Finite-size scaling of energy spectra along boundary RG flows in one-dimensional critical systems with open boundaries

The exact finite-size scaling theory for the energy spectra of gapless one-dimensional systems with open boundaries (Liu, Shimizu, Ueda, Oshikawa, 2024) describes the lattice spectrum at conformally invariant boundary conditions: it organizes corrections from irrelevant bulk and boundary perturbations, absorbs the boundary energy-momentum-tensor perturbation into a non-universal effective length shift, and separates non-universal boundary energies from universal contributions. When a relevant boundary perturbation — for example a boundary longitudinal magnetic field in the critical Ising or three-state Potts chain — is applied, the boundary condition undergoes a renormalization-group (RG) flow between conformal fixed points, and the existing theory covers only the fixed points and the leading order in the perturbation. The open problem, stated explicitly in the source paper's discussion of future directions, is to develop a formulation of the finite-size spectrum valid along the entire boundary RG flow: treating higher-order and non-perturbative effects, consistently renormalizing the effective length shift and the boundary energy, both of which flow under the boundary RG, and delivering universal crossover functions, so that boundary critical data can be extracted from finite-size spectra away from the fixed points.

Background

A one-dimensional quantum system at a continuous phase transition is described at long wavelengths by a two-dimensional conformal field theory (CFT), a scale-invariant field theory characterized by its central charge $c$, a number measuring the effective amount of long-wavelength degrees of freedom. Finite-size scaling relates the energy levels of a critical chain of length $L$ to the CFT data: on a periodic ring the universal part of the spectrum is $\frac{2\pi v}{L}(x_n-c/12)$, where $v$ is the characteristic velocity and $x_n$ the scaling dimension of the operator creating the state. On an open chain (a strip in Euclidean space-time), Cardy's boundary conformal field theory (BCFT) relates the spectrum to boundary operators, the local operators living on a conformally invariant boundary, through

$E_n(L)=\mathfrak{e}_0 L+\mathfrak{E}_\alpha+\mathfrak{E}_\beta+\frac{\pi v}{L}\left(h_n-\frac{c}{24}\right),$

where $h_n$ is the boundary scaling dimension (when the boundary conditions $\alpha$ and $\beta$ at the two ends differ, the state is created by a boundary-condition-changing operator), $\mathfrak{e}_0$ is the non-universal bulk energy density, and $\mathfrak{E}_\alpha,\mathfrak{E}_\beta$ are non-universal boundary energies. This formula underlies level spectroscopy: the extraction of conformal and boundary critical data from spectra computed by exact diagonalization or by the density matrix renormalization group (DMRG), a variational method for one-dimensional quantum systems.

A lattice model is described by the CFT only up to perturbations: the effective field theory contains the CFT plus irrelevant bulk operators (scaling dimension above 2, whose couplings diminish at long distances) and boundary operators at the two ends. For periodic chains the systematic theory of the resulting finite-size corrections is well developed and underlies precision determinations of critical points. For open chains, Liu, Shimizu, Ueda and Oshikawa (SciPost Phys. 17, 099 (2024)) derived the general expression: corrections to first order in bulk couplings, through bulk-boundary operator product expansions (the short-distance expansion of a bulk operator near a boundary into boundary operators) and conformal blocks (the universal building blocks of conformal correlation functions), and to second order in boundary couplings, through boundary operator product expansion coefficients, with careful treatment of ultraviolet (short-distance) divergences. A central mechanism is that every CFT has a boundary operator of scaling dimension 2, identifiable with the holomorphic energy-momentum tensor $T$, the generator of conformal transformations; its leading effect is exactly a renormalization of the effective system size, $L\to\tilde L=L+\delta L_\alpha+\delta L_\beta$, where the length shifts $\delta L$ (also called extrapolation lengths) are non-universal constants fixed by the boundary conditions. This absorbs the ubiquitous $O(L^{-2})$ corrections, and the boundary energy $\mathfrak{E}$ must likewise be redefined to absorb ultraviolet-divergent constants before universal terms can be isolated. The theory was verified against the exact solution of the critical transverse-field Ising chain and against DMRG spectra of the critical three-state Potts chain (the $q=3$ generalization of the Ising chain).

A relevant boundary perturbation — a boundary operator of scaling dimension below 1, whose coupling grows under the renormalization group (RG) — drives the system from one conformally invariant boundary condition to another: a boundary RG flow, a concept introduced by Cardy in 1989. On the lattice, the canonical realization is a longitudinal magnetic field on the end spins of a critical Ising or three-state Potts chain, which drives the free boundary condition at zero field to a fixed boundary condition at strong field. Along such flows the boundary entropy (the $g$-factor, the universal part of the boundary free energy) decreases monotonically, by the $g$-theorem. The 2024 theory applies only at the conformal fixed points; for the boundary field it treats just the leading perturbative order (second order in the field $\zeta$, in the scaling limit with $\zeta L^{1/2}$ fixed), verified against the exact Ising solution. The paper's discussion of future directions identifies the missing step treated here: going to higher orders and to the non-perturbative level is difficult because the non-universal quantities $\delta L$ and $\mathfrak{E}$ are themselves renormalized along the flow, and the authors hope for a new formulation covering the entire boundary RG flow.

Several model- or method-specific results exist beyond the fixed points but do not provide the general formulation: exact finite-size crossover functions for the free-fermionic Ising chain with boundary fields (Campostrini, Pelissetto and Vicari, 2015); non-perturbative boundary-flow crossover data for massive integrable field theories via the boundary thermodynamic Bethe ansatz (an exact finite-size integral-equation technique, e.g. Nepomechie and Ahn, 2002); higher-order boundary conformal perturbation theory and truncated-conformal-space (a numerical Hamiltonian truncation) or variational studies of boundary flows in minimal models (the exactly solvable conformal field theories with a finite spectrum of fields; Gaberdiel, Konechny and Schmidt-Colinet, 2009; Graham, Runkel and Watts, 2000; Konechny, 2023); and exact finite-size spectra from the Bethe ansatz (an exact technique that solves certain one-dimensional models by reducing the eigenvalue problem to algebraic equations) for open chains with integrable boundary conditions that do not flow (Frahm, Gehrmann, Nepomechie and Retore, 2025). None of these tracks the renormalization of $\delta L$ and $\mathfrak{E}$ along a boundary RG flow in the finite-size spectrum of a generic interacting critical chain.

Problem Statement

Extend the exact finite-size scaling theory for the energy spectra of gapless one-dimensional systems with open boundaries, which is valid at conformally invariant boundary conditions, to the boundary renormalization group (RG) flow triggered by a relevant boundary perturbation.

Specifically: for a critical open chain whose boundary perturbation (for example, a boundary longitudinal magnetic field in the critical Ising or three-state Potts chain) drives the boundary condition from an ultraviolet (weak-coupling) fixed point to an infrared (strong-coupling) fixed point, develop a formulation that predicts the finite-size energy levels along the entire flow, with the same rigor that the boundary-conformal-perturbation-theory formulation of Liu, Shimizu, Ueda and Oshikawa achieves at the fixed points. The formulation should:

1. give the energy levels as functions of the system size and the running boundary coupling, covering higher-order perturbative corrections and the non-perturbative strong-coupling regime of the flow, not only the leading order in the boundary field;
2. handle consistently the renormalization of the non-universal effective length shift $\delta L$ and the boundary energy $\mathfrak{E}$ under the boundary RG, so that their flow does not contaminate the universal content of the spectrum;
3. identify the universal crossover functions of the flow (as functions of the scaling variable combining the boundary coupling and the system size) in the finite-size spectrum, including their approach to both fixed points;
4. reduce, at the ultraviolet and infrared fixed points and at leading order in the boundary perturbation, to the established fixed-point results.

No restriction to a particular model class is intended; the Ising and Potts chains are the canonical testbeds. A complete answer would allow boundary critical data — boundary scaling dimensions, boundary entropies, and boundary exponents — to be extracted from finite-size spectra away from the conformal fixed points.

References

1. Yifan Liu, Haruki Shimizu, Atsushi Ueda, Masaki Oshikawa, "Finite-size corrections to the energy spectra of gapless one-dimensional systems in the presence of boundaries", SciPost Physics 17, 099 (2024), DOI: 10.21468/SciPostPhys.17.4.099, arXiv:2405.06891, https://arxiv.org/abs/2405.06891
2. Yifan Liu, Natalia Chepiga, Yoshiki Fukusumi, Masaki Oshikawa, "Boundary critical phenomena in the quantum Ashkin-Teller model", SciPost Physics 20, 130 (2026), DOI: 10.21468/SciPostPhys.20.5.130, arXiv:2601.16951, https://arxiv.org/abs/2601.16951
3. Yifan Liu, Haruki Shimizu, Dongchang Liu, Kohei Kawabata, "Extracting Boundary Conformal Data from Periodic Non-Hermitian Critical Chains", arXiv:2606.16785 (2026), https://arxiv.org/abs/2606.16785
4. Massimo Campostrini, Andrea Pelissetto, Ettore Vicari, "Quantum Ising chains with boundary fields", Journal of Statistical Mechanics P11015 (2015), DOI: 10.1088/1742-5468/2015/11/P11015, arXiv:1507.08391, https://arxiv.org/abs/1507.08391
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