Hall conductance in a weakly time-reversal invariant open system

This paper demonstrates that a non-equilibrium open system with weak time-reversal symmetry can exhibit non-quantized Hall conductance without an external magnetic field, as interactions with bosonic degrees of freedom and an external reservoir induce a time-reversal-breaking self-energy in the fermionic subsystem, a phenomenon that requires wave-function renormalization effects beyond a simple mass term.

The Gist

Here is an explanation of the paper using simple language and creative analogies.

The Big Idea: A Hall Effect Without a Magnet

Imagine you are trying to make a river flow sideways. In the world of physics, this "sideways flow" of electricity is called the Hall Effect.

Usually, to get electricity to flow sideways, you need a strong magnetic field (like a giant magnet pushing the electrons) or you need to break a fundamental rule of nature called Time-Reversal Symmetry (TRI). Think of TRI like a movie playing in reverse: if you play the movie backward, the laws of physics should look exactly the same. In standard Hall effects, the magnetic field makes the movie look different when played backward, which allows the sideways flow to happen.

The Surprise:
This paper says, "Wait a minute! We found a way to make electricity flow sideways without a magnet and without breaking the time-reversal rule for the whole system."

How? By creating a system that is open and out of balance, like a busy highway where cars are constantly entering and leaving, rather than a closed loop.

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The Setup: The "Dissipative" Dance Floor

To understand their discovery, let's use an analogy of a crowded dance floor.

1. The Dancers (Fermions): These are the electrons (or fermions) we are interested in. They want to dance.
2. The Music (Bosons): These are vibrations or waves in the material (like sound waves or phonons) that the dancers interact with.
3. The Doorman (The Reservoir): This is the outside world. In a normal physics experiment, the system is "closed"—no one leaves or enters. But here, the authors imagine a "doorman" who constantly kicks dancers off the floor and lets new ones in. This is called a reservoir.

The Magic Trick: Weak Symmetry vs. Strong Symmetry

In a normal, closed system, the rules of the dance floor are strict. If you play the dance video backward, it looks perfect. This is Strong Time-Reversal Symmetry.

In this paper's system, the "Doorman" (the reservoir) is chaotic. He kicks people out and brings people in at random rates.

• The Whole System: If you look at the entire building (the dance floor + the doorman + the outside world), the rules are still fair. If you play the whole movie backward, the doorman just swaps "kicking out" with "letting in." The total system is still symmetric.
• The Dancers Only: However, if you only look at the dancers on the floor, they are being pushed and pulled by the doorman. To them, the rules look broken. They feel a "wind" blowing them one way.

The authors call this Weak Symmetry. The system is symmetric as a whole, but the part we care about (the electrons) feels a broken symmetry because of the constant exchange with the outside world.

The Mechanism: The "Self-Energy" Mass

Because the dancers are constantly being kicked out and brought in, they start to feel heavy and sluggish. In physics, we call this a mass term.

Usually, if you give electrons a "mass" (like making them heavy), and you are in a normal, quiet room (equilibrium), they will start flowing sideways (Hall effect) only if the room itself is already biased (broken symmetry).

The Twist:
The authors found that in this noisy, open room, the "mass" created by the doorman isn't enough to create the sideways flow. You need something else.

They discovered that the friction and noise from the doorman (specifically, the imaginary part of the "self-energy") are crucial. It's not just that the dancers are heavy; it's that the way they lose energy and get replaced creates a specific kind of "twist" in their movement.

The Result: A Non-Quantized Hall Effect

In the famous Quantum Hall Effect (the Nobel Prize-winning discovery), the sideways flow is quantized. This means it comes in perfect, discrete steps, like climbing a ladder where you can only stand on the rungs, never between them. It's a very precise, integer number.

In this new "open system" discovered by the authors:

1. Sideways Flow Happens: The electrons do flow sideways, even without a magnet.
2. It's Messy (Non-Quantized): The flow is not a perfect integer step. It's a smooth, continuous value that depends on how strong the doorman is (the coupling constants).
3. Why it Matters: This proves that you don't need a perfect, isolated, magnetic world to get Hall physics. You can get it in messy, real-world systems where energy is constantly leaking in and out.

Summary Analogy

Imagine a merry-go-round (the electrons).

• Normal Hall Effect: You need a giant magnet to push the horses sideways.
• This Paper's Effect: You don't use a magnet. Instead, you have a chaotic crowd of people constantly jumping on and off the merry-go-round (the reservoir).
  • Because people are jumping on and off at different rates, the horses get "jittery" and "heavy."
  • This jitteriness, combined with the specific way they jump on and off, causes the whole merry-go-round to drift sideways.
  • The drift isn't a perfect, rigid step (quantized); it's a wobbly, fluid drift that depends on how crazy the crowd is.

Why Should We Care?

This is a big deal for Topological Physics. For decades, we thought topological effects (like the Hall effect) were fragile things that only happened in perfect, isolated, zero-temperature labs.

This paper shows that topology can survive in the messy, noisy, real world. It opens the door to designing new electronic devices that work using "open" systems, potentially leading to new types of sensors or quantum computers that don't need to be perfectly isolated from the environment.

The Bottom Line: You can create a "Hall Effect" without a magnet, as long as you have a system that is constantly exchanging energy with its surroundings, effectively tricking the electrons into flowing sideways through the chaos of the outside world.

Technical Summary

Here is a detailed technical summary of the paper "Hall conductance in a weakly time-reversal invariant open system" by Fagerlund, Ekman, and Arouca.

1. Problem Statement

The Quantum Hall Effect (QHE) and the Quantum Anomalous Hall Effect (QAHE) are traditionally understood to require the explicit breaking of Time-Reversal Invariance (TRI), typically achieved via strong magnetic fields or magnetic ordering. In equilibrium systems, a non-zero Hall conductance is linked to a non-zero Chern number, which necessitates TRI breaking.

The authors investigate a counter-intuitive scenario: Can Hall physics arise in a system that remains Time-Reversal Symmetric (TRI) in the sense of "weak symmetries," even without an external magnetic field? Specifically, they explore an open quantum system where the total action respects TRI, but the fermionic subsystem, due to interactions with a bosonic environment and an external reservoir, develops a self-energy that breaks TRI effectively.

2. Methodology

The study employs the Schwinger-Keldysh formalism, which is essential for describing non-equilibrium open quantum systems where the system is not close to thermal equilibrium.

• System Model: A 2D relativistic system containing:
  • Fermions: A Weyl fermion doublet ($\Psi = (\psi_\uparrow, \psi_\downarrow)^T$).
  • Bosons: A relativistic real scalar field ($\phi$).
  • Reservoir: Coupled to the system via Lindblad jump operators.
• Dissipation Mechanism: The coupling to the reservoir is modeled using Lindblad operators ($\ell_\alpha, \ell'_\alpha$) that add or remove fermions in a chirality-dependent manner, modulated by the scalar field. This introduces gain ($g_G$) and loss ($g_L$) terms.
• Symmetry Analysis:
  • The total action is constructed to be TRI invariant. Under time reversal, the gain and loss coefficients are exchanged ($g_G \leftrightarrow g_L$), and quantum fields pick up specific signs, ensuring the total action remains invariant.
  • However, the authors distinguish between strong symmetry (commuting with both Hamiltonian and jump operators) and weak symmetry (commuting only with the full Lindbladian). The system possesses weak TRI but lacks strong TRI.
• Calculations:
  1. Self-Energy ($\Sigma$): The authors compute the one-loop self-energy for the fermions arising from interactions with the bosons and the dissipative reservoir. This results in a matrix-valued self-energy in the Keldysh basis ($\Sigma^R, \Sigma^A, \Sigma^K$).
  2. Dressed Propagator: Using the Dyson equation ($G = G_0 + G_0 \Sigma G$), they derive the dressed fermion propagators.
  3. Polarization Tensor ($\Pi_{\mu\nu}$): They calculate the retarded polarization tensor using the dressed propagators to determine the electromagnetic response.
  4. Chern-Simons Term: They isolate the topological term in the effective action to extract the Hall conductivity.

3. Key Contributions

• Weak Symmetry Breaking: The paper demonstrates that a system can exhibit Hall physics while maintaining TRI at the level of the full action (weak symmetry), provided the fermionic subsystem acquires a TRI-breaking self-energy through non-equilibrium interactions.
• Role of Wave-Function Renormalization: A critical finding is that the imaginary part of the self-energy (associated with wave-function renormalization and finite quasiparticle lifetime) is essential for generating the Hall conductance. In the equilibrium limit or if one ignores the imaginary part (setting external frequency $\nu \to 0$), the Hall conductance vanishes, even if a mass term is present.
• Non-Quantized Hall Effect: Unlike the integer or half-integer quantized Hall conductance found in equilibrium topological phases, the Hall conductance in this open system is non-quantized. It depends continuously on the coupling constants ($g_S, g_B$) and the gain/loss imbalance.

4. Key Results

• Self-Energy Structure: The self-energy takes the form $\Sigma = (\Sigma_r - i\Sigma_i)\sigma_z$ (for the advanced component), where:
  • $\Sigma_r$ is the real part (mass term).
  • $\Sigma_i$ is the imaginary part (related to dissipation/decay).
  • Both depend on the gain/loss imbalance parameter $g_- = (g_G - g_L)/2$.
• Vanishing Conductance without Renormalization: If the calculation is performed neglecting the frequency dependence of the self-energy (i.e., setting $\nu=0$, which kills $\Sigma_i$ and $\Sigma^K$), the resulting Chern-Simons level $k$ is zero. This implies that a mass term alone is insufficient to generate a Hall effect in this non-equilibrium setting; the wave-function renormalization is mandatory.
• Non-Zero Conductance with Renormalization: When the full frequency-dependent self-energy is included, the polarization tensor yields a non-zero Chern-Simons term:
  $$\sigma_{xy} = -\frac{e^2 g_S g_B}{64\pi^2 v_F^4}$$
  This result is linear in the coupling constants and is not quantized.
• Parity Anomaly Analogue: The system exhibits a version of the parity anomaly. However, unlike the equilibrium case where the anomaly leads to half-quantized conductance ($k = \pm 1/2$), here the coefficient is unquantized and vanishes if the couplings are turned off or if the system returns to equilibrium (where gain equals loss).

5. Significance and Implications

• New Mechanism for Hall Physics: The work establishes a new pathway for generating Hall conductance that does not rely on magnetic fields or intrinsic magnetic ordering, but rather on non-equilibrium dissipation and reservoir coupling.
• Limitations of Effective Hamiltonians: The results highlight that standard effective non-Hermitian Hamiltonian approaches (which often ignore the Keldysh component of the Green's function or the density matrix evolution) fail to capture this phenomenon. The full Keldysh formalism is required to account for the change in the system's density matrix.
• Topological Invariants in Open Systems: The non-quantized nature of the Hall conductance suggests that topological invariants in open, dissipative systems behave differently than in closed systems. The lack of quantization poses challenges for defining the theory on closed manifolds (like a torus) due to gauge invariance issues, though it remains consistent for 2D planar geometries relevant to experiments.
• Experimental Relevance: The authors suggest potential experimental realizations involving fermions in a 2D material coupled to a bosonic bath (e.g., acoustic phonons) with specific chiral jump operators, although engineering the specific jump operators remains a significant challenge.

In summary, the paper reveals that non-equilibrium effects can induce a Hall response in a system that is globally time-reversal symmetric, provided one accounts for the full Keldysh structure of the self-energy, specifically the interplay between the real (mass) and imaginary (dissipative) parts. This challenges the conventional wisdom that TRI breaking is strictly necessary for Hall physics and underscores the unique topological properties of open quantum systems.