MagTop researchers and collaborators have extended the traditional Fermi-Dirac distribution to non-Hermitian systems. This new formalism provides a general framework to compute quantum many-body observables in equilibrium systems coupled to dissipative environments.
The Fermi-Dirac distribution function describes the statistical distribution of particles over energy states in systems that obey the Pauli exclusion principle. Traditionally, this distribution is framed within the quantum mechanics paradigm, where the Hamiltonian of the quantum system is Hermitian. However, many physical systems, particularly those coupled to dissipative environments, are better described by non-Hermitian Hamiltonians. Typically, these operators exhibit complex energy eigenvalues tied to novel technological applications, like improved sensing capabilities and effective diode functionalities.
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Figure: Artistic visualization of the non-Hermitian Fermi-Dirac distribution (shown in the background 3D plot) and its impact on persistent current transport in a superconducting-normal-superconducting junction (blue-green-blue). The junction is coupled to an external environment (violet), showcasing the dissipative nature of the quantum transport process.
We have developed a theoretical framework allowing to determine a non-Hermitian Femi-Dirac distribution (nHFDd) (see Figure). To verify its effectiveness, we examined the issue of persistent current transport in superconducting-normal-superconducting junctions and normal metallic rings. Persistent currents, which are equilibrium currents that can flow indefinitely without an applied voltage, can be considerably influenced by external environments through dephasing processes. By utilizing the nHFDd function, we derived an analytical formula for the persistent current that relies exclusively on the system’s complex energy spectrum. It is worthwhile to remark that this new approach successfully corrects inaccuracies shown in previous methodologies, particularly at exceptional points—singularities in the parameter space where two or more eigenvalues and their corresponding eigenstates merge. At these points, traditional methods usually yield anomalies, but the new approach ensures that physical observables remain continuous and well-defined. This continuity at exceptional points is crucial for accurately describing quantum transport and other phenomena in non-Hermitian systems. Temperature effects and many-body interactions were also studied in this Letter. It was shown that the current amplitude is suppressed by thermal fluctuations and electron-electron scattering. Due to the analytic characteristics of the nHFDd, exceptional points cannot be identified in static observables, such as the current. However, they become apparent in dynamical quantities, such as the current susceptibility that is routinely measured in experiment. Thus, our work has significant implications for future research in quantum transport and non-Hermitian thermodynamics, as well as for the development of advanced quantum devices that can exploit non-Hermitian physics for enhancing their functionality.
[1] P.-X. Shen, Z. Lu, J. L. Lado, and M. Trif, Non-Hermitian Fermi-Dirac Distribution in Persistent Current Transport, Phys. Rev. Lett. 133, 086301 (2024) (Editors’ Suggestion).
Making coupling to the reservoir a resource rather than an obstacle emerges as one of the most promising roads in topological quantum computing. MagTop’s researchers proposed a 1D non-Hermitian model, in which they revealed the presence of a hidden Chern number. They used this model to describe lasing in a polariton system and, most recently, to examine a chain of transmon devices, the qubits of the most mature quantum computers and an optomechanical superlattice. The dynamics of this system with dissipation, examined employing the third quantization methods, revealed the presence of controllable long-range quantum entanglement between distant end states of the chain.
To open a new avenue in exploration of quantum effects in non-Hermitian topological systems, we proposed to utilize the mature superconducting quantum device technologies and introduced a transmon chain where the spatially-dependent dissipation, the ABBA-like pattern, is realized by tunable quantum circuit refrigerators [1] (see Figure). By solving the many-body Lindblad master equation using a combination of the density matrix renormalization group and Prosen-Seligman third quantization approaches, we show that the topological end modes and the associated phase transition are visible in simple reflection measurements with experimentally realistic parameters. Most importantly, we demonstrated the possibility to generate genuine long-range quantum entanglement between the topologically protected end states. This can be done by time-evolving a Fock state with fixed number of bosons in the central site. The developed formalism is quite general and can be adapted to, for instance, optomechanical chains, for which we have demonstrated [2] that mechanical and optical end states are entangled and that such entanglement is robust with respect to external perturbations, paving the way for entanglement in topological lasers.
The crucial ingredient of those recent developments is the so-called hidden Chern number in one-dimensional non-Hermitian topological systems proposed in our earlier work [3]. This Chern number manifests itself as topologically protected in-gap end states at zero real part of the energy. We show that the bulk-boundary correspondence coming from the hidden Chern number is robust and immune to the non-Hermitian skin effect. We introduce a minimal model Hamiltonian supporting topologically nontrivial phases in this symmetry class, derive its topological phase diagram, and calculate the end states originating from the hidden Chern number. This formalism was then used to describe lasing in a polariton system [4].
Fig.: Non-Hermitian Bose-Hubbard transmon ABBA chain (A – blue, B – red) with the on-site and inter-site energies Ui and Ji, respectively. The dissipation strength is tuned by the coupling to the measurement circuit ki and loss caused by the quantum circuit refrigerator gi (QCR) (after [1]).
[1] W. Brzezicki, M. Silveri, M. Płodzień, F. Massel, T. Hyart, Non-Hermitian topological quantum states in a reservoir-engineered transmon chain, Phys. Rev. B 107, 115146 (2023).
[2] W. Brzezicki, T. Hyart, F. Massel, Non-hermitian topology and entanglement in an optomechanical superlattice, Phys. Rev. Research 7, 013089 (2025).
[3] W. Brzezicki, T. Hyart, Hidden Chern number in one-dimensional non-Hermitian chiral-symmetric systems, Phys. Rev. B 100, 161105(R) (2019).
[4] P. Comaron, V. Shahnazaryan, W. Brzezicki, T. Hyart, M. Matuszewski, Non-Hermitian topological end-mode lasing in polariton systems, Phys. Rev. Research 2, 022051(R) (2020).