🔬 mesoscale physics

Nanoscale defects as probes of time reversal symmetry breaking

This paper proposes using Nitrogen Vacancy (NV) centers as sensitive probes to detect time-reversal symmetry breaking in low-dimensional conductors by measuring the difference in spin relaxation rates, a technique that can determine Hall viscosity and distinguish chiral superconducting states, such as those in twisted BSCCO flakes.

Original authors: Suman Jyoti De, Tami Pereg-Barnea, Kartiek Agarwal

Published 2026-02-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Suman Jyoti De, Tami Pereg-Barnea, Kartiek Agarwal

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to listen to a secret conversation happening inside a crowded, noisy room. Usually, to hear the conversation, you might have to shout questions (apply external fields) or stick a giant microphone right in the middle of the crowd (which disturbs the room).

This paper proposes a much sneakier, more elegant way to listen: using a tiny, invisible "ear" called a Nitrogen-Vacancy (NV) center. Think of this NV center as a microscopic spy, a single atom with a magnetic personality, hovering just above the material you want to study.

Here is the breakdown of what the scientists are doing, using simple analogies:

1. The Spy and the "Time-Travel" Rule

Most materials follow a rule called Time-Reversal Symmetry. Imagine playing a video of a ball bouncing; if you play it backward, it looks perfectly normal. The physics works the same way forward and backward.

However, some exotic materials (like certain superconductors or quantum Hall systems) break this rule. If you played a video of the electrons moving inside them backward, it would look weird. This is called Time-Reversal Symmetry Breaking (TRSB).

The problem is, detecting this "weirdness" is hard. Traditional methods often require poking the system with strong magnets, which changes the very thing you are trying to measure.

2. The "Left-Handed vs. Right-Handed" Spin

The authors realized that in these "time-broken" materials, the electromagnetic noise (the jiggling of electrons) isn't the same in all directions. It has a handedness, like a screw or a spiral.

Imagine the electrons are spinning like a corkscrew.
In a normal material, they spin left and right equally.
In a TRSB material, they prefer to spin one way (say, clockwise) more than the other.

The NV center is special because it can tell the difference between a "left-handed" spin and a "right-handed" spin.

3. The "Relaxation" Game

Here is the clever trick the paper describes:

The NV center has two "up" states (let's call them Spin-Up and Spin-Down) and one "down" state (Ground).
The scientists prepare the NV center in Spin-Up and watch how fast it falls to the Ground state. Then, they prepare it in Spin-Down and watch how fast that falls.
The Analogy: Imagine the NV center is a surfer.
- If the surfer is facing the "Right-Handed" waves, the ocean might be calm, and they relax slowly.
- If the surfer is facing the "Left-Handed" waves, the ocean might be turbulent, and they crash (relax) quickly.

In a normal material, the ocean looks the same from both sides, so the relaxation speed is identical. But in a TRSB material, the ocean looks different depending on which way the surfer faces.

The Key Discovery: If the relaxation speed is different when the NV center is "Up" versus "Down," you have found a material that breaks time-reversal symmetry. You don't need to shout or poke; you just listen to the difference in how fast the spy gets tired.

4. Why This Matters: The "Hall Viscosity"

The paper goes deeper. They show that by measuring this difference, you can calculate something called Hall Viscosity.

Viscosity is usually how "thick" a fluid is (like honey vs. water).
Hall Viscosity is a weird, invisible property of fluids that only exists when time symmetry is broken. It's like the fluid has a "twist" to it.
Knowing the Hall Viscosity helps scientists figure out exactly what kind of exotic state the material is in. For example, is it a "Fractional Quantum Hall" state (a very strange state of matter)? Or is it a "Chiral Superconductor" (a superconductor that spins in a specific direction)?

5. The Real-World Test: Twisted Sandwiches

The authors tested their theory on a specific material: BSCCO (a type of superconductor).

Imagine taking two slices of this superconductor and stacking them, but twisting one slice by 45 degrees.
This "twisted sandwich" is suspected to be a time-broken superconductor.
The paper predicts that if you hover your NV spy over this sandwich, you will see a specific "Hebel-Slichter peak" (a spike in activity) just below the temperature where it becomes a superconductor.
Crucially, this spike will only appear if the material is "chiral" (twisted/spinning). If it's a normal superconductor, the spike disappears.

Summary

Think of this paper as a new stethoscope for the quantum world.

Old way: Poke the patient with a needle (apply magnetic fields) to see how they react.
New way: Place a tiny, sensitive ear (the NV center) near the patient.
The trick: Listen to the difference in the heartbeat (relaxation rate) depending on which way the ear is facing.
The result: If the heartbeat sounds different from the left vs. the right, you know the patient has a "twisted" heart (Time-Reversal Symmetry Breaking). This allows scientists to diagnose exotic quantum states without disturbing them, potentially leading to better quantum computers and new materials.

1. Problem Statement

Detecting Time-Reversal Symmetry Breaking (TRSB) in condensed matter systems (such as Quantum Hall states and chiral superconductors) is a significant challenge. Traditional methods often rely on external fields (e.g., Polar Kerr effect, NMR), which can perturb the delicate quantum states of the material or fail to distinguish between competing orders. Furthermore, global transport measurements often average out local, wave-vector-dependent phenomena that are crucial for identifying specific topological states (e.g., distinguishing between different fractional quantum Hall states or determining pairing angular momentum in superconductors).

The authors propose using Nitrogen-Vacancy (NV) centers in diamond as non-invasive, nanoscale probes to detect TRSB by measuring the asymmetry in magnetic fluctuation spectra.

2. Methodology

The paper develops a theoretical framework connecting the relaxation dynamics of an NV center to the electromagnetic fluctuation spectrum of a nearby material.

Physical Setup: An NV center is placed at a distance $z_{NV}$ above a 2D material. The NV center acts as a spin-1 qubit with a ground state $|m_s=0\rangle$ and excited states $|m_s=\pm 1\rangle$ .
Relaxation Mechanism: The relaxation rate ( $\Gamma$ ) from the excited states to the ground state is driven by magnetic fluctuations orthogonal to the NV dipole axis.
Chiral Probing: The core insight is that TRSB materials exhibit circular dichroism in their magnetic fluctuation spectra. The spectrum of left-circularly polarized ( $b_+$ $b_{+}$ ) and right-circularly polarized ( $b_-$ $b_{-}$ ) magnetic fields differs.
- The relaxation rate starting from $|m_s = +1\rangle$ ( $\Gamma_{+\hat{z}}$ ) depends on the spectral weight of $b_+$ at frequency $\omega_{NV}$ and $b_-$ at $-\omega_{NV}$ .
- The relaxation rate starting from $|m_s = -1\rangle$ ( $\Gamma_{-\hat{z}}$ ) depends on $b_-$ at $\omega_{NV}$ and $b_+$ at $-\omega_{NV}$ .
The Diagnostic Metric: The authors argue that the difference in relaxation rates, $\Delta \Gamma = \Gamma_{+\hat{z}} - \Gamma_{-\hat{z}}$ $ΔΓ = Γ_{+ \overset{z}{^}} - Γ_{- \overset{z}{^}}$ , isolates the TRSB contribution.
- In a time-reversal symmetric system, detailed balance ensures these rates are related such that the difference vanishes (or is negligible).
- In a TRSB system, the cross-correlation term $\text{Im}[N_{xy}]$ (related to the imaginary part of the off-diagonal susceptibility) is non-zero, leading to a measurable $\Delta \Gamma$ .
Theoretical Derivation:
- The authors solve Maxwell's equations for a 2D conductive sheet sandwiched between dielectrics, incorporating a Hall conductivity $\sigma_H(q, \omega)$ .
- They derive the magnetic response function $\chi_{yx}$ and relate it to the relaxation rates.
- They show that $\Gamma_{\pm \hat{z}} \propto \text{Re}[\sigma_T \mp i\sigma_H]$ , where $\sigma_T$ is the transverse conductivity and $\sigma_H$ is the Hall conductivity.
- Crucially, the difference $\Gamma_{+\hat{z}} - \Gamma_{-\hat{z}}$ is directly proportional to $\text{Im}[\sigma_H(q, \omega)]$ .

3. Key Contributions & Results

A. Quantum Hall Systems and Hall Viscosity

Quantum Limit: In the Quantum Hall limit ( $\omega_c \tau \gg 1$ ), the authors show a stark asymmetry. If the NV dipole is anti-aligned with the external magnetic field, the relaxation rate approaches zero (dominated only by vacuum fluctuations). If aligned, it remains finite.
Hall Viscosity Connection: The paper establishes a novel link between the imaginary part of the Hall conductivity and Hall viscosity ( $\eta_H$ $η_{H}$ ).
- While Hall viscosity is traditionally associated with the real part of the Hall conductivity at finite wavevectors ( $O(q\ell)^2$ ), the authors demonstrate that in the imaginary part (dissipative component), the Hall viscosity manifests as a distinct pole at frequency $\omega = 2\omega_c$ (twice the cyclotron frequency).
- This allows for the direct inference of Hall viscosity from NV relaxation dynamics, potentially distinguishing candidate fractional quantum Hall states (e.g., Pfaffian vs. Anti-Pfaffian) based on their angular momentum.

B. TRSB Superconductors

Hebel-Slichter Enhancement: The authors calculate relaxation rates for both conventional $d$ $d$ -wave and chiral superconductors. They find that the average relaxation rate $[\Gamma_{+\hat{z}} + \Gamma_{-\hat{z}}]/2$ $[Γ_{+ \overset{z}{^}} + Γ_{- \overset{z}{^}}] /2$ exhibits a Hebel-Slichter-like peak just below the critical temperature $T_c$ $T_{c}$ .
- Significance: Unlike standard NMR, which often misses this peak in $d$ -wave superconductors due to gap nodes and sign changes, NV relaxometry is sensitive to small wavevectors ( $q \sim 1/z_{NV}$ ), preserving the peak.
Chiral Detection: The difference rate $\Delta \Gamma$ $ΔΓ$ is zero for conventional superconductors but becomes non-zero for chiral superconductors (e.g., chiral $d$ $d$ -wave or $p$ $p$ -wave).
- $\Delta \Gamma$ is directly proportional to the pairing angular momentum ( $l$ ) of the Cooper pairs.
Case Study (Twisted BSCCO): The authors provide concrete estimates for stacked twisted Bismuth Strontium Calcium Copper Oxide (BSCCO) flakes (twisted at $\approx 45^\circ$ $\approx 4 5^{\circ}$ ), a system recently suggested to host TRSB.
- They predict a measurable differential relaxation rate below $T_c$ (approx. 90K) for an NV center at $z_{NV} \approx 20$ nm.
- The peak in $\Delta \Gamma$ would confirm the chiral nature of the superconducting state.

4. Significance and Impact

Non-Invasive Local Probe: Unlike NMR or Kerr rotation, this method requires no external magnetic or optical fields that could alter the material's state. It is purely passive, sensing the intrinsic thermal fluctuations.
Wave-Vector Resolution: The technique probes conductivity and susceptibility at specific wavevectors $q \sim 1/z_{NV}$ . By varying the probe distance, one can map the $q$ -dependence of the Hall conductivity, a feature inaccessible to bulk transport measurements.
Access to Hall Viscosity: The paper provides a practical pathway to measure Hall viscosity, a quantity that is notoriously difficult to detect experimentally but is a key topological invariant for distinguishing quantum Hall states.
Resolving Superconducting Order: The method offers a definitive test for chiral superconductivity (distinguishing $l \neq 0$ pairing) in complex materials like twisted cuprates, where other probes have yielded ambiguous results.
Experimental Feasibility: The authors estimate that for typical parameters (conductivity $\sim e^2/h$ , distance $\sim 10-100$ nm), the relaxation rates are on the order of 1 Hz or faster, which is well within the detection capabilities of current cryogenic NV center setups.

Conclusion

The paper establishes a robust theoretical protocol for using NV centers to detect Time-Reversal Symmetry Breaking. By exploiting the chirality of magnetic fluctuations, the technique isolates the Hall conductivity's imaginary part, enabling the measurement of Hall viscosity in Quantum Hall systems and the determination of pairing angular momentum in chiral superconductors. This offers a powerful, non-invasive tool for characterizing topological phases of matter.