Finite-size corrections to the energy spectra of gapless one-dimensional systems in the presence of boundaries

The paper has a systematic discussion of the subleading finite size terms in lattice theories matching them to conformal perturbation theory. It raises a number of interesting issues.

The authors attempt to systematically take into account the subleading finite size correction terms
in lattice theories with boundary by matching them to the leading corrections in conformal perturbation theory from bulk and boundary irrelevant perturbations. The general expressions are obtained and for a few concrete models they are matched to numerics. A general discussion of divergences and non-universal terms is given, in particular of the ambiguity in defining the continuum system size and, related to it, boundary energy contributions.

This is an interesting paper which deserves publication. I would only suggest some minor corrections.

1. Although the authors state in several places that the continuum system size should scale as $L= (N+ O(1)) a$so that $L$ and $a$ are dimensionful, in some formulae the length $L$ appears to be dimensionless, e.g. in formulae (26), (28), (29), (30). I suggest the authors stick with the convention in which $L$ is dimensional and is measured in the units of lattice spacing $a$ and restore the relevant powers of $a$. Powers of $a$ are also omitted in the non-universal terms stemming from UV divergences such as formula (12) where the last term should contain a power of $a$. If think for consistency reasons and also to avoid confusion the explicit powers of $a$ should be put in for such terms as well.

2. Although the $T\bar T$ perturbation is discussed in section (3.4) it is not mentioned in the initial general discussion of irrelevant perturbations in section 2.1.2 and in the discussions of the concrete models, e.g. in section 2.2. I suggest the authors mention this perturbation earlier on in the paper and also comment on its absence from the effective actions of the models they consider as examples.

3. Related to the previous point, I’d like to bring to the attention of the authors the following two papers in which the $T\bar T$ term is discussed in connection with the effective continuum Hamiltonian describing the leading finite size corrections to the XXZ model.
S. Lukyanov, Low energy effective Hamiltonian for the XXZ spin chain, Nucl. Phys. B522 (1998) 533-549; arXiv:cond-mat/9712314.

S. Lukyanov and V. Terras, Long-distance asymptotics of spin-spin correlation functions for the XXZ spin chain, Nucl. Phys. B654 (2003) 323-356; arXiv:hep-th/0206093.

4. I found the discussion in section 4 quite important. The authors emphasise that for the perturbative corrections calculated in conformal perturbation theory to match with the lattice theories expansions it is crucial to fix the ambiguity in the definition of the continuum theory size $L$. I do find however the discussion missing some definitive answer. It left me wondering whether any general theoretical prescription for fixing the ambiguity is possible or the authors believe no such general prescription is possible and one needs to proceed in a model by model fashion trying to guess the correct definition and match the expansions? I think that section would benefit from some summary of the situation.

This is a very high quality paper which should be published. I only suggest a number of small corrections listed in the report which I hope will improve the quality of the paper.

Ask for minor revision

The authors study the corrections to finite-size scaling in a critical
(1+1) dimensional quantum system with open boundaries due to
deviations of the low energy effective theory from a boundary CFT in
the conformal limit.
Knowing the effect of such perturbations on the finite-size spectrum
is the basis of numerical studies of critical lattice systems (which
are naturally constrained to a finite number of sites).

In Section 3.3 the subleading corrections to scaling due to
(marginally) irrelevant bulk and boundary perturbations to the fixed
point Hamiltonian on a segment of finite length $L$ is studied using
perturbation theory. To relate the results to the spectrum of an
$N$-site lattice model it is essential to correctly identify the
dependence of $L$ on the number of lattice sites, Eq. (105). With the
energy-momentum tensor any boundary CFT always has a operator of
boundary scaling dimension 2. Therefore, the authors propose to
renormalize $L$ the system size such that the resulting $1/L^2$
correction in the finite-size spectrum is removed.

The authors verify their results from conformal perturbation theory by
comparison with the exact solution of the transverse field Ising model
and numerical DMRG-calculations for the 3-state Potts model at their
critical points subject to several boundary conditions.

The paper provides an explicit result for the corrections to
finite-size scaling due to perturbations of a boundary CFT arising in
the scaling limit of lattice models. This can extend the scope of
numerical studies of the ciritical properties of (1+1)-dimensional
lattice models. The manuscript should be accepted for publication in
SciPost Physics, provided that the authors address the following
points:

- a central results of the paper appears to be that the
renormalization of the system size can be fixed by requiring that
the 1/L^2 corrections to the ground state energy vanishes. The
authors give the finite-size energies of the first three excited
states for the TF Ising model where this critereon appears to work
for the excitations, too. Can the authors provide an argument that
this is true for any system, i.e. that the renormalization of $L$
based on this critereon is independent of the state? At least the
perturbative corrections to scaling (6) do depend on the boundary
OPE coefficients.

- the amplitude of the universal $(1/N)$ term in Eq. (40) would imply
$c=1$. Is that correct?

Publish (easily meets expectations and criteria for this Journal; among top 50%)