happens there may be corrections from the off-diagonal terms in the
Hamiltonian which would compete with the constant terms from the spectator
boundary condition if $\Delta_{1,2}<1$ where $\Delta_{1,2}$ is the lowest
weight of the boundary condition changing fields linking the two infrared
boundary conditions. We note that there are no such fields in this
case161616This is actually the case for all $\psi_{1,3}$ flows in the A-series
minimal models.

IR UV |  (d) | (0+) | (-0)
---|---|---|---
(d) |  $e(\hat{1})=e(\hat{\epsilon}^{\prime\prime})=e(\hat{\epsilon}^{\prime})=0$; $e(\hat{\epsilon})=-3.63$ | $e(\hat{\sigma}^{\prime})=e(\hat{\sigma})=0$ |  $e(\hat{\sigma}^{\prime})=0$; $e(\hat{\sigma})=-2.80$
(0+) |  $e(\hat{\sigma})=e(\hat{\sigma}^{\prime})=0$ | $e(\hat{1})=e(\hat{\epsilon}^{\prime})=0$ | $e(\hat{\epsilon})=e(\hat{\epsilon}^{\prime\prime})=0$
(-0) |  $e(\hat{\sigma})=e(\hat{\sigma}^{\prime})=0$ |  $e(\hat{\epsilon}^{\prime\prime})=0$; $e(\hat{\epsilon})=-3.63$ |  $e(\hat{1})=0$; $e(\hat{\epsilon}^{\prime})=-0.24$
(0) |  $e(\hat{\epsilon})=0$, $e(\hat{\epsilon}^{\prime})=-0.82$ | $e(\hat{\sigma})={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-9.32}$ | $e(\hat{\sigma})=0$
(+) |  $e(\hat{\sigma})={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-9.32}$ | $e(\hat{\epsilon})=0$ | $e(\hat{\epsilon}^{\prime})=0$
(-) |  $e(\hat{\sigma})={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-9.32}$ | $e(\hat{\epsilon}^{\prime})=-0.82$ | $e(\hat{\epsilon})={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-12.11}$
Table 2: Variational energy factors for $\psi_{1,3}$ flows with positive coupling given to the second decimal place. Here the columns are labeled by UV boundary conditions while the rows correspond to tentative IR fixed points. The smallest values are highlighted in red.  IR UV |  (d) | (0+) | (-0)
---|---|---|---
(d) |  $e(\hat{1})=e(\hat{\epsilon}^{\prime\prime})=e(\hat{\epsilon})=0$; $e(\hat{\epsilon}^{\prime})=-0.24$ |  $e(\hat{\sigma}^{\prime})=0$; $e(\hat{\sigma})=-2.80$ | $e(\hat{\sigma}^{\prime})=e(\hat{\sigma})=0$
(0+) |  $e(\hat{\sigma}^{\prime})=0$; $e(\hat{\sigma})=-2.80$ |  $e(\hat{1})=0$; $e(\hat{\epsilon}^{\prime})=-0.24$ |  $e(\hat{\epsilon}^{\prime\prime})=0$; $e(\hat{\epsilon})=-3.63$
(-0) |  $e(\hat{\sigma}^{\prime})=0$; $e(\hat{\sigma})=-2.80$ | $e(\hat{\epsilon}^{\prime\prime})=e(\hat{\epsilon})=0$ | $e(\hat{1})=e(\hat{\epsilon}^{\prime})=0$
(0) |  $e(\hat{\epsilon}^{\prime})=0$, $e(\hat{\epsilon})={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-12.11}$ | $e(\hat{\sigma})=0$ | $e(\hat{\sigma})={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-9.32}$
(+) |  $e(\hat{\sigma})=0$ | $e(\hat{\epsilon})={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-12.11}$ | $e(\hat{\epsilon}^{\prime})=-0.82$
(-) |  $e(\hat{\sigma})=0$ | $e(\hat{\epsilon}^{\prime})=0$ | $e(\hat{\epsilon})=0$
Table 3: Variational energy factors for $\psi_{1,3}$ flows with negative
coupling given to the second decimal place. Here the columns are labeled by UV
boundary conditions while the rows correspond to tentative IR fixed points.
The smallest values are highlighted in red.

Besides confirming the blue arrows on the diagram on Figure 6 we also obtain
the RG operators for these flows which we summarise in the following diagrams
where we put the RG operator above the arrow designating the flow.

$(+)\oplus(-)\,\mathop{\xleftarrow{\hskip
42.67912pt}}^{{\hat{\sigma}\oplus\hat{\sigma}}}\,(d)\,\mathop{\xrightarrow{\hskip
42.67912pt}}^{{\hat{\epsilon}}}\,(0)$ (5.50) $(0)\,\mathop{\xleftarrow{\hskip
42.67912pt}}^{{\hat{\sigma}}}\,(0+)\,\mathop{\xrightarrow{\hskip
42.67912pt}}^{{\hat{\epsilon}}}\,(+)$ (5.51) $(-)\,\mathop{\xleftarrow{\hskip
42.67912pt}}^{{\hat{\epsilon}}}\,(-0)\,\mathop{\xrightarrow{\hskip
42.67912pt}}^{{\hat{\sigma}}}\,(0)$ (5.52)

Here the left arrows correspond to the flows with $\lambda>0$ and the right
arrows to the flows with $\lambda<0$. We observe that these assignments
satisfy our general conjecture 2 as well as the mapping of RG flows induced by
the $X_{\sigma^{\prime}}$ topological defect.

###  5.7 $\psi_{1,2}$ flows

The $\psi_{1,2}$ boundary field is only present on the $(d)$ boundary
condition. The factors $e(\hat{\psi})$ for the variational energy are
presented in Table 4. The smallest energies correspond to the two flows which
are drawn in red on Figure 6. We also represent these two flows on the diagram
below indicating the RG operators above the arrows.

$(-)\,\mathop{\xleftarrow{\hskip
42.67912pt}}^{{\hat{\sigma}}}\,(d)\,\mathop{\xrightarrow{\hskip
42.67912pt}}^{{\hat{\sigma}}}\,(+)$ (5.53)

IR UV |  (d), $\lambda>0$ |  (d), $\lambda<0$
---|---|---
(d) | $e(\hat{1})=e(\hat{\epsilon})=e(\hat{\epsilon}^{\prime})=e(\hat{\epsilon}^{\prime\prime})=0$ | $e(\hat{1})=e(\hat{\epsilon})=e(\hat{\epsilon}^{\prime})=e(\hat{\epsilon}^{\prime\prime})=0$
(0+) | $e(\hat{\sigma}^{\prime})=e(\hat{\sigma})=0$ |  $e(\hat{\sigma}^{\prime})=0$; $e(\hat{\sigma})=-0.69$
(-0) |  $e(\hat{\sigma}^{\prime})=0$; $e(\hat{\sigma})=-0.69$ | $e(\hat{\sigma}^{\prime})=e(\hat{\sigma})=0$
(0) | $e(\hat{\epsilon})=e(\hat{\epsilon}^{\prime})=0$ | $e(\hat{\epsilon})=e(\hat{\epsilon}^{\prime})=0$
(+) | $e(\hat{\sigma})=0$ | $e(\hat{\sigma})={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-1.17}$
(-) | $e(\hat{\sigma})={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-1.17}$ | $e(\hat{\sigma})=0$
Table 4: Variational energy factors for $\psi_{1,2}$ flows given to the second
decimal place. The smallest values are highlighted in red.

Regarding our conjecture 3 we observe that the leading irrelevant operator for
the $(d)\to~{}(+)\oplus~{}(-)$ flow is believed to be the boundary condition
changing operator $(\epsilon^{\prime\prime})^{[+,-]}$. This operator is
present in the OPE $\hat{\psi}^{[\rm ir,uv]}\hat{\psi}^{[\rm uv,ir]}$ of the
RG operator with itself with our identification of its components as the
primaries: $\hat{\sigma}^{[d,+]}$, $\hat{\sigma}^{[d,-]}$. The leading
irrelevant operator for the $(d)\to(0)$ flow is the stress energy tensor which
is present in the relevant OPE. While the leading IR operators for the pure
$\psi_{1,2}$ flows are not known they can be only composites of the stress-
energy tensor which are always present in the OPE of interest.

###  5.8 Superpositions of trial states

As we discussed in section 5.2 the off-diagonal matrix elements of the
Hamiltonian between trial states built upon different IR BCFTs are suppressed
by powers of $\tau/L$. However there may be $L$-independent corrections in the
average of the perturbation for the same IR BCFT but different RG operators.
We find that this is precisely the case for the trial states linking $(d)$
with $(0)$ and $(d)$ with $(0+)$ or $(-0)$. In the first case we have two
trial states:

$\hat{\epsilon}^{[d,0]}(-\tau)|0\rangle_{(0)}\,,\qquad(\hat{\epsilon}^{\prime})^{[d,0]}(-\tau)|0\rangle_{(0)}$
(5.54)

while in the second:

$\hat{\sigma}^{[d,0+]}(-\tau)|0\rangle_{(0+)}\,,\qquad(\hat{\sigma}^{\prime})^{[d,0+]}(-\tau)|0\rangle_{(0+)}\,$
(5.55)

and similarly for $(0-)$. The perturbing operators $\psi_{1,3}$ and
$\psi_{1,2}$ each have off-diagonal matrix elements in the above states. We
can thus introduce more general trial states

$|\theta,\tau\rangle_{(0)}=\cos(\theta)\frac{(2\tau)^{\hat{\Delta}_{\epsilon}}}{\sqrt{C^{1}_{\hat{\epsilon}\hat{\epsilon}}}}\hat{\epsilon}^{[d,0]}(-\tau)|0\rangle_{(0)}+\sin(\theta)\frac{(2\tau)^{\hat{\Delta}^{\prime}_{\epsilon}}}{\sqrt{C^{1}_{\hat{\epsilon}^{\prime}\hat{\epsilon}^{\prime}}}}(\hat{\epsilon}^{\prime})^{[d,0]}(-\tau)|0\rangle_{(0)}\,,$
(5.56)
$|\varphi,\tau\rangle_{(0+)}=\cos(\varphi)\frac{(2\tau)^{\hat{\Delta}_{\sigma}}}{\sqrt{C^{1}_{\hat{\sigma}\hat{\sigma}}}}\hat{\sigma}^{[d,0+]}(-\tau)|0\rangle_{(0+)}+\sin(\varphi)\frac{(2\tau)^{\hat{\Delta}^{\prime}_{\sigma}}}{\sqrt{C^{1}_{\hat{\sigma}^{\prime}\hat{\sigma}^{\prime}}}}(\hat{\sigma}^{\prime})^{[d,0+]}(-\tau)|0\rangle_{(0+)}$
(5.57)

where

$\hat{\Delta}_{\epsilon}=\Delta_{1,2}=\frac{1}{10}\,,\quad\hat{\Delta}^{\prime}_{\epsilon}=\Delta_{1,3}=\frac{3}{5}\,,\quad\hat{\Delta}_{\sigma}=\Delta_{2,2}=\frac{3}{80}\,,\quad\hat{\Delta}_{\sigma}^{\prime}=\Delta_{2,1}=\frac{7}{16}\,.$
(5.58)

The states (5.56), (5.57) are normalised up to finite size suppressed
corrections. As far as the components of these trial vectors are concerned the
contribution from the operator of larger dimension: $\hat{\epsilon}^{\prime}$
or $\hat{\sigma}^{\prime}$ is suppressed because it contains an extra positive
power of $\tau$ that at the minimum of energy is proportional to a negative
power of a coupling. Thus the terms with the higher dimension operator can be
considered as an additional correction to the RG operators $\hat{\epsilon}$,
$\hat{\sigma}$ in addition to the corrections coming from the $\tau$-shift
(which are a series in derivatives).

The variational energy averages in the above states receive contributions from
the off-diagonal matrix elements which at the leading order are expressed via
OPE coefficients. The general form of these contributions can be written as

${}_{\rm ir}\langle
0|(\hat{\psi}^{\prime})^{[0,d]}(-\tau)\psi_{i}(0)\hat{\psi}^{[d,0]}(-\tau)|0\rangle_{\rm
ir}\sim\frac{C_{i\hat{\psi}}^{\hat{\psi}^{\prime}}C_{\hat{\psi}^{\prime}\hat{\psi}^{\prime}}^{1}}{2^{\hat{\Delta}_{\psi}+\hat{\Delta}^{\prime}_{\psi}-\Delta_{i}}\tau^{\hat{\Delta}_{\psi}+\hat{\Delta}^{\prime}_{\psi}+\Delta_{i}}}\,.$
(5.59)

For completeness we calculate the variational energies in the states (5.56),
(5.57) for the generic perturbation

$H=H_{0}-\lambda_{1,3}\psi_{1,3}-\lambda_{1,2}\psi_{1,2}\,.$ (5.60)

It is convenient to rescale the couplings as

$\displaystyle\nu_{12}$ $\displaystyle=$
$\displaystyle\lambda_{12}\,2^{1/10}\left(F_{(2,2),(1,1)}\left[\begin{array}[]{cc}(2,2)&(2,2)\\\
(1,2)&(1,2)\end{array}\right]\right)^{-1/2}\,,$ (5.63) $\displaystyle\nu_{13}$
$\displaystyle=$
$\displaystyle\lambda_{13}\,2^{3/5}\left(F_{(2,2),(1,1)}\left[\begin{array}[]{cc}(2,2)&(2,2)\\\
(1,3)&(1,3)\end{array}\right]\right)^{-1/2}\,.$ (5.66)

We find the following averages

$E^{(+)}(\tau)=\frac{3}{80\tau}-\frac{\nu_{13}}{\tau^{3/5}}+\frac{\nu_{12}}{\tau^{0.1}}\,,$
(5.67)
$E^{(0)}(\tau,\theta)=\frac{1}{20\tau}(7-5\cos(2\theta))+\frac{2\nu_{13}\cos(2\theta)}{\tau^{3/5}}-\frac{\nu_{12}\sqrt{2/3}\sin(2\theta)}{\tau^{0.1}}\,,$
(5.68) $\displaystyle
E^{(0+)}(\tau,\varphi)=\frac{1}{\tau}\left(\frac{3}{80}\cos^{2}(\varphi)+\frac{7}{16}\sin^{2}(\varphi)\right)+\cos^{2}(\varphi)\frac{\sqrt{5}-1}{2}\left(\frac{\nu_{13}}{\tau^{3/5}}+\frac{\nu_{12}}{\tau^{0.1}}\right)$
$\displaystyle-\sin(2\varphi)\left(\frac{b\nu_{12}}{\tau^{0.1}}+\frac{a\nu_{13}}{\tau^{3/5}}\right)$
(5.69)

where

$\displaystyle a=F_{(2,2),(2,1)}\left[\begin{array}[]{cc}(2,2)&(1,3)\\\
(1,3)&(2,2)\end{array}\right]\left(\frac{F_{(2,2),(1,1)}\left[\begin{array}[]{cc}(1,3)&(1,3)\\\
(2,1)&(2,1)\end{array}\right]}{F_{(2,2),(1,1)}\left[\begin{array}[]{cc}(1,3)&(1,3)\\\
(2,2)&(2,2)\end{array}\right]}\right)^{1/2}$ (5.76)
$\displaystyle=-\frac{15\Gamma(6/5)\sqrt{(3-\sqrt{5})\Gamma(4/5)}}{2\Gamma^{3/2}(2/5)}\approx-1.9658$
(5.77)

and

$\displaystyle b=F_{(2,2),(2,1)}\left[\begin{array}[]{cc}(2,2)&(1,3)\\\
(1,2)&(2,2)\end{array}\right]\left(\frac{F_{(2,2),(1,1)}\left[\begin{array}[]{cc}(1,3)&(1,3)\\\
(2,1)&(2,1)\end{array}\right]}{F_{(2,2),(1,1)}\left[\begin{array}[]{cc}(1,3)&(1,3)\\\
(2,2)&(2,2)\end{array}\right]}\right)^{1/2}$ (5.84)
$\displaystyle=\frac{2^{\frac{1}{20}}((5-\sqrt{5})\pi)^{3/4}\sqrt{-3\Gamma(-3/10)}}{25\Gamma(7/10)\Gamma(7/5)}\approx
0.6552$ (5.85)

For completeness we also included above $E^{(+)}(\tau)$ – the variational
energy linking $(d)$ and $(+)$ that does not receive any additional terms
because there is a unique boundary condition changing primary.

To minimise the energy one can either minimise the above expressions in both
$\tau$ and $\theta$ or find the eigenvalues of the corresponding $2\times 2$
matrix and minimise them in $\tau$. Here we focus on the pure $\psi_{1,3}$ and
pure $\psi_{1,2}$ perturbations to compare with the results obtained in
sections 5.6, 5.7.

As the new ansatze (5.56), (5.57) contain the old trial vectors the new
variational energies will be lower in comparison to the ones tabulated in
Tables 2, 3, 4. We define the dimensionless factors for the new trial energies
similarly to (5.49) as

$E_{\rm
min}^{(0)}=e(\hat{\epsilon},\hat{\epsilon}^{\prime})|\lambda|^{\frac{1}{1-\Delta^{\rm
uv}}}\,,\quad E_{\rm
min}^{(0+)}=e(\hat{\sigma},\hat{\sigma}^{\prime})|\lambda|^{\frac{1}{1-\Delta^{\rm
uv}}}$ (5.86)

where for $\lambda$ we take $\lambda_{13}$ for pure $\psi_{1,3}$ flows and
$\lambda_{12}$ for pure $\psi_{1,3}$ flows. If there is no local minimum we
set the corresponding factor to zero. For the pure $\psi_{1,3}$ flows we
obtain the same values for $e(\hat{\epsilon},\hat{\epsilon}^{\prime})$ as
before while

$e(\hat{\sigma},\hat{\sigma}^{\prime})=-2.39\,,\quad\mbox{for
}\lambda_{13}>0\,,\qquad
e(\hat{\sigma},\hat{\sigma}^{\prime})={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-12.64}\,,\quad\mbox{for
}\lambda_{13}<0$ (5.87)

with two decimal places retained. The corresponding factors for $E_{\rm
min}^{(-0)}$ are the same. The value of
$e(\hat{\sigma},\hat{\sigma}^{\prime})$ for the negative coupling has changed
by a large amount from $e(\sigma)=-2.80$. Moreover, it is now lower than the
value for the correct endpoint $(0)$ which is $e(\hat{\epsilon})=-12.11$. This
presents a problem. We do know from TBA and TCSA numerics that the correct end
point for this flow is $(0)$. We also know from the TCSA results presented in
section 6 that $\hat{\epsilon}$ is the correct RG operator. It may be that
further local corrections to the interface operator are needed to be taken
into account, e.g, the $(L_{-2}\hat{\psi})$ descendant field, or that non-
local corrections are present. We plan to investigate this further in future
work.

For the pure $\psi_{1,2}$ flows we obtain
$e(\hat{\epsilon},\hat{\epsilon}^{\prime})=-0.73$ for $\lambda_{12}\neq 0$ and

$e(\hat{\sigma},\hat{\sigma}^{\prime})=-0.35\,,\quad\mbox{for
}\lambda_{12}>0\,,\qquad
e(\hat{\sigma},\hat{\sigma}^{\prime})=-1.06\,,\quad\mbox{for
}\lambda_{12}<0\,.$ (5.88)

(The two values are swapped for $(-0)$.) Comparing these new values to the
ones in Table 4 we see that the winning trial state remains the same.

It would be very interesting to locate the black lines on the diagramme on
Figure 6. We have done this numerically using TCSA approach and will present
the results elsewhere. As far as the variational method is concerned we feel
that one needs first to understand the above mentioned problem with the
$(d)\rightarrow(0)$ flow and the trial state (5.57) before trying to apply the
method to the genuine two-coupling flows. However, even without doing the
variational calculations, we can say that if the flow exists the corresponding
RG operator must be the operator $\hat{\sigma}^{[0+,d]}$. This is because
there are only two primaries linking the two fixed points: $\sigma$ and
$\sigma^{\prime}$, and the $\sigma^{\prime}$ primary OPE with itself only
produces the identity and the $\epsilon^{\prime\prime}$ fields.

###  5.9 The eigenvector equation

Some insight into the general structure of the vacuum state can be gained by
looking at the eigenvector equation. We discussed such equations in section 2.
The eigenvector equation in the perturbed theory can be written as

$\lim_{\epsilon\to\,+\,0}\Bigl{[}H_{0}-\lambda\psi(\epsilon)+\mbox{Counterterms}\Bigr{]}\hat{\psi}^{[0,\lambda]}(0)|{\cal
E}_{I}\rangle_{\lambda}={\cal E}_{I}\hat{\psi}^{[0,\lambda]}(0)|{\cal
E}_{I}\rangle_{\lambda}\,.$ (5.89)

In the variational method we try to approximate the vacuum state
$\hat{\psi}^{[0,\lambda]}(0)|0\rangle_{\lambda}$ by a state in the IR BCFT.
This approximation can be written as

$\hat{\psi}^{[0,\lambda]}(0)|0\rangle_{\lambda}=\hat{\psi}^{[\rm
uv,ir]}(0)|0\rangle_{\rm ir}+C|\lambda|^{\frac{1}{\Delta^{\rm
uv}-1}}\partial_{\tau}\hat{\psi}^{[\rm uv,ir]}(0)|0\rangle_{\rm ir}+\dots$
(5.90)

where $C$ is a constant and the omitted terms all contain higher powers of the
inverse correlation length and may contain local operators such as higher
Virasoro descendants of the RG operator as well as non-local operators such as
Virasoro modes (that act on $|0\rangle_{\rm ir}$ and thus give different
states in the IR BCFT on which the local boundary condition changing operators
act). When we act on such a state by the perturbed Hamiltonian
$H_{0}-\lambda\psi(0)$ the divergences come from collision of $\psi$ with the
local operators present in the above expansion. Assuming the divergences
correspond to the divergence in the vacuum energy ${\cal E}_{0}$ (as can be
demonstrated perturbatively) they should be cancelled by counterterms
proportional to the eigenvector itself. This may be possible only if
$\hat{\psi}^{[\rm uv,ir]}$ is a primary. If it wasn’t then the most singular
terms in the OPE with $\psi$ would come with the primary underlying
$\hat{\psi}^{[\rm uv,ir]}$ which would not be in the eigenvector itself.

The eigenvector equation also tells us that if the OPE of $\psi$ with
$\hat{\psi}^{[\rm uv,ir]}$ contains other primary states which are singular
then the corresponding primary must also be present in the expansion (5.90).
This is what we tried to incorporate in the extended variational ansatze
(5.56), (5.57). We also note that the RG operators identified for the pure
$\psi_{1,3}$ and pure $\psi_{1,2}$ flows in sections 5.5, 5.6, 5.7 are all
closed under the action of the relevant perturbing operator that is the OPE
contains only terms from the same primary tower. In general the requirement
that the OPE of the perturbing operator $\psi$, which is a primary or a linear
combination of different primaries, with the RG operator must contain the RG
operator itself is essentially equivalent to our conjecture 2. The insight
based on the eigenvector equation even suggests a stronger version of that
conjecture: the term in the OPE

$\psi_{i}(0)\hat{\psi}^{[\rm
uv,ir]}(-\tau)\sim\frac{D_{i}}{\tau^{\Delta_{i}^{\rm uv}}}\hat{\psi}^{[\rm
uv,ir]}(-\tau)+\dots$ (5.91)

containing the RG operator must be the most singular term.

##  6 Some numerical results

The truncated conformal space approach (TCSA) is a numerical method put
forward in [36], [37] which allows one to find approximately the spectrum of
perturbed 2D CFTs. It was adapted to boundary RG flows in [38]. The
eigenvalues and their degeneracies in particular allow one to identify the IR
fixed point of the flow. Here we investigate the boundary flows in TIM
focusing on the numerical eigenvectors for the vacuum and the first excited
state. We choose the spectator boundary condition on the strip to be $(-)$
that corresponds to the identity primary state. The state space of the UV BCFT
then has a single Virasoro tower of states. Let $|0\rangle_{\rm uv}$ and
$|0\rangle_{\rm ir}$ denote the vacuum states on the strip in the UV and the
IR state spaces respectively. Assuming that $|0\rangle_{\rm uv}$ is a primary
of positive weight, which with the $(-)$ spectator is true when the UV BCFT
has a relevant operator and thus must be one of $(d)$, $(0+)$, $(-0)$, the
image of the IR vacuum in the UV state space can be expanded in components of
increasing weights as

$\hat{\psi}^{\rm[uv,ir]}(0)|0\rangle_{\rm ir}=C_{0}|0\rangle_{\rm
uv}+C_{1}L_{-1}|0\rangle_{\rm uv}+D_{1}L_{-1}^{2}|0\rangle_{\rm
uv}+D_{2}L_{-2}|0\rangle_{\rm uv}+\dots$ (6.1)

While the normalisations of these states depend on the normalisation of the RG
operator, the following ratios of the lowest components are independent of the
overall normalisation171717Of course one can also consider $D_{1}/C_{0}$ it is
just a matter of convenience that dictated our choice of the ratios for which
we present results.:

$\Gamma^{1}_{\rm vac}=\frac{C_{1}}{C_{0}}\,,\qquad\Gamma^{2}_{\rm
vac}=\frac{D_{2}}{C_{0}}$ (6.2)

Given the RG operator we can calculate the values of these ratios using the
mapping of the RG field from the half plane to the strip. To find the matrix
elements of $\hat{\psi}^{\rm[ir,uv]}(0)$ we note that the conformal
transformations that preserve the insertion point are generated by

$l_{n}=L_{n}-L_{0}\,.$ (6.3)

For a primary $\hat{\psi}^{\rm[ir,uv]}$ inserted at $z=1$ on the upper half
plane the commutation relations with the generators $l_{n}$ are

$l_{n}^{\rm ir}\hat{\psi}^{\rm[ir,uv]}(1)-\hat{\psi}^{\rm[ir,uv]}(1)l_{n}^{\rm
uv}=n\hat{\Delta}\hat{\psi}^{\rm[ir,uv]}(1)\,.$ (6.4)

This equation can be interpreted as a gluing condition for the UV and IR
generators preserving the position of the interface. The term on the right
hand side of (6.4) means that the generators are glued up to a central
element181818This neat interpretation holds only when the RG operator is a
primary which we believe is always the case.. Using (6.4) we find

$\Gamma^{1}_{\rm vac}=\frac{\Delta^{\rm uv}_{0}-\Delta^{\rm
ir}_{0}+\hat{\Delta}}{2\Delta^{\rm uv}_{0}}\,,$ (6.5) $\Gamma^{2}_{\rm
vac}=-3\frac{(\hat{\Delta}-\Delta^{\rm ir}_{0}+\Delta^{\rm
uv}_{0})(\hat{\Delta}-\Delta^{\rm ir}_{0}+\frac{1}{3}(1-\Delta^{\rm
uv}_{0}))-\frac{2}{3}\hat{\Delta}(2\Delta^{\rm uv}_{0}+1)}{c(1+2\Delta^{\rm
uv}_{0})+2\Delta^{\rm uv}_{0}(17\Delta^{\rm uv}_{0}-5)}$ (6.6)

where $c$ is the central charge of the bulk CFT which is equal to $7/10$ for
TIM.

We can also consider ratios of components in the excited states. If
$|0\rangle_{\rm ir}$ is a primary with a positive weight then the first
excited state in the IR theory is $|1\rangle_{\rm ir}=L_{-1}|0\rangle_{\rm
ir}$ and its image in the UV theory is

$\hat{\psi}^{\rm[uv,ir]}(0)|1\rangle_{\rm ir}=C_{0}^{(1)}|0\rangle_{\rm
uv}+C_{1}^{(1)}L_{-1}|0\rangle_{\rm uv}+\dots$ (6.7)

We define a ratio

$\Gamma^{1}_{\rm 1}=\frac{C_{1}^{(1)}}{C_{0}^{(1)}}$ (6.8)

for which the theoretical value is

$\Gamma^{1}_{\rm 1}=\Gamma^{1}_{\rm vac}+\frac{1}{2\Delta^{\rm
uv}_{0}}\left(\frac{2\Delta^{\rm ir}_{0}}{\Delta^{\rm
ir}_{0}+\hat{\Delta}-\Delta^{\rm uv}_{0}}-1\right)\,.$ (6.9)

In the case when $|0\rangle_{\rm ir}$ has zero weight, which happens e.g. for
a flow into the $(-)$ boundary condition, the first excited state is
$|1\rangle_{\rm ir}=L_{-2}|0\rangle_{\rm ir}$ and the predicted value of the
ratio is

$\Gamma^{1}_{\rm 1}=\frac{\hat{\Delta}+\Delta^{\rm uv}_{0}-2}{2\Delta^{\rm
uv}_{0}}\,.$ (6.10)

In TCSA the infinite dimensional state space of a BCFT is truncated to a
finite dimensional subspace of states with the conformal weight being less
than a cutoff value. Empirically one finds that the lowest energy eigenstates
are the best approximated in this sheme. Moreover the lowest weight components
of the numerical eigenvectors are the most reliable. Hence the above ratios
are the observables we hope are best approximated numerically. On the other
hand the theoretical values of the ratios change significantly when we change
the tentative RG operators. Hence the numerical values can be used to confirm
the RG operators of particular flows. A similar strategy was used in [12] to
identify the conformal boundary states giving the RG interfaces for massive
bulk flows. For the boundary flows the RG operator should be independent of
the choice of the spectator that can be also tested numerically. The plots of
the vacuum ratios versus the coupling for 3 different boundary flows in TIM
are presented on Figures 7, 8, 9.

Figure 7: The component ratios in the vacuum state for the $(d)\rightarrow(0)$
flow. The red dashed lines mark the predicted values: $\Gamma_{\rm
vac}^{1}=-4$, $\Gamma_{\rm vac}^{2}=\frac{640}{1361}\approx 0.47$. The
dimension of the truncated state space is indicated in the legend.

As in the bulk case, where similar ratios were discussed in [12], we observe a
rather slow power-like convergence in the coupling. At the very large coupling
one expects the truncation errors to be large, however empirically TCSA tends
to have bounded errors even for very large couplings. On the plots we can see
that for some flows the ratios get closer to the predicted values in the
asymptotic region for example for the $(d)\rightarrow(0)$ flow, while for the
$\Gamma_{\rm vac}^{2}$ ratio in the $(d)\rightarrow(+)\oplus(-)$ flow the best
fit happens near a local minimum after which the ratio diverges linearly from
the theoretical value. This fits well with the observation that there is a
”flow beyond” present in the second case but not in the first one. The TCSA
numerics in the second flow behaves as if the theory flows back to the UV
fixed point. This phenomenon for boundary flows was first observed in [39] and
discussed more recently in [40]. For the $\psi_{1,2}$ flows the asymptotic
behaviour is monotonic with the approch to theoretical values being even
slower than for the $\psi_{1,3}$ flows. The dimensions of the truncated state
spaces are indicated in the legend for each plot. For all quantities we see
the movement towards the theoretical values for increased size of the
truncated state space. For the $\psi_{1,2}$ flows this improvement is only
visible for very large values of the coupling.

Figure 8: The component ratios in the vacuum state for the
$(d)\rightarrow(+)\oplus(-)$ flow. The red dashed lines mark the predicted
values: $\Gamma_{\rm vac}^{1}=1$, $\Gamma_{\rm vac}^{2}=0$.

Figure 9: The component ratios in the vacuum state for the $(d)\rightarrow(-)$
flow. The red dashed lines mark the predicted values: $\Gamma_{\rm
vac}^{1}=1$, $\Gamma_{\rm vac}^{2}=0$.

On Figures 10, 11 we present plots of the ratio of the first two components in
the first excited energy eigenvector. They have similar features to the plots
of the vacuum ratios.

Figure 10: The component ratio $\Gamma_{1}^{1}$ of the first excited state for
the $(d)\rightarrow(0)$ (left) and $(d)\rightarrow~{}(+)\oplus(-)$ (right)
flows. The red dashed lines mark the predicted values: $\Gamma_{1}^{1}=6$ and
$\Gamma_{1}^{1}=-19$. For the second flow the first excited IR state is the
$\epsilon^{\prime\prime}$ primary.  Figure 11: The component ratio
$\Gamma_{1}^{1}$ of the first excited state for the $(d)\rightarrow(-)$ flow.
The red dashed line marks the predicted value
$\Gamma_{1}^{1}=-\frac{77}{3}\approx-25.6$.

##  7 Concluding remarks

Our main non-perturbative tool for using RG operators has been the variational
method we developed in section 5. The method assumes a particular ansatz for
the perturbed theory vacuum state based on the RG operator. We think that to
make further progress with RG operators the method needs to be developed
further. The main technical advantage of the method is that its key ingredient
is a local operator and we could use OPE to derive variational energies. It
seems to be important to understand to what extent such local terms capture
the true vacuum and how one can systematically improve such an ansatz. Such
general understanding should clarify the puzzle with the superpositions we
pointed at in section 5.8. The same remarks go for the variational method of
[13] where a similar puzzle was also noted for the case of bulk massive flows
in the Ising field theory. Provided that we understand better the variational
method and know how to improve it, it would be interesting to use it to gain
insight into the special critical lines like the ones designated by the black
arrows on the diagram on Figure 6. Such critical lines appear to be important
features of the spaces of RG flows. We hope to return to these questions in
future work.

Another setup in which it would be interesting to consider the RG operators is
that of spliced flows. Suppose we have pairwise flows between 3 fixed point as
on Figure 12. We assume further that there is a continuous family of flows
between ${\rm BCFT_{1}}$ and ${\rm BCFT_{3}}$ such that in a limit one obtains
a spliced flow that is a concatenation of flow from ${\rm BCFT_{1}}$ to ${\rm
BCFT_{2}}$ with the flow from ${\rm BCFT_{2}}$ to ${\rm BCFT_{3}}$. Such a
spliced flow is not a real RG flow but it does belong to the boundary of the
space of true flows (see e.g. [44] for a nice exposition of such matters). Let
$\hat{\psi}^{[2,1]}$ and $\hat{\psi}^{[3,2]}$ be the RG operator for each
component flow in the spliced flow. Assume further that all flows in the
family of flows between ${\rm BCFT_{1}}$ and ${\rm BCFT_{3}}$ have the same RG
operator: $\hat{\psi}^{[3,1]}$. (For the theories along these flows near the
IR fixed point the interface operators can differ in the subleading components
as in our discussion of superpositions in section 5.8.)

${\rm BCFT_{1}}$${\rm BCFT_{2}}$${\rm
BCFT_{3}}$$\dots$$\hat{\psi}^{[2,1]}$$\hat{\psi}^{[3,2]}$$\hat{\psi}^{[3,1]}$
Figure 12: A space of RG flows between 3 fixed points. The associated RG
operators are put next to the arrows. Here we assume that there is a family of
flows between ${\rm BCFT_{1}}$ and ${\rm BCFT_{3}}$ for which the spliced flow
that passes through ${\rm BCFT_{2}}$ appears as a singular limit.

It is tempting to conjecture then that given such a setup the OPE of the first
two RG operators must contain the third:

$\hat{\psi}^{[3,2]}(0)\hat{\psi}^{[2,1]}(0)\sim
C_{32,21}^{31}\tau^{\hat{\Delta}_{31}-\hat{\Delta}_{32}-\hat{\Delta}_{21}}\hat{\psi}^{[3,1]}(0)+\dots\qquad
C_{32,21}^{31}\neq 0\,.$ (7.1)

The situation depicted on Figure 12 is realised in the space of boundary flows
in TIM with the role of ${\rm BCFT_{1}}$ played by $(d)$, the role of ${\rm
BCFT_{2}}$ by $(0+)$ and that of ${\rm BCFT_{3}}$ by $(0)$. As argued in the
end of section 5.8 the RG operator for the flow from $(d)$ to $(0+)$ (which we
were able to locate numerically using TCSA) must be $\hat{\sigma}^{[0+,d]}$.
For the other two flows we found the RG operators in section 5.6, they are
$\hat{\epsilon}^{[0,d]}$ and $\hat{\sigma}^{[0,0+]}$ and the rule (7.1) indeed
holds. It would be interesting to investigate this conjecture further.

Although we did discuss the bulk RG interfaces in section 3 the focus of the
paper is on the boundary RG interfaces. Our conjectures 2 and 3 can be
generalised to the bulk case as follows. Instead of fusing the RG interface
line with itself, that would be the direct analogue of the OPEs in conjectures
2 and 3, we can place the perturbing operator close to the RG interface and
perform the bulk to boundary OPE. It is natural then to require, especially
using the intuition from the eigenvector equation considered in section 5.9,
that this OPE contains the identity operator. Equivalently we can say that the
UV operator must have a non-vanishing one-point function in the presence of
the RG interface. A similar condition holds for the massive flows as follows
from Cardy’s variational method in which the interaction term comes with the
one point function of the perturbing operator in the conformal boundary
condition giving the vacuum state. We can also formulate the same condition
for the leading IR operator bringing it close to the interface on the IR side.
The principle obstacle in making these conjectures useful in the bulk case is
our poor knowledge of conformal interfaces. It is possible though that in the
context of topological or supersymmetric QFTs (in two dimensions or higher)
these conjectures can lead to some interesting insights. The RG interfaces for
such theories have been studied in [43], [41], [42].

Acknowledgements

The TCSA numerical results presented in this paper have been obtained using
Wolfram’s Mathematica package (version 10.2.0.0). I thank Ingo Runkel for
sharing his Mathematica code for calculating fusion matrices. I also thank
Matthew Buican, Marco Meineri, Cornelius Schmidt-Colinet and Gerard Watts for
stimulating discussions.

## References

  * [1] A. Zamolodchikov, “Irreversibility” of the flux of the renormalization group in a 2D field theory,
  * [2] C. Vafa, c-Theorem and the topology of 2D QFTs, Phys. Lett. B212 (1988) 28.
  * [3] S. Gukov, Counting RG flows, JHEP01 (2016) 020; arXiv:1503.01474.
  * [4] S. Gukov, RG Flows and Bifurcations, Nucl.Phys. B919 (2017) 583; arXiv:1608.06638.
  * [5] I. Brunner and D. Roggenkamp, Defects and bulk perturbations of boundary Landau-Ginzburg orbifolds, JHEP 0804 (2008) 00; arXiv:0712.0188.
  * [6] S. Fredenhagen and T. Quella, Generalised permutation branes, JHEP 11 (2005) 004; arXiv:hep-th/0509153.
  * [7] D. Gaiotto, Domain walls for two-dimensional renormalization group flows, JHEP 12 (2012) 103; arXiv:1201.0767.
  * [8] A. B. Zamolodchikov, Renormalization group and perturbation theory about fixed points in two-dimensional field theory, Sov. J. Nucl. Phys. 46 (1987) 1090.
  * [9] R. Poghossian, Two dimensional renormalization group flows in next to leading order, JHEP 01 (2014)167; arXiv:1303.3015.
  * [10] A. Poghosyan, H. Poghosyan, Mixing with descendant fields in perturbed minimal CFT models, JHEP 10 (2013)131; arXiv:1305.6066.
  * [11] M. Oshikawa and I. Affleck, Boundary conformal field theory approach to the critical two-dimensional Ising model with a defect line, Nucl. Phys. B495 (1997) 533; cond-mat/9612187.
  * [12] A. Konechny, RG boundaries and interfaces in Ising field theory, J. Phys. A50 (2017) 145403; arXiv:1610.07489.
  * [13] J. Cardy, Bulk Renormalization Group Flows and Boundary States in Conformal Field Theories, SciPost Phys. 3, 011 (2017); arXiv:1706.01568.
  * [14] I. Runkel, Boundary structure constants for the A-series Virasoro minimal models, Nucl. Phys. B549 (1999) 563; arXiv:hep-th/9811178.
  * [15] D. Lewellen, Sewing constraints for conformal field theories on surfaces with boundaries, Nucl. Phys. B372 (1992) 654.
  * [16] K. Graham, On Perturbations of Unitary Minimal Models by Boundary Condition Changing Operators, JHEP 03 (2002) 028; arXiv:hep-th/0111205.
  * [17] D. Friedan and A. Konechny, Curvature formula for the space of 2-d conformal field theories, JHEP 09 (2012) 113; arXiv:1206.1749.
  * [18] J. Cardy, Boundary Conditions, Fusion Rules and the Verlinde Formula, Nucl. Phys. B324 (1989) 581-596.
  * [19] R. Behrend, P. Pearce, V. Petkova, and J. -B. Zuber, On the Classification of Bulk and Boundary Conformal Field Theories, Phys. Lett. B444 (1998) 163; arXiv:hep-th/9809097.
  * [20] R. Behrend, P. Pearce, V. Petkova, and J. -B. Zuber, Boundary Conditions in Rational Conformal Field Theories, Nucl. Phys. B579 (2000) 707; arXiv:hep-th/9908036.
  * [21] I. Affleck, Edge Critical Behaviour of the 2-Dimensional Tri-critical Ising Model, J. Phys. A33 (2000) 6473; arXiv:cond-mat/0005286.
  * [22] A. Konechny, Renormalisation group defects for boundary flows, J. Phys. A46 (2013) 145401; arXiv:1211.3665.
  * [23] A. Konechny, Critical Ising Model with Boundary Magnetic Field: RG Interface and Effective Hamiltonians, JHEP 04 (2019) 001; arXiv:1811.07599.
  * [24] F. A. Berezin, The method of second quantization, Academic Press, 1966.
  * [25] F. Constantinescu and R. Flume, The convergence of strongly relevant perturbations of $d=2$ conformal field theories, Phys. Lett. B326 (1994) 101.
  * [26] A. Recknagel, D. Roggenkamp and V. Schomerus, On relevant boundary perturbations of unitary minimal models, Nucl. Phys. B588 (2000) 552; arXiv:hep-th/0003110.
  * [27] F. Lesage, H. Saleur and P. Simonetti, Boundary flows in minimal models, Phys. Lett. B427 (1998) 85; arXiv:hep-th/9802061.
  * [28] K. Graham and G. M. T. Watts, Defect Lines and Boundary Flows, JHEP 04 (2004) 019; arXiv:hep-th/0306167.
  * [29] R. Guida and N. Magnoli, All order I.R. finite expansion for short distance behavior of massless theories perturbed by a relevant operator, Nucl. Phys. B471 (1996) 361-388; arXiv: hep-th/9511209.
  * [30] A. Cappelli and J. I. Latorre, Perturbation theory of higher-spin conserved currents off criticality, Nucl. Phys. B340 (1990) 659.
  * [31] K. Graham, I. Runkel and G. Watts, Minimal model boundary flows and c=1 CFT, Nucl. Phys. B608 (2001) 527; arXiv:hep-th/0101187.
  * [32] I. Affleck and A.W.W. Ludwig, Universal non integer ‘ground state degeneracy’ in critical quantum systems, Phys. Rev. Lett 67 (1991) 161.
  * [33] D. Friedan and A. Konechny, Boundary entropy of one-dimensional quantum systems at low temperature, Phys. Rev. Lett. 93 (2004) 030402; arXiv:hep-th/0312197.
  * [34] I. Runkel, Boundary Problems in Conformal Field Theory, PhD Thesis, https://www.math.uni-hamburg.de/home/runkel/PDF/phd.pdf.
  * [35] P. Giokas, Mean field theory for boundary Ising and tricritical Ising models, arXiv:1106.1843.
  * [36] V. Yurov and Al. Zamolodchikov, Truncated Conformal Space Approach To Scaling Lee-yang Model, Int. J. Mod. Phys. A5 (1990) 3221-3246.
  * [37] V. Yurov and Al. Zamolodchikov, Truncated fermionic space approach to the critical 2-D Ising model with magnetic field, Int. J. Mod. Phys. A6 (1991) 4557-4578.
  * [38] P. Dorey, A. Pocklington, R. Tateo, and G. Watts, TBA and TCSA with boundaries and excited states, Nucl. Phys. B525 (1998) 641; arXiv:hep-th/9712197.
  * [39] G. Watts, On the renormalisation group for the boundary Truncated Conformal Space Approach, Nucl. Phys. B859 (2012) 177-206; arXiv:1104.0225.
  * [40] A. Konechny and D. McAteer, On asymptotic behaviour in truncated conformal space approach JHEP 09 (2019) 044; arXiv:1904.09616.
  * [41] F. Klos and D. Roggenkamp, Realizing IR theories by projections in the UV , JHEP 01 (2020)097; arXiv:1907.12339.
  * [42] F. Klos and D. Roggenkamp, Complementary Projection Defects and Decompositions, arXiv:2006.08961.
  * [43] T. Dimofte, D. Gaiotto, and R. van der Veen, RG Domain Walls and Hybrid Triangulations, Adv. Theor. Math. Phys. 19 (2015) 137; arXiv:1304.6721.
  * [44] R. L. Cohen, K. Iga, and P. Norbury, Topics in Morse theory: lecture notes, unpublished, available online at:
http://math.stanford.edu/ ralph/morsecourse/biglectures.pdf