Ising model in a boundary magnetic field with random discontinuities

Anatoly Konechny

Department of Mathematics, Heriot-Watt University
Edinburgh EH14 4AS, United Kingdom
and
Maxwell Institute for Mathematical Sciences
Edinburgh, United Kingdom
E-mail<EMAIL_ADDRESS>

We consider a two-dimensional Ising field theory on a space with boundary in
the presence of a piecewise constant boundary magnetic field which is allowed
to change value discontinuously along the boundary. We assume zero magnetic
field in the bulk. The positions of discontinuities are averaged over as in
the annealed disorder. This model is described by a boundary field theory in
which a superposition of the free spin boundary condition is perturbed by a
collection of boundary condition changing operators. The corresponding
boundary couplings give the allowed constant values of the magnetic field as
well as the fugacities for the transitions between them. We show that when the
value of the magnetic field is allowed to take only two different values which
are the same in magnitude but have different signs the model can be described
by a quadratic Lagrangian. We calculate and analyse the exact reflection
matrix for this model. We also calculate the boundary entropy and study in
detail the space of RG flows in a three-parameter space and with four
different infrared fixed points. We discuss the likely breakdown of
integrability in the extended model which allows for two generic values of the
boundary magnetic field, backing it by some calculations.

##  1 Introduction

The two-dimensional Ising model on a half-plane with a boundary magnetic field
was considered in [1] and later in [2]. In the continuum limit this model was
first studied in [3] where it was written as a free fermion field theory with
a particular boundary interaction term. As the boundary term is also quadratic
in the bulk and the boundary fermion fields many quantities can be calculated
exactly. Various aspects of this boundary field theory were subsequently
studied in [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19].

The critical Ising model with a boundary magnetic field can be described as a
perturbed boundary conformal field theory (BCFT). The critical Ising model is
described as the Virasoro minimal model ${\cal M}(3,4)$ and it has three
conformal boundary conditions corresponding to the free or fixed boundary
spins in the underlying lattice model. The free boundary condition has a
relevant boundary operator $\sigma_{B}$ of dimension 1/2 that describes the
boundary spin. Let $S_{\rm BCFT}$ be the Euclidean action that describes the
critical Ising model with free spin boundary condition on the upper half plane
and let $z=x+iy$, $\bar{z}=x-iy$ be the complex coordinates. Switching on the
boundary magnetic field amounts to deforming the action as

$S_{h}=S_{\rm
BCFT}+h\\!\int\limits_{-\infty}^{\infty}\\!\\!\sigma_{B}(x)dx\,.$ (1.1)

Here $h$ stands for the value of the boundary magnetic field and is a relevant
coupling. The RG flow interpolates between the free and fixed spin boundary
conditions.

The model (1.1) can be generalised to a perturbation of a superposition of
$N>1$ free boundary conditions. Let $i=1,\dots,N$ be the index labelling a
copy of the free boundary condition. The boundary primaries on the
superposition are dimension 1/2 fields $\sigma_{B}^{[ij]}$ and dimension zero
fields $\chi^{[ij]}$ with $i,j=1,\dots,N$. Our conventions are such that the
field insertion $\sigma_{B}^{[ij]}(x)$ has the $i$-th copy of the boundary
condition to the right of $x$ and the $j$-th copy to the left, and similarly
for $\chi^{[ij]}$. We can then consider the most general relevant deformation

$S_{h_{ij},m_{ij}}=S_{\rm
BCFT}+\sum_{i,j=1}^{N}[h_{ij}\\!\\!\int\limits_{-\infty}^{\infty}\\!\\!\sigma_{B}^{[ij]}(x)dx+m_{ij}\\!\\!\int\limits_{-\infty}^{\infty}\\!\\!\chi^{[ij]}(x)dx]\,.$
(1.2)

Here the coupling constants are arranged into two $N\times N$ matrices:
$h_{ij}$ and $m_{ij}$. For the perturbed theory to be unitary these matrices
have to be hermitian.

The physical interpretation of the fields $\sigma_{B}^{[ij]}$ and
$\chi^{[ij]}$ may not be immediately clear. By taking the $N$ copies of the
free boundary condition we have now endowed the boundary with additional
degrees of freedom. As in the Kondo model we could try to interpret these
degrees of freedom as those of an impurity particle located at the boundary.
Thus in the BCFT approach to Kondo model (see e.g. [20]) the fields
$\chi^{[ij]}$ describe the impurity spin degrees of freedom. As far as we can
see this interpretation does not help with giving a physical meaning to (1.2).
Instead we are offering a different kind of physical interpretation in terms
of a non-constant boundary magnetic field with a fluctuating profile. As we
will explain below the eigenvalues of $h_{ij}$ give possible values of the
boundary magnetic field while the values of the couplings $m_{ij}$ are related
to statistical weights of different profiles.

Let us consider a deformed correlation function corresponding to (1.2) with
some local insertions and some choice of vacua at $x\to\pm\infty$

$\langle\dots\rangle_{h,m;IJ}=\langle\exp(-S_{h_{ij},m_{ij}})\dots\rangle_{0;IJ}\,.$
(1.3)

Here the ellipsis stands for some local operator insertions and
$\langle\enspace\rangle_{0;IJ}$ denotes the correlator in the unperturbed BCFT
with the $I$-th copy of the boundary condition at $x\to\infty$ and the $J$-th
copy at $x\to-\infty$. We can expand the exponential to obtain a series of
boundary conformal perturbation theory. For boundary condition changing
operators it was considered in [21]. We choose the normalisations of the
$\chi^{[ij]}$ fields so that they satisfy the OPEs

$\displaystyle\chi^{[ij]}(x_{2})\chi^{[kl]}(x_{1})$ $\displaystyle=$
$\displaystyle\delta_{kj}\chi^{[il]}(x_{1})\,,$
$\displaystyle\sigma_{B}^{[ij]}(x_{2})\chi^{[kl]}(x_{1})$ $\displaystyle=$
$\displaystyle\delta_{kj}\sigma_{B}^{[il]}(x_{1})\,,$
$\displaystyle\chi^{[ij]}(x_{2})\sigma_{B}^{[kl]}(x_{1})$ $\displaystyle=$
$\displaystyle\delta_{kj}\sigma_{B}^{[il]}(x_{1})$ (1.4)

where $x_{1}<x_{2}$. We also have a selection rule

$\sigma_{B}^{[ij]}(x_{2})\sigma_{B}^{[kl]}(x_{1})=\delta_{jk}[\sigma_{B}(x_{2})\sigma_{B}(x_{1}]^{[il]}\,,\quad
x_{1}<x_{2}$ (1.5)

where the expression on the right hand side means that we should take the OPE
in a single copy of the free spin boundary condition and then add the Chan-
Paton indices $il$ to every term. The above OPEs considered inside the
conformal perturbation expansion imply that the series can be rewritten as a
path-ordered exponential

$\langle\dots\rangle_{h,m;IJ}=\langle{\rm Pexp}(-S_{\rm
BCFT}-[\int\limits_{-\infty}^{\infty}\\!\\!H(x)\sigma_{B}(x)dx+\int\limits_{-\infty}^{\infty}\\!\\!M(x)dx])_{IJ}\dots\rangle_{0}\,.$
(1.6)

where $H(x)$ and $M(x)$ stand for the $h_{ij}$ and $m_{ij}$ matrices inserted
at $x$ and the path ordering applies to these insertions. The correlator on
the right hand side: $\langle\dots\rangle_{0}$ now refers to the VEV in a
single copy of the free spin boundary condition.

The unperturbed BCFT possesses the ${\rm U}(N)$ Chan-Paton symmetry generated
by the charges $\chi^{[ij]}$. This symmetry is explicitly broken by the
perturbation (1.2) that results in emergence of some redundant boundary
operators $\partial_{x}\chi^{[ij]}$ (see e.g. appendix A in [22] for a more
detailed discussion). Perturbations related by unitary rotations of the
coupling matrices are related by adding redundant operators to the action and
are thus equivalent. We can use this symmetry of the coupling space to
diagonalise one of the coupling matrices. Note that the operator
$\sum_{i=1}\chi^{[ii]}$ commutes with all operators so that without loss of
generality we can assume that $m_{ij}$ is traceless. One useful choice is to
diagonalise $h_{ij}$ in which case $m_{ij}$ is generically some non-diagonal
traceless hermitian matrix. Suppose the eigenvalues of $h_{ij}$ are the
numbers $h_{1},h_{2},\dots,h_{N}$ and the index is the same as the index
labelling the copies of the boundary condition.Then it is not hard to see that
the perturbed theory (1.2) describes the critical Ising model on a half plane
in the presence of a fluctuating piece-wise constant boundary magnetic field.
To get a precise picture consider the perturbed correlation function (1.3).
Expanding in the powers of the $m_{ij}$ couplings we can consider terms with
insertions

$\prod_{i=1}^{m}\chi^{[i_{m+1},i_{m}]}(x_{m})\dots\chi^{[i_{3},i_{2}]}(x_{2})\chi^{[i_{2},i_{1}]}(x_{1})$
(1.7)

where $x_{1}<x_{2}<\dots<x_{m}$. The terms with boundary magnetic fields in
the perturbation expansion are then selected according to OPEs (1), (1.5) so
that, after summing up the series in $h_{i}$, we obtain the boundary magnetic
field with the value $h_{i_{k}}$ between $x_{k-1}$ and $x_{k}$ for $2\leq
k\leq m$, and with the values $h_{i_{1}}$ and $h_{i_{m+1}}$ for $x<x_{1}$ and
$x>x_{m}$ respectively. A sample profile of this kind is depicted on Fig. 1.

$x$$h$$h_{i_{1}}$$x_{1}$$h_{i_{2}}$$x_{2}$$h_{i_{3}}$$x_{3}$$h_{i_{4}}$ Figure
1: Sample profile of the boundary magnetic field

The summation over the indices $i_{1},\dots,i_{m+1}$ and integration over the
positions of the discontinuities $x_{1},\dots,x_{m}$ describes an annealed
disorder ensemble in which each coupling $m_{ij}$ with $i\neq$ plays the role
of a fugacity variable for the discontinuity of the boundary magnetic field
allowed to jump from the value $h_{j}$ to the value $h_{i}$. The couplings
$m_{ii}$, $i=1,\dots,N$ correspond to shifts of the vacuum energy density in
the corresponding sectors of the undeformed theory. They are additional simple
deformations of the constant boundary magnetic field model. For $N=1$ such a
perturbation drops out as it is proportional to the identity field but for
$N>1$ and $m_{i}\neq m_{j}$, $i\neq j$ they lead to some non-trivial effects.
We note that the quenched disorder case of the boundary magnetic field model
was previously considered in [23], [24], [9].

To fully specify the model on a half plane we need to fix the boundary
conditions at infinity that can be done for instance as in (1.3), (1.6) i.e.
by fixing the boundary components in each limit. This does not treat all of
the values $h_{i}$ on equal footing. Alternatively we can consider averaging
over the different values of $I$ and $J$ in (1.3), (1.6). One natural way to
do it is by putting the classical theory on a cylinder that is equivalent to
considering the quantum theory at finite temperature $T$. Allowing all values
$h_{i}$ to appear on a circle corresponds to taking a trace over the vacua
labels $I$, $J$ so that the thermal partition function has the form

$Z_{\rm annealed}=C_{N}{\rm Tr}\langle{\rm Pexp}(-S_{\rm
BCFT}-\int\limits_{0}^{\beta}\\!\\!H(\tau)\sigma_{B}(\tau)d\tau-\int\limits_{0}^{\beta}\\!\\!M(\tau)d\tau)\rangle_{0}$
(1.8)

which is similar to a Wilson line. Here $\tau$ is the Euclidean time
coordinate that parameterises a circle of circumference $\beta=1/T$, $C_{N}$
is a normalisation constant. If we choose $C_{N}=1/N$ the partition function
(1.8) describes a single copy of the free boundary condition perturbed by a
non-constant boundary magnetic field whose profile fluctuates along the
boundary according to Fig. 1 (adopted to a circle). The same model can be
still described as a superposition of $N$ copies of the boundary boundary
perturbed by a collection of boundary fields with constant couplings as in
(1.2). The $g$-theorem for boundary RG flows [25], [26] was proven for
boundary theories with constant couplings and thus applies to the latter kind
of description. In that description the $g$-factors are given by the boundary
partition functions which are canonically normalised by the Cardy condition
[27] applied to the irreducible components so that the normalisation constant
$C_{N}=1$. In sections 4, 5 when discussing the boundary RG flows we will
choose this normalisation.

As shown in [3] the model described by (1.1) is integrable. Moreover one can
deform the bulk theory by the energy density operator which corresponds to
shifting the temperature away from critical value in the classical lattice
model. The corresponding two-coupling model is still integrable. In the
present paper we focus on the model described by (1.2) with $N=2$. In addition
to the critical case we will also consider (1.2) with an additional bulk
deformation by the energy density. In section 2 we will show that the model
can be described by a Lagrangian theory containing the usual Majorana field
and 3 real boundary fermions. Generically this Lagrangian contains a four-
Fermi interaction term but when $h_{1}=\pm h_{2}$ it is Gaussian. The case
$h_{1}=-h_{2}$ is non-trivial and to the best of our knowledge has not been
studied before (see however [11], [12] for a closely related work on the
lattice). In section 3 we derive and study the exact boundary reflection
amplitudes describing the corresponding boundary integrable model that
contains 4 parameters. In section 4 we calculate the disc partition function
for the same model in the bulk critical case. In section 5 we use the disc
partition function to study the details of the 3-coupling family of the RG
boundary flows described by the $h_{1}=-h_{2}$ model critical in the bulk. In
particular we elucidate the IR asymptotics of these flows. In section 6 we
work out the all orders infrared effective action. We finish the paper with
some discussion in section 7. The appendix contains details of calculations
demonstrating the absence of conserved currents of spins 2 and 4 in the
generic $N=2$ classical model.

##  2 Free fermion description

Let us consider the Ising field theory with zero magnetic field on a complex
plane with coordinates $z=x+iy$, $\bar{z}=x-iy$. In the free fermion
description the bulk action is111We use the conventions of [28] in which
$\epsilon=i:\\!\\!\psi\bar{\psi}\\!\\!:$, $C_{\sigma\sigma\epsilon}=1/2$ and
$m>0$ corresponds to the low temperature ordered phase.

$S_{\rm
bulk}=\frac{1}{2\pi}\\!\iint\\![\psi\bar{\partial}\psi+\bar{\psi}\partial\bar{\psi}+im\psi\bar{\psi}]dxdy$
(2.1)

where

$\partial\equiv\frac{\partial}{\partial
z}=\frac{1}{2}\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial
y}\right)\,,\quad\bar{\partial}\equiv\frac{\partial}{\partial\bar{z}}=\frac{1}{2}\left(\frac{\partial}{\partial
x}+i\frac{\partial}{\partial y}\right)\,,$ (2.2)

and the coupling $m$ is proportional to $T_{c}-T$. We will assume that $m>0$
and the system is in the low temperature phase with two degenerate ground
states.

Suppose now that we have a simply connected region ${\cal B}$ on the complex
plane with boundary ${\cal D}=\partial{\cal B}$ that is connected and is
described as a curve $z=Z(t)$, $\bar{z}=\bar{Z}(t)$. Following [4] we assume
that the parameterisation is chosen in such a way that

$e(t)\bar{e}(t)=1\,,\quad\mbox{where
}e(t)=\frac{dZ}{dt}(t)\,,\enspace\bar{e}(t)=\frac{d\bar{Z}}{dt}(t)$ (2.3)

and the unit normal vector $(ie,-i\bar{e})$ points inside ${\cal B}$. With
these conventions a suitably normalised boundary spin operator can be
introduced as [4]

$\sigma_{B}(t)=ia(t)(e^{1/2}(t)\psi(t)+\bar{e}^{1/2}\bar{\psi}(t))$ (2.4)

where $\psi(t)\equiv\psi(Z(t))$, $\bar{\psi}(t)\equiv\bar{\psi}(\bar{Z}(t))$
and $a(t)$ is the boundary fermion that is a real fermionic boundary field
with the two-point function

$\langle a(t)a(t^{\prime})\rangle=\frac{1}{2}{\rm sign}(t-t^{\prime})\,.$
(2.5)

The Ising field theory on ${\cal B}$ with a boundary magnetic field on ${\cal
D}$ is then described by the action

$S=\frac{1}{2\pi}\\!\iint\limits_{\cal
B}\\![\psi\bar{\partial}\psi+\bar{\psi}\partial\bar{\psi}+im\psi\bar{\psi}]dxdy+\int[\frac{1}{2}a\dot{a}-\frac{i}{4\pi}\psi\bar{\psi}(t)]dt+h\\!\int\\!\sigma_{B}(t)dt$
(2.6)

where $\dot{a}=\frac{da}{dt}$. For the upper half plane we can choose
$Z(t)=t$, $\bar{Z}(t)=t$ and $e(t)=\bar{e}(t)=1$. The action (2.6) then gives
the standard bulk equations of motion and the boundary conditions

$\displaystyle\psi(t)-\bar{\psi}(t)=4\pi ha(t)\,,$
$\displaystyle\dot{a}=-ih(\psi(t)+\bar{\psi}(t))\,.$ (2.7)

Combining the two equations we can obtain a closed boundary condition on the
$\psi$ and $\bar{\psi}$ fields:

$\partial_{t}\psi(t)+i\lambda\psi(t)=\partial_{t}\bar{\psi}(t)-i\lambda\bar{\psi}(t)$
(2.8)

where

$\lambda=4\pi h^{2}\,.$ (2.9)

As first noticed in [4] this boundary condition means that the field
$\chi(z)\equiv(\partial+i\lambda)\psi(z)$ can be extended to a holomorphic
field on the entire complex plane. The fact that $\chi(z)$ depends only on the
square of the coupling means that the gluing condition remains the same in the
presence of discontinuities in the boundary magnetic field at which it changes
the sign. It is not hard to show that in this case the field $\chi(z)$ is
meromorphic on the plane with simple poles at the discontinuities. This
singles out this type of discontinuity and suggests integrability of the
theory with such a fixed profile. On the lattice such models have been worked
out in terms of free fermions in [11], [12]. In this paper we are mainly
interested in the case when the discontinuities are dynamic in the sense of an
annealed ensemble in which the discontinuities are governed by a fugacity
coupling. We proceed next to a free fermion description of perturbations of
the form (1.2) defined on superpositions.

The use of boundary fermions to describe Chan-Paton interactions and Wilson
loops in string theory is a well known trick [29] , [30]. From the point of
view of the conformal field theory both the bulk fermionic fields $\psi(z)$
and $\bar{\psi}(\bar{z})$ and the boundary fermion $a(t)$ can be described in
terms of topological defects [31], [32]. The critical Ising model has three
topological defects labeled by the primary fields. They can end on the
conformal boundary conditions. The $\epsilon$-defect is the spin reversal
symmetry defect. The bulk free fermion fields $\psi$ and $\bar{\psi}$ are
located at the tip of the $\epsilon$ defect. This defect can also end on the
free boundary condition forming either a topological junction that (with an
appropriate choice of normalisation) corresponds to the boundary fermion field
$a$ or via a non-topological junction given by a weight $1/2$ fermionic field
that (up to normalisation) is given by the $\psi$ (or $\bar{\psi}$) field
inserted on the boundary222Recall that any holomorphic or anti-holomorphic
field cannot have singularities at the boundary, with a conformal boundary
condition, and thus can be put on the boundary. That gives a boundary field of
the same conformal weight.. Given two such junctions: one topological and one
non-topological, we can merge them together. In the bulk this results in
fusing the $\epsilon$-defect with itself that gives the identity defect and on
the boundary that gives a (bosonic) “ordinary” boundary field of dimension
1/2. This boundary field is proportional to the boundary spin field
$\sigma_{B}$. This process depicted on Figure 2 explains formula (2.4). Such a
relation along with other bulk-to-boundary and boundary OPEs involving
fermionic fields has been worked out in [32] from first principles.

$\epsilon$$\epsilon$$\bullet$$a$$\psi$$\bullet$$=$$\bullet$$\alpha\sigma_{B}$
Figure 2: Merger of two defect junctions producing a boundary magnetic field
operator. $\alpha\neq 0$ is a normalisation related constant.

Let us consider now a superposition of $N>1$ copies of the free spin boundary
conditions. The $\epsilon$-defect has a topological and a non-topological
junction between any pair of copies. This gives rise to the junction fields
$a^{[ij]}$ and $\psi^{[ij]}$ which have dimension 0 and dimension 1/2
respectively. Like for boundary fields our notation is such that $a^{[ij]}$
when inserted at a boundary point of the upper half plane has the $j$-th copy
of the boundary condition to the left and the $i$-th copy to the right, and
similarly for $\psi^{[ij]}$. Choosing an appropriate normalisation of the
$\psi^{[ij]}$ fields we now have

$\lim_{z\to\tau\in{\mathbb{R}}}\psi(z)=\sum_{i=1}^{N}\psi^{[ii]}(\tau)$ (2.10)

so that only the combination on the right hand side of this equation can be
moved from the boundary into the bulk. We choose the normalisations of the
$a^{[ij]}$ fields so that merging of the topological junctions between
themselves is given by the following (product) OPE rule

$a^{[ij]}a^{[kl]}=\delta_{jk}\chi^{[il]}\,.$ (2.11)

Furthermore we choose the normalisations of the $\sigma_{B}^{[ij]}$ fields in
such a way that

$\sigma_{B}^{[ij]}(t)=ia^{[ik]}(t)(e^{1/2}(t)\psi^{[kj]}(t)+\bar{e}^{1/2}\bar{\psi}^{[kj]}(t))\quad\mbox{for
any }k\,.$ (2.12)

This allows us to write

$\sigma_{B}^{[ij]}(t)=ia^{[ij]}(t)(e^{1/2}(t)\psi(t)+\bar{e}^{1/2}\bar{\psi}(t))=-i(e^{1/2}(t)\psi(t)+\bar{e}^{1/2}\bar{\psi}(t))a^{[ij]}(t)$
(2.13)

where the $\psi(t)$ and $\bar{\psi}(t)$ fields on the right hand side are
given by (2.10). To obtain a functional integral representation for the theory
described by (1.2) we need a set of anti-commuting boundary fields. Although
one can choose such a set for any value of $N$ from now on we will specialise
to the $N=2$ case on which the present paper is focused. Consider the
combinations

$\displaystyle a$ $\displaystyle=$ $\displaystyle a^{[11]}-a^{[22]}\,,$
$\displaystyle b$ $\displaystyle=$ $\displaystyle i(a^{[21]}-a^{[12]})\,,$
$\displaystyle c$ $\displaystyle=$ $\displaystyle a^{[21]}+a^{[12]}\,.$ (2.14)

Using (2.11) we check that these fields satisfy

$\\{a,b\\}=\\{a,c\\}=\\{b,c\\}=0\,,\qquad
a^{2}=b^{2}=c^{2}=\chi^{[11]}+\chi^{[22]}\equiv{\bf 1}\,.$ (2.15)

We also find the following relations

$\displaystyle iab=\chi^{[12]}+\chi^{[21]}\,,\quad
ibc=\chi^{[11]}-\chi^{[22]}\,,\quad ac=\chi^{[12]}-\chi^{[21]}\,,$
$\displaystyle abc=-i(a^{[11]}+a^{[22]})$ (2.16)

and

$a^{[11]}=\frac{1}{2}(a+iabc)\,,\quad a^{[22]}=\frac{1}{2}(-a+iabc)\,.$ (2.17)

We note that the fields $a,b,c$ are hermitian that follows from the
conjugation relations

$(\chi^{[ij]})^{\dagger}=\chi^{[ji]}\,.$ (2.18)

The above relations mean that in the gauge in which $h_{ij}={\rm
diag}(h_{1},h_{2})$ we can represent the $N=2$ theory (1.2) as a functional
integral

$\int e^{-S}D[\psi]D[\bar{\psi}]D[a]D[b]D[c]$ (2.19)

with the action functional

$\displaystyle S=\frac{1}{2\pi}\\!\iint\limits_{\cal
B}\\!(\psi\bar{\partial}\psi+\bar{\psi}\partial\bar{\psi}+im\psi\bar{\psi})dxdy+\int\limits_{-\infty}^{\infty}\\!\\!\Bigl{[}-\frac{i}{4\pi}\psi\bar{\psi}+\frac{1}{2}a\dot{a}+\frac{1}{2}b\dot{b}+\frac{1}{2}c\dot{c}$
$\displaystyle+(ih_{-}a+h_{+}abc)(e^{1/2}\psi+\bar{e}^{1/2}\bar{\psi})+m_{1}iab+m_{2}ibc\Bigr{]}dt$
(2.20)

Here

$h_{+}=-\frac{h_{1}+h_{2}}{2}\,,\quad h_{-}=\frac{h_{1}-h_{2}}{2}\,.$ (2.21)

and333Here without loss of generality we assume that $m_{1}$ is real as that
can be achieved by a residual unitary transformation that preserves the
diagonal matrix $h_{ij}$.

$(m_{ij})=\left(\begin{array}[]{rr}m_{2}&m_{1}\\\
m_{1}&-m_{2}\end{array}\right)\,.$ (2.22)

We immediately observe that when $h_{+}=0$ the theory (2) is Gaussian. From
now on we will assume that $h_{+}=0$ postponing the discussion of the general
$h_{-}h_{+}\neq 0$ case until section 7. When $h_{+}=0$, considering the model
on the upper half plane we obtain the following bulk equations of motion and
boundary conditions

$\bar{\partial}\psi+\frac{im}{2}\bar{\psi}=0\,,\qquad\partial\bar{\psi}-\frac{im}{2}\psi=0\,,$
(2.23) $\displaystyle\psi(t)-\bar{\psi}(t)$ $\displaystyle=$ $\displaystyle
4\pi h_{-}a(t)\,,$ $\displaystyle\dot{a}(t)$ $\displaystyle=$ $\displaystyle-
ih_{-}(\psi(t)+\bar{\psi}(t))-m_{1}ib(t)\,,$ $\displaystyle\dot{b}(t)$
$\displaystyle=$ $\displaystyle m_{1}ia(t)-m_{2}ic(t)\,,$
$\displaystyle\dot{c}(t)$ $\displaystyle=$ $\displaystyle m_{2}ib(t)\,.$
(2.24)

By taking more derivatives of the equations in (2) we can obtain a closed
boundary condition on the $\psi$ and $\bar{\psi}$ fields that looks as follows

$\partial_{t}^{3}\psi+i\lambda\partial_{t}^{2}\psi-(m_{1}^{2}+m_{2}^{2})\partial_{t}\psi-
im_{2}^{2}\lambda\psi=\partial_{t}^{3}\bar{\psi}-i\lambda\partial_{t}^{2}\bar{\psi}-(m_{1}^{2}+m_{2}^{2})\partial_{t}\bar{\psi}+im_{2}^{2}\lambda\bar{\psi}$
(2.25)

where

$\lambda\equiv\lambda_{-}=4\pi h_{-}^{2}\,.$ (2.26)

##  3 Exact boundary reflection amplitudes

To find the boundary reflection amplitude corresponding to (2) we put the
model onto a left half-plane: ${\rm Re}z\leq 0$, and parameterise the boundary
by $t=y={\rm Im}z$. This gives $e=\omega$, $\bar{e}=\bar{\omega}$ where

$\omega=e^{i\frac{\pi}{4}}$ (3.1)

and the boundary conditions are given by equations (2), (2.25) in which $\psi$
and $\bar{\psi}$ are replaced by $\omega\psi$ and $\bar{\omega}\bar{\psi}$
respectively. The canonical quantisation expansion of the fields $\psi$ and
$\bar{\psi}$ corresponding to (2.23) has the form

$\displaystyle\psi(x,y)=\int\limits_{-\infty}^{\infty}\\!\\!d\theta\Bigl{[}\omega
e^{\theta/2}A(\theta)e^{-my\cosh\theta+imx\sinh\theta}+\bar{\omega}e^{\theta/2}A^{\dagger}(\theta)e^{my\cosh\theta-
imx\sinh\theta}\Bigr{]}\,,$
$\displaystyle\bar{\psi}(x,y)=\int\limits_{-\infty}^{\infty}\\!\\!d\theta\Bigl{[}\omega
e^{-\theta/2}A(\theta)e^{-my\cosh\theta+imx\sinh\theta}-\bar{\omega}e^{-\theta/2}A^{\dagger}(\theta)e^{my\cosh\theta-
imx\sinh\theta}\Bigr{]}$ (3.2)

where $y$ is the Euclidean time, $\theta$ is the rapidity variable and
$A(\theta)$, $A^{\dagger}(\theta)$ are the creation and annihilation operators
satisfying

$\\{A(\theta),A^{\dagger}(\theta^{\prime})\\}=\delta(\theta-\theta^{\prime})\,.$
(3.3)

Following [3] the boundary can be treated as an infinitely heavy particle
located at $x=0$. The vacuum state for the $t$-quantisation is formally given
by the action of a boundary creation operator $B$ on the bulk theory vacuum:

$|0\rangle_{B}=B|0\rangle\,.$ (3.4)

The reflection coefficient $R(\theta)$ is then defined by the relation

$A^{\dagger}(\theta)B=R(\theta)A^{\dagger}(-\theta)B\,.$ (3.5)

Substituting the expansions (3) into the boundary condition (2.25) we obtain
the reflection matrix

$R(\theta)=i\frac{p^{0}((p^{0})^{2}-m_{1}^{2}-m_{2}^{2})\cosh\left(\frac{\theta}{2}-i\frac{\pi}{4}\right)-i\lambda((p^{0})^{2}-m_{2}^{2})\sinh\left(\frac{\theta}{2}-i\frac{\pi}{4}\right)}{p^{0}((p^{0})^{2}-m_{1}^{2}-m_{2}^{2})\sinh\left(\frac{\theta}{2}-i\frac{\pi}{4}\right)+i\lambda((p^{0})^{2}-m_{2}^{2})\cosh\left(\frac{\theta}{2}-i\frac{\pi}{4}\right)}$
(3.6)

where

$p^{0}=m\cosh(\theta)\,.$ (3.7)

Another useful form for this amplitude is

$R(\theta)=i\tanh\left(\frac{\theta}{2}-i\frac{\pi}{4}\right)\frac{(\cosh^{2}(\theta)-a_{1}-a_{2})(\sinh(\theta)+i)-ib(\cosh^{2}(\theta)-a_{2})}{(\cosh^{2}(\theta)-a_{1}-a_{2})(\sinh(\theta)-i)+ib(\cosh^{2}(\theta)-a_{2})}$
(3.8)

where we introduced dimensionless parameters

$a_{1}=\frac{m_{1}^{2}}{m^{2}}\,,\quad a_{2}=\frac{m_{2}^{2}}{m^{2}}\,,\quad
b=\frac{\lambda}{m}\,.$ (3.9)

It is straightforward to check that the reflection amplitude given in (3.6),
(3.8) satisfies the unitarity and boundary crossing symmetry:

$R(\theta)R(-\theta)=1\,,\qquad
R\left(i\frac{\pi}{2}-\theta\right)=-R\left(i\frac{\pi}{2}+\theta\right)\,.$
(3.10)

Of particular interest are poles of $R(\theta)$ located in the physical
region: $\theta=i\xi$, $\xi\in[0,\pi]$. They correspond to boundary bound
states. To find them it is convenient to set $\sinh(\theta)=ix$ then the
denominator in (3.8) up to a factor of $-i$ is given by the polynomial

$q(x)=x^{3}+x^{2}(b-1)-x(1-a_{1}-a_{2})+a_{2}b+1-a_{1}-a_{2}-b\,.$ (3.11)

This polynomial vanishes when we have

$(x-1+b)(x^{2}-1+a_{2})=a_{1}(1-x)\,.$ (3.12)

From this representation we see that, assuming $a_{1}>0$, $a_{2}>0$, $b>0$, we
always have a single bound state, that is a pole in the physical interval
$x\in[0,1]$, when $a_{1}>(1-a_{2})(1-b)$. Also when $a_{1}<(1-a_{2})(1-b)$ and
$a_{2}<1$ we have two distinct bound states and when $a_{1}<(1-a_{2})(1-b)$
and $a_{2}>1$ we have no bound states. When $a_{1}=(1-a_{2})(1-b)>0$ there is
no bound state unless $a_{2}<1$ and $b<1$ in which case there is a single
bound state (note that the zero of the denominator of $R$ at $x=0$ is
cancelled by the zero of the numerator). As argued in [3] (see also
discussions in [8] and [15]) the boundary bound states which are present for
sufficiently small boundary couplings correspond to the deformed degenerate
ground states. In the undeformed model we are considering the vacuum state on
a half-plane is four times degenerate. It is interesting to note then that we
can only have at most two boundary bound states rather than three expected
naively (plus the vacuum). It would be interesting to obtain an intuitive
physical picture explaining this.

The remaining special cases when one of the three couplings vanishes can be
described as follows. When $b=0$ the reflection coefficient (3.8) reduces to
the one for the free boundary condition:

$R_{\rm free}(\theta)=i\coth\left(\frac{\theta}{2}-i\frac{\pi}{4}\right)\,.$
(3.13)

This is because in this case we only have the boundary identity fields
switched on which decouple from the bulk fermion field. When $a_{1}=0$ the
model factorises into two decoupled boundary conditions with the boundary
magnetic field given by $h_{-}$ and $-h_{-}$ and additional shifts of vacuum
energy given by $m_{2}$ and $-m_{2}$. The vacuum energy shifts decouple from
the bulk fermion and we have the reflection amplitude of the usual boundary
magnetic field model [3]

$R_{h}(\theta)=-i\tanh\left(\frac{\theta}{2}-i\frac{\pi}{4}\right)\frac{1-b-i\sinh(\theta)}{1-b+i\sinh(\theta)}\,.$
(3.14)

When $a_{2}=0$ we obtain a two coupling model with reflection amplitude that
can be written as

$R_{a,b}(\theta)=i\coth\left(\frac{\theta}{2}-i\frac{\pi}{4}\right)\frac{\sinh^{2}(\theta)-ib\sinh(\theta)+1-a_{1}-b}{\sinh^{2}(\theta)+ib\sinh(\theta)+1-a_{1}-b}\,.$
(3.15)

For $a_{1}\neq 0$ this amplitude has a pole at $\theta=i\frac{\pi}{2}$. This
pole is absent when all three couplings are switched on (we actually get a
zero at $\theta=i\frac{\pi}{2}$ in that case). As discussed in [3] such a
pole, which is present for the free spin boundary condition, means that the
boundary state has a contribution proportional to the momentum zero one-
particle state. For the free spin boundary condition this is interpreted as
the spin reversal symmetry defect attached to the boundary. In our case we can
interpret this one-particle state as a superposition of two spin-reversal
symmetry defects attached to two copies of the free boundary condition as
described by the junction field

$c=a^{[12]}+a^{[21]}\,.$ (3.16)

As can be seen from (2.15) this junction field commutes with the perturbation
when $m_{2}=0$.

In addition to the above pole, the reflection amplitude $R_{a,b}$ has another
pole444As usual there is also the crossing symmetric pole at
$\theta=i(\pi-\xi_{*})$. at $\theta=i\xi_{*}$ with

$\xi_{*}=\arcsin\left(-\frac{b}{2}+\frac{1}{2}\sqrt{b^{2}+4(1-b-a_{1})}\right)$
(3.17)

which is located on the physical interval $\xi\in[0,1]$ provided $a_{1}+b<1$.
We plan to discuss correlation functions and the physical interpretation of
the boundary bound states we found in this section in a separate publication.

##  4 Exact disc partition functions in the critical case

In this section we calculate the disc partition function for the three-
coupling model (2) in the case of conformal bulk that is when $m=0$. We aim to
find the boundary entropy and investigate the boundary RG flows.

We start by putting the model (2) on a cylinder with complex coordinate
$z=x+i\tau$ where $x\geq 0$ is the coordinate along the cylinder,
$\tau\in[0,\beta]$ is the Euclidean time, $\beta$ is the inverse temperature.
To calculate the partition function on a disc we assume that on the cylinder
the boundary condition at $x\to\infty$ corresponds to the bulk theory
conformal vacuum and thus we take all fermionic fields: $\psi$, $\bar{\psi}$,
$a,b,c$ in the NS sector that is antiperiodic in $\tau$. At the beginning we
will treat the functional integral formally introducing a regularization and
renormalisation later. The integral over $\psi$ and $\bar{\psi}$ with fixed
boundary values is Gaussian and hence localises on the solutions to the bulk
equations of motion decaying in the bulk. We can write these solutions as the
following mode expansions

$\psi(z)=\sum\limits_{k\in
1/2+{\mathbb{Z}};k<0}\psi_{k}e^{\frac{2\pi}{\beta}kz}\,,\quad\bar{\psi}(\bar{z})=\sum\limits_{k\in
1/2+{\mathbb{Z}};k>0}\bar{\psi}_{k}e^{-\frac{2\pi}{\beta}k\bar{z}}\,.$ (4.1)

When restricted to the boundary $x=0$ these solutions give anti-periodic
functions

$\psi_{b}(\tau)\equiv\psi(i\tau)\,,\qquad\bar{\psi}_{b}(\tau)\equiv\bar{\psi}(-i\tau)\,.$
(4.2)

We further use the Fourier expansions

$a(\tau)=\sum\limits_{k\in
1/2+{\mathbb{Z}}}a_{k}e^{i\frac{2\pi}{\beta}k\tau}\,,\qquad
b(\tau)=\sum\limits_{k\in
1/2+{\mathbb{Z}}}b_{k}e^{i\frac{2\pi}{\beta}k\tau}\,,$ (4.3)
$c(\tau)=\sum\limits_{k\in
1/2+{\mathbb{Z}}}c_{k}e^{i\frac{2\pi}{\beta}k\tau}\,.$ (4.4)

Integrating over the fermions $\psi$ and $\bar{\psi}$ with the fixed values on
the boundary we obtain up to the divergent determinant factor the following
fermionic path integral

$\displaystyle Z_{\rm disc}=\int
D[\psi_{b}]D[\bar{\psi}_{b}]D[a]D[b]D[c]e^{-S_{a,b,c}}$
$\displaystyle\times\exp\left(-i\frac{\beta}{4\pi}\sum_{k<0}\psi_{k}\bar{\psi}_{-k}+ih\beta\sum_{k<0}(\omega\psi_{k}a_{-k}+\bar{\omega}\bar{\psi}_{-k}a_{k})\right)$
(4.5)

where

$S_{a,b,c}=\int
d\tau\Bigl{[}\frac{1}{2}a\dot{a}+\frac{1}{2}b\dot{b}+\frac{1}{2}c\dot{c}++m_{1}iab+m_{2}ibc\Bigr{]}\,.$
(4.6)

Removing the determinant factor here amounts to subtracting the bulk CFT
vacuum energy density so that what remains corresponds to the disc partition
function. Integrating out the boundary functions $\psi_{b}(\tau)$ and
$\bar{\psi}_{b}(\tau)$ gives

$Z_{\rm disc}=D_{1}\int
D[a]D[b]D[c]e^{-S_{a,b,c}}\exp\left(i\sum_{k<0}a_{-k}a_{k}(\lambda\beta-2\pi
k)\right)$ (4.7)

where

$D_{1}=\prod_{n=1}^{\infty}\left(-i\frac{\beta}{4\pi}\right)$ (4.8)

is a formal expression for the functional determinant. As this factor does not
depend on any coupling constants we remove it from the renormalised partition
function. What remains now is a path integral over the boundary fermions:
$a,b,c$. We normalise this path integral in such a way that in the Hamiltonian
quantisation it amounts to taking the trace over the degenerate vacuum space.
The remaining Gaussian integrations now produce the following formal
expression

$Z_{\rm
disc}=C\prod\limits_{m=0}^{\infty}\Bigl{[}\left(1+\frac{\alpha}{m+1/2}\right)\left(1+\frac{\nu_{2}}{(m+1/2)^{2}}\right)+\frac{\nu_{1}}{(m+1/2)^{2}}\Bigr{]}$
(4.9)

where

$\alpha=\frac{\lambda\beta}{2\pi}=2\beta
h_{-}^{2}\,,\qquad\nu_{1}=\frac{\beta^{2}m_{1}^{2}}{4\pi^{2}}\,,\quad\nu_{2}=\frac{\beta^{2}m_{3}^{2}}{4\pi^{2}}$
(4.10)

and the factor

$C=\Bigl{[}\prod_{k=0}^{\infty}(-2\pi i)(k+1/2)\Bigr{]}^{3}$ (4.11)

comes from the kinetic terms of the boundary fermions. As this factor should
give the trace of the identity in the vacuum space we set it equal to 2:

$C_{\rm ren}=2\,.$ (4.12)

The remaining infinite product in (4.9) we regularise as follows

$\displaystyle Z_{\rm disc}^{(\epsilon)}=2[(1+2\alpha)(1+4\nu_{2})+4\nu_{1}]$
$\displaystyle\times\prod\limits_{m=1}^{\infty}\Bigl{[}\left(1+\frac{\alpha}{m+1/2}\right)\left(1+\frac{\nu_{2}}{(m+1/2)^{2}}\right)+\frac{\nu_{1}}{(m+1/2)^{2}}\Bigr{]}e^{\frac{\alpha}{m}(e^{-\epsilon
m}-1)}$ (4.13)

where $\epsilon>0$ is a regulator. The regulated expression can be written in
a manifestly finite form

$Z_{\rm
disc}^{(\epsilon)}=\frac{2\pi^{3/2}e^{-\alpha\ln(1-e^{-\epsilon})}}{\Gamma(1/2-k_{1})\Gamma(1/2-k_{2})\Gamma(1/2-k_{3})}$
(4.14)

where $k_{1}$, $k_{2}$ and $k_{3}$ are roots of the polynomial

$p(z)\equiv z^{3}+\alpha
z^{2}+(\nu_{1}+\nu_{2})z+\alpha\nu_{2}=(z-k_{1})(z-k_{2})(z-k_{3})$ (4.15)

and are thus radical functions of the couplings. The regularised expression
(4.14) has a logarithmic divergence in the exponential in the limit
$\epsilon\to 0$. In perturbation theory this divergence emerges at the second
order when we integrate the two-point function
$\langle\sigma_{B}(\tau)\sigma_{B}(\tau^{\prime})\rangle$ of the boundary spin
operator. Subtracting this divergence we obtain a renormalised value of the
partition function

$Z_{\rm
disc,ren}=\frac{2\pi^{3/2}e^{\alpha\ln(\beta\mu)}}{\Gamma(1/2-k_{1})\Gamma(1/2-k_{2})\Gamma(1/2-k_{3})}$
(4.16)

where $\mu$ is the subtraction scale which has dimension of mass. When the
couplings $\alpha$, $\nu_{1}$, $\nu_{2}$ are all zero we have
$k_{1}=k_{2}=k_{3}=0$ and $Z_{\rm disc,ren}=2$ that is the correct $g$-factor
of two copies of the free spin boundary condition. The subtraction scale $\mu$
is in principle arbitrary but can be fixed by requiring that the partition
function tends to a finite limit in the far infrared where $\beta\to\infty$.
For example when $\nu_{2}=\nu_{1}=0$ this requirement fixes the subtraction
point to be

$\mu\beta=\frac{\alpha}{e}\,.$ (4.17)

In this case the model essentially reduces to the boundary magnetic field
model. Up to a factor of 2 the partition function is the same as the one
computed in [5]

$Z_{h}=\frac{\sqrt{2\pi}}{\Gamma(\alpha+1/2)}\left(\frac{\alpha}{e}\right)^{\alpha}\,.$
(4.18)

In the generic case the infrared asymptotic of (4.16) is

$Z_{\rm disc,ren}\sim{\rm
Const}\exp\left[\alpha\ln(\beta\mu)+\alpha+\sum_{i}k_{i}{\rm
Log}\left(\frac{1}{2}-k_{i}\right)\right]$ (4.19)

and we can set

$\mu\beta=\exp\left(-1-\sum_{i}\frac{k_{i}}{\alpha}{\rm
Log}(-k_{i})\right)\,.$ (4.20)

Here ${\rm Log}(z)$ stands for the principal branch of the complex natural
logarithm function. The above expressions are real as the roots $k_{i}$ are
either all real and negative or are comprised of a complex pair and one real
negative root.

In the following we will be interested in the boundary entropy rather than the
partition function itself. In the boundary entropy the subtraction scale $\mu$
drops out so we do not need to assume it is fixed as in (4.20) or otherwise.

##  5 RG flows

The model (2) describes a relevant perturbation of the superposition of two
copies of the free spin boundary condition $(f)$. Its $g$-factor [25] equals
$2$. The fixed spin conformal boundary conditions $(+)$ and $(-)$ each have
the $g$-factor $g_{+}=g_{-}=\frac{1}{\sqrt{2}}$. The $g$ theorem [25], [26]
implies that the boundary entropy monotonically decreases and thus the RG flow
starting with $2(f)$ can end up in any of the 8 fixed points: $(f)\oplus(+)$,
$(f)\oplus(-)$, $(+)\oplus(-)$, $2(+)$, $2(-)$, $(f)$, $(+)$, $(-)$. We show
below that the three parameter subspace of flows with $h_{+}=0$ ends on half
of these possible fixed points: $(+)\oplus(-)$, $(f)$, $(+)$, $(-)$.

The boundary entropy of (2) with $h_{+}=0$ is given by

$s=\left(1-\beta\frac{\partial}{\partial\beta}\right)\ln\\!Z_{\rm disc,ren}$
(5.1)

where $Z_{\rm disc,ren}$ is given in (4.16). The large $\beta$ asymptotic of
$s$ depends on the values of the couplings. Since
$k_{1}k_{2}k_{3}=-\alpha\nu_{2}$, when $\alpha\nu_{2}\neq 0$ the roots $k_{i}$
are all non-vanishing. It also follows from (4.15) that

$\beta\frac{\partial k_{i}}{\partial\beta}=k_{i}$ (5.2)

and hence in the case at hand each $k_{i}$ tends to infinity homogeneously in
$\beta$. Using the asymptotic expansion555This expansion can be obtained by
combining formulae (1.18.12), (1.13.14), (1.12.2) and (1.12.21) from [33].

$\ln\Gamma(1/2+z)=z\ln(z)-z+\frac{1}{2}\ln(2\pi)+\sum_{n=1}^{\infty}\frac{B_{2n}(2^{1-2n}-1)}{2n(2n-1)z^{2n-1}}$
(5.3)

where $B_{m}$ denotes the Bernoulli numbers, we obtain the asymptotic
expansion

$s=-\frac{1}{2}\ln
2+\sum_{i=1}^{3}\sum_{n=1}^{\infty}\frac{B_{2n}(1-2^{1-2n})}{(1-2n)k_{i}^{2n-1}}\,.$
(5.4)

This gives the infrared fixed point $g$-factor to be

$g_{\rm IR}=\frac{1}{\sqrt{2}}$ (5.5)

that is the value of the $(+)$ or $(-)$ boundary condition. On physical
grounds, which of the two fixed points the flow approaches depends on the sign
of the product $h_{-}m_{2}$. (The coupling $m_{2}$ shifts the ground energy
density by $-m_{2}$ for the portion of the boundary with the magnetic field
$h_{-}$ and by the opposite value the portion perturbed by $-h_{-}$. This
favours the value for which the shift is negative.)

We next consider the cases when $\alpha\nu_{2}=0$. When $\nu_{2}=0$,
$\alpha\neq 0$, $\nu_{1}\neq 0$ the polynomial (4.15) takes the form

$p(z)=z(z^{2}+\alpha z+\nu_{1})$ (5.6)

and we can set

$k_{1}=0\,,\quad
k_{2}=-\frac{\alpha}{2}+\frac{1}{2}\sqrt{\alpha^{2}-4\nu_{1}}\,,\quad
k_{3}=-\frac{\alpha}{2}-\frac{1}{2}\sqrt{\alpha^{2}-4\nu_{1}}\,.$ (5.7)

The boundary entropy in this case has the small temperature asymptotic
expansion

$s=\sum_{i=1}^{2}\sum_{n=1}^{\infty}\frac{B_{2n}(1-2^{1-2n})}{(1-2n)k_{i}^{2n-1}}$
(5.8)

that corresponds to

$g_{\rm IR}=1$ (5.9)

and hence the boundary condition flows to the (single copy of) free spin
boundary condition.

When $\nu_{1}=\nu_{2}=0$, $\alpha\neq 0$ we have two factorised boundary
magnetic field flows from free to fixed spin and the endpoint of the flow is
the superposition $(+)\oplus(-)$.The corresponding boundary entropy asymptotic
expansion is

$s=\frac{1}{2}\ln
2+\sum_{n=1}^{\infty}\frac{B_{2n}(1-2^{1-2n})}{(2n-1)\alpha^{2n-1}}\,.$ (5.10)

When $\alpha=0$ we have a perturbation by dimension zero operators only and
the partition function reduces to

$Z_{m_{1},m_{2}}=2\cosh\left(\pi\sqrt{\nu_{1}+\nu_{2}})\right)\,.$ (5.11)

The boundary entropy is

$s=\ln
2+\ln[\cosh\left(\pi\sqrt{\nu_{1}+\nu_{2}})\right)]-\pi\sqrt{\nu_{1}+\nu_{2}}\tanh\left(\pi\sqrt{\nu_{1}+\nu_{2}})\right)$
(5.12)

that asymptotically decreases as

$s\approx 2\pi\sqrt{\nu_{1}+\nu_{2}}e^{-2\pi\sqrt{\nu_{1}+\nu_{2}}}\,.$ (5.13)

This means that the model approaches a single copy of the free spin boundary
condition along an infinitely irrelevant direction. This is always the case
for boundary perturbations by dimension zero operators. The corresponding flow
can be called component flows as the end point is always some component in the
original superposition. The approach to the infrared fixed point via an
infinitely irrelevant direction is similar to bulk perturbations generating a
mass gap for some of the fields.

The above flows act throughout a three-dimensional space and it is hard to
depict them all on a single diagram. On Figure 3 and Figure 4 we show diagrams
representing the $m_{1}=0$ and $m_{2}=0$ slices. These diagrams include
special lines along which the theory flows to intermediate fixed points as
well as the asymptotic flow lines that start at those intermediate fixed
points and which lie on the boundary of the space of flows from the original
UV fixed point.

$\boldsymbol{(-)}$$\boldsymbol{(+)}$$\boldsymbol{(f)\oplus(f)}$$\boldsymbol{(f)}$$\boldsymbol{(+)\oplus(-)}$
Figure 3: The diagram of flows for the $m_{1}=0$ slice. The red flow line
(vertical, upwards) depicts the $h_{-}=0$ flow. The blue flow line (vertical,
downwards) depicts the $m_{2}=0$ flow. The solid lines correspond to generic
$h_{-}m_{2}\neq 0$ flows. The dashed lines depict the asymptotic flows from
the fixed point $(f)$ generated by $\sigma_{B}$ while the dotted lines show
the asymptotic flows from the fixed point $(+)\oplus(-)$ generated by the
identity fields (the component flows).
$\boldsymbol{(f)\oplus(f)}$$\boldsymbol{(f)}$$\boldsymbol{(+)\oplus(-)}$
Figure 4: The diagram of flows for the $m_{2}=0$ slice. The blue flow line
(vertical, downwards) depicts the $m_{1}=0$ flow. The solid lines correspond
to generic $m_{1}\neq 0$ flows (that include the vertical upward flow with
$h_{-}=0$). The dotted lines show the asymptotic flows from the fixed point
$(+)\oplus(-)$ generated by the boundary condition changing field
$\sigma_{B}^{[+-]}$.

##  6 Infrared effective action

Using the asymptotic expansions of the boundary entropy we can also obtain the
irrelevant operators along which the flows arrive at the infrared fixed
points. Thus, for the pure boundary magnetic field flow ($\nu_{1}=\nu_{2}=0$)
we get from (5.10)

$s=\frac{1}{2}\ln 2+\frac{1}{12\alpha}-\frac{7}{720\alpha^{3}}+\dots$ (6.1)

As the leading correction in this expansion scales as $1/\beta$ this means
that the leading irrelevant perturbation near the infrared fixed point has
dimension 2 and thus must be proportional to the stress-energy tensor $T$.
Comparing the expectation value

$\langle T\rangle=-\frac{\pi^{2}}{12\beta^{2}}$ (6.2)

with the coefficient in (6.1) we see that the leading term in the infrared
effective action is

$S_{\rm
IR}=\frac{1}{4\pi^{2}h^{2}_{-}}\int\limits_{0}^{\beta}\\!\\!d\tau\,T(\tau)+\dots$
(6.3)

The same leading term was obtained in [13], [18] by other methods. Such a
perturbation would normally be expected to contribute to the free energy (and
thus to the boundary entropy) at higher orders in the effective coupling via
the integrated connected $n$-point functions. For a perturbing operator ${\cal
O}$ of dimension $\Delta$ the second order correction is proportional to the
integral

$\int\limits_{0}^{\beta}\\!\\!d\tau_{1}\int\limits_{0}^{\beta}\\!\\!d\tau_{2}\langle{\cal
O}(\tau_{1}){\cal
O}(\tau_{2})\rangle_{c;\beta}=(2\beta)^{2(1-\Delta)}\frac{\Gamma(1/2-\Delta)}{4\sqrt{\pi}\Gamma(1-\Delta)}$
(6.4)

defined via analytic continuation in dimension (see e.g. [34]). Here
$\langle\dots\rangle_{c;\beta}$ stands for a connected correlator on a
cylinder of circumference $\beta$. The integral (6.4) vanishes for any
positive integer valued $\Delta$. This explains the absence of a term of order
$1/\alpha^{2}$ in (6.1). Furthermore, an integrated correlator of three
primary fields is given by [34]

$\displaystyle\int\limits_{0}^{\beta}\\!\\!d\tau_{1}\int\limits_{0}^{\beta}\\!\\!d\tau_{2}\int\limits_{0}^{\beta}\\!\\!d\tau_{3}\langle{\cal
O}_{1}(\tau_{1}){\cal O}_{2}(\tau_{2}){\cal
O}_{3}(\tau_{3})\rangle_{c;\beta}=\frac{1}{2}(C_{123}+C_{213})(2\beta)^{3-\Delta_{1}-\Delta_{2}-\Delta_{3}}$
$\displaystyle\times\frac{\Gamma\left(\frac{2-\Delta_{1}-\Delta_{2}-\Delta_{3}}{2}\right)\Gamma\left(\frac{1+\Delta_{1}+\Delta_{2}-\Delta_{3}}{2}\right)\Gamma\left(\frac{1+\Delta_{1}+\Delta_{3}-\Delta_{2}}{2}\right)\Gamma\left(\frac{1+\Delta_{3}+\Delta_{2}-\Delta_{1}}{2}\right)}{(4\pi)^{3/2}\Gamma(1-\Delta_{1})\Gamma(1-\Delta_{2})\Gamma(1-\Delta_{3})}\,.$
(6.5)

We see that in the absence of a resonance $1+\Delta_{i}+\Delta_{j}=\Delta_{k}$
this quantity vanishes when at least one of the fields ${\cal O}_{i}$ has an
integer dimension. In particular this implies that for the perturbation (6.3)
there is no contribution of order $1/\alpha^{3}$ to the boundary entropy.
(Although $T$ is only a quasiprimary it mixes under general conformal
transformations only with the identity field that drops out in the connected
correlators.) This means that the third term in the expansion (6.1) must come
from an additional perturbation by an operator of dimension 4. The only
operator available is the fermion bilinear

$T_{4}(\tau)\equiv:\\!\psi\partial_{z}^{3}\psi\\!:\Bigr{|}_{z=i\tau}$ (6.6)

for which the expectation value on a cylinder is

$\langle T_{4}\rangle=-\frac{7}{60}\left(\frac{\pi}{\beta}\right)^{4}\,.$
(6.7)

This implies that the next to leading term in the effective action is

$S_{\rm
IR}=\frac{1}{4\pi^{2}h^{2}_{-}}\int\limits_{0}^{\beta}\\!\\!d\tau\,T(\tau)-\frac{1}{24(2\pi)^{4}h^{6}_{-}}\int\limits_{0}^{\beta}\\!\\!d\tau\,T_{4}(\tau)+\dots$
(6.8)

Using formulae666The connected three point function $\langle
T\,T\,T_{4}\rangle$ on the boundary of a half-cylinder can be obtained via a
mapping to the plane. The operator $T_{4}$ is represented on the plane by a
combination containing the identity, $T$ and $\partial T$. Nevertheless the
desired three-point function can be represented as a linear combination of
3-point functions of the type standing in the integrand of (6). (6.4), (6) we
can show that the two terms in (6.8) do not generate a correction of order
$1/\alpha^{5}$ to the boundary entropy and hence a new term with dimension 6
operator has to be added which has to be777Note that the lowest dimension
operator quartic in the free fermion field is
$:\\!\psi\partial_{\tau}\psi\partial^{2}_{\tau}\psi\partial^{3}_{\tau}\psi\\!:$
that has dimension 8. proportional to
$T_{6}=:\\!\psi\partial_{z}^{5}\psi\\!:$. Given that the theory at hand is
Gaussian it is natural to assume that all higher order operators in the
effective action are fermion bilinears

$T_{2n}(\tau)\equiv:\\!\psi\partial_{z}^{2n-1}\psi\\!:\Bigr{|}_{z=i\tau}$
(6.9)

and that the complete effective action has the form

$S_{\rm
IR}=\lambda_{2}\int\limits_{0}^{\beta}\\!\\!d\tau\,T(\tau)+\sum_{n=2}^{\infty}\lambda_{2n}\int\limits_{0}^{\beta}\\!\\!d\tau\,T_{2n}(\tau)$
(6.10)

where $\lambda_{2n}=\lambda_{2n}(h_{-})$ are the effective couplings with

$\lambda_{2}=\frac{1}{4\pi^{2}h^{2}_{-}}\,,\qquad\lambda_{4}=-\frac{1}{24(2\pi)^{4}h^{6}_{-}}\,.$
(6.11)

The expectation values of the fermion bilinears on a cylinder can be obtained
in a closed form using point splitting regularisation

$\langle T_{2n}\rangle=-\left(\frac{\pi}{\beta}\right)^{2n}\lim_{z\to
0}\frac{\partial^{2n-1}}{\partial
z^{2n-1}}\left(\frac{1}{\sinh(z)}-\frac{1}{z}\right)=\left(\frac{\pi}{\beta}\right)^{2n}\frac{(2^{2n-1}-1)}{n}B_{2n}\,.$
(6.12)

The same answer was obtained by other methods in [35], [36]. To make a
complete analysis of the infrared effective action we need to have the
integrals of higher order correlators under control. We present now a general
argument that shows that all such integrals vanish. The integrands in the
integrals of interest are connected correlation functions of the operators $T$
and $T_{2n}$, $n\geq 2$ taken on the cylinder and restricted to a circle. They
are thus restrictions of meromorphic functions defined on the entire cylinder.
Using this we regularise the integrals by shifting the integration contours
away from this circle for example by considering888This regularisation was
considered in [37], [38] for perturbations by marginal operators. One can make
it more symmetric by symmetrising over permutations of the operator insertions
as well as over $\epsilon\to-\epsilon$.

$\int\limits_{0}^{\beta}\\!\\!d\tau_{1}\dots\int\limits_{0}^{\beta}\\!\\!d\tau_{k}\langle
T_{2n_{1}}(\tau_{1})T_{2n_{2}}(\tau_{2}-i\epsilon)T_{2n_{3}}(\tau_{3}-2i\epsilon)\dots
T_{n_{k}}(\tau_{k}-(k-1)i\epsilon)\rangle_{c;\beta}\,.$ (6.13)

Clearly in the limit $\epsilon\to 0$ we recover the original integrand. On the
other hand the outer contour of integration parameterised by $\tau_{k}$ can
now be deformed to infinity. Since at infinity the connected correlation
function decays exponentially the integral vanishes. This extends to all
orders the vanishing of the second and cubic order corrections observed above.
Therefore the boundary entropy corresponding to (6.10 ) can be obtained
entirely from the expectation values (6.12). Matching the resulting expression
with the complete expansion (5.10) we obtain

$\lambda_{2n}=-\frac{1}{(2h_{-}^{2})^{2n-1}(2\pi)^{2n}(2n-1)}\,,\quad n\geq
2\,.$ (6.14)

The same kind of analysis of the infrared effective action can be repeated in
the presence of other couplings. Thus, in the generic case $\alpha\nu_{2}\neq
0$ the two leading terms in the expansion of $s$ are

$s=-\frac{1}{2}\ln
2+\frac{\nu_{1}+\nu_{2}}{12\alpha\nu_{2}}+7\frac{3\nu_{1}\nu_{2}\alpha^{2}-(\nu_{1}+\nu_{2})^{3}}{720\alpha^{3}\nu_{2}^{3}}+\dots$
(6.15)

that gives the first two couplings in the effective action (6.10 )

$\lambda_{2}=\frac{m_{1}^{2}+m_{2}^{2}}{(2\pi)^{2}h_{-}^{2}m_{2}^{2}}\,,\quad\lambda_{4}=\frac{1}{48\pi^{4}}\left(\frac{6\pi
m_{1}^{2}}{h_{-}^{2}m_{2}^{4}}-\frac{1}{8h_{-}^{6}}\left(1+\frac{m_{1}^{2}}{m_{2}^{2}}\right)^{2}\right)\,.$
(6.16)

We clearly see that this couplings are singular when $\alpha\nu_{2}$ tends to
zero. This reflects the obvious fact that the $\beta\to\infty$ limit does not
commute with the $\alpha\nu_{2}\to 0$ limit in the coupling space. It would be
interesting to investigate further this singularity in particular in
conjunction with the RG operators discussed in [39]. It is straightforward to
obtain all other couplings in the effective action. They can be expressed in
terms of symmetrised sums of odd inverse powers of the roots $k_{i}$. We
discuss the effective action (6.10) a bit more in the last section of the
paper.

##  7 Concluding remarks

In this section we would like to point some directions in which the present
work can be extended. Firstly we would like to note that the three-coupling
model (2) with $h_{+}=0$ can be further investigated. It would be interesting
to calculate some local correlation functions, in particular the local
magnetisation that would further elucidate the boundary bound states and the
phase structure of the model. Secondly, this model presents a nice toy model
of RG flows in a multi-coupling space with several fixed points present. It
would be interesting to investigate the singularities in the space of flows
further by finding the RG operators and investigating the transport of local
operators along the flows as discussed in [39]. The complete infrared
effective action (6.10) that we obtained should be a useful tool here. It is
interesting to note its much greater simplicity in comparison to the infrared
expansion obtained in [18] using the mode truncation regularisation and the
Schriffer-Wolf method.

Furthermore, there are several directions in which the present work could go
beyond the three-coupling Gaussian model. Integrable boundary conditions in
the presence of bulk four-fermion interactions were found in [40] at the
classical level. It is foreseeable then that for some restricted values of the
couplings the four-coupling model (2) is integrable in the presence of the
boundary four-fermion interaction governed by $h_{+}$. To study this question
systematically we searched for classically integrable boundary conditions in
the model (2). We found that the dimension 4 current $(T_{4},\bar{T}_{4})$ and
the dimension 6 current $(T_{6},\bar{T}_{6})$ are each classically conserved
if and only if $m_{1}h_{+}h_{-}=0$. Some details of these calculations are
presented in the appendix. This suggests that if the discontinuities are
present at all the four coupling model is integrable only if $h_{1}=\pm
h_{2}$. It may be possible to analyse the generic $N=2$ model approximately
using the Hubbard-Stratonovich transformation. We plan to pursue this
direction in future work.

For $N\geq 3$ we can represent the boundary perturbation (1.2) in terms of a
larger number of boundary fermions similar to (2). The quadratic models do not
seem to contain any new interesting cases. It would be interesting to extend
the integrability analysis done in the appendix to this case to see whether
there are any new integrable models with non-trivial four-fermion boundary
interaction.

Although we relied heavily on the free fermion representation we find our
results suggestive that there may be interesting integrable boundary theories
which are obtained by deforming a superposition of boundary conditions in a
minimal model by a collection of boundary condition changing operators with
weights zero and $h_{(1,3)}$ that are known to give integrable deformations
for irreducible Cardy boundary conditions. As the exact partition function
(4.16) we obtained in the present model satisfies non-trivial factorisation
conditions, one way to search for new integrable deformations could be using
the results of [41] and generalising them to deformations in superpositions.

One of the motivations for the present work was the problem of obtaining some
description of a complete set of boundary RG flows in the Ising model. This
space includes all flows generated by relevant boundary fields starting from
an arbitrary superposition of the free and fixed spin conformal boundary
conditions. Unlike any space of bulk RG flows, in which there always could be
flows from an arbitrary UV fixed point with a high enough central charge
ending in a given IR CFT, a space of boundary flows in a fixed bulk CFT is
much more contained. The bulk CFT does not change and all conformal boundary
conditions may be known as is the case in the Ising model or more generally in
any Virasoro minimal model. With these observations in mind we could try a
bottom-up approach to describing the space of such flows. An irrelevant
boundary deformation in a theory where all conformal boundary conditions are
known, in the deep UV region would either lead to a known conformal boundary
condition or not to lead to any local boundary condition at all. Starting with
[42] an investigation into the structure of the space of $T\bar{T}$ and more
general irrelevant integrable deformations has been going on. In [43], [44]
the question of finding additions of higher dimension irrelevant operators
that ensure a UV completion was considered. A similar question can be posed
for boundary deformations – find integrable deformations by irrelevant
operators that have a UV completion. Note that the boundary deformation by the
stress-energy operator $T$ has many similarities to the $T\bar{T}$ bulk
deformation. As the stress-tensor operator is universal the deformation can be
unambiguously defined at any value of the coupling. It is in some sense
topological as the perturbation contour of integration can be moved away from
the boundary using holomorphy. The perturbation’s behaviour depends on the
sign of the coupling with one direction being singular. This singularity is
signified by the divergence of a cylinder partition function999This
singularity of the boundary $T$-perturbation has been discussed in detail in
[45]. at a critical cylinder length that is similar to the Hagedorn behaviour
of the partition function in the $T\bar{T}$ deformation case. The effective
action (6.10) with the couplings (6.14) give an explicit example of how adding
an infinite sequence of irrelevant boundary interactions to a $T$-deformation
leads to a UV-complete theory. The perturbing terms in (6.10) are the KdV
charges and the cylinder partition function can be built from the chiral
traces which were recently considered in [36] in connection with generalised
Gibbs ensembles. We believe this connection needs to be investigated further
and may lead to interesting results on he space of all boundary flows in the
Ising model. We leave this to future work.

Acknowledgements

The author thanks R. Weston for useful discussions and the anonymous referees
for valuable comments on the first version of the paper.

##  Appendix A Classical integrability of the generic four coupling model

In this appendix we investigate classical integrability of the theory defined
in (2). For simplicity we consider the conformal point $m=0$. The boundary
conditions in the generic $h_{+}h_{-}\neq 0$ case read

$\displaystyle\psi-\bar{\psi}$ $\displaystyle=$ $\displaystyle 4\pi
h_{-}a-4\pi ih_{+}abc\,,$ $\displaystyle\dot{a}$ $\displaystyle=$
$\displaystyle-ih_{-}(\psi+\bar{\psi})-h_{+}bc(\psi+\bar{\psi})-m_{1}ib\,,$
$\displaystyle\dot{b}$ $\displaystyle=$ $\displaystyle
h_{+}ac(\psi+\bar{\psi})+im_{1}a-im_{2}c\,,$ $\displaystyle\dot{c}$
$\displaystyle=$ $\displaystyle-h_{+}ab(\psi+\bar{\psi})+im_{2}b\,$ (A.1)

where for brevity we suppress the argument $t$ in all functions.

The bulk currents whose conservation we investigate are

$T(z)=-\frac{1}{2}:\\!\psi\partial_{z}\psi\\!:\,,\qquad
T_{2n}(z)=:\\!\psi\partial^{2n-1}_{z}\psi\\!:\,,\enspace n\geq 2$ (A.2)

and their anti-holomorphic counterparts in which $\partial_{z}$ is replaced by
$\partial_{\bar{z}}$. For the stress energy tensor the boundary conservation
equation on a half plane is

$(T-\bar{T})\Bigr{|}_{z=\bar{z}=t}=-2\pi i\partial_{t}\theta(t)$ (A.3)

where $\theta(t)$ is the boundary part of the stress-energy tensor in standard
normalisation (see e.g. [26] ). A higher spin current $T_{2n}$ is conserved if

$(T_{2n}-\bar{T}_{2n})\Bigr{|}_{z=\bar{z}=t}=\partial_{t}\theta_{2n}(t)$ (A.4)

for some boundary operator $\theta_{2n}(t)$. As a warm up we first derive
$\theta$ and then investigate the conservation of $T_{4}$ and $T_{6}$. We will
be using the classical boundary conditions (2) in which all fermionic fields
are considered to be anti-commuting.

We start by noting the following expressions for derivatives of dimension zero
operators that follows from (A)

$\displaystyle\partial_{t}(ab)$ $\displaystyle=$ $\displaystyle
ih_{-}b(\psi+\bar{\psi})-im_{2}ac\,,$ $\displaystyle\partial_{t}(ac)$
$\displaystyle=$ $\displaystyle ih_{-}c(\psi+\bar{\psi})-im_{1}bc+im_{2}ab\,,$
$\displaystyle\partial_{t}(bc)$ $\displaystyle=$ $\displaystyle im_{1}ac\,,$
$\displaystyle\partial_{t}(abc)$ $\displaystyle=$ $\displaystyle-
ih_{-}bc(\psi+\bar{\psi})\,.$ (A.5)

To find $\theta$ we differentiate the first equation in (A) to obtain

$\partial_{t}\psi-\partial_{t}\bar{\psi}=-i4\pi
h_{-}^{2}(\psi+\bar{\psi})-8\pi h_{-}h_{+}bc(\psi+\bar{\psi})-4\pi
h_{-}m_{1}ib\,.$ (A.6)

Multiplying this equation by $\psi$ and separately by $\bar{\psi}$ on the left
we obtain the following two equations

$\displaystyle\psi\partial_{t}\psi-\psi\partial_{t}\bar{\psi}$
$\displaystyle=$ $\displaystyle-i4\pi h_{-}^{2}\psi\bar{\psi}-8\pi
h_{-}h_{+}\psi\bar{\psi}bc-4\pi h_{-}m_{1}i\psi b\,,$
$\displaystyle\bar{\psi}\partial_{t}\psi-\bar{\psi}\partial_{t}\bar{\psi}$
$\displaystyle=$ $\displaystyle-i4\pi h_{-}^{2}\bar{\psi}\psi-8\pi
h_{-}h_{+}\bar{\psi}\psi bc-4\pi h_{-}m_{1}i\bar{\psi}b$ (A.7)

Adding these two equations we obtain

$\psi\partial_{t}\psi-\bar{\psi}\partial_{t}\bar{\psi}=\partial_{t}(\psi\bar{\psi})-4\pi
h_{-}m_{1}i(\psi+\bar{\psi})b\,.$ (A.8)

From (A) we obtain

$m_{1}\partial_{t}(ba)+m_{2}\partial_{t}(cb)=ih_{-}m_{1}(\psi+\bar{\psi})b\,.$
(A.9)

The last formula allows us to rewrite (A.8) as

$\psi\partial_{t}\psi-\bar{\psi}\partial_{t}\bar{\psi}=4\pi\partial_{t}\Bigl{[}\frac{1}{4\pi}\psi\bar{\psi}+m_{1}ab+m_{2}bc\Bigr{]}\,.$
(A.10)

Comparing this with (A.3) we find that

$\theta(t)=-i\Bigl{[}\frac{1}{4\pi}\psi\bar{\psi}+m_{1}ab+m_{2}bc\Bigr{]}$
(A.11)

that can be also rewritten as

$\theta(t)=-\Bigl{[}\frac{ih_{-}}{2}a(\psi+\bar{\psi})+\frac{h_{+}}{2}abc(\psi+\bar{\psi})+im_{1}ab+im_{2}bc\Bigr{]}$
(A.12)

that has the standard form of the sum of perturbing operators multiplied by
their beta functions.

We next take up the dimension 4 currents: $T_{4}$ and $\bar{T}_{4}$. To find
their difference on the boundary we differentiate (A.6). From that, using (A)
and (A), we obtain

$\displaystyle\partial_{t}^{2}\psi-\partial_{t}^{2}\bar{\psi}$
$\displaystyle=-4\pi ih_{-}^{2}(\partial_{t}\psi+\partial_{t}\bar{\psi})-8\pi
h_{-}h_{+}bc(\partial_{t}\psi+\partial_{t}\bar{\psi})$ (A.13)
$\displaystyle+24\pi ih_{-}h_{+}ca(\psi+\bar{\psi})+4\pi m_{1}^{2}h_{-}a-4\pi
m_{1}m_{2}h_{-}c$

Differentiating one more time we get

$\displaystyle\partial_{t}^{3}\psi-\partial_{t}^{3}\bar{\psi}=-4\pi
ih_{-}^{2}(\partial_{t}^{2}\psi+\partial_{t}^{2}\bar{\psi})-8\pi
h_{-}h_{+}bc(\partial_{t}^{2}\psi+\partial_{t}^{2}\bar{\psi})$
$\displaystyle+20\pi
h_{-}h_{+}m_{1}ica(\partial_{t}\psi+\partial_{t}\bar{\psi})+16\pi
m_{1}^{2}h_{-}h_{+}cb(\psi+\bar{\psi})$ $\displaystyle+16\pi
m_{1}m_{2}h_{-}h_{+}ab(\psi+\bar{\psi})-4\pi
m_{1}^{2}h_{-}^{2}i(\psi+\bar{\psi})-4\pi m_{1}h_{-}(m_{1}^{2}+m_{2}^{2})ib$
(A.14)

Next we multiply both sides of this equation by $\psi+\bar{\psi}$ from the
left to obtain

$\displaystyle(T_{4}-\bar{T}_{4}))\Bigr{|}_{z=\bar{z}=t}=\psi\partial_{t}^{3}\psi-\bar{\psi}\partial_{t}^{3}\bar{\psi}=X_{1}-4\pi
ih_{-}^{2}X_{2}-8\pi h_{-}h_{+}bcX_{2}$ $\displaystyle+20\pi
h_{-}h_{+}m_{1}icaX_{3}-4\pi
m_{1}h_{-}(m_{1}^{2}+m_{2}^{2})i(\psi+\bar{\psi})b$ (A.15)

where

$X_{1}\equiv\psi\partial_{t}^{3}\bar{\psi}-\bar{\psi}\partial_{t}^{3}\psi=\partial_{t}[(\partial_{t}\bar{\psi})\partial_{t}\psi+\psi\partial_{t}^{2}\bar{\psi}-\bar{\psi}\partial_{t}^{2}\psi]\,,$
(A.16)
$X_{2}\equiv(\psi+\bar{\psi})(\partial_{t}^{2}\psi+\partial_{t}^{2}\bar{\psi})=\partial_{t}[\psi\partial_{t}\psi+\bar{\psi}\partial_{t}\bar{\psi}+\psi\partial_{t}\bar{\psi}+\bar{\psi}\partial_{t}\psi]$
(A.17)
$X_{3}\equiv(\psi+\bar{\psi})(\partial_{t}\psi+\partial_{t}\bar{\psi})\,.$
(A.18)

Note that

$X_{2}=\partial_{t}X_{3}\,.$ (A.19)

Since by (A.9) the term containing $(\psi+\bar{\psi})b$ is also a derivative
the problematic terms are

$\Xi\equiv-8\pi h_{-}h_{+}bcX_{2}+20\pi h_{-}h_{+}m_{1}icaX_{3}\,.$ (A.20)

We can rewrite this as

$\displaystyle\Xi=-8\pi h_{-}h_{+}bc\partial_{t}X_{3}+40\pi
h_{-}h_{+}m_{1}icaX_{3}$ $\displaystyle=\partial_{t}(-8\pi
h_{-}h_{+}bcX_{3})+8\pi h_{-}h_{+}\partial_{t}(bc)X_{3}+20\pi
h_{-}h_{+}m_{1}icaX_{3}$ $\displaystyle=\partial_{t}(-8\pi
h_{-}h_{+}bcX_{3})+(20\pi-8\pi)h_{-}h_{+}m_{1}icaX_{3}$
$\displaystyle=\partial_{t}(-8\pi h_{-}h_{+}bcX_{3})+12\pi
h_{-}h_{+}m_{1}icaX_{3}$ (A.21)

and the obstruction for the difference $T_{4}-\bar{T}_{4}$ to be a derivative
now looks very compact. If $h_{-}h_{+}m_{1}\neq 0$ then to show that the
dimension 4 current is not conserved we have to prove that

${\cal O}\equiv
caX_{3}=ca(\psi+\bar{\psi})(\partial_{t}\psi+\partial_{t}\bar{\psi})=4ca\psi\partial_{t}\psi-8\pi
im_{1}h_{-}abc\psi$ (A.22)

cannot be written as a derivative of another operator. Although applying the
derivative always rises the dimension by one we cannot use the dimension
grading as the couplings also carry the natural dimension and there could be
divisions by the couplings present. Instead we can rely on the grading
provided by the number of $\psi$-fields and their derivatives present in a
composite operator. As no derivative of $\psi$ or $\bar{\psi}$ is present in
the boundary conditions (A) any operator can be represented as a finite linear
combination of operators of the form

$f(a,b,c)\partial_{t}^{k_{1}}\psi\dots\partial_{t}^{k_{n}}\psi\,,\enspace
k_{i}\geq 0$ (A.23)

where $f(a,b,c)$ is a polynomial in boundary fermions. The derivative of such
an operator is a linear combination of operators of the same form.
Schematically we have

$\displaystyle\partial_{t}[f(a,b,c)\partial_{t}^{k_{1}}\psi\dots\partial_{t}^{k_{n}}\psi]=f_{1}(a,b,c)\partial_{t}^{k_{1}}\psi\dots\partial_{t}^{k_{n}}\psi$
$\displaystyle+f_{2}(a,b,c)\psi\partial_{t}^{k_{1}}\psi\dots\partial_{t}^{k_{n}}\psi+f(a,b,c)\partial_{t}(\partial_{t}^{k_{1}}\psi\dots\partial_{t}^{k_{n}}\psi)$
(A.24)

where $f_{1}$ and $f_{2}$ are some new polynomials. This means that when we
take the derivative of (A.23) we obtain only operators with either $n$ or
$n+1$ $\psi$-fields and the total number of derivatives acting on them can
increase only by one. This immediately implies that there are only 4 possible
operator types whose derivative could give ${\cal O}$. Schematically we can
write them as

$\displaystyle P=p(a,b,c)\psi\partial_{t}\psi\,,\qquad
Q=q(a,b,c)\partial_{t}\psi\,$ $\displaystyle R=r(a,b,c)\psi\,,\qquad
S=s(a,b,c)$ (A.25)

where $p,q,r,s$ are some polynomials in boundary fermions whose coefficients
can depend on the couplings . In the notation introduced in (A) the
derivatives of these operators can be written as

$\partial_{t}P=p_{1}(a,b,c)\psi\partial_{t}\psi+p(a,b,c)\psi\partial_{t}^{2}\psi\,,$
(A.26)
$\partial_{t}Q=q_{1}(a,b,c)\partial_{t}\psi+q_{2}(a,b,c)\psi\partial_{t}\psi+q(a,b,c)\partial_{t}^{2}\psi\,,$
(A.27) $\partial_{t}R=r_{1}(a,b,c)\psi+r(a,b,c)\partial_{t}\psi\,,$ (A.28)
$\partial_{t}S=s_{1}(a,b,c)+s_{2}(a,b,c)\psi\,.$ (A.29)

From these expressions we see that the only way to ensure
$\psi\partial_{t}\psi$ is present is to include at least one of the operators
$P$ and $Q$. But the first one necessarily generates
$\psi\partial_{t}^{2}\psi$ while the second generates $\partial_{t}^{2}\psi$
which are not present in ${\cal O}$. We conclude that the operator ${\cal O}$
cannot be written as a derivative and the current $(T_{4},\bar{T}_{4})$ is
conserved if and only if $h_{-}h_{+}m_{1}=0$.

In a similar fashion we also analysed the conservation of the current
$(T_{6},\bar{T}_{6})$. Omitting the details, we found that

$\displaystyle(T_{6}-\bar{T}_{6})\Bigr{|}_{z=\bar{z}=t}=$ $\displaystyle 20\pi
im_{1}h_{+}h_{-}[(m_{1}^{2}+m_{2}^{2}){\cal
O}+4ac\partial_{t}\psi\partial_{t}^{2}\psi+16\pi
ih_{-}^{2}ac\psi\partial_{t}^{2}\psi$ (A.30) $\displaystyle-8\pi
im_{1}h_{-}abc\partial_{t}^{2}\psi]+\partial_{t}Y$

where $Y$ is some local operator and ${\cal O}$ is given in (A.22). Analysing
the expression in the square brackets as above we find that it cannot be
represented as a derivative. Hence the current $(T_{6},\bar{T}_{6})$ is
conserved if and only if $h_{-}h_{+}m_{1}=0$. Moreover, the conclusion holds
even if we take a linear combination of $T_{4}$ and $T_{6}$ which removes the
${\cal O}$ field from (A.30). The essential obstruction in (A) and (A.30) for
being represented as derivatives is the presence of operators
$ac\psi\partial_{t}\psi$ and $ac\psi\partial_{t}^{3}\psi$ which are the
operators with the highest number of derivatives in each expression.

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