SFB TRR 352 Mathematics of Many-Body Quantum Systems and Their Collective Phenomena
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B8 Local Stable Gaps and Response in Interacting Many-Body Quantum Systems

Project Leaders

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JProf. Dr. Ángela Capel, Prof. Dr. Stefan Teufel

Researchers

Dr. Barbara Roos, Tom Wessel, Paul Hege, Cornelia Vogel, Marius Wesle


Summary

The theoretical understanding of the quantum Hall effect has driven important developments in mathematics and mathematical physics over the past three decades and continues to do so. The most important aspects are the quantization of Hall conductivity, the validity of the linear response formalism, the role of random impurities, and the correspondence between bulk and boundaries. Only recently have important advances
concerning quantization and linear response been made specifically for models of interacting fermions at zero temperature either on torus geometries or in the thermodynamic limit.

For systems without boundaries the starting point for the mathematical analysis of quantum Hall systems is a many-body Hamiltonian with a gapped ground state. In particular, such systems display exponential decay of correlations [HK06]. While establishing a spectral gap for a given many-body Hamiltonian is an important and difficult problem in itself (see for example Projects A7 and A8), for us the central question will be the local
stability and the local response of gapped ground states resp. thermal states: starting from an extended fermionic gapped system describing the electrons in a Hall insulator on a torus geometry, the introduction of edges can be viewed as a local perturbation of the Hamiltonian near the new edges. Because of the appearance of edge states, such a perturbation closes the spectral gap above the ground state energy and introduces
long-range correlations along the boundary.
 
In our project, we aim to further develop the mathematical tools necessary to understand adiabatic and linear response for systems with boundaries. To this end, we explore mathematical questions whose relevance is not limited to quantum Hall systems, but extends, for example, to quantum information. Conversely, we also intend to use new approaches recently developed in quantum information to solve these problems motivated from mathematical physics. The main targeted advances are:

(1) To extend recent results [BDF18; MT19*; T20*; MO20; HT22*a; HT22*b] on adiabatic approximations for gapped systems to a theory of adiabatic and linear response in the bulk. We formulate a “local dynamically stable gap” (LDSG) condition that guarantees that in regions where this condition is satisfied the adiabatic and linear response to perturbations supported in these regions is the same as for a gapped system.
 
(2) To show that also for weakly interacting gapped fermion systems a strong LPPL-principle for the ground state (“local perturbations perturb locally”) [BDDF21; HTW22*] holds, i.e., that local but not necessarily small perturbations (like the introduction of edges) change the ground state only locally.

(3) To establish a strong LPPL-principle for the Gibbs state uniformly for all temperatures in systems that satisfy a strong LPPL-principle for the ground state. It is known that above a critical temperature the Gibbs state of a quantum lattice system with finite-range interactions is determined by the local Hamiltonian [KGKRE14], as well as at every temperature for 1-dimensional systems [BCP22*].
 
(4) To extend the generalized bulk-edge correspondence recently established for systems of noninteracting fermions [CMT21*] to systems of interacting fermions above the critical temperature. Successful completion of items (1)–(4) will allow us to formulate and address the problem of quantized Hall conductivity in integer quantum Hall samples with boundaries. Currently, we see this as an intermediate-term goal for the second period of the CRC. The ultimate long-term goal would be a similar theory also for models including long-range Coulomb repulsion, where screening effects near boundaries become important, and for fractional quantum Hall systems, i.e., for systems of strongly interacting fermions with degenerate ground states.

[HK06] M. B. Hastings and T. Koma. Spectral gap and exponential decay of correlations. Commun. Math. Phys. 265 (2006), 781–804.

[BDF18] S. Bachmann, W. De Roeck, and M. Fraas. The adiabatic thm. and linear response theory for extended quantum systems. Commun. Math. Phys. 361 (2018), 997–1027.

[MT19*] D. Monaco and S. Teufel. Adiabatic currents for interacting fermions on a lattice. Rev. Math. Phys. 31 (2019), 1950009.

[T20*] S. Teufel. Non-equilibrium almost-stationary states and linear response for gapped quantum systems. Commun. Math. Phys. 373 (2020), 621–653.

[MO20] A. Moon and Y. Ogata. Microscopic Conductivity of Lattice Fermions at Equilibrium. Part II: Interacting Particles. J. Func. Anal. 278 (2020), 108422.

[HT22*a] J. Henheik and S. Teufel. Adiabatic theorem in the thermodynamic limit: Systems with a uniform gap. J. Math. Phys. 63 (2022), 011901.

[HT22*b] J. Henheik and S. Teufel. Adiabatic theorem in the thermodynamic limit: Systems with a gap in the bulk. Forum Mathematics Sigma 10 (2022), 1–35.

[BDDF21] S. Bachmann, W. De Roeck, B. Donvil, and M. Fraas. Stability against large perturbations of invertible, frustration-free ground states. arXiv e-prints (2021). arXiv: 2110. 11194.

[HTW22*] J. Henheik, S. Teufel, and T. Wessel. Local stability of ground states in locally gapped and weakly interacting quantum spin systems. Lett. Math. Phys. 112 (2022), 1–12.

[KGKRE14] M. Kliesch, C. Gogolin, M. Kastoryano, A. Riera, and J. Eisert. Locality of temperature. Phys. Rev. X 4 (2014), 031019.

[BCP22*] A. Bluhm, Á. Capel, and A. Pérez-Hernández. Exponential decay of mutual information for Gibbs states of local Hamiltonians. Quantum 6 (2022), 650.

[CMT21*] H. D. Cornean, M. Moscolari, and S. Teufel. General bulk-edge correspondence at positive temperature. arXiv e-prints (2021). arXiv: 2107.13456.