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arXiv:2510.07279 · Physical Review Letters

Tangent-Space Krylov Computation of Real-Frequency Spectral Functions: Influence of Density-Assisted Hopping on 2D Mott Physics

Oleksandra I. Kovalska, Jan von Delft, Andreas Gleis

Published Aug 19, 2026• 0 citations
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Abstract

We present a tangent-space Krylov method for efficient computation of zero-temperature real-frequency spectral functions on top of ground-state matrix product states obtained from the density matrix renormalization group. It relies on projecting the Hamiltonian to the tangent space of the ground-state matrix product states, where resolvents can be efficiently represented using Krylov space techniques. This enables direct computations of excitation energies and the corresponding spectral weights. We demonstrate the accuracy and efficiency of the tangent-space Krylov approach by showcasing spectral data for various models. These include the 1D Haldane-Shastry and Heisenberg models as benchmarks. As an interesting application, we study the Hubbard model on a cylinder at half filling, augmented by a density-assisted hopping (DAH) term. We find that DAH leads to particle-hole asymmetric single-particle mobilities and lifetimes in the resulting Mott insulator, and identify the responsible scattering processes. Furthermore, we find that DAH influences the dispersion of Green’s function zeros beyond its range, which has a frustrating effect on the Mott insulator studied here.

DOI
10.1103/6f9c-gb69

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We present a tangent-space Krylov method for efficient computation of zero-temperature real-frequency spectral functions on top of ground-state matrix product states obtained from the density matrix renormalization group. It relies on projecting the Hamiltonian to the tangent space of the ground-state matrix product states, where resolvents can be efficiently represented using Krylov space techniques. This enables direct computations of excitation energies and the corresponding spectral weights. We demonstrate the accuracy and efficiency of the tangent-space Krylov approach by showcasing spectral data for various models. These include the 1D Haldane-Shastry and Heisenberg models as benchmarks. As an interesting application, we study the Hubbard model on a cylinder at half filling, augmented by a density-assisted hopping (DAH) term. We find that DAH leads to particle-hole asymmetric single-particle mobilities and lifetimes in the resulting Mott insulator, and identify the responsible scattering processes. Furthermore, we find that DAH influences the dispersion of Green’s function zeros beyond its range, which has a frustrating effect on the Mott insulator studied here.
Frequency-dependent correlations, such as the spectral function and the dynamical structure factor, help illustrate condensed matter experiments. Within the density matrix renormalization group (DMRG) framework, an accurate method for calculating spectral functions directly in frequency is the correction-vector method. The correction vector can be computed by solving a linear equation or by minimizing a functional. This paper proposes an alternative to calculate the correction vector: to use the Krylov-space approach. This paper then studies the accuracy and performance of the Krylov-space approach, when applied to the Heisenberg, the t-J, and the Hubbard models. The cases studied indicate that the Krylov-space approach can be more accurate and efficient than the conjugate gradient, and that the error of the former integrates best when a Krylov-space decomposition is also used for ground state DMRG.
Accurate simulations of the two-dimensional (2D) Hubbard model constitute one of the most challenging problems in condensed matter and quantum physics. Here we develop a tangent space tensor renormalization group (tanTRG) approach for the calculations of the 2D Hubbard model at finite temperature. An optimal evolution of the density operator is achieved in tanTRG with a mild O(D^{3}) complexity, where the bond dimension D controls the accuracy. With the tanTRG approach we boost the low-temperature calculations of large-scale 2D Hubbard systems on up to a width-8 cylinder and 10×10 square lattice. For the half-filled Hubbard model, the obtained results are in excellent agreement with those of determinant quantum Monte Carlo (DQMC). Moreover, tanTRG can be used to explore the low-temperature, finite-doping regime inaccessible for DQMC. The calculated charge compressibility and Matsubara Green's function are found to reflect the strange metal and pseudogap behaviors, respectively. The superconductive pairing susceptibility is computed down to a low temperature of approximately 1/24 of the hopping energy, where we find d-wave pairing responses are most significant near the optimal doping. Equipped with the tangent-space technique, tanTRG constitutes a well-controlled, highly efficient and accurate tensor network method for strongly correlated 2D lattice models at finite temperature.
We investigate the application of the density matrix renormalization group (DMRG) to the Hubbard model in momentum space. We treat the one-dimensional models with dispersion relations corresponding to nearest-neighbor hopping and 1/r1/r1/r hopping and the two-dimensional model with isotropic nearest-neighbor hopping. By comparing with the exact solutions for both one-dimensional models and with exact diagonalization in two dimensions, we first investigate the convergence of the ground-state energy. We find variational convergence of the energy with the number of states kept for all models and parameter sets. In contrast to the real-space algorithm, the accuracy becomes rapidly worse with increasing interaction and is not significantly better at half filling. We compare the results for different dispersion relations at fixed interaction strength over bandwidth and find that extending the range of the hopping in one dimension has little effect, but that changing the dimensionality from one to two leads to lower accuracy at weak-to-moderate interaction strength. In the one-dimensional models at half filling, we also investigate the behavior of the single-particle gap, the dispersion of spinon excitations, and the momentum distribution function. For the single-particle gap, we find that proper extrapolation in the number of states kept is important. For the spinon dispersion, we find that good agreement with the exact forms can be achieved at weak coupling if the large momentum-dependent finite-size effects are taken into account for nearest-neighbor hopping. For the momentum distribution, we compare with various weak-coupling and strong-coupling approximations and discuss the importance of finite-size effects as well as the accuracy of the DMRG.
We propose a method to compute spectral functions of generic Hamiltonians using the density matrix renormalization group (DMRG) algorithm directly in the frequency domain, based on a modified Krylov-space decomposition to compute the correction vectors. Our approach entails the calculation of the root-NNN (N=2N=2N=2 is the standard square root) of the Hamiltonian propagator using Krylov-space decomposition and repeating this procedure NNN times to obtain the actual correction vector. We show that our method greatly alleviates the burden of keeping a large bond dimension at large target frequencies, a problem found with conventional correction-vector DMRG, whereas achieving better computational performance at large NNN. We apply our method to spin and charge spectral functions of $t\text{\ensuremath{-}}J$ and Hubbard models in the challenging two-leg ladder geometry and provide evidence that the root-NNN approach reaches a much improved spectral resolution compared to the conventional correction vector.
Frequency-dependent correlations, such as the spectral function and the dynamical structure factor, help understand condensed matter experiments. Within the DMRG framework, an accurate method for calculating spectral functions directly in frequency is the correction-vector method. The correction-vector can be computed by solving a linear equation or by minimizing a functional. This paper proposes an alternative to calculate the correction vector: to use the Krylov-space approach. This paper then studies the accuracy and performance of the Krylov-space approach, when applied to the Heisenberg, the t-J, and the Hubbard models. The cases studied indicate that Krylov-space approach can be more accurate and efficient than conjugate gradient, and that the error of the former integrates best when a Krylov-space decomposition is also used for ground state DMRG.
We introduce an alternative approach for obtaining smooth finite-temperature spectral functions of quantum impurity models using the numerical renormalization group (NRG) technique. It is based on calculating first the Green's function on the imaginary-frequency axis, followed by an analytic continuation to the real-frequency axis using Pad\'e approximants. The arbitrariness in choosing a suitable kernel in the conventional broadening approach is thereby removed and, furthermore, we find that the Pad\'e method is able to resolve fine details in spectral functions with less artifacts on the scale of $\ensuremath{\omega}\ensuremath{\sim}T$. We discuss the convergence properties with respect to the NRG calculation parameters (discretization $\ensuremath{\Lambda}$, zzz-averaging, truncation cutoff) and the number of Matsubara points taken into account in the analytic continuation. We test the technique on the single-impurity Anderson model and the Hubbard model (within the dynamical mean-field theory). For the Anderson impurity model, we discuss the shape of the Kondo resonance and its temperature dependence. For the Hubbard model, we discuss the inner structure of the Hubbard bands in metallic and insulating solutions at half-filling, as well as in the doped Mott insulator. Based on these test cases, we conclude that the Pad\'e approximant approach provides improved results for spectral functions at low-frequency scales of $\ensuremath{\omega}\ensuremath{\lesssim}T$ and that it is capable of resolving sharp spectral features also at high frequencies.