When a one-dimensional Bose-Hubbard model with a period-two optical superlattice is driven by a linear-in-time quench of the superlattice amplitude at interaction strength U>UcU > U_c, the system crosses a Mott-insulator-to-superfluid critical point, traverses an extended gapless superfluid region, and exits through a second critical point into a superlattice-induced Mott insulator. Tensor-network simulations show that the residual energy in an intermediate window follows a power law ΔE∼τ−κ\Delta E \sim \tau^{-\kappa} with a nontrivial exponent: κ≈0.95\kappa \approx 0.95 at U=8U = 8, κ≈1.58\kappa \approx 1.58 for a superfluid-to-superlattice-Mott-insulator quench at U=2.5U = 2.5, and, in the Figure 2(b) inset, a rather complex dependence of κ\kappa on UU over U≈5U \approx 5–2525, with values spanning κ≈0.75\kappa \approx 0.75–0.950.95 as the crossed superfluid region narrows; the direction of κ(U)\kappa(U) is left ambiguous by the paper's prose. The Kibble-Zurek mechanism, which predicts quench exponents from equilibrium critical scaling at an isolated critical point, is not predictive for this extended crossing, and no later theory supplies the missing scaling. The open problem is a predictive account of defect production for quenches across an extended gapless region: the dependence of κ\kappa on the interaction, the width of the crossed region, and the quench path; the thermodynamic-limit ordering of system size and quench time; the asymptotic quench-rate (in)dependence of the exponent; a mapping onto equilibrium measures; and an experimental test with ultracold atoms in optical superlattices.

Background

The Bose-Hubbard model describes interacting bosons hopping between the sites of a lattice with amplitude JJ and repelling each other on-site with strength UU; it is the standard model of bosonic atoms trapped in optical lattices. In one dimension its two archetypal ground-state phases are the superfluid (SF), in which bosons are delocalized and the single-particle correlation function decays algebraically (quasi-long-range order), and the Mott insulator (MI), in which repulsion pins an integer number of bosons per site and a finite energy gap suppresses particle-number fluctuations. Superimposing a second laser standing wave with double period produces an optical superlattice: a periodic modulation of the site energies, of amplitude λ\lambda, alternating between even and odd sites. In the resulting superlattice Bose-Hubbard model a third phase appears, the superlattice-induced Mott insulator (SLMI): a gapped, incompressible state stabilized by the modulation, in which the density pattern locks to the superlattice period instead of staying uniform. At unit filling and zero temperature the λ\lambda-UU phase diagram, mapped by density-matrix renormalization group (DMRG, a variational matrix-product-state method for one-dimensional systems), contains a superfluid strip whose width decreases monotonically with increasing UU, separating the MI from the SLMI for U>Uc≈3.3U > U_c \approx 3.3 (the critical interaction of the unmodulated chain at λ=0\lambda = 0); for U<UcU < U_c only an SF-to-SLMI transition occurs.

Dhar, Rossini, and Das studied what happens when this system is driven slowly across the gapless strip, by a linear quench of the superlattice amplitude λ(t)=λ0+(λf−λ0)t/τ\lambda(t) = \lambda_0 + (\lambda_f - \lambda_0)t/\tau over a quench time τ\tau, simulated with time-evolving block decimation (TEBD, a tensor-network wave-function method for one-dimensional many-body dynamics) on chains of up to L=200L = 200 sites. The excitation produced is quantified by the residual (excess) energy,

ΔE(τ)=⟨ψ(τ)∣H(τ)∣ψ(τ)⟩−EGS(τ),\Delta E(\tau) = \langle\psi(\tau)|H(\tau)|\psi(\tau)\rangle - E_{\mathrm{GS}}(\tau),

the excess of the final energy expectation value over the instantaneous ground-state energy. For a quench across a single quantum critical point, the Kibble-Zurek mechanism (KZM) predicts a power law ΔE∼τ−κ\Delta E \sim \tau^{-\kappa} with κ\kappa fixed by the equilibrium critical exponents: the argument splits the evolution into adiabatic and impulsive regimes by comparing the closing excitation gap with the quench rate. Three regimes were found instead in the superlattice model: a fast-quench regime where ΔE\Delta E saturates; an intermediate regime with a clean power law, κ≈0.95(1)\kappa \approx 0.95(1) at U=8U = 8 for the MI-SF-SLMI crossing and κ≈1.58\kappa \approx 1.58 for an SF-to-SLMI quench at U=2.5U = 2.5; and a slow regime with κ=2\kappa = 2, explained by the Landau-Zener treatment of an avoided two-level crossing and necessarily a finite-size effect, since the gap in the superfluid region closes as 1/L1/L and vanishes in the thermodynamic limit. Within the intermediate regime, the Figure 2(b) inset of the source paper reports the fitted exponent as a function of UU for MI-SF-SLMI quenches over U≈5U \approx 5–2525, with values spanning κ≈0.75\kappa \approx 0.75–0.950.95 (axis ticks), a dependence the paper describes as 'rather complex … on UU, and hence on the width of the intermediate SF region'; its prose says that κ\kappa 'decreases monotonically as a function of the time during which the system is crossing the gapless region', a time that shrinks as UU grows, so the direction of κ(U)\kappa(U) is left ambiguous by the paper's extractable text. Also, κ\kappa decreases slightly when the quench spends a larger fraction of its time in the gapped phases (at U=5U = 5, L=100L = 100: κ=0.9219,0.8742,0.8762,0.8089\kappa = 0.9219, 0.8742, 0.8762, 0.8089 for (λi,λf)=(3,7),(2.5,7.5),(2,8),(3,9)(\lambda_i, \lambda_f) = (3,7), (2.5,7.5), (2,8), (3,9)). The final-state correlation length is non-monotonic in τ\tau, first growing in the intermediate regime and then shrinking as the adiabatic regime sets in.

The failure of standard KZM scaling here has identifiable structural causes. At the MI boundary the gap closes in the Berezinskii-Kosterlitz-Thouless (BKT) manner, exponentially slowly in the distance from the transition, which is how the correlation length diverges at this transition type; Dziarmaga and Zurek showed for the unmodulated one-dimensional Bose-Hubbard model that this makes the apparent power-law exponent depend on the quench rate. Inside the strip the system is a Luttinger liquid, the gapless universality class of one-dimensional interacting bosons with continuously varying exponents, so there is no single set of critical exponents to insert into the KZM argument. Gardas, Dziarmaga, and Zurek found that for quenches of the unmodulated model from the superfluid into the Mott insulator the gapless superfluid itself generates little excitation, with defects appearing only after the critical point is crossed; in the superlattice setting, by contrast, the entire gapless interval is crossed between two gapped phases and substantial excitation is produced. Two limits also do not commute, as the source paper notes: taking L→∞L \to \infty first, the evolution is never adiabatic at any finite τ\tau and defects are necessarily produced, while taking τ→∞\tau \to \infty first the evolution is adiabatic at any finite LL.

Problem Statement

Give a predictive, quantitatively verified account of defect production when a linear quench crosses an extended gapless region, in the concrete setting of the one-dimensional superlattice Bose-Hubbard model at unit filling (period-two superlattice amplitude λ\lambda, on-site repulsion UU). The source paper's open questions decompose into five linked components of one objective, a controlled theory of the excitation generated inside the gapless superfluid strip, validated against the published numerics and new simulations or experiments:

(1) Scaling theory: derive, or rigorously rule out, an intermediate-regime power law ΔE∼τ−κ\Delta E \sim \tau^{-\kappa} for the residual energy after a linear quench λ(t)=λ0+(λf−λ0)t/τ\lambda(t) = \lambda_0 + (\lambda_f - \lambda_0)t/\tau that crosses the superfluid strip, with κ\kappa predicted as a function of UU (equivalently, of the width of the crossed strip, which shrinks as UU grows), of the quench endpoints (λ0,λf)(\lambda_0, \lambda_f), and of whether one or both critical points are crossed. The theory must account for the established numerical anchors — intermediate-window effective fit values of the published numerics, which a "rule out" outcome would have to explain as effective fits rather than asymptotic exponents: κ≈0.95(1)\kappa \approx 0.95(1) at U=8U = 8 for the MI-SF-SLMI crossing (legend fits κ=0.9602,0.9596,0.9476,0.9563,0.9564\kappa = 0.9602, 0.9596, 0.9476, 0.9563, 0.9564 for L=20,60,100,150,200L = 20, 60, 100, 150, 200), κ≈1.58\kappa \approx 1.58 (fit 1.57541.5754) at U=2.5U = 2.5 for the SF-SLMI quench, the slow-regime κ=2\kappa = 2 Landau-Zener behavior as a finite-size effect, and the slight decrease of κ\kappa when the quench spans more of the gapped phases (at U=5U = 5, L=100L = 100: κ=0.9219,0.8742,0.8762,0.8089\kappa = 0.9219, 0.8742, 0.8762, 0.8089 for (λi,λf)=(3,7),(2.5,7.5),(2,8),(3,9)(\lambda_i, \lambda_f) = (3,7), (2.5,7.5), (2,8), (3,9)). The Figure 2(b) inset reports κ(U)\kappa(U) for MI-SF-SLMI quenches over U≈5U \approx 5–2525, with axis ticks spanning κ≈0.75\kappa \approx 0.75–0.950.95, but its per-point values are not recoverable from the paper's extractable text and its prose leaves the direction of κ(U)\kappa(U) ambiguous; determining the true trend of κ(U)\kappa(U) — as the width of the crossed strip shrinks with increasing UU — is therefore itself part of this component rather than a fixed anchor the theory must match.

(2) Thermodynamic limit: determine whether the power law with a size-independent κ\kappa survives the correct limit ordering. The slow-regime crossover to κ=2\kappa = 2 is a finite-size Landau-Zener effect; taking L→∞L \to \infty first, the evolution is never adiabatic at any finite τ\tau because the superfluid is gapless, while taking τ→∞\tau \to \infty first it is adiabatic at any finite LL. What is the true asymptotic scaling of ΔE\Delta E (or the defect density) in the thermodynamic limit, and over what window in τ\tau is the intermediate power law the correct description?

(3) Asymptotic rate dependence: establish whether κ\kappa tends to a rate-independent constant as τ→∞\tau \to \infty, or whether, as is known for quenches through a single Berezinskii-Kosterlitz-Thouless critical point, the exponentially slow gap closure at the Mott-insulator boundary forces effective exponents that drift with the quench rate, in which case state what the fitted intermediate-regime κ\kappa actually converges to (or that it does not converge).

(4) Equilibrium mapping: determine whether κ(U)\kappa(U), and more generally the excitation generated inside the gapless strip, can be mapped quantitatively onto equilibrium properties of the superfluid region, such as its width in the λ\lambda-UU plane, the varying Luttinger parameter along the path, and the correlation lengths at its boundaries, and use the theory to predict the observed non-monotonic dependence of the final-state correlation length on τ\tau.

(5) Experimental test: specify, and ideally realize, the cold-atom test proposed in the source, quenching the superlattice amplitude of trapped bosonic atoms over a range of durations and reading out the final correlation length (hence the defect content) from time-of-flight images, with enough dynamic range in τ\tau to identify or exclude the intermediate power-law regime.

The five components jointly target the extended gapless crossing of the one-dimensional superlattice Bose-Hubbard model; a treatment restricted to an isolated critical point, or to the unmodulated Bose-Hubbard model, addresses a different question.

References

  1. A. Dhar, D. Rossini, and B. P. Das, 'Quasiadiabatic dynamics of ultracold bosonic atoms in a one-dimensional optical superlattice', Physical Review A 92, 033610 (2015), DOI: 10.1103/PhysRevA.92.033610, arXiv:1510.00144, https://arxiv.org/abs/1510.00144.
  2. F. Pellegrini, S. Montangero, G. E. Santoro, and R. Fazio, 'Adiabatic quenches through an extended quantum critical region', Physical Review B 77, 140404(R) (2008), DOI: 10.1103/PhysRevB.77.140404, arXiv:0801.4475, https://arxiv.org/abs/0801.4475.
  3. U. Divakaran, A. Dutta, and D. Sen, 'Quenching along a gapless line: A different exponent for defect density', Physical Review B 78, 144301 (2008), DOI: 10.1103/PhysRevB.78.144301, arXiv:0805.3328, https://arxiv.org/abs/0805.3328.
  4. J. Dziarmaga and W. H. Zurek, 'Quench in the 1D Bose-Hubbard model: Topological defects and excitations from the Kosterlitz-Thouless phase transition dynamics', Scientific Reports 4, 5950 (2014), DOI: 10.1038/srep05950, arXiv:1312.5139, https://arxiv.org/abs/1312.5139.
  5. B. Gardas, J. Dziarmaga, and W. H. Zurek, 'Dynamics of the quantum phase transition in the one-dimensional Bose-Hubbard model: Excitations and correlations induced by a quench', Physical Review B 95, 104306 (2017), DOI: 10.1103/PhysRevB.95.104306, arXiv:1612.05084, https://arxiv.org/abs/1612.05084.
  6. A. Dhar, D. Jaschke, and L. D. Carr, 'Dynamics for the Haldane Phase in the Bilinear-Biquadratic Model', Physical Review B 105, 094309 (2022), DOI: 10.1103/PhysRevB.105.094309, arXiv:2012.11479, https://arxiv.org/abs/2012.11479.
  7. N. Chepiga and F. Mila, 'Kibble-Zurek exponent and chiral transition of the period-4 phase of Rydberg chains', Nature Communications 12, 414 (2021), DOI: 10.1038/s41467-020-20641-y, arXiv:2001.06698, https://arxiv.org/abs/2001.06698.
  8. J. Soto Garcia and N. Chepiga, 'Resolving chiral transitions in one-dimensional Rydberg arrays with quantum Kibble-Zurek mechanism and finite-time scaling', Physical Review B 110, 125113 (2024), DOI: 10.1103/PhysRevB.110.125113, arXiv:2403.03081, https://arxiv.org/abs/2403.03081.