Research highlights:
- Dynamic response of strongly-correlated quantum systems
- Tensor network state methods for strongly-correlated systems in D≥2 dimensions
- Investigating quantum matter with entanglement renormalization on quantum computers
- Scaling and evolution of entanglement in many-body systems
- Nonequilibrium dynamics and driven-dissipative systems
- Stochastic dynamics in network systems and epidemic outbreaks
Dynamic response of strongly-correlated quantum systems.
Tensor network state methods for strongly-correlated systems in D≥2 dimensions.
Investigating quantum matter with entanglement renormalization on quantum computers.
Scaling and evolution of entanglement in many-body systems. Entanglement entropies quantify the complexity of quantum matter and its utility for quantum information processing. The computation costs in tensor network simulations are intimately related to entanglement properties of the studied systems. Also, one can use the entanglement structure to optimize the network geometry. In addition to such practical issues, the information-theoretic lens on quantum many-body systems leads to a deeper understanding of the physics. We are trying to gain a comprehensive understanding of the entanglement scaling in many-body systems: - In [1], we derived a conformal field theory (CFT) formula for subsystem Rényi entropies and the boundary entropy in ground states of critical 1D systems.
- In [2], we established the log-area law for critical fermionic systems with a finite Fermi surface, where the proportionality constant is predicted by the Widom conjecture. We showed that there is now logarithmic correction for 2D critical fermions with a zero-dimensional Fermi surface, and confirmed the area law for critical 2D bosons.
- For collective spin models in the thermodynamic limit, we derived the logarithmic scaling of the groundstate entanglement entropy at criticality and developed a perturbative method for 1/
*N*expansions of entanglement entropies [3,4]. - In [5], we elucidated entanglement properties of excited states in Bethe-ansatz integrable systems.
- Employing the eigenstate thermalization hypothesis (ETH), we found that the entanglement entropy of (almost) all energy eigenstates is described by a single crossover function [6,7,8]. These functions capture the full crossover from the groundstate entanglement regime at low energies and small subsystem size (area or log-area law) to the extensive volume-law regime at high energies or large subsystem size. For quantum-critical regimes, the crossover functions have universal scaling properties.
- In the context of tensor network states (TNS), we proved, for example, that MERA states in D≥22 dimensions obey the entanglement area law and are, actually a subclass of PEPS [9]. In [10], I determined the scaling of entanglement properties in thermofield double states for typical 1D systems at finite temperatures. Thermofield double states are employed in TNS simulations for at nonzero temperatures. The results of field-theoretical arguments and numerical examples, show that the cost for the classical simulation at finite temperatures grows only polynomially with the inverse temperature and is system-size independent -- even for quantum critical systems.
Nonequilibrium dynamics and driven-dissipative systems.
- We have used Trotter decompositions and Lieb-Robinson bound techniques to prove that Markovian dynamics can be simulated efficiently on unitary quantum computers and with system-size independent costs on classical computers [6,7]. We employ tensor network methods that rest on these findings to simulate large open quantum systems.
- We showed, for example, that the internal interactions of a dissipative system can hinder decoherence. And, under certain circumstances, the interplay of the internal interactions and dissipation processes can lead to divergent decoherence times, i.e., a critical system with an algebraic coherence decay [8].
- In [9], we discussed how block-triangular structures in the Liouvillian super-operator, corresponding to (possibly hidden) dynamical constraints, can make driven-dissipative phase transitions impossible.
- Based on work by Prosen and Seligman, we laid out a formalism for the solution of quasi-free and of quadratic open fermion and boson systems [10].
- In [11], we used this formalism to elucidate fundamental differences concerning criticality in open and closed systems, and found that, without symmetry constraints beyond invariance under single-particle basis and particle-hole transformations, all gapped quadratic Liouvillians belong to the same phase. We are currently employing Keldysh field theory and renormalization group analyses to investigate driven-dissipative phase transitions in interacting systems with a focus on fundamental differences to closed systems [5].
Stochastic dynamics in network systems and epidemic outbreaks.
Somewhat surprisingly, we found that tensor network approximations and ideas from quantum Monte Carlo can be adapted to study stochastic nonequilibrium dynamics in - While tensor networks are used to encode entangled states in quantum many-body systems, we found that they can be employed to efficiently encode conditional probabilities for state trajectories in stochastic network dynamics [1,2], reducing the time complexity from exponential to polynomial. With simulations of Glauber-Ising dynamics on Boolean networks, we have established that the new tensor network technique has substantial advantages over MCMC methods. It gives access to precise temporal correlations, which are otherwise very difficult to assess in systems with bidirectional dynamics. There are still some limitations: The new tensor network technique is based on the dynamic cavity method which is an extension of Mézard's cavity method for equilibrium systems and, hence, restricted to locally tree-like networks. Also, computation costs increase strongly with increasing vertex degrees.
- Recently, I have developed an alternative Monte Carlo algorithm that is not restricted to tree-like graphs and allows us to sample stochastic dynamics conditioned on specific (rare) events. New efficient simulation techniques are needed to properly understand the preconditions of rare catastrophic events and to improve mitigation strategies. For example, large epidemic outbreaks often emerge via a confluence of rare, stochastic events. Potentially dangerous spillover infections from animals into humans are frequent but rarely take off: The introduced infectious disease typically dies quickly or spreads to a small number of local cases and fizzles out. Only a small percentage of these local outbreaks manage to pass bottlenecks in the heterogeneous contact network, to expand, and become widespread epidemics.
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