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Gero Friesecke

Publications and source records attributed to Gero Friesecke.

At least 19 recordsLinked to original sources
arXiv2026

Block entropy area based non-local fermionic mode optimization with gradient disentanglers

We introduce a systematic block entropy area based mode optimization algorithm for many-body quantum states of interacting fermions represented by matrix product states. From the gradient of a global cost function, the block entropy area, a long-ranged, non-interacting effective disentangler Hamiltonian is formed. We then simulate the time-dependent Schrödinger equation driven by the disentangler Hamiltonian by employing the time-dependent variational principle based on projector splitting, and minimize the cost function. The combination of the density matrix renormalization group with this gradient-based entanglement minimization forms an efficient low-rank iterative ground-state algorithm that also provides an optimized single-particle basis for matrix product state representation. We demonstrate the method on two-dimensional lattice models of interacting fermions and the Fe${_4}$S${_4}$ cluster, and show its robustness and superiority over earlier protocols using nearest-neighbor mode rotations and reorderings.

cond-mat.str-el↗
arXiv2026

Global fermionic mode optimization via swap gates

An optimal choice of single-particle modes leads to an optimal tensor network-based compression of the many-body wave function for systems of indistinguishable particles, which compression can be essential for large, close-to-critical systems. The proposed mode optimization relies on the minimization of the block entropy area, a global quantity that measures entanglement of the wave function along the one-dimensional chain of modes. Extension of the two-site DMRG algorithm by nearest-neighbor mode optimizations is straightforward, but it performs poorly without systematic reordering of modes. Moreover, finding a stationary optimal point in the combination of the fixed-rank MPS manifold and the unitary group of mode rotations requires optimization for every generator, i.e., for every two-mode pair. Here, a systematic joint optimization protocol over the MPS manifold and the full unitary group is presented in which every two-mode pair is accessed by a carefully constructed sequence of two-qubit operations, including swap-gate controlled permutations. Large-scale DMRG simulations of strongly correlated two-dimensional fermionic lattice models and the multireference Fe$_4$S$_4$ transition metal cluster demonstrate rapid convergence towards the optimum, achieving low energy and low entanglement.

cond-mat.str-el↗
arXiv2026

Partial regularity of optimal transport with Coulomb cost

We prove that for two-marginal optimal transport with Coulomb cost on $\mathbb{R}^d$, the optimal map is a $C^{1,α}$ diffeomorphism outside a closed set of Lebesgue measure zero provided the marginals are $α$-Hölder continuous, bounded, and strictly positive. Excluding a set of measure zero is necessary as optimal maps for the Coulomb cost have long been known to exhibit jump singularities across codimension $1$ surfaces (even for smooth marginals on convex domains).

math.AP↗
arXiv2026

Sampling Boltzmann distributions via normalizing flow approximation of transport maps

In a celebrated paper \cite{noe2019boltzmann}, Noé, Olsson, Köhler and Wu introduced an efficient method for sampling high-dimensional Boltzmann distributions arising in molecular dynamics via normalizing flow approximation of transport maps. Here, we place this approach on a firm mathematical foundation. We prove the existence of a normalizing flow between the reference measure and the true Boltzmann distribution up to an arbitrarily small error in the Wasserstein distance. This result covers general Boltzmann distributions from molecular dynamics, which have low regularity due to the presence of interatomic Coulomb and Lennard-Jones interactions. The proof is based on a rigorous construction of the Moser transport map for low-regularity endpoint densities and approximation theorems for neural networks in Sobolev spaces. Numerical simulations for a simple model system and for the alanine dipeptide molecule confirm that the true and generated distributions are close in the Wasserstein distance. Moreover we observe that the RealNVP architecture does not just successfully capture the equilibrium Boltzmann distribution but also the metastable dynamics.

cs.LG↗
arXiv2026

Mass splitting in the time-discrete generalized Euler equations and non-Monge solutions in multi-marginal optimal transport

The time-discretized, spatially continuous generalized Euler equations are a prototype example of multi-marginal optimal transport, yet the question whether they exhibit mass-splitting (or equivalently, whether they have solutions that are not of Monge form) has remained open. Here we resolve this question by giving a mass-splitting example in one spatial dimension. Moreover we present a related and very simple fully discrete example of mass-splitting which reveals a transparent underlying mechanism.

math.AP↗
arXiv2025

Copula methods for modeling pair densities in density functional theory

We propose a new approach towards approximating the density-to-pair-density map based on copula theory from statistics. We extend the copula theory to multi-dimensional marginals, and deduce that one can describe any (exact or approximate) pair density by the single-particle density and a copula. We present analytical formulas for the exact copula in scaling limits, numerically compute the copula for dissociating systems with two to four particles in one dimension, and propose accurate approximations of the copula between equilibrium and dissociation for two-particle systems.

physics.comp-ph↗
arXiv2024

$p$-Wasserstein barycenters

We study barycenters of $N$ probability measures on $\mathbb{R}^d$ with respect to the $p$-Wasserstein metric ($1<p<\infty$). We prove that -- $p$-Wasserstein barycenters of absolutely continuous measures are unique, and again absolutely continuous -- $p$-Wasserstein barycenters admit a multi-marginal formulation -- the optimal multi-marginal plan is unique and of Monge form if the marginals are absolutely continuous, and its support has an explicit parametrization as a graph over any marginal space. This extends the Agueh--Carlier theory of Wasserstein barycenters [SIAM J. Math. Anal. 43 (2011), no.2, 904--924] to exponents $p\neq 2$. A key ingredient is a quantitative injectivity estimate for the (highly non-injective) map from $N$-point configurations to their $p$-barycenter on the support of an optimal multi-marginal plan. We also discuss the statistical meaning of $p$-Wasserstein barycenters in one dimension.

math.AP↗
arXiv2024

$h$-Wasserstein barycenters

We generalize the notion and theory of Wasserstein barycenters introduced by Agueh and Carlier (2011) from the quadratic cost to general smooth strictly convex costs $h$ with non-degenerate Hessian. We show the equivalence between a coupled two-marginal and a multi-marginal formulation and establish that the multi-marginal optimal plan is unique and of Monge form. To establish the latter result we introduce a new approach which is not based on explicitly solving the optimality system, but instead deriving a quantitative injectivity estimate for the (highly non-injective) map from $N$-point configurations to their $h$-barycenter on the support of an optimal multi-marginal plan.

math.AP↗
arXiv2023

Robust self-assembly of nonconvex shapes in 2D

We present fast simulation methods for the self-assembly of complex shapes in two dimensions. The shapes are modeled via a general boundary curve and interact via a standard volume term promoting overlap and an interpenetration penalty. To efficiently realize the Gibbs measure on the space of possible configurations we employ the hybrid Monte Carlo algorithm together with a careful use of signed distance functions for energy evaluation. Motivated by the self-assembly of identical coat proteins of the tobacco mosaic virus which assemble into a helical shell, we design a particular nonconvex 2D model shape and demonstrate its robust self-assembly into a unique final state. Our numerical experiments reveal two essential prerequisites for this self-assembly process: blocking and matching (i.e., local repulsion and attraction) of different parts of the boundary; and nonconvexity and handedness of the shape.

physics.comp-ph↗
arXiv2023

Convergence proof for the GenCol algorithm in the case of two-marginal optimal transport

The recently introduced Genetic Column Generation (GenCol) algorithm has been numerically observed to efficiently and accurately compute high-dimensional optimal transport plans for general multi-marginal problems, but theoretical results on the algorithm have hitherto been lacking. The algorithm solves the OT linear program on a dynamically updated low-dimensional submanifold consisting of sparse plans. The submanifold dimension exceeds the sparse support of optimal plans only by a fixed factor $β$. Here we prove that for $β\geq 2$ and in the two-marginal case, GenCol always converges to an exact solution, for arbitrary costs and marginals. The proof relies on the concept of c-cyclical monotonicity. As an offshoot, GenCol rigorously reduces the data complexity of numerically solving two-marginal OT problems from $O(\ell^2)$ to $O(\ell)$ without any loss in accuracy, where $\ell$ is the number of discretization points for a single marginal. At the end of the paper we also present some insights into the convergence behavior in the multi-marginal case.

math.NA↗
arXiv2023

The density-density response function in time-dependent density functional theory: mathematical foundations and pole shifting

We establish existence and uniqueness of the solution to the Dyson equation for the density-density response function in time-dependent density functional theory (TDDFT) in the random phase approximation (RPA). We show that the poles of the RPA density-density response function are forward-shifted with respect to those of the non-interacting response function, thereby explaining mathematically the well known empirical fact that the non-interacting poles (given by the spectral gaps of the time-independent Kohn-Sham equations) underestimate the true transition frequencies. Moreover we show that the RPA poles are solutions to an eigenvalue problem, justifying the approach commonly used in the physics community to compute these poles.

math-ph↗
arXiv2023

Predicting the FCI energy of large systems to chemical accuracy from restricted active space density matrix renormalization group calculations

We theoretically derive and validate with large scale simulations a remarkably accurate power law scaling of errors for the restricted active space density matrix renormalization group (DMRG-RAS) method [arXiv:2111.06665] in electronic structure calculations. This yields a new extrapolation method, DMRG-RAS-X, which reaches chemical accuracy for strongly correlated systems such as the Chromium dimer, dicarbon up to a large cc-pVQZ basis, and even a large chemical complex like the FeMoco with significantly lower computational demands than previous methods. The method is free of empirical parameters, performed robustly and reliably in all examples we tested, and has the potential to become a vital alternative method for electronic structure calculations in quantum chemistry, and more generally for the computation of strong correlations in nuclear and condensed matter physics.

physics.chem-ph↗
arXiv2023

Next-order correction to the Dirac exchange energy of the free electron gas in the thermodynamic limit and generalized gradient approximations

We derive the next order correction to the Dirac exchange energy for the free electron gas in a box with zero boundary conditions in the thermodynamic limit. The correction is of the order of the surface area of the box, and comes from three different contributions: (i) a real-space boundary layer, (ii) a boundary-condition-induced small shift of Fermi momentum and bulk density, and (iii) a long-range electrostatic finite-size correction. Moreover we show that the LDA, in addition to capturing the bulk term exactly, also produces a correction of the correct order but not the correct size. GGA corrections are found to be capable of capturing the surface term exactly, provided the gradient enhancement factor satisfies a simple explicit integral constraint. For current GGAs such as B88 and PBE we find that the new constraint is not satisfied and the size of the surface correction is overestimated by about ten percent. The new constraint might thus be of interest for the design of future exchange functionals.

physics.chem-ph↗
arXiv2022

On the closedness and geometry of tensor network state sets

Tensor network states (TNS) are a powerful approach for the study of strongly correlated quantum matter. The curse of dimensionality is addressed by parametrizing the many-body state in terms of a network of partially contracted tensors. These tensors form a substantially reduced set of effective degrees of freedom. In practical algorithms, functionals like energy expectation values or overlaps are optimized over certain sets of TNS. Concerning algorithmic stability, it is important whether the considered sets are closed because, otherwise, the algorithms may approach a boundary point that is outside the TNS set and tensor elements diverge. We discuss the closedness and geometries of TNS sets, and we propose regularizations for optimization problems on non-closed TNS sets. We show that sets of matrix product states (MPS) with open boundary conditions, tree tensor network states (TTNS), and the multiscale entanglement renormalization ansatz (MERA) are always closed, whereas sets of translation-invariant MPS with periodic boundary conditions (PBC), heterogeneous MPS with PBC, and projected entangled-pair states (PEPS) are generally not closed. The latter is done using explicit examples like the W state, states that we call two-domain states, and fine-grained versions thereof.

quant-ph↗
arXiv2022

Density Functionals based on the mathematical structure of the strong-interaction limit of DFT

While in principle exact, Kohn-Sham density functional theory -- the workhorse of computational chemistry -- must rely on approximations for the exchange-correlation functional. Despite staggering successes, present-day approximations still struggle when the effects of electron-electron correlation play a prominent role. The limit in which the electronic Coulomb repulsion completely dominates the exchange-correlation functional offers a well-defined mathematical framework that provides insight for new approximations able to deal with strong correlation. In particular, the mathematical structure of this limit, which is now well-established thanks to its reformulation as an optimal transport problem, points to the use of very different ingredients (or features) with respect to the traditional ones used in present approximations. We focus on strategies to use these new ingredients to build approximations for computational chemistry and highlight future promising directions.

physics.chem-ph↗
arXiv2022

Exact matrix product state representation and convergence of a fully correlated electronic wavefunction in the infinite basis limit

In this article we present the exact representation of a fully correlated electronic wavefunction as the single-particle basis approaches completeness. It consists of a half-infinite chain of matrices of exponentially increasing size. The complete basis limit is illustrated numerically using the density matrix renormalization group method by computing the core-valence entanglement in the C$_2$ ground state in increasing subsets of cc-pVTZ and pVQZ bases until convergence is reached.

quant-ph↗
arXiv2022

The strong-interaction limit of density functional theory

This is a comprehensive review of the strong-interaction limit of density functional theory. It covers the derivation of the limiting strictly correlated electrons (SCE) functional from exact Hohenberg-Kohn DFT, basic aspects of SCE physics such as the nonlocal dependence of the SCE potential on the density, equivalent formulations and the mathematical interpretation as optimal transport with Coulomb cost, rigorous results (including exactly soluble cases), approximations, numerical methods, integration into Kohn-Sham DFT (KS SCE), and applications to molecular systems, an example being that KS SCE, unlike the local density approximation or generalized gradient approximations, dissociates H$_2$ correctly. We have made an effort to make this review accessible to a broad audience of physicists, chemists, and mathematicians.

physics.chem-ph↗