An exact decimation renormalization group (RG) acting on the transfer matrices of the Loschmidt amplitude controls the dynamical quantum phase transitions (DQPTs) of the one-dimensional quantum QQ-state clock and Potts chains after a quench from the infinite-transverse-field paramagnet to the ferromagnetic Hamiltonian. For Q=2,4,6Q=2,4,6 the RG flow launched at the critical times of the Loschmidt rate function terminates on unstable fixed points (leading Jacobian eigenvalue 2), and for Q=3Q=3 it oscillates on a period-2 cycle; the corresponding singularities are periodic linear cusps. For Q=5Q=5 and, generically, Q>6Q>6, the sequence of critical times is aperiodic - traced arithmetically to the irrationality of cos⁡(2πm/Q)\cos(2\pi m/Q) for QQ outside {2,3,4,6}\{2,3,4,6\} - and the RG flow at those times is chaotic. The source paper explicitly defers the question of why these two phenomena occur together, provides no criterion predicting whether the flow at a critical time converges to a fixed point, a cycle, or a chaotic transient, and establishes the escape-time law of the chaotic transients only semi-empirically (observed slope 1.439 vs 1/log⁡2=1.44271/\log 2=1.4427). The problem is to construct the missing dynamical-systems theory of this exact RG map: explain the co-occurrence and determine whether the fate of the flow is fixed by QQ and symmetry class.

Background

Dynamical quantum phase transitions (DQPTs) are non-analyticities in time of the Loschmidt rate function l(t)=−1Llog⁡∣G(t)∣2l(t)=-\frac{1}{L}\log|G(t)|^2, where the Loschmidt amplitude G(t)=⟨ψ0∣e−iH1t∣ψ0⟩G(t)=\langle\psi_0|e^{-iH_1t}|\psi_0\rangle is the overlap of the initial state with its time-evolved image after a sudden quench from H0H_0 to H1H_1. They were discovered in the transverse-field Ising chain, and their scaling and universality were later explained by a renormalization group (RG) acting on the quenched Hamiltonian, whose two fixed points (stable infinite-temperature and unstable zero-temperature) control the cusp singularities of l(t)l(t). Transfer-matrix diagonalization had also yielded DQPTs of the quantum Potts chain.

The source paper (Y. Wu, Phys. Rev. B 101, 014305 (2020)) develops a different, exactly solvable RG for the quantum QQ-state clock chain (and the DQPT-equivalent quantum Potts chain) in one dimension. The clock model has, per site, a QQ-state Hilbert space on which σi\sigma_i multiplies the mm-th basis state by ωm\omega^m with ω=e2πi/Q\omega=e^{2\pi i/Q} and τi\tau_i cyclically permutes the states, acting as a transverse field. For the exactly solvable quench protocol - the paramagnetic product state (ground state at infinite transverse field) evolving under the pure ferromagnetic Hamiltonian H=−∑iJi(σi†σi+1+σi+1†σi)H=-\sum_i J_i(\sigma_i^\dagger\sigma_{i+1}+\sigma_{i+1}^\dagger\sigma_i) - the Loschmidt amplitude equals the partition function of a one-dimensional classical model with commuting transfer matrices Tmimi+1(i)=Em(i)T^{(i)}_{m_im_{i+1}}=E^{(i)}_m that depend only on the difference m=(mi+1−mi) mod Qm=(m_{i+1}-m_i)\bmod Q, with Em(i)=1Qe2itJicos⁡(2πm/Q)E^{(i)}_m=\frac{1}{Q}e^{2itJ_i\cos(2\pi m/Q)}. Because these matrices commute, G(t)G(t) can be coarse-grained exactly: decimation (multiplying adjacent transfer matrices, block size b=2b=2) followed by division by the first nonzero matrix element gives an exact RG map on the coupling constants EmE_m, whose fixed points and Jacobian eigenvalues determine the DQPTs. In equilibrium RG language, the Jacobian eigenvalue λ=by\lambda=b^y at an unstable fixed point fixes the cusp exponent of l(τ)∼∣τ∣d/yl(\tau)\sim|\tau|^{d/y} near a critical time tct_c, τ=t−tc\tau=t-t_c.

The fixed-point structure found is: for Q=2Q=2, three isolated fixed points, the unstable one having leading eigenvalue 2 (linear cusp); for Q=3Q=3, the critical-time flows oscillate between a pair of fixed points - a period-2 cycle of the RG map whose composed Jacobian has eigenvalue 4=b24=b^2, still giving a linear cusp; for Q=4Q=4, fifteen fixed points with the same qualitative structure; for Q=6Q=6, periodic behavior l(t)=l(t+2π)l(t)=l(t+2\pi) with unstable fixed points of leading eigenvalue 2 at the critical times. Within this clock-symmetric manifold the fixed points are isolated; a continuous manifold of fixed points (marginal directions) arises generically for transfer matrices without the symmetry, a case analyzed in the author's companion paper (Phys. Rev. B 101, 064427 (2020)), which identifies a DQPT universality class with Loschmidt-rate critical exponent 1/21/2. These are the two transfer-matrix symmetry classes that the questions below range over: the clock-ZQ\mathbb{Z}_Q-symmetric class of the quench protocol, whose fixed points are isolated, and the generic non-symmetric class, whose fixed points form a marginal manifold onto which the flow terminates - a fixed-point fate with marginal directions, and not a finite cycle or a chaotic transient. For the pure Potts chain the same construction collapses to a one-dimensional rational map on a single coupling, whose DQPTs - which occur only for Q<4Q<4 - are identical to the clock chain's; the chaotic-flow phenomena at issue below are therefore properties of the clock-chain map.

For Q=5Q=5, and generically for Q>6Q>6, the situation changes qualitatively: the sequence of critical times has no simple structure (aperiodic), and the RG flow launched at those critical times is not attracted to any fixed point or cycle but is chaotic. The aperiodicity of the rate function is explained arithmetically - the exponents cos⁡(2πm/Q)\cos(2\pi m/Q) in the initial couplings are rational for all mm only when Q∈{2,3,4,6}Q\in\{2,3,4,6\} - but the source states verbatim that it remains to be understood why the aperiodicity of the rate function and the chaos of the RG flow occur together, deferring the question to future study. Quantitatively, for Q=5Q=5 near a critical time one can define the escape time ne(τ)n_e(\tau): the number of RG iterations before the chaotic flow settles near a (meta)stable fixed point. A scaling argument combining RG invariance of the singular part of the rate function with a power-law cusp ansatz l(τ)∼∣τ∣αl(\tau)\sim|\tau|^\alpha yields the linear law ned=−αlog⁡blog⁡∣τ∣+cn_e d=-\frac{\alpha}{\log b}\log|\tau|+c, checked numerically (fitted slope −1.439-1.439 vs 1/log⁡2=1.44271/\log 2=1.4427 for α=d=1\alpha=d=1, b=2b=2), and the source regards this mean law as established by its RG analysis; the deeper dynamical question it defers is why the flow at the critical times is chaotic at all and how that chaos is tied to the aperiodicity of the critical-time sequence.

Problem Statement

For the exact transfer-matrix RG of the quench-driven QQ-state clock chain - the decimation map with block size b=2b=2 and normalization by the first nonzero matrix element, acting on the coupling constants EmE_m and launched at a critical time from Em(tc)=1Qe2itcJcos⁡(2πm/Q)E_m(t_c)=\frac{1}{Q}e^{2it_cJ\cos(2\pi m/Q)} - construct a quantitative dynamical-systems theory that answers two linked questions:

  1. Explain why the aperiodicity of the sequence of critical times of the Loschmidt rate function and the chaotic character of the RG flow at those times occur together (they co-occur for Q=5Q=5 and generically for Q>6Q>6, while the periodic cases Q∈{2,3,4,6}Q\in\{2,3,4,6\} have fixed-point or finite-cycle flows), given that the known arithmetic fact - rationality of cos⁡(2πm/Q)\cos(2\pi m/Q) for all mm only for Q∈{2,3,4,6}Q\in\{2,3,4,6\} - explains the aperiodicity but not the chaos.

  2. Determine whether the fate of the RG flow launched at a critical time is fixed by QQ and the transfer-matrix symmetry class, and if so give the criterion: whether the flow converges to an unstable fixed point (as for Q=2,4,6Q=2,4,6), enters a finite cycle (as for Q=3Q=3), or behaves as a chaotic transient (as for Q=5Q=5 and generically Q>6Q>6). A criterion means a derived rule that determines this fate for every QQ - covering the Q>6Q>6 cases the Background only calls generic as well as the enumerated ones - and not a restatement of the cases listed above. A resolution establishing that the fate is not fixed by these invariants, so that no such rule exists and only a generic classification is possible, also completes this question.

These are two manifestations of one objective, a theory of the dynamics of the exact RG map at DQPT critical times: the co-occurrence question is its explanatory core, and the fate of the flow at a critical time its predictive content. Both questions are required independently: a complete solution addresses each, and resolving one does not by itself resolve the task. The mean escape-time law ne(τ)n_e(\tau) of the chaotic transients - derived semi-empirically in the source (RG invariance plus a cusp ansatz) and numerically confirmed, and treated there as established - is retained above as background context for the dynamics, not as a required completion component; the question the source defers to future study is the co-occurrence of aperiodicity and chaos.

References

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