SciPost Physics Lecture Notes

Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy)

Johannes Hauschild, Frank Pollmann

SciPost Phys. Lect. Notes 5 (2018) · published 8 October 2018

• doi: 10.21468/SciPostPhysLectNotes.5
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Abstract

Tensor product state (TPS) based methods are powerful tools to efficiently simulate quantum many-body systems in and out of equilibrium. In particular, the one-dimensional matrix-product (MPS) formalism is by now an established tool in condensed matter theory and quantum chemistry. In these lecture notes, we combine a compact review of basic TPS concepts with the introduction of a versatile tensor library for Python (TeNPy) [https://github.com/tenpy/tenpy]. As concrete examples, we consider the MPS based time-evolving block decimation and the density matrix renormalization group algorithm. Moreover, we provide a practical guide on how to implement abelian symmetries (e.g., a particle number conservation) to accelerate tensor operations.

• Sompet et al., Realizing the symmetry-protected Haldane phase in Fermi–Hubbard ladders
  Nature 606, 484 (2022) [Crossref]
• Rodriguez-Nieva et al., Transverse instability and universal decay of spin spiral order in the Heisenberg model
  Phys. Rev. B 105, L060302 (2022) [Crossref]
• Su et al., Observation of many-body scarring in a Bose-Hubbard quantum simulator
  Phys. Rev. Research 5, 023010 (2023) [Crossref]
• Roy et al., The quantum sine-Gordon model with quantum circuits
  Nuclear Physics B 968, 115445 115445 (2021) [Crossref]
• Kottmann et al., Supersolid-superfluid phase separation in the extended Bose-Hubbard model
  Phys. Rev. B 104, 174514 (2021) [Crossref]
• Coissard et al., Imaging tunable quantum Hall broken-symmetry orders in graphene
  Nature 605, 51 (2022) [Crossref]
• Wouters et al., Phase diagram of an extended parafermion chain
  SciPost Phys. Core 5, 008 (2022) [Crossref]
• Alsallom et al., Fate of the non-Hermitian skin effect in many-body fermionic systems
  Phys. Rev. Research 4, 033122 (2022) [Crossref]
• Morera et al., Quantum droplets of bosonic mixtures in a one-dimensional optical lattice
  Phys. Rev. Research 2, 022008 (2020) [Crossref]
• Rzepkowski et al., DMRG and Monte Carlo studies of CrI3 magnetic phases and the phase transition
  Physica E: Low-dimensional Systems and Nanostructures 146, 115520 115520 (2023) [Crossref]
• Gao et al., Automatic transformation of irreducible representations for efficient contraction of tensors with cyclic group symmetry
  SciPost Phys. Codebases, 10 (2023) [Crossref]
• Kupczyński et al., DMRG analysis of magnetic order in the zigzag edges of hexagonal CrN nanoribbons
  Phys. Rev. B 107, 205418 (2023) [Crossref]
• Cookmeyer et al., Four-Spin Terms and the Origin of the Chiral Spin Liquid in Mott Insulators on the Triangular Lattice
  Phys. Rev. Lett. 127, 087201 (2021) [Crossref]
• Zechmann et al., Fractonic Luttinger liquids and supersolids in a constrained Bose-Hubbard model
  Phys. Rev. B 107, 195131 (2023) [Crossref]
• Fishman et al., The ITensor Software Library for Tensor Network Calculations
  SciPost Phys. Codebases, 4 (2022) [Crossref]
• [undetermined], Publisher’s Note
  Annals of Physics 431, 168552 168552 (2021) [Crossref]
• Villa et al., Quench spectroscopy of a disordered quantum system
  Phys. Rev. A 104, L021301 (2021) [Crossref]
• Andrews et al., Stability, phase transitions, and numerical breakdown of fractional Chern insulators in higher Chern bands of the Hofstadter model
  Phys. Rev. B 104, 125107 (2021) [Crossref]
• Li et al., Numerical assessment for accuracy and GPU acceleration of TD-DMRG time evolution schemes
  J. Chem. Phys. 152, 024127 (2020) [Crossref]
• Cardarelli et al., Accessing the topological Mott insulator in cold atom quantum simulators with realistic Rydberg dressing
  Quantum Sci. Technol. 8, 025018 (2023) [Crossref]
• Iraola et al., Towards a topological quantum chemistry description of correlated systems: The case of the Hubbard diamond chain
  Phys. Rev. B 104, 195125 (2021) [Crossref]
• Roy et al., Entanglement Hamiltonian of the 1 + 1-dimensional free, compactified boson conformal field theory
  J. Stat. Mech. 2020, 083104 (2020) [Crossref]
• Bibo et al., Fractional corner charges in a two-dimensional superlattice Bose-Hubbard model
  Phys. Rev. B 102, 041126 (2020) [Crossref]
• Morera et al., Universal Dimerized Quantum Droplets in a One-Dimensional Lattice
  Phys. Rev. Lett. 126, 023001 (2021) [Crossref]
• Ge et al., Quantum simulation of lattice gauge theories on superconducting circuits: Quantum phase transition and quench dynamics
  Chinese Phys. B 31, 020304 (2021) [Crossref]
• Jin et al., Matrix product states for Hartree-Fock-Bogoliubov wave functions
  Phys. Rev. B 105, L081101 (2022) [Crossref]
• Devakul et al., Quantum Anomalous Hall Effect from Inverted Charge Transfer Gap
  Phys. Rev. X 12, 021031 (2022) [Crossref]
• Peña et al., Stable Many-Body Resonances in Open Quantum Systems
  Symmetry 14, 2562 (2022) [Crossref]
• Huang et al., Coupled dimer and bond-order-wave order in the quarter-filled one-dimensional Kondo lattice model
  Phys. Rev. B 102, 245143 (2020) [Crossref]
• Feldmeier et al., Emergent tracer dynamics in constrained quantum systems
  Phys. Rev. B 106, 094303 (2022) [Crossref]
• Verresen et al., Unifying Kitaev Magnets, Kagomé Dimer Models, and Ruby Rydberg Spin Liquids
  Phys. Rev. X 12, 041029 (2022) [Crossref]
• Roy et al., Quantum phases of a one-dimensional Majorana-Bose-Hubbard model
  Phys. Rev. B 101, 075419 (2020) [Crossref]
• Dong et al., Spin-1 Kitaev-Heisenberg model on a honeycomb lattice
  Phys. Rev. B 102, 121102 (2020) [Crossref]
• Zechmann et al., Tunable transport in the mass-imbalanced Fermi-Hubbard model
  Phys. Rev. B 106, 075115 (2022) [Crossref]
• Scholl et al., Quantum simulation of 2D antiferromagnets with hundreds of Rydberg atoms
  Nature 595, 233 (2021) [Crossref]
• Wang et al., Calculating Rényi entropies with neural autoregressive quantum states
  Phys. Rev. A 102, 062413 (2020) [Crossref]
• Haller et al., Quantum skyrmion lattices in Heisenberg ferromagnets
  Phys. Rev. Research 4, 043113 (2022) [Crossref]
• Stéphan, Free fermions at the edge of interacting systems
  SciPost Phys. 6, 057 (2019) [Crossref]
• Zhang et al., SU(4) Chiral Spin Liquid, Exciton Supersolid, and Electric Detection in Moiré Bilayers
  Phys. Rev. Lett. 127, 247701 (2021) [Crossref]
• Moca et al., Simulating Lindbladian evolution with non-Abelian symmetries: Ballistic front propagation in the SU(2) Hubbard model with a localized loss
  Phys. Rev. B 105, 195144 (2022) [Crossref]
• Schiffer et al., Adiabatic Spectroscopy and a Variational Quantum Adiabatic Algorithm
  PRX Quantum 3, 020347 (2022) [Crossref]
• Tang et al., Measurement-induced phase transition: A case study in the nonintegrable model by density-matrix renormalization group calculations
  Phys. Rev. Research 2, 013022 (2020) [Crossref]
• Rigobello et al., Entanglement generation in (1+1)D QED scattering processes
  Phys. Rev. D 104, 114501 (2021) [Crossref]
• Argüello-Luengo et al., Tuning Long-Range Fermion-Mediated Interactions in Cold-Atom Quantum Simulators
  Phys. Rev. Lett. 129, 083401 (2022) [Crossref]
• Doggen et al., Many-body localization in a tilted potential in two dimensions
  Phys. Rev. B 105, 134204 (2022) [Crossref]
• Tada, Quantum phase transition and criticality in quasi-one-dimensional spinless Dirac fermions
  Phys. Rev. B 100, 125145 (2019) [Crossref]
• Haldar et al., Signatures of Quantum Phase Transitions after Quenches in Quantum Chaotic One-Dimensional Systems
  Phys. Rev. X 11, 031062 (2021) [Crossref]
• Kuno et al., Topological pump and bulk-edge-correspondence in an extended Bose-Hubbard model
  Phys. Rev. B 104, 125146 (2021) [Crossref]
• Kottmann et al., Variational quantum anomaly detection: Unsupervised mapping of phase diagrams on a physical quantum computer
  Phys. Rev. Research 3, 043184 (2021) [Crossref]
• Kraus et al., Quantum phases of dipolar bosons in one-dimensional optical lattices
  Phys. Rev. B 106, 035144 (2022) [Crossref]
• You et al., Higher-order entanglement and many-body invariants for higher-order topological phases
  Phys. Rev. Research 2, 033192 (2020) [Crossref]
• Chen et al., Continuous symmetry breaking in a two-dimensional Rydberg array
  Nature 616, 691 (2023) [Crossref]
• Wybo et al., Entanglement dynamics of a many-body localized system coupled to a bath
  Phys. Rev. B 102, 064304 (2020) [Crossref]
• Peña et al., Fractional resonances and prethermal states in Floquet systems
  Phys. Rev. B 106, 064307 (2022) [Crossref]
• Araz et al., Quantum-inspired event reconstruction with Tensor Networks: Matrix Product States
  J. High Energ. Phys. 2021, 112 (2021) [Crossref]
• Cataldi et al., Hilbert curve vs Hilbert space: exploiting fractal 2D covering to increase tensor network efficiency
  Quantum 5, 556 556 (2021) [Crossref]
• Wang et al., Measurable signatures of bosonic fractional Chern insulator states and their fractional excitations in a quantum-gas microscope
  SciPost Phys. 12, 095 (2022) [Crossref]
• Morera et al., Universality of quantum liquids and droplets in one dimension
  Phys. Rev. Research 4, L042024 (2022) [Crossref]
• González-Cuadra, Higher-order topological quantum paramagnets
  Phys. Rev. B 105, L020403 (2022) [Crossref]
• Kottmann et al., Unsupervised Phase Discovery with Deep Anomaly Detection
  Phys. Rev. Lett. 125, 170603 (2020) [Crossref]
• Zhang et al., An improved quantum network communication model based on compressed tensor network states
  Quantum Inf Process 21, 253 (2022) [Crossref]
• Zhang et al., Transformer quantum state: A multipurpose model for quantum many-body problems
  Phys. Rev. B 107, 075147 (2023) [Crossref]
• González-Cuadra et al., Robust Topological Order in Fermionic Z2 Gauge Theories: From Aharonov-Bohm Instability to Soliton-Induced Deconfinement
  Phys. Rev. X 10, 041007 (2020) [Crossref]
• Hartmann et al., Intersublattice entanglement entropy as an extensive property in antiferromagnets
  Phys. Rev. B 104, 064436 (2021) [Crossref]
• Huang et al., Quantum phase diagram and chiral spin liquid in the extended spin- 12 honeycomb XY model
  Phys. Rev. B 103, L041108 (2021) [Crossref]
• Villa et al., Finding the phase diagram of strongly correlated disordered bosons using quantum quenches
  Phys. Rev. A 104, 023323 (2021) [Crossref]
• Jobst et al., Finite-depth scaling of infinite quantum circuits for quantum critical points
  Phys. Rev. Research 4, 033118 (2022) [Crossref]
• Koepsell et al., Microscopic evolution of doped Mott insulators from polaronic metal to Fermi liquid
  Science 374, 82 (2021) [Crossref]
• Kadow et al., Hole spectral function of a chiral spin liquid in the triangular lattice Hubbard model
  Phys. Rev. B 106, 094417 (2022) [Crossref]
• Li et al., Cluster dynamics in two-dimensional lattice gases with intersite interactions
  Phys. Rev. A 103, 043331 (2021) [Crossref]
• Lerose et al., Scaling of temporal entanglement in proximity to integrability
  Phys. Rev. B 104, 035137 (2021) [Crossref]
• Hauschild et al., Finding purifications with minimal entanglement
  Phys. Rev. B 98, 235163 (2018) [Crossref]
• Borla et al., Confined Phases of One-Dimensional Spinless Fermions Coupled to Z2 Gauge Theory
  Phys. Rev. Lett. 124, 120503 (2020) [Crossref]
• You et al., Fracton critical point at a higher-order topological phase transition
  Phys. Rev. B 106, 235130 (2022) [Crossref]
• Kuno et al., Plateau transitions of a spin pump and bulk-edge correspondence
  Phys. Rev. B 104, 045113 (2021) [Crossref]
• Lerose et al., Influence Matrix Approach to Many-Body Floquet Dynamics
  Phys. Rev. X 11, 021040 (2021) [Crossref]
• Zhang, Doping a Mott insulator with excitons in a moiré bilayer: Fractional superfluid, neutral Fermi surface, and Mott transition
  Phys. Rev. B 106, 195120 (2022) [Crossref]
• Szasz et al., Phase diagram of a bilinear-biquadratic spin-1 model on the triangular lattice from density matrix renormalization group simulations
  Phys. Rev. B 106, 115103 (2022) [Crossref]
• Birnkammer et al., Prethermalization in one-dimensional quantum many-body systems with confinement
  Nat Commun 13, 7663 (2022) [Crossref]
• Robertson et al., Quenches to the critical point of the three-state Potts model: Matrix product state simulations and conformal field theory
  Phys. Rev. B 105, 195103 (2022) [Crossref]
• Klein et al., Interacting stochastic topology and Mott transition from light response
  Phys. Rev. B 103, 035114 (2021) [Crossref]
• Fraxanet et al., Topological Quantum Critical Points in the Extended Bose-Hubbard Model
  Phys. Rev. Lett. 128, 043402 (2022) [Crossref]
• Ronetti et al., Fractional spin excitations and conductance in the spiral-staircase Heisenberg ladder
  Phys. Rev. B 105, 134413 (2022) [Crossref]
• Soejima et al., Efficient simulation of moiré materials using the density matrix renormalization group
  Phys. Rev. B 102, 205111 (2020) [Crossref]
• Huang et al., Coexistence of non-Abelian chiral spin liquid and magnetic order in a spin-1 antiferromagnet
  Phys. Rev. B 105, 155104 (2022) [Crossref]
• Tada, Quantum criticality of magnetic catalysis in two-dimensional correlated Dirac fermions
  Phys. Rev. Research 2, 033363 (2020) [Crossref]
• Bohrdt et al., Dominant Fifth-Order Correlations in Doped Quantum Antiferromagnets
  Phys. Rev. Lett. 126, 026401 (2021) [Crossref]
• González-Cuadra et al., Symmetry-breaking topological insulators in the Z2 Bose-Hubbard model
  Phys. Rev. B 99, 045139 (2019) [Crossref]
• González-Cuadra et al., Dynamical Solitons and Boson Fractionalization in Cold-Atom Topological Insulators
  Phys. Rev. Lett. 125, 265301 (2020) [Crossref]
• Rogerson et al., Entanglement entropy and negativity in the Ising model with defects
  J. High Energ. Phys. 2022, 165 (2022) [Crossref]
• Zhang et al., Pair-density-wave superconductor from doping Haldane chain and rung-singlet ladder
  Phys. Rev. B 106, 045103 (2022) [Crossref]
• Lee et al., Detection of a quantum phase transition in a spin-1 chain through multipartite high-order correlations
  Phys. Rev. A 106, 042432 (2022) [Crossref]
• Brenig, Spinless fermions in a Z2 gauge theory on a triangular ladder
  Phys. Rev. B 105, 245105 (2022) [Crossref]
• Claeys et al., Absence of Superdiffusion in Certain Random Spin Models
  Phys. Rev. Lett. 128, 246603 (2022) [Crossref]
• Tindall et al., Quantum physics in connected worlds
  Nat Commun 13, 7445 (2022) [Crossref]
• Wang et al., Robust and ultrafast state preparation by ramping artificial gauge potentials
  New J. Phys. 23, 063017 (2021) [Crossref]
• Bello et al., Spin Many-Body Phases in Standard- and Topological-Waveguide QED Simulators
  PRX Quantum 3, 010336 (2022) [Crossref]
• Morera et al., Entanglement entropy in low-energy field theories at a finite chemical potential
  Phys. Rev. Research 2, 033016 (2020) [Crossref]
• Grusdt et al., Microscopic spinon-chargon theory of magnetic polarons in the t−J model
  Phys. Rev. B 99, 224422 (2019) [Crossref]
• González-Cuadra et al., Rotor Jackiw-Rebbi Model: A Cold-Atom Approach to Chiral Symmetry Restoration and Charge Confinement
  PRX Quantum 1, 020321 (2020) [Crossref]
• Jones et al., Skeleton of matrix-product-state-solvable models connecting topological phases of matter
  Phys. Rev. Research 3, 033265 (2021) [Crossref]
• Vecsei et al., Lee-Yang theory of the two-dimensional quantum Ising model
  Phys. Rev. B 106, 054402 (2022) [Crossref]
• Parker et al., Local matrix product operators: Canonical form, compression, and control theory
  Phys. Rev. B 102, 035147 (2020) [Crossref]
• Hidaka et al., Gapless symmetry-protected topological phase of quantum antiferromagnets on anisotropic triangular strip
  Phys. Rev. B 106, 144436 (2022) [Crossref]
• Tsitsishvili et al., Phase diagram of Rydberg-dressed atoms on two-leg square ladders: Coupling supersymmetric conformal field theories on the lattice
  Phys. Rev. B 105, 155159 (2022) [Crossref]
• Claeys et al., Emergent quantum state designs and biunitarity in dual-unitary circuit dynamics
  Quantum 6, 738 738 (2022) [Crossref]
• Bauer et al., Mechkit: A continuum mechanics toolkit in Python
  JOSS 7, 4389 (2022) [Crossref]
• Kebrič et al., Confinement induced frustration in a one-dimensional Z2 lattice gauge theory
  New J. Phys. 25, 013035 (2023) [Crossref]
• Chung et al., Tunable Single-Ion Anisotropy in Spin-1 Models Realized with Ultracold Atoms
  Phys. Rev. Lett. 126, 163203 (2021) [Crossref]
• Wouters et al., Interrelations among frustration-free models via Witten's conjugation
  SciPost Phys. Core 4, 027 (2021) [Crossref]
• Fromholz et al., Phase diagram of Rydberg-dressed atoms on two-leg triangular ladders
  Phys. Rev. B 106, 155411 (2022) [Crossref]
• Sim et al., Nonlinear spectroscopy of bound states in perturbed Ising spin chains
  Phys. Rev. B 107, L100404 (2023) [Crossref]
• East et al., AKLT-States as ZX-Diagrams: Diagrammatic Reasoning for Quantum States
  PRX Quantum 3, 010302 (2022) [Crossref]
• Repellin et al., Ferromagnetism in Narrow Bands of Moiré Superlattices
  Phys. Rev. Lett. 124, 187601 (2020) [Crossref]
• Štrkalj et al., Many-body localization in the interpolating Aubry-André-Fibonacci model
  Phys. Rev. Research 3, 033257 (2021) [Crossref]
• Kuno et al., Interaction-induced topological charge pump
  Phys. Rev. Research 2, 042024 (2020) [Crossref]
• Garratt et al., Measurements Conspire Nonlocally to Restructure Critical Quantum States
  Phys. Rev. X 13, 021026 (2023) [Crossref]
• Villa et al., Local quench spectroscopy of many-body quantum systems
  Phys. Rev. A 102, 033337 (2020) [Crossref]
• Slagle, Fast tensor disentangling algorithm
  SciPost Phys. 11, 056 (2021) [Crossref]
• Bohrdt et al., Dynamical formation of a magnetic polaron in a two-dimensional quantum antiferromagnet
  New J. Phys. 22, 123023 (2020) [Crossref]
• Wybo et al., Quantum sine-Gordon dynamics in coupled spin chains
  Phys. Rev. B 106, 075102 (2022) [Crossref]
• Scherg et al., Observing non-ergodicity due to kinetic constraints in tilted Fermi-Hubbard chains
  Nat Commun 12, 4490 (2021) [Crossref]
• Sous et al., Phonon-induced disorder in dynamics of optically pumped metals from nonlinear electron-phonon coupling
  Nat Commun 12, 5803 (2021) [Crossref]
• Doggen et al., Generalized quantum measurements with matrix product states: Entanglement phase transition and clusterization
  Phys. Rev. Research 4, 023146 (2022) [Crossref]
• de Felice et al., DisCoPy: Monoidal Categories in Python
  Electron. Proc. Theor. Comput. Sci. 333, 183 (2021) [Crossref]
• Jin et al., Bosonization of the interacting Su-Schrieffer-Heeger model
  Phys. Rev. B 107, L201111 (2023) [Crossref]
• Darkwah Oppong et al., Probing Transport and Slow Relaxation in the Mass-Imbalanced Fermi-Hubbard Model
  Phys. Rev. X 12, 031026 (2022) [Crossref]
• Kloss et al., Spin transport in disordered long-range interacting spin chain
  Phys. Rev. B 102, 060201 (2020) [Crossref]
• Kishony et al., Converting electrons into emergent fermions at a superconductor–Kitaev spin liquid interface
  Phys. Rev. B 104, 235118 (2021) [Crossref]
• Bhakuni et al., Interaction-driven phase transition in one dimensional mirror-symmetry protected topological insulator
  SciPost Phys. Core 5, 048 (2022) [Crossref]
• Rai et al., Bulk topological signatures of a quasicrystal
  Phys. Rev. B 104, 184202 (2021) [Crossref]
• Jepsen et al., Transverse Spin Dynamics in the Anisotropic Heisenberg Model Realized with Ultracold Atoms
  Phys. Rev. X 11, 041054 (2021) [Crossref]
• González-Cuadra et al., Intertwined topological phases induced by emergent symmetry protection
  Nat Commun 10, 2694 (2019) [Crossref]
• Seifert et al., Fractionalized Fermionic Quantum Criticality in Spin-Orbital Mott Insulators
  Phys. Rev. Lett. 125, 257202 (2020) [Crossref]
• Cervia et al., Collective neutrino oscillations with tensor networks using a time-dependent variational principle
  Phys. Rev. D 105, 123025 (2022) [Crossref]
• Dborin et al., Matrix product state pre-training for quantum machine learning
  Quantum Sci. Technol. 7, 035014 (2022) [Crossref]
• Kohlert et al., Exploring the Regime of Fragmentation in Strongly Tilted Fermi-Hubbard Chains
  Phys. Rev. Lett. 130, 010201 (2023) [Crossref]
• Kadow et al., Isometric tensor network representations of two-dimensional thermal states
  Phys. Rev. B 107, 205106 (2023) [Crossref]
• Hirthe et al., Magnetically mediated hole pairing in fermionic ladders of ultracold atoms
  Nature 613, 463 (2023) [Crossref]
• Denner et al., Efficient learning of a one-dimensional density functional theory
  Phys. Rev. Research 2, 033388 (2020) [Crossref]
• Sonner et al., Characterizing many-body localization via exact disorder-averaged quantum noise
  Phys. Rev. B 105, L020203 (2022) [Crossref]
• Bohrdt et al., Strong pairing in mixed-dimensional bilayer antiferromagnetic Mott insulators
  Nat. Phys. 18, 651 (2022) [Crossref]
• Birnkammer et al., Characterizing topological excitations of a long-range Heisenberg model with trapped ions
  Phys. Rev. B 105, L241103 (2022) [Crossref]
• Bohrdt et al., Rotational Resonances and Regge-like Trajectories in Lightly Doped Antiferromagnets
  Phys. Rev. Lett. 127, 197004 (2021) [Crossref]
• Hubig et al., Evaluation of time-dependent correlators after a local quench in iPEPS: hole motion in the t-J model
  SciPost Phys. 8, 021 (2020) [Crossref]
• Thorngren et al., Intrinsically gapless topological phases
  Phys. Rev. B 104, 075132 (2021) [Crossref]
• Ogino et al., Ground-state phase diagram of a spin- 12 frustrated XXZ ladder
  Phys. Rev. B 106, 155106 (2022) [Crossref]
• He et al., Superconductivity of repulsive spinless fermions with sublattice potentials
  Phys. Rev. Research 5, L012009 (2023) [Crossref]
• Quella, Symmetry-protected topological phases beyond groups: The q -deformed bilinear-biquadratic spin chain
  Phys. Rev. B 103, 054404 (2021) [Crossref]
• Lee et al., Bell-Type Correlation at Quantum Phase Transitions in Spin-1 Chain
  Entropy 22, 1282 (2020) [Crossref]
• Andrews et al., Abelian topological order of ν=2/5 and 3/7 fractional quantum Hall states in lattice models
  Phys. Rev. B 103, 075132 (2021) [Crossref]
• He et al., Conductance of correlated many-fermion systems from charge fluctuations
  Phys. Rev. B 105, 165120 (2022) [Crossref]
• Christandl et al., Optimization at the boundary of the tensor network variety
  Phys. Rev. B 103, 195139 (2021) [Crossref]
• Borla et al., Gauging the Kitaev chain
  SciPost Phys. 10, 148 (2021) [Crossref]
• González-Cuadra et al., Zn solitons in intertwined topological phases
  Phys. Rev. B 102, 245137 (2020) [Crossref]
• Evenbly, A Practical Guide to the Numerical Implementation of Tensor Networks I: Contractions, Decompositions, and Gauge Freedom
  Front. Appl. Math. Stat. 8, 806549 (2022) [Crossref]
• Tschirhart et al., Phase diagram of spin-1 chains with Dzyaloshinskii-Moriya interaction
  Phys. Rev. B 100, 195111 (2019) [Crossref]
• Schuckert et al., Probing finite-temperature observables in quantum simulators of spin systems with short-time dynamics
  Phys. Rev. B 107, L140410 (2023) [Crossref]
• Bañuls et al., From spin chains to real-time thermal field theory using tensor networks
  Phys. Rev. Research 2, 033301 (2020) [Crossref]
• Kiese et al., TMDs as a platform for spin liquid physics: A strong coupling study of twisted bilayer WSe2
  APL Materials 10, 031113 (2022) [Crossref]
• Korbmacher et al., Lattice control of nonergodicity in a polar lattice gas
  Phys. Rev. A 107, 013301 (2023) [Crossref]
• Pang et al.,
  , 1 (2020) [Crossref]
• Vovrosh et al., Confinement-induced impurity states in spin chains
  Phys. Rev. B 105, L100301 (2022) [Crossref]
• Lin et al., Efficient simulation of dynamics in two-dimensional quantum spin systems with isometric tensor networks
  Phys. Rev. B 106, 245102 (2022) [Crossref]
• Julià-Farré et al., Revealing the topological nature of the bond order wave in a strongly correlated quantum system
  Phys. Rev. Research 4, L032005 (2022) [Crossref]
• Tada, Robust orbital diamagnetism in correlated Dirac fermions
  New J. Phys. 24, 093018 (2022) [Crossref]
• Štrkalj et al., Coexistence of localization and transport in many-body two-dimensional Aubry-André models
  Phys. Rev. B 106, 184209 (2022) [Crossref]
• Mahyaeh et al., Phase diagram of the $\mathbb{Z}_3$-Fock parafermion chain with pair hopping
  SciPost Phys. Core 3, 011 (2020) [Crossref]
• Doggen et al., Stark many-body localization: Evidence for Hilbert-space shattering
  Phys. Rev. B 103, L100202 (2021) [Crossref]
• Bachmayr et al., Particle number conservation and block structures in matrix product states
  Calcolo 59, 24 (2022) [Crossref]
• Zheng et al., Ground-state properties via machine learning quantum constraints
  Phys. Rev. Research 4, L032043 (2022) [Crossref]
• Buessen et al., Prethermalization via self driving and external driving of extensive subsystems
  Phys. Rev. B 106, 174417 (2022) [Crossref]
• Méndez-Córdoba et al., From edge to bulk: Cavity-induced displacement of topological nonlocal qubits
  Phys. Rev. B 107, 125104 (2023) [Crossref]