arXiv subjects

Jan von Delft

Publications and source records attributed to Jan von Delft.

At least 37 records · Page 2Linked to original sources

arXiv2024

Learning tensor networks with tensor cross interpolation: new algorithms and libraries

The tensor cross interpolation (TCI) algorithm is a rank-revealing algorithm for decomposing low-rank, high-dimensional tensors into tensor trains/matrix product states (MPS). TCI learns a compact MPS representation of the entire object from a tiny training data set. Once obtained, the large existing MPS toolbox provides exponentially fast algorithms for performing a large set of operations. We discuss several improvements and variants of TCI. In particular, we show that replacing the cross interpolation by the partially rank-revealing LU decomposition yields a more stable and more flexible algorithm than the original algorithm. We also present two open source libraries, xfac in Python/C++ and TensorCrossInterpolation.jl in Julia, that implement these improved algorithms, and illustrate them on several applications. These include sign-problem-free integration in large dimension, the superhigh-resolution quantics representation of functions, the solution of partial differential equations, the superfast Fourier transform, the computation of partition functions, and the construction of matrix product operators.

physics.comp-ph↗

arXiv2024

Nonequilibrium steady-state thermoelectrics of Kondo-correlated quantum dots

The transport across a Kondo-correlated quantum dot coupled to two leads with independent temperatures and chemical potentials is studied using a controlled non-perturbative, and in this sense exact numeric treatment based on a hybrid numerical renormalization group combined with time-dependent density matrix renormalization group (NRG-tDMRG). We find a peak in the conductance at finite voltage bias vs. the temperature gradient $ΔT = T_R - T_L$ across left and right lead. We then focus predominantly on zero voltage bias but finite $ΔT$ far beyond linear response. We reveal the dependence of the characteristic zero-bias conductance on the individual lead temperatures. We find that the finite-$ΔT$ data behaves quantitatively similar to linear response with an effective equilibrium temperature derived from the different lead temperatures. The regime of sign changes in the Seebeck coefficient, signaling the presence of Kondo correlations, and its dependence on the individual lead temperatures provide a complete picture of the Kondo regime in the presence of finite temperature gradients. The results from the zero-bias conductance and Seebeck coefficient studies unveil an approximate `Kondo circle' in the $T_L/T_R$ plane as the regime within which the Kondo correlations dominate. We also study the heat current and the corresponding heat conductance vs. finite $ΔT$. We provide a polynomial fit for our numerical results for the thermocurrent as a function of the individual lead temperatures which may be used to fit experimental data in the Kondo regime.

cond-mat.mes-hall↗

arXiv2024

KeldyshQFT: A C++ codebase for real-frequency multiloop functional renormalization group and parquet computations of the single-impurity Anderson model

We provide a detailed exposition of our computational framework designed for the accurate calculation of real-frequency dynamical correlation functions of the single-impurity Anderson model (AM) in the regime of weak to intermediate coupling. Using quantum field theory within the Keldysh formalism to directly access the self-energy and dynamical susceptibilities in real frequencies, as detailed in our recent publication (https://doi.org/10.1103/PhysRevB.109.115128), the primary computational challenge is the full three-dimensional real-frequency dependence of the four-point vertex. Our codebase provides a fully MPI+OpenMP parallelized implementation of the functional renormalization group (fRG) and the self-consistent parquet equations within the parquet approximation. It leverages vectorization to handle the additional complexity imposed by the Keldysh formalism, using optimized data structures and highly performant integration routines. Going beyond the results shown in the previous publication, the code includes functionality to perform fRG calculations in the multiloop framework, at arbitrary loop order, including self-consistent self-energy iterations. Moreover, implementations of various regulators, such as hybridization, interaction, frequency, and temperature are supplied.

cond-mat.str-el↗

arXiv2024

Analytic continuation of multipoint correlation functions

Conceptually, the Matsubara formalism (MF), using imaginary frequencies, and the Keldysh formalism (KF), formulated in real frequencies, give equivalent results for systems in thermal equilibrium. The MF has less complexity and is thus more convenient than the KF. However, computing dynamical observables in the MF requires the analytic continuation from imaginary to real frequencies. The analytic continuation is well-known for two-point correlation functions (having one frequency argument), but, for multipoint correlators, a straightforward recipe for deducing all Keldysh components from the MF correlator had not been formulated yet. Recently, a representation of MF and KF correlators in terms of formalism-independent partial spectral functions and formalism-specific kernels was introduced by Kugler, Lee, and von Delft [Phys. Rev. X 11, 041006 (2021)]. We use this representation to formally elucidate the connection between both formalisms. We show how a multipoint MF correlator can be analytically continued to recover all partial spectral functions and yield all Keldysh components of its KF counterpart. The procedure is illustrated for various correlators of the Hubbard atom.

cond-mat.str-el↗

arXiv2024

Dynamical scaling and Planckian dissipation due to heavy-fermion quantum criticality

We study dynamical scaling associated with a Kondo-breakdown quantum critical point (KB-QCP) of the periodic Anderson model, treated by two-site cellular dynamical mean-field theory (2CDMFT). In the quantum critical region, the staggered spin exhibits SYK-like slow dynamics and its dynamical susceptibility shows $ω/T$ scaling. We propose a scaling Ansatz that describes this behavior. It also implies Planckian dissipation for the longest-lived excitations. The current susceptibility follows the same scaling ansatz, leading to strange-metal scaling. This demonstrates that the KB-QCP described by 2CDMFT is an intrinsic (i.e., disorder-free) strange-metal fixed point. Surprisingly, the SYK-like dynamics and scaling are driven by strong vertex contributions to the susceptibilities. Our results for the optical conductivity match experimental observations on YbRh${}_2$Si${}_2$ and CeCoIn${}_5$.

cond-mat.str-el↗

arXiv2024

Real-frequency quantum field theory applied to the single-impurity Anderson model

A major challenge in the field of correlated electrons is the computation of dynamical correlation functions. For comparisons with experiment, one is interested in their real-frequency dependence. This is difficult to compute, as imaginary-frequency data from the Matsubara formalism require analytic continuation, a numerically ill-posed problem. Here, we apply quantum field theory to the single-impurity Anderson model (AM), using the Keldysh instead of the Matsubara formalism with direct access to the self-energy and dynamical susceptibilities on the real-frequency axis. We present results from the functional renormalization group (fRG) at one-loop level and from solving the self-consistent parquet equations in the parquet approximation. In contrast to previous Keldysh fRG works, we employ a parametrization of the four-point vertex which captures its full dependence on three real-frequency arguments. We compare our results to benchmark data obtained with the numerical renormalization group and to second-order perturbation theory. We find that capturing the full frequency dependence of the four-point vertex significantly improves the fRG results compared to previous implementations, and that solving the parquet equations yields the best agreement with the NRG benchmark data, but is only feasible up to moderate interaction strengths. Our methodical advances pave the way for treating more complicated models in the future.

cond-mat.str-el↗

arXiv2024

Overcomplete intermediate representation of two-particle Green's functions and its relation to partial spectral functions

Two-particle response functions are a centerpiece of both experimental and theoretical quantum many-body physics. Yet, due to their size and discontinuity structure, they are challenging to handle numerically. Recently, two advances were made to tackle this problem: first, the overcomplete intermediate representation (OIR), which provides a highly efficient compression of Green's functions in imaginary frequency, and second, partial spectral functions (PSFs), which allow for an efficient evaluation in real frequency. We show that there is a two-to-one correspondence between PSFs and OIR coefficients and exploit this fact to construct the OIR for three-or-more-particle propagators. We then use OIR to fit and compress imaginary-frequency data obtained from the numerical renormalization group (NRG), reaching a compression ratio of more than 400. Finally, we attempt to match the OIR data to partial Green's functions from NRG.Due to the overcompleteness, we achieve only qualitative agreement.

cond-mat.str-el↗

arXiv2024

Quantics Tensor Cross Interpolation for High-Resolution, Parsimonious Representations of Multivariate Functions in Physics and Beyond

Multivariate functions of continuous variables arise in countless branches of science. Numerical computations with such functions typically involve a compromise between two contrary desiderata: accurate resolution of the functional dependence, versus parsimonious memory usage. Recently, two promising strategies have emerged for satisfying both requirements: (i) The quantics representation, which expresses functions as multi-index tensors, with each index representing one bit of a binary encoding of one of the variables; and (ii) tensor cross interpolation (TCI), which, if applicable, yields parsimonious interpolations for multi-index tensors. Here, we present a strategy, quantics TCI (QTCI), which combines the advantages of both schemes. We illustrate its potential with an application from condensed matter physics: the computation of Brillouin zone integrals.

physics.comp-ph↗

arXiv2024

Symmetric improved estimators for multipoint vertex functions

Multipoint vertex functions, and the four-point vertex in particular, are crucial ingredients in many-body theory. Recent years have seen significant algorithmic progress toward numerically computing their dependence on multiple frequency arguments. However, such computations remain challenging and are prone to suffer from numerical artifacts, especially in the real-frequency domain. Here, we derive estimators for multipoint vertices that are numerically more robust than those previously available. We show that the two central steps for extracting vertices from correlators, namely the subtraction of disconnected contributions and the amputation of external legs, can be achieved accurately through repeated application of equations of motion, in a manner that is symmetric with respect to all frequency arguments and involves only fully renormalized objects. The symmetric estimators express the core part of the vertex and all asymptotic contributions through separate expressions that can be computed independently, without subtracting the large-frequency limits of various terms with different asymptotic behaviors. Our strategy is general and applies equally to the Matsubara formalism, the real-frequency zero-temperature formalism, and the Keldysh formalism. We demonstrate the advantages of the symmetric improved estimators by computing the Keldysh four-point vertex of the single-impurity Anderson model using the numerical renormalization group.

cond-mat.str-el↗

arXiv2024

Finite-size subthermal regime in disordered SU(N)-symmetric Heisenberg chains

SU(N) symmetry is incompatible with the many-body localized (MBL) phase, even when strong disorder is present. However, recent studies have shown that finite-size SU(2) systems exhibit non-ergodic, subthermal behavior, characterized by the breakdown of the eigenstate thermalization hypothesis, and by the excited eigenstates entanglement entropy that is intermediate between area and volume law. In this work, we extend previous studies of the SU(2)-symmetric disordered Heisenberg model to larger systems, using the time-dependent density matrix renormalization group (tDMRG) method. We simulate quench dynamics from weakly entangled initial states up to long times, finding robust subthermal behavior at stronger disorder. Although we find an increased tendency towards thermalization at larger system sizes, the subthermal regime persists at intermediate time scales, nevertheless, and therefore should be accessible experimentally. At weaker disorder, we observe signatures of thermalization, however, entanglement entropy exhibits slow sublinear growth, in contrast to conventional thermalizing systems. Furthermore, we study dynamics of the SU(3)-symmetric disordered Heisenberg model. Similarly, strong disorder drives the system into subthermal regime, albeit thermalizing phase is broader compared to the SU(2) case. Our findings demonstrate the robustness of the subthermal regime in spin chains with non-Abelian continuous symmetry, and are consistent with eventual thermalization at large system sizes and long time scales, suggested by previous works.

quant-ph↗

arXiv2023

MatsubaraFunctions.jl: An equilibrium Green's function library in the Julia programming language

The Matsubara Green's function formalism stands as a powerful technique for computing the thermodynamic characteristics of interacting quantum many-particle systems at finite temperatures. In this manuscript, our focus centers on introducing MatsubaraFunctions.jl, a Julia library that implements data structures for generalized n-point Green's functions on Matsubara frequency grids. The package's architecture prioritizes user-friendliness without compromising the development of efficient solvers for quantum field theories in equilibrium. Following a comprehensive introduction of the fundamental types, we delve into a thorough examination of key facets of the interface. This encompasses avenues for accessing Green's functions, techniques for extrapolation and interpolation, as well as the incorporation of symmetries and a variety of parallelization strategies. Examples of increasing complexity serve to demonstrate the practical utility of the library, supplemented by discussions on strategies for sidestepping impediments to optimal performance.

cond-mat.str-el↗

arXiv2023

Emergent Properties of the Periodic Anderson Model: a High-Resolution, Real-Frequency Study of Heavy-Fermion Quantum Criticality

We study paramagnetic quantum criticality in the periodic Anderson model (PAM) using cellular dynamical mean-field theory, with the numerical renormalization group (NRG) as an impurity solver. The PAM describes an itinerant $c$ band hybridizing with a localized $f$ band. At $T=0$, it exhibits a hybridization tuned Kondo breakdown quantum critical point (KB-QCP) from a Kondo to an RKKY phase. At the KB-QCP, the $f$ band changes character from itinerant to mainly localized, while the $c$ band remains itinerant. We elucidate its nature in detail by performing a high-resolution, real-frequency study of dynamical quantities. NRG allows us to study the quantum critical non-Fermi-liquid (NFL) regime located between $T_{FL}<T_{NFL}$. Surprisingly, self-consistency is essential to stabilize the NFL and the QCP. The Fermi-liquid (FL) scale $T_{FL}$ decreases towards and vanishes at the QCP. At $T=0$, we find the following properties. The $f$ quasiparticle (QP) weight $Z_f$ decreases continuously as the QCP is approached from either side, vanishing only at the QCP. Therefore, $Z_f$ is nonzero in both the Kondo and the RKKY phase; hence, the FL QP comprise $c$ and $f$ electrons in both phases. The Fermi surface (FS) volumes in the two phases differ. Whereas the large-FS Kondo phase has a usual two-band structure, the small-FS RKKY phase has an unexpected three-band structure. We provide a detailed analysis of quasiparticle properties of both the Kondo and the RKKY phase. The FS reconstruction is accompanied by the appearance of a Luttinger surface (LS) on which the $f$ self-energy diverges. The FS and LS volumes are related to the density by a generalized Luttinger sum rule. We interpret the small FS volume and the emergent LS as evidence for $f$-electron fractionalization in the RKKY phase. Our Hall coefficient and specific heat are in good qualitative agreement with experiment.

cond-mat.str-el↗

arXiv2023

Frustration-Induced Superconductivity in the $t$-$t'$ Hubbard Model

The two-dimensional (2D) Hubbard model is widely believed to capture key ingredients of high-$T_c$ superconductivity in cuprate materials. However, compelling evidence remains elusive. In particular, various magnetic orders may emerge as strong competitors of superconducting orders. Here, we study the ground state properties of the doped 2D $t$-$t'$ Hubbard model on a square lattice via the infinite Projected Entangled-Pair State (iPEPS) method with $\mathrm{U}(1)$ or $\mathrm{SU}(2)$ spin symmetry. The former is compatible with antiferromagnetic orders, while the latter forbids them. Therefore, we obtain by comparison a detailed understanding of the magnetic impact on superconductivity. Moreover, an additional $t'$ term accommodates the particle-hole asymmetry, which facilitates studies on the discrepancies between electron- and hole-doped systems. We demonstrate that (i) a positive $t'/t$ significantly amplifies the strength of superconducting orders; (ii) at sufficiently large doping levels, the $t$-$t'$ Hubbard model favors a uniform state with superconducting orders instead of stripe states with charge and spin modulations; and (iii) the enhancement of magnetic frustration, by increasing either the strength of NNN interactions or the charge doping, impairs stripe orders and helps stabilize superconductivity.

cond-mat.str-el↗

arXiv2023

Projector formalism for kept and discarded spaces of matrix product states

Any matrix product state $|Ψ\rangle$ has a set of associated kept and discarded spaces, needed for the description of $|Ψ\rangle$, and changes thereof, respectively. These induce a partition of the full Hilbert space of the system into mutually orthogonal spaces of irreducible $n$-site variations of $|Ψ\rangle$. Here, we introduce a convenient projector formalism and diagrammatic notation to characterize these $n$-site spaces explicitly. This greatly facilitates the formulation of MPS algorithms that explicitly or implicitly employ discarded spaces. As an illustration, we derive an explicit expression for the $n$-site energy variance and evaluate it numerically for a model with long-range hopping. We also describe an efficient algorithm for computing low-lying $n$-site excitations above a finite MPS ground state.

quant-ph↗

arXiv2023

Controlled bond expansion for DMRG ground state search at single-site costs

DMRG ground state search algorithms employing symmetries must be able to expand virtual bond spaces by adding or changing symmetry sectors if these lower the energy. Traditional single-site DMRG does not allow bond expansion; two-site DMRG does, but at much higher computational costs. We present a controlled bond expansion (CBE) algorithm that yields two-site accuracy and convergence per sweep, at single-site costs. Given a matrix product state $Ψ$ defining a variational space, CBE identifies parts of the orthogonal space carrying significant weight in $HΨ$ and expands bonds to include only these. CBE-DMRG uses no mixing parameters and is fully variational. Using CBE-DMRG, we show that the Kondo-Heisenberg model on a width 4 cylinder features two distinct phases differing in their Fermi surface volumes.

cond-mat.str-el↗

arXiv2023

Tangle of Spin Double Helices in the Honeycomb Kitaev-$Γ$ Model

We investigate the ground-state nature of the honeycomb Kitaev-$Γ$ model in the material-relevant parameter regime through a combination of classical and quantum simulations. The classical model is imprinted with a tangle of highly structured spin double helices. This helix tangle exhibits $18$ inequivalent helical axes and features a spontaneous periodicity anisotropy and a ${\rm sgn}(Γ)$-determined chirality pattern. Infinite PEPS simulations with clusters up to $36$ sites identify hallmarks of this many-body order in the quantum spin-$1/2$ model. Our findings provide a fresh perspective of the Kitaev-$Γ$ model and enrich the physics of Kitaev magnetism.

cond-mat.str-el↗

arXiv2022

Time-dependent variational principle with controlled bond expansion for matrix product states

We present a controlled bond expansion (CBE) approach to simulate quantum dynamics based on the time-dependent variational principle (TDVP) for matrix product states. Our method alleviates the numerical difficulties of the standard, fixed-rank one-site TDVP integrator by increasing bond dimensions on the fly to reduce the projection error. This is achieved in an economical, local fashion, requiring only minor modifications of standard one-site TDVP implementations. We illustrate the performance of CBE--TDVP with several numerical examples on finite quantum lattices.

cond-mat.str-el↗

arXiv2022

Nonequilibrium spintronic transport through Kondo impurities

In this work we analyze the nonequilibrium transport through a quantum impurity (quantum dot or molecule) attached to ferromagnetic leads by using a hybrid numerical renormalization group-time-dependent density matrix renormalization group thermofield quench approach.For this, we study the bias dependence of the differential conductance through the system, which shows a finite zero-bias peak, characteristic of the Kondo resonance and reminiscent of the equilibrium local density of states. In the non-equilibrium settings, the resonance in the differential conductance is also found to decrease with increasing the lead spin polarization. The latter induces an effective exchange field that lifts the spin degeneracy of the dot level. Therefore as we demonstrate, the Kondo resonance can be restored by counteracting the exchange field with a finite external magnetic field applied to the system. Finally, we investigate the influence of temperature on the nonequilibrium conductance, focusing on the split Kondo resonance. Our work thus provides an accurate quantitative description of the spin-resolved transport properties relevant for quantum dots and molecules embedded in magnetic tunnel junctions.

cond-mat.mes-hall↗