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Department of Applied Mathematics and Theoretical Physics

I am a postdoctoral research assistant in the group of Benjamin Béri at the Department of Applied Mathematics and Theoretical Physics (DAMTP), Centre for Mathematical Sciences. I am a visitor of the Theory of Condensed Matter (TCM) Group, Cambridge, and a Trevelyan Research Associate of Selwyn College. I received my PhD in the group of Ignacio Cirac, Max Planck Institute of Quantum Optics, Garching, Germany. I previously worked as a postdoctoral research assistant at the Rudolf Peierls Centre for Theoretical Physics, University of Oxford, in the group of Steve Simon. My research is focused on the application of tensor network states to open problems in Condensed Matter Physics and to some extent also High Energy Physics. 

In general, the complexity of wave functions of many quantum particles grows exponentially with the particle number. However, due to the locality of interactions found in nature, quantum many-body systems do not fully occupy the exponentially large Hilbert space they have access to but are instead constrained to a polynomially large submanifold. For ground states of gapped local Hamiltonians, tensor network states (TNS) have been shown to efficiently approximate this submanifold. As a result, TNS have emerged as the most powerful description of one-dimensional lattice systems and have also provided qualitatively new insights into important two-dimensional lattice problems. I use TNS both analytically and numerically to describe intriguing quantum many-body effects, such as many-body localisation and the anomalous quantum Hall effect. I have also applied TNS to Lattice Gauge Theories at finite fermionic density in two dimensions.


Many-body localisation

Non-interacting metallic systems in one and two dimensions (without spin-orbit coupling or spontaneously broken time reversal symmetry) are fragile to arbitrarily weak disorder, i.e., disorder induces localisation of all single-particle orbitals - a phenomenon known as Anderson localisation. If one includes interactions, one might anticipate that they destroy localisation as they lead to mixing of all single-particle orbitals. However, it turns out that in one dimension localisation survives for sufficiently strong disorder, which has been dubbed many-body localisation [Ann. Phys. 321, 1126 (2006); Phys. Rev. Lett. 95, 206603 (2005)]. Many-body localised systems do not thermalise, i.e., they retain a memory of their initial state for arbitrarily long times. All eigenstates of such systems fulfil the area law of entanglement and as a result can be efficiently approximated by tensor network states [Phys. Rev. Lett. 114, 170505 (2015)].

Building on a previous approach to represent all eigenstates as a single tensor network [Phys. Rev. B 94, 041116 (2016)], we devised an ansatz for such a tensor network whose approximation error decreases inversely polynomially with computational time [Phys. Rev. X 7, 021018 (2017)]. I work on applying and extending our scheme in order to describe important many-body localisation effects, such as the emergence of conserved, effective spin degrees of freedom [Phys. Rev. B, 184201 (2018); Phys. Rev. B 99, 104201 (2019)], localisation-protected topological features of all eigenstates [Phys. Rev. B 98, 054204 (2018); J. Phys.: Cond. Mat. 32, 305601 (2020); Phys. Rev. B 102, 014205 (2020); Phys. Rev. Research 2, 033099 (2020); Phys. Rev. B 105, 144205 (2022)] and the (presumably metastable) phenomenon of many-body localisation in two dimensions [Nat. Phys. 15 (2019); arXiv:2108.08268; arXiv:2212.09775], where exact diagonalisation is unfeasible. I have also shown that two-dimensional many-body localisation can be employed to stabilise topologically ordered time crystals, a new dynamical phase of matter [arXiv:2105.09694].

Recently, I also proved rigorous upper bounds on the classical simulation time of quantum simulations using tensor networks [arXiv:2208.01498].


Behavior of 1-bits near the many-body localization transition
AK Kulshreshtha, A Pal, TB Wahl, SH Simon
– Physical Review B
Tensor networks demonstrate the robustness of localization and symmetry-protected topological phases
TB Wahl
– Physical Review B
Efficient Representation of Fully Many-Body Localized Systems Using Tensor Networks
TB Wahl, A Pal, SH Simon
– Physical Review X
Projected Entangled Pair States with non-Abelian gauge symmetries: An SU(2) study
E Zohar, TB Wahl, M Burrello, JI Cirac
– Annals of Physics
Fermionic projected entangled pair states and localU(1)gauge theories
E Zohar, M Burrello, TB Wahl, JI Cirac
– Annals of Physics
Chiral Projected Entangled-Pair State with Topological Order
S Yang, TB Wahl, H-H Tu, N Schuch, JI Cirac
– Physical review letters
Symmetries and boundary theories for chiral projected entangled pair states
TB Wahl, ST Hassler, H-H Tu, JI Cirac, N Schuch
– Physical Review B
Projected entangled-pair states can describe chiral topological states.
TB Wahl, H-H Tu, N Schuch, JI Cirac
– Phys Rev Lett
Matrix product states with long-range localizable entanglement
TB Wahl, D Perez-Garcia, JI Cirac
– Physical Review A
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Research Group

Centre for Quantum Information and Foundations