Isospectrality and Operator Complexity

This paper demonstrates that free and interacting many-body systems can be isospectral yet exhibit fundamentally distinct phase structures and operator complexity dynamics, linked by a nonlocal unitary transformation that maps local operators to extended many-body strings.

The Gist

Imagine you have two different musical instruments. One is a simple, pure flute (the "quadratic" model), and the other is a complex, heavy drum kit with many interacting parts (the "interacting" model).

Usually, if you play a specific note on the flute, it sounds very different from a note on the drum kit. But in this paper, the researchers discovered a magical trick: they found a way to tune the drum kit so that every single note it produces is exactly the same pitch and volume as the flute. In physics terms, they are "isospectral"—they share the exact same energy spectrum.

However, the paper reveals a mind-bending twist: Even though they sound the same, they behave completely differently when you try to play a melody or change the music.

Here is the breakdown of their discovery using simple analogies:

1. The Magic Translator (The Unitary Transformation)

The researchers found a "translator" (a mathematical transformation) that turns the complex drum kit into the simple flute without changing the notes.

• The Catch: This translator is "nonlocal." Imagine if to play a single note on the flute, you had to hit a specific sequence of drums across the entire room, involving dozens of other drums at once.
• The Result: In the simple flute world, a "local" action (hitting one key) stays local. But in the complex drum world, that same action gets stretched out into a giant, tangled string of interactions across the whole system.

2. Different Landscapes, Same Map

Because they share the same notes (energy spectrum), you might think they represent the same "landscape" or phase of matter.

• The Flute (Quadratic Model): It behaves like a standard topological material. It has clean, simple edges (like Majorana modes) that are easy to describe.
• The Drum Kit (Interacting Model): Even though it has the same notes, it lives in a totally different "phase." Depending on how you tune it, it can become a "charge-density wave" (like a checkerboard pattern) or a "density-polarized" state.
• The Lesson: Just because two systems have the same "menu" of energy levels doesn't mean they serve the same "meal." The structure of the ingredients (the operators) matters just as much as the final taste.

3. The Speed of Information (OTOCs)

The researchers looked at how fast information travels through these systems (like a ripple spreading in a pond).

• The Front: Both systems have a "speed limit" for how fast a ripple can move. This speed is determined by the notes (the spectrum), so both the flute and the drum kit have the same speed limit.
• The Inside: However, what happens inside the ripple is different.
  • In the flute, the ripple is smooth and predictable.
  • In the drum kit, because the "translator" stretched the local action into a giant string, the ripple develops a complex, interference pattern. It's like the difference between a clean laser beam and a beam of light passing through a kaleidoscope. The light travels at the same speed, but the pattern inside is chaotic and complex.

4. The Complexity of Growth (Krylov Complexity)

Finally, they looked at how "complex" the system gets over time. Imagine you are trying to describe the state of the system.

• The Flute: To describe the state, you only need a few simple words. The complexity stays low and bounded. It's like writing a haiku; it's short and contained.
• The Drum Kit: To describe the state, you need to keep adding more and more words, connecting more and more parts of the system. The complexity grows steadily (like the square root of time). It's like writing a novel that keeps getting longer and more intricate the more you think about it.

The Big Takeaway

The paper proves a fundamental point in quantum physics: You cannot judge a system solely by its energy levels (its spectrum).

Two systems can be perfect twins in terms of their energy "notes," but if you look at how their parts interact and how information spreads, they can be as different as a flute and a drum kit. The "soul" of the system (its dynamics and complexity) is hidden in the structure of its operators, not just in its energy spectrum.

In short: Same notes, different song. Same energy, different complexity.

Technical Summary

Technical Summary: Isospectrality and Operator Complexity

Problem Statement
A central question in quantum many-body physics concerns the extent to which the many-body energy spectrum encodes the physical properties of a quantum system. While spectral information traditionally underpins the understanding of equilibrium phenomena—such as thermodynamics, response functions, and phase classifications—recent developments in non-equilibrium dynamics suggest that dynamical properties (e.g., quantum chaos, operator scrambling, and Krylov complexity) depend critically on the structure of operators under time evolution, not merely on spectral data. This work addresses the open question of whether two local Hamiltonians with identical many-body spectra can exhibit sharply different phase structures, operator growth, scrambling dynamics, and dynamical complexity.

Methodology
The authors study a pair of exactly solvable, isospectral fermion chains related by a nonlocal, nonlinear unitary transformation.

1. The Models:
   • Interacting Model ($H_{\text{int}}$): A nearest-neighbor chain of spinless fermions featuring a quartic density-density interaction term, $H_{\text{nn}} = \sum (2n_j - 1)(2n_{j+1} - 1)$, added to a quadratic Kitaev chain.
   • Quadratic Model ($H_{\text{ff}}$): A quadratic Kitaev-type Hamiltonian with tunable hopping and pairing amplitudes dependent on the interaction strength $\lambda$.
2. The Transformation: The authors utilize a specific nonlocal, nonlinear fermion-to-fermion unitary transformation $U$ (introduced in Ref. [12]) that maps the interacting Hamiltonian $H_{\text{int}}(\lambda)$ exactly onto the quadratic Hamiltonian $H_{\text{ff}}(\lambda)$. This transformation preserves the entire many-body spectrum (isospectrality) but reorganizes the local operator algebra, mapping local fermion operators into extended many-body strings.
3. Analytical Tools:
   • Exact Solvability: By mapping the interacting model to a quadratic dual, the authors derive exact Pfaffian representations for observables determined by the propagator of the dual quadratic model.
   • Dynamical Diagnostics: The study employs ground-state density correlations, Out-of-Time-Ordered Correlators (OTOCs), and Krylov complexity (via Liouvillian Lanczos coefficients) to compare the dynamics of the two models.

Key Contributions and Results

• Exact Isospectrality: Numerical verification confirms that the many-body spectra of $H_{\text{int}}$ and $H_{\text{ff}}$ agree level-by-level for system sizes up to $N=12$, with spectral differences vanishing to numerical precision.
• Distinct Phase Structures: Despite sharing the same bulk critical points at $\lambda = \pm 1$, the two models realize fundamentally different phases:
  • The quadratic model ($H_{\text{ff}}$) is a BDI symmetry class Kitaev chain with topological phases characterized by winding numbers $\nu = \pm 1$ and simple Majorana edge modes.
  • The interacting model ($H_{\text{int}}$) exhibits three distinct phases: a charge-density-wave (CDW) order for $\lambda > 1$, a Symmetry Protected Topological (SPT) phase for $-1 < \lambda < 1$, and a density-polarized regime for $\lambda < -1$. Crucially, the zero modes in the interacting SPT phase are highly nonlocal many-body strings, contrasting with the simple Majorana operators of the quadratic model.
• Operator Spreading and OTOCs:
  • Both models exhibit a ballistic light cone with velocity $v_{\text{max}} = 4$ at $\lambda=1$, dictated by the common single-particle spectrum.
  • However, the interior of the light cone differs significantly. The quadratic model shows simple propagation, while the interacting model displays rich interference patterns and substantial weight throughout the cone. This arises because physical density operators in the interacting model map to extended Majorana strings, leading to nontrivial operator spreading despite the free-fermion spectrum.
• Krylov Complexity and Lanczos Growth:
  • Quadratic Model: Local operators evolve within a closed linear subspace (the Majorana sector), resulting in bounded Lanczos coefficients ($b_n$) that approach a period-two form.
  • Interacting Model: Repeated commutators generate increasingly higher-body operators. The Lanczos coefficients exhibit asymptotic growth $b_n \propto \sqrt{n}$, characteristic of interacting systems. This demonstrates that the operator complexity grows unboundedly in the interacting model, even though the spectrum is identical to the bounded-complexity quadratic model.

Significance and Claims
The paper demonstrates that free many-body spectra and interacting operator dynamics are fundamentally compatible. The authors conclude that spectral equivalence alone is insufficient to determine:

1. The phase structure of a system.
2. The nature of the local operator algebra.
3. The dynamical complexity and operator growth (scrambling) of the system.

The results highlight that while the spectrum dictates propagation velocities (light cones), the operator algebra determines the internal structure of dynamics and the rate of operator complexity growth. The study establishes that a nonlocal unitary transformation can preserve the energy spectrum while radically altering the physical interpretation of local observables and their time evolution.