When a gapped free-fermion spin chain — the isotropic XY chain with a staggered magnetic field, or with dimerized hopping — is quenched from a domain-wall magnetization state, it reaches a current-carrying nonequilibrium steady state (NESS) in which spin-spin correlations decay algebraically, even though the ground-state correlations of the same models decay exponentially. Lancaster and Godoy (Phys. Rev. Research 1, 033104 (2019)) derived the oscillatory power-law forms of the NESS correlators, but the transverse correlation functions are determinants of structured, row-alternating non-Toeplitz matrices to which the Szegő/Fisher-Hartwig asymptotic theory does not apply: the power and oscillation pattern were obtained by extrapolating Fisher-Hartwig logic outside its domain of validity, and the amplitude functions were fitted numerically. This statement records the open problem: (i) a rigorous asymptotic theory for these position-dependent determinants, including analytic expressions for the amplitude and phase functions; (ii) whether the algebraic NESS correlations survive interactions, weak integrability breaking, and finite-size/experimental conditions; and (iii) whether the halving of the Brillouin zone (unit-cell doubling, gap at k=±π/2k=\pm\pi/2) both suffices for and is required by the effect. No known work to date settles any of these strands.

Background

The isotropic XY spin chain, H=−J∑j(SjxSj+1x+SjySj+1y)H=-J\sum_j(S^x_jS^x_{j+1}+S^y_jS^y_{j+1}), maps under the Jordan-Wigner transformation (an exact rewriting of spin-1/2 operators in terms of spinless lattice fermions) to free fermions, so its dynamics from any initial state is exactly computable: by Wick's theorem, many-body correlation functions reduce to determinants or Pfaffians of the fermionic two-point function. A standard way to drive such a chain far from equilibrium is a domain-wall quench: the initial state ∣Ψ0⟩=∣⋯↑↑↑↓↓↓⋯ ⟩|\Psi_0\rangle=|\cdots\uparrow\uparrow\uparrow\downarrow\downarrow\downarrow\cdots\rangle (spins fully polarized up on the left half, down on the right half) is not an eigenstate of HH, and at long times a nonequilibrium steady state (NESS) forms in the central region, with zero local magnetization and a saturated persistent spin current. For translation-invariant final Hamiltonians, local NESS observables depend only on an effective momentum distribution G(p)G(p); breaking translation invariance makes it position dependent, G(p)→G(j)(p)G(p)\to G^{(j)}(p), interpretable as a Wigner (phase-space) distribution.

Lancaster and Godoy (2019) studied two gapped versions of the XY chain: a staggered transverse field, Hs=H+m∑j(−1)jSjzH_s=H+m\sum_j(-1)^jS^z_j, and a dimerization of the hopping, J→J(1−(−1)jδ)J\to J(1-(-1)^j\delta). Both perturbations break one-site translation invariance with period two, doubling the unit cell, halving the first Brillouin zone, and opening an energy gap at momenta k=±π/2k=\pm\pi/2. In the ground state of both models all spin-spin correlations decay exponentially with distance, e.g. in the staggered case C0xx(n)∼A0(m~)(m~+m~2+1)−nC^{xx}_0(n)\sim\mathcal{A}_0(\tilde m)(\tilde m+\sqrt{\tilde m^2+1})^{-n} with m~=m/J\tilde m=m/J and correlation length ℓ=1/sinh⁡−1m~\ell=1/\sinh^{-1}\tilde m. The paper's central finding is that in the NESS after the domain-wall quench the correlations instead decay algebraically: the longitudinal correlator behaves as CNESSzz(n)∼−(πn)−2C^{zz}_{\mathrm{NESS}}(n)\sim-(\pi n)^{-2} (with different oscillatory prefactors on the even and odd sublattices), and the transverse correlator behaves as CNESSxx(n)∼[A(m)+B(m)cos⁡(πn/4)]cos⁡(πn/2)/n\mathcal{C}^{xx}_{\mathrm{NESS}}(n)\sim[\mathcal{A}(m)+\mathcal{B}(m)\cos(\pi n/4)]\cos(\pi n/2)/n for the staggered chain, with an analogous primed form [A′(δ)+B′(δ)cos⁡(πn/4)]cos⁡(πn/2)/n[\mathcal{A}'(\delta)+\mathcal{B}'(\delta)\cos(\pi n/4)]\cos(\pi n/2)/n for the dimerized chain. The transverse correlations vanish for odd separations and split into two amplitude branches for separations divisible by 4 versus only by 2. Thus quasi-long-range order persists in the NESS of a gapped system, in contrast to its own ground state.

The mathematical obstacle is that the transverse correlator is the determinant of a matrix whose entries depend on both the separation and the absolute site index. For Toeplitz matrices (entries Gil=gl−iG_{il}=g_{l-i} depending only on the difference of indices), the Szegő limit theorem and the Fisher-Hartwig conjecture give the large-dimension asymptotics of the determinant from the factorization of the generating function (the Fourier series of gg) into a smooth nonzero part, zeros with exponents αl\alpha_l, and jump discontinuities with exponents βl\beta_l. Here the position dependence G(j)(p)G^{(j)}(p) inherited from the broken translation invariance makes the matrix non-Toeplitz, and the source shows that formally applying Fisher-Hartwig logic anyway gives the correct n−1n^{-1} power but a spurious exponential decay factor that is absent from the exact numerics (in the dimerized ground-state case it even produces the wrong correlation length as δ→1\delta\to1). Consequently, the algebraic forms above are effectively ansätze verified numerically, and the amplitude functions A(m),B(m),A′(δ),B′(δ)\mathcal{A}(m),\mathcal{B}(m),\mathcal{A}'(\delta),\mathcal{B}'(\delta) were fixed only by fitting to direct numerical evaluation of the determinants.

Two controls sharpen the puzzle. In the gapped anisotropic XY model (nonzero anisotropy γ\gamma, whose gap does not double the unit cell), the same domain-wall quench yields exponentially decaying NESS transverse correlations, CNESSxx(n)∼F(γ)cos⁡(πn/2)(1+γ)−n\mathcal{C}^{xx}_{\mathrm{NESS}}(n)\sim\mathcal{F}(\gamma)\cos(\pi n/2)(1+\gamma)^{-n} (there the matrix is genuinely Toeplitz and Fisher-Hartwig applies). And when the initial state is the homogeneous ground state of the gapless XY model instead of the domain wall, the transverse NESS correlations also decay exponentially, although the longitudinal ones remain algebraic. The authors therefore hypothesize that the doubling of the unit cell — mathematically, the halving of the Brillouin zone — is closely tied to the algebraic transverse NESS decay, but this is not proved, and all results are for free (quadratic) fermion Hamiltonians; the robustness to interactions, the behavior in finite systems where the gap slows domain-wall spreading, and a rigorous derivation of the asymptotics are all left open.

Problem Statement

Quench the infinite isotropic XY chain from the domain-wall state ∣Ψ0⟩=∣⋯↑↑↑↓↓↓⋯ ⟩|\Psi_0\rangle=|\cdots\uparrow\uparrow\uparrow\downarrow\downarrow\downarrow\cdots\rangle to a final Hamiltonian that is gapped either by a staggered transverse field, Hs=−J∑j(SjxSj+1x+SjySj+1y)+m∑j(−1)jSjzH_s=-J\sum_j(S^x_jS^x_{j+1}+S^y_jS^y_{j+1})+m\sum_j(-1)^jS^z_j (with m~=m/J<1\tilde m=m/J<1), or by dimerized hopping J→J(1−(−1)jδ)J\to J(1-(-1)^j\delta) with 0<δ<10<\delta<1 (the endpoint δ=0\delta=0 being the gapless isotropic point), and let ⟨⋅⟩NESS=lim⁡t→∞⟨⋅⟩t\langle\cdot\rangle_{\mathrm{NESS}}=\lim_{t\to\infty}\langle\cdot\rangle_t at fixed sites denote the current-carrying steady state in the central region. In this setting, give a mathematically controlled characterization of the persistence of power-law spin-spin correlations in the NESS, in the following three linked strands posed by the source.

(1) Rigorous determinant asymptotics with closed amplitudes. The transverse NESS correlator is given by a free-fermion Pfaffian (a Pfaffian being the determinant analogue for antisymmetric matrices) that here reduces to CNESSxx=14det⁡Q\mathcal{C}^{xx}_{\mathrm{NESS}}=\frac14\det\mathbf{Q} because one of its constituent fermionic contractions vanishes identically in the NESS; the antisymmetric matrix Q\mathbf{Q} has entries Qjl=ql−j(j)Q_{jl}=q^{(j)}_{l-j} for l>jl>j (antisymmetry fixing the entries on and below the diagonal), built from the row-dependent kernel qn(j)=∫−ππdp2π e−ipn cos⁡p−(−1)jm~cos⁡2p+m~2 σ(p),σ(p)=sgn(p) sgn ⁣(π2−p)sgn ⁣(π2+p),q^{(j)}_{n}=\int_{-\pi}^{\pi}\frac{dp}{2\pi}\,e^{-ipn}\,\frac{\cos p-(-1)^{j}\tilde m}{\sqrt{\cos^{2}p+\tilde m^{2}}}\,\sigma(p),\qquad\sigma(p)=\mathrm{sgn}(p)\,\mathrm{sgn}\!\left(\frac{\pi}{2}-p\right)\mathrm{sgn}\!\left(\frac{\pi}{2}+p\right), together with its dimerized analogue, obtained by the identical construction from the dimerized chain's NESS Wigner distribution Gδ(j)(p)=12[1+(cos⁡ϕp−i(−1)jsin⁡ϕp)σ(p)]G^{(j)}_{\delta}(p)=\frac12[1+(\cos\phi_p-i(-1)^j\sin\phi_p)\sigma(p)] with tan⁡ϕp=δtan⁡p\tan\phi_p=\delta\tan p. Develop a rigorous asymptotic technique — a generalization of the Szegő/Fisher-Hartwig theory to these structured, non-Toeplitz, row-alternating matrices, or an alternative method — that derives rather than conjectures: the n−1n^{-1} power-law decay, the oscillatory prefactors of periods 4 and 8, the vanishing of the correlations at odd separations, the two amplitude branches for separations divisible by 4 versus by 2, and analytic expressions for the amplitude functions A(m)\mathcal{A}(m), B(m)\mathcal{B}(m), A′(δ)\mathcal{A}'(\delta), B′(δ)\mathcal{B}'(\delta), which are currently known only by numerical fitting.

(2) Robustness to interactions, integrability breaking, and finite size. Determine whether the algebraic NESS correlations survive when the final Hamiltonian is not quadratic — treating as two distinct required target classes the addition of short-range interactions (the genuinely non-quadratic terms) and weak integrability-breaking perturbations — and establish how the predicted algebraic signatures manifest in finite systems, where the energy gap slows the spreading of the domain wall so that finite-size effects can obscure the NESS before the long-time limit is reached.

(3) The unit-cell-doubling mechanism. Settle both directions of the hypothesis that the doubling of the unit cell (the gap at k=±π/2k=\pm\pi/2 arising from the halved Brillouin zone) causes the algebraic decay of the transverse NESS correlations after a domain-wall quench of a gapped chain: prove sufficiency for the full class of gapped free-fermion chains whose unit cell is doubled by a period-2 modulation, their gap opening at k=±π/2k=\pm\pi/2 — a class that must properly contain both period-2 gap mechanisms above, so that a class consisting only of those two models does not count — and settle necessity, that is, whether algebraic transverse decay can occur without unit-cell doubling, where exactly solvable counterexamples count only for the direction they refute. A result settling a single direction is partial progress on strand (3) rather than a resolution of it; the source offers only two control data points: the anisotropic-XY gap without unit-cell doubling gives exponential NESS decay, and a homogeneous initial state also gives exponential transverse decay.

The three strands share one scientific objective — establishing whether and why quasi-long-range order persists in the nonequilibrium steady state of a gapped quantum spin chain after a domain-wall quench — with strand (1) supplying the exact analytic backbone, strand (2) testing its physical robustness, and strand (3) isolating the band-structure mechanism; an answer should make the relationship among them explicit. The full scientific objective encompasses all three strands, so a result settling a single strand is partial progress toward it rather than a complete solution; answers may address the strands jointly or individually, and any addressed strand must be treated at the full generality stated for that strand (for strands (1) and (3): all m~<1\tilde m<1, all dimerizations 0<δ<10<\delta<1, both gap mechanisms, and the full NESS double limit of long time at fixed separation followed by large separation; for strand (2): the perturbation classes named in it).

References

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