Universal finite-size gap scaling of the quantum Ising chain beyond the scaling limit: irrelevant-operator corrections, continuum-parameter extraction, and general Kitaev-chain defects

The transverse-field quantum Ising chain with a one-parameter family of $\mathbb{Z}_2$-symmetric 'defect' boundary conditions — which interpolates among open, antiperiodic, and periodic chains — possesses a universal lowest-gap scaling function $\Delta(L,m,\chi)$ describing the finite-size crossover between its ordered and disordered phases, where $m$ is the Majorana mass and $\chi$ the defect scattering phase shift (M. Oshikawa, arXiv:1910.06353, 2019). That function was derived in the scaling limit, which neglects corrections from irrelevant operators, presumes the continuum parameters are known, and treats general Kitaev-chain defects only at the critical point. This record states the open problem of completing that theory: (i) quantifying the subleading corrections from irrelevant operators to the gap scaling function across the ordered–disordered crossover, including its defect-bound-state and fermion-parity-reversal structure; (ii) establishing controlled protocols to extract the continuum parameters (mass, defect phase shift, velocity, and correction amplitudes) for microscopic realizations lacking an exact lattice solution; and (iii) deriving the finite-size gap scaling for general Kitaev-chain defects with independent hopping and pairing and complex defect phases.

Background

The transverse-field Ising chain (TFIM) is a chain of spin-$1/2$ degrees of freedom with an Ising coupling along one axis and a magnetic field transverse to it; it is the paradigmatic exactly solvable example of a second-order quantum phase transition. Pfeuty (1970) solved it by the Jordan–Wigner transformation, which maps spins to spinless fermions whose real and imaginary parts (Majorana fermions, particles that are their own antiparticles) render the Hamiltonian bilinear. A quantum critical point separates a disordered (paramagnetic) phase from an ordered (ferromagnetic) phase with a twofold degenerate ground state in the thermodynamic limit, reflecting spontaneous breaking of the $\mathbb{Z}_2$ spin-flip symmetry. Near the critical point the low-energy physics is the relativistic Majorana field theory in $1+1$ dimensions, with mass $m$ proportional to the deviation from criticality and a spin-wave (Fermi) velocity $v$.

On a finite chain of $L$ sites the lowest excitation gap $\Delta$ becomes a universal function: at criticality this is Cardy's conformal finite-size scaling, and away from criticality the Lüscher-type finite-volume quantization of a massive relativistic particle. Boundary conditions matter decisively. Cardy's boundary conformal field theory shows that they change the operator content (the set of scaling operators that characterize the finite geometry) of the theory, and a defect or boundary acts on the bulk fermion as a scattering problem whose phase shifts correct the momentum quantization. For the Ising model, Oshikawa and Affleck (1996, 1997) showed that a defect line at criticality is characterized by the scattering phase shift of the Majorana fermion; Abraham, Ko and Švrakić (1989) had earlier obtained the exact transfer-matrix spectrum of the classical finite-width Ising model with adjustable boundary conditions, including the asymptotic degeneracy in the ordered phase.

Oshikawa (arXiv:1910.06353) combined the exact lattice solution with continuum scattering to derive the universal lowest-gap scaling function $\Delta(L,m,\chi)$ of the quantum Ising chain with a one-parameter family of $\mathbb{Z}_2$-symmetric defect boundary conditions (parametrized by a defect coupling $b$, with open, periodic, and antiperiodic chains as special cases, and by the induced scattering phase shift $\chi$), valid across the ordered–disordered crossover. Its striking feature is an asymmetry under $m\to-m$: the Majorana dispersion is symmetric under mass inversion — a consequence of the Kramers–Wannier self-duality of the Ising model, which exchanges the ordered and disordered phases — yet the actual lowest gap must vanish as $L\to\infty$ throughout the ordered phase to realize the ground-state degeneracy. This is achieved by (i) a bound state localized at the defect (for all boundary conditions except periodic and antiperiodic) and (ii) an effective reversal of the fermion number parity — the $\mathbb{Z}_2$ grading of the fermionic Hilbert space — in one parity sector (for all except the open chain). The result is a Heaviside step in the scaling function and a nontrivial crossover 'phase diagram' in the plane of defect strength and mass.

The Jordan–Wigner fermion of the Ising chain coincides with Kitaev's one-dimensional $p$-wave superconducting wire (the Kitaev chain, with hopping $t$ and pairing $\Delta_{\mathrm{sc}}$; the Ising chain corresponds to $t=\Delta_{\mathrm{sc}}$), which hosts unpaired Majorana zero modes at its open ends. The Kitaev chain admits more general defects than the Ising mapping allows: independent $t$ and $\Delta_{\mathrm{sc}}$, and complex phase factors $\phi_1,\phi_2$ at the defect bond. Twisted Kitaev chains were shown by Kawabata, Kobayashi, Wu and Katsura (2017) to host exact Majorana zero modes even at nonzero defect coupling; Oshikawa rederives this as the vanishing of the continuum transmission amplitude at the critical point, $t\cos\phi_1+\Delta_{\mathrm{sc}}\cos\phi_2=0$, and shows that $t\neq\Delta_{\mathrm{sc}}$ adds extra evanescent (bound-state-like) eigenstates absent at $t=\Delta_{\mathrm{sc}}$.

All of this holds in the scaling limit $L\to\infty$ with $mL$ fixed, where corrections from irrelevant operators — operators suppressed under the renormalization group whose effects vanish asymptotically but generate subleading $1/L$ corrections with nonuniversal amplitudes at finite size — are discarded. Those corrections matter in practice: finite-size spectra of near-critical chains are a standard diagnostic (level spectroscopy, the technique of reading off critical points, boundary conditions, and phase shifts from the structure of finite-size energy levels), and the source explicitly notes they may be important when fitting numerical or experimental data. Likewise, for a microscopic realization in the Ising universality class that is not exactly solvable, the continuum parameters $m$, $\chi$, and $v$ are not known a priori. The open problem is to complete the theory on these three fronts.

Problem Statement

Complete the universal finite-size gap theory of the quantum Ising chain with $\mathbb{Z}_2$-symmetric defect boundary conditions derived at scaling-limit accuracy by Oshikawa (arXiv:1910.06353), in three linked directions that the source leaves open.

(1) Subleading corrections. Quantify the corrections from irrelevant operators (in the renormalization-group sense) to the universal lowest-gap scaling function $\Delta(L,m,\chi)$ of the defect Ising chain: derive the leading corrections in the $1/L$ expansion — from bulk operators such as $T\bar T$ (the leading bulk irrelevant operator of the two-dimensional theory, the product of the chiral and antichiral stress tensors) and from boundary irrelevant operators, with their dependence on the microscopic defect coupling — valid across the ordered–disordered crossover, including the regime in which the lowest gap is carried by the defect bound state and the loci where the bound-state energy crosses zero and the effective fermion-parity reversal switches sectors.

(2) Parameter extraction. Establish a controlled protocol by which the continuum parameters entering the corrected scaling functions — the Majorana mass $m$ (its proportionality to the microscopic tuning parameter), the defect phase shift $\chi$ (the mapping from the microscopic defect coupling), the spin-wave (Fermi) velocity, and the nonuniversal correction amplitudes from (1) — are determined for microscopic realizations in the Ising universality class that lack an exact lattice solution.

(3) General Kitaev-chain defects. Extend the finite-size gap analysis to the general Kitaev-chain defect with independent hopping $t$ and pairing $\Delta_{\mathrm{sc}}$ and complex defect phase factors $\phi_1,\phi_2$: incorporate the extra evanescent (bound-state-like) eigenstates existing at $t\neq\Delta_{\mathrm{sc}}$ into the finite-size quantization conditions at finite mass, and determine the resulting lowest-gap scaling behavior. The source already derives the continuum transmission and reflection amplitudes for the $t=\Delta_{\mathrm{sc}}$ Kitaev defect with complex phase factors $\phi_1,\phi_2$ at finite momentum and mass, and, in the scaling limit near the critical point, the transmission amplitude at $t\neq\Delta_{\mathrm{sc}}$ — reproducing the vanishing-transmission condition $t\cos\phi_1+\Delta_{\mathrm{sc}}\cos\phi_2=0$; what remains unknown is the finite-size spectrum and lowest-gap scaling for the general defect at $t\neq\Delta_{\mathrm{sc}}$.

These components jointly serve one objective: a quantitatively predictive theory of the lowest gap of finite defect chains. The problem is complete when the lowest gap of a microscopic defect chain (Ising-mapped or general Kitaev) can be predicted throughout the near-critical scaling regime in which the source's scaling function applies, at the accuracy of the first subleading corrections derived in (1), with the continuum parameters and correction amplitudes determined by the procedure of (2); consistency with the scaling-limit function of the source and with the known special cases (periodic, antiperiodic, and open chains; the critical-point spectra with the defect phase shift; the critical-point Kitaev-defect transmission amplitude) is required throughout.

References

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