systems

Andrew Stasiuk

In generality, perfect predictions of the structure and dynamics of multi-body quantum systems are few and far between. As experimental design advances and becomes more refined, experimentally probing the interactions of multiple quantum systems has become commonplace. Predicting this behavior is not a ``one size fits all" problem, and has lead to the inception of a multitude of successful theoretical techniques which have made precise and verifiable predictions through, in many cases, clever approximations and assumptions. As the state-of-the-art pushes the quantum frontier to new experimental regimes, many of the old techniques become invalid, and there is often no tractable methodology to fall back on. \n \nThis work focuses on expanding the theoretical techniques for making predictions in newly accessible experimental regimes. The transport of quantum information in a room-temperature dipolar spin network is veritably diffusive in nature, but much less is known about the transport properties of such a sample at low temperatures. This work presupposes that diffusion is still a good model for incoherent transport at low temperatures, and proposes a new method to calculate its diffusion coefficient. The diffusion coefficient is reported as a function of the temperature of the ensemble. Further, the interaction of an i.i.d. spin ensemble with a quantized electromagnetic field has long been analyzed via restriction to the Dicke subspace implicit in the Holstein--Primakoff approximation, as well as other within other approximations. This work reanalyzes the conditions under which such a restriction is valid. In regimes where it is shownt that restricting to the Dicke subspace would be invalid, the Hamiltonian structure is thoroughly analyzed. Various predictions can be made by appealing to a reduction in effective dimensionality via a direct sum decomposition. \n \nThe main theme of the techniques utilized throughout this work is to appeal to a reduction in difficulty via various theoretical tools in order to prepare for an otherwise intractable computational analysis. Computational insights due to this technique have then gone on to motivate directly provable theoretical results, which might otherwise have remained hidden behind the complexity of the structure and dynamics of a multi-body quantum system.

•Mar 31, 2021•0 citations

Density dynamics in translationally invariant spin- 12 chains at high temperatures: A current-autocorrelation approach to finite time and length scales

Robin Steinigeweg, Jochen Gemmer B

We investigate transport in several translationally invariant spin-$\frac{1}{2}$ chains in the limit of high temperatures. We concretely consider spin transport in the anisotropic Heisenberg chain, the pure Heisenberg chain within an alternating field, and energy transport in an Ising chain which is exposed to a tilted field. Our approach is essentially based on a connection between the evolution of the variance of an inhomogeneous nonequilibrium density and the current-autocorrelation function at finite times. Although this relationship is not restricted to the case of diffusive transport, it allows to extract a quantitative value for the diffusion constant in that case. By means of numerically exact diagonalization we indeed observe diffusive behavior in the considered spin chains for a range of model parameters and confirm the diffusion coefficients which were obtained for these systems from nonequilibrium bath scenarios.

arXiv:0906.4457•Physical Review B•Nov 2, 2009•68 citations

Burnett coefficients in quantum many-body systems

Robin Steinigeweg, Tomaž Prosen

The Burnett coefficient B is investigated for transport in one-dimensional quantum many-body systems. Extensive numerical computations in spin-1/2 chains suggest a linear growth with time, B(t)~t, for nonintegrable chains exhibiting diffusive transport. For integrable spin chains in the metallic regime, on the other hand, we find a cubic growth with time, B(t)~-D(m)(2)t(3), with the proportionality constant being simply a square of the Drude weight D(m). The results are corroborated with additional studies in noninteracting quantum chains and in the classical limit of large-spin chains.

arXiv:1302.4879•Physical Review E•May 30, 2013•5 citations

Scaling of diffusion constants in perturbed easy-axis Heisenberg spin chains

Markus Kraft, Mariel Kempa, Jiaozi Wang +2 more

Understanding the physics of the integrable spin-1/2 XXZ chain has witnessed substantial progress, due to the development and application of sophisticated analytical and numerical techniques. In particular, infinite-temperature magnetization transport has turned out to range from ballistic, over superdiffusive, to diffusive behavior in different parameter regimes of the anisotropy. Since integrability is rather the exception than the rule, a crucial question is the change of transport under integrability-breaking perturbations. This question includes the stability of superdiffusion at the isotropic point and the change of diffusion constants in the easy-axis regime. In our work, we study this change of diffusion constants by a variety of methods and cover both, linear response theory in the closed system and the Lindblad equation in the open system, where we throughout focus on periodic boundary conditions. In the closed system, we compare results from the recursion method to calculations for finite systems and find evidence for a continuous change of diffusion constants over the full range of perturbation strengths. In the open system weakly coupled to baths, we find diffusion constants in quantitative agreement with the ones in the closed system in a range of nonweak perturbations, but disagreement in the limit of weak perturbations. Using a simple model in this limit, we point out the possibility of a diverging diffusion constant in such an open system.

arXiv:2410.22586•Physical review. B./Physical review. B•Jul 23, 2025•2 citations

Boltzmann-equation approach to transport in finite modular quantum systems

Mehmet Kadiroḡlu, Jochen Gemmer B

We investigate the transport behavior of finite modular quantum systems. Such systems have recently been analyzed by different methods. These approaches indicate diffusive behavior even, and especially, for finite systems. Inspired by these results we analytically and numerically analyze if and in which sense the dynamics of those systems are in agreement with an appropriate Boltzmann equation. We find that the transport behavior of a certain type of finite modular quantum systems may indeed be described in terms of a Boltzmann equation. However, the applicability of the Boltzmann equation appears to be rather limited to a very specific type of model.

arXiv:0706.3607•Physical Review B•Jul 13, 2007•8 citations

Fourier's law and many-body quantum systems

Christian B. Mendl

The topic of this article are transport properties of many-body quantum chains accompanying DMRG-type simulations of the Bose–Hubbard model. To set the stage, we first provide a very brief introduction to many-body quantum theory and tensor network approximations. Transport properties are studied via dynamical density correlation functions, in line with linear response theory. We observe diffusive behavior at “infinite temperature” T → ∞ . Finally, we mention other approaches to study transport, e.g., explicitly imposing a temperature gradient via thermal reservoirs at the boundary.

Comptes Rendus Physique•Jul 1, 2019•1 citation
Understanding the transport behavior of quantum many-body systems constitutes an important physical endeavor, both experimentally and theoretically. While a reliable classification into normal and anomalous dynamics is known to be notoriously difficult for a given microscopic model, even the seemingly simpler evaluation of transport coefficients in diffusive systems continues to be a hard task in practice. This fact has motivated the development and application of various sophisticated methods and is also the main issue of this paper. We particularly take a barely used strategy, which is based on the recursion method, and demonstrate that this strategy allows for accurate calculation of diffusion constants for different paradigmatic examples, including magnetization transport in nonintegrable spin-1/21/21/2 chains and ladders as well as energy transport in the mixed-field Ising model in one dimension.
In generality, perfect predictions of the structure and dynamics of multi-body quantum systems are few and far between. As experimental design advances and becomes more refined, experimentally probing the interactions of multiple quantum systems has become commonplace. Predicting this behavior is not a ``one size fits all" problem, and has lead to the inception of a multitude of successful theoretical techniques which have made precise and verifiable predictions through, in many cases, clever approximations and assumptions. As the state-of-the-art pushes the quantum frontier to new experimental regimes, many of the old techniques become invalid, and there is often no tractable methodology to fall back on. \n \nThis work focuses on expanding the theoretical techniques for making predictions in newly accessible experimental regimes. The transport of quantum information in a room-temperature dipolar spin network is veritably diffusive in nature, but much less is known about the transport properties of such a sample at low temperatures. This work presupposes that diffusion is still a good model for incoherent transport at low temperatures, and proposes a new method to calculate its diffusion coefficient. The diffusion coefficient is reported as a function of the temperature of the ensemble. Further, the interaction of an i.i.d. spin ensemble with a quantized electromagnetic field has long been analyzed via restriction to the Dicke subspace implicit in the Holstein--Primakoff approximation, as well as other within other approximations. This work reanalyzes the conditions under which such a restriction is valid. In regimes where it is shownt that restricting to the Dicke subspace would be invalid, the Hamiltonian structure is thoroughly analyzed. Various predictions can be made by appealing to a reduction in effective dimensionality via a direct sum decomposition. \n \nThe main theme of the techniques utilized throughout this work is to appeal to a reduction in difficulty via various theoretical tools in order to prepare for an otherwise intractable computational analysis. Computational insights due to this technique have then gone on to motivate directly provable theoretical results, which might otherwise have remained hidden behind the complexity of the structure and dynamics of a multi-body quantum system.
We investigate transport in several translationally invariant spin-12\frac{1}{2}21​ chains in the limit of high temperatures. We concretely consider spin transport in the anisotropic Heisenberg chain, the pure Heisenberg chain within an alternating field, and energy transport in an Ising chain which is exposed to a tilted field. Our approach is essentially based on a connection between the evolution of the variance of an inhomogeneous nonequilibrium density and the current-autocorrelation function at finite times. Although this relationship is not restricted to the case of diffusive transport, it allows to extract a quantitative value for the diffusion constant in that case. By means of numerically exact diagonalization we indeed observe diffusive behavior in the considered spin chains for a range of model parameters and confirm the diffusion coefficients which were obtained for these systems from nonequilibrium bath scenarios.
The Burnett coefficient B is investigated for transport in one-dimensional quantum many-body systems. Extensive numerical computations in spin-1/2 chains suggest a linear growth with time, B(t)~t, for nonintegrable chains exhibiting diffusive transport. For integrable spin chains in the metallic regime, on the other hand, we find a cubic growth with time, B(t)~-D(m)(2)t(3), with the proportionality constant being simply a square of the Drude weight D(m). The results are corroborated with additional studies in noninteracting quantum chains and in the classical limit of large-spin chains.
Understanding the physics of the integrable spin-1/2 XXZ chain has witnessed substantial progress, due to the development and application of sophisticated analytical and numerical techniques. In particular, infinite-temperature magnetization transport has turned out to range from ballistic, over superdiffusive, to diffusive behavior in different parameter regimes of the anisotropy. Since integrability is rather the exception than the rule, a crucial question is the change of transport under integrability-breaking perturbations. This question includes the stability of superdiffusion at the isotropic point and the change of diffusion constants in the easy-axis regime. In our work, we study this change of diffusion constants by a variety of methods and cover both, linear response theory in the closed system and the Lindblad equation in the open system, where we throughout focus on periodic boundary conditions. In the closed system, we compare results from the recursion method to calculations for finite systems and find evidence for a continuous change of diffusion constants over the full range of perturbation strengths. In the open system weakly coupled to baths, we find diffusion constants in quantitative agreement with the ones in the closed system in a range of nonweak perturbations, but disagreement in the limit of weak perturbations. Using a simple model in this limit, we point out the possibility of a diverging diffusion constant in such an open system.
We investigate the transport behavior of finite modular quantum systems. Such systems have recently been analyzed by different methods. These approaches indicate diffusive behavior even, and especially, for finite systems. Inspired by these results we analytically and numerically analyze if and in which sense the dynamics of those systems are in agreement with an appropriate Boltzmann equation. We find that the transport behavior of a certain type of finite modular quantum systems may indeed be described in terms of a Boltzmann equation. However, the applicability of the Boltzmann equation appears to be rather limited to a very specific type of model.
The topic of this article are transport properties of many-body quantum chains accompanying DMRG-type simulations of the Bose–Hubbard model. To set the stage, we first provide a very brief introduction to many-body quantum theory and tensor network approximations. Transport properties are studied via dynamical density correlation functions, in line with linear response theory. We observe diffusive behavior at “infinite temperature” T → ∞ . Finally, we mention other approaches to study transport, e.g., explicitly imposing a temperature gradient via thermal reservoirs at the boundary.