DMRG Preprints

We investigate a non-Abelian SU$(2)$ quantum link model in 2+1 dimensions on a hexagonal lattice using tensor network methods. We determine the static quark potential for a wide range of bare coupling values and find that the theory is confining. We also probe the existence of a Luescher term and find a clear signal, however, the value of the dimensionless constant $γ$ strongly deviates from the expected universal value $-π/24$ for almost all values of the coupling $g^2$ we investigated. The width of the strings scales logarithmically with the string length again for all $g^2$-values, providing evidence for a rough string, with no indication for a roughening transition.

We report an extremely rich ground state phase diagram of two spin-1 Haldane chains frustrated with a three-site exchange and coupled by the antiferromagnetic Heisenberg interaction on a zig-zag ladder. A particular feature of the phase diagram is the extended quantum phase transition in the Ashkin-Teller universality class that separates the plaquette phase, which spontaneously breaks translation symmetry, and the uniform disordered phase. The former is connected to the Haldane phase, stabilized by large inter-chain coupling, via the topological Gaussian transition. Upon decreasing the inter-chain interactions, this intermediate disorder phase vanishes, giving place to a dimerized phase separated from the plaquette phase on one side via a non-magnetic Ising transition and from the Haldane phase on the other side by a topological weak first-order transition. Finally, in the limit of two decoupled chains, we recover a quantum critical point that corresponds to two copies of the Wess-Zumino-Witten $\mathrm{SU(2)}_2$ criticality with a total central charge $c=3$.

We study when a density-matching Kohn--Sham (KS) description can reproduce a many-body Berry phase in a correlated insulator, despite the fact that geometric phases are functionals of the wave function. Focusing on the one-dimensional SSH--Hubbard chain on a ring as a controlled interacting topological model, we introduce a $U(1)$ twist $θ$ (flux insertion). The many-body ground state along the full twist cycle is computed by the density-matrix renormalization group (DMRG), while the onsite interaction $U$ is tuned from the noninteracting to the strong-coupling regime. At half filling in the inversion-symmetric gapped regime, our DMRG calculations show that the density remains constant within numerical accuracy over the entire $(θ,U)$ range studied. Thus, the density has no dependence on either the flux $θ$ or the interaction strength $U$. Accordingly, the symmetry-preserving density constraint collapses the KS reference to an SSH-type quadratic representative with $U$-independent geometric diagnostics. Nevertheless, the many-body wave function exhibits a nontrivial geometric response: the quantum metric associated with the $θ$-parametrized ground-state manifold depends on $θ$ at intermediate $U$ and is strongly suppressed at large $U$, consistent with the charge fluctuation freezing. Intriguingly, the KS and many-body Berry phases coincide throughout the gapped regime as $U$ is tuned from weak to strong coupling. We show that this agreement is best understood as symmetry-enforced $ℤ_2$ sector matching, rather than as evidence that the density encodes the many-body Berry connection.

We construct simple qubit-regularized Hamiltonian lattice gauge theories formulated in the monomer--dimer--tensor-network (MDTN) basis that are free of sign problems in the pure gauge sector. These models naturally realize both confined and deconfined phases. Using classical Monte Carlo methods, we investigate the associated finite-temperature phase transitions and show that they exhibit the expected universality classes of conventional SU(N) lattice gauge theories in various spacetime dimensions. Furthermore, we argue that second-order quantum phase transitions separating the confined and deconfined phases are likely to exist. Such critical points would provide a nonperturbative route to defining continuum limits of qubit-regularized gauge theories, potentially allowing Yang--Mills theory and related continuum gauge theories to emerge from finite-dimensional lattice constructions.

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.

Representing the time-evolution operator as a tensor network constitutes a key ingredient in several algorithms for studying quantum lattice systems at finite temperature or in a non-equilibrium setting. For a Hamiltonian composed of strictly short-ranged interactions, the Suzuki-Trotter decomposition is the main technique for obtaining such a representation. In [B.~Vanhecke, L.~Vanderstraeten and F.~Verstraete, Physical Review A, L020402 (2021)], an alternative strategy, the cluster expansion, was introduced. This approach naturally preserves internal and lattice symmetries and can more easily be extended to higher-order representations or longer-ranged interactions. We extend the cluster expansion to two-dimensional fermionic systems, and employ it to construct projected entangled-pair operator (PEPO) approximations of Gibbs states. We also discuss and benchmark different truncation schemes for multiplying layers of PEPOs together. Applying the resulting framework to a two-dimensional spinless fermion model with attractive interactions, we resolve a clear phase boundary at finite temperature.

Open quantum systems host mixed-state phases that go beyond the symmetry-protected topological and spontaneous symmetry-breaking paradigms established for closed, pure-state systems. Developing a unified and physically transparent classification of such phases remains a central challenge. In this work, we introduce a purification-based framework that systematically characterizes all mixed-state phases in one-dimensional systems with $ℤ_2^σ \times ℤ_2^τ$ symmetry. By introducing an ancillary $κ$ chain and employing decorated domain-wall constructions, we derive eight purified fixed-point Hamiltonians labeled by topological indices $(μ_{στ},μ_{τκ},μ_{κσ}) \in \{\pm1\}^3$. Tracing out the ancilla recovers the full structure of mixed-state phases, including symmetric, strong-to-weak spontaneous symmetry breaking, average symmetry-protected topological phases, and their nontrivial combinations. Interpolations between the eight fixed points naturally define a three-dimensional phase diagram with a cube geometry. The edges correspond to elementary transitions associated with single topological indices, while the faces host intermediate phases arising from competing domain-wall decorations. Along the edges, we identify a class of critical behavior that connects distinct strong-to-weak symmetry-breaking patterns associated with distinct strong subgroups, highlighting a mechanism unique to mixed-state settings. Large-scale tensor-network simulations reveal a rich phase structure, including pyramid-shaped symmetry-breaking regions and a fully symmetry-broken phase at the cube center. Overall, our purification approach provides a geometrically transparent and physically complete classification of mixed-state phases, unified with a single $ℤ_2^σ \times ℤ_2^τ \times ℤ_2^κ$ model.

The numerical solution of algebraic tensor equations is a largely open and challenging task. Assuming that the operator is symmetric and positive definite, we propose two new gradient-descent type methods for tensor equations that generalize the recently proposed Subspace Conjugate Gradient (SS-CG), D. Palitta et al, SIAM J. Matrix Analysis and Appl (2025). As our interest is mainly in a modest number of tensor modes, the Tucker format is used to efficiently represent low-rank tensors. Moreover, mixed-precision strategies are employed in certain subtasks to improve the memory usage, and different preconditioners are applied to enhance convergence. The potential of our strategies is illustrated by experimental results on tensor-oriented discretizations of three-dimensional partial differential equations with separable coefficients. Comparisons with the state-of-the-art Alternating Minimal Energy (AMEn) algorithm confirm the competitiveness of the proposed strategies.

Fast matrix multiplication can be described as searching for low-rank decompositions of the matrix--multiplication tensor. We design a neural architecture, \textsc{StrassenNet}, which reproduces the Strassen algorithm for $2\times 2$ multiplication. Across many independent runs the network always converges to a rank-$7$ tensor, thus numerically recovering Strassen's optimal algorithm. We then train the same architecture on $3\times 3$ multiplication with rank $r\in\{19,\dots,23\}$. Our experiments reveal a clear numerical threshold: models with $r=23$ attain significantly lower validation error than those with $r\le 22$, suggesting that $r=23$ could actually be the smallest effective rank of the matrix multiplication tensor $3\times 3$. We also sketch an extension of the method to border-rank decompositions via an &acaron;repsilon--parametrisation and report preliminary results consistent with the known bounds for the border rank of the $3\times 3$ matrix--multiplication tensor.