#  $OSp(n|2m)$ quantum chains with free boundaries

Holger Frahm  Institut für Theoretische Physik, Leibniz Universität Hannover,
Appelstraße 2, 30167 Hannover, Germany  Márcio J. Martins  Departamento de
Física, Universidade Federal de São Carlos, C.P. 676, 13565-905 São Carlos
(SP), Brazil

###### Abstract

In this paper we investigate the spectrum of $OSp(n|2m)$ quantum spin chains
with free boundary conditions. We compute the surface free energy of these
models which, similar to other properties in the thermodynamic limit including
the effective central charge of the underlying conformal field theory, depends
on $n-2m$ only. For several models in the regime $n-2m<2$ we have studied the
finite-size properties including the subleading logarithmic corrections to
scaling. As in the case of periodic boundary conditions we find the existence
of a tower of states with the same conformal dimension as the identity
operator. As expected the amplitudes of the corresponding logarithmic
corrections differ from those found previously for the models with periodic
boundary conditions. We point out however the existence of simple relations
connecting such amplitudes for free and periodic boundaries. Based on our
findings we formulate a conjecture on the long distance behaviour of the bulk
and surface watermelon correlators.

##  I Introduction

In recent years there has been some interest in the study of the critical
properties of integrable one-dimensional quantum spin chains based on
supergroup symmetries because of their mathematical and physical implications.
For instance, the staggered $sl(2|1)$ superspin chain with spins alternating
between the fundamental and dual representations may be of relevance for the
description of properties of fermions in the presence of random potentials
Gade (1999); Essler  _et al._ (2005). Yet another example are the spin chains
invariant by the fundamental vector representation of the $OSp(n|2m)$
superalgebra which can be related to an intersecting loop model on the square
lattice with fugacity $z=n-2m$ Martins  _et al._ (1998). This loop model
describes the motion of particles through randomly fixed scatterers in such
way that path intersections are allowed. For $n-2m<2$ the crossing of the
loops appears to become a relevant perturbation and model properties have been
argued to be those of the Goldstone phase of the $O(z)$ sigma model Jacobsen
_et al._ (2003). The spectrum of these superspin chains for large number of
sites $L$ present some distinguished features when compared to that of spin
chains based on ordinary Lie groups. For example, it was observed that several
of the scaled gaps appear to produce the same conformal weight implying a
macroscopic degeneracy of the respective state in the thermodynamic limit
Martins  _et al._ (1998); Frahm and Martins (2018). A similar observation in
the staggered $sl(2|1)$ superspin chain has been interpreted as signature for
a continuous component to the spectrum of conformal weight resulting from a
non-compact target space of the conformal field theory associated to these
superspin chains Essler  _et al._ (2005); Saleur and Schomerus (2007). At this
point we remark that such scenario has also been found in other families of
staggered vertex models Ikhlef  _et al._ (2008); Frahm and Martins (2011,
2012); Frahm and Hobuß (2017) and quantum deformations of superspin chains
Vernier  _et al._ (2014a, b, 2016); Frahm  _et al._ (2019).

We further motivate this work by mentioning some of our earlier findings
concerning the eigenspectrum behaviour of $OSp(n|2m)$ superspin chains with
periodic boundary conditions Frahm and Martins (2015, 2018): in these models
there exist towers of low energy excitations over the ground state which for
large system sizes leads to the same effective central charge
$c_{\mathrm{eff}}$. If we denote the energies of such set of states by
$E_{k}(L)$ we have found that the behaviour for ${L}\rightarrow\infty$ is

$E_{k}(L)=Le_{\infty}+\frac{2\pi
v_{F}}{L}\left(-\frac{c_{\mathrm{eff}}}{12}+\frac{\beta(k)}{\log
L}\right)\,,\quad k=0,1,2,\cdots,k_{\infty}$ (1)

where the integer $k_{\infty}$ is limited by system size $\mathrm{L}$,
$e_{\infty}$ refers to the ground state energy per site in the thermodynamic
limit and $v_{F}$ denotes the velocity of the low-lying excitations. We next
observe that in the regime $n-2m<2$ certain correlation functions of the
related loop model can be rewritten in terms of the subleading logarithmic
amplitudes $\beta_{k}$. These ’watermelon correlators’ measure the probability
of $k$ distinct loop segments connecting two arbitrary lattice points $x$ and
$y$ which for large distances $r=|x-y|$ which has been argued to decrease
logarithmically with $r$ Polyakov (1975); Nahum  _et al._ (2013). As we have
pointed out in Ref. Frahm and Martins (2018) this behaviour of bulk
correlation functions can be re-written in terms of the finite-size
logarithmic amplitudes as follows

$G^{(b)}_{k}(r)\sim 1/\ln(r)^{2(\beta(k)-\beta(k_{0}))}\,$ (2)

for a suitable choice of the $k_{0}$ state.

The purpose of the present paper is to investigate the effect of boundary
conditions on the spectrum of conformal weights of the $OSp(n|2m)$ superspin
chains. Generally, knowing the properties of a critical system under various
boundary conditions is a prerequisite for the identification of the full
operator content of a given universality class Cardy (1984, 1986a). Moreover,
recent studies of the staggered six-vertex model have revealed that open
boundary conditions may change its low energy properties significantly
Robertson  _et al._ (2020, 2021); Frahm and Gehrmann (2022). Here we shall
present evidence that the tower of low energy states over the identity
operator present in the $OSp(n|2m)$ models with periodic boundary conditions
continues to exist in the presence of free boundary conditions. As we shall
argue these states have the following finite-size structure as $L\to\infty$:

$E_{k}(L)=Le_{\infty}+f_{\infty}+\frac{\pi
v_{F}}{L}\left(-\frac{c_{\mathrm{eff}}}{24}+\frac{\alpha(k)}{\log
L}\right)\,,\quad k=0,1,2,\cdots,k_{\infty}$ (3)

where $f_{\infty}$ is the surface energy resulting from the free boundary
conditions. The leading finite-size term in (3) is in accordance with the
predictions for conformally invariant theories with free boundary conditions
Blöte  _et al._ (1986). Even for conventional conformal theories, however, the
subleading corrections are expected to depend on the boundary terms Cardy
(1986b); Affleck and Qin (1999). Indeed, we find that the amplitudes
$\alpha_{k}$ and $\beta_{k}$ differ. For $n-2m<2$ we present evidences that
they appear to obey the rather simple relation,

$\alpha(k)-\alpha({k_{0}})=2(\beta(k)-\beta({k_{0}}))\,,\quad n-2m<2$ (4)

for the similar choice of the state $k_{0}$ for both free and periodic
boundary conditions.

It is now tempting to use the above relationship among logarithmic amplitudes
and the asymptotic behaviour of correlators to infer about the behaviour of
surface watermelon correlators for large distances. Recall here that free
boundary conditions play the role of Dirichlet boundary conditions in which
the order parameters entering the correlators are expected to vanish on the
boundary. Let us denote the surface watermelon correlator by
$G^{(s)}_{k}(\rho)$ where $\rho$ is the distance between to points $x$ and $y$
parallel the half-plane boundary. Considering that the asymptotic behaviour of
such correlators should be governed instead by the surfaces amplitudes
$\alpha_{k}$ one obtains,

$G^{(s)}_{k}(\rho)\sim 1/\ln(\rho)^{4(\beta({k})-\beta({k_{0}}))}\,$ (5)

and hence a faster logarithmic surface decay as compared with the bulk
behaviour by a factor two. Note that this dependence on $\rho$ is very
different from that of polymers, i.e. loops without intersections Duplantier
and Saleur (1986).

##  II The open $OSp(n|2m)$ spin chain properties

In this section we describe the thermodynamic limit properties of spin chains
based on the vector representation of the $OSp(n|2m)$ superalgebra with free
boundary conditions. The model Hamiltonian can be represented in terms of
generators of a braid-monoid algebra which underpins a square lattice loop
model admitting intersections between the polygon configurations Martins  _et
al._ (1998). The Hamiltonian of the spin chain in an one-dimensional lattice
of size $L$ is given by

$\mathrm{H}=\epsilon\sum_{i=1}^{L-1}\left[\mathrm{P}_{i,i+1}+\frac{2}{2-z}\mathrm{E}_{i,i+1}\right]\,,$
(6)

where we chose $\epsilon$ to select the anti-ferromagnetic regime of the
model, i.e. $\epsilon=-1$ $(+1)$ for $n-2m<2$ ($>2$). Note that Eq. (6)
describes the superspin chain with free boundary conditions. The fugacity $z$
of the related intersecting loop model is realized in the spin chain as the
difference between the number of the bosonic and fermionic degrees of freedom
$z=n-2m$.

The braid $\mathrm{P}_{i,i+1}$ turns out to the graded permutation operator
whose expression is,

$\mathrm{P}_{i,i+1}=\sum_{\alpha,\beta=1}^{n+2m}(-1)^{p_{\alpha}p_{\beta}}e_{\alpha\beta}\otimes
e_{\beta\alpha}$ (7)

where $p_{\alpha}$ are the Grassmann parities for the $n$ bosonic
($p_{\alpha}=0$) and the $2m$ fermionic ($p_{\alpha}=1$) degrees of freedom.
The matrices $e_{\alpha\beta}$ have only one non-vanishing element with value
1 at row $\alpha$ and column $\beta$. The operator $\mathrm{E}_{i,i+1}$ is a
generator of the Temperley-Lieb algebra weighted by the fugacity $z$. It can
be represented by the expression,

$\mathrm{E}_{i,i+1}=\sum_{\alpha,\beta,\gamma,\delta=1}^{n+2m}A_{\alpha\beta}A^{-1}_{\gamma\delta}e_{\alpha\gamma}\otimes
e_{\beta\delta}$ (8)

where the non-zero matrix elements $A_{\alpha\beta}$ are $\pm 1$ such that
their matrix positions depend on the grading ordering of the basis. For
explicit matrix representations of the Temperley-Lieb generator see for
instance Martins and Ramos (1997).

Before proceeding we remark that the quantum integrability of the Hamiltonian
(6) can be established within the double row transfer matrix framework devised
by Sklyanin for the Heisenberg chain Sklyanin (1988). In this method the
Hamiltonian boundary terms depend on the certain one-body scattering matrices
on the half-line. In the specific case of free boundary conditions considered
in this paper these reflecting matrices are trivial being proportional to the
identity operators. For the details about the technical points concerning this
construction for the open $OSp(n|2m)$ spin chain see for instance Arnaudon
_et al._ (2003, 2004).

We have studied the eigenspectrum properties of open $OSp(n|2m)$ spin chain
(6) for some values of the numbers $n$ of bosonic and $2m$ of fermionic
degrees of freedom. Our numerical results for small lattice sizes suggest we
have the following sequence of spectral inclusions,

$\mathrm{Spec}[OSp(n|2m)]\subset\mathrm{Spec}[OSp(n+2|2(m+1))]\subset\mathrm{Spec}[OSp(n+4|2(m+2))]\subset\dots$
(9)

similar to what happens for periodic conditions Frahm and Martins (2018);
Granet  _et al._ (2019). As a consequence the basic properties of the
Hamiltonian (6) are expected to depend solely on the fugacity $z=n-2m$ in the
thermodynamic limit. In addition to that it is known that the low-lying
excitations of the $OSp(n|2m)$ with periodic boundary conditions are gapless
Martins  _et al._ (1998); Frahm and Martins (2018). This feature is not
expected to depend on the boundary conditions. As a consequence of that the
ground state energy of the Hamiltonian (6) should scale with the lattice size
$L$ as Blöte  _et al._ (1986),

$E_{0}(L)\simeq Le_{\infty}+f_{\infty}-\frac{\pi
v_{F}c_{\mathrm{eff}}}{24L}\,,$ (10)

where $c_{\mathrm{eff}}$ is the effective central charge of the respective
conformal field theory. This invariant is expected to be the same as the one
underlying the model with periodic boundary conditions Martins  _et al._
(1998); Frahm and Martins (2018)

$c_{\mathrm{eff}}=\begin{cases}z/2&\mathrm{for}~{}~{}~{}z\geq 2\\\
z-1&\mathrm{for}~{}~{}~{}z<2\\\ \end{cases}\,.$ (11)

The parameters $e_{\infty}$ and $v_{F}$ denote the bulk ground state energy
and the Fermi velocity of the elementary excitations. Again, these bulk
quantities are expected not to depend of the boundary conditions, hence their
values are known to be Martins  _et al._ (1998); Frahm and Martins (2018)
given by

$e_{\infty}=-\frac{2}{|2-z|}\left[\psi\left(\frac{1}{2}+\frac{1}{|2-z|}\right)-\psi\left(\frac{1}{|2-z|}\right)+2\ln(2)\right]+1\,,$
(12)

where $\psi(x)$ is the Euler $\mathrm{psi}$ function, and the speed of sound
is $v_{F}=2\pi/|2-z|$.

By way of contrast the surface energy $f_{\infty}$ depends on the boundary
conditions which are imposed on the spin chain. Using the root density method
Yang and Yang (1969) we compute this quantity for the $OSp(1|2m)$, $OSp(2|2m)$
and various $O(n)$ models in Appendix A. Together with our numerical results
for the the $OSp(3|2)$ model this leads us to conjecture the expression of the
surface energy for the generic $OSp(n|2m)$ superspin chain with free boundary
conditions to be

$\displaystyle f_{\infty}=-\frac{1}{2-z}$
$\displaystyle\left[\psi\left(1+\frac{1}{2(2-z)}\right)-\psi\left(\frac{1}{2}+\frac{1}{2(2-z)}\right)-\psi\left(\frac{3}{4}+\frac{1}{2(2-z)}\right)\right.$
(13)
$\displaystyle\left.+\psi\left(\frac{1}{4}+\frac{1}{2(2-z)}\right)+2\ln(2)-\pi\right]+1,\qquad\mathrm{for~{}}z<2$

and

$\displaystyle f_{\infty}=-\frac{1}{z-2}$
$\displaystyle\left[\psi\left(1+\frac{1}{2(z-2)}\right)-\psi\left(\frac{1}{2}+\frac{1}{2(z-2)}\right)+\psi\left(\frac{3}{4}+\frac{1}{2(z-2)}\right)\right.$
(14)
$\displaystyle\left.-\psi\left(\frac{1}{4}+\frac{1}{2(z-2)}\right)+2\ln(2)-\pi\right]+1,\qquad\mathrm{for~{}}z>2$

Note the difference in signs of the last two Euler psi functions between the
regimes $z<2$ and $z>2$: among the non-universal quantities describing the
thermodynamics of the models the bulk energy and Fermi velocity of the
$OSp(2|2(m-1))$ and $O(2m)$ spin chains coincide while their surface energies
differ. The same is true for the effective central charge (11) characteristic
for the universal critical behaviour described by the underlying conformal
field theory.

The model with $z=2$ has to be dealt with separately since the Hamiltonian (6)
becomes dominated by the Temperley-Lieb operator. The simplest realization of
this model is that with $n=2$ and $m=0$ which corresponds to the isotropic
spin-$1/2$ Heisenberg model,

$\mathrm{H}=\frac{1}{2}\sum_{i=1}^{L-1}\left[\sigma_{i}^{x}\sigma_{i+1}^{x}+\sigma_{i}^{y}\sigma_{i+1}^{y}+\sigma_{i}^{z}\sigma_{i+1}^{z}-\mathrm{I}_{i,i+1}\right]$
(15)

where $\sigma_{i}^{x},\sigma_{i}^{y},\sigma_{i}^{z}$ are Pauli matrices acting
on the $i$-th lattice site and $\mathrm{I}_{i,i+1}$ is the $4\times 4$
identity matrix. It turns out that the bulk and the surface energies of this
model may be obtained by considering the limit $z\rightarrow 2$ in Eqs. (12)
and (13). The coefficients proportional to $\mathcal{O}(1/\varepsilon)$ in the
expansion around $z=2(1-\varepsilon)$ turns out to be the respective values
for $e_{\infty}$ and $f_{\infty}$ associated to the Heisenberg chain (15)
Hulthén (1939); Hamer  _et al._ (1987). The Fermi velocity of massless
excitations in this model is $v_{F}=\pi$.

In Table 1 we present the parameters data characterizing the thermodynamic
limit of some of the $OSp(n|2m)$ models including the ones whose critical
properties we are going to analyze further below.

$z$ | $e_{\infty}$ | $f_{\infty}$ | $v_{F}$ | $c_{\mathrm{eff}}$
---|---|---|---|---
$-2$ | $-\frac{\pi}{2}-\ln(2)+1$ | $\frac{\pi}{4}(1+2\sqrt{2})-\frac{\ln(2)}{2}-1$ | $\frac{\pi}{2}$ | $-3$
$-1$ | $-\frac{4\pi\sqrt{3}}{9}+1$ | $\pi+\frac{2\pi}{3\sqrt{3}}-\frac{2}{\sqrt{3}}\ln(2+\sqrt{3})-1$ | $\frac{2\pi}{3}$ | $-2$
$0$ |  $-4\ln(2)$+1 | $\pi-1$ | $\pi$ | $-1$
$1$ | $-3$ | $3$ | $2\pi$ | $0$
$2$ | $-2\ln(2)$ | $\frac{\pi}{2}-\ln(2)$ | $\pi$ | $1$
$3$ | $-3$ | $2\pi-5$ | $2\pi$ | $\frac{3}{2}$
$4$ | $-4\ln(2)+1$ | $\pi-2\ln(2)-1$ | $\pi$ | $2$
$5$ | $-\frac{4\sqrt{3}}{9}\pi+1$ | $\frac{2\pi}{\sqrt{3}}-\frac{\pi}{3}+\frac{2}{\sqrt{3}}\ln(2+\sqrt{3})-1$ | $\frac{2\pi}{3}$ | $\frac{5}{2}$
Table 1: The bulk and surface energies, the Fermi velocity as well as the
effective central charge for some values of the fugacity.

##  III Finite-size spectrum

We now turn to the analysis of the finite-size spectrum for the spin chains
with fugacity $z<2$ exhibited in Table 1. The leading terms appearing in the
finite-size scaling of low energy levels with quantum numbers
$Q=\\{q_{1},q_{2},\dots\\}$ of a critical model in $1+1$ dimensions are given
by conformal invariance Blöte  _et al._ (1986); Affleck (1986); Cardy (1986c):
for periodic boundary conditions they are given as

$E_{Q}(L)\simeq Le_{\infty}+\frac{2\pi
v_{F}}{L}\left(-\frac{c_{\mathrm{eff}}}{12}+X_{Q}+\dots\right)\,,$ (16)

while one has

$E_{Q}(L)\simeq Le_{\infty}+f_{\infty}+\frac{\pi
v_{F}}{L}\left(-\frac{c_{\mathrm{eff}}}{24}+X_{Q}+\dots\right)\,.$ (17)

in models with open boundary conditions. Here $c_{\mathrm{eff}}$ is the
effective central charge characterizing the universality class of the critical
point and $X_{Q}$ are the (surface) critical dimensions describing the decay
of correlations in the bulk and along the boundary, respectively.

For the $OSp(n|2m)$ spin chains the effective central charges are given by
(11) Martins  _et al._ (1998); Frahm and Martins (2015, 2018). Therefore, the
conformal weights (and possible subleading corrections to scaling) appearing
in the models with free boundaries can be extracted from (17) by extrapolation
of

$X_{\mathrm{eff},Q}(L)=\frac{L}{\pi
v_{F}}\left(E_{Q}-Le_{\infty}-f_{\infty}\right)\equiv-\frac{c_{\mathrm{eff}}}{24}+X_{Q}(L)\,.$
(18)

Based on the perturbative RG analysis the model flows to weak coupling and our
previous work on the periodic chains we expect logarithmic corrections to
scaling, i.e.

$X_{Q}(L)\simeq X_{Q}+\frac{\alpha(Q)}{\log L}+\dots\,,$ (19)

with integer conformal weights $X_{Q}$. In the following we are particularly
interested in the amplitudes $\alpha(Q)$ for the tower of levels over the
identity operator with $X_{Q}=0$ and their relation to the corresponding ones
found for the periodic spin chain Frahm and Martins (2018).

###  III.1 $z=-2$: the $OSp(2|4)$ superspin chain

The Bethe equations for the $OSp(2|4)$ model in the grading $ffbbff$ read

$\displaystyle\left[f_{1/2}\left(\lambda_{j}^{(1)}\right)\right]^{2L}=\prod_{k\neq
j}^{L-n_{1}}f_{1}\left(\lambda_{j}^{(1)}-\lambda_{k}^{(1)}\right)\,f_{1}\left(\lambda_{j}^{(1)}+\lambda_{k}^{(1)}\right)\times$
(20)
$\displaystyle\qquad\times\prod_{\sigma=\pm}\left(\prod_{k=1}^{L/2-n_{\sigma}}f_{-1/2}\left(\lambda_{j}^{(1)}-\lambda_{k}^{(\sigma)}\right)\,f_{-1/2}\left(\lambda_{j}^{(1)}+\lambda_{k}^{(\sigma)}\right)\right)\,,\quad
j=1,\dots,L-n_{1}\,,$
$\displaystyle\prod_{k=1}^{L-n_{1}}f_{1/2}\left(\lambda_{j}^{(\pm)}-\lambda_{k}^{(1)}\right)\,f_{1/2}\left(\lambda_{j}^{(\pm)}+\lambda_{k}^{(1)}\right)=\prod_{k=1}^{L/2-n_{\mp}}f_{1}\left(\lambda_{j}^{(\pm)}-\lambda_{k}^{(\mp)}\right)\,f_{1}\left(\lambda_{j}^{(\pm)}+\lambda_{k}^{(\mp)}\right)\,,$
$\displaystyle\qquad\qquad j=1,\dots,\frac{1}{2}L-n_{\pm}\,$

where we have defined

$f_{s}(x)=\frac{x+is}{x-is}\,.$ (21)

The eigenvalues of the conserved $U(1)$ charges from the Cartan subalgebra are
determined by the numbers of Bethe roots. The energy of a state parameterized
by a solution of (20) is

$E=L-1-\sum_{j=1}^{L-n_{1}}a_{1/2}\left(\lambda_{j}^{(1)}\right)\,.$ (22)

with $a_{s}(x)=i\partial_{x}\ln f_{s}(x)=2s/(x^{2}+s^{2})$. In the
thermodynamic limit, $L\to\infty$, the root configurations corresponding to
the ground state and many low energy excitations are found to consist of reals
with finite $n_{1}$, $n_{\pm}$. The ground state of even length chains is
realized in the sector with $n_{1}=n_{\pm}=1$. Solving the Bethe equations
numerically and extrapolating the finite size energies assuming a rational
dependence of the effective scaling dimension on $1/\log L$ we find

$X_{\mathrm{eff},0}^{(2|4)}(L)\simeq\frac{1}{8}-\frac{7}{16}\,\frac{1}{\log
L}\,$ (23)

corresponding to an effective central charge $c_{\mathrm{eff}}=-3$, as
expected from Eq. (11).

The lowest excitations appear in the sectors $n_{1}=1$, $n_{\pm}=1\pm k/2$
with $|k|=1,2,3,\dots\sim L\mod 2$. They, too, are parameterized by real
solutions to the Bethe equations (20). The leading finite size scaling of
these states coincides with that of the ground state, see Figure 1.

Figure 1:  Corrections to the effective scaling dimensions
$X_{\mathrm{eff}}=\frac{1}{8}+n$ with $n=0$ (in black) and $1,2$ (in grey) for
some of the low-lying states of the $OSp(2|4)$ chain. Data for even (odd)
length are presented by filled (open) symbols, data shown in red correspond to
a descendent state. Dashed lines are extrapolations to $L\to\infty$.

The subleading logarithmic corrections to scaling, however, vanish with
amplitudes depending on $|k|$ as

$\alpha^{(2|4)}(k)=\frac{1}{4}k^{2}-\frac{7}{16}\,.$ (24)

Note that these can be related to the corresponding amplitudes observed in
spectrum of the periodic $OSp(2|4)$ model, $\beta^{(2|4)}(k)=(k^{2}-1)/8$
Frahm and Martins (2018), by

$\alpha^{(2|4)}(k)=2\beta^{(2|4)}(k)-\frac{3}{16}\,.$ (25)

Similar groups of excitations corresponding to primaries with scaling
dimension $X=1$ ($2$) appear in the sectors with $n_{1}=2$ and $n_{\pm}=(3\pm
k)/2$ for $|k|=0,1,2,\dots\sim(L+1)\mod 2$ ($n_{\pm}=(2\pm k/2)$ for
$|k|=0,1,2,\dots\sim L\mod 2$).

In addition we have identified the root configuration for an excitation of the
even length superspin chain in the sector $n_{1}=n_{\pm}=1$: apart from the
real roots it contains a pair of complex conjugate roots
$\lambda_{c\pm}^{(1)}\simeq\lambda_{0}\pm i/2$ with
$\lambda_{0}\in\mathbb{R}^{+}$ on the first level and imaginary roots
$\lambda_{c}^{(+)}=-\lambda_{c}^{(-)}\simeq i/2$ (or $-i/2$) on the second and
third level. Extrapolation of the finite size data gives scaling dimensions
$X=1$, see Figure 1, indicating that this is a descendent of the ground state.

###  III.2 $z=-1$: the $OSp(1|2)$ and $OSP(3|4)$ superspin chains

####  $OSp(1|2)$.

Solutions of the Bethe equations

$\displaystyle\left[f_{1/2}\left(\lambda_{j}\right)\right]^{2L}=\prod_{k\neq
j}^{L-2n}f_{1}\left(\lambda_{j}-\lambda_{k}\right)f_{1}\left(\lambda_{j}+\lambda_{k}\right)f_{-1/2}\left(\lambda_{j}-\lambda_{k}\right)f_{-1/2}\left(\lambda_{j}+\lambda_{k}\right)\,,\quad
j=1,\dots,L-2n\,,$ (26)

parameterize highest weight states for $(4n+1)$-dimensional
$OSp(1|2)$-multiplets with superspin $J=n$ where $2n$ is a non-negative
integer. The energy of this state is

$E=(L-1)-\sum_{j=1}^{L-n}a_{1/2}\left(\lambda_{j}\right)\,.$ (27)

The lowest levels in each sector with given superspin $J>0$ are given in terms
of positive roots of the Bethe equations (26). Among these is the ground state
of the $OSp(1|2)$ chain with both even and odd length in the $J=1/2$ triplet
sector. From the extrapolation of the finite size energies of this state we
reproduce the known central charge $c_{\mathrm{eff}}=-2$ for this model. Up to
subleading corrections to scaling this state is degenerate with the $OSp(1|2)$
$J=0$ singlet with a root configuration consisting of $L-2$ positive
rapidities and a two-string of complex conjugate ones,
$\lambda_{c\pm}\simeq\lambda_{0}\pm i/2$ with $\lambda_{0}\in\mathbb{R}^{+}$.
Complemented with results from the finite size analysis of the ground states
in the sectors $J>1/2$, we find the conformal weights corresponding to the
lowest states with superspin $J$ to be

$X^{(1|2)}_{J}(L)\simeq J(2J-1)+\frac{\alpha^{(1|2)}(J)}{\log L}\,,\quad
J=0,\frac{1}{2},1,\frac{3}{2},\dots\,..$ (28)

We have identified the lowest excitation in the $J=1/2$ and $1$ sectors: the
triplet excitation has a root configuration similar to the $J=0$ ground state
described above and corresponds to an operator with conformal weight $X=1$.
The excitation on top of the lowest $J=1$ state is given in terms of real
roots with a particle-hole pair at the Fermi point giving $X=2$.

The subleading corrections to scaling of some of these states have been
studied in Ref. Granet  _et al._ (2019). For the lowest states with superspin
$J>0$ they are

$\alpha^{(1|2)}(J)=-\frac{2}{3}J(J+1)+\frac{5}{24}\,,\quad
J=\frac{1}{2},1,\frac{3}{2},\dots\,.$ (29)

####  $OSp(3|4)$.

According to (9) these energies do appear in the spectrum of the $OSp(3|4)$
model. The Bethe equations for the latter (in grading $bffbffb$) are

$\displaystyle\left[f_{1/2}\left({\lambda_{j}^{(1)}}\right)\right]^{2L}=\prod_{k=1}^{N_{2}}f_{1/2}\left({\lambda_{j}^{(1)}-\lambda_{k}^{(2)}}\right)\,f_{1/2}\left({\lambda_{j}^{(1)}+\lambda_{k}^{(2)}}\right)\,,\quad
j=1\ldots N_{1}\,,$ (30)
$\displaystyle\prod_{k=1}^{N_{1}}f_{1/2}\left(\lambda_{j}^{(2)}-\lambda_{k}^{(1)}\right)\,f_{1/2}\left(\lambda_{j}^{(2)}+\lambda_{k}^{(1)}\right)\,\prod_{k=1}^{N_{3}}f_{1/2}\left({\lambda_{j}^{(2)}-\lambda_{k}^{(3)}}\right)\,f_{1/2}\left({\lambda_{j}^{(2)}+\lambda_{k}^{(3)}}\right)\,$
$\displaystyle\qquad=\prod_{k\neq
j}^{N_{2}}f_{1}\left({\lambda_{j}^{(2)}-\lambda_{k}^{(2)}}\right)\,f_{1}\left({\lambda_{j}^{(2)}+\lambda_{k}^{(2)}}\right)\,,\quad
j=1\ldots N_{2}\,,$
$\displaystyle\prod_{k=1}^{N_{2}}f_{1/2}\left(\lambda_{j}^{(3)}-\lambda_{k}^{(2)}\right)\,f_{1/2}\left(\lambda_{j}^{(3)}+\lambda_{k}^{(2)}\right)\,$
$\displaystyle\qquad=\prod_{k\neq
j}^{N_{3}}f_{1}\left({\lambda_{j}^{(3)}-\lambda_{k}^{(3)}}\right)\,f_{1}\left({\lambda_{j}^{(3)}+\lambda_{k}^{(3)}}\right)\,f_{-1/2}\left({\lambda_{j}^{(3)}-\lambda_{k}^{(3)}}\right)\,f_{-1/2}\left({\lambda_{j}^{(3)}+\lambda_{k}^{(3)}}\right)\,,\quad
j=1\dots N_{3}\,.$

The energy of a state corresponding to a root configuration of (30) is

$E=-(L-1)+\sum_{j=1}^{L-n}a_{1/2}\left(\lambda_{j}\right)\,.$ (31)

The root densities for the ground state and low energy excitations of the
$OSp(3|4)$ superspin chain are $N_{i}/L\to 1$ in the thermodynamic limit. As
in Ref. Frahm and Martins (2018) we label the charge sectors of this model by
quantum numbers $(n_{1},n_{2},n_{3})=(N_{1}-N_{2}+1,N_{2}-N_{3}+1,L-N_{1}-2)$.
For the low energy states most Bethe roots are arranged in complex conjugate
pairs as

$\lambda^{(1)}_{\pm}\simeq\lambda^{(1)}\pm\frac{5i}{4}\,,\quad\lambda^{(2)}_{\pm}\simeq\lambda^{(2)}\pm\frac{3i}{4}\,,\quad\lambda^{(3)}_{\pm}\simeq\lambda^{(3)}\pm\frac{i}{4}\,,\quad\lambda^{a}\in\mathbb{R}^{+}\,.$
(32)

The states with lowest energies states are parameterized by configurations
with $N_{a}=(L-k-2)/2$ of these strings, i.e. found in the sectors
$(n_{1},n_{2},n_{3})=(1,1,k)$ with $k=0,1,2,\dots\sim L\mod 2$. Among them the
energies of the $k=0,1$ levels coincide with those of the $J=1/2$ ground state
and the lowest $J=0$ state of the $OSp(1|2)$ chain. In the thermodynamic limit
all of these states are degenerate giving conformal weights
$\lim_{L\to\infty}X^{(3|4)}_{(1,1,k)}(L)=0$, see Figure 2.

Figure 2:  Corrections to the effective scaling dimensions
$X_{\mathrm{eff}}=\frac{1}{12}+n$ for the lowest states of the $OSp(3|4)$
superspin chain. The inset shows the lowest levels of the $OSp(1|2)$ model.
See Fig. 1 for the meaning of symbols and colors.

From our numerical finite size data we find that this degeneracy is lifted for
finite $L$ by subleading corrections to scaling depending on $k$ as

$X^{(3|4)}_{(1,1,k)}(L)\simeq\frac{\alpha^{(3|4)}(k)}{\log
L}\,,\quad\alpha^{(3|4)}(k)=\frac{1}{3}k(k+1)-\frac{7}{24}\,,\quad
k=0,1,2,\dots\,.$ (33)

Note that $\alpha^{(3|4)}(k=0)=\alpha^{(1|2)}(J=\frac{1}{2})$. Comparing these
amplitudes to those for the model with periodic boundary conditions Frahm and
Martins (2018) we find

$\alpha^{(3|4)}(k)=2\beta^{(3|4)}(k)-\frac{1}{8}\,.$ (34)

A group of energies extrapolating to $X^{(3|4)}=1$ is found in the sectors
$(n_{1},n_{2},n_{3})=(2,1,k)$ with $k=0,1,2,\dots\sim L+1\mod 2$. Here one of
the $N_{1}=L-2-k$ roots on the first level of (30) is real. The energy of the
$k=0$ level coincides with the lowest $J=1$ level of the $OSp(1|2)$ chain. The
next group of excitations with $X^{(3|4)}=2$ is observed in the sectors
$(n_{1},n_{2},n_{3})=(2,2,k)$, $k=0,1,2,\dots\sim L\mod 2$. A summary of the
finite size spectrum of the $OSp(1|2)$ and $OSp(3|4)$ is shown in Figure 2.

###  III.3 $z=0$: the $OSp(2|2)$ superspin chain

The Bethe equations for the $OSp(2|2)$ model (grading $fbbf$) are

$\displaystyle\left[f_{1/2}\left({\lambda_{j}^{(+)}}\right)\right]^{2L}=\prod_{k=1}^{N_{-}}f_{1}\left({\lambda_{j}^{(+)}-\lambda_{k}^{(-)}}\right)\,f_{1}\left({\lambda_{j}^{(+)}+\lambda_{k}^{(-)}}\right)\,,\quad
j=1\ldots N_{+}\,,$ (35)
$\displaystyle\left[f_{1/2}\left({\lambda_{j}^{(-)}}\right)\right]^{2L}=\prod_{k=1}^{N_{+}}f_{1}\left({\lambda_{j}^{(-)}-\lambda_{k}^{(+)}}\right)\,f_{1}\left({\lambda_{j}^{(-)}+\lambda_{k}^{(+)}}\right)\,,\quad
j=1\ldots N_{-}\,.$

Each solution to these equations parameterizes an eigenstate of the superspin
chain with energy

$E=(L-1)-\sum_{j=1}^{N_{+}}a_{1/2}\left(\lambda_{j}^{(+)}\right)-\sum_{j=1}^{N_{-}}a_{1/2}\left(\lambda_{j}^{(-)}\right)\,.$
(36)

The root configurations for the ground state and low energy excitations of the
model consist of real roots $\lambda_{j}^{(\pm)}>0$ with densities
$N_{\pm}/L\to 1/2$ in the thermodynamic limit. Using quantum numbers
$(n_{1},n_{2})=(L-N_{+}-N_{-},N_{+}-N_{-})$ for the $U(1)$ charges we find
that the ground state of the model is realized in the $(n_{1},n_{2})=(1,0)$
sector of the superspin chain with odd length. The effective central charge of
the model is known to be $c_{\mathrm{eff}}=-1$. In the thermodynamic limit
this state degenerates with the lowest levels in the sectors
$(n_{1},n_{2})=(1,k)$ for $k=1,2,3,\dots\sim L+1\mod 2$, all of them giving a
conformal weight $X_{(1,k)}^{(1|2)}=0$. For finite $L$ the degeneracy is
lifted by logarithmic corrections to scaling

$X^{((2|2)}_{(1,k)}(L)\simeq\frac{\alpha^{(2|2)}(k)}{\log
L}\,,\quad\alpha^{(2|2)}(k)=\frac{1}{2}k^{2}-\frac{5}{16}\,.$ (37)

This expression can be related to that for the periodic $OSp(2|2)$ chain Frahm
and Martins (2018) as follows

$\alpha^{(2|2)}(k)=2\beta^{(2|2)}(k)-\frac{1}{16}\,.$ (38)

A similar tower of excitations giving conformal weight $X^{(2|2)}=1$ up to
logarithmic corrections exists in the sectors $(n_{1},n_{2})=(2,k)$ for
$k=0,1,2,\dots\sim L\mod 2$. The finite size scaling behaviour of the states
we have analyzed is presented in Figure 3.

Figure 3:  Corrections to the effective scaling dimensions
$X_{\mathrm{eff}}=\frac{1}{24}+n$ for the lowest states of the $OSp(2|2)$
chain. See Fig. 1 for the meaning of symbols and colors.

###  III.4 $z=1$: the $OSp(3|2)$ superspin chain

To study the finite size spectrum of the $OSp(3|2)$ superspin chain we make
use of the Bethe equations in two different gradings: the Bethe equations for
the model with open boundaries in the grading $fbbbf$ read

$\displaystyle\left[f_{1/2}\left(\lambda_{j}^{(1)}\right)\right]^{{2L}}=\prod_{k=1}^{{L-n_{1}-n_{2}}}f_{1/2}\left({\lambda_{j}^{(1)}-\lambda_{k}^{(2)}}\right)\,f_{1/2}\left(\lambda_{j}^{(1)}+\lambda_{k}^{(2)}\right)\,,\quad
j=1,\cdots,{L-n_{1}},$ (39)
$\displaystyle\prod_{k=1}^{{L-n_{1}}}f_{1/2}\left(\lambda_{j}^{(2)}-\lambda_{k}^{(1)}\right)\,f_{1/2}\left(\lambda_{j}^{(2)}+\lambda_{k}^{(1)}\right)=$
$\displaystyle\qquad=\prod_{k\neq
j}^{{L-n_{1}-n_{2}}}f_{1/2}\left(\lambda_{j}^{(2)}-\lambda_{k}^{(2)}\right)\,f_{1/2}\left(\lambda_{j}^{(2)}+\lambda_{k}^{(2)}\right)\,,\quad
j=1,\cdots,{L-n_{1}-n_{2}}\,.$

The corresponding energy is given in terms of the Bethe roots from the first
level as

$E_{fbbbf}=(L-1)-\sum_{j=1}^{L-n_{1}}a_{1/2}\left(\lambda_{j}^{(1)}\right)\,.$
(40)

Choosing the grading $bfbfb$ the spectrum of the open superspin chain is
parameterized by solutions to the Bethe equations:

$\displaystyle\left[f_{1/2}\left(\lambda_{j}^{(1)}\right)\right]^{{2L}}=\prod_{k=1}^{{L-n_{1}-n_{2}}}f_{1/2}\left({\lambda_{j}^{(1)}-\lambda_{k}^{(2)}}\right)\,f_{1/2}\left(\lambda_{j}^{(1)}+\lambda_{k}^{(2)}\right)\,,\quad
j=1,\cdots,{L-n_{2}-1},$ (41)
$\displaystyle\prod_{k=1}^{{L-n_{2}-1}}f_{1/2}\left(\lambda_{j}^{(2)}-\lambda_{k}^{(1)}\right)\,f_{1/2}\left(\lambda_{j}^{(2)}+\lambda_{k}^{(1)}\right)=$
$\displaystyle\qquad=\prod_{k\neq
j}^{{L-n_{1}-n_{2}}}f_{-1/2}\left(\lambda_{j}^{(2)}-\lambda_{k}^{(2)}\right)\,f_{-1/2}\left(\lambda_{j}^{(2)}+\lambda_{k}^{(2)}\right)\
f_{1}\left(\lambda_{j}^{(2)}-\lambda_{k}^{(2)}\right))\,f_{1}\left((\lambda_{j}^{(2)}+\lambda_{k}^{(2)}\right)\,,$
$\displaystyle\qquad\qquad j=1,\cdots,{L-n_{1}-n_{2}}\,,$

and the corresponding energy eigenvalue is

$E_{bfbfb}=-L+1+\sum_{j=1}^{L-n_{2}-1}a_{1/2}\left(\lambda_{j}^{(1)}\right)\,.$
(42)

Solutions to the Bethe equations (39) and (41) parameterize highest weight
states of $OSp(3|2)$ in the irreducible representations $(p;q)$ appearing in
the tensor product $(0;\frac{1}{2})^{\otimes L}$ of local spins der Jeugt
(1984); Frahm and Martins (2015). In terms of the number of Bethe roots the
quantum numbers $p$ and $q$ are given as

$p=n_{1}-1\,,\quad q=(n_{2}+1)/2\,.$ (43)

Exact diagonalization of the $OSp(3|2)$ Hamiltonian shows that the ground
state is a $(p;q)=(0;0)$ singlet ($(0;\frac{1}{2})$ quintet) for $L$ even
(odd). Its energy is $E_{0}=Le_{\infty}+f_{\infty}\equiv-3(L-1)$ without any
finite size corrections – similar to the model with periodic boundary
conditions – giving the effective central charge $c_{\mathrm{eff}}=0$.. The
$fbbbf$ Bethe root configuration for $L$ odd contains $(L-1)/2$ pairs of
complex conjugate rapidities $\lambda^{(a)}_{j\pm}\simeq\lambda^{(a)}_{j}\pm
i/4$ with positive $\lambda^{(a)}_{j}$ on each level $a=1,2$. The root
configuration for even $L$ contains degenerate roots.

As for the models considered above the finite size spectrum of the $OSp(3|2)$
superspin chain can be grouped into sets extrapolating to the same integer
conformal weight in the thermodynamic limit. Specifically, the lowest states
in the sectors $(0;q)$ with $2q=0,1,2,\dots$ (or $(n_{1},n_{2})=(1,k)$ with
integer $k$) become degenerate with the ground state, see Figure 4.

Figure 4:  Corrections to the effective scaling dimensions
$X_{\mathrm{eff}}=n$ for the lowest excitations of the $OSp(3|2)$ chain in the
sectors $(0;q)$ and $(1;q)$ (the ground state in the sectors $(0;0)$,
$(0;\frac{1}{2})$ with $X_{\mathrm{eff}}\equiv 0$ independent of $L$ is not
shown). See Fig. 1 for the meaning of symbols and colors.

At large but finite $L$ this degeneracy is lifted by logarithmic corrections
to scaling

$X^{(3|2)}_{(0;q)}(L)\simeq\frac{\alpha^{(3|2)}(q)}{\log
L}\,,\quad\alpha^{(3|2)}(q)=2q(2q-1)\,.$ (44)

We note that the amplitudes $\alpha^{(3|2)}$ are twice of those found for the
periodic $OSp(3|2)$ chain Frahm and Martins (2015), i.e.
$\alpha^{(3|2)}(q)=2\beta^{(3|2)}(q)$.

Finite size data for the lowest states in the sectors $(p;q)=(1;q)$,
$q=\frac{1}{2},1,\frac{3}{2}$, and some descendents states are also shown in
Figure 4.

##  IV Discussion

In this paper we have investigated the finite-size properties of the spectrum
of the $OSp(n|2m)$ superchain chain with free boundary conditions. We perform
this analysis by solving numerically the corresponding Bethe equations for
large systems sizes. This study made it possible to identify the corresponding
operator content and to extract the amplitudes associated to the subleading
corrections to the asymptotic behaviour.

For $z=n-2m<2$ we find that the surface exponents are built out of a set of
integer numbers. Similar as in the case of periodic boundary conditions this
was to be expected based on the perturbative RG analysis of the model. The
surface exponents turn out to be exactly the same as the bulk exponents which
is a peculiarity of the underlying universality class. The spectra contain an
abundance of states with null conformal dimension whose degeneracy is lifted
by subleading logarithmic corrections. We find that the amplitudes of such
corrections are different for periodic and free boundary conditions. From our
numerical analysis we conjecture a simple relation among these amplitudes to
be

$\alpha(k)=2\beta(k)+\frac{z-1}{16}\,,\qquad z<2\,,$ (45)

where $\alpha(k)$ and $\beta(k)$ correspond to the amplitudes associated to
free and periodic boundaries, respectively.

We now can use this result to infer about the asymptotic behaviour of the
surface watermelon correlators associated to the respective loop model. The
above relation tells us that we can use the same reference state $k_{0}$
associated to the smallest non-negative amplitudes for both free and periodic
boundaries. Proceeding in analogy as has already been explained for periodic
boundary Frahm and Martins (2018) we conjecture that the surface correlators
should behave as

$G^{(s)}_{k}(\rho)\sim
1/\ln(\rho)^{2\gamma(k)}\,,\qquad\gamma(k)=\frac{k(k+z-2)}{2-z}\,,$ (46)

where $\rho$ denotes the distance among two points close to the boundary.

We conclude by recalling some existing results on the finite-size properties
in the regime $z\geq 2$ for free boundary conditions. For the $O(2)$
(spin-$1/2$ Heisenberg) model of even length it has been argued that all
conformal dimensions are given by the identity conformal tower Alcaraz  _et
al._ (1988); Eggert and Affleck (1992). The logarithmic corrections to scaling
for this model have been computed in Affleck and Qin (1999). In particular the
gap between the ground state and the lowest (triplet) excitation is given by

$E_{1}^{O(2)}(L)-E_{0}^{O(2)}(L)\simeq\frac{\pi
v_{F}}{L}\left(1-1/\ln(L)\right)\,,$ (47)

corresponding to conformal weight $X=1$ where the amplitude of the logarithmic
correction to scaling is determined by the quadratic Casimir of the underlying
algebra. The spectrum of the $O(4)$ chain can be composed from the
eigenenergies of two decoupled Heisenberg chains Martins and Ramos (1997).
This can be used to infer the corresponding behaviour, in particular (note
that the Fermi velocities of the $O(2)$ and the $O(4)$ model coincide, see
Table 1)

$E_{1}^{O(4)}(L)-E_{0}^{O(4)}(L)\simeq\frac{\pi
v_{F}}{L}\left(1-1/\ln(L)\right)\,.$ (48)

Based on our previous work on the $OSp(5|2)$ superspin chain with periodic
boundary conditions Frahm and Martins (2018) we expect towers of levels with
the same conformal weight to be present in the regime $z=n-2m\geq 2$ but
$m>0$. We hope to investigate how these degeneracies are lifted by subleading
corrections to scaling in a forthcoming paper.

###### Acknowledgements.

Partial support for the work of HF is provided by the Deutsche
Forschungsgemeinschaft under grant No. Fr 737/9-2 within the research unit
_Correlations in Integrable Quantum Many-Body Systems_ (FOR2316). The work of
MJM was supported in part by the Brazilian Research Council CNPq through the
grants 304758/2017-7.

##  Appendix A The surface energy

To employ Eq. (3) for the analysis of the finite size spectrum of the open
$OSp(n|2m)$ superspin chains with free boundary conditions the corresponding
surface energy is needed as an input. As a consequence of the spectral
inclusion (9) it is sufficient to consider the cases based on the
superalgebras $OSp(1|2m)$, $OSp(2|2m)$ and $OSp(3|2)$ for the regime
$z=n-2m<2$ considered in the main text. Here the last of these is special due
to the singular Bethe root configuration for the ground state, see Section
III.4. Based on our numerical results for $OSp(3|2)$ superspin chains of
finite length, however, we conclude that the surface energy for this model is
$f_{\infty}=3$. The $\mathcal{O}(L^{0})$ contribution to the ground state
energy for the other two series of models can be computed using the root
density method Yang and Yang (1969); Hamer  _et al._ (1987). We start our
discussion below by presenting the main steps of this approach for the
$OSp(2|2)$ superspin chain which, as discussed in Section III.3, is solved by
a nested Bethe ansatz with real roots for the low energy states. Using similar
arguments we then apply this method to derive the surface energies for the
spin chains based on $OSp(2|2m)$, $OSp(1|2m)$.

For sake of completeness we also compute the surface energies for the spin
chains in the regime $z>2$ (the case $z=2$ can be represented by the isotropic
spin-$1/2$ Heisenberg magnet (15) as discussed in the main text). For the
models based on the ordinary Lie algebras $O(2n)$ the root density approach
can be applied as before. In the case of the $O(2n+1)$ spin chains it has to
be modified slightly due to the presence of complex roots in the ground state
configuration. For the $O(3)$ and $O(5)$ model we will show at the end of this
Appendix, that this can be dealt with using the so-called string hypothesis.

The results obtained in this appendix are summarized in Eqs. (13) and (14).

###  A.1 $OSp(2|2)$ in the grading $fbbf$

We start by taking the logarithm of the Bethe ansatz equations (35) associated
to the $OSp(2|2)$ model for configurations of real roots
$\lambda_{j}^{(\pm)}$. As a result we find

$\displaystyle
2L\phi_{1/2}\left(\lambda_{j}^{(+)}\right)=2\pi\mathrm{Q}_{j}^{(+)}+\sum_{k=1}^{N_{-}}\left[\phi_{1}\left(\lambda_{j}^{(+)}-\lambda_{k}^{(-)}\right)+\phi_{1}\left(\lambda_{j}^{(+)}+\lambda_{k}^{(-)}\right)\right],~{}~{}~{}j=1,\dots,N_{+}\,,$
(49) $\displaystyle
2L\phi_{1/2}\left(\lambda_{j}^{(-)}\right)=2\pi\mathrm{Q}_{j}^{(-)}+\sum_{k=1}^{N_{+}}\left[\phi_{1}\left(\lambda_{j}^{(-)}-\lambda_{k}^{(+)}\right)+\phi_{1}\left(\lambda_{j}^{(-)}+\lambda_{k}^{(+)}\right)\right],~{}~{}~{}j=1,\dots,N_{-}\,,$

a1where $\phi_{s}(x)\equiv 2\arctan(x/s)$ and the numbers
$\mathrm{Q}_{j}^{(\pm)}$ are positive integers characterizing the possible
branches of the logarithm.

From the above Bethe equations the so-called counting functions Yang and Yang
(1969)

$\displaystyle
z_{L}^{(+)}(\lambda)=\frac{\phi_{1/2}(\lambda)}{\pi}-\frac{1}{2\pi
L}\sum_{k=1}^{N_{-}}\left[\phi_{1}\left(\lambda-\lambda_{k}^{(-)}\right)+\phi_{1}\left(\lambda+\lambda_{k}^{(-)}\right)\right]\,,$
(50) $\displaystyle
z_{L}^{(-)}(\lambda)=\frac{\phi_{1/2}(\lambda)}{\pi}-\frac{1}{2\pi
L}\sum_{k=1}^{N_{+}}\left[\phi_{1}\left(\lambda-\lambda_{k}^{(+)}\right)+\phi_{1}\left(\lambda+\lambda_{k}^{(+)}\right)\right]\,,$

take values
$z_{L}^{(\pm)}\left(\lambda_{j}^{(\pm)}\right)=\mathrm{Q}_{j}^{(\pm)}/L$ for
$j=1,\dots,N_{\pm}$. For the lowest states of the towers considered in Sect.
III.3 we have $N_{+}=N_{-}\lesssim L/2$ with uniformly spaced quantum numbers
$\mathrm{Q}_{j}^{(\pm)}=j$. Hence densities of the Bethe roots
$\lambda_{j}^{(\pm)}$ in these states can be derived from the counting
functions as

$\rho_{L}^{(\pm)}(\lambda)=\frac{d}{d\lambda}z_{L}^{(\pm)}(\lambda)=\frac{a_{1/2}(\lambda)}{\pi}+\frac{1}{2\pi
L}a_{1}(\lambda)-\frac{1}{2\pi
L}\sum_{k=-N_{\mp}}^{N_{\mp}}a_{1}\left(\lambda-\lambda_{k}^{(\mp)}\right)\,$
(51)

where we have symmetrized the sums by extending the sets of roots to
$\\{\lambda_{j}^{(\pm)}\\}\cup\\{0\\}\cup\\{\lambda_{-j}^{(\pm)}\equiv-\lambda_{j}^{(\pm)}\\}$.
Note that these relations are similar to those obtained for periodic boundary
conditions except for the presence of the additional boundary terms
$a_{1}(\lambda)/(2\pi L)$. We anticipate that these terms are responsible to
provide the surface contribution to the ground state energy.

For $L\gg 1$ the extended set of Bethe roots for the ground state tends to a
continuous distribution on the entire real axis with densities
$\rho_{0}^{(\pm)}(\lambda)$ and the sums in (51) can be replaced by integrals

$\displaystyle\rho_{0}^{(+)}(\lambda)\simeq\frac{a_{1/2}(\lambda)}{\pi}+\frac{1}{2\pi
L}a_{1}(\lambda)-\frac{1}{2\pi}\int_{-\infty}^{+\infty}\mathrm{d}\mu\,a_{1}\left(\lambda-\mu\right)\rho_{0}^{(-)}(\mu)\,,$
(52)
$\displaystyle\rho_{0}^{(-)}(\lambda)\simeq\frac{a_{1/2}(\lambda)}{\pi}+\frac{1}{2\pi
L}a_{1}(\lambda)-\frac{1}{2\pi}\int_{-\infty}^{+\infty}\mathrm{d}\mu\,a_{1}\left(\lambda-\mu\right)\rho_{0}^{(+)}(\mu)\,.$

These integral equations can be solved order by order in powers of $L^{-1}$ by
elementary Fourier techniques resulting in
$\rho^{(\pm)}(\omega)=\sigma_{0}(\omega)+\tau_{0}(\omega)/L$ with

$\sigma_{0}(\omega)=\frac{1}{\cosh(\omega/2)}\,,\quad\tau_{0}(\omega)=\frac{\exp(-|\omega|)}{1+\exp(-|\omega|)}\,.$
(53)

Similarly, we rewrite the ground state energy (36) as

$E_{0}/L\simeq
1-\frac{1}{2}\int_{-\infty}^{\infty}a_{1/2}(\lambda)\left(\rho_{0}^{(+)}(\lambda)+\rho_{0}^{(-)}(\lambda)\right)+\frac{1}{L}\left(-1+a_{1/2}(0)\right)\,.$
(54)

Using (53) we reproduce the known result $\epsilon_{\infty}=1-4\ln 2$ for the
bulk energy density and obtain

$f_{\infty}=-1+a_{1/2}(0)-2\int_{0}^{\infty}\mathrm{d}\lambda\,a_{1/2}(\lambda)\tau_{0}(\lambda)=3-\int_{-\infty}^{\infty}\mathrm{d}\omega\,\mathrm{e}^{-|\omega|/2}\tau_{0}(\omega)=\pi-1\,$
(55)

for the surface energy of the $OSp(2|2)$ spin chain as shown in Table 1.

###  A.2 $OSp(2|2m)$ on the grading $f\dots fbbf\dots f$

For this model the ground state and the low-lying excitations are described in
terms real roots in the $f\dots fbbf\dots f$ basis ordering. The Bethe
equations are given by

$\displaystyle\delta_{\ell,1}(2L)\phi_{1/2}\left(\lambda_{j}^{(\ell)}\right)=$
$\displaystyle 2\pi\mathrm{Q}_{j}^{(\ell)}+\sum_{\stackrel{{\scriptstyle
k=1}}{{k\neq
j}}}^{N_{\ell}}\left[\phi_{1}\left(\lambda_{j}^{(\ell)}-\lambda_{k}^{(\ell)}\right)+\phi_{1}\left(\lambda_{j}^{(\ell)}+\lambda_{k}^{(\ell)}\right)\right]$
(56) $\displaystyle-\sum_{\alpha=\ell\pm
1}\sum_{k=1}^{N_{\alpha}}\left[\phi_{1/2}\left(\lambda_{j}^{(\ell)}-\lambda_{k}^{(\alpha)}\right)+\phi_{1/2}\left(\lambda_{j}^{(\ell)}+\lambda_{k}^{(\alpha)}\right)\right],\quad\ell=1,\dots,m-2\,,$
$\displaystyle\delta_{m,2}(2L)\phi_{1/2}\left(\lambda_{j}^{(m-1)}\right)=$
$\displaystyle 2\pi\mathrm{Q}_{j}^{(m-1)}+\sum_{\stackrel{{\scriptstyle
k=1}}{{k\neq
j}}}^{N_{m-1}}\left[\phi_{1}\left(\lambda_{j}^{(m-1)}-\lambda_{k}^{(m-1)}\right)+\phi_{1}\left(\lambda_{j}^{(m-1)}+\lambda_{k}^{(m-1)}\right)\right]$
$\displaystyle-\sum_{{\alpha=m-2,\pm}}\sum_{k=1}^{N_{\alpha}}\left[\phi_{1/2}\left(\lambda_{j}^{(m-1)}-\lambda_{k}^{(\alpha)}\right)+\phi_{1/2}\left(\lambda_{j}^{(m-1)}+\lambda_{k}^{(\alpha)}\right)\right]\,,$
$\displaystyle\delta_{m,1}(2L)\phi_{1/2}\left(\lambda_{j}^{(\ell)}\right)=$
$\displaystyle
2\pi\mathrm{Q}_{j}^{(\pm)}+\sum_{k=1}^{N_{\mp}}\left[\phi_{1}\left(\lambda_{j}^{(\ell)}-\lambda_{k}^{(\mp)}\right)+\phi_{1}\left(\lambda_{j}^{(\ell)}+\lambda_{k}^{(\mp)}\right)\right]$
$\displaystyle-\sum_{k=1}^{N_{m-1}}\left[\phi_{1/2}\left(\lambda_{j}^{(\ell)}-\lambda_{k}^{(m-1)}\right)+\phi_{1/2}\left(\lambda_{j}^{(\ell)}+\lambda_{k}^{(m-1)}\right)\right],\quad\ell=\pm\,,$

and the corresponding energy is

$E=L-1-\sum_{j}a_{1/2}(\lambda^{(1)}_{j})\,.$ (57)

The ground state is parameterized by $N_{1}=\dots=N_{m-1}=L$ and $N_{\pm}=L/2$
roots distributed on the positive real axis in the thermodynamic limit. Hence,
by proceeding as for the $OSp(2|2)$ model above we obtain the $L^{-1}$
boundary contributions $\tau_{0}^{(\ell)}(\lambda)$, $\ell=1,\dots,m-1,\pm$ to
the densities. The energy (57) of the $OSp(2|2m)$ superspin chain is given in
terms of the first level roots, $\ell=1$. The Fourier representation of their
boundary density is

$\tau_{0}^{(1)}(\omega)=-2\mathrm{e}^{-(m-1)|\omega|/4}\,\frac{\sinh(m|\omega|/4)\cosh(\omega/4)}{\cosh(m\omega/2)}\,.$
(58)

From (57) the resulting surface energy is given in terms of $\tau_{0}^{(1)}$
as

$f_{\infty}=-1+\frac{1}{2}a_{1/2}(0)-\int_{0}^{\infty}\mathrm{d}\lambda\,a_{1/2}(\lambda)\tau_{0}^{(1)}(\lambda)=1-\int_{0}^{\infty}\mathrm{d}\omega\,\mathrm{e}^{-|\omega|/2}\tau_{0}^{(1)}(\omega)$
(59)

After some manipulations with the help of the identity

$\int_{0}^{+\infty}\frac{\exp(-\mu
x)}{\cosh(x)}dx=\frac{1}{2}\Big{(}\psi(\mu/2+1/2)-\psi(\mu/2)\Big{)}\,,$ (60)

we can rewrite the surface free energy of the $OSp(2|2m)$ superspin chain in
terms of the Euler $\psi$ function as presented in the main text (13) with
$z=2-2m$.

###  A.3 $OSp(1|2m)$ on the grading $f\dots fbf\dots f$

In the $f\dots fbf\dots f$ grading the ground state and the low-lying
excitations of this model are described in terms of positive rapidities
satisfying the Bethe equations

$\displaystyle\delta_{\ell,1}(2L)\phi_{1/2}\left(\lambda_{j}^{(\ell)}\right)=$
$\displaystyle 2\pi\mathrm{Q}_{j}^{(\ell)}+\sum_{\stackrel{{\scriptstyle
k=1}}{{k\neq
j}}}^{N_{\ell}}\left[\phi_{1}\left(\lambda_{j}^{(\ell)}-\lambda_{k}^{(\ell)}\right)+\phi_{1}\left(\lambda_{j}^{(\ell)}+\lambda_{k}^{(\ell)}\right)\right]$
(61) $\displaystyle-\sum_{\alpha=\ell\pm
1}\sum_{k=1}^{N_{\alpha}}\left[\phi_{1/2}\left(\lambda_{j}^{(\ell)}-\lambda_{k}^{(\alpha)}\right)+\phi_{1/2}\left(\lambda_{j}^{(\ell)}+\lambda_{k}^{(\alpha)}\right)\right]\,,\quad\ell=1,\dots,m-1\,,$
$\displaystyle\delta_{m,1}(2L)\phi_{1/2}\left(\lambda_{j}^{(m)}\right)=$
$\displaystyle 2\pi\mathrm{Q}_{j}^{(m)}+\sum_{\stackrel{{\scriptstyle
k=1}}{{k\neq
j}}}^{N_{m}}\left[\phi_{1}\left(\lambda_{j}^{(m)}-\lambda_{k}^{(m)}\right)+\phi_{1}\left(\lambda_{j}^{(m)}+\lambda_{k}^{(m)}\right)\right]$
$\displaystyle-\sum_{\stackrel{{\scriptstyle k=1}}{{k\neq
j}}}^{N_{m}}\left[\phi_{1/2}\left(\lambda_{j}^{(m)}-\lambda_{k}^{(m)}\right)+\phi_{1/2}\left(\lambda_{j}^{(m)}+\lambda_{k}^{(m)}\right)\right]$
$\displaystyle-\sum_{k=1}^{N_{m-1}}\left[\phi_{1/2}\left(\lambda_{j}^{(m)}-\lambda_{k}^{(m-1)}\right)+\phi_{1/2}\left(\lambda_{j}^{(m)}+\lambda_{k}^{(m-1)}\right)\right]\,.$

The energy associated to a solution is given again by (57). In the
thermodynamic limit we can introduce densities to describe the root
configuration. Solving the corresponding integral equations as above the
Fourier expression for the boundary contribution $\tau_{0}^{(1)}$ to the
density of first level roots is found to be

$\tau_{0}^{(1)}(\omega)=-2\mathrm{e}^{-(2m-1)|\omega|/8}\,\frac{\sinh((2m+1)|\omega|/8)\cosh(\omega/4)}{\cosh((2m+1)\omega/4)}\,.$
(62)

The surface energy of the $OSp(1|2m)$ superspin chain can be computed from
(59) which, using (60), can be brought into the form (13) with $z=1-2m$.

###  A.4 $O(2n)$

As has been discussed in the main text the thermodynamical properties
including the surface energy of the $O(2)$ model are known from the studies of
the isotropic spin $s=1/2$ Heisenberg model (15). Similarly, the spectrum of
the $O(4)$ spin chain can be derived by composing the eigenenergies of two
decoupled Heisenberg spin chains Martins and Ramos (1997). From this
identification we obtain that the surface free energy of the $O(4)$ spin chain
is

$f_{\infty}^{O(4)}=2f_{\infty}^{O(2)}-1=\pi-2\ln(2)-1\,.$ (63)

From now on we shall concentrate our analysis for the models with $n\geq 3$.
The corresponding Bethe equations are given by

$\displaystyle\delta_{\ell,1}(2L)\phi_{1/2}\left(\lambda_{j}^{(\ell)}\right)=$
$\displaystyle 2\pi\mathrm{Q}_{j}^{(\ell)}+\sum_{\stackrel{{\scriptstyle
k=1}}{{k\neq
j}}}^{N_{l}}\left[\phi_{1}\left(\lambda_{j}^{(\ell)}-\lambda_{k}^{(\ell)}\right)+\phi_{1}\left(\lambda_{j}^{(\ell)}+\lambda_{k}^{(\ell)}\right)\right]$
(64) $\displaystyle-\sum_{\alpha=l\pm
1}\sum_{k=1}^{N_{\alpha}}\left[\phi_{1/2}\left(\lambda_{j}^{(\ell)}-\lambda_{k}^{(\alpha)}\right)+\phi_{1/2}\left(\lambda_{j}^{(\ell)}+\lambda_{k}^{(\alpha)}\right)\right]\,,\quad\ell=1,\dots,n-3\,,$
$\displaystyle\delta_{n,3}(2L)\phi_{1/2}\left(\lambda_{j}^{(n-2)}\right)=$
$\displaystyle 2\pi\mathrm{Q}_{j}^{(n-2)}+\sum_{\stackrel{{\scriptstyle
k=1}}{{k\neq
j}}}^{N_{l}}\left[\phi_{1}\left(\lambda_{j}^{(n-2)}-\lambda_{k}^{(n-2)}\right)+\phi_{1}\left(\lambda_{j}^{(n-2)}+\lambda_{k}^{(n-2)}\right)\right]$
$\displaystyle-\sum_{\alpha=n-3,\pm}\sum_{k=1}^{N_{\alpha}}\left[\phi_{1/2}\left(\lambda_{j}^{(n-2)}-\lambda_{k}^{(\alpha)}\right)+\phi_{1/2}\left(\lambda_{j}^{(n-2)}+\lambda_{k}^{(\alpha)}\right)\right]\,,$
$\displaystyle\delta_{n,2}(2L)\phi_{1/2}\left(\lambda_{j}^{(\ell)}\right)=$
$\displaystyle 2\pi\mathrm{Q}_{j}^{(\pm)}+\sum_{\stackrel{{\scriptstyle
k=1}}{{k\neq
j}}}^{N_{\pm}}\left[\phi_{1}\left(\lambda_{j}^{(\ell)}-\lambda_{k}^{(\ell)}\right)+\phi_{1}\left(\lambda_{j}^{(\ell)}+\lambda_{k}^{(\ell)}\right)\right]$
$\displaystyle-\sum_{k=1}^{N_{n-2}}\left[\phi_{1/2}\left(\lambda_{j}^{(\ell)}-\lambda_{k}^{(n-2)}\right)+\phi_{1/2}\left(\lambda_{j}^{(\ell)}+\lambda_{k}^{(n-2)}\right)\right]\,,\quad\ell=\pm\,.$

The energy of the corresponding eigenstate of the $O(2n)$ model is given again
in terms of the first level roots by the expression (57). The ground state and
low lying excitations are parameterized by real roots $\lambda_{j}^{(\ell)}>0$
with total densities $n_{\ell}=N_{\ell}/L=1$ and $n_{\pm}=N_{\pm}/L=1/2$ in
the thermodynamic limit. Proceeding as for the models discussed in the
previous sections we obtain the Fourier expression for the boundary
contribution $\tau_{0}^{(1)}$ to the density of roots $\lambda_{j}^{(1)}$:

$\tau_{0}^{(1)}(\omega)=-\mathrm{e}^{-(n-2)|\omega|/4}\,\frac{\sinh(n|\omega|/4)-\cosh((n-2)\omega/4)}{\cosh((n-1)\omega/2)}\,.$
(65)

Using this expression in (59) we find that the surface energy of the $O(2n)$
spin chain is given by (14) with $z=2n$.

###  A.5 $O(3)$

The Bethe equations for the integrable $O(3)$ spin chain (or, equivalently,
the spin $S=1$ Takhtajan-Babujian model Takhtajan (1982); Babujian (1982))
read

$\left[f_{1/2}(\lambda_{j})\right]^{2L}=\prod_{k\neq
j}^{L-n}f_{1/2}(\lambda_{j}-\lambda_{k})f_{1/2}(\lambda_{j}+\lambda_{k})\,,\quad
j=1,\dots,L-n\,.$ (66)

The corresponding energy eigenvalue is given as

$E=L-1-\sum_{j}^{L-n}a_{1/2}(\lambda_{j})\,.$ (67)

The ground state of the model for even $L$ is parametrized by a solution of
(66) in the sector $n=0$ containing $L/2$ two-strings
$x_{j,\pm}\simeq\xi_{j}\pm i/4$, $\xi_{j}>0$. Neglecting corrections to the
imaginary parts the Bethe equations can be rewritten in terms of the
coordinates of the string centers. Taking the logarithm we obtain

$\displaystyle
2L\left(\phi_{3/4}(\xi_{j})+\phi_{1/4}(\xi_{j})\right)=2\pi\mathrm{Q}_{j}-\phi_{1/2}(\xi_{j})$
(68)
$\displaystyle\qquad-\sum_{k=1}^{L/2}\left[2\left(\phi_{1/2}(\xi_{j}-\xi_{k})+\phi_{1/2}(\xi_{j}+\xi_{k})\right)+\phi_{1}(\xi_{j}-\xi_{k})+\phi_{1}(\xi_{j}+\xi_{k})\right]$

with quantum numbers $\mathrm{Q}_{j}=1,2,\dots,L/2$. Similarly, the energy
(67) becomes

$E_{0}=L-1-\sum_{j=1}^{L/2}\left(a_{1/4}(\xi_{j})+a_{3/4}(\xi_{j})\right)\,.$
(69)

Proceeding as in Appendix A.1 we obtain an integral equation for the density
of strings in the ground state for $L\gg 1$

$\displaystyle\rho_{0}(\xi)\simeq$
$\displaystyle\frac{1}{\pi}\left(a_{3/4}(\xi)+a_{1/4}(\xi)\right)+\frac{1}{2\pi
L}\left(3a_{1/2}(\xi)+a_{1}(\xi)\right)$ (70)
$\displaystyle-\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{d}\xi^{\prime}\,\left[2a_{1/2}(\xi-\xi^{\prime})+a_{1}(\xi-\xi^{\prime})\right]\rho_{0}(\xi^{\prime})\,.$

Solving this integral equation by Fourier methods we obtain

$\tau_{0}(\omega)=\frac{3+\exp(-|\omega|/2)}{4\cosh^{2}(\omega/4))}$ (71)

for the $\mathcal{O}(L^{-1})$ contribution to the density. Hence we have

$\displaystyle f_{\infty}$
$\displaystyle=-1+\frac{1}{2}\left(a_{1/4}(0)+a_{3/4}(0)\right)-\int_{0}^{\infty}\mathrm{d}\xi\,\left(a_{1/4}(\xi)+a_{3/4}(\xi)\right)\tau_{0}(\xi)$
(72)
$\displaystyle=\frac{13}{3}-\int_{0}^{\infty}\mathrm{d}\omega\,\left(\mathrm{e}^{-|\omega|/4}+\mathrm{e}^{-3|\omega|/4}\right)\tau_{0}(\omega)=2\pi-5\,.$

which coincides with (14) for $z=3$ (the same result has recently been
obtained using a slightly different method in Zheng  _et al._ (2022)).

###  A.6 $O(5)$

The Bethe equations for the $O(5)$ model read

$\displaystyle\left[f_{1/2}(\lambda^{(1)}_{j})\right]^{2L}=\prod_{k\neq
j}^{L-n_{1}}f_{1}(\lambda^{(1)}_{j}-\lambda^{(1)}_{k})f_{1}(\lambda^{(1)}_{j}+\lambda^{(1)}_{k})\prod_{k=1}^{L-n_{1}-n_{2}}f_{-1}(\lambda^{(1)}_{j}-\lambda^{(2)}_{k})f_{-1}(\lambda^{(1)}_{j}+\lambda^{(2)}_{k})\,,$
(73) $\displaystyle\qquad\qquad j=1,\dots,L-n_{1}\,,$
$\displaystyle\prod_{k\neq
j}^{L-n_{1}-n_{2}}f_{1/2}(\lambda^{(2)}_{j}-\lambda^{(2)}_{k})f_{1/2}(\lambda^{(2)}_{j}+\lambda^{(2)}_{k})=\prod_{k=1}^{L-n_{1}}f_{1/2}(\lambda^{(2)}_{j}-\lambda^{(1)}_{k})f_{1/2}(\lambda^{(2)}_{j}+\lambda^{(1)}_{k})\,,$
$\displaystyle\qquad\qquad j=1,\dots,L-n_{1}-n_{2}\,.$

The ground state for even length spin chains is in the sector
$(n_{1},n_{2})=(0,0)$ with $\lambda^{(1)}_{j}\in\mathbb{R}^{+}$ and
$\lambda^{(2)}_{j}$ arranged in 2-strings $\lambda^{(2)}_{j}\simeq\xi_{j}\pm
i/4$, $\xi_{j}>0$. The corresponding energy is again given by (57). Following
the same steps as for the $O(3)$ model above we obtain Bethe equations for the
string coordinates from (73)

$\displaystyle
2L\phi_{1/2}(\lambda^{(1)}_{j})=2\pi\mathrm{Q}_{j}-\phi_{1/2}(\lambda^{(1)}_{j})+\sum_{k=1}^{L}\left(\phi_{1}(\lambda^{(1)}_{j}-\lambda^{(1)}_{k})+\phi_{1}(\lambda^{(1)}_{j}+\lambda^{(1)}_{k})\right)$
(74)
$\displaystyle\qquad-\sum_{k=1}^{L/2}\left[\phi_{3/4}(\lambda^{(1)}_{j}-\xi_{k})+\phi_{3/4}(\lambda^{(1)}_{j}+\xi_{k})+\phi_{1/4}(\lambda^{(1)}_{j}-\xi_{k})+\phi_{1/4}(\lambda^{(1)}_{j}+\xi_{k})\right]\,,\quad
j=1,\dots,L\,,$
$\displaystyle\sum_{k=1}^{L}\left[\phi_{3/4}(\xi_{j}-\lambda^{(1)}_{k})+\phi_{3/4}(\xi_{j}+\lambda^{(1)}_{k})+\phi_{1/4}(\xi_{j}-\lambda^{(1)}_{k})+\phi_{1/4}(\xi_{j}+\lambda^{(1)}_{k})\right]$
$\displaystyle\qquad=2\pi\overline{\mathrm{Q}}_{j}-\phi_{1/2}(\xi_{j})+\sum_{k=1}^{L/2}\left[\phi_{1}(\xi_{j}-\xi_{k})+\phi_{1}(\xi_{j}+\xi_{k})+2\left(\phi_{1/2}(\xi_{j}-\xi_{k})+\phi_{1/2}(\xi_{j}+\xi_{k})\right)\right]\,,$
$\displaystyle\qquad\qquad j=1,\dots,L/2\,.$

For $L\gg 1$ the ground state densities $\rho_{0}(\lambda)$ of real roots from
the first level Bethe equations and $\overline{\rho}_{0}(\xi)$ of two-strings
from the second one are given in terms of the integral equations

$\displaystyle\rho_{0}(\lambda)=$
$\displaystyle\frac{1}{\pi}a_{1/2}(\lambda)+\frac{1}{2\pi
L}\left(a_{1}(\lambda)-a_{3/4}(\lambda)+a_{1/2}(\lambda)-a_{1/4}(\lambda)\right)$
(75)
$\displaystyle-\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{d}\lambda^{\prime}\,a_{1}(\lambda-\lambda^{\prime})\rho(\lambda^{\prime})+\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{d}\xi^{\prime}\,\left[a_{3/4}(\lambda-\xi^{\prime})+a_{1/4}(\lambda-\xi^{\prime})\right]\overline{\rho}_{0}(\xi)\,,$
$\displaystyle\overline{\rho}_{0}(\xi)=$ $\displaystyle\frac{1}{2\pi
L}\left(a_{1}(\xi)-a_{3/4}(\xi)+3a_{1/2}(\xi)-a_{1/4}(\xi)\right)+\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{d}\lambda^{\prime}\,\left[a_{3/4}(\xi-\lambda^{\prime})+a_{1/4}(\xi-\lambda^{\prime})\right]\rho_{0}(\lambda^{\prime})$
$\displaystyle-\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{d}\xi^{\prime}\,\left[2a_{1/2}(\xi-\xi^{\prime})+a_{1}(\xi-\xi^{\prime})\right]\overline{\rho}_{0}(\xi^{\prime})\,.$

Solving these equations the boundary contribution to the density $\rho_{0}$ of
first level roots is found to be

$\tau_{0}(\omega)=-\mathrm{e}^{-|\omega|/8}\,\frac{\sinh(5|\omega|/8)-\cosh(\omega/8)}{\cosh(3\omega/4)}\,.$
(76)

Using this expression in (59) yields the resulting surface energy of the
$O(5)$ spin chain (and the $OSp(n|2m)$ superspin chains with $n-2m=5$)

$f_{\infty}=-1+\frac{2\sqrt{3}}{9}\pi+\frac{\pi}{3}+\frac{1}{3}\left(\psi\left(\frac{5}{12}\right)-\psi\left(\frac{11}{12}\right)\right)\,$
(77)

in agreement with Eq. (14) for $z=5$.

## References

  * Gade (1999) R. M. Gade, “An integrable $sl(2|1)$ vertex model for the spin quantum Hall critical point,” J. Phys. A: Math. Gen. 32, 7071–7082 (1999).
  * Essler  _et al._ (2005) F. H. L. Essler, H. Frahm, and H. Saleur, “Continuum limit of the integrable $sl(2/1)$ $3$-$\bar{3}$ superspin chain,” Nucl. Phys. B 712 [FS], 513–572 (2005), cond-mat/0501197 .
  * Martins  _et al._ (1998) M. J. Martins, B. Nienhuis, and R. Rietman, “An Intersecting Loop Model as a Solvable Super Spin Chain,” Phys. Rev. Lett. 81, 504–507 (1998), cond-mat/9709051 .
  * Jacobsen  _et al._ (2003) J. L. Jacobsen, N. Read, and H. Saleur, “Dense Loops, Supersymmetry, and Goldstone Phases in Two Dimensions,” Phys. Rev. Lett. 90, 090601 (2003), cond-mat/0205033 .
  * Frahm and Martins (2018) H. Frahm and M. J. Martins, “The fine structure of the finite-size effects for the spectrum of the $OSp(n|2m)$ spin chain,” Nucl. Phys. B 930, 545–562 (2018), arXiv:1802.05191 .
  * Saleur and Schomerus (2007) H. Saleur and V. Schomerus, “On the $SU(2|1)$ WZW model and its statistical mechanics applications,” Nucl. Phys. B 775, 312–340 (2007), hep-th/0611147 .
  * Ikhlef  _et al._ (2008) Y. Ikhlef, J. L. Jacobsen, and H. Saleur, “A staggered six-vertex model with non-compact continuum limit,” Nucl. Phys. B 789, 483–524 (2008), cond-mat/0612037 .
  * Frahm and Martins (2011) H. Frahm and M. J. Martins, “Finite size properties of staggered ${U}_{q}[sl(2|1)]$ superspin chains,” Nucl. Phys. B 847, 220–246 (2011), arXiv:1012.1753 .
  * Frahm and Martins (2012) H. Frahm and M. J. Martins, “Phase Diagram of an Integrable Alternating ${U}_{q}[sl(2|1)]$ Superspin Chain,” Nucl. Phys. B 862, 504–552 (2012), arXiv:1202.4676 .
  * Frahm and Hobuß (2017) H. Frahm and K. Hobuß, “Spectral flow for an integrable staggered superspin chain,” J. Phys. A: Math. Theor. 50, 294002 (2017), arXiv:1703.08054 .
  * Vernier  _et al._ (2014a) É. Vernier, J. L. Jacobsen, and H. Saleur, “Non compact conformal field theory and the $a_{2}^{(2)}$ (Izergin-Korepin) model in regime III,” J. Phys. A: Math. Theor 47, 285202 (2014a), arXiv:1404.4497 .
  * Vernier  _et al._ (2014b) É. Vernier, J. L. Jacobsen, and H. Saleur, “Non compact continuum limit of two coupled Potts models,” J. Stat. Mech. , P10003 (2014b), arXiv:1406.1353 .
  * Vernier  _et al._ (2016) E. Vernier, J. L. Jacobsen, and H. Saleur, “The continuum limit of $a_{N-1}^{(2)}$ spin chains,” Nucl. Phys. B 911, 52–93 (2016), arXiv:1601.01559 .
  * Frahm  _et al._ (2019) H. Frahm, K. Hobuß, and M. J. Martins, “On the critical behaviour of the integrable $q$-deformed ${OSp(3|2)}$ superspin chain,” Nucl. Phys. B 946, 114697 (2019), arXiv:1906.00655 .
  * Frahm and Martins (2015) H. Frahm and M. J. Martins, “Finite-size effects in the spectrum of the ${OS}p(3|2)$ superspin chain,” Nucl. Phys. B 894, 665–684 (2015), arXiv:1502.05305 .
  * Polyakov (1975) A. M. Polyakov, “Interaction of goldstone particles in two dimensions. Applications to ferromagnets and massive Yang-Mills fields,” Phys. Lett. B 59, 79–81 (1975).
  * Nahum  _et al._ (2013) A. Nahum, P. Serna, A. M. Somoza, and M. Ortuño, “Loop models with crossings,” Phys. Rev. B 87, 184204 (2013), arXiv:1303.2342 .
  * Cardy (1984) J. L. Cardy, “Conformal invariance and surface critical behavior,” Nucl. Phys. B 240 [FS12], 514–532 (1984).
  * Cardy (1986a) J. L. Cardy, “Effect of boundary conditions on the operator content of two-dimensional conformally invariant theories,” Nucl. Phys. B 275, 200–218 (1986a).
  * Robertson  _et al._ (2020) N. F. Robertson, M. Pawelkiewicz, J. L. Jacobsen, and H. Saleur, “Integrable boundary conditions in the antiferromagnetic potts model,” JHEP 2020, 144 (2020), arXiv:2003.03261 .
  * Robertson  _et al._ (2021) N. F. Robertson, J. L. Jacobsen, and H. Saleur, “Lattice regularisation of a non-compact boundary conformal field theory,” JHEP 2102, 180 (2021), arXiv:2012.07757 .
  * Frahm and Gehrmann (2022) H. Frahm and S. Gehrmann, “Finite size spectrum of the staggered six-vertex model with $u_{q}(\mathfrak{sl}(2))$-invariant boundary conditions,” JHEP , 01, 070 (2022), arXiv:2111.00850 .
  * Blöte  _et al._ (1986) H. W. J. Blöte, J. L. Cardy, and M. P. Nightingale, “Conformal invariance, the central charge and universal finite-size amplitudes at criticality,” Phys. Rev. Lett. 56, 742–745 (1986).
  * Cardy (1986b) J. L. Cardy, “Logarithmic corrections to finite-size scaling in strips,” J. Phys. A: Math. Gen. 19, L1093–L1098 (1986b).
  * Affleck and Qin (1999) I. Affleck and S. Qin, “Logarithmic Corrections in Quantum Impurity Problems,” J. Phys. A: Math. Gen. 32, 7815–7826 (1999), cond-mat/9907284 .
  * Duplantier and Saleur (1986) B. Duplantier and H. Saleur, “Exact surface and wedge exponents for polymers in two dimensions,” Phys. Rev. Lett. 57, 3179–3182 (1986).
  * Martins and Ramos (1997) M. J. Martins and P. B. Ramos, “The algebraic Bethe ansatz for rational braid-monoid lattice models,” Nucl. Phys. B 500, 579–620 (1997), hep-th/9703023 .
  * Sklyanin (1988) E. K. Sklyanin, “Boundary conditions for integrable quantum systems,” J. Phys. A: Math. Gen. 21, 2375–2389 (1988).
  * Arnaudon  _et al._ (2003) D. Arnaudon, J. Avan, N. Crampe’, A. Doikou, L. Frappat, and E. Ragoucy, “Classification of reflection matrices related to (super) Yangians and application to open spin chain models,” Nucl. Phys. B 668, 469–505 (2003), math/0304150 .
  * Arnaudon  _et al._ (2004) D. Arnaudon, J. Avan, N. Crampe, A. Doikou, L. Frappat, and E. Ragoucy, “Bethe Ansatz equations and exact ${S}$ matrices for the ${osp(M|2n)}$ open super spin chain,” Nucl. Phys. B 687, 257–278 (2004), math-ph/0310042 .
  * Yang and Yang (1969) C. N. Yang and C. P. Yang, “Thermodynamics of a one-dimensional system of bosons with repulsive delta-function interaction,” J. Math. Phys. 10, 1115–1122 (1969).
  * Hulthén (1939) L. Hulthén, “Über das Austauschproblem eines Kristalles,” Arkiv Mat. Astron. Fys. 26A, 1–105 (1939).
  * Hamer  _et al._ (1987) C. J. Hamer, G. R. W. Quispel, and M. T. Batchelor, “Conformal anomaly and surface energy for Potts and Ashkin-Teller quantum chains,” J. Phys. A: Math. Gen. 20, 5677–5693 (1987).
  * Affleck (1986) I. Affleck, “Universal term in the free energy at a critical point and and the conformal anomaly,” Phys. Rev. Lett. 56, 746–748 (1986).
  * Cardy (1986c) J. L. Cardy, “Operator content of two-dimensional conformally invariant theories,” Nucl. Phys. B 270, 186–204 (1986c).
  * Granet  _et al._ (2019) E. Granet, J. L. Jacobsen, and H. Saleur, “Spontaneous symmetry breaking in 2D supersphere sigma models and applications to intersecting loop soups,” J. Phys. A: Math. Theor 52, 345001 (2019), arXiv:1810.07807 .
  * der Jeugt (1984) J. V. der Jeugt, “Finite- and infinite-dimensional representations of the orthosymplectic superalgebra OSP(3,2),” J. Math. Phys. 25, 3334–3349 (1984).
  * Alcaraz  _et al._ (1988) F. C. Alcaraz, M. Baake, U. Grimmn, and V. Rittenberg, “Operator content of the XXZ chain,” J. Phys. A: Math. Gen. 21, L117–L120 (1988).
  * Eggert and Affleck (1992) S. Eggert and I. Affleck, “Magnetic impurities in half-integer-spin Heisenberg antiferromagnetic chains,” Phys. Rev. B 46, 10866–10883 (1992).
  * Takhtajan (1982) L. Takhtajan, “The picture of low–lying excitations in the isotropic Heisenberg chain of arbitrary spins,” Phys. Lett. A 87, 479–482 (1982).
  * Babujian (1982) H. M. Babujian, “Exact solution of the one–dimensional isotropic Heisenberg chain with arbitrary spin ${S}$,” Phys. Lett. A 90, 479–482 (1982).
  * Zheng  _et al._ (2022) Z. Zheng, P. Sun, X. Xu, T. Yang, J. Cao, and W.-L. Yang, “Thermodynamic limit and boundary energy of the spin-1 Heisenberg chain with non-diagonal boundary fields,” SciPost Phys. 12, 71 (2022).


# BrainPainter: A software for the visualisation of brain structures,
biomarkers and associated pathological processes

Răzvan V. Marinescu  Arman Eshaghi  Daniel C. Alexander  Polina Golland

###### Abstract

We present BrainPainter, a software that automatically generates images of
highlighted brain structures given a list of numbers corresponding to the
output colours of each region. Compared to existing visualisation software
(i.e. Freesurfer, SPM, 3D Slicer), BrainPainter has three key advantages: (1)
it does not require the input data to be in a specialised format, allowing
BrainPainter to be used in combination with any neuroimaging analysis tools,
(2) it can visualise both cortical and subcortical structures and (3) it can
be used to generate movies showing dynamic processes, e.g. propagation of
pathology on the brain. We highlight three use cases where BrainPainter was
used in existing neuroimaging studies: (1) visualisation of the degree of
atrophy through interpolation along a user-defined gradient of colours, (2)
visualisation of the progression of pathology in Alzheimer’s disease as well
as (3) visualisation of pathology in subcortical regions in Huntington’s
disease. Moreover, through the design of BrainPainter we demonstrate the
possibility of using a powerful 3D computer graphics engine such as Blender to
generate brain visualisations for the neuroscience community. Blender’s
capabilities, e.g. particle simulations, motion graphics, UV unwrapping,
raster graphics editing, raytracing and illumination effects, open a wealth of
possibilities for brain visualisation not available in current neuroimaging
software. BrainPainter is customisable, easy to use, and can run straight from
the web browser: https://brainpainter.csail.mit.edu, as well as from source-
code packaged in a docker container: https://github.com/mrazvan22/brain-
coloring. It can be used to visualise biomarker data from any brain imaging
modality, or simply to highlight a particular brain structure for e.g. anatomy
courses.

##  1 Introduction

Efficient visualisation of brain structure, function and pathology is crucial
for understanding the mechanisms underlying neurodegenerative diseases and
eases the interpretation of results in neuroimaging studies. This is
especially important in populations studies, where two or more populations are
compared for group differences in biomarkers derived from e.g. Magnetic
Resonance Imaging, Positron Emission Tomography (PET) or Computer Tomography
(CT). The results are best visualised as brain images, where regions-of-
interest (ROIs) are highlighted based on the magnitude of the difference
between the two groups. These visualisation are generally done by the same
software that performs the registration, segmentation and statistical
analysis. However, for traumatic brain injury or less common neurodegenerative
diseases such as Parkinson’s disease and Multiple Sclerosis, visualisations of
statistical results is sometimes not performed due to the inability to
register images to a common template or lack of robust registration software.
Therefore, many studies such as (Coughlin et al. (2015); Schoonheim et al.
(2012)) only report differences between patients and controls in tables or as
box plots. There is therefore a lack of visualisation tools that can highlight
neuroimaging findings for these complex diseases.

When registration to a common population template is possible, e.g. in
Alzheimer’s disease (AD), excellent 3D visualisation software exists which
allows interactive visualisation of population differences – e.g. 3D Slicer
(Pieper et al. (2004)), Freesurfer (Fischl (2012)) or SPM (Penny et al.
(2011)). However, they have several inherent limitations. First, such software
– e.g. Freesurfer – generally require inputs in their proprietary data format,
which is usually difficult and time-consuming to create without using their
pipeline. While creating these proprietary data formats is necessary when
users need to display voxelwise visualisations, often users only need to
highlight entire ROIs – in this simpler case the user could only provide a
list of RGB colors for each ROI in a csv file, removing the need to create
input data in a specialised format. Another limitation of existing
visualisation software is their difficulty in highlighting complex patterns of
pathology in a single slice of a 3D volumetric image. To overcome this, some
authors show multiple slices (sometimes up to 8 slices, e.g. Migliaccio et al.
(2015)), although this takes too much space on the academic paper being
published. While Freesurfer solves this problem using a cortical surface-based
representation that captures most of the complexity of pathology patterns in a
single image, this surface representation is not supported for subcortical
structures such as the hippocampus. Third, current visualisation software
cannot be easily used to generate e.g. a movie showing a dynamic process, e.g.
propagation of pathology within the human brain, as most of them have been
intended for interactive visualisation and have no application programming
interface (API) that allows automatic generation of hundreds of images using
pre-defined settings.

We present BrainPainter, a software for easy visualisation of structures,
pathology and biomarkers in the brain. As opposed to previous visualisation
software, the input data is very simple: a generic .csv file containing
numbers for each ROI, each number mapping to a different colour to be assigned
to that ROI – such a simple input allows BrainPainter to be used in
conjunction with any other neuroimaging analysis software. Secondly,
BrainPainter can visualise both cortical and subcortical structures using a
surface representation, removing the need to show multiple slices of the same
3D scan. Third, the images are generated automatically from pre-defined view-
points, and can be easily used to create a movie showing e.g. the propagation
of pathology, without the need to write any extra software code or interface
with an API. Written in Python, BrainPainter is open-source and can be
extended in many ways. In this paper we describe the capabilities of the
software, and showcase three use cases where it was used in neuroimaging
studies.

##  2 Design

Figure 2: BrainPainter website interface at http://brainpainter.csail.mit.edu,
showing how it can be easily customised. Here, the user selects the brain
type, colours and resolution, and finally uploads the input .csv file with ROI
biomarkers. The server then generates the output images, which can be
downloaded by the user. More customisation features will be added in future
versions.

BrainPainter has a very simple yet effective workflow. Given an input csv file
with biomarkers for each region, it produces high-quality visualisations of
cortical and subcortical structures. For this, it uses Blender as a rendering
engine, and loads 3D meshes from a template brain (one 3D mesh for each ROI),
which are then coloured according to the input numbers. Instead of providing a
list of RGB colours for each ROI, we choose a simpler interface of providing
one number for each ROI which maps to an RGB color using a user-defined color
gradient. For example, the gradient can range from white $\rightarrow$ yellow
$\rightarrow$ orange $\rightarrow$ red, as in the example from Fig. 1. In this
case, the input numbers for each ROI need to be in the range [0,3], where a
value of 1.3 would interpolate between colour 1 (yellow) and color 2 (orange).

BrianPainter uses open-source software Blender as the rendering engine. We
chose Blender for three reasons. First, it is open-source, allowing us to
distribute it already integrated with BrainPainter, thus requiring no further
installation. Secondly, Blender is a powerful 3D graphics software, which
allowed us to create realistic lightning conditions and handle transparency
required for the glass-brain. Third, it also supports creating movies of
complex temporal processes such as pathology spread along the brain. The
software also supports a variety of object formats for the brain template,
including the popular .obj mesh format. As BrainPainter is written in Python,
it allows interfacing with any Blender function.

The software is able to colour and visualise regions belonging to a pre-
defined atlas. Currently, we support three widely-used atlases: (i) the
Desikan-Killiany (DK) atlas (Desikan et al. (2006)), (ii) the Destrieux atlas
(Destrieux et al. (2010)) and (iii) the Tourville atlas (Klein and Tourville
(2012)). However, a custom atlas can also be used by mapping those regions to
any of the three atlases currently supported, through the modification a
simple mapping in the main configuration file.
111https://github.com/mrazvan22/brain-coloring/blob/master/config.py

##  3 Customisation

BrainPainter can be easily customised in several ways, as shown in Fig 2.
First of all, the colours assigned to each region can be changed by modifying
both the control points of the color gradient and the input numbers selecting
colors along the gradient. The background colour and image resolution can also
be changed.

The 3D structures being visualised can also be customised. We currently
support three atlases (Desikan-Killiany, Destrieux and Tourville) as well as
two types of brain surfaces: inflated, which is a brain surface that is
smoothed out and where no gyri appear, and also pial, the standard brain
surface with gyri. The software allows one to remove some 3D structures – for
example, Fig. 5 shows the subcortical structures with the cerebellum removed
from the visualisation – contrast this with Fig. 4.

BrainPainter also support two types of surfaces, cortical and subcortical
structures. For the cortical surface, we only show the left hemisphere
(although the right hemisphere can also be added), and provide two default
viewing angles (front and back). For the subcortical structures, we show them
for both hemispheres and also show the right hemisphere as a glass brain, for
reference.

More complex settings such as the viewing angle and luminosity can also be
customised, but currently require minor modifications to the source code. In
the future, we plan to enable these customisations from the main configuration
file.

Figure 3: Demonstration of BrainPainter for showing extent of pathology on a
vertical bar, where colours towards blue show increased severity. Source:
Young et al. (2018). Figure 4: Demonstration of our software for showing the
temporal progression of pathology, as a sequence of snapshot at different
stages. Images used by Young et al. (2018).

##  4 First use case: Visualising the degree of pathology

In the first use case, we want to visualise the degree of pathology in
Alzheimer’s disease. During the progression of Alzheimer’s disease, some
regions of the brain such as the hippocampus and temporal lobes will be more
affected compared to other regions of the brain such as the occipital lobe.
Visualisation of pathology in AD is important in order to understand its
underlying mechanisms and generate new hypotheses.

The notion of pathology here is abstract, and can refer to atrophy as measured
by volume loss or cortical thinning, white matter degradation as measured by
diffusion tensor imaging (DTI) changes in fractional anisotropy (FA), or the
level of abnormal conformations of proteins such as amyloid-beta or tau as
measured by Positron Emission Tomography. However, BrainPainter is agnostic to
the meaning of these biomarkers and can be used with any imaging modality,
including markers derived from several modalities together.

Fig. 3 shows an application of BrainPainter by (Young et al. (2018)) to
highlight the degree of atrophy in Alzheimer’s disease. Regions with no
atrophy are coloured in white, while regions with severe atrophy are coloured
in blue. The gradient on the right shows, for every color, the number of
standard deviations away from controls.

##  5 Second use case: Visualising the temporal progression of
neurodegenerative diseases

In the second use case, we would like to visualise the temporal progression of
Alzheimer’s disease (AD). Alzheimer’s disease is characterised by a slow,
continuous progression – while it’s mechanisms are still not fully understood,
it is currently believed that initial abnormalities in the amyloid and tau
proteins cause a cascade of events that eventually lead to axonal degradation,
neural death and cognitive decline (Mudher and Lovestone (2002)). Therefore,
being able to visualise the progression of these events, including their
timing and speed, is crucial for understanding the mechanisms of Alzheimer’s
disease.

Fig. 4, reproduced and adapted from Young et al. (2018), demonstrates the
ability of BrainPainter to visualise the evolution of atrophy in two subtypes
of Alzheimer’s disease – _cortical_ and _subcortical_ – characterised by
prominent atrophy in the cortical and subcortical regions respectively. This
study done by Young et al. (2018) used data from the Alzheimer’s disease
Neuroimaging Initiative to disentangle the heterogeneity of AD into subtypes
with different progression. Here, visualisations provided by BrainPainter were
able to characterise not only the degree of atrophy in each region (white/red
to blue colors), but also the timing of atrophy events. For example, even in
the _cortical_ subtype, the hippocampus becomes affected by stage 13, while
similarly, in the _subcortical_ subtype the temporal lobe becomes affected by
stage 13.

##  6 Third use case: Visualising pathology in subcortical structures

The ability to visualise subcortical structures is crucial for
neurodegenerative diseases that cause damage to these regions. Apart from
Alzheimer’s disease, Huntington’s disease (HD) is also known for targetting
subcortical regions (Douaud et al. (2009); Wijeratne et al. (2018)). The
neurodegeneration in HD is believed to begin in the striatum and pallidum, and
later followed by other subcortical and cortical regions (Douaud et al.
(2009)).

Fig. 5, reproduced and adapted with permission from Wijeratne et al. (2018),
shows visualisations generated by BrainPainter of atrophy progression in
subcortical areas, for Huntington’s disease. The images show early involvement
of the putamen, caudate and pallidum in the progression of Huntington’s
disease, and demonstrate the potential of BrainPainter in visualising
pathology dynamics in subcortical regions using parsimonious glass-brain
images.

(a) Stage 0

(b) Stage 3

(c) Stage 7

(d) Stage 10

Figure 5: Progression of pathology in subcortical regions within a glass
brain, using images generated with our method. Images used by Wijeratne et al.
(2018).

##  7 Conclusion

We presented BrainPainter, an open-source software that can be used to
visualise structures, biomarkers and pathologies in the human brain. The
visualisations generated by BrainPainter can be used to significantly enhance
the interpretation of neuroimaging research and can be easily embedded by
researchers into scientific articles. While not demonstrated here,
BrainPainter can also easily generate movies showing dynamic processes, e.g.
propagation of brain pathology.

Our software has several limitations that can be addressed in future versions.
First, it can currently only highlight entire regions-of-interest from an
atlas. However, this was a design choice, as it removes the need for users to
create specialised input files with voxelwise measurements, thus increasing
usability. Nevertheless, in future versions we might add the ability to
highlight fine-grained patterns of pathology. Yet another limitation of
BrainPainter is that it cannot visualise more complex structures such as
white-matter tracts, although we plan to add such functionality in future
releases.

The use of the powerful Blender engine opens numerous avenues not possible
with current neuroimaging software: motion graphics can be used to generate
realistic movies showing e.g. the evolution of biomarkers, particle
simulations can be used to visualise toxic proteins accumulating in certain
regions, soft-body simulations can be used to model brain deformations due to
head trauma, while camera-based rendering allows the creation of educational
videos.

##  8 Acknowledgements

RVM was supported by the NIH grants NIBIB NAC P41EB015902 and NINDS
R01NS086905, as well as the EPSRC Centre For Doctoral Training in Medical
Imaging with grant EP/L016478/1. AE received a McDonald Fellowship from the
Multiple Sclerosis International Federation (MSIF, www.msif.org), and the
ECTRIMS – MAGNIMS Fellowship. DCA was supported by EuroPOND, which is an EU
Horizon 2020 project, and by EPSRC grants J020990, M006093 and M020533. PG was
supported by NIH grants NIBIB NAC P41EB015902 and NINDS R01NS086905.

We are also particularly grateful to Anderson Winkler for creating the 3D
brain templates for all three atlases, which are used in this
work.222https://brainder.org/research/brain-for-blender/

##  9 References

## References

  * Coughlin et al. (2015) Jennifer M Coughlin, Yuchuan Wang, Cynthia A Munro, Shuangchao Ma, Chen Yue, Shaojie Chen, Raag Airan, Pearl K Kim, Ashley V Adams, Cinthya Garcia, et al.  Neuroinflammation and brain atrophy in former NFL players: an in vivo multimodal imaging pilot study.  _Neurobiology of disease_ , 74:58–65, 2015.
  * Desikan et al. (2006) Rahul S Desikan, Florent Ségonne, Bruce Fischl, Brian T Quinn, Bradford C Dickerson, Deborah Blacker, Randy L Buckner, Anders M Dale, R Paul Maguire, Bradley T Hyman, et al.  An automated labeling system for subdividing the human cerebral cortex on MRI scans into gyral based regions of interest.  _Neuroimage_ , 31(3):968–980, 2006.
  * Destrieux et al. (2010) Christophe Destrieux, Bruce Fischl, Anders Dale, and Eric Halgren.  Automatic parcellation of human cortical gyri and sulci using standard anatomical nomenclature.  _Neuroimage_ , 53(1):1–15, 2010.
  * Douaud et al. (2009) Gwenaëlle Douaud, Timothy E Behrens, Cyril Poupon, Yann Cointepas, Saâd Jbabdi, Véronique Gaura, Narly Golestani, Pierre Krystkowiak, Christophe Verny, Philippe Damier, et al.  In vivo evidence for the selective subcortical degeneration in Huntington’s disease.  _Neuroimage_ , 46(4):958–966, 2009.
  * Fischl (2012) Bruce Fischl.  Freesurfer.  _Neuroimage_ , 62(2):774–781, 2012.
  * Klein and Tourville (2012) Arno Klein and Jason Tourville.  101 labeled brain images and a consistent human cortical labeling protocol.  _Frontiers in neuroscience_ , 6:171, 2012.
  * Migliaccio et al. (2015) Raffaella Migliaccio, Federica Agosta, Katherine L Possin, Elisa Canu, Massimo Filippi, Gil D Rabinovici, Howard J Rosen, Bruce L Miller, and Maria Luisa Gorno-Tempini.  Mapping the progression of atrophy in early-and late-onset alzheimer’s disease.  _Journal of Alzheimer’s Disease_ , 46(2):351–364, 2015.
  * Mudher and Lovestone (2002) Amritpal Mudher and Simon Lovestone.  Alzheimer’s disease–do tauists and baptists finally shake hands?  _Trends in neurosciences_ , 25(1):22–26, 2002\.
  * Penny et al. (2011) William D Penny, Karl J Friston, John T Ashburner, Stefan J Kiebel, and Thomas E Nichols.  _Statistical parametric mapping: the analysis of functional brain images_.  Elsevier, 2011.
  * Pieper et al. (2004) Steve Pieper, Michael Halle, and Ron Kikinis.  3d slicer.  In _2004 2nd IEEE international symposium on biomedical imaging: nano to macro (IEEE Cat No. 04EX821)_ , pages 632–635. IEEE, 2004.
  * Schoonheim et al. (2012) Menno M Schoonheim, Veronica Popescu, Fernanda C Rueda Lopes, Oliver T Wiebenga, Hugo Vrenken, Linda Douw, Chris H Polman, Jeroen JG Geurts, and Frederik Barkhof.  Subcortical atrophy and cognition: sex effects in multiple sclerosis.  _Neurology_ , 79(17):1754–1761, 2012.
  * Wijeratne et al. (2018) Peter A Wijeratne, Alexandra L Young, Neil P Oxtoby, Razvan V Marinescu, Nicholas C Firth, Eileanoir B Johnson, Amrita Mohan, Cristina Sampaio, Rachael I Scahill, Sarah J Tabrizi, et al.  An image-based model of brain volume biomarker changes in Huntington’s disease.  _Annals of clinical and translational neurology_ , 5(5):570–582, 2018.
  * Young et al. (2018) Alexandra L Young, Razvan V Marinescu, Neil P Oxtoby, Martina Bocchetta, Keir Yong, Nicholas C Firth, David M Cash, David L Thomas, Katrina M Dick, Jorge Cardoso, et al.  Uncovering the heterogeneity and temporal complexity of neurodegenerative diseases with Subtype and Stage Inference.  _Nature communications_ , 9(1):4273, 2018.