The kicked two-dimensional quantum Ising model (2D KI) is among the simplest strongly interacting, nonintegrable periodically driven many-body systems. Kukuljan and Prosen (J. Stat. Mech. 2016) developed a numerically exact method, based on Baxter's corner transfer matrices (CTM), for the Fourier coefficients of the Floquet quasienergy level density, ρk=tr Uk/N\rho_k=\mathrm{tr}\,\mathcal{U}^k/\mathcal{N}, reaching lattices of 104×10410^4\times 10^4 spins. The method is renormalizable — convergent in the thermodynamic limit — only in certain regions of the coupling/kick parameter plane (h,J)(h,J); in the complementary central region the CTM spectra are flat, i.e. effectively of full rank, and the method fails. The physical origin of this partition is unknown. It is conjectured that the steepness of the CTM spectra reflects the entanglement entropy of an associated auxiliary many-body state, so that the renormalizable/non-renormalizable partition would be an area-law-to-volume-law entanglement transition, but this has not been proven; renormalizability of the scheme for other 2D Floquet systems is likewise open. No published work resolving these questions is known.

Background

A periodically driven (Floquet) quantum system has a Hamiltonian periodic in time, H(t+nτ)=H(t)H(t+n\tau)=H(t); its dynamics over one period is captured by the unitary Floquet propagator U\mathcal{U}, whose eigenvalues e−iϕne^{-i\phi_n} define quasienergies ϕn∈[0,2π)\phi_n\in[0,2\pi), the analogue of energy levels for driven systems. The simplest spectral observable of a many-body Floquet system is the level density ρ(ϕ)\rho(\phi) of the quasienergy spectrum, equivalently its Fourier coefficients

ρk=1N tr Uk,N=2∣Λ∣,\rho_k=\frac{1}{\mathcal{N}}\,\mathrm{tr}\,\mathcal{U}^{k},\qquad \mathcal{N}=2^{|\Lambda|},

where ∣Λ∣|\Lambda| is the number of spins and kk counts drive periods. Random matrix theory — specifically the circular orthogonal ensemble (COE), the ensemble of symmetric unitary matrices appropriate to time-reversal-invariant propagators — predicts that a chaotic system has an exactly flat level density in the thermodynamic limit, with ρk\rho_k decaying universally as 2−∣Λ∣2^{-|\Lambda|}, independently of the system's parameters.

The 2D kicked Ising model places spins 1/21/2 on a square lattice with nearest-neighbor Ising couplings Jσm,nzσm+1,nzJ\sigma^z_{m,n}\sigma^z_{m+1,n} and Jσm,nzσm,n+1zJ\sigma^z_{m,n}\sigma^z_{m,n+1}, driven by periodic δ\delta-kicks of a transverse magnetic field of strength hh; the propagator factorizes into one- and two-spin unitary gates. Exact-diagonalization studies of small lattices (up to 5×55\times 5, Pineda, Prosen and Villase~nor, 2014) found Fourier coefficients ρk(h,J)\rho_k(h,J) with rich, strongly non-universal parameter dependence, but could not decide whether these structures survive in the thermodynamic limit.

Kukuljan and Prosen (2016) answered that question with a tensor-network method. A tensor network is an arrangement of indexed arrays whose full contraction (summation over all shared indices) equals the desired scalar; for this model, tr Uk\mathrm{tr}\,\mathcal{U}^k reduces, after decomposing each two-spin gate and contracting the physical spins, to a two-dimensional network of rank-4 tensors on a diamond-shaped lattice with bond dimension 2k2^k. To contract it in the thermodynamic limit they used Baxter's corner transfer matrix: the lattice is divided into four quadrants, and the CTM — the matrix of contraction weights accumulated by one quadrant, effectively a renormalized half-space density matrix — is grown row by row while its spectrum is truncated to the χ\chi dominant eigenvalues. In equilibrium classical models this iteration is the corner transfer matrix renormalization group. The method is called renormalizable at a given parameter point when the CTM eigenvalue spectrum decays steeply, so that a fixed effective rank χ\chi suffices at any lattice size and the computed ρk\rho_k converge as N→∞N\to\infty (truncation is then the only error source); it fails when the spectrum is flat, i.e. all eigenvalues are comparable and the effective rank grows without bound. With this method the authors treated lattices up to 10000×1000010000\times 10000 and found: (i) generically the level density does become flat, but with a non-universal decay

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},

refuting the universal COE prediction μk≡1\mu_k\equiv 1; and (ii) the parameter plane partitions into regions where the method is renormalizable and a central region where the CTM spectra are flat and non-renormalizable — a partition that survives increases of the truncation and of the initial lattice, since the flat spectra merely rescale self-similarly.

The natural interpretation appeals to entanglement. In equilibrium integrable systems a CTM is, up to normalization, a reduced density matrix of half of the associated one-dimensional quantum system, and CTM spectra literally encode entanglement entropies and entanglement spectra — an established tool in integrable lattice models. By analogy, one expects the CTM of the Floquet network to be the reduced density matrix of an auxiliary many-body state living on the cut, and the steepness of its spectrum to be set by that state's entanglement entropy: an area law (entropy bounded independently of system size) gives a steep spectrum and renormalizability, while a volume law (entropy growing in proportion to system size) gives a flat spectrum and non-renormalizability — the same heuristic that explains when the density matrix renormalization group, the standard one-dimensional ground-state method, succeeds or fails. Whether this analogy is valid for the auxiliary states of a driven, strongly nonintegrable 2D network is exactly what has not been established.

Problem Statement

Explain the renormalizability structure that the corner-transfer-matrix computation reveals in the kicked 2D quantum Ising model, and determine its physical meaning, as posed at the close of Kukuljan and Prosen (2016). Concretely: (1) What is the physical origin of the flat corner transfer matrix spectra — spectra whose eigenvalues are all of comparable magnitude, i.e. effectively full rank at any truncation — in the central region of the parameter plane (h,J)(h,J), which makes the CTM method non-renormalizable there, while in the surrounding regions the spectra decay steeply and the method converges in the thermodynamic limit? (2) Prove or refute the conjecture that the steepness of the CTM spectra (equivalently, their effective rank) is determined by the entanglement entropy of the auxiliary many-body state associated with the CTM, so that the renormalizable/non-renormalizable partition of the parameter plane corresponds to an area-law-to-volume-law entanglement transition of that state. (3) Establish whether the CTM contraction scheme is renormalizable for other two-dimensional Floquet systems and other driving protocols. These subquestions share one scientific objective: to identify what governs the computational renormalizability of CTM contractions of two-dimensional Floquet propagators — entanglement scaling of the auxiliary state, many-body quantum chaos of the drive, or another mechanism — and thereby to decide whether the method's domain of validity reflects a physical phase structure of the driven system. All three sub-questions are in scope and each is independently resolving: a definite answer to any one of them — the mechanism governing the partition (1), a proof or a refutation of the conjecture (2), or a general renormalizability criterion for the CTM scheme demonstrated on at least one other periodically driven two-dimensional lattice system or driving protocol (3) — suffices to resolve the task.

References

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