The Kibble–Zurek (KZ) mechanism predicts that defect production under a slow quench through an isolated quantum critical point follows a power law in the quench time, with an exponent fixed by equilibrium critical exponents. When a quench instead traverses an extended gapless region bounded by two quantum phase transitions — as when a one-dimensional Bose–Hubbard system in an optical superlattice is driven from a Mott insulator through a gapless superfluid region into a superlattice-induced Mott insulator — the standard KZ argument loses its predictive basis: no single critical point controls the freeze-out, and numerics find an intermediate-regime power law whose exponent depends nontrivially on interaction strength, gapless-region width, and quench endpoints. The problem is to settle whether and in what sense KZ universality survives such crossings in this setting: establish a predictive scaling account of defect formation for the superlattice quench, determine whether such an exponent exists and, if so, its parameter dependence, establish the intensivity and universal scaling function of the defect measure, and fix the ordering of the thermodynamic limit versus the long-quench-time limit.

Background

The Kibble–Zurek (KZ) mechanism originated in cosmology and condensed-matter physics as the observation that a system driven through a continuous phase transition cannot stay ordered on arbitrarily large scales, so topological defects are produced with a density set by the quench rate (Kibble 1976; Zurek 1985). In its quantum version, a parameter of a many-body Hamiltonian is varied linearly in time across a quantum critical point (QCP) over a quench time τ\tau, and the mechanism predicts a power law for the defect density, n∼τ−dν/(zν+1)n \sim \tau^{-d\nu/(z\nu+1)}, where ν\nu is the correlation-length exponent and zz the dynamical critical exponent of the equilibrium transition; the argument splits the evolution into an adiabatic regime far from criticality and an impulsive 'freeze-out' regime near it, where the closing gap makes relaxation impossible (reviews: Dziarmaga 2010; Polkovnikov et al. 2011). In cold-atom systems the defect density is commonly gauged by the excess (residual) energy ΔE\Delta E, the energy of the time-evolved state above the instantaneous ground state at the end of the ramp, or by its intensive version, the excess energy per site ΔE/L\Delta E/L.

The standard KZ exponent formula presumes an isolated critical point at which the gap closes as a power law. Known deviations already arise at the boundaries of this assumption: quenching through a multicritical point or along a gapless line modifies the defect exponent (Divakaran et al. 2008, 2009), and at a Kosterlitz–Thouless (KT) transition — where the correlation length diverges exponentially rather than as a power law — the effective exponent depends on the quench rate (Dziarmaga and Zurek 2014). Crossing an extended gapless region probes further beyond this assumption, and whether a single endpoint controls the scaling need not be uniform: Pellegrini et al. (2008) studied quenches of the spin-1/2 XXZ chain across the extended quantum critical region — a Luttinger liquid, the generic gapless state of one-dimensional interacting quantum systems, with sound-like collective modes and scale-invariant correlations rather than a single critical point — and found that the outcome depends on the quench type: for an antiferromagnetic →\to ferromagnetic quench crossing the full critical region, the intermediate residual-energy power law τQ−1/4\tau_Q^{-1/4} is explained by a Landau–Zener argument localized at the endpoint Δ=1\Delta = 1, where the finite-size gap closes quadratically, in very good agreement with the numerics; whereas a critical →\to antiferromagnetic quench starting inside the gapless region yields an extrapolated exponent 0.78±0.020.78 \pm 0.02 instead of the single-point KZ/LZ value 0.50.5, an evolution not well described by previously proposed single-critical-point models.

The motivating instance is ultracold bosons in a one-dimensional optical superlattice (two superposed standing waves creating a period-doubled lattice). The system is described by a Bose–Hubbard model — bosons with hopping amplitude JJ, on-site repulsion UU, plus a superlattice potential of depth λ\lambda — whose phase diagram at unit filling (one boson per lattice site) contains a gapped Mott insulator (MI), a gapless superfluid (SF), and a gapped superlattice-induced Mott insulator (SLMI); the MI–SF boundary is of KT type with critical interaction Uc≈3.3U_c \approx 3.3. Dhar, Rossini, and Das (2015) simulated a linear quench λ(t)=λ0+(λf−λ0)t/τ\lambda(t) = \lambda_0 + (\lambda_f - \lambda_0)t/\tau using time-dependent density-matrix renormalization group (DMRG, a variational method for one-dimensional quantum many-body states) on chains up to L=200L = 200 sites. For U>UcU > U_c the quench MI →\to SF →\to SLMI crosses two QPTs and an extended gapless region; for U<UcU < U_c it crosses only the SF →\to SLMI transition. They resolved three dynamical regimes: a saturated regime for fast quenches, an intermediate power law ΔE∼τ−κ\Delta E \sim \tau^{-\kappa} with κ≈0.95\kappa \approx 0.95 at U=8U = 8 (double crossing) and κ≈1.58\kappa \approx 1.58 at U=2.5U = 2.5 (single crossing), and a slow τ−2\tau^{-2} regime that they identify as a finite-size Landau–Zener effect (transitions at the avoided level crossings of a finite system, suppressed as the ramp slows through the finite minimum gap), since in the thermodynamic limit the gap closes throughout the SF region and the evolution is non-adiabatic for any finite τ\tau. Crucially, κ\kappa depends nontrivially on UU, on the width of the crossed SF region, and even on the chosen insulating endpoints — dependence that the equilibrium exponents of either boundary transition cannot capture, leading the authors to state that KZ scaling 'cannot be regarded as ultimately predictive' for extended critical regions.

Problem Statement

For a slow, linear-in-time quench of the one-dimensional Bose–Hubbard model with a period-two optical superlattice at unit filling — driving the superlattice depth as λ(t)=λ0+(λf−λ0)t/τ\lambda(t) = \lambda_0 + (\lambda_f - \lambda_0)t/\tau from a Mott insulator, through the gapless superfluid region, into a superlattice-induced Mott insulator, crossing two quantum phase transitions for U>Uc≈3.3U > U_c \approx 3.3 and only the SF →\to SLMI transition for U<UcU < U_c — establish a predictive scaling account of defect formation, replacing the standard Kibble–Zurek argument whose exponent is fixed by the critical exponents of a single isolated critical point. Concretely, the following are jointly required, with (iv) a synthesis of (i)–(iii): (i) determine whether the intensive defect measure (excess energy per site ΔE/L\Delta E/L, or an equivalent defect density) obeys an asymptotic power law ΔE/L∼τ−κ\Delta E/L \sim \tau^{-\kappa} in the thermodynamic limit, and either derive κ\kappa — including its dependence on the on-site interaction UU, the width of the crossed gapless region, and the quench endpoints λ0,λf\lambda_0, \lambda_f — from equilibrium properties or from a generalized non-equilibrium scaling argument, or show that no such KZ-type universality exists and identify what replaces it; (ii) establish whether the defect measure is intensive with a universal scaling function over a range of quench times, as required to interpret finite-size numerics as bulk scaling; (iii) determine whether the thermodynamic limit L→∞L \to \infty and the long-quench-time limit τ→∞\tau \to \infty commute and, if not, which ordering governs the observed exponent; and (iv) through these, determine which critical point — if any — controls the out-of-equilibrium power law when the quench crosses two quantum critical points and a finite-width critical region, thereby settling the domain of validity of Kibble–Zurek universality for such trajectories. A negative answer to the exponent question in (i) settles only that item: the intensivity and scaling-function question (ii) and the limit-ordering question (iii) must still be resolved.

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