Topological spectral form factor reveals emergent non-Hermitian single-particle $\mathcal{PT}$ transitions from many-body quantum chaos

This paper introduces the topological spectral form factor (TopSFF) as a non-perturbative probe that maps the dynamics of 1D many-body chaotic systems with topological defects to an emergent non-Hermitian single-particle problem, revealing a $\mathcal{PT}$ symmetry breaking transition at a critical interaction strength that governs the system-size scaling behavior of the TopSFF.

The Gist

The Big Picture: Listening to the Chaos

Imagine a crowded, chaotic dance floor where thousands of people (quantum particles) are moving randomly. In physics, we usually try to understand this chaos by looking at the average behavior of the crowd. But sometimes, the most interesting things happen in the exceptions—the rare moments when the chaos reveals a hidden pattern.

This paper introduces a new way to "listen" to this quantum dance floor. The authors created a special tool called the Topological Spectral Form Factor (TopSFF). Think of it as a high-tech microphone that doesn't just record the music; it records the music played by two identical bands at the same time, but with a twist: one band is playing the song forward, and the other is playing it backward, and they are forced to swap partners in a very specific, weird way.

The Main Discovery: A "Phase Transition" in the Chaos

The most exciting finding is that when the authors used this tool, they discovered that the chaotic dance floor behaves like a light switch.

Usually, we think of quantum chaos as just being "messy." But this paper shows that as you turn up the "interaction strength" (how much the dancers bump into each other), the system suddenly flips into a completely different state.

• State A (The "Unbroken" Phase): The dancers move in a predictable, steady rhythm. If you measure the chaos, it grows or shrinks smoothly, like a balloon inflating or deflating.
• State B (The "Broken" Phase): The dancers start to wobble and oscillate. The measurement doesn't just grow; it starts to vibrate or oscillate up and down as the system gets bigger.
• The Switch (The Exceptional Point): There is a precise moment right between these two states where the system behaves strangely. It's like a door that is stuck halfway open; the math describing the system breaks down in a specific way, creating a unique "glitch" that signals the transition.

The Secret Ingredient: The "Mickey Mouse" Map

How did they figure this out? They had to simplify the incredibly complex math of billions of particles. They did this by folding the problem in half (like folding a piece of paper) and looking at the "loops" the particles make.

They discovered that the most important patterns look like Mickey Mouse heads.

• Imagine a diagram with one big loop (the head) and two smaller loops (the ears).
• In their math, these "Mickey Mouse" shapes represent Temporal Domain Walls (tDW).
• Think of a "Domain Wall" as a fence line separating two different types of weather. On one side of the fence, the weather is "Gaussian" (calm, standard); on the other side, it's "Non-Gaussian" (wild, unusual).
• The "Mickey Mouse" diagram is the fence itself. The paper shows that these fences can exist in two states: calm or wild.

The "PT" Transition: A Mirror Game

The paper describes a phenomenon called a PT transition.

• P (Parity): Imagine looking at the dance floor in a mirror.
• T (Time): Imagine playing the dance floor video backward.
• PT Symmetry: Usually, if you look in the mirror and play the video backward, the scene looks different. But in this specific "unbroken" state, the system is so perfectly balanced that the mirror-image-backward version looks exactly the same as the original.

The paper proves that as the dancers interact more strongly, this perfect balance breaks. The system stops being its own mirror image and starts oscillating. This is the "PT transition."

The "Jordan Block" Glitch

At the exact moment the switch flips (the "Exceptional Point"), the math gets weird. Normally, you can describe the system with two distinct modes (like a high note and a low note). But at the switch point, these two notes merge into one.

The authors found that at this exact moment, the system doesn't just sit there; it gets a "boost." It's like a car that, instead of just accelerating, suddenly gets a burst of speed that grows linearly with the size of the car. This is a mathematical signature called a Jordan block, and it's the smoking gun that proves the system has hit that critical transition point.

Why Does This Matter?

The authors show that this isn't just a trick of the math. They tested it on different computer models of quantum chaos, and the "Mickey Mouse" patterns and the "light switch" behavior appeared every time.

They also looked at "time-traveling" defects (defects that stretch across time rather than space) and found that the energy cost of these defects follows a universal rule, similar to how the cost of stretching a rubber band depends only on its length, not on what the rubber band is made of.

Summary

In short, the paper says:

1. We built a new tool (TopSFF) to look at quantum chaos through a "topological" lens (looking at the shape of the paths particles take).
2. We found that this chaos has a hidden "light switch" (a PT transition) that flips the system from a smooth, steady state to an oscillating, vibrating state.
3. This transition is driven by "Temporal Domain Walls" (fences in time) that look like "Mickey Mouse" diagrams in the math.
4. At the exact moment of the switch, the system shows a unique mathematical "glitch" (Jordan block) that confirms the transition is real.

This work bridges the gap between the messy, complex world of many interacting particles and the cleaner, simpler world of single-particle physics, showing that even in total chaos, there are universal rules waiting to be found.

Technical Summary

Technical Summary: Topological Spectral Form Factor Reveals Emergent Non-Hermitian Single-Particle PT Transitions from Many-Body Quantum Chaos

Problem and Motivation
In equilibrium physics, topological defect insertions in partition functions serve as non-perturbative probes for phase transitions and global properties beyond local observables. In non-equilibrium quantum dynamics, the spectral form factor (SFF), defined as $K(t) = |Z(it)|^2$, acts as a minimal universal probe of quantum chaos, exhibiting characteristic "bump-ramp-plateau" behavior. While the ramp and plateau are signatures of random matrix theory (RMT), the "bump" regime arises from intrinsically many-body phenomena, specifically spatial domain walls between locally distinct pairings of Feynman paths.

This work addresses the need for a probe that isolates these many-body topological structures. The authors define the Topological Spectral Form Factor (TopSFF) by inserting a topological defect that acts non-trivially on the "doubled" partition function (the two copies required for the SFF). Unlike standard twisted boundary conditions which preserve world-sheet topology, this asymmetric insertion creates mismatched spacetime world-sheet topologies (e.g., a Klein bottle and a torus), inducing a topological defect in the pairing structure of Feynman paths. The goal is to map the dynamics of these defects in generic 1D many-body chaotic systems to an effective single-particle problem.

Methodology
The authors analyze generic 1D quantum chaotic circuits, specifically the Random Phase Model (RPM) and the Random Hamiltonian Model (RHM), with local Hilbert space dimension $q$. They focus on a minimal non-trivial $Z_2$ defect implemented by a global swap operator $\hat{S}$ acting on one copy of the doubled evolution operator, $\hat{U} \otimes \hat{U}^*$.

1. Path Integral Formulation: By folding the world-sheets along parity inversion axes and unwrapping them in the time direction, the ensemble-averaged TopSFF is mapped to a path integral of Temporal Domain Walls (tDWs). These tDWs propagate along the spatial direction as emergent non-Hermitian single particles.
2. Diagrammatic Analysis: Using Haar-random unitary averaging (Weingarten calculus), the authors identify that in the large-$q$ limit, the leading local diagrams belong to the "Mickey Mouse" topological equivalence class. These diagrams are characterized by two species of tDWs:
   • Gaussian tDWs ($a=0$): Cross physical worldlines (single lines).
   • Non-Gaussian tDWs ($a=1$): Cross unitary double lines (matrix indices contracted separately).
     Crucially, these two species carry opposite Weingarten signs ($+1$ and $-1$), introducing a relative phase factor of $i$ in their conversion amplitudes.
3. Effective Hamiltonian: The TopSFF is expressed as a discrete path integral of a transfer matrix $\tilde{B}$ acting on the two-species tDW basis. The matrix elements depend on momentum sectors $(k_b, k_c, k_d)$ and interaction strength $\epsilon$.
4. Symmetry Analysis: The transfer matrix $\tilde{B}$ is shown to possess an emergent antiunitary $PT$ symmetry (where $P=\sigma_z$ acts on the species index and $T$ is complex conjugation) in specific momentum sectors where $k_b + k_c + k_d \equiv 0 \pmod \pi$.

Key Results
The analysis reveals an emergent $PT$ symmetry breaking transition in the effective single-particle dynamics of the tDWs, controlled by the interaction strength $\epsilon$.

• PT Unbroken Phase ($\epsilon < \epsilon_{EP}$): The eigenvalues of the transfer matrix are real. The leading eigenmode is a superposition of Gaussian and non-Gaussian tDWs, polarized predominantly into one sector. Consequently, the TopSFF exhibits monotonic exponential scaling with system size $L$.
• PT Broken Phase ($\epsilon > \epsilon_{EP}$): The eigenvalues form a complex-conjugate pair. The tDW sectors hybridize, and the leading modes acquire opposite phases during propagation. This results in oscillatory behavior in the TopSFF as a function of $L$, modulated by an exponential envelope.
• Exceptional Point ($\epsilon = \epsilon_{EP}$): At a finite critical interaction strength, the eigenvalues coalesce, and the transfer matrix becomes non-diagonalizable (Jordan block). This produces a linear-in-system-size enhancement ($L$) in the TopSFF, distinct from the exponential scaling in other phases.
• Universal Scaling for Temporally Extended Defects: For defects extended in time (probing spatial propagation), the authors derive exact universal scaling forms for the TopSFF free energy in systems with time-reversal symmetry (TRS) and time-translation symmetry. These forms capture the physics of spatial domain walls (sDWs) and identify the Thouless length $L_{Th}$ as the sole remaining length scale, bridging the 0D RMT regime and the many-body chaotic regime.

Significance and Claims
The paper claims to establish the TopSFF as a distinct non-local observable that captures intrinsically many-body phenomena and universal structures beyond RMT. Its primary contributions are:

1. Emergent Non-Hermitian Physics: It demonstrates a direct bridge between unitary many-body quantum chaos and non-Hermitian single-particle physics, specifically revealing an emergent $PT$ transition within the chaotic regime.
2. Defect-Resolved Observables: By isolating the "bump" regime via topological defects, the TopSFF provides access to universal non-perturbative signatures (such as the $PT$ transition and Jordan-block enhancements) that are not resolved by conventional observables like the standard SFF.
3. Universality: The derived scaling forms for temporally extended defects are shown to be universal across different chaotic models (RPM and RHM) and persist from the large-$q$ limit to finite $q$, as verified by numerical simulations.
4. Theoretical Framework: The work provides an exact analytic framework (via the Mickey Mouse diagrammatic class and transfer matrix formalism) to describe the interplay between unitary time evolution and non-unitary spatial propagation in quantum chaotic systems.

The authors note that while the $PT$ transition is robust in time-translation symmetric models, it appears as an asymptotic transition in models without this symmetry, highlighting the role of symmetry in the competition between detuning and conversion terms. The work suggests that TopSFF could be experimentally accessible by adapting existing measurement protocols to incorporate non-trivial defect insertions in doubled evolution.