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Claudius Hubig

Postdoc in the group of Ignacio Cirac at the Max Planck Institute of Quantum Optics in Garching, working on the development, implementation and application of tensor network algorithms.

Contact Details

Max-Planck-Institut für Quantenoptik
Hans-Kopfermann-Str. 1
85748 Garching
arXiv author page · at MPQ


I will be leaving academia in February 2020. If you have any questions, queries etc., please get in touch.

Current Research Interests

Research is focused on numerical and algorithmic improvements to tensor network methods in condensed matter physics and related fields such as quantum chemistry. Examples include advances in convergence helpers for ground-state search methods, a detailled study of the errors in time evolution methods and the implementation of automatic differenation as used in gradient descent methods in tensor network applications. When applied to physical problem settings, both one-dimensional matrix-product state methods and two-dimensional infinite projected entangled pair states are utilised.

Work on the tensor network toolkit SyTen has been ongoing since 2015. SyTen is capable of handling arbitrary-rank tensors with named indices, non-abelian symmetries such as SU(2)-Spin, automatic handling of fermionic contractions and automatic differentiation while providing a high degree of parallelisation. Standard MPS and iPEPS tools are implemented atop of the basic tensor library, with further extensions facilitated by Python bindings and well-documented code.


  1. Evaluation of time-dependent correlators after a local quench in iPEPS: hole motion in the t-J model
    Claudius Hubig, Annabelle Bohrdt, Michael Knap, Fabian Grusdt, J. Ignacio Cirac

    In standard iPEPS calculation, the evaluation of time-dependent correlators and the simulation of local quenches are not possible. Here, we show that by introducing auxiliary states on each site, we can simulate both local excitations and evaluate non-equal-time correlators in an iPEPS setting under real-time evolution. We showcase the method by simulating the t-J model after a single hole has been placed in the half-filled antiferromagnetic background and evaluating both return probabilities and spin correlation functions, as accessible in quantum gas microscopes.

  2. Open quantum systems in thermal non-ergodic environments
    Carlos A. Parra-Murillo, Max Bramberger, Claudius Hubig, Inés de Vega

    We show that for thermal non-Harmonic environments the correlation function of this environment may not decay to zero but to an offset. The presence of such an offset is determined by the environment eigenstate structure and whether it fulfills the eigenstate thermalization hypothesis. Furthermore, it may render the weak coupling approximation inaccurate and prevent the open system to thermalize. For a realistic environment of dye molecules, we show the emergence of the offset by using matrix product states (MPS).

  3. Absence of superconductivity in the pure two-dimensional Hubbard model
    Mingpu Qin, Chia-Min Chung, Hao Shi, Ettore Vitali, Claudius Hubig, Ulrich Schollwöck, Steven R. White, Shiwei Zhang

    A combined study of the nearest-neighbour Hubbard model at filling n=7/8 and U=4 and U=8 using MPS-DMRG, system-environment DMRG and AFQMC shows the absence of long-range superconducting order in this model.

  4. Imaginary-time matrix product state impurity solver in a real material calculation: Spin-orbit coupling in Sr2RuO4
    Nils-Oliver Linden, Manuel Zingl, Claudius Hubig, Olivier Parcollet, Ulrich Schollwöck

    MPS-based DMFT impurity solver applied to a real material while incorporating spin-orbit coupling at zero temperature, a regime where previous CTQMC-based solvers encountered a severe sign problem.

  5. Use and implementation of autodifferentiation in tensor network methods with complex scalars
    Claudius Hubig

    Sketches the implementation of reverse-mode autodifferentiation in tensor network toolkits and stresses that this is not too hard. Also discusses the extension of autodifferentiation to calculations based on complex scalars.

  6. Gaussian TDVP for the Bose-Hubbard model
    Tommaso Guaita, Lucas Hackl, Tao Shi, Claudius Hubig, Eugene Demler, J. Ignacio Cirac

    We systematically extend Bogoliubov theory beyond the mean field approximation of the Bose- Hubbard model in the superfluid phase. Our approach is based on the time dependent variational principle applied to the family of all Gaussian states (i.e., Gaussian TDVP). First, we find the best ground state approximation within our variational class using imaginary time evolution in 1d, 2d and 3d. We benchmark our results by comparing to Bogoliubov theory and DMRG in 1d. Second, we compute the approximate 1- and 2-particle excitation spectrum as eigenvalues of the linearized projected equations of motion (linearized TDVP).

    Phys. Rev B 100, 094529 arXiv:1907.04837
  7. Dynamical topological quantum phase transitions in nonintegrable models
    Imre Hagymási, Claudius Hubig, Örs Legeza, Ulrich Schollwöck

    We consider sudden quenches across quantum phase transitions in the S=1 XXZ model starting from the Haldane phase. We demonstrate that dynamical phase transitions may occur during these quenches that are identified by nonanalyticities in the rate function for the return probability. In addition, we show that the temporal behavior of the string order parameter is intimately related to the subsequent dynamical phase transitions. We furthermore find that the dynamical quantum phase transitions can be accompanied by enhanced two-site entanglement.

    arXiv:1904.00867 Phys. Rev Lett. 122, 250601
  8. Quantum phases and topological properties of interacting fermions in one-dimensional superlattices
    Leo Stenzel, Andrew L. C. Hayward, Claudius Hubig, Ulrich Schollwöck, Fabian Heidrich-Meisner

    A very extensive study of the possible quantum phases of interacting fermions in effective one-dimensional superlattices. For certain filling factors, a series of different phases from band insulators to other topological insulators is observed, which are then linked to topological transitions observed in the Chern numbers of the whole manifold of states.

    arXiv:1903.06108 Phys. Rev A 99, 053614
  9. Time-evolution methods for matrix-product states
    Sebastian Paeckel, Thomas Köhler, Andreas Swoboda, Salvatore R. Manmana, Ulrich Schollwöck and Claudius Hubig

    An overview and detailed description of time-evolution methods for matrix-product states. Discusses TEBD, the MPO WI,II method, the global Krylov method, the local Krylov method and the time-dependent variational principle. Also includes a lengthy section on additional tricks to be combined with any of the methods described before and four exemplary applications in condensed-matter physics to test the advantages of each individual method.

    Spectrum of a XXZ Heisenberg chain obtained using 2TDVP

    Exemplary excitation spectrum of a near-Ising XXZ Heisenberg chain obtained using two-site TDVP-MPS within a few CPU hours.

    arXiv:1901.05824 Annals of Physics 411, 167998
  10. Time-dependent study of disordered models with infinite projected entangled pair states
    Claudius Hubig and J. Ignacio Cirac

    Applies real-time evolution operators to an infinite projected entangled pair state to study the averaged dynamics of the Néel state under the Heisenberg square lattice Hamiltonian with bi-valued disorder. The primary challenge was in the stability of the corner transfer matrix used for the more precise full update mechanism here. However, careful choice of parameters allowed us to obtain times beyond the reach of MPS, ED or NLCE calculations and demonstrate a slow-down of the dynamics in the presence of the bi-valued disorder field (implemented by an ancilla Hilbert space).

    Evolution of local observables during 2D iPEPS calculations

    Evolution of ⟨sz(t)⟩ starting from a Néel state under the Heisenberg XXX Hamiltonian with increasingly stronger disorder. MPS calculations at zero disorder exhibit finite-size effects at t < 1, while time-dependent iPEPS can obtain longer times and also demonstrate a marked slowdown of dynamics, which may hint towards many-body localisation.

    arXiv:1812.03801 SciPost Phys. 6, 031
  11. Thermal control of spin excitations in the coupled Ising-chain material RbCoCl3
    Mattia Mena, Nora Hänni, Simon Ward, Eva Hirtenlechner, Robert Bewley, Claudius Hubig, Ulrich Schollwöck, Bruce Normand, Karl W. Krämer, Des F. McMorrow and Christian Rüegg

    Using finite-temperature time-dependent MPS (with TDVP), it was possible to obtain excitation spectra which precisely matched the data experimentally obtained using neutron scattering. In particular, it was possible to derive the temperature-dependence of the spectrum essentially from first principles and model it without free parameters, which allows for the understanding of two phase transitions present in the material at finite temperature.

  12. Density-matrix embedding theory study of the one-dimensional Hubbard-Holstein model
    Teresa E. Reinhard, Uliana Mordovina, Claudius Hubig, Joshua S. Kretchmer, Ulrich Schollwöck, Heiko Appel, Michael A. Sentef and Angel Rubio

    An extension of DMET to handle mixed bosonic/fermionic systems and solution of the subsequent impurity problem using DMRG. The resulting method is applied to the Hubbard-Holstein model in one dimension to obtain the single-particle excitation gap and by extension allow for a study of the phase diagram.

    arXiv:1811.00048 J. Chem. Theory Comput. 15 (4), 2221
  13. Interaction quench and thermalization in a one-dimensional topological Kondo insulator
    Imre Hagymási, Claudius Hubig and Ulrich Schollwöck

    Study of the non-equilibrium dynamics of a one-dimensional topological Kondo insulator modelled by a p-wave Anderson lattice model. The system is quenched by a change of the on-site interaction strength and topological quantities are studied in a real-time evolution after the quench. Up to a certain interaction strength, observables can be captured well by a thermal ensemble while topological properties are preserved.

    arXiv:1810.09799 Phys. Rev. B 99, 075145
  14. Abelian and non-abelian symmetries in infinite projected entangled pair states
    Claudius Hubig

    Studies the effect of implementing (that is, preserving exactly) abelian and non-abelian symmetries at the example of the Heisenberg model on the square and kagome lattice in two dimensions.
    It was found that (a) using the symmetries greatly improves computational efficiency and (b) implementing the symmetry also tends to force the state into a symmetry-preserving phase, leading to marked signals in the variational energy obtained at phase transitions.

    Detection of spontaneous symmetry breaking in 2D

    Symmetry breaking detection in the XXZ Heisenberg model: at Δ<1, the U(1) symmetry is spontaneously broken, resulting in a higher energy if tensors are restricted to be U(1) symmetric (“U(1)”) while the energy curve for unrestricted tensors (“None”) shows a kink. Monte Carlo data at Δ=1 is given as a reference (“MC”).

    1808.10804 SciPost Phys. 5, 047
  15. Error estimates for extrapolations with matrix-product states
    Claudius Hubig, Jutho Haegeman and Ulrich Schollwöck

    Using single-site DMRG, much larger bond dimensions and much lower variational energies are obtainable than with two-site DMRG due to computational speed-ups and reduced memory usage. However, single-site DMRG does not produce a reliable error measure which allows extrapolation of the observables to the zero-error limit. Here, we introduce an approximation to the variance ⟨H²⟩-⟨H⟩² which allows for the calculation of a reliable error measure even in challenging cases and hence extrapolation of the variational data to the zero-error case.

    Comparison of error estimates for extrapolations in 1D

    Comparison of the two-site variance approximation introduced here and the 2DMRG truncation error applied to a 10×4 Hubbard cylinder. Both methods can extrapolate their variational results to comparable ground-state energy estimates, but by using single-site DMRG and only the two-site variance, much larger bond dimensions and hence lower variational energies are obtained.

    1711.01104 Phys. Rev. B 97, 045125
  16. Doped Kondo chain, a heavy Luttinger liquid
    Ilia Khait, Patrick Azaria, Claudius Hubig, Ulrich Schollwöck and Assa Auerbach

    A study of the one-dimensional Kondo lattice model using SU(2)-symmetric DMRG beyond the analytically approachable large-N limit. Found signatures of a heavy Tomonaga-Luttinger liquid phase and divergent charge and spin susceptibilities at the large Fermi surface.

    1710.04847 PNAS 115, 5140
  17. Spinon confinement in a quasi one dimensional anisotropic Heisenberg magnet
    Anup K. Bera, Bella Lake, Fabian H. L. Essler, Laurens Vanderstraeten, Claudius Hubig, Ulrich Schollwöck, A. T. M. Nazmul Islam, Astrid Schneidewind and Diana L. Quintero-Castro

    A study of spinon confinement in a one-dimensional Heisenberg chain in the presence of an external staggered field. Ground-state DMRG results on the gap and time-dependent MPS data of the spectrum could qualitatively reproduce the experimental results with quantitative agreement upon use of a tangent-space MPS method.

    1705.01259 Phys. Rev. B 96, 054423
  18. Generic construction of efficient matrix product operators
    Claudius Hubig, Ian P. McCulloch and Ulrich Schollwöck

    Presents a generic method to algorithmically construct efficient matrix-product operators (MPOs). Apart from established MPO arithmetic, two lossless MPO compression schemes are presented which ensure optimality of the resulting MPO. Furthermore may serve as an introduction to the use of MPOs with some explicit examples provided.

    1611.02498 Phys. Rev. B 95, 035129
  19. Universal long-time behavior of stochastically driven interacting quantum systems
    Zi Cai, Claudius Hubig and Ulrich Schollwöck

    A systematic study the long-time relaxation dynamics in stochastically driven interacting quantum systems. Finds that even though the stochastic forces will inevitably drive the systems into a featureless infinite temperature state, the way to approach the steady state can be highly nontrivial and exhibit rich universal dynamical behavior determined by the interplay between the stochastic driving and quantum many-body effects.

    1609.08518 Phys. Rev. B 96, 054303
  20. Strictly single-site DMRG algorithm with subspace expansion
    Claudius Hubig, Ian P. McCulloch, Ulrich Schollwöck and F. Alexander Wolf

    Introduces a strictly single-site DMRG variant using subspace expansion to adapt the auxiliary basis size. This variant improves both runtime and memory usage over the traditional two-site DMRG and the centermatrix wavefunction approach.

    The DMRG method introduced in this paper is the default method used in the SyTen toolkit.

    Comparison of the new DMRG3S and the CWF approach

    Error in energy and runtime for the introduced single site variant (“DMRG3S”) and the previous centermatrix wavefunction approach (“CWF“) when applied to free fermions on a one-dimensional chain. Traditional two-site DMRG was slower by a factor of two to three than the CWF approach shown here.

    1501.05504 Phys. Rev. B 91, 155115


  1. Symmetry-Protected Tensor Networks
    PhD thesis submitted to the Faculty of Physics at LMU Munich on 2017-08-31, supervised by Ulrich Schollwöck

    A detailed introduction to symmetry-protection in tensor networks in Chapter 2, which is useful as an overview for the SyTen tensor network implementation and also discusses matrix-product states and operators (cf.~paper 3 above). Subsequent chapters 3 and 4 summarise papers 1 and 4 above. The last chapter 5 sums up my work on the two-dimensional Hubbard model using MPS-DMRG on finite-size cylinders.

    Direct link at LMU
  2. Spin Dynamics in Adiabatic Transport through Quantum Dots
    BSc thesis submitted to the Faculty of Physics at TU Berlin on 2012-07-03, supervised by Anja Metelmann and Tobias Brandes

    Direct link

Workshops Co-organised





Summer 2017
Supervision of Bachelor thesis project: Study of a Sampling Method for Singular Values
Summer 2016
Tutoring for Numerische Mathematik (BSc, LMU)
Summer 2016
Supervision of Bachelor thesis project: Parallelized Time Evolution on the Heisenberg Spin Chain with Matrix Product States
Summer 2015
Tutoring for Advanced Statistical Physics (MSc, LMU)
Winter 2014/15
Tutoring for Thermodynamik und Statistische Physik (BSc, LMU)
Summer 2014
Tutoring for Advanced Statistical Physics (MSc, LMU)


since 11/2017
Postdoc in the group of Ignacio Cirac, Max Planck Institute of Quantum Optics, Garching, Germany.
10/2013 - 10/2017
PhD Student at the Chair of Uli Schollwöck, LMU Munich, Germany. Funded by the ExQM Graduate School and the Nanosystems Initiative Munich. Title of thesis: Symmetry-Protected Tensor Networks.
09/2012 - 07/2013
Read Applied Mathematics (Part III) at Hughes Hall, University of Cambridge, United Kingdom. Graduated with Masters of Advanced Studies in Applied Mathematics.
10/2009 - 07/2012
Read Physics at TU Berlin. Graduated with Bachelor of Science in Physics, title of thesis: Spin Dynamics in Adiabatic Transport through Quantum Dots. Partially funded by Deutschlandstipendium in 2011/2012.