arXiv subjects

Jan Naumann

Publications and source records attributed to Jan Naumann.

8 recordsLinked to original sources

arXiv2025

Derivative of the truncated singular value and eigen decomposition

Recently developed applications in the field of machine learning and computational physics rely on automatic differentiation techniques, that require stable and efficient linear algebra gradient computations. This technical note provides a comprehensive and detailed discussion of the derivative of the truncated singular and eigenvalue decomposition. It summarizes previous work and builds on them with an extensive description of how to derive the relevant terms. A main focus is correctly expressing the derivative in terms of the truncated part, despite lacking knowledge of the full decomposition.

math.NA↗

arXiv2025

Variational optimization of projected entangled-pair states on the triangular lattice

We introduce a general corner transfer matrix renormalization group algorithm tailored to projected entangled-pair states on the triangular lattice. By integrating automatic differentiation, our approach enables direct variational energy minimization on this lattice geometry. In contrast to conventional approaches that map the triangular lattice onto a square lattice with diagonal next-nearest-neighbour interactions, our native formulation yields improved variational results at the same bond dimension. This improvement stems from a more faithful and physically informed representation of the entanglement structure in the tensor network and an increased number of variational parameters. We apply our method to the antiferromagnetic nearest-neighbour Heisenberg model on the triangular and kagome lattice, and benchmark our results against previous numerical studies.

cond-mat.str-el↗

arXiv2025

Triangular lattice models of the Kalmeyer-Laughlin spin liquid from coupled wires

Chiral spin liquids (CSLs) are exotic phases of interacting spins in two dimensions, characterized by long-range entanglement and fractional excitations. We construct a local Hamiltonian on the triangular lattice that stabilizes the Kalmeyer-Laughlin CSL without requiring fine-tuning. Our approach employs coupled-wire constructions and introduces a lattice duality to construct a solvable chiral sliding Luttinger liquid, which is driven toward the CSL phase by generic perturbations. By combining symmetry analysis and bosonization, we make sharp predictions for the ground states on quasi-one-dimensional cylinders and tori, which exhibit a fourfold periodicity in the circumference. Extensive tensor network simulations demonstrating ground-state degeneracies, fractional quasiparticles, nonvanishing long-range order parameters, and entanglement signatures confirm the emergence of the CSL in the lattice Hamiltonian.

cond-mat.str-el↗

arXiv2025

Variationally optimizing infinite projected entangled-pair states at large bond dimensions: A split corner transfer matrix renormalization group approach

Projected entangled-pair states (PEPS) have become a powerful tool for studying quantum many-body systems in the condensed matter and quantum materials context, particularly with advances in variational energy optimization methods. A key challenge within this framework is the computational cost associated with the contraction of the two-dimensional lattice, crucial for calculating state vector norms and expectation values. The conventional approach, using the corner transfer matrix renormalization group (CTMRG), involves combining two tensor network layers, resulting in significant time and memory demands. In this work, we introduce an alternative "split-CTMRG" algorithm, which maintains separate PEPS layers and leverages new environment tensors, reducing computational complexity while preserving accuracy. Benchmarks on quantum lattice models demonstrate substantial speedups for variational energy optimization, rendering this method valuable for large-scale PEPS simulations.

cond-mat.str-el↗

arXiv2024

An introduction to infinite projected entangled-pair state methods for variational ground state simulations using automatic differentiation

Tensor networks capture large classes of ground states of phases of quantum matter faithfully and efficiently. Their manipulation and contraction has remained a challenge over the years, however. For most of the history, ground state simulations of two-dimensional quantum lattice systems using (infinite) projected entangled pair states have relied on what is called a time-evolving block decimation. In recent years, multiple proposals for the variational optimization of the quantum state have been put forward, overcoming accuracy and convergence problems of previously known methods. The incorporation of automatic differentiation in tensor networks algorithms has ultimately enabled a new, flexible way for variational simulation of ground states and excited states. In this work we review the state-of-the-art of the variational iPEPS framework, providing a detailed introduction to automatic differentiation, a description of a general foundation into which various two-dimensional lattices can be conveniently incorporated, and demonstrative benchmarking results.

cond-mat.str-el↗

arXiv2024

An extended non-magnetic phase in the spin-1/2 Heisenberg antiferromagnet from the ruby to the maple-leaf lattice

The spin-1/2 Heisenberg antiferromagnet on the two-dimensional ruby and maple-leaf lattices provides a stringent test case for frustrated quantum magnetism, where semiclassical magnetic order competes closely with quantum-paramagnetic states. We study the generalized maple-leaf model along the axis interpolating between the isotropic ruby lattice and the isotropic maple-leaf lattice, using two independent variational infinite projected entangled-pair state ans\"atze: one defined on the mapped square lattice and one directly on the native triangular lattice. Our lowest-energy variational states show no magnetic order throughout the region connecting the two isotropic limits, and the local bond correlations recover the $120^\circ$ lattice rotation symmetry without imposing it explicitly. Magnetic-field simulations at the two endpoints further indicate a finite zero-magnetization plateau in both cases, with a gradual onset of magnetization on the ruby lattice and a sharper onset on the maple-leaf lattice. These results establish a broad gapped non-magnetic regime in a minimal nearest-neighbor Heisenberg model on lattices of direct relevance to both frustrated quantum magnets and programmable ruby-lattice simulators.

cond-mat.str-el↗

arXiv2023

Hamiltonian truncation tensor networks for quantum field theories

Understanding the equilibrium properties and out of equilibrium dynamics of quantum field theories are key aspects of fundamental problems in theoretical particle physics and cosmology. However, their classical simulation is highly challenging. In this work, we introduce a tensor network method for the classical simulation of continuous quantum field theories that is suitable for the study of low-energy eigenstates and out-of-equilibrium time evolution. The method is built on Hamiltonian truncation and tensor network techniques, bridging the gap between two successful approaches. One of the key developments is the exact construction of matrix product state representations of global projectors, crucial for the implementation of interacting theories. Despite featuring a relatively high computational effort, our method dramatically improves predictive precision compared to exact diagonalisation-based Hamiltonian truncation, allowing the study of so far unexplored parameter regimes and dynamical effects. We corroborate trust in the accuracy of the method by comparing it with exact theoretical results for ground state properties of the sine-Gordon model. We then proceed with discussing $(1+1)$-dimensional quantum electrodynamics, the massive Schwinger model, for which we accurately locate its critical point and study the growth and saturation of momentum-space entanglement in sudden quenches.

quant-ph↗

arXiv2022

Tensor network study of the spin-1/2 Heisenberg anti-ferromagnet on the Shuriken lattice

We investigate the ground state of the spin $S=1/2$ Heisenberg anti-ferromagnet on the Shuriken lattice, also in the presence of an external magnetic field. To this end, we employ two-dimensional tensor network techniques based on infinite projected entangled pair and simplex states considering states with different sizes of the unit cells. We show that a valence bond crystal with resonances over length six loops emerges as the ground state (at any given finite bond dimension) yielding the lowest reported estimate of the ground state energy $E_0/J = -0.4410 \pm 0.0001$ for this model, estimated in the thermodynamic limit. We also study the model in the presence of an external magnetic field and find the emergence of $0$, $1/3$ and $2/3$ magnetization plateaus with states respecting translation and point group symmetries that feature loop-four plaquette resonances instead.

cond-mat.str-el↗