Finite-size scaling theory for boundary renormalization-group flows in gapless one-dimensional quantum systems

The finite-size scaling theory of the energy spectra of gapless one-dimensional quantum systems with open boundaries is established at and near conformally invariant boundary conditions: once the non-universal effective length shift $\delta L$ and boundary energy $\mathfrak{E}$ are accounted for, the spectrum follows boundary conformal field theory with controlled perturbative corrections. When a relevant boundary perturbation is switched on, the system undergoes a boundary renormalization-group (RG) flow between conformally invariant boundary conditions, and this theory fails in the crossover regime. Liu, Shimizu, Ueda and Oshikawa (SciPost Phys. 17, 099 (2024)) stated as an open problem the extension of their theory to such flows: treating higher-order and non-perturbative effects of the boundary perturbation while consistently renormalizing $\delta L$ and $\mathfrak{E}$ along the flow, so that the entire flow region — the universal crossover between the ultraviolet and infrared boundary conformal spectra — becomes quantitatively accessible to lattice spectroscopy.

Background

Gapless one-dimensional quantum systems at criticality — spin chains such as the transverse-field Ising (TF-Ising) model, the three-state Potts chain, or the Ashkin-Teller chain — are described at long wavelengths by a conformal field theory (CFT), a quantum field theory invariant under local scale transformations. With open ends, the continuum limit is a CFT on a finite segment, that is, a boundary conformal field theory (BCFT) with a conformally invariant boundary condition at each end. Cardy showed that the finite-size spectrum then obeys $E_n - E_0 = \pi v x_n / L$, where $v$ is the sound velocity, $L$ the system length, and $x_n$ the scaling dimension of a boundary operator, so that finite-size spectroscopy directly measures boundary conformal data. The lattice spectrum, however, contains non-universal pieces: with $N$ the number of sites and $a$ the lattice spacing, the ground-state energy behaves as $E_0 = e_\infty L + \mathfrak{E} - \pi v c/(24L) + \cdots$, where $c$ is the central charge, $\mathfrak{E}$ is a non-universal boundary energy, and the length entering these formulae is an effective system size $L = Na + \delta L$ shifted by a non-universal extrapolation length $\delta L$ of order $a$. Liu, Shimizu, Ueda and Oshikawa (2024) completed this finite-size scaling theory: they derived the general structure of finite-size corrections from bulk and boundary perturbations, showed that a boundary perturbation by the energy-momentum tensor precisely renormalizes the effective system size (thereby eliminating the ubiquitous $O(1/N^2)$ corrections), and verified the theory against the exact solution of the critical TF-Ising chain and against density-matrix renormalization group (DMRG, the standard numerical method for one-dimensional quantum systems) simulations of the critical three-state Potts chain.

A relevant boundary perturbation — one whose boundary scaling dimension satisfies $h < 1$, so that its coupling grows under coarse-graining — drives a boundary RG flow: for example, a boundary longitudinal magnetic field in the Ising chain takes the free boundary condition into the fixed one, and by the g-theorem of Affleck and Ludwig the Affleck–Ludwig boundary entropy $g_{\mathrm{AL}}$ decreases monotonically along the flow. Near the ultraviolet (weak-coupling) end, the flow is accessible by boundary conformal perturbation theory; Liu et al. verified agreement with the exact solution of the critical Ising chain with boundary longitudinal fields at this leading (second) order in the boundary field, where the universal $\log L$ term comes with a non-universal boundary spin renormalization factor $c_\sigma$ fixed by matching. In the interior of the flow, where the natural scaling variable combining the boundary coupling $g$ with the system size, $g L^{1-h}$ (for an Ising boundary field $\zeta$ of dimension $h = 1/2$, the variable $\zeta_b = \zeta L^{1/2}$), is of order one, no theory of the finite-size spectrum exists: perturbation theory in the boundary coupling fails, and the non-universal quantities $\delta L$ and $\mathfrak{E}$ are themselves renormalized along the flow. Special cases are understood — the free-fermion TF-Ising chain with boundary fields was solved exactly by Campostrini, Pelissetto and Vicari; thermodynamic Bethe ansatz methods (an exact technique computing finite-volume energies of integrable models from their scattering data) yield finite-size ground-state energies along boundary flows in integrable massive quantum field theories; and boundary conformal perturbation theory has been developed beyond leading order — but none of these provides the general finite-size scaling formulation for generic gapless lattice chains that the source paper calls for.

Problem Statement

Construct a finite-size scaling theory for the energy spectra of gapless one-dimensional quantum systems with open boundaries that remains valid along boundary renormalization-group flows — flows driven by relevant boundary perturbations $g\,\Psi$ of boundary scaling dimension $h < 1$, which connect one conformally invariant boundary condition (ultraviolet) to another (infrared). The formulation must extend the existing leading-order perturbative treatment to higher orders and to the non-perturbative level, remaining controlled in the crossover regime where the scaling variable $g L^{1-h}$ is of order one and no expansion in the boundary coupling is available; it must consistently account for the renormalization of the non-universal quantities along the flow — the effective length shift $\delta L$ defining the effective system size entering the finite-size formulae, and the boundary energy $\mathfrak{E}$ — so that they can be disentangled from the universal content; and it must describe the entire boundary RG flow, so that the universal crossover of the finite-size spectrum between the ultraviolet and infrared conformal boundary spectra is quantitatively predicted as a function of system size and boundary coupling. These components serve one objective — a flow-resolved finite-size scaling theory — whose completion would let boundary critical data and boundary exponents be extracted reliably from finite-size spectra of lattice chains in the crossover regime, not only near the fixed points. The problem is the general formulation, not the solution of any single model.

References

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