arXiv subjects

Jan von Delft

Publications and source records attributed to Jan von Delft.

At least 19 recordsLinked to original sources

arXiv2026

Parallelized contraction of tensor trains or matrix product operators

Tensor Trains (TT), also known as Matrix Product States (MPS) and Matrix Product Operators (MPO), provide a compact and structured representation for high-dimensional data and operators. One of the most expensive manipulations involving tensor trains is the contraction of two MPOs. A popular and accurate method for mitigating this cost is the fit algorithm. However, it is still comparatively costly since it involves 2-site updates. Moreover, the parallelization of the fit algorithm when used for MPO-MPO contractions has received comparatively little attention. In this work, we present two strategies for accelerating the fit algorithm, usable in combination: (1) We use MPI-based distributed-memory parallelization tailored for MPO-MPO contractions, employing one of two MPO gauge choices: (1a) the inverse canonical gauge, which yields near-ideal parallelization speedup across all problem sizes; and (1b) the site-canonical gauge, which avoids the inversion of singular values but requires extra computations to ensure global consistency, thus yielding excellent parallelization speedup only for large problems requiring several sweeps before convergence. (2) We use randomized projections to reduce the cost of local updates from 2-site to 1-site costs while retaining 2-site accuracy, and to speed up contractions of environment tensors with MPO tensors.

physics.comp-ph↗

arXiv2026

Quantum criticality in the two-dimensional Hubbard model

We study the normal-state, doping-driven phase diagram of the square-lattice Hubbard model using the dynamical cluster approximation combined with the numerical renormalization group as a cluster solver, which gives direct access to real-frequency dynamics at essentially zero temperature. In a parameter regime relevant for cuprates, $U=7t$ and $t'=-0.3t$, we find a critical doping $p^{\ast}$ that marks a continuous quantum phase transition between a pseudogap metal and a normal Fermi liquid. The transition is identified by a continuous collapse, from both sides, of the Fermi-liquid scale extracted from charge, spin, and $d_{x^2-y^2}$-wave pairing susceptibilities. This collapse produces a non-Fermi-liquid regime at intermediate energy scales, which appears to extend to arbitrarily low scales at $p^{\ast}$. As $p^{\ast}$ is crossed from the normal Fermi liquid at $p>p^{\ast}$ into the pseudogap metal at $p<p^{\ast}$, the coherent low-energy spectral weight in the antinodal region is lost and replaced by a narrow, metallic pseudogap, while the nodal region evolves smoothly and remains comparatively coherent. This gives rise to Fermi arcs in the pseudogap metal at $p<p^{\ast}$, since the zero-frequency spectral weight remains large in the nodal region but is strongly suppressed in the antinodal region.

cond-mat.str-el↗

arXiv2026

Dynamical scaling near the pseudogap quantum critical point of the two-dimensional Hubbard model

We study dynamical scaling in the quantum-critical fan of the pseudogap-metal to Fermi-liquid transition of the two-dimensional Hubbard model. Using a four-patch dynamical cluster approximation with the numerical renormalization group as a cluster impurity solver, we access real-frequency dynamics over several decades at arbitrary temperatures. Close to the critical doping, the local spin and cluster-current susceptibility spectra exhibit $x=\omega/T$ scaling of the form $\chi''(\omega,T)\sim \tanh(x/2)$, and the cluster contribution to the optical conductivity obeys $T\sigma'_{\mathrm{cl}}(\omega,T) \sim \tanh(x/2)/x$, implying a $1/T$ cluster dc conductivity. In the scaling regime, the vertex contribution to the cluster optical response is much larger than the bubble contribution. We further find evidence for a marginal-Fermi-liquid nodal self-energy. This, together with the $1/T$ vertex contribution to the conductivity, implies strange-metal optical transport in the quantum critical region. Our results describe several qualitative aspects of several experimental observations.

cond-mat.str-el↗

arXiv2026

Symmetric estimator for discrete self-energy of discrete many-body systems

We derive a discrete spectral representation of the single-particle self-energy using a discrete evaluation of Kugler's symmetric improved estimator. Our construction can be used on both the real and the complex (Matsubara) frequency axis. It is guaranteed to remain causal at the numerical level, in contrast to standard approaches that may generate unphysical negative spectral weight or require additional broadening. Our representation can be used for any Hamiltonian; here we apply it to quantum impurity models and in dynamical mean-field theory. The latter is formulated with a discrete hybridization function throughout its self-consistency loop. In both cases and across various numerical methods, we obtain significantly improved accuracy for a range of impurity properties.

cond-mat.str-el↗

arXiv2026

Revisiting the $J_1$-$J_2$ Heisenberg Model on a Triangular Lattice: Quasi-Degenerate Ground States and Phase Competition

It is generally believed that the spin-$\tfrac{1}{2}$ triangular-lattice $J_1$-$J_2$ Heisenberg model hosts a quantum spin liquid in the intermediate regime between the $120^\circ$ and stripe ordered phases. Density matrix renormalization group studies on cylinders have consistently found two nearly degenerate ground states, commonly interpreted as distinct topological sectors. Using state-of-the-art matrix product state simulations on YC6 cylinders, we compare the static and dynamical properties of these two sectors at $J_2/J_1 = 0.125$. Noticeable differences appear already in static correlations; moreover, high-resolution dynamical structure factors reveal qualitatively distinct low-energy excitations. These results suggest that the two ground states cannot be understood as merely topologically distinct sectors of a gapped $\mathbb{Z}_2$ spin liquid.

cond-mat.str-el↗

arXiv2026

Adaptive Patching for Tensor Train Computations

Quantics Tensor Train (QTT) operations such as matrix product operator contractions are prohibitively expensive for large bond dimensions. We propose an adaptive patching scheme that exploits block-sparse QTT structures to reduce costs through divide-and-conquer, adaptively partitioning tensors into smaller patches with reduced bond dimensions. We demonstrate substantial improvements for sharply localized functions and show efficient computation of bubble diagrams and Bethe-Salpeter equations, opening the door to practical large-scale QTT-based computations previously beyond reach.

physics.comp-ph↗

arXiv2026

Liouvillian interpolation of the self-energy of cluster dynamical mean-field theories

Two widely-used non-local extensions of dynamical mean field theory (DMFT), cellular DMFT (CDMFT) and the dynamical cluster approximation (DCA), both yield self-energies marred by having some unphysical properties: CDMFT yields real-space self-energies that are not translationally invariant, and DCA yields momentum-space self-energies with discontinuities in their momentum dependence. It is often desirable to remove these flaws by post-processing cluster DMFT results, using strategies called periodization for CDMFT and interpolation for DCA -- for brevity, we refer to both cases as interpolation. However, traditional interpolation approaches struggle to capture intricate structures such as hole pockets in the hole-doped square-lattice Hubbard model, as highlighted in Phys. Rev. B 105, 35117 (2022). Further, these approaches interpolate frequency-dependent functions, which may lead to causality violations. Here, we propose Liouvillian interpolation, a novel, intuitive, and robust scheme for interpolating cluster DMFT results. Our key idea is to interpolate frequency-independent matrix elements of the single-particle irreducible part of the Liouvillian, obtained from a continued-fraction expansion of the cDMFT self-energy. We demonstrate that the ingredients of such an expansion possess a more local Fourier expansion than the functions involved in traditional interpolation schemes, and that Liouvillian interpolation inherently conserves causality. We illustrate our method for the one-dimensional Hubbard model using CDMFT, and for the two-dimensional Hubbard model using four-patch DCA. For the latter, we find that L-interpolation can (depending on doping) yield Fermi and Luttinger arcs which together form a closed surface.

cond-mat.str-el↗

arXiv2025

Connecting single-layer $t$-$J$ to Kondo lattice models: Exploration with cold atoms

The Kondo effect, a hallmark of many-body physics, emerges from the antiferromagnetic coupling between localized spins and conduction fermions, leading to a correlated many-body singlet state. Here we propose to use the mixed-dimensional (mixD) bilayer Hubbard geometry as a platform to study Kondo lattice physics with current ultracold atom experiments. At experimentally feasible temperatures, we predict that key features of the Kondo effect can be observed, including formation of the Kondo cloud around a single impurity and the competition of singlet formation with Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions for multiple impurities, summarized in the Doniach phase diagram. Moreover, we show that the mixD platform provides a natural bridge between the Doniach phase diagram of the Kondo lattice model, relevant to heavy-fermion materials, and the phase diagram of cuprate superconductors as described by a single-layer Zhang-Rice type $t$-$J$ model: It is possible to continuously tune between the two regimes by changing the interlayer Kondo coupling. Our findings demonstrate that the direct connection between high-temperature superconductivity and heavy-fermion physics can be experimentally studied using currently available quantum simulation platforms.

cond-mat.quant-gas↗

arXiv2025

Interpretable Artificial Intelligence (AI) Analysis of Strongly Correlated Electrons

Artificial Intelligence (AI) has become an exceptionally powerful tool for analyzing scientific data. In particular, attention-based architectures have demonstrated a remarkable capability to capture complex correlations and to furnish interpretable insights into latent, otherwise inconspicuous patterns. This progress motivates the application of AI techniques to the analysis of strongly correlated electrons, which remain notoriously challenging to study using conventional theoretical approaches. Here, we propose novel AI workflows for analyzing snapshot datasets from tensor-network simulations of the two-dimensional (2D) Hubbard model over a broad range of temperature and doping. The 2D Hubbard model is an archetypal strongly correlated system, hosting diverse intriguing phenomena including Mott insulators, anomalous metals, and high-$T_c$ superconductivity. Our AI techniques yield fresh perspectives on the intricate quantum correlations underpinning these phenomena and facilitate universal omnimetry for ultracold-atom simulations of the corresponding strongly correlated systems.

cond-mat.str-el↗

arXiv2025

Finite-Temperature Study of the Hubbard Model via Enhanced Exponential Tensor Renormalization Group

The two-dimensional (2D) Hubbard model has long attracted interest for its rich phase diagram and its relevance to high-$T_c$ superconductivity. However, reliable finite-temperature studies remain challenging due to the exponential complexity of many-body interactions. Here, we introduce an enhanced $1\text{s}^+$ eXponential Tensor Renormalization Group algorithm that enables efficient finite-temperature simulations of the 2D Hubbard model. By exploring an expanded space, our approach achieves two-site update accuracy at the computational cost of a one-site update, and delivers up to 50% acceleration for Hubbard-like systems, which enables simulations down to $T\!\approx\!0.004t$. This advance permits a direct investigation of superconducting order over a wide temperature range and facilitates a comparison with zero-temperature infinite Projected Entangled Pair State simulations. Finally, we compile a comprehensive dataset of snapshots spanning the relevant region of the phase diagram, providing a valuable reference for Artificial Intelligence-driven analyses of the Hubbard model and a comparison with cold-atom experiments.

cond-mat.str-el↗

arXiv2025

Compressing local vertex functions from the multipoint numerical renormalization group using quantics tensor cross interpolation

The multipoint numerical renormalization group (mpNRG) is a powerful impurity solver that provides accurate spectral data useful for computing local, dynamic correlation functions in imaginary or real frequencies non-perturbatively across a wide range of interactions and temperatures. It gives access to a local, non-perturbative four-point vertex in imaginary and real frequencies, which can be used as input for subsequent computations such as diagrammatic extensions of dynamical mean--field theory. However, computing and manipulating the real-frequency four-point vertex on large, dense grids quickly becomes numerically challenging when the density and/or the extent of the frequency grid is increased. In this paper, we compute four-point vertices in a strongly compressed quantics tensor train format using quantics tensor cross interpolation, starting from discrete partial spectral functions obtained from mpNRG. This enables evaluations of the vertex on frequency grids with resolutions far beyond the reach of previous implementations. We benchmark this approach on the four-point vertex of the single-impurity Anderson model across a wide range of physical parameters, both in its full form and its asymptotic decomposition. For imaginary frequencies, the full vertex can be represented to an accuracy on the order of $2\cdot 10^{-3}$ with maximum bond dimensions not exceeding 120. The more complex full real-frequency vertex requires maximum bond dimensions not exceeding 170 for an accuracy of $\lesssim 2\%$. Our work marks another step toward tensor-train-based diagrammatic calculations for correlated electronic lattice models starting from a local, non-perturbative mpNRG vertex.

cond-mat.str-el↗

arXiv2025

Tangent space Krylov computation of real-frequency spectral functions: Influence of density-assisted hopping on 2D Mott physics

We present a tangent-space Krylov (TaSK) method for efficient computation of zero-temperature real-frequency spectral functions on top of ground state (GS) matrix product states (MPS) obtained from the Density Matrix Renormalization Group. It relies on projecting resolvents to the tangent space of the GS-MPS, where they can be efficiently represented using Krylov space techniques. This allows for a direct computation of spectral weights and their corresponding positions on the real-frequency axis. We demonstrate the accuracy and efficiency of the TaSK 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. Further, 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.

cond-mat.str-el↗

arXiv2025

Testing the parquet equations and the U(1) Ward identity for real-frequency correlation functions from the multipoint numerical renormalization group

Recently, it has become possible to compute real-frequency four-point correlation functions of quantum impurity models using a multipoint extension of the numerical renormalization group (mpNRG). In this work, we perform several numerical consistency checks of the output of mpNRG by investigating exact relations between two- and four-point functions. This includes the Bethe-Salpeter equations and the Schwinger-Dyson equation from the parquet formalism, which we evaluate in two formally identical but numerically nonequivalent ways. We also study the first-order U(1) Ward identity between the vertex and the self-energy, which we derive for the first time in full generality in the real-frequency Keldysh formalism. We generally find good agreement of all relations, often up to a few percent, both at weak and at strong interaction.

cond-mat.str-el↗

arXiv2025

Magnetic polarons at finite temperature: One-hole spectroscopy study

The physics of strongly correlated fermions described by Hubbard or $t$-$J$ models in the underdoped regime -- relevant for high-temperature superconductivity in cuprate compounds -- remains a subject of ongoing debate. In particular, the nature of charge carriers in this regime is poorly understood, in part due to the unusual properties of their spectral function. In this Letter, we present unbiased numerical results for the one-hole spectral function in a $t$-$J$ model at finite temperatures. Our study provides valuable insights into the underlying physics of magnetic (or spin-) polaron formation in a doped antiferromagnet (AFM). For example, we find how the suppression of spectral weight outside the magnetic Brillouin zone -- a precursor of Fermi arc formation -- disappears with increasing temperature, revealing nearly-deconfined spinon excitations of the undoped AFM. The pristine setting we consider can be directly explored using quantum simulators. Our calculations demonstrate that coherent quasiparticle peaks associated with magnetic polarons can be observed up to temperatures $T>J$ above the spin-exchange $J$, routinely obtained in such experiments. This paves the way for future studies of the fate of magnetic polarons in the pseudogap phase.

cond-mat.str-el↗

arXiv2025

Finite-temperature real-time properties of magnetic polarons in two-dimensional quantum antiferromagnets

Due to significant progress in quantum gas microscopy in recent years, there is a rapidly growing interest in real-space properties of single mobile dopands created in correlated antiferromagnetic (AFM) Mott insulators. However, a detailed numerical description remains challenging, even for simple toy models. As a consequence, previous numerical simulations for large systems were largely limited to $T=0$. To provide guidance for cold-atom experiments, numerical calculations at finite temperature are required. Here, we numerically study the real-time properties of a single mobile hole in the 2D $t$-$J$ model at finite temperature and draw a comparison to features observed at $T=0$. We find that a three-stage process of hole motion, which was reported at $T=0$, is valid even at finite temperature. However, already at low temperatures, the average hole velocity at long times is not simply proportional to the spin coupling, contrary to the $T=0$ behavior. Comparing our finite-temperature numerical results with the experimental data from quantum gas microscopy we find a qualitative disagreement: in experiment, hole spreading speeds up with increasing $J/t$, while in our numerics it slows down. The latter is consistent with the numerical findings previously reported at $T=0$.

cond-mat.quant-gas↗

arXiv2025

Two-particle calculations with quantics tensor trains: Solving the parquet equations

We present the first application of quantics tensor trains (QTTs) and tensor cross interpolation (TCI) to the solution of a full set of self-consistent equations for multivariate functions, the so-called parquet equations. We show that the steps needed to evaluate the equations (Bethe--Salpeter equations, parquet equation and Schwinger--Dyson equation) can be decomposed into basic operations on the QTT-TCI (QTCI) compressed objects. The repeated application of these operations does not lead to a loss of accuracy beyond a specified tolerance and the iterative scheme converges even for numerically demanding parameters. As examples we take the Hubbard model in the atomic limit and the single impurity Anderson model, where the basic objects in parquet equations, the two-particle vertices, depend on three frequencies, but not on momenta. The results show that this approach is able to overcome major computational bottlenecks of standard numerical methods. The applied methods allow for an exponential increase of the number of grid points included in the calculations leading to an exponentially improving computational error for a linear increase in computational cost.

cond-mat.str-el↗

arXiv2025

Subleading logarithmic behavior in the parquet formalism

The Fermi-edge singularity in x-ray absorption spectra of metals is a paradigmatic case of a logarithmically divergent perturbation series. Prior work has thoroughly analyzed the leading logarithmic terms. Here, we investigate the perturbation theory beyond leading logarithms and formulate self-consistent equations to incorporate all leading and next-to-leading logarithmic terms. This parquet solution of the Fermi-edge singularity goes beyond the previous first-order parquet solution and sheds new light on the parquet formalism regarding logarithmic behavior. We present numerical results in the Matsubara formalism and discuss the characteristic power laws. We also show that, within the single-boson exchange framework, multi-boson exchange diagrams are needed already at the leading logarithmic level.

cond-mat.str-el↗

arXiv2025

Reply to comment on "Controlled bond expansion for Density Matrix Renormalization Group ground state search at single-site costs"

We reply to McCulloch and Osborne's recent comment on our manuscript (Phys. Rev. Lett. 130, 246402 (2023)) on controlled bond expansion (CBE) for density matrix renormalization group (DMRG) ground state search. We appreciate their suggestion to consider randomized SVD and address their constructive critique on the variational properties of CBE-DMRG. However, we strongly disagree with their proposal to omit the projection to the 2-site tangent space and explain its importance for efficient bond expansion. In particular, in the context of CBE applied to the time-dependent variational principle (TDVP), we show that omitting this projection can lead to avoidable errors. Lastly, we emphasize the complementary roles of 3S mixing and CBE, reiterating our recommendation from Phys. Rev. Lett. 130, 246402 (2023) to combine both methods (CBE+$α$). We provide examples to demonstrate the superior efficiency and robustness of CBE+$α$.

cond-mat.str-el↗