Estimation of conserved charges for a one dimensional system with inhomogeneous hopping

This paper demonstrates that integrable matrix theory can effectively estimate conserved charges in a one-dimensional single-particle system with inhomogeneous hopping, revealing that the number of such charges serves as a quantitative measure of quantum integrability across the chaotic-to-integrable crossover.

The Gist

Imagine a crowded dance floor where people (particles) are trying to move around. In a perfectly chaotic party, everyone bumps into each other randomly, and the energy levels of the room are all mixed up and unpredictable. This is what physicists call a "chaotic" system.

On the other hand, imagine a very orderly ballroom where everyone follows a strict, predictable pattern. They move in perfect sync, and the energy levels are distinct and uncorrelated. This is an "integrable" system.

The paper by Triparna Mondal explores a middle ground: a dance floor where the rules of movement are a bit messy and uneven. Specifically, the author studies a one-dimensional line of dancers where the "hopping" (how easily they move to the next spot) is random and uneven. The goal is to figure out: How do we measure if this messy system is becoming more orderly (integrable) or staying chaotic?

The "Secret Handshakes" (Conserved Charges)

In physics, an "integrable" system is special because it has hidden rules, or "conserved charges." Think of these as secret handshakes that every dancer knows.

• In a chaotic system, there are no secret handshakes; everyone is doing their own thing.
• In a perfectly integrable system, there are as many secret handshakes as there are dancers. Everyone is locked into a rigid, predictable pattern.

The paper uses a mathematical tool called Integrable Matrix Theory (IMT) to try to count these "secret handshakes." The theory suggests that if you can find these handshakes, you can prove the system is orderly.

The Experiment: Tuning the Chaos

The author creates a computer model of this 1D line of dancers. They introduce a "knob" (a parameter called $\gamma$) that controls how uneven the hopping is.

• Turning the knob one way: The hopping becomes very random and strong. The system acts chaotic.
• Turning the knob the other way: The hopping becomes weak and uneven. The system starts to look more orderly.

The author then tries to count the "secret handshakes" (conserved charges) as they turn this knob.

What They Found

1. The "Handshake" Count Increases: As the system moves from chaotic toward orderly, the number of detectable "secret handshakes" increases. When the system is fully orderly, the number of handshakes equals the number of dancers (the size of the system).
2. A Strange Twist: The author noticed something weird. When they turned the knob too far (making the hopping extremely weak), the "handshake" counting method got confused.
   • The energy levels (the music of the party) started acting chaotic again.
   • But the dancers themselves (the wave functions) stayed perfectly frozen in place (localized).
   • Because the dancers were frozen, the math said there were no handshakes to count using their specific method, even though the system was technically "integrable" (frozen).
3. The Conclusion: The number of conserved charges is a great way to measure how "integrable" a system is, but it has limits. It works perfectly when the system is transitioning from chaos to order. However, if the system becomes too frozen, the method struggles to count them, even though the system is technically fully ordered.

The Big Picture

The paper demonstrates that counting these "secret handshakes" (conserved charges) is a valid way to tell if a quantum system is chaotic or orderly. It confirms that as a system becomes more integrable, it gains more of these hidden rules.

However, the study also highlights a quirk: if you push the system to the extreme limit where movement stops completely, the standard way of counting these rules breaks down, even though the system is technically in a state of perfect order. This helps physicists understand the boundaries of how we measure order in the quantum world.

Technical Summary

Technical Summary: Estimation of Conserved Charges for a One-Dimensional System with Inhomogeneous Hopping

Problem Statement
Quantum integrability is traditionally characterized by the existence of a large number of conserved charges. While identifying these charges in generic quantum systems is challenging, Integrable Matrix Theory (IMT) offers a unified framework for specific classes of systems. A critical open question addressed in this work is the efficacy of conserved charges as a measure of quantum integrability in one-dimensional (1D) systems where nearest-neighbor hopping is inhomogeneous. Standard models, such as the 1D Anderson ensemble with homogeneous hopping and random onsite disorder, exhibit integrable behavior and Poissonian spectral statistics. However, the introduction of inhomogeneity in hopping parameters complicates the landscape, particularly in the crossover region between chaotic and integrable limits. The authors aim to formulate a 1D system with random, inhomogeneous hopping and utilize IMT to estimate conserved charges across the chaotic–integrable transition.

Methodology
The study employs a random matrix model defined on a 1D finite-sized lattice with $N$ sites. The Hamiltonian is constructed within the IMT framework as $H = V + xT$, where $V$ represents the onsite potential and $T$ represents the hopping terms.

• Model Formulation: The matrix $V$ is chosen to be a diagonal matrix with eigenvalues following Wigner-Dyson statistics (specifically from a Gaussian Orthogonal Ensemble, GOE, context). The matrix $T$ is a tridiagonal real symmetric matrix with zero diagonal elements, representing nearest-neighbor hopping.
• Inhomogeneity: The off-diagonal hopping elements $t_{i,i+1}$ are sampled from a $\chi$-distribution with degrees of freedom determined by a tunable parameter $\beta$. The authors introduce a parameter $\gamma$ such that $\beta = 1/(N\gamma)$, allowing them to tune the system from a chaotic limit ($\gamma \to 0$) to an integrable limit ($\gamma \to 1$).
• Conserved Charge Estimation: Using IMT, conserved charges $H_i$ are formulated as infinite series $H_i = \sum P^i_m$, where coefficients $P^i_m$ are determined via a recursion relation involving the eigenvalues of $V$ and the matrix elements of $T$. The existence of a conserved charge is confirmed by the convergence of the coefficients $\eta^{(m)}$ in the recursion relation as the order $m$ increases.
• Statistical Analysis: The authors analyze the spectral statistics (nearest-neighbor spacing distribution $P(s)$, average spacing ratio $\langle \tilde{r} \rangle$) and eigenstate statistics (Inverse Participation Ratio, fractal dimension $D_2$) to benchmark the system's behavior against known limits (GOE, GUE, GSE, Poisson).

Key Results

1. Spectral and Eigenstate Statistics:

   • As $\gamma$ increases from 0 to 1, the spectral statistics transition from Wigner-Dyson (chaotic) toward Poissonian (integrable). However, for finite system sizes, the system does not fully reach the Poisson limit at $\gamma=1$.
   • For $\gamma > 1$, the spectral statistics unexpectedly revert toward Wigner-Dyson behavior, while the eigenstate localization (measured by fractal dimension $D_2$) continues to increase, approaching complete localization. This decoupling of spectral and eigenstate statistics is attributed to the specific construction of the Hamiltonian where the diagonal elements (obeying GOE) dominate as off-diagonal terms vanish.
   • The system exhibits a non-ergodic-extended phase in the crossover region, distinct from the standard $\beta$-ensemble behavior.

2. Estimation of Conserved Charges:

   • The convergence of the recursion coefficients $\eta$ serves as a proxy for the existence of conserved charges. The rate of convergence increases with $\gamma$, correlating with increased localization.
   • The number of conserved charges, $n$, is estimated by counting the number of independent charges for which the slope of $\log|\eta|$ vs. $m$ is negative.
   • Crossover Behavior: The number of conserved charges $n$ increases as $\gamma$ goes from 0 to 1, indicating a transition from a chaotic to a near-integrable regime. At $\gamma \approx 1$, the number of charges approaches the system size $N$ (specifically $N-1$ for a fully integrable system).
   • Finite Size Effects: The ratio $n/N$ is dependent on system size; larger systems approach the fully integrable limit ($n/N \to 1$) more closely than smaller ones.
   • High $\gamma$ Regime: For $\gamma > 1$, the numerical calculation of higher-order terms becomes difficult as hopping terms vanish. Theoretically, a diagonal Hamiltonian implies $N$ conserved charges (complete integrability), but the numerical method based on slopes $\Delta_\gamma$ shows a decrease in the calculated number of charges, highlighting a limitation of the slope-based metric in the extreme diagonal limit.

Significance and Claims
The paper claims that the number of conserved charges, estimated using Integrable Matrix Theory, serves as a viable measure of quantum integrability for 1D systems with inhomogeneous hopping. The primary contribution is the demonstration that this metric successfully tracks the chaotic–integrable crossover, showing a near-linear increase in the number of charges as the system approaches the integrable limit ($\gamma \approx 1$).

The authors note that while the exact analytical formulation of the conserved charges remains non-trivial, the numerical estimation of their count provides a robust indicator of integrability strength. The study highlights that for finite-sized systems, the transition is not sharp, and finite-size effects significantly influence the observed statistics. The work extends the application of conserved charge analysis beyond standard Anderson localization to systems with inhomogeneous hopping, offering a new perspective on the non-ergodic-extended phase. The paper concludes that the IH (Inhomogeneous Hopping) ensemble is an appropriate model for studying the delocalization-to-localization transition through the lens of conserved charges.