Researchers Define Interacting Open Systems Using non-Hermitian Hamiltonians and Distinguish Them from Conventional Physics

The behaviour of interacting systems that lose and gain energy simultaneously presents a fundamental challenge to conventional physics, and researchers are increasingly exploring non-Hermitian approaches to describe them. Aaron Kleger and Rufus Boyack, both from Dartmouth College, investigate the physical limits of these non-Hermitian systems, revealing crucial constraints on how they can accurately model interacting particles. Their work demonstrates that a common technique for incorporating energy loss and gain into calculations is actually incompatible with established theories of interaction, and instead proposes that these systems operate under the principles of pseudo-Hermitian mechanics. By establishing a consistent framework for understanding non-Hermitian interactions and characterising the behaviour of particles within them, this research provides a vital step towards unlocking the potential of these exotic systems, with implications for understanding materials that actively dissipate or amplify energy.

Non-Hermitian Systems and Open Quantum Dynamics

This document presents a comprehensive overview of research into non-Hermitian quantum mechanics and open quantum systems, areas exploring how quantum systems interact with their environment and exhibit behaviors beyond those predicted by standard quantum theory. It details investigations into how these systems can be mathematically described, focusing on concepts like dissipation, decoherence, and the emergence of novel topological phases. Researchers are actively investigating open quantum systems, which describe the interaction between a quantum system and its surroundings, leading to energy loss and a reduction in quantum coherence. Non-Hermitian Hamiltonians, mathematical descriptions that don’t adhere to traditional symmetry requirements, are used to model these interactions.

Pseudo-Hermiticity and PT-symmetry provide frameworks for dealing with these non-Hermitian Hamiltonians, allowing for the calculation of real energy levels, and reveal the emergence of unique phases of matter with unusual properties. Investigations explore pseudo-Hermiticity, a mathematical technique where a non-Hermitian Hamiltonian is transformed into a Hermitian one using a metric operator, allowing for a consistent quantum mechanical description with real energy eigenvalues. Researchers are also utilizing path integral formulations to adapt quantum mechanics to non-Hermitian scenarios. The Lindblad master equation is a fundamental tool for describing the time evolution of open quantum systems, accounting for dissipation and decoherence, while completely positive semigroups provide the mathematical framework for ensuring probabilities remain positive.

Non-Hermitian topological insulators and Weyl semimetals represent novel phases of matter exhibiting topological protection alongside non-Hermitian features. Bulk Fermi arcs, gapless surface states connecting the bulk projections of Weyl nodes, are characteristic of non-Hermitian Weyl semimetals. Researchers are also studying Hall conductance, which exhibits unique behavior in non-Hermitian systems, and applying the Kubo formula to calculate conductivity and statistical mechanics to understand many-body non-Hermitian systems at finite temperatures. Field theory and linear response theory provide powerful frameworks for describing many-body systems, extending to non-Hermitian scenarios. This research demonstrates connections between pseudo-Hermiticity and open quantum systems, where pseudo-Hermiticity provides a mathematical framework for describing effective non-Hermitian Hamiltonians. Exceptional points play a crucial role in the emergence of novel topological phases, and many-body interactions significantly modify the behavior of non-Hermitian systems, leading to new phenomena.

Pseudo-Hermitian Mapping for Unitary Time Evolution

Researchers have developed a new approach to understanding non-Hermitian quantum mechanics by formulating it within the framework of pseudo-Hermitian mechanics. This method requires a carefully defined inner product involving a positive-definite metric, allowing for a consistent physical description of non-Hermitian systems by relating them to conventional interacting physics. The team demonstrated that observables evolve unitarily in time, establishing an equivalence between Hermitian and pseudo-Hermitian representations of observables. To apply this method to many-body systems, scientists computed observables using the Matsubara formalism, defining a pseudo-Hermitian Matsubara Green’s function.

This allows for calculations similar to those in Hermitian systems and enables the analytical continuation to obtain retarded and advanced Green’s functions. Researchers contrasted this approach with standard methods used when non-Hermitian behavior arises from self-energy effects, revealing key differences in the resulting Green’s functions. The investigation revealed that both open and interacting systems can be described using retarded, advanced, and Keldysh Green’s functions, but the standard relationship between these functions does not hold in pseudo-Hermitian quantum mechanics. Instead, researchers discovered a new relationship between causal Green’s functions, highlighting a fundamental distinction from standard interacting theory. Calculations for a dissipative interacting system revealed that the Matsubara Green’s function differs in the pseudo-Hermitian approach, demonstrating consistency with a quantum-metric-based approach.

Unified Framework Resolves Non-Hermitian Quantum Ambiguity

Researchers are developing a more unified understanding of non-Hermitian quantum systems, which describe phenomena beyond standard quantum mechanics. The team demonstrates that seemingly disparate approaches to describing these systems can be reconciled through a consistent framework centered on a modified inner product. This work addresses ambiguity in calculating expectation values and defining equilibrium states within these complex systems. The investigation reveals that the key to consistency lies in recognizing the role of a “pseudo-metric operator,” which modifies the standard mathematical rules for calculating inner products.

By applying this operator, researchers establish a direct connection between non-Hermitian Hamiltonians and their corresponding Hermitian counterparts, allowing for a consistent treatment of observables and Green’s functions. This approach resolves inconsistencies found in previous methods, particularly regarding the relationship between causal Green’s functions and the modification of the inner product. The team applied this framework to the (1+1)-dimensional “Tachyonic” non-Hermitian Dirac model, calculating its optical conductivity. This calculation provides a benchmark for comparing different theoretical approaches and demonstrates the power of the unified framework for predicting physical properties. The results offer a pathway towards a more general and consistent description of many-body non-Hermitian systems.

Pseudo-Hermitian Mechanics and Open Quantum Systems

This research investigates non-Hermitian Hamiltonians, increasingly used to model open quantum systems and dissipative interactions. The team demonstrates that common approaches to incorporating non-Hermitian terms into calculations are inconsistent with standard descriptions of interacting systems, unless interpreted within the framework of pseudo-Hermitian mechanics. They establish that these approaches implicitly rely on a modified inner product defined by a pseudo-metric operator, offering a consistent physical description for these systems. The findings reveal a crucial distinction between systems described by pseudo-Hermitian quantum mechanics and standard interacting systems, particularly concerning their Green’s functions and electromagnetic responses.

Specifically, researchers show that a gauge-invariant system described using a non-Hermitian Hamiltonian within the pseudo-Hermitian framework can exhibit an electromagnetic current different from that of an equivalent Hermitian system. This difference provides a potential experimental signature to distinguish between these two types of systems, despite having identical energy spectra. The authors acknowledge that their analysis relies on certain approximations and that further investigation is needed to fully.