[Paper Review] Simulability and regularity of complex quantum systems

Hannah Venzl, Andrew J. Daley+2 more|ArXiv.org|Aug 28, 2008

Quantum Mechanics and Applications2 references3 citations

TL;DR

This paper demonstrates that the transition from regular to chaotic spectral statistics in interacting quantum many-body systems is unambiguously signaled by the distribution of dynamically generated Schmidt coefficients, which directly indicates the breakdown of t-DMRG simulability. The key result is that chaotic regimes require exponentially more resources due to widespread Schmidt coefficient contributions, rendering standard t-DMRG inefficient beyond short timescales.

ABSTRACT

We show that the transition from regular to chaotic spectral statistics in interacting many-body quantum systems has an unambiguous signature in the distribution of Schmidt coefficients dynamically generated from a generic initial state, and thus limits the efficiency of the t-DMRG algorithm.

Motivation & Objective

• To identify a universal, computationally accessible signature of quantum chaos in many-body systems that predicts t-DMRG simulation efficiency.
• To investigate whether the efficiency of time-dependent Density Matrix Renormalization Group (t-DMRG) methods is limited by underlying spectral statistics.
• To determine whether the distribution of Schmidt coefficients can serve as a reliable indicator of dynamical complexity in quantum systems.
• To establish a connection between spectral statistics (Poisson vs. Wigner-Dyson) and the numerical simulability of many-body quantum dynamics.
• To explore the generality of these findings across different interaction and tilt parameters in the Bose-Hubbard model.

Proposed method

• The study employs the tilted Bose-Hubbard model as a prototypical many-body quantum system with tunable spectral statistics via the parameter $F/J$.
• Spectral statistics are analyzed using nearest-neighbor spacing distributions, comparing against Poissonian and Wigner-Dyson ensembles to classify regular vs. chaotic regimes.
• The time evolution of a generic initial state is simulated using t-DMRG with a fixed bond dimension $\chi = 100$, tracking the Schmidt coefficients across bipartite cuts.
• The distribution of Schmidt coefficients is quantified by counting how many exceed a threshold $\epsilon = 0.01$, serving as a proxy for Hilbert space truncation efficiency.
• The von Neumann entropy of the reduced density matrix is computed to measure entanglement growth, which correlates with dynamical complexity.
• Statistical measures such as the mean square deviation $\Delta^2$ between observed and expected level spacing distributions are used to validate spectral classification.

Experimental results

Research questions

• RQ1 Does the distribution of Schmidt coefficients reflect the transition from regular to chaotic spectral statistics in many-body quantum systems?
• RQ2 To what extent does the efficiency of t-DMRG simulations break down in chaotic regimes due to the widespread distribution of Schmidt coefficients?
• RQ3 Can the number of significant Schmidt coefficients serve as a reliable indicator of dynamical complexity independent of system-specific details?
• RQ4 Is the breakdown of t-DMRG efficiency in chaotic regimes a universal feature of quantum many-body systems with complex spectral structures?
• RQ5 How do spectral statistics (Poisson vs. Wigner-Dyson) correlate with entanglement growth and Schmidt coefficient distributions in time-evolved states?

Key findings

• For $U/J = 1$ and $F/J \lesssim 1.3$, the system exhibits Wigner-Dyson statistics, indicating chaotic spectral behavior, which correlates with a broad distribution of Schmidt coefficients.
• In the chaotic regime ($U/J = 1$, $F/J \lesssim 1.3$), over 80% of Schmidt coefficients exceed the threshold $\epsilon = 0.01$, indicating that no effective basis truncation is possible.
• In contrast, for $U/J = 10$ or $U/J = 1$ with $F/J \gtrsim 2$, the system follows Poissonian statistics and fewer than 20% of Schmidt coefficients exceed $\epsilon = 0.01$, enabling efficient t-DMRG simulation.
• The von Neumann entropy $S$ saturates early in regular regimes due to truncation artifacts, while it continues to grow in chaotic regimes, reflecting uncontrolled entanglement growth.
• The transition from regular to chaotic dynamics is clearly marked by a sharp change in the number of significant Schmidt coefficients, with a threshold at $F/J \approx 1.3$.
• The same qualitative behavior in Schmidt coefficient distributions is observed in larger systems (20 particles on 20 sites), confirming the universality of the observed simulability breakdown in chaotic regimes.