Kukuljan and Prosen (2016) introduced a numerically exact corner-transfer-matrix (CTM) method for computing the quasienergy level density of the two-dimensional kicked quantum Ising (2D KI) Floquet system and found that the method is renormalizable only in part of the parameter space: in a central region of the coupling-field plane the corner spectra are flat, the effective rank grows with system size, and truncation fails. The paper asks for the physical origin of this non-renormalizability, conjectures by analogy with the density-matrix renormalization group that the spectral steepness is governed by the entanglement of an associated auxiliary state (possibly an area-law to volume-law transition), leaves open whether CTMs are renormalizable for general 2D periodically driven systems, and leaves open whether the non-flat level density indicated at boundary points where the fitted decay exponent tends to zero survives the thermodynamic limit. The objective is to determine the physical origin of the flat CTM spectra, the role of auxiliary-state entanglement in the CTM spectral steepness, the generality of CTM renormalizability beyond the 2D KI model, and the thermodynamic-limit status of the level density at boundary points where the fitted exponent μk\mu_k vanishes.

Background

Periodically driven (Floquet) quantum many-body systems are described by the Floquet operator UU, the unitary time-evolution propagator over one drive period; its eigenphases ϕn\phi_n are called quasienergies and form the analogue of the energy spectrum for time-periodic dynamics. The one-point spectral statistic is the level density ρ(ϕ)=1N∑nδ(ϕ−ϕn)\rho(\phi)=\frac{1}{\mathcal{N}}\sum_n\delta(\phi-\phi_n), with N\mathcal{N} the Hilbert-space dimension, whose Fourier coefficients are ρk=tr⁡(Uk)/N\rho_k=\operatorname{tr}(U^k)/\mathcal{N}: the kk-th coefficient is the normalized trace of the evolution over kk drive periods. For a strongly nonintegrable system with time-reversal symmetry, the quantum-chaos conjecture predicts the statistics of Dyson's circular orthogonal ensemble (COE), a random-matrix ensemble of symmetric unitary matrices; the COE prediction is ⟨ρk⟩=δk,0\langle\rho_k\rangle=\delta_{k,0} with typical fluctuations ρk∝2−∣Λ∣\rho_k\propto 2^{-|\Lambda|}, where ∣Λ∣|\Lambda| is the number of spins, i.e. an exponential decay with a parameter-independent rate.

The two-dimensional kicked quantum Ising model (2D KI) is arguably the simplest nontrivial strongly coupled driven 2D quantum system: Ising spins on a lattice with nearest-neighbor coupling JJ evolve under H(t)=HIsing+Hkick∑jδ(t−jτ)H(t)=H_{\rm Ising}+H_{\rm kick}\sum_j\delta(t-j\tau), where HkickH_{\rm kick} is the Zeeman coupling to a homogeneous field h⃗=(hcos⁡θ,0,hsin⁡θ)\vec{h}=(h\cos\theta,0,h\sin\theta) pulsed once per period, so the Floquet operator is U=exp⁡(−iHIsing)exp⁡(−iHkick)U=\exp(-iH_{\rm Ising})\exp(-iH_{\rm kick}). Pineda, Prosen and Villasenor (2014) computed ρk\rho_k for small lattices and found nontrivial phase diagrams ρk(h,J)\rho_k(h,J), suggesting anomalously slow convergence of the level density to the flat COE prediction.

Kukuljan and Prosen (2016) turned the trace tr⁡(Uk)\operatorname{tr}(U^k) into the contraction of a two-dimensional tensor network on a diamond-shaped lattice with ∣Λ∣=N2|\Lambda|=N^2 spins and evaluated it using Baxter's corner transfer matrices (CTMs) — the matrices representing single quadrants of a two-dimensional tensor network, whose eigenvalue spectra control how much of each quadrant must be kept when the contraction is performed with truncation tolerance ε\varepsilon. This is the same truncation philosophy as the density-matrix renormalization group (DMRG), which compresses a quantum state by discarding eigenvectors with small eigenvalues of a reduced density matrix and succeeds precisely when the state's entanglement entropy obeys an area law (entropy bounded by a region's boundary) rather than a volume law (entropy proportional to its size). A CTM contraction is called renormalizable when the corner spectra are steep enough that, for any system size, only finitely many eigenvalues exceed any fixed tolerance — the CTMs have effectively finite rank. The corner transfer matrix acts on the "auxiliary" spins sitting on the boundary of its quadrant of the lattice, and in the DMRG analogy its eigenvalue spectrum plays the role of the spectrum of a reduced density matrix of an auxiliary many-body quantum state; it is the entanglement scaling of that auxiliary state which the authors conjecture to control the spectral steepness — area-law entanglement where the method works, volume-law entanglement where it fails.

The method computed ρk\rho_k for lattices up to 104×10410^4\times 10^4 and showed that the small-lattice phases are strictly finite-size effects: everywhere the method is reliable, ρk(N,h,J)=ck(h,J) 2−μk(h,J)N2−νk(h,J)N\rho_k(N,h,J)=c_k(h,J)\,2^{-\mu_k(h,J)N^2-\nu_k(h,J)N} with a leading volume term and subleading surface term, but with decay rates μk(h,J)\mu_k(h,J) that depend nontrivially on the parameters, in contrast to the universal random-matrix value μk≡1\mu_k\equiv 1. The most striking discovery is a partition of the parameter space: in a central region the corner spectra are flat, the spectral magnitudes (sums of eigenvalue magnitudes) grow exponentially with the CTM environment size N0N_0, truncation cuts away non-small eigenvalues, and the algorithm is, in the authors' words, fundamentally non-renormalizable; increasing N0N_0 merely rescales the self-similar flat spectra. At isolated boundary points where μk→0\mu_k\to 0 (where small-NN ρk\rho_k peak, all in rather degenerate limits of the model), the fit indicates that the level density may stay non-flat in the thermodynamic limit.

Modern adjacent literature studies a closely related question under different names — the growth of temporal entanglement (entanglement of the space-time dual or influence operator) and the compressibility of influence-matrix representations of chaotic dynamics — and universal corner spectra are known for equilibrium tensor networks; but none of that literature addresses renormalizability of the CTM contraction for 2D Floquet spectra or the entanglement conjecture for the CTM auxiliary state.

Problem Statement

For the corner-transfer-matrix computation of the quasienergy level density of 2D strongly coupled Floquet systems introduced by Kukuljan and Prosen (2016), the following connected questions are open. (1) What is the physical origin of the flat CTM spectra that make the method non-renormalizable in the central region of the 2D KI parameter space (h,J)(h,J)? A satisfactory answer characterizes which property of the strongly coupled driven dynamics produces full-rank CTMs whose spectral magnitudes grow exponentially with the CTM environment size, and accounts for the observed location of the boundary between working and non-working regions. (2) Establish or refute the paper's conjecture that the steepness of the CTM spectra, or the effective CTM rank (the number of eigenvalues above a fixed truncation tolerance), is governed by the entanglement of the associated auxiliary quantum state. This question carries two sub-claims of independent standing: (2a) the steepness or effective CTM rank is governed by the entanglement of the associated auxiliary quantum state; (2b) the working and non-working regions are separated by a transition from area-law to volume-law entanglement scaling of that state. Each sub-claim is independently a partial resolution of (2): settling (2a) with a smooth crossover settles (2a) but leaves (2b) open, and refuting (2b) alone leaves (2a) open. (3) Determine whether the CTM method is renormalizable for general 2D periodically driven strongly coupled many-body systems and drive protocols, beyond the 2D KI model. (4) Determine the thermodynamic-limit status of the level density at the boundary points where the fitted exponent μk\mu_k tends to zero: genuinely non-flat as N→∞N\to\infty, or a finite-size effect or artifact of the method. The four questions are separately attackable rather than logically independent, and an answer to any one is a partial resolution of the overall problem: resolving (2) supplies the entangling mechanism behind (1) and the general criterion behind (3), so (1) additionally requires accounting for the observed location of the boundary between the working and non-working regions, and (3) additionally requires evidence beyond the 2D KI model. The full objective, a physical and structural understanding of when, and why, corner-transfer-matrix contraction of 2D Floquet dynamics is renormalizable (connecting computational compressibility, entanglement scaling, and many-body quantum chaos), requires all four questions to be answered.

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