Hall Effect Without a Magnet: A New Route to Dissipative Conductivity

Author: Denis Avetisyan

Researchers have discovered a mechanism for generating a Hall effect in two-dimensional systems driven by dissipation and interactions, bypassing the need for external magnetic fields or equilibrium conditions.

This work demonstrates that a Chern-Simons term, induced by non-equilibrium physics and described using the Keldysh formalism, can lead to Hall conductivity in weakly time-reversal invariant open systems.

The conventional quantum Hall effect demands robust time-reversal symmetry breaking, a condition often achieved through strong magnetic fields; however, this work, ‘Hall conductance in a weakly time-reversal invariant open system’, demonstrates that Hall physics can emerge even in systems with only weak time-reversal symmetry and without applied magnetic fields. Specifically, we show that non-equilibrium effects and interactions induce a non-quantized Hall conductance via a Chern-Simons term arising from a fermionic subsystem coupled to bosonic degrees of freedom and an external reservoir. Unlike equilibrium scenarios, a simple mass term is insufficient; wave-function renormalization and dissipation are crucial for generating this effect. Could this mechanism offer a pathway to realizing topologically non-trivial phases and novel electronic transport phenomena in systems lacking strong symmetry constraints?

The Illusion of Equilibrium: Why Static Models Fail

Conventional Green’s functions, a cornerstone of many-body physics, are inherently designed to analyze systems at equilibrium – static, time-independent states where properties remain constant. However, this framework proves inadequate when confronted with the realities of driven or dissipative systems. These systems, constantly exchanging energy and matter with their surroundings, exhibit dynamics fundamentally divorced from equilibrium. For instance, a material absorbing light or a chemical reaction releasing heat cannot be accurately described by static Green’s functions. The time-dependent nature of these processes, coupled with the influence of external forces or energy loss, necessitates a more versatile formalism capable of tracking the evolution of quantum states and accounting for the continual flow of energy – a capability beyond the reach of traditional, equilibrium-focused methods. Consequently, extending or replacing these established techniques is crucial for modeling a vast range of physical phenomena, from the behavior of electrons in a time-varying electromagnetic field to the intricate dynamics of open quantum systems.

The behavior of interacting many-body systems – from complex materials to biological entities – is rarely governed by static, equilibrium conditions. Realistic environments introduce continuous driving forces and dissipation, demanding theoretical tools that move beyond traditional approaches. These systems constantly exchange energy and matter with their surroundings, creating a dynamic interplay that dictates their properties and responses. Consequently, a formalism capable of accurately describing non-equilibrium dynamics is not merely a refinement, but a necessity. Such a framework must account for the time evolution of the system, the memory effects arising from past interactions, and the influence of the environment on the collective behavior – effectively capturing how these systems respond while they are changing, rather than simply describing their final, settled state. This shift in perspective is crucial for understanding phenomena ranging from the transport properties of nanoscale devices to the emergent behavior of living organisms.

A comprehensive understanding of many-body systems invariably requires acknowledging their inherent connection to the surrounding environment. Traditional theoretical frameworks, such as those built upon equilibrium Green’s functions, often operate under the assumption of isolated systems, effectively neglecting the crucial influence of external degrees of freedom. However, realistic physical scenarios routinely involve strong coupling to reservoirs – be they thermal baths, electromagnetic fields, or other interacting entities. This interaction manifests as dissipation, decoherence, and the continuous exchange of energy and information, fundamentally altering the system’s dynamics. Consequently, accurately modeling these systems necessitates incorporating environmental effects, not merely as perturbations, but as integral components of the theoretical description – demanding formalisms that move beyond the limitations of equilibrium considerations and embrace the complexities of open quantum systems and non-equilibrium statistical mechanics.

A Contour Through Time: The Schwinger-Keldysh Formalism

The Schwinger-Keldysh formalism addresses systems experiencing dissipation and driven by external time-dependent forces by extending standard field theory. Unlike equilibrium scenarios, these systems require a method to handle initial conditions and time-dependent perturbations. The formalism achieves this by considering fields defined on a closed time contour – extending from an initial time $t_i$ to a final time $t_f$ and back to $t_i$ . This contour ordering allows for the consistent treatment of time-dependent interactions and the calculation of response functions, correlation functions, and transport coefficients relevant to dissipative processes. The framework is applicable across a range of physical systems, including quantum optics, condensed matter physics, and cosmology, where non-equilibrium dynamics are prevalent.

The Schwinger-Keldysh formalism addresses non-equilibrium scenarios by employing a closed time contour, denoted as $\mathcal{C}$ . This contour extends from an initial time $t_i$ to a final time $t_f$ and back to $t_i$ , effectively considering time evolution both forward and backward. This approach allows for the simultaneous treatment of influences from both the past and the future, which is crucial for describing systems subject to dissipation or external driving where the future state can affect the present behavior. Traditional methods based on a single time evolution operator are insufficient for these systems as they assume time-translation invariance and cannot account for the history dependence inherent in non-equilibrium processes. The closed time contour allows calculation of response functions and correlation functions relevant to dissipative systems by considering both retarded and advanced Green’s functions.

The Keldysh rotation is a specific linear transformation applied to fields in the Schwinger-Keldysh formalism, redefining them in terms of retarded and advanced Green’s functions. This rotation, typically expressed as $\phi \rightarrow \frac{1}{\sqrt{2}} (\phi_{+} + \phi_{-})$ where $\phi_{+}$ and $\phi_{-}$ represent the forward and backward time components respectively, effectively decouples the contour-ordered correlation functions into those solely dependent on the difference between times. This decoupling simplifies calculations of response functions and dissipation by allowing the use of standard Feynman diagram techniques, avoiding direct evaluation of complex contour integrals and providing a clear separation between causal and anti-causal contributions to the system’s dynamics.

Tracing the Echo of Interaction: Self-Energy and the Dressed Propagator

The self-energy, denoted as $Σi(ω)$ , represents the modifications to a particle’s propagation due to interactions with its surrounding environment. This environmental influence alters the particle’s energy and momentum, effectively shifting its properties from those predicted by a free particle model. Specifically, calculations reveal a linear frequency dependence for the self-energy, mathematically expressed as $Σi(ω) = ωΣ̄i$ , where $Σ̄i$ is a frequency-independent constant characterizing the strength of the environmental interaction. This linear relationship indicates that the environmental correction to the particle’s propagation is directly proportional to its energy, ω.

The Dyson equation is a functional equation used to calculate the dressed propagator, $G(ω)$ , given the bare propagator, $G_0(ω)$ , and the self-energy, $Σ(ω)$ . Specifically, the equation takes the form $G(ω) = G_0(ω) + G_0(ω)Σ(ω)G(ω)$ . This integral equation is solved iteratively or through direct inversion to determine $G(ω)$ , which then accurately represents the particle’s propagation including all environmental corrections encapsulated within the self-energy. The resulting dressed propagator replaces the bare propagator in calculations, providing a more realistic description of particle behavior within the interacting many-body system.

The dressed propagator represents the full Green’s function, incorporating the effects of interactions with the surrounding medium. Specifically, it accounts for scattering events arising from both bosonic and fermionic fields, which modify the propagation of the particle under consideration. This is achieved by summing infinite diagrams representing all possible interactions, effectively replacing the bare propagator with a modified form that includes the self-energy Σ. Consequently, the dressed propagator, often denoted as $G$ , relates initial and final states while accurately reflecting the particle’s behavior within the complex many-body system, providing a more realistic description than the bare propagator alone.

From Microscopic Whispers to Macroscopic Conductance

The macroscopic electrical response of a material isn’t simply an emergent property, but a direct consequence of interactions occurring at the microscopic level. These interactions, meticulously described by the $\text{dressed propagator}$ , which accounts for quantum fluctuations and many-body effects, are mathematically linked to the material’s overall behavior through the $\text{polarization tensor}$ . This tensor effectively ‘sums up’ how individual particles respond to external fields, translating those microscopic responses into measurable macroscopic quantities like conductivity. Understanding this connection allows researchers to predict and control a material’s electrical properties by manipulating its underlying microscopic structure and interactions; it bridges the gap between the quantum world of electrons and the observable electrical behavior of materials, providing a fundamental framework for materials design.

The Hall effect, typically understood as arising from magnetic fields deflecting charge carriers, can also emerge from topological properties within a material, specifically through a term known as the Chern-Simons term. This term doesn’t require broken time-reversal symmetry to manifest; instead, it’s intrinsically linked to the interactions between fermions – the material’s constituent particles – and bosons. The strength of this topological Hall effect is directly proportional to the coupling constants $g_B$ and $g_S$ , which quantify the degree of interaction between these particles. Consequently, the Hall conductivity – a measure of how effectively the material conducts electricity in the presence of an applied field – becomes sensitive to these microscopic couplings, offering a pathway to manipulate conductivity through topological considerations and particle interactions.

Recent investigations, leveraging the Schwinger-Keldysh formalism, reveal a pathway for the emergence of the Hall effect even in the absence of traditional time-reversal symmetry breaking. This unconventional Hall effect stems from intricate interactions between fermions and bosons, captured within a topological framework and expressed through the Chern-Simons term. The resulting Hall conductivity isn’t a fixed property, but rather a nuanced response dictated by the specific parameters of the system-including the strength of fermion-boson couplings and, crucially, a momentum cutoff Λ. This dependence on Λ highlights the influence of high-energy physics on macroscopic transport phenomena, suggesting a deeper connection between microscopic interactions and observable electrical behavior than previously understood, and offering potential avenues for materials design with tailored Hall responses.

The exploration of Hall conductance, as detailed in this study, reveals the delicate interplay between dissipation and topological phases. It’s a humbling reminder that even seemingly fundamental properties, like the Hall effect, can arise from mechanisms beyond conventional understanding. As Marie Curie once observed, “Nothing in life is to be feared, it is only to be understood.” This sentiment echoes the spirit of inquiry driving this research – a willingness to confront complexity and unravel the hidden connections within physical systems. The discovery of a Chern-Simons term emerging from non-equilibrium conditions highlights how models, much like maps, inevitably simplify a far more intricate reality. The bending of established expectations, much like light around a massive object, serves as a potent reminder of the limits of current theories.

Where Do the Currents Flow?

The demonstration of a Hall effect divorced from conventional magnetic fields, generated instead by the subtle interplay of dissipation and interactions, feels less like a triumph and more like a careful peeling back of layers. It reveals how easily the foundations of established intuition can dissolve. The system studied here, though simplified, suggests that a great deal of what is presently understood as ‘law’ regarding charge transport may be contingent-dependent on a specific stillness, a lack of drive-that rarely exists in any truly interesting material. This work implies that complexity isn’t simply noise obscuring a clear signal; it is the signal.

A natural extension of this lies in exploring the limits of the Keldysh formalism itself. How robust is this approach when confronted with stronger interactions, or systems further from equilibrium? The current model relies on certain approximations; the point at which these break down, and the qualitative changes that emerge, remain largely uncharted territory. Moreover, the question arises: could such a mechanism contribute to, or even explain, anomalies observed in more complex materials – perhaps even hinting at novel states of matter hidden within seemingly disordered systems?

Discovery isn’t a moment of glory; it’s realizing how little is actually known. The event horizon isn’t a boundary of space, but of understanding. The currents flow where they will, regardless of the neat pictures drawn to contain them.

Original article: https://arxiv.org/pdf/2603.11186.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

The Illusion of Equilibrium: Why Static Models Fail

A Contour Through Time: The Schwinger-Keldysh Formalism

Tracing the Echo of Interaction: Self-Energy and the Dressed Propagator

From Microscopic Whispers to Macroscopic Conductance

Where Do the Currents Flow?

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