Excited-state g-functions from the truncated conformal space: convergence control, coupling renormalisation, bulk perturbations, and defect applications

In a two-dimensional boundary quantum field theory, a perturbed conformal boundary condition is characterised by its boundary state, whose overlaps with the finite-energy bulk states are the excited-state g-functions. Takács and Watts (JHEP 02 (2012) 082) introduced a method to compute these overlaps with a variation of the truncated conformal space approach (TCSA): one-point functions of bulk fields on a disk, mapped to a Schrödinger-type evolution in a Hilbert space truncated at a conformal-energy cutoff with a time-dependent boundary coupling. Applied to the boundary-perturbed Lee-Yang model, the method reproduced the ground state and the first two excited g-functions accurately, with the empirical extrapolation extending accurate results to the third and fourth excited g-functions, but the authors documented two unresolved methodological limitations of the overlap computation and proposed two further extensions of it: (i) direct truncation estimates oscillate and diverge for operators of total conformal weight above about 2, and the empirical Padé/Wynn resummation used instead has no a priori convergence or error control; (ii) the truncation-induced renormalisation of the effective boundary coupling is not incorporated into the evolution equations, only removed a posteriori by taking g-function ratios or extrapolating; (iii) the extension to massive bulk perturbations, needed to validate or falsify the conjectural TBA equations for bulk excited-state g-functions with their infinite series of bulk corrections, was not carried out; (iv) the application to defect matrix elements and to the structure and endpoints of defect flows was suggested but not implemented. The overlap-method convergence theory, a renormalised evolution, the combined bulk-plus-boundary-perturbed case, and defect matrix elements computed as overlaps all remain open.

Background

A two-dimensional conformal field theory (CFT) defined on a domain with a boundary requires a choice of boundary condition. Each conformal boundary condition is encoded in a boundary state: a non-normalisable vector in the closed-channel Hilbert space, uniquely fixed by its overlaps with the normalisable finite-energy bulk states. The overlap with the ground state defines the boundary g-function, Affleck and Ludwig's universal 'boundary degeneracy' (the exponential of the boundary entropy), which decreases monotonically along boundary renormalisation-group flows (the g-theorem) and is a central observable of boundary quantum field theory (QFT).

If the conformal boundary condition is perturbed by the integral of a local boundary field along the boundary, the resulting boundary condition is still described by a boundary state, but its overlaps with the bulk states are no longer fixed by symmetry. The overlaps with excited bulk states are called excited-state g-functions. When the perturbation is integrable, these overlaps can in principle be computed from the thermodynamic Bethe ansatz (TBA), the integral-equation machinery for finite-volume observables of integrable QFT; excited states are reached by analytically continuing the TBA equations in the source term (Dorey and Tateo 1996; Bazhanov, Lukyanov and Zamolodchikov 1997). The truncation-independent benchmark is therefore available in integrable cases only.

The truncated conformal space approach (TCSA), introduced by Yurov and Zamolodchikov for the scaling Lee-Yang model, approximates a perturbed CFT by keeping only the unperturbed CFT states below a conformal-energy cutoff $N$ and diagonalising the perturbed Hamiltonian in that finite space. Dorey, Pocklington, Tateo and Watts (1998) extended it to boundary perturbations, and Dorey, Runkel, Tateo and Watts (2000) extracted the ground-state g-function from the strip-width scaling of the cylinder partition function; this route, however, could not access excited-state overlaps. Takács and Watts (2012) filled that gap with a different construction inspired by Zamolodchikov's calculation of the sphere partition function of the scaling Lee-Yang model: the overlap of a bulk state with the perturbed boundary state is a one-point function of a bulk field on a disk; conformally mapping the disk to the upper half-plane turns it into the expectation value of the bulk field in a state generated by a boundary perturbation with a time-dependent coupling, and the truncated version of the resulting Schrödinger equation is integrated numerically. In the boundary Lee-Yang model (the non-unitary minimal CFT of central charge $c=-22/5$ with its single relevant integrable boundary perturbation), where the TBA for the boundary overlaps is known by analytic continuation, the direct truncation estimates of the ground-state and first two excited g-functions converge, the empirical extrapolation described below extends accurate results to the third and fourth excited g-functions, and the method outperforms both second-order perturbation theory and the older cylinder-partition-function estimate of the ground-state g-function.

The same paper, and the boundary-TCSA renormalisation-group analysis of Watts (2012, Nucl. Phys. B 859) it relies on, identified the following limitations. First, the direct estimates at truncation level $N$ converge for the states associated with operators of total conformal weight up to about 2 but oscillate and diverge for higher weights; the cure applied was empirical: the level-$N$ estimate can be reorganised into two polynomial families $P_N(\lambda)$ and $Q_N(\lambda)$, which weight the matrix elements between the excitation-level-$m$ and excitation-level-$n$ components of the truncated state by $\lambda^{\max(m,n)}$ and by $\lambda^{m+n}$ respectively, and whose values at $\lambda=1$ are extrapolated to $N\to\infty$ using Padé approximants (rational-function interpolants of a truncated series) computed with Wynn's epsilon algorithm (a standard algorithm for accelerating the convergence of a sequence). The naive Euler, Cesàro and Padé resummations of the direct estimates give results correct to a few percent for the third and fourth excited g-functions $g_3$ and $g_4$ but all fail on the fifth, $g_5$; the reorganised $Q$-polynomials work effectively for all states considered (the $g_i/g_0$ ratios agree with the TBA up to the coupling at which the strip TCSA Hamiltonian itself stops resolving the fifth excited state) — but with no a priori control of convergence or error. Second, the dominant residual error is a truncation-dependent renormalisation of the effective boundary coupling shifting the extensive boundary free energy; the authors avoided it by taking ratios of g-functions or extrapolating in $N$, not by building the renormalisation into the differential equations. Third, the method was formulated only for purely boundary perturbations with a conformal bulk; including massive bulk perturbations is needed to test the conjectural TBA equations for bulk excited-state g-functions, which contain an infinite series of bulk corrections of the type appearing in the exact ground-state g-function of Dorey, Fioravanti, Rim and Tateo (2004), proved by Pozsgay (2010). Fourth, the method should extend to defect matrix elements — for a conformal or perturbed line defect, reflection and transmission coefficients (Quella, Runkel and Watts 2007) — via the 'mirror trick', which folds the theory with a defect into a boundary problem on a doubled theory, and thereby clarify the endpoints of defect flows (Kormos, Runkel and Watts 2009).

Later work has built neighbouring tools rather than completing this program: a more direct TCSA overlap computation for massive theories with conformal boundaries (Bajnok and Tompa 2021), a defect TCSA for spectra along defect flows (Bajnok, Holló and Watts 2014), systematic counterterm renormalisation of TCSA spectra (Lencsés and Takács 2014), truncation-error asymptotics for boundary flows (Konechny and McAteer 2019), and a mature analytic theory of the ground-state g-function with bulk corrections (Kostov, Serban and Vu 2019; Vu, Kostov and Serban 2019; Caetano and Komatsu 2020; He and Jiang 2024).

Problem Statement

Turn the truncated-conformal-space computation of excited-state g-functions into a controlled, systematically improvable method. The scientific objective is a reliable determination of boundary-state overlaps (excited-state g-functions) by truncated-space methods, with quantified truncation errors, sufficient to test the corresponding TBA and exact predictions item by item. Concretely, for a two-dimensional conformal field theory perturbed on the boundary and treated by the TCSA evolution with time-dependent boundary coupling, the open methodological core consists of the two limitations the source paper documented but did not resolve:

1. Truncation-error control: replace the empirical $P_N/Q_N$ plus Padé/Wynn extrapolation by an extrapolation scheme for the truncation-level sequences of overlap estimates that carries an a priori convergence or error estimate, and make it work for operators of total conformal weight above about 2, where the direct estimates oscillate and diverge.

2. Renormalised evolution: incorporate the truncation-induced renormalisation of the effective boundary coupling (which shifts the extensive boundary free-energy term) systematically into the truncated evolution equations, instead of eliminating it a posteriori by taking g-function ratios or fitting.

The same paper also proposed two further applications of this overlap method, which remain unimplemented by this route. They indicate the range of settings in which a controlled overlap method is meant to operate and the exact predictions it should be able to confront; they are not additional deliverables demanded on top of the methodological core:

3. Bulk perturbations: extend the overlap computation to theories with both massive bulk and boundary perturbations, and use it to validate or falsify the conjectural TBA equations for bulk excited-state g-functions, including their infinite series of bulk corrections - beyond the ground state and beyond the conformal-boundary, one-particle cases already tested by other routes.

4. Defect observables: apply the method to defect matrix elements via the mirror trick (folding the defect theory to a boundary problem on the doubled theory), the route proposed in the source paper, computing reflection and transmission coefficients of perturbed defects and clarifying the detailed structure and endpoints of defect renormalisation-group flows.

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