Dynamical spin-nematic order in a transverse field Ising chain with non-Hermitian Gamma interaction

This paper investigates a transverse field Ising chain with non-Hermitian Gamma interaction, demonstrating that parity-time symmetry breaking induces a gapless phase with long-range and dynamical spin-nematic order, thereby revealing a rich quantum phase diagram and offering a scheme for generating such order through non-equilibrium dynamics.

The Gist

Imagine a long line of tiny magnets (spins) sitting next to each other, like a row of dominoes. In the world of quantum physics, these dominoes don't just sit still; they wiggle, flip, and interact with their neighbors in complex ways. This paper explores what happens when we shake up this line of dominoes with three specific ingredients: a magnetic push, a standard "neighborly" interaction, and a very strange, new ingredient called non-Hermitian Gamma interaction.

Here is the story of what they found, explained without the heavy math.

1. The Setup: The Domino Line

Think of the Transverse Field Ising Chain as a line of dominoes that want to align in a specific pattern.

• The Standard Interaction: Usually, these dominoes want to be opposite to their neighbors (like a checkerboard). This is the "Ising interaction."
• The Push: Scientists apply a "transverse field," which is like a strong wind blowing on the dominoes, trying to knock them over and make them chaotic.
• The New Ingredient (Gamma): This is the star of the show. In real-world materials (like certain crystals), the way these dominoes interact isn't just simple; it's twisted and directional. The "Gamma interaction" adds a twist to the rules.

2. The Twist: Introducing "Loss and Gain" (Non-Hermitian Physics)

In the old days, physicists thought "dissipation" (losing energy to the environment, like friction) was a bad thing that ruined experiments. They tried to eliminate it.

This paper treats dissipation as a superpower.

• The Analogy: Imagine a game of musical chairs. In a normal game (Hermitian physics), the music stops, and everyone sits down. In this new game (Non-Hermitian), some players are secretly being removed from the game (loss), while others are being added (gain).
• The "Gamma interaction" in this paper acts like a specific rule for this game: it creates a situation where the system is constantly losing and gaining energy in a balanced, yet strange way. This is called PT Symmetry (Parity-Time symmetry).

3. The Discovery: A Hidden "Ghost" Phase

When the scientists turned up the "Gamma" knob, they expected the dominoes to either line up perfectly or fall into chaos. Instead, they found a third, mysterious state.

• The Normal States:
  • Ferromagnetic: All dominoes point the same way.
  • Paramagnetic: The wind is too strong; they are all jumbled up.
• The New State (The Gapless Phase):
  • In a normal world, if the dominoes are jumbled (gapless), they usually only have "short-range" order. It's like a crowd of people whispering; you can hear your neighbor, but you can't hear the person across the room.
  • The Surprise: In this new "Gamma" world, even though the system is jumbled, the dominoes start whispering to each other across the entire line. They develop a Long-Range Spin-Nematic Order.
  • The Metaphor: Imagine a stadium full of people. Usually, if they are chaotic, no one hears anyone else. But in this new state, even though no one is standing up in a uniform row, the direction of their whispers creates a perfect, synchronized pattern that stretches from one end of the stadium to the other. This is the "Spin-Nematic Order."

4. The "Ghost" Connection: Why Does This Happen?

This long-range order appears because the system breaks a fundamental rule called PT Symmetry.

• The Analogy: Think of a mirror (Parity) and a movie playing backward (Time). In a normal world, if you look in the mirror and play the movie backward, it looks the same.
• In this new phase, the "mirror" and the "backward movie" stop working together. The system enters a state where the "loss" and "gain" become unbalanced in a way that creates this exotic, long-range connection. It's as if the dominoes suddenly realize they are part of a single, giant organism, even though they aren't lined up in a row.

5. The Dynamic Experiment: The "Quench"

To prove this wasn't just a static trick, the scientists did a "Quench."

• The Analogy: Imagine the dominoes are in a calm state (no Gamma interaction). Suddenly, they slam the "Gamma" switch on.
• The Result: The dominoes start dancing. In normal physics, this dance would be a chaotic wobble that averages out to nothing. But in this new state, the dance settles into a rhythmic, non-zero pattern.
• The Takeaway: By watching how the system moves after the switch is flipped, they could map out exactly where this new "Spin-Nematic" phase exists. It's like diagnosing a disease by watching how a patient runs, rather than just looking at them sitting still.

Summary: Why Does This Matter?

This paper shows that by intentionally adding "loss" and "gain" (dissipation) to a quantum system, we can create brand new states of matter that don't exist in the natural, friction-free world.

• The Big Picture: We used to think dissipation was just noise to be removed. This paper shows that dissipation is a tool. By tuning it, we can force quantum materials to organize themselves in ways we never thought possible, creating "long-range whispers" in a chaotic crowd.
• The Future: This could help us design better quantum computers or new types of lasers that use these "lossy" states to do things normal materials can't.

In short: By breaking the rules of energy conservation in a controlled way, the scientists unlocked a hidden, synchronized dance in a line of quantum magnets.

Technical Summary

1. Problem Statement

The paper investigates the interplay between non-Hermitian physics and quantum magnetic order in one-dimensional spin chains. Specifically, it addresses the following challenges:

• Non-Hermitian Effects on Phase Transitions: How does the introduction of non-Hermitian interactions (specifically the symmetric off-diagonal Gamma interaction, $\Gamma$) alter the phase diagram and quantum phase transitions (QPTs) of the standard transverse field Ising chain?
• Emergence of Exotic Orders: Can non-Hermitian dissipation induce new types of magnetic orders, such as spin-nematic order, that are absent or different in Hermitian systems?
• Dynamical Characterization: How can non-equilibrium dynamics (specifically after a quantum quench) be used to characterize and generate these exotic phases, particularly in the gapless regime where traditional energy gaps may not fully describe the system's criticality?

2. Methodology

The authors employ an exact analytical and numerical approach based on free fermion techniques:

• Model Hamiltonian: They study a transverse field Ising chain with an added non-Hermitian Gamma interaction:
  $$\hat{H} = J \sum_{j=1}^N \hat{\sigma}^x_j \hat{\sigma}^x_{j+1} + h \sum_{j=1}^N \hat{\sigma}^z_j - i\Gamma \sum_{j=1}^N (\hat{\sigma}^x_j \hat{\sigma}^y_{j+1} + \hat{\sigma}^y_j \hat{\sigma}^x_{j+1})$$
  Here, $J$ is the Ising coupling, $h$ is the transverse field, and $\Gamma$ represents the strength of the imaginary symmetric off-diagonal interaction (derived from the no-click limit of correlated quantum jump trajectories).

• Diagonalization:

  • The Hamiltonian is mapped to a quadratic fermionic form using the Jordan-Wigner transformation.
  • A Fourier transformation is applied to convert the system into momentum space.
  • The resulting Hamiltonian is expressed in the Bogoliubov-de Gennes (BdG) form and diagonalized using a complex Bogoliubov transformation. This yields the energy dispersion relation $\epsilon_k$.

• Symmetry Analysis: The authors analyze the Parity-Time (PT) symmetry of the system. They demonstrate that while the Hamiltonian respects PT symmetry, increasing $\Gamma$ can lead to PT symmetry breaking, resulting in complex energy eigenvalues.

• Correlation Functions:

  • Static: They calculate the spin-spin correlation function ($C^{xx}_r$) and the spin-nematic correlation function ($Q^{xy}_r$) using Wick's theorem and the Pfaffian method for multi-point fermionic correlations.
  • Dynamic: They perform a quantum quench simulation. The system is initialized in the ground state of a Hermitian Hamiltonian ($\Gamma=0$) and evolved under the non-Hermitian Hamiltonian ($\Gamma \neq 0$). They compute the time-evolution of the spin-nematic order parameter and its time average.

3. Key Contributions and Results

A. Phase Diagram and Energy Gap

The study reveals a rich phase diagram in the $(\Gamma, h)$ plane consisting of three distinct regions:

1. Gapped Ferromagnetic Phase: Occurs at low $\Gamma$ and low $h$.
2. Gapped Paramagnetic Phase: Occurs at high $h$.
3. Gapless Phase (PT-Broken): A novel phase emerging at high $\Gamma$ and low $h$.
   • Unlike Hermitian gapless phases (which typically exhibit quasi-long-range order with power-law decay), this non-Hermitian gapless phase exhibits true long-range order.
   • The transition lines are defined by the vanishing of the real part of the energy gap ($\text{Re}(\epsilon_k) = 0$).
   • The PT-symmetric region (real spectrum) transitions to a PT-broken region (complex spectrum) via exceptional points.

B. Static Spin-Nematic Order

• PT Symmetry Breaking as a Driver: The authors show that spin-nematic order (characterized by the correlation $Q^{xy}_r = \langle \sigma^x_m \sigma^y_n + \sigma^y_m \sigma^x_n \rangle$) is strictly zero in the PT-symmetric phase.
• Long-Range Order: Upon crossing into the PT-broken phase (where $\Gamma$ is sufficiently large), the system develops long-range spin-nematic order for $h < J$.
• Contrast with Hermitian Systems: In standard Hermitian gapless spin chains, true long-range order is forbidden by the Mermin-Wagner theorem (or manifests only as quasi-long-range order). The non-Hermitian nature here stabilizes true long-range order in the gapless regime.

C. Dynamical Spin-Nematic Order

• Quench Dynamics: The authors simulate a quench where the system starts in a Hermitian ground state and evolves under the non-Hermitian Hamiltonian.
• Suppression of Oscillations: In PT-symmetric regions, the spin-nematic correlation oscillates rapidly due to dynamical phase factors, leading to a vanishing time average.
• Emergence of Order: In the PT-broken region, the imaginary components of the energy dispersion suppress these oscillations. Consequently, the time-averaged spin-nematic order remains finite and non-zero.
• Characterization: This dynamical behavior provides a robust method to characterize the spin-nematic phase diagram using non-equilibrium dynamics, effectively distinguishing the PT-broken phase from the PT-symmetric phase.

4. Significance and Implications

• New Quantum Phases: The work demonstrates that non-Hermitian interactions can stabilize exotic quantum phases (specifically long-range spin-nematic order) that are inaccessible in Hermitian systems, particularly in gapless regimes.
• Mechanism for Order Generation: It proposes a scheme for generating spin-nematic order in spin chains by engineering dissipation (via the Gamma interaction) to induce PT symmetry breaking.
• Dynamical Diagnostics: The paper establishes that non-equilibrium dynamics (specifically the time-averaged order after a quench) serve as a powerful diagnostic tool for identifying PT-symmetry breaking and mapping phase diagrams, offering an alternative to static energy gap analysis.
• Experimental Relevance: The findings are relevant for experimental platforms involving open quantum systems, such as superconducting qubits, cold atoms, or photonic systems, where dissipation can be engineered to simulate non-Hermitian Hamiltonians.

In conclusion, the paper provides a comprehensive theoretical framework showing that the competition between Ising interactions, transverse fields, and non-Hermitian Gamma interactions leads to a unique gapless phase with long-range spin-nematic order, driven by PT symmetry breaking and detectable through dynamical protocols.