Nonreciprocal current induced by dissipation in time-reversal symmetric systems

This paper demonstrates that dissipation-induced interband processes generate a nonreciprocal current in time-reversal symmetric, noncentrosymmetric crystals, a phenomenon inversely proportional to the lifetime and governed by the shift vector, particularly in minigap systems like the Rice-Mele model.

The Gist

The Big Idea: Breaking the "Fairness" of Electricity

Imagine you are pushing a shopping cart. In a perfect, frictionless world (a "clean" system), if you push the cart forward, it moves forward. If you push it backward, it moves backward. The distance it travels depends only on how hard you push, not the direction. This is reciprocity: the system treats both directions equally.

In physics, we usually expect electricity to behave the same way. If you apply a voltage to the left, current flows left. If you apply it to the right, current flows right with the same ease. This is especially true in crystals that look the same when you flip them inside out (time-reversal symmetric systems).

The Breakthrough:
This paper discovers a new way to break this "fairness." The authors show that if you introduce friction (dissipation) into the system, you can make electricity flow much easier in one direction than the other, even without using magnets or breaking the crystal's symmetry.

The Analogy: The Slippery Hill vs. The Sticky Mud

To understand how this works, let's use two different scenarios:

1. The Old Way: The Slippery Hill (Clean Systems)

Imagine a perfectly smooth, icy hill. If you slide down, you go fast. If you try to slide up, you can't. But in a "time-reversal symmetric" crystal, the hill looks the same upside down. So, sliding left is just like sliding right.

• The Problem: In the past, scientists thought you needed a magnetic field (like a giant magnet) to tilt the hill so that sliding left was easier than sliding right. Without the magnet, the hill was perfectly symmetrical.

2. The New Way: The Sticky Mud (Dissipative Systems)

Now, imagine the hill is covered in thick, sticky mud. This represents dissipation (energy loss, friction, or "noise").

• The Mechanism: When you push a ball through mud, it doesn't just slide; it gets stuck, wiggles, and sometimes jumps out of the mud into the air (exciting to a higher energy state) before landing back in the mud.
• The Twist: The authors found that in certain crystals, the "mud" interacts with the crystal's internal geometry in a weird way. When the ball (an electron) gets stuck and then jumps, it doesn't land in the same spot it would have if it were just sliding.
• The Result: Because of this "jumping and landing" process (which requires energy loss/dissipation), the ball finds it easier to roll one way than the other. The friction itself creates a preferred direction.

The Key Ingredients

The paper identifies three main ingredients needed for this trick to work:

1. No Center of Symmetry: The crystal must be lopsided (like a boot, not a sphere). This is the "shape" of the hill.
2. Time-Reversal Symmetry: No magnets allowed. The laws of physics look the same if you play the movie backward.
3. Dissipation (The "Mud"): This is the most important part. The electrons must lose energy (have a finite "lifetime" before they scatter).
   • Analogy: Think of a dancer. If the floor is perfectly smooth, they spin the same way left or right. But if the floor is sticky, and they have to hop over a puddle (dissipation), their movement becomes asymmetric. The "hop" changes the outcome.

The "Shift Vector": The Secret Map

The paper mentions a geometric quantity called the Shift Vector.

• Analogy: Imagine you are walking through a forest. If you take a step forward, you expect to land exactly where your foot was. But in this quantum forest, when an electron "jumps" from one energy level to another (due to the electric field), it doesn't land directly above where it started. It lands slightly to the side.
• This "sideways landing" is the Shift Vector. In a normal world, these sideways steps cancel out. But with the "mud" (dissipation), the steps don't cancel out perfectly. They add up to a net push in one direction.

Why Does This Matter?

1. It's a New Kind of Diode:
A diode is a device that lets electricity flow in one direction but blocks it in the other. Usually, you need complex materials or magnets to make one. This paper suggests we can make "diode-like" behavior just by using materials that are naturally lopsided and slightly "noisy" (dissipative).

2. It Works in "Minigap" Systems:
The effect is strongest in materials where the energy gap (the height of the hill) is similar to the strength of the friction (the stickiness of the mud).

• Real-world candidates: The authors suggest looking at Moiré materials (like layers of graphene stacked at a slight angle) or Weyl semimetals. These are materials where electrons behave like massless particles and have very specific, weird geometries.

The Bottom Line

For a long time, physicists thought that to get electricity to flow differently in opposite directions, you needed to break the symmetry of time (using magnets).

This paper says: "No, you don't need magnets. You just need friction."

By understanding how electrons "slip and slide" in a sticky, lopsided environment, we can create new types of electronic devices that control the direction of current without needing external magnetic fields. It turns a nuisance (energy loss/dissipation) into a useful tool for controlling electricity.

Technical Summary

1. Problem Statement

The paper addresses a fundamental question in condensed matter physics: Can a nonreciprocal current (directional asymmetry in response to an electric field) exist in systems that preserve time-reversal symmetry (TRS)?

• Context: Nonreciprocal transport is well-established in TRS-broken systems (e.g., magnetochiral anisotropy) and relies on $k$-asymmetric band structures ($\epsilon_k \neq \epsilon_{-k}$). In TRS-preserving systems, standard Bloch electron theory with finite lifetime suggests that nonreciprocal currents are impossible because the scattering matrix remains unitary ($|t_{LR}| = |t_{RL}|$).
• The Gap: While nonreciprocal optical responses (like shift current) exist in TRS systems via interband transitions, it was unclear if a DC nonreciprocal current could arise solely from dissipation (finite lifetime) in TRS systems without external magnetic fields or interactions. Previous mechanisms (nonlinear Drude, quantum metric dipole) require TRS breaking.

2. Methodology

The authors employ a Green's function formalism to derive the nonlinear conductivity in the presence of dissipation.

• Theoretical Framework:
  • They define the nonreciprocal current as the static limit ($\omega \to 0$) of the second-order conductivity $\sigma_{\mu\alpha\beta}$.
  • They utilize the retarded ($G^R$) and advanced ($G^A$) Green's functions with a finite relaxation rate $\Gamma$ (where $\Gamma = \hbar / 2\tau$, and $\tau$ is the lifetime).
  • The derivation involves calculating the second-order response function $K_{\mu\alpha\beta}(\omega, -\omega)$ and taking the limit $\omega \to 0$.
• Key Distinction: The authors explicitly separate contributions from intraband scattering (within a single band) and interband scattering (between different bands).
  • Intraband processes in TRS systems yield zero nonreciprocal current in the clean limit.
  • The focus is on interband processes induced by dissipation, where the finite lifetime allows virtual or real transitions between bands even in the DC limit.
• Model System: To validate the theory, they perform numerical simulations using the 1D Rice-Mele model, a two-band model that breaks inversion symmetry but preserves time-reversal symmetry.

3. Key Contributions and Derivations

A. General Formula for Nonreciprocal Current

The authors derive a general expression for the nonreciprocal conductivity $\sigma_{\alpha\alpha\alpha}$ in TRS systems (Eq. 5):
$$\sigma_{\alpha\alpha\alpha} = \frac{e^3}{\hbar} \int_{BZ} \frac{d^dk}{(2\pi)^d} \left[ \sum_{a,b} R_{ab} |A_{ab}|^2 D^{(2)}_{ab} + \sum_{a,b,c} \text{Re}[A_{ab}A_{bc}A_{ca}] D^{(3)}_{abc} \right]$$

• $R_{ab}$: The Shift Vector, a geometric quantity representing the real-space displacement of a wave packet during an interband transition.
• $A_{ab}$: The interband Berry connection.
• $D^{(n)}$ terms: Functions dependent on the relaxation rate $\Gamma$, band gaps $\Delta_{ab}$, and derivatives of the Fermi distribution.

B. Role of Dissipation (Lifetime $\tau$)

The paper establishes that the nonreciprocal current is inversely proportional to the lifetime (or directly proportional to the relaxation rate $\Gamma$) in the relevant regimes:

• Insulating Case (Low Temp, $\beta\Gamma \gg 2\pi$): The leading order contribution scales as $O(\Gamma^2)$ (or $O(\tau^{-2})$).
• Metallic Case / High Temp: The leading order contribution scales as $O(\Gamma)$ (or $O(\tau^{-1})$).
• Physical Mechanism: Dissipation breaks the unitarity of the scattering process. In the DC limit, a strong electric field combined with finite $\Gamma$ induces interband excitations (analogous to Landau-Zener tunneling but driven by scattering). The geometric phase (shift vector) associated with these transitions creates a directional asymmetry in the current.

C. Comparison with Existing Mechanisms

The authors clarify the hierarchy of nonreciprocal currents:

• TRS Broken Systems: Dominated by the Nonlinear Drude term and Quantum Metric Dipole (independent of $\tau$ in the clean limit).
• TRS Symmetric Systems: Dominated by the Dissipation-induced Interband term (vanishes as $\tau \to \infty$). This term is of order $\tau^{-1}$ or lower, making it the only dominant contribution in TRS systems with finite lifetime.

4. Results (Numerical Simulation)

Using the Rice-Mele model ($H(k) = t_0 \cos k \sigma_x + \delta t \sin k \sigma_y + m \sigma_z$):

• Dependence on Chemical Potential ($\mu$): The nonreciprocal current is maximized when the chemical potential is near the band gap ($\mu \approx 0$) and the relaxation rate is comparable to the band gap ($\Gamma \sim \Delta$).
• Dependence on Temperature and $\Gamma$:
  • Insulating Regime ($\mu=0$): At low temperatures, the current shows a quadratic dependence on $\Gamma$ for small $\Gamma$. At high temperatures, it transitions to a linear dependence.
  • Metallic Regime ($\mu \neq 0$): The current exhibits a linear dependence on $\Gamma$ across all temperatures.
• Sign Changes: The current changes sign near the band edges, consistent with the derivative of the Fermi distribution and the geometric factors involved.

5. Significance and Implications

• Theoretical Breakthrough: The paper resolves the question of whether DC nonreciprocity is possible in TRS systems. It proves that dissipation is not just a nuisance but a necessary driver for this specific type of nonreciprocal transport, mediated by interband geometric quantities (shift vector).
• Material Candidates: The authors propose that this effect is most observable in:
  • Moiré Heterostructures: Graphene/hBN and TMD (e.g., WS$_2$/WSe$_2$) bilayers, which possess tunable band gaps and significant lifetime broadening.
  • Nonmagnetic Weyl Semimetals: Materials like TaAs, TaP, and MoTe$_2$ where inversion symmetry is broken but TRS is preserved.
• Experimental Accessibility: The mechanism suggests that materials with "short" scattering times (high dissipation) and small direct band gaps are ideal candidates, contrasting with the clean-limit requirements for other nonlinear Hall effects.

In summary, this work identifies a new class of nonreciprocal transport driven by the interplay of dissipation, interband transitions, and quantum geometry (shift vector), expanding the understanding of nonlinear transport in time-reversal symmetric quantum materials.