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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
MaxPlanckInstitut für Quantenoptik
HansKopfermannStr. 1
85748 Garching
claudius.hubig@mpq.mpg.de
arXiv author page · at MPQ
Future
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 groundstate 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 onedimensional matrixproduct state methods and twodimensional infinite projected entangled pair states are utilised.
Work on the tensor network toolkit SyTen has been ongoing since 2015. SyTen is capable of handling arbitraryrank tensors with named indices, nonabelian 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 welldocumented code.
Papers

Evaluation of timedependent correlators after a local quench in iPEPS: hole
motion in the tJ model
Claudius Hubig, Annabelle Bohrdt, Michael Knap, Fabian Grusdt, J. Ignacio CiracIn standard iPEPS calculation, the evaluation of timedependent 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 nonequaltime correlators in an iPEPS setting under realtime evolution. We showcase the method by simulating the tJ model after a single hole has been placed in the halffilled antiferromagnetic background and evaluating both return probabilities and spin correlation functions, as accessible in quantum gas microscopes.
arXiv:1911.01159 
Open quantum systems in thermal nonergodic environments
Carlos A. ParraMurillo, Max Bramberger, Claudius Hubig, Inés de VegaWe show that for thermal nonHarmonic 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).
arXiv:1910.10496 
Absence of superconductivity in the pure twodimensional Hubbard
model
Mingpu Qin, ChiaMin Chung, Hao Shi, Ettore Vitali, Claudius Hubig, Ulrich Schollwöck, Steven R. White, Shiwei ZhangA combined study of the nearestneighbour Hubbard model at filling n=7/8 and U=4 and U=8 using MPSDMRG, systemenvironment DMRG and AFQMC shows the absence of longrange superconducting order in this model.
arXiv:1910.08931 
Imaginarytime matrix product state impurity solver in a real material
calculation: Spinorbit coupling in Sr2RuO4
NilsOliver Linden, Manuel Zingl, Claudius Hubig, Olivier Parcollet, Ulrich SchollwöckMPSbased DMFT impurity solver applied to a real material while incorporating spinorbit coupling at zero temperature, a regime where previous CTQMCbased solvers encountered a severe sign problem.
arXiv:1909.02503 
Use and implementation of autodifferentiation in
tensor network methods with complex scalars
Claudius HubigSketches the implementation of reversemode 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.
arXiv:1907.13422 
Gaussian TDVP for the BoseHubbard model
Tommaso Guaita, Lucas Hackl, Tao Shi, Claudius Hubig, Eugene Demler, J. Ignacio CiracWe 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 2particle excitation spectrum as eigenvalues of the linearized projected equations of motion (linearized TDVP).
Phys. Rev B 100, 094529 arXiv:1907.04837 
Dynamical topological quantum phase transitions in nonintegrable models
Imre Hagymási, Claudius Hubig, Örs Legeza, Ulrich SchollwöckWe 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 twosite entanglement.
arXiv:1904.00867 Phys. Rev Lett. 122, 250601 
Quantum phases and topological properties of interacting fermions in onedimensional
superlattices
Leo Stenzel, Andrew L. C. Hayward, Claudius Hubig, Ulrich Schollwöck, Fabian HeidrichMeisnerA very extensive study of the possible quantum phases of interacting fermions in effective onedimensional 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 
Timeevolution methods for matrixproduct states
Sebastian Paeckel, Thomas Köhler, Andreas Swoboda, Salvatore R. Manmana, Ulrich Schollwöck and Claudius HubigAn overview and detailed description of timeevolution methods for matrixproduct states. Discusses TEBD, the MPO W^{I,II} method, the global Krylov method, the local Krylov method and the timedependent 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 condensedmatter physics to test the advantages of each individual method.
Exemplary excitation spectrum of a nearIsing XXZ Heisenberg chain obtained using twosite TDVPMPS within a few CPU hours.
arXiv:1901.05824 Annals of Physics 411, 167998 
Timedependent study of disordered models with infinite projected entangled pair
states
Claudius Hubig and J. Ignacio CiracApplies realtime 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 bivalued 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 slowdown of the dynamics in the presence of the bivalued disorder field (implemented by an ancilla Hilbert space).
Evolution of ⟨s^{z}(t)⟩ starting from a Néel state under the Heisenberg XXX Hamiltonian with increasingly stronger disorder. MPS calculations at zero disorder exhibit finitesize effects at t < 1, while timedependent iPEPS can obtain longer times and also demonstrate a marked slowdown of dynamics, which may hint towards manybody localisation.
arXiv:1812.03801 SciPost Phys. 6, 031 
Thermal control of spin excitations in the coupled Isingchain 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 RüeggUsing finitetemperature timedependent 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 temperaturedependence 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.
arXiv:1811.07178 
Densitymatrix embedding theory study of the onedimensional HubbardHolstein model
Teresa E. Reinhard, Uliana Mordovina, Claudius Hubig, Joshua S. Kretchmer, Ulrich Schollwöck, Heiko Appel, Michael A. Sentef and Angel RubioAn 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 HubbardHolstein model in one dimension to obtain the singleparticle excitation gap and by extension allow for a study of the phase diagram.
arXiv:1811.00048 J. Chem. Theory Comput. 15 (4), 2221 
Interaction quench and thermalization in a onedimensional topological Kondo
insulator
Imre Hagymási, Claudius Hubig and Ulrich SchollwöckStudy of the nonequilibrium dynamics of a onedimensional topological Kondo insulator modelled by a pwave Anderson lattice model. The system is quenched by a change of the onsite interaction strength and topological quantities are studied in a realtime 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 
Abelian and nonabelian symmetries in infinite projected entangled pair
states
Claudius HubigStudies the effect of implementing (that is, preserving exactly) abelian and nonabelian 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 symmetrypreserving phase, leading to marked signals in the variational energy obtained at phase transitions.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 
Error estimates for extrapolations with matrixproduct states
Claudius Hubig, Jutho Haegeman and Ulrich SchollwöckUsing singlesite DMRG, much larger bond dimensions and much lower variational energies are obtainable than with twosite DMRG due to computational speedups and reduced memory usage. However, singlesite DMRG does not produce a reliable error measure which allows extrapolation of the observables to the zeroerror 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 zeroerror case.
Comparison of the twosite 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 groundstate energy estimates, but by using singlesite DMRG and only the twosite variance, much larger bond dimensions and hence lower variational energies are obtained.
1711.01104 Phys. Rev. B 97, 045125 
Doped Kondo chain, a heavy Luttinger liquid
Ilia Khait, Patrick Azaria, Claudius Hubig, Ulrich Schollwöck and Assa AuerbachA study of the onedimensional Kondo lattice model using SU(2)symmetric DMRG beyond the analytically approachable largeN limit. Found signatures of a heavy TomonagaLuttinger liquid phase and divergent charge and spin susceptibilities at the large Fermi surface.
1710.04847 PNAS 115, 5140 
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. QuinteroCastroA study of spinon confinement in a onedimensional Heisenberg chain in the presence of an external staggered field. Groundstate DMRG results on the gap and timedependent MPS data of the spectrum could qualitatively reproduce the experimental results with quantitative agreement upon use of a tangentspace MPS method.
1705.01259 Phys. Rev. B 96, 054423 
Generic construction of efficient matrix product operators
Claudius Hubig, Ian P. McCulloch and Ulrich SchollwöckPresents a generic method to algorithmically construct efficient matrixproduct 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 
Universal longtime behavior of stochastically driven interacting quantum systems
Zi Cai, Claudius Hubig and Ulrich SchollwöckA systematic study the longtime 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 manybody effects.
1609.08518 Phys. Rev. B 96, 054303 
Strictly singlesite DMRG algorithm with subspace expansion
Claudius Hubig, Ian P. McCulloch, Ulrich Schollwöck and F. Alexander WolfIntroduces a strictly singlesite DMRG variant using subspace expansion to adapt the auxiliary basis size. This variant improves both runtime and memory usage over the traditional twosite DMRG and the centermatrix wavefunction approach.
The DMRG method introduced in this paper is the default method used in the SyTen toolkit.
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 onedimensional chain. Traditional twosite DMRG was slower by a factor of two to three than the CWF approach shown here.
1501.05504 Phys. Rev. B 91, 155115
Theses

SymmetryProtected Tensor Networks
PhD thesis submitted to the Faculty of Physics at LMU Munich on 20170831, supervised by Ulrich SchollwöckA detailed introduction to symmetryprotection in tensor networks in Chapter 2, which is useful as an overview for the SyTen tensor network implementation and also discusses matrixproduct 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 twodimensional Hubbard model using MPSDMRG on finitesize cylinders.
Direct link at LMU 
Spin Dynamics in Adiabatic Transport through Quantum Dots
BSc thesis submitted to the Faculty of Physics at TU Berlin on 20120703, supervised by Anja Metelmann and Tobias Brandes
Workshops Coorganised

01/2019: Quantum Emitters in
NonConventional Baths at MPQ Garching
together with Ignacio Cirac and Eduardo SanchezBurillo 
06/2016: First International ExQM Workshop at Lake Chiemsee
with the ExQM graduate school
Awards
 Arnold Sommerfeld PhD Prize, 2017
Arnold Sommerfeld Center, LMU Munich  Deutschlandstipendium, 2011/2012
TU Berlin & Robert Bosch GmbH
Talks
 09/2019: Korrelationstage 2019, Dresden: Realtime evolution of tensor networks in one and two dimensions
 05/2019: Exploring Quantum Matter Seminar, Garching: Recent developments in tensor networks
 04/2019: Condensed Matter Theory Seminar, Paul Scherrer Institut, Villigen: Recent developments in tensor networks
 03/2019: Physics Seminar, University of Queensland, Brisbane: Recent developments in tensor networks
 11/2018: Tensor Network based approaches to Quantum ManyBody Systems, Dresden: Symmetries in Tensor Networks [slides]
 11/2018: Arnold Sommerfeld Center, Munich: PhD Prize Colloquium: Evolving Tensor Networks
 07/2018: Seminar of the Institute for Theoretical Physics, Göttingen: Applications of Tensor Networks in Ab Initio Methods
 06/2018: Optimising, Renormalising, Evolving and Quantising Tensor Networks, Dresden: Symmetry implementations in tensor networks
 03/2018: Frühjahrstagung der DPG, Berlin: Error estimates for extrapolations with matrixproduct states
 02/2018: Entanglement in Strongly Correlated Systems, Benasque: Error estimates for extrapolations with matrixproduct states [slides] [local copy]
 11/2017: 2017 School, European Tensor Network, Ghent: Tutorial on DMRG with Subspace Expansion; Python script
 09/2017: Arnold Sommerfeld School on Numerical Methods for Strongly Correlated ManyBody Systems, Munich: Tutorial & HandsOn Session on DMRG and MPS Methods
 04/2017: Theory Department, MaxPlanckInstitut für Struktur und Dynamik, Hamburg: Time Evolution with MatrixProduct States
 03/2017: Theory Seminar, MaxPlanckInstitut für Quantenoptik, Garching: DMRG with Subspace Expansion on SymmetryProtected Tensor Networks
 03/2017: Frühjahrstagung der DPG, Dresden: Generic Construction of Efficient Matrix Product Operators
 01/2017: Theory Winter School, MagLab, Tallahassee: Tutorial & HandsOn Session on DMRG and DMRG+DMFT
 03/2016: Frühjahrstagung der DPG, Regensburg: DMRG on Binary Tree Tensor Networks
 01/2016: Fritz Haber Institut der MaxPlanckGesellschaft, Berlin: Symmetries in Tensor Networks and Subspace Expansion with DMRG
Posters
 09/2017: Tensor Computation Workshop, Flatiron Institute, New York City: Claudius Hubig and Ulrich Schollwöck: Error Estimates for MatrixProduct States
 09/2017: Arnold Sommerfeld School on Numerical Methods for Strongly Correlated ManyBody Systems, Munich: Claudius Hubig and Ulrich Schollwöck: Error Estimates for MatrixProduct States
 05/2017: Munich Quantum Center Workshop: Andreas Swoboda, NilsOliver Linden, Ulrich Schollwöck and Claudius Hubig: Time Evolution for MatrixProduct States with the Krylov Subspace Method
 10/2016: Munich Quantum Symposium: Claudius Hubig, Ian P. McCulloch and Ulrich Schollwöck: A Generic Algorithm for the Construction of Efficient Matrix Product Operators
 06/2016: International Summer School on Computational Quantum Materials, Sherbrooke: Claudius Hubig, Ian P. McCulloch and Ulrich Schollwöck: Generic Algorithm for the Construction of Efficient Matrix Product Operators
 02/2016: Entanglement in Strongly Correlated Systems, Benasque: Claudius Hubig and Ulrich Schollwöck: DMRG on Binary Tree Tensor Networks
 11/2015: WEHWorkshop Isolated Quantum ManyBody Systems out of Equilibrium, Bad Honnef: Claudius Hubig, Fabian H. L. Essler and Ulrich Schollwöck: Spin Excitations in a Staggered Magnetic Field
 02/2015: Advanced Computational Methods for Strongly Correlated Quantum Systems, Würzburg: Claudius Hubig, Ian P. McCulloch, Ulrich Schollwöck and F. Alexander Wolf: A Strictly SingleSite DMRG Algorithm with Subspace Expansion
 09/2014: Munich Quantum Day: Claudius Hubig, F. Alexander Wolf and Ulrich Schollwöck: Algorithmic Advancements in DMRG
Teaching
 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)
CV
 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: SymmetryProtected 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.