For the two-dimensional kicked quantum Ising (2D KI) model, Kukuljan and Prosen (J. Stat. Mech. 2016) introduced a renormalizable, numerically exact method, based on Baxter's corner transfer matrices (CTMs), for computing the Fourier coefficients of the Floquet quasienergy level density on lattices up to 10000 × 10000. Remarkably, the method is renormalizable — the CTMs have effectively finite rank so that truncation converges in the thermodynamic limit — only in certain regions of the parameter space, while in the central region the CTM spectra are flat (effectively full rank for every lattice size) and the method fails. The open problem is to determine the physical origin of these flat spectra; in particular, to prove or refute the conjectured identification of CTM spectral steepness with entanglement scaling of the auxiliary states, under which the renormalizable/non-renormalizable partition would correspond to an area-law-to-volume-law entanglement transition; to establish whether the method is renormalizable for other 2D Floquet systems and driving protocols; and to settle whether the boundary parameter points where the fitted decay coefficients tend to zero carry a genuinely non-flat level density in the thermodynamic limit.

Background

A Floquet system is a quantum system driven by a time-periodic Hamiltonian; its stroboscopic dynamics over one period is generated by a unitary Floquet operator, whose eigenphases (defined modulo 2π) are called quasienergies. The model at issue is the kicked two-dimensional quantum Ising (2D KI) model on a square lattice, with Hamiltonian H(t)=HIsing+Hkick∑jδ(t−jτ)H(t) = H_{\mathrm{Ising}} + H_{\mathrm{kick}}\sum_{j} \delta(t - j\tau), where HIsing=J∑(σm,nzσm+1,nz+σm,nzσm,n+1z)H_{\mathrm{Ising}} = J\sum (\sigma^{z}_{m,n}\sigma^{z}_{m+1,n} + \sigma^{z}_{m,n}\sigma^{z}_{m,n+1}) couples nearest-neighbour spins and Hkick=h∑(σm,nxcos⁡θ+σm,nzsin⁡θ)H_{\mathrm{kick}} = h\sum(\sigma^{x}_{m,n}\cos\theta + \sigma^{z}_{m,n}\sin\theta) is a spatially homogeneous magnetic-field pulse applied once per period. The quasienergy level density ρ(ϕ)\rho(\phi) is expanded in Fourier modes ρk=tr Uk/N\rho_k = \mathrm{tr}\, \mathscr{U}^{k}/\mathscr{N} (normalized traces of powers of the Floquet operator, N=2∣Λ∣\mathscr{N} = 2^{|\Lambda|} the Hilbert-space dimension), which measure deviations of the spectrum from a uniform distribution. For a strongly chaotic, nonintegrable system, the standard conjecture of quantum chaos predicts that spectra are described by random matrix theory — for a unitary, time-reversal-symmetric operator, the circular orthogonal ensemble — which predicts a flat level density in the thermodynamic limit, with all ρk\rho_k decaying at a universal rate. Pineda, Prosen and Villaseñor (New J. Phys. 16, 123044 (2014)) found instead rich, parameter-dependent finite-size phase diagrams ρk(h,J)\rho_k(h,J) with sharp peaks, raising the question of the thermodynamic-limit behaviour. Kukuljan and Prosen (J. Stat. Mech. (2016) 043305) answered this for generic parameters by a new method: the trace defining ρk\rho_k is represented as the contraction of a three-dimensional tensor network (one layer of small tensors per kick), evaluated by a recursion of corner transfer matrices — the objects introduced by Rodney J. Baxter in the theory of exactly solved two-dimensional classical lattice models, which act here as a numerical renormalization group. At each renormalization step, eigenvalues of the CTMs below a threshold ϵ\epsilon are discarded, with a cap of 2k(N0−1)2^{k(N_0-1)} kept eigenvalues (N0N_0 sets the initial strip width, kk the number of kicks). The method is called renormalizable where the CTMs have effectively finite rank for every lattice size, so that the truncation error stays controlled and lattices up to 10000 × 10000 are accessible (finite lattices are taken diamond-shaped with open boundary conditions, NN denoting their linear size); the computed decay follows ρk(N)=ck 2−μkN2−νkN\rho_k(N) = c_k\, 2^{-\mu_k N^2 - \nu_k N} with highly nontrivial exponents μk(h,J)\mu_k(h,J). The central discovery, however, is that in the central region of the (h,J)(h,J) parameter space the CTM spectra are flat — effectively full rank at all sizes — so that non-small eigenvalues must be cut and the method becomes non-renormalizable; increasing N0N_0 only rescales the flat spectra self-similarly. By analogy with the density matrix renormalization group (DMRG), where the spectrum of the truncated reduced density matrix is tied to the entanglement of the simulated state, the authors conjectured that the steepness of the CTM spectra reflects the entanglement of the corresponding auxiliary boundary states generated by the recursion: an area law (entanglement of a spatial subregion scaling with its boundary, as in gapped ground states) in the renormalizable regions and a volume law (scaling with the subregion volume, as in highly excited or infinite-temperature states) in the central region — possibly signalling a new type of dynamical entanglement transition in the driven 2D system. This conjecture, the physical origin of the flat spectra, the extension of the method to other 2D Floquet systems, and the status of the boundary points where μk→0\mu_k \to 0 (which the paper notes are degenerate limits of the model and may retain a non-flat level density) were all left open.

Problem Statement

Determine the physical origin of the flat corner transfer matrix spectra — and hence the non-renormalizability of the CTM level-density method — in the central region of the (h,J)(h,J) parameter space of the kicked 2D quantum Ising model, and establish what this implies about the driven system. Concretely, as posed in the conclusions of Kukuljan and Prosen (2016), the problem comprises the following jointly linked questions: (i) prove or refute the conjecture that the steepness, i.e., the effective rank, of the CTM spectra reflects the entanglement scaling of the auxiliary states generated by the CTM renormalization — the auxiliary many-body state that controls the CTM contraction and is generated by the recursion, which Ref. [1] invokes by analogy with the density matrix renormalization group but does not construct, so that the state, its bipartition and the entanglement measure must be specified explicitly, independently of the CTM spectrum, before the identification can be tested — under which the observed renormalizable/non-renormalizable partition of the parameter space would correspond to an area-law-to-volume-law entanglement transition of those states, with the characterization of that transition (whether it is sharp or a crossover, its location in the (h,J)(h,J) parameter space, and its scaling properties) required on the branch where the identification holds and void on the branch where it is refuted, so that a refutation of the identification alone resolves (i); (ii) establish whether the CTM method is renormalizable for other two-dimensional locally interacting Floquet systems and other driving protocols — a general criterion for renormalizability, demonstrated on at least one representative 2D Floquet system or driving protocol beyond the 2D KI model, counts as resolving (ii); an exhaustive survey of such systems is not required — and thereby characterize the general relationship between computational renormalizability, entanglement scaling, and many-body quantum chaos in driven quantum lattice systems; and (iii) settle whether the boundary parameter points where the fitted leading decay coefficients μk(h,J)→0\mu_k(h,J) \to 0 — degenerate limits of the 2D KI model — genuinely exhibit a non-flat quasienergy level density in the thermodynamic limit, as indicated by the CTM computation, or whether the apparent non-flatness is an artifact of the degenerate limits or of the method. The single overarching objective is an explanatory, physically grounded account of when and why the CTM contraction of the spacetime tensor network of a two-dimensional Floquet system is renormalizable; items (ii) and (iii) test and delimit the account obtained in (i), and a complete answer addresses them under that common objective.

References

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