# Quasispins of vacancy defects in Ising chains with nearest- and next-to-
nearest-neighbour interactions

Shijun Sun  Physics Department, University of California, Santa Cruz,
California 95064, USA  Arthur P. Ramirez  Physics Department, University of
California, Santa Cruz, California 95064, USA  Sergey Syzranov  Physics
Department, University of California, Santa Cruz, California 95064, USA

###### Abstract

Motivated by frustrated magnets and quasi-one-dimensional magnetic materials,
we study the magnetic properties of 1D Ising chains with nearest-neighbour
(NN) and weaker next-to-nearest neighbour (NNN) interactions in the presence
of vacancy defects. The effect of a vacancy on the magnetic susceptibility of
a spin chain is two-fold: it reduces the length of the chain by an effective
“vacancy size” and may also act as a free spin, a “quasispin”, with a Curie-
type $\chi_{\text{quasi}}=\langle S^{2}\rangle/T$ contribution to the
susceptibility. In chains with antiferromagnetic short-range order, the
susceptibility of vacancy-free chains is exponentially suppressed at low
temperatures, and quasispins dominate the effect of impurities on the chains’
magnetic properties. For chains with antiferromagnetic NN interactions, the
quasispin matches the value $\langle S^{2}\rangle=1$ of the Ising spins in the
chain for ferromagnetic NNN interactions and vanishes for antiferromagnetic
NNN interactions. For chains with ferromagnetic short-range order, quasispin
effects are insignificant due to exponentially large low-temperature
susceptibilities, and the dominant effect of a vacancy is effectively changing
the length of the chain.

Impurities in magnetic materials are often seen as detrimental to magnetic
order and collective quantum coherent phenomena such as quantum spin liquids.
Non-magnetic impurities can dilute the spins in a magnetic material, thus
inhibiting long-range magnetic order. In materials with strong geometrical
frustration [1], small amounts of impurities lead to the formation of a spin
glass state.

Despite their detrimental effects in some situations, impurities can also be
used as a powerful probe of the low-temperature physics of pure materials. For
example, an analysis of the spin-glass freezing temperature in geometrically
frustrated magnets (GFMs) as a function of vacancy concentration suggests the
existence of a “hidden energy scale” [2], at least an order of magnitude lower
than the Weiss constant, at which random spin freezing occurs in very clean
samples.

Vacancy defects can also have a significant effect on the magnetic
susceptibility. In vacancy-free GFMs, the susceptibility above the hidden
energy scale is described by the Curie-Weiss law
$\chi(T)\propto(T+|\theta_{W}|)^{-1}$, where $\theta_{W}$ is the Weiss
constant. Introducing vacancies can be naturally expected to re-scale the
susceptibility by a factor of $\propto\left(N-N_{\text{imp}}\right)/N$ where
$N_{\text{imp}}$ is the number of vacancies and $N$ is the total number of
sites at which vacancies can be located. At the same time, introducing
vacancies in a GFM leads to another dramatic effect [3, 4]: it gives rise to a
Curie-like contribution $\propto 1/T$ to the susceptibility. This suggests
that, in addition to diluting the bulk magnetic susceptibility in GFMs,
vacancies lead to the emergence of degrees of freedom that behave as free
spins.

Such a Curie-like contribution has been attributed in Ref. [3] to free “orphan
spins”, i.e. spins disconnected from the other spins by quenched disorder (not
necessarily vacancies). The density of such orphan spins would, however, be
too low [4] in real materials to explain the experimentally observed
susceptibility. It has also been shown theoretically [5, 6, 7] that “half-
orphan” spins, i.e. spins located next to pairs of vacancies and thus not
fully connected to the others, have the susceptibility of free fractional
spins.

The $\propto 1/T$ behaviour of the vacancy-induced magnetic susceptibility
can, however, be generically associated with individual vacancy defects.
Indeed, for a system whose magnetisation $M_{z}$ along a certain direction
commutes with the Hamiltonian, the fluctuation-dissipation relation
$\chi_{zz}(T)=\left<M_{z}^{2}\right>/T$ suggests the same temperature
dependence as for the susceptibility of a free, possibly fractional spin if
$\left<M_{z}^{2}\right>\neq 0$ at $T=0$.

The existence of vacancy-associated magnetisation has also been theoretically
demonstrated for defects in 2D Heisenberg antiferromagnets [8, 9, 10, 11, 12,
13, 14], classical vector spins [13, 14, 15, 16] and several-legged Heisenberg
ladders [8]. We note that the previous theoretical studies of vacancies in
magnetic systems in dimensions $d>1$ focussed on magnetically ordered states.

For one-dimensional (1D) systems, analytical solutions are easily available,
which often allows one to gain valuable insights into many-body phenomena.
Vacancies in 1D chains with nearest-neighbour (NN) interactions also give rise
to the $\propto 1/T$ contribution to the susceptibility [17, 18, 19, 20, 21].
In realistic magnetic systems, next-to-nearest-neighbour (NNN) interactions
may significantly alter spin correlations around the vacancy and thus affect
the susceptibility. In addition to that, the presence of the NNN interactions,
in general, leads to frustration.

NNN NN |  Antiferromagnetic ($J_{1}>0$) |  Ferromagnetic ($J_{1}<0$)
---|---|---
Antiferro- magnetic ($J_{2}>0$) |  Quasispin $\langle S^{2}\rangle=0$; length increased by $-b\approx e^{(2J_{1}-6J_{2})/T}$ |  Length reduced by $b\approx e^{\left(2|J_{1}|-4J_{2}\right)/T}$
Ferro- magnetic ($J_{2}<0$) |  Quasispin $\langle S^{2}\rangle=1$; length reduced by $b\approx e^{(2J_{1}-2J_{2})/T}$ |  Length reduced by $b\approx e^{\left(2|J_{1}|-2J_{2}\right)/T}$
Table 1:  The leading effects of a single vacancy on an Ising chain with the
nearest-neighbour (NN) coupling $J_{1}$ and next-to-nearest-neighbour (NNN)
coupling $J_{2}$. The vacancy effectively reduces the length of the chain by
the “size” $b(T)$. In addition to that, a significant effect in chains with
antiferromagnetic NN interactions and ferromagnetic NNN interactions is the
emergence of vacancy “quasispins”.

In this paper, we study the effect of vacancy defects on the magnetism of
Ising chains with both NN and NNN interactions. The presence of both NN and
NNN interactions allows us to reveal some of the effects of vacancies on
frustrated materials. Our results can also be used to describe quasi-1D
magnetic materials, such as $\text{CoNb}_{2}\text{O}_{6}$[22, 23, 24] and
$\text{CoCl}_{2}\cdot\text{D}_{2}\text{O}$[25] (with ferromagnetic ground
states) and BCVO[26, 27, 28] (with an antiferromagnetic ground state).
Although defects in these systems have not been studied yet, the results
presented here would be applicable. We compute analytically the magnetic
susceptibility of such chains, revealing the contributions of both vacancies
(“quasispins”), the bulk spins and their interplay.

Summary of results. Due to a rapid decay of exchange interactions with
distance, we focus on chains with ratios of NN to NNN couplings
$|J_{1}/J_{2}|>2$. We find that the magnetic susceptibility of Ising chains
with dilute vacancies is described by the formula

$\displaystyle\chi(T)=\frac{\langle
S^{2}\rangle}{T}N_{\text{imp}}+\frac{N-b(T)N_{\text{imp}}}{N}\chi_{0}(T),$ (1)

where $\chi_{0}(T)$ is the susceptibility of a vacancy-free chain, and the
length $b(T)$ is the effective “vacancy size”.

As Eq. (1) shows, the effect of vacancies is two-fold. In general, a vacancy
effectively reduces the length of the chain by the effective “vacancy size”
$b(T)$ and also acts as a “quasispin” of magnitude $\sqrt{\langle
S^{2}\rangle}$. Our results are summarised in Table 1. Such behaviour of
magnetic susceptibility captures the physics of vacancy defects in GFMs. The
empirical dependence for the “orphan” spins proposed in Ref. [3] corresponds
to $b(T)=0$ and the susceptibility $\chi_{0}(T)$ described by the Curie-Weiss
law.

In vacancy-free chains with antiferromagnetic NN interactions, the
susceptibility $\chi_{0}(T)$ is exponentially suppressed and non-singular at
$T=0$. As a result, quasispin susceptibility, the first term in Eq. (1),
dominates the change $\partial\chi(T)/\partial N_{\text{imp}}$ of the total
susceptibility with vacancy concentration [29, 2]. We find that for
ferromagnetic NNN interactions, $\langle S^{2}\rangle=1$, while for
antiferromagnetic NNN interactions, $\langle S^{2}\rangle=0$.

In chains with ferromagnetic NN interactions, the susceptibility $\chi_{0}(T)$
is exponentially large at $T\rightarrow 0$ and more singular than the
quasispin contribution. In that regime, the quasispins are insignificant. We
compute the effective “vacancy” size $b(T)$ in all regimes.

Qualitative interpretation. The obtained quasispin values can be understood as
follows. An infinite chain with a vacancy is equivalent to two half-infinite
chains whose open ends are coupled by the NNN interaction at the vacancy. The
end of a half-infinite chain may, in general, be expected to have its own
magnetic moment $\mu$. When the two ends are connected by ferromagnetic NNN
interaction, their magnetic moments add up, resulting in the magnetic moment
$2\mu$ of the vacancy. By contrast, antiferromagnetic NNN interaction makes
the magnetic moments of the open ends antialign, resulting in a zero magnetic
moment of the vacancy, in accordance with the results of a direct calculation
(see Table 1).

To complement this qualitative argument, in Appendix E, we evaluate the
magnetic moment of a free end of the spin chain explicitly and obtain
$\mu=\frac{1}{2}$, in accordance with vacancy quasispin
$\left<S^{2}\right>=1$.

Generic expressions for magnetic susceptibility. For the sufficiently fast
decay of exchange interactions with distance we consider, the ground state is
determined by the NN interactions: all spins are aligned for $J_{1}<0$ and
neighbouring spins are antialigned for $J_{1}>0$ regardless of the sign of the
NNN coupling $J_{2}$.

We assume the temperature $T$ of the chain to be low and significantly
exceeded by both the couplings $J_{1}$ and $J_{2}$, as well as the energy

$\displaystyle E_{D}=2|J_{1}|-4J_{2}>0$ (2)

of a “domain wall”, i.e. the energy of a state obtained from the ground state
by flipping all spins to the right of a given site.

According to the fluctuation-dissipation theorem, the susceptibility of chains
with and without vacancies is given by

${\chi}(T)=\frac{\langle
M^{2}\rangle}{T}=\frac{1}{T}\sum_{i,j}\langle\sigma_{i}\sigma_{j}\rangle,$ (3)

where $M=\sum_{i}\sigma_{i}$ is the total magnetisation of the system;
$\langle\ldots\rangle$ is averaging with respect to the thermal state of the
system at temperature $T$; and the summation with respect to the indices $i$
and $j$ runs over all sites with spins. In Eq. 3, we used that the average
magnetisation $\langle\sigma_{i}\rangle$ vanishes at each site.

To compute the contribution of one vacancy to the susceptibility, we utilise
Eq. (3) to obtain the susceptibilities $\chi(T)$ and $\chi_{0}(T)$ of the
chains with and without a vacancy, as shown in Fig. 1, and consider the
difference

$\displaystyle\chi(T)-\chi_{0}(T)=$ $\displaystyle
2\beta\left(\langle\sigma_{-1}\sigma_{1}\rangle-\langle\sigma_{-1}\sigma_{1}\rangle_{0}\right)$
$\displaystyle+4\beta\sum_{j=2}^{+\infty}\left(\langle\sigma_{1}\sigma_{j}\rangle-\langle\sigma_{1}\sigma_{j}\rangle_{0}\right)$
$\displaystyle+4\beta\sum_{j=2}^{+\infty}\left(\langle\sigma_{-1}\sigma_{j}\rangle-\langle\sigma_{-1}\sigma_{j}\rangle_{0}\right)$
$\displaystyle+2\beta\sum_{i=-2}^{-\infty}\sum_{j=2}^{+\infty}\left(\langle\sigma_{i}\sigma_{j}\rangle-\langle\sigma_{i}\sigma_{j}\rangle_{0}\right),$
(4)

where the averages $\langle\cdots\rangle$ and $\langle\cdots\rangle_{0}$ are
for the chains, respectively, with and without the vacancy, and

$\chi_{0}(T)\approx NT^{-1}e^{\mp E_{D}/T}$ (5)

is the susceptibility of the vacancy-free chain; “$-$” and “$+$” correspond,
respectively, to ferromagnetic and antiferromagnetic NN interactions. In each
of the sums in Eq. (4), we extended one of the summation limits to infinity,
assuming that the vacancy is located far (significantly further than the
correlation length $\xi$) from the ends of the chain.

Figure 1: Spin configurations in the ground states of Ising chains with
antiferromagnetic NN interactions and ferromagnetic NNN interactions. (a) A
chain without a vacancy. (b) A chain with a vacancy. The labeling of the spins
allows to conveniently compare the magnetic susceptibilities of the two chains
and thus obtain the susceptibility of the vacancy (“quasispin”).

Spin correlations in a gas of domain walls. The spin correlators in Eq. (4)
can be conveniently evaluated by mapping the chain to a one-dimensional ideal
gas of domain walls. Indeed, the density of the domain walls (per site) can be
evaluated as $n_{D}\approx\exp\left({-E_{D}/T}\right)\ll 1$. This density
significantly exceeds the density $n_{DD}\approx
n_{D}\exp\left(-2|J_{1}|/T\right)$ of double domain walls (domain walls on
neighbouring sites). The approximation of an ideal gas of domain walls is,
therefore, well justified in the considered limit of law temperatures
$T\ll|J_{1}|,|J_{2}|,E_{D}$.

The mapping to a gas of domain walls allows us to compute the correlators of
magnetisation in Eq. (4). In the case of antiferromagnetic NN interactions, we
obtain (see Appendix A for details)

$\displaystyle\langle\sigma_{-1}\sigma_{1}\rangle$
$\displaystyle=\frac{1-e^{2\beta J_{2}}}{1+e^{2\beta J_{2}}},$ (6a)
$\displaystyle\langle\sigma_{1}\sigma_{j}\rangle$
$\displaystyle\approx\frac{e^{-\beta E_{D}-2\beta J_{2}}-1}{1+e^{-\beta
E_{D}-2\beta J_{2}}}\left(\frac{e^{-\beta E_{D}}-1}{1+e^{-\beta
E_{D}}}\right)^{j-2},$ (6b) $\displaystyle\langle\sigma_{-1}\sigma_{j}\rangle$
$\displaystyle\approx\langle\sigma_{-1}\sigma_{1}\rangle\langle\sigma_{1}\sigma_{j}\rangle,$
(6c) $\displaystyle\langle\sigma_{i}\sigma_{j}\rangle$
$\displaystyle\approx\langle\sigma_{i}\sigma_{-1}\rangle\langle\sigma_{-1}\sigma_{1}\rangle\langle\sigma_{1}\sigma_{j}\rangle,$
(6d)

for $i\leq-2$ and $j\geq 2$, where the spins in the chains with and without
the vacancy are labelled as shown, respectively, in Figs. 1a and 1b. In
accordance with the chain’s symmetry, the correlators satisfy the relation

$\langle\sigma_{i}\sigma_{j}\rangle=\langle\sigma_{-i}\sigma_{-j}\rangle$ (7)

for all $i$ and $j$.

For antiferromagnetic NN and ferromagnetic NNN interactions ($J_{1}>0$,
$J_{2}<0$), Eqs. (6a)-(6d) and (3) give the magnetic susceptibility

$\displaystyle\chi(T)-\chi_{0}(T)=T^{-1}$
$\displaystyle-T^{-1}e^{-2|J_{2}|/T}$
$\displaystyle+o\left(T^{-1}e^{-2|J_{2}|/T}\right),$ (8)

of the vacancy in the limit of low temperatures. In accordance with Eq. (1),
the first term is the free-spin-type contribution corresponding to the
quasispin $\langle S^{2}\rangle=1$, and the second term describes an effective
shortening of the chain by the “vacancy size” $b(T)\approx
e^{(2J_{1}-2J_{2})/T}$.

In the case of other signs of the couplings $J_{1}$ and $J_{2}$, the
correlators $\langle\sigma_{i}\sigma_{j}\rangle$ can be computed similarly. We
present the details of the calculations of the susceptibility of the chain
with a vacancy for the cases $J_{1}>0,J_{2}>0$; $J_{1}<0;J_{2}>0$ and
$J_{1}<0;J_{2}<0$ in, respectively, Appendices B, C and D. The results for the
leading effects of the vacancy on the susceptibility are summarised in Table
1.

We emphasise that for ferromagnetic chains ($J_{1}<0$), the susceptibility is
exponentially large in temperature, $\chi(T)\propto\exp\left(E_{D}/T\right)$.
In this case, the main effect of the vacancy is changing the length of the
chain by an effective vacancy size $b(T)$ (see Table 1 for the values of this
length).

Conclusion. We have studied the effects of vacancy defects on the magnetic
susceptibility of Ising chains with NN and (weaker) NNN interactions. We have
shown, using analytical calculations, that the susceptibility of a chain with
a vacancy is given, in general, by a combination of two contributions: the
contribution of the bulk spins and the Curie-like contribution of vacancy
“quasispins”.

For chains with antiferromagnetic NN interactions, we found the quasispin of a
vacancy to be, in the units of Ising spins, one (zero) for ferromagnetic
(antiferromagnetic) NNN interactions. In the case of ferromagnetic NN
interactions, the leading effect of a vacancy defect is the effective change
of the chain length.

The susceptibilities of individual vacancies accurately describe the
properties of magnetic materials in the limit of dilute vacancies, when the
average distance between the defects significantly exceeds the correlation
length $\xi$ of a clean material. At larger vacancy concentrations, quasispin-
quasispin interactions may become essential. The qualitative effect of such
interactions may be particularly important in the case of zero (or very small)
quasispins of individual defects [16]: effective interactions may lead to a
finite (large) collective magnetisation of the defects.

Another interesting aspect of quasispins that deserves investigation is their
behaviour beyond linear response in magnetic field and behaviour in
inhomogeneous magnetic fields. A cluster of spins in a GF system of a size
smaller than the correlation length $\xi$ can be expected to behave similarly
to a free spin whose magnetisation is encoded by distinct ground states of the
cluster. Unlike a conventional spin on a single site of the lattice, such a
quasispin will be protected from local weak fluctuations of the field due to
the collective character of its magnetisation.

Acknowledgements. This work has been supported by the NSF grant DMR-2218130.

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##  Appendix A Chains with antiferromagnetic NN and ferromagnetic NNN
interactions $J_{1}>0$, $J_{2}<0$

###  A.1 Mapping to a gas of domain walls

In the ground state of a chain with antiferromagnetic NN and ferromagnetic NNN
interactions, neighbouring spins are antiparallel. Each excited state can be
considered as a sequence of domain walls, i.e. pairs of neighbouring parallel
spins, separating antiferromagnetic domains. Correlations between spins can be
conveniently found by mapping the spin chain to a gas of domain walls.

In a system of size $r$, the partition function of the domain-wall excitations
is given by

$Z=1+C_{r}^{1}e^{-\beta E_{D}}+\left(C_{r}^{2}-C_{r-1}^{1}\right)e^{-2\beta
E_{D}}+C_{r-1}^{1}e^{-2\beta E_{D}-4\beta J_{2}}+\cdots,$ (1)

where

$\displaystyle E_{D}=2J_{1}-4J_{2}$ (2)

is the energy of a single domain wall. The second term in the right-hand side
(rhs) of Eq. (1) is the contribution of a single domain wall that can be
located anywhere. The third term comes from two domain walls that are not
located next to each other, with $C_{r}^{2}-C_{r-1}^{1}$ being the number of
such configurations. The fourth term comes from two domain walls next to each
other, where $4J_{2}$ is the effective interaction energy of such walls.

In the limit of low temperatures, the domain-wall excitations are very sparse,
and the system may be considered as an almost ideal gas of domain walls. In
this limit, configurations with domain walls located next to each other can be
neglected. Because the value of the product $\sigma_{i}\sigma_{j}$ of two
spins distance $|i-j|=r$ apart is determined by the number of domain walls
between them, the correlator of those spins can be found as

$\langle\sigma_{i}\sigma_{j}\rangle_{0}\approx(-1)^{r}\,\frac{1-C_{r}^{1}e^{-\beta
E_{D}}+C_{r}^{2}e^{-2\beta E_{D}}-\cdots}{1+C_{r}^{1}e^{-\beta
E_{D}}+C_{r}^{2}e^{-2\beta E_{D}}+\cdots}=\left(\frac{e^{-\beta
E_{D}}-1}{1+e^{-\beta E_{D}}}\right)^{r}.$ (3)

According to Eq. (3), the correlation length in the spin chain is given by

$\xi\approx\left[\log\left(\frac{1+e^{-\beta E_{D}}}{1-e^{-\beta
E_{D}}}\right)\right]^{-1}\approx\frac{1}{2}e^{\beta E_{D}}.$ (4)

In this paper, we assume that the length of the chain $N$ significantly
exceeds the correlation length $\xi$.

Equation (3) together with the fluctuation-dissipation relation (3) give the
magnetic susceptibility

$\chi_{0}(T)=\beta\sum_{i,j}\langle\sigma_{i}\sigma_{j}\rangle_{0}\approx\beta
N+2\beta N\,\frac{e^{-\beta E_{D}}-1}{1+e^{-\beta
E_{D}}}\,\left({1-\frac{e^{-\beta E_{D}}-1}{1+e^{-\beta
E_{D}}}}\right)^{-1}=\beta Ne^{-\beta E_{D}}=\beta Ne^{-2\beta J_{1}+4\beta
J_{2}}.$ (5)

###  A.2 Magnetic susceptibility of a vacancy

In this subsection, we compute the magnetic susceptibility of a chain with a
vacancy. The vacancy is assumed to be far away from the chain ends, with the
distance much larger than the correlation length $\xi$. In what follows, we
label the spins as shown in Fig. 1b.

For ferromagnetic NNN interactions considered here, the ground-state spin
configuration in the chain with the vacancy is identical to that in the
absence of the vacancy, as shown in Figs. 1a and b. For excited states, the
presence of the vacancy modifies the domain-wall energies near and at the
vacancy. A domain wall at the location of the vacancy, i.e. an excitation
corresponding to anti-aligned spins at sites $-1$ and $1$, has an energy of
$2|J_{2}|$. A domain-wall excitation between sites $1$ and $2$ or between
sites $-1$ and $-2$ has an energy of $2J_{1}-2J_{2}$. The energy of domain
walls at the other locations is unchanged by the vacancy,
$E_{D}=2J_{1}-4J_{2}$.

Using these excitation energies, we obtain the correlators of spins in the
chain with a vacancy:

$\displaystyle\langle\sigma_{-1}\sigma_{1}\rangle$
$\displaystyle=\frac{1-e^{2\beta J_{2}}}{1+e^{2\beta J_{2}}},$ (6a)
$\displaystyle\langle\sigma_{1}\sigma_{j}\rangle$
$\displaystyle\approx\frac{e^{-2\beta J_{1}+2\beta J_{2}}-1}{1+e^{-2\beta
J_{1}+2\beta J_{2}}}\left(\frac{e^{-\beta E_{D}}-1}{1+e^{-\beta
E_{D}}}\right)^{j-2},$ (6b) $\displaystyle\langle\sigma_{-1}\sigma_{j}\rangle$
$\displaystyle\approx\frac{1-e^{2\beta J_{2}}}{1+e^{2\beta
J_{2}}}\,\frac{e^{-2\beta J_{1}+2\beta J_{2}}-1}{1+e^{-2\beta J_{1}+2\beta
J_{2}}}\left(\frac{e^{-\beta E_{D}}-1}{1+e^{-\beta E_{D}}}\right)^{j-2},$ (6c)
$\displaystyle\langle\sigma_{i}\sigma_{j}\rangle$
$\displaystyle\approx\frac{1-e^{2\beta J_{2}}}{1+e^{2\beta
J_{2}}}\left(\frac{e^{-2\beta J_{1}+2\beta J_{2}}-1}{1+e^{-2\beta J_{1}+2\beta
J_{2}}}\right)^{2}\left(\frac{e^{-\beta E_{D}}-1}{1+e^{-\beta
E_{D}}}\right)^{j-i-4},$ (6d)

for $i\leq-2$, $j\geq 2$. For $i,j\leq-2$ and $i,j\geq 2$, the correlators are
unaltered by the presence of the vacancy and are given by Eq. (3). All the
correlators satisfy the symmetry relation
$\langle\sigma_{i}\sigma_{j}\rangle=\langle\sigma_{-i}\sigma_{-j}\rangle$.
When obtaining the correlators (6a)-(6d), we used the approximation of low
temperature and, similar to the case of a vacancy-free chain, neglected the
probabilities of having two domain walls next to each other.

The magnetic susceptibility of the chain is given by

$\displaystyle\chi(T)$ $\displaystyle=\beta\sum_{i,j\neq
0}\langle\sigma_{i}\sigma_{j}\rangle$ $\displaystyle=\beta\sum_{i\neq
0}\langle\sigma_{i}^{2}\rangle+2\beta\left(\langle\sigma_{-1}\sigma_{1}\rangle+2\sum_{j=2}^{\infty}\langle\sigma_{1}\sigma_{j}\rangle+2\sum_{j=2}^{\infty}\langle\sigma_{-1}\sigma_{j}\rangle+\sum_{i=-2}^{-\infty}\sum_{j=2}^{\infty}\langle\sigma_{i}\sigma_{j}\rangle+2\sum_{i,j=2,\,i\neq
j}^{\infty}\langle\sigma_{i}\sigma_{j}\rangle\right).$ (7)

The magnetic susceptibility of the vacancy is given by the difference in the
susceptibilities of the chain with and without the vacancy:

$\displaystyle\chi(T)-\chi_{0}(T)$
$\displaystyle=2\beta\left(\langle\sigma_{-1}\sigma_{1}\rangle-\langle\sigma_{-1}\sigma_{1}\rangle_{0}\right)+4\beta\sum_{j=2}^{\infty}\left(\langle\sigma_{1}\sigma_{j}\rangle-\langle\sigma_{1}\sigma_{j}\rangle_{0}\right)$
$\displaystyle\quad+4\beta\sum_{j=2}^{\infty}\left(\langle\sigma_{-1}\sigma_{j}\rangle-\langle\sigma_{-1}\sigma_{j}\rangle_{0}\right)+2\beta\sum_{i=-2}^{-\infty}\sum_{j=2}^{\infty}\left(\langle\sigma_{i}\sigma_{j}\rangle-\langle\sigma_{i}\sigma_{j}\rangle_{0}\right),$
(8)

where the correlators $\langle\sigma_{i}\sigma_{j}\rangle_{0}$ in a vacancy-
free system are given by Eq. (3) and, in terms of the labels chosen in this
subsection [cf. Fig. 1] can be rewritten as

$\displaystyle\langle\sigma_{i}\sigma_{j}\rangle_{0}$
$\displaystyle\approx\left(\frac{e^{-\beta E_{D}}-1}{1+e^{-\beta
E_{D}}}\right)^{j-i-1},\quad\text{for}\,\,i\leq-1,\,\,j\geq 1;$ (9a)
$\displaystyle\langle\sigma_{i}\sigma_{j}\rangle_{0}$
$\displaystyle\approx\left(\frac{e^{-\beta E_{D}}-1}{1+e^{-\beta
E_{D}}}\right)^{|j-i|},\quad\text{for}\,\,i,j\leq-1\,\,\text{and}\,\,i,j\geq
1.$ (9b)

Utilising Eqs. (6a)-(6d), (9a)-(9b) and (8), we obtain the magnetic
susceptibility of the vacancy:

$\displaystyle\chi(T)-\chi_{0}(T)$ $\displaystyle\approx
2\beta\left(\frac{1-e^{2\beta J_{2}}}{1+e^{2\beta J_{2}}}-\frac{e^{-\beta
E_{D}}-1}{1+e^{-\beta E_{D}}}\right)$
$\displaystyle\quad+4\beta\left(\frac{1}{1+e^{2\beta J_{2}}}\frac{e^{-2\beta
J_{1}+2\beta J_{2}}-1}{1+e^{-2\beta J_{1}+2\beta J_{2}}}-\frac{e^{-\beta
E_{D}}}{1+e^{-\beta E_{D}}}\frac{e^{-\beta E_{D}}-1}{1+e^{-\beta
E_{D}}}\right)\left(1+e^{-\beta E_{D}}\right)$
$\displaystyle\quad+\frac{\beta}{2}\left[\frac{1-e^{2\beta J_{2}}}{1+e^{2\beta
J_{2}}}\left(\frac{e^{-2\beta J_{1}+2\beta J_{2}}-1}{1+e^{-2\beta J_{1}+2\beta
J_{2}}}\right)^{2}-\left(\frac{e^{-\beta E_{D}}-1}{1+e^{-\beta
E_{D}}}\right)^{3}\right]\left(1+e^{-\beta E_{D}}\right)^{2}.$ (10)

At low temperatures, $T\ll 2J_{1},\,2|J_{2}|$,

$\chi(T)-\chi_{0}(T)=\beta-\beta e^{-2\beta|J_{2}|}+6\beta e^{-2\beta
J_{1}-2\beta|J_{2}|}+o\left(\beta e^{-2\beta J_{1}-2\beta|J_{2}|}\right)$ (11)

The leading order contributions give equation (8) in the main text and
corresponds to the vacancy quasispin $\langle S^{2}\rangle=1$ and the vacancy
size $b(T)\approx e^{2\beta J_{1}-2\beta J_{2}}$.

##  Appendix B Chains with antiferromagnetic NN and NNN interactions,
$J_{1}>0$, $J_{2}>0$

Figure 1: The ground state of chain with a vacancy antiferromagnetic NN and
NNN interactions. Half of the spins are flipped relative to the chain’s
vacancy-free counterpart shown in Fig. 1a.

This section deals with the case of a chain with antiferromagnetic NN and
antiferromagnetic NNN interactions ($J_{1}<0$, $J_{2}>0$). Due to strong
antiferromagnetic NN interactions, the ground state of such a chain is still
antiferromagnetic, with neighbouring spins antiparallel. Low-temperature spin
correlators in the absence of a vacancy are given by

$\langle\sigma_{i}\sigma_{j}\rangle_{0}\approx\left(\frac{e^{-\beta
E_{D}}-1}{1+e^{-\beta E_{D}}}\right)^{|j-i|},$ (12)

and the energy of a domain wall given by

$E_{D}=2J_{1}-4J_{2}.$ (13)

The magnetic susceptibility of a vacancy-free chain is given by

$\chi_{0}(T)=\beta\sum_{i,j}\langle\sigma_{i}\sigma_{j}\rangle_{0}\approx\beta
Ne^{-2\beta J_{1}+4\beta J_{2}}.$ (14)

Similar to the case considered in Appendix A, to find the susceptibility of a
vacancy, we consider a chain with a vacancy located sufficiently far away from
the ends. The spins are labeled as shown in Fig. 1.

The ground state has half of the spins flipped on one side of the vacancy
relative to the other side, as shown in Figs. 1a and 1. A domain wall
excitation at the location of the vacancy, i.e. an excitation corresponding to
parallel spins at sites $-1$ and $1$, has an energy of $2J_{2}$. A domain wall
excitation between sites $1$ and $2$ or between sites $-1$ and $-2$ has an
energy of $2J_{1}-2J_{2}$. The energy of domain walls at other locations is
unaltered by the vacancy, $E_{D}=2J_{1}-4J_{2}$.

Spins correlators in the chain with a vacancy are given by

$\displaystyle\langle\sigma_{-1}\sigma_{1}\rangle$
$\displaystyle=\frac{e^{-2\beta J_{2}}-1}{1+e^{-2\beta J_{2}}},$ (15a)
$\displaystyle\langle\sigma_{1}\sigma_{j}\rangle$
$\displaystyle\approx\frac{e^{-2\beta J_{1}+2\beta J_{2}}-1}{1+e^{-2\beta
J_{1}+2\beta J_{2}}}\left(\frac{e^{-\beta E_{D}}-1}{1+e^{-\beta
E_{D}}}\right)^{j-2},$ (15b)
$\displaystyle\langle\sigma_{-1}\sigma_{j}\rangle$
$\displaystyle\approx\frac{e^{-2\beta J_{2}}-1}{1+e^{-2\beta
J_{2}}}\,\frac{e^{-2\beta J_{1}+2\beta J_{2}}-1}{1+e^{-2\beta J_{1}+2\beta
J_{2}}}\left(\frac{e^{-\beta E_{D}}-1}{1+e^{-\beta E_{D}}}\right)^{j-2},$
(15c) $\displaystyle\langle\sigma_{i}\sigma_{j}\rangle$
$\displaystyle\approx\frac{e^{-2\beta J_{2}}-1}{1+e^{-2\beta
J_{2}}}\left(\frac{e^{-2\beta J_{1}+2\beta J_{2}}-1}{1+e^{-2\beta J_{1}+2\beta
J_{2}}}\right)^{2}\left(\frac{e^{-\beta E_{D}}-1}{1+e^{-\beta
E_{D}}}\right)^{j-i-4},$ (15d)

for $i\leq-2$, $j\geq 2$. For $i,j\leq-2$ and $i,j\geq 2$,
$\langle\sigma_{i}\sigma_{j}\rangle$, the correlators are unchanged by the
presence of the vacancy and are given by (12). All the correlators satisfy
symmetry relation
$\langle\sigma_{i}\sigma_{j}\rangle=\langle\sigma_{-i}\sigma_{-j}\rangle$.
When obtaining the correlators (15a)-(15d), we have again used the
approximation of low temperature and neglected the probabilities of having two
domain walls next to each other. These results are the same as in the case of
ferromagnetic NNN interactions [cf. Eqs. (6a)-(6d)].

The magnetic susceptibility of the vacancy is given by

$\displaystyle\chi(T)-\chi_{0}(T)$
$\displaystyle=2\beta\left(\langle\sigma_{-1}\sigma_{1}\rangle-\langle\sigma_{-1}\sigma_{1}\rangle_{0}\right)+4\beta\sum_{j=2}^{\infty}\left(\langle\sigma_{1}\sigma_{j}\rangle-\langle\sigma_{1}\sigma_{j}\rangle_{0}\right)$
$\displaystyle\quad+4\beta\sum_{j=2}^{\infty}\left(\langle\sigma_{-1}\sigma_{j}\rangle-\langle\sigma_{-1}\sigma_{j}\rangle_{0}\right)+2\beta\sum_{i=-2}^{-\infty}\sum_{j=2}^{\infty}\left(\langle\sigma_{i}\sigma_{j}\rangle-\langle\sigma_{i}\sigma_{j}\rangle_{0}\right),$
$\displaystyle\approx 2\beta\left(\frac{e^{-2\beta J_{2}}-1}{1+e^{-2\beta
J_{2}}}-\frac{e^{-\beta E_{D}}-1}{1+e^{-\beta E_{D}}}\right)$
$\displaystyle\quad+4\beta\left(\frac{e^{-2\beta J_{2}}}{1+e^{-2\beta
J_{2}}}\,\frac{e^{-2\beta J_{1}+2\beta J_{2}}-1}{1+e^{-2\beta J_{1}+2\beta
J_{2}}}-\frac{e^{-\beta E_{D}}}{1+e^{-\beta E_{D}}}\frac{e^{-\beta
E_{D}}-1}{1+e^{-\beta E_{D}}}\right)\left(1+e^{-\beta E_{D}}\right)$
$\displaystyle\quad+\frac{\beta}{2}\left[\frac{e^{-2\beta
J_{2}}-1}{1+e^{-2\beta J_{2}}}\left(\frac{e^{-2\beta J_{1}+2\beta
J_{2}}-1}{1+e^{-2\beta J_{1}+2\beta J_{2}}}\right)^{2}-\left(\frac{e^{-\beta
E_{D}}-1}{1+e^{-\beta E_{D}}}\right)^{3}\right]\left(1+e^{-\beta
E_{D}}\right)^{2}.$ (16)

At low temperatures, $T\ll 2J_{2},\,2J_{1}-4J_{2}$,

$\chi(T)-\chi_{0}(T)=\beta e^{-2\beta J_{2}}+o\left(\beta e^{-2\beta
J_{2}}\right),$ (17)

The exponential suppression of the $\chi(T)-\chi_{0}(T)$ corresponds to a
vanishing pseudospin of the vacancy, $\langle S^{2}\rangle=0$. The leading
term corresponds to an increase in the chain length by $-b(T)\approx e^{2\beta
J_{1}-6\beta J_{2}}$.

##  Appendix C Chains with ferromagnetic NN and antiferromagnetic NNN
interactions, $J_{1}<0$, $J_{2}>0$

In this section, we provide the details of the calculation of the magnetic
susceptibility of a chain with ferromagnetic NN interactions and
antiferromagnetic NNN interactions ($J_{1}<0$, $J_{2}>0$), following similar
analysis in in Appendix. A for a chain with $J_{1}>0$ and $J_{2}<0$.

In the ground state, all spins are parallel, owing to large NN interactions,
$|J_{1}|>2J_{2}$. In excited states, the system splits into ferromagnetic
domains, with domain walls corresponding to pairs of antiparallel neighbouring
spins. Low-temperature correlators of spins can be obtained similarly to those
in Appendix. A and are given by

$\langle\sigma_{i}\sigma_{j}\rangle_{0}\approx\frac{1-C_{r}^{1}e^{-\beta
E_{D}}+C_{r}^{2}e^{-2\beta E_{D}}-\cdots}{1+C_{r}^{1}e^{-\beta
E_{D}}+C_{r}^{2}e^{-2\beta E_{D}}+\cdots}=\left(\frac{1-e^{-\beta
E_{D}}}{1+e^{-\beta E_{D}}}\right)^{|i-j|}.$ (18)

where

$E_{D}=-2J_{1}-4J_{2}$ (19)

is the energy of a domain wall.

Equation (18) together with the fluctuation-dissipation relation (3) gives the
magnetic susceptibility

$\chi_{0}(T)=\beta\sum_{i,j}\langle\sigma_{i}\sigma_{j}\rangle_{0}\approx\beta
N+2\beta N\,\frac{1-e^{-\beta E_{D}}}{1+e^{-\beta
E_{D}}}\,\left({1-\frac{1-e^{-\beta E_{D}}}{1+e^{-\beta
E_{D}}}}\right)^{-1}=\beta Ne^{\beta E_{D}}=\beta Ne^{-2\beta J_{1}-4\beta
J_{2}}.$ (20)

The magnetic susceptibility for a ferromagnetic chain is exponentially large.

Figure 2: The ground state of chain with strong ferromagnetic NN and
antiferromagnetic NNN interactions with a vacancy.

For a chain with antiferromagnetic NNN interactions considered here, the
ground-state spin configuration in a chain with a vacancy has half of the
spins on one side of the vacancy antiparallel to the spins on the other half,
as shown in Figs. 2 and 3. For excited states, the presence of the vacancy
modifies the domain-wall energies near and at the vacancy. A domain wall
excitation at the location of the vacancy, i.e. an excitation corresponding to
aligned spins at sites $-1$ and $1$, has an energy of $2J_{2}$. A domain-wall
excitation between sites $1$ and $2$ or between sites $-1$ and $-2$ has an
energy of $-2J_{1}-2J_{2}$. The energy of domain walls at other locations is
unchanged by the vacancy, $E_{D}=-2J_{1}-4J_{2}$.

These excitation energies result in the following spin correlators:

$\displaystyle\langle\sigma_{-1}\sigma_{1}\rangle$
$\displaystyle=\frac{e^{-2\beta J_{2}}-1}{1+e^{-2\beta J_{2}}},$ (21a)
$\displaystyle\langle\sigma_{1}\sigma_{j}\rangle$
$\displaystyle\approx\frac{1-e^{2\beta J_{1}+2\beta J_{2}}}{1+e^{2\beta
J_{1}+2\beta J_{2}}}\left(\frac{1-e^{-\beta E_{D}}}{1+e^{-\beta
E_{D}}}\right)^{j-2},$ (21b)
$\displaystyle\langle\sigma_{-1}\sigma_{j}\rangle$
$\displaystyle\approx\frac{e^{-2\beta J_{2}}-1}{1+e^{-2\beta
J_{2}}}\,\frac{1-e^{2\beta J_{1}+2\beta J_{2}}}{1+e^{2\beta J_{1}+2\beta
J_{2}}}\left(\frac{1-e^{-\beta E_{D}}}{1+e^{-\beta E_{D}}}\right)^{j-2},$
(21c) $\displaystyle\langle\sigma_{i}\sigma_{j}\rangle$
$\displaystyle\approx\,\frac{e^{-2\beta J_{2}}-1}{1+e^{-2\beta
J_{2}}}\left(\frac{1-e^{2\beta J_{1}+2\beta J_{2}}}{1+e^{2\beta J_{1}+2\beta
J_{2}}}\right)^{2}\left(\frac{1-e^{-\beta E_{D}}}{1+e^{-\beta
E_{D}}}\right)^{j-i-4},$ (21d)

for $i\leq-2$, $j\geq 2$. For $i,j\leq-2$ and $i,j\geq 2$,
$\langle\sigma_{i}\sigma_{j}\rangle$, the correlators are unchanged by the
presence of the vacancy and are given by (18). All the correlators satisfy
symmetry relation
$\langle\sigma_{i}\sigma_{j}\rangle=\langle\sigma_{-i}\sigma_{-j}\rangle$.

The magnetic susceptibility of the vacancy is given by

$\displaystyle\chi(T)-\chi_{0}(T)$
$\displaystyle=2\beta\left(\langle\sigma_{-1}\sigma_{1}\rangle-\langle\sigma_{-1}\sigma_{1}\rangle_{0}\right)+4\beta\sum_{j=2}^{\infty}\left(\langle\sigma_{1}\sigma_{j}\rangle-\langle\sigma_{1}\sigma_{j}\rangle_{0}\right)$
$\displaystyle\quad+4\beta\sum_{j=2}^{\infty}\left(\langle\sigma_{-1}\sigma_{j}\rangle-\langle\sigma_{-1}\sigma_{j}\rangle_{0}\right)+2\beta\sum_{i=-2}^{-\infty}\sum_{j=2}^{\infty}\left(\langle\sigma_{i}\sigma_{j}\rangle-\langle\sigma_{i}\sigma_{j}\rangle_{0}\right).$
(22)

where the correlators $\langle\sigma_{i}\sigma_{j}\rangle_{0}$ in a vacancy-
free system are given by Eq. (18) and, in terms of the labels chosen in this
subsection [cf. Fig. 3] can be rewritten as

Figure 3: Ising chain with ferromagnetic NN interactions and antiferromagnetic
NNN interactions. The chain has no vacancy and has the same number of spins as
the chain with a vacancy. We compare the magnetic susceptibility of this chain
to the chain with a vacancy Fig. 2.

$\displaystyle\langle\sigma_{i}\sigma_{j}\rangle_{0}$
$\displaystyle\approx\left(\frac{1-e^{-\beta E_{D}}}{1+e^{-\beta
E_{D}}}\right)^{j-i-1},\quad\text{for}\,\,i\leq-1,\,\,j\geq 1;$ (23a)
$\displaystyle\langle\sigma_{i}\sigma_{j}\rangle_{0}$
$\displaystyle\approx\left(\frac{1-e^{-\beta E_{D}}}{1+e^{-\beta
E_{D}}}\right)^{|j-i|},\quad\text{for}\,\,i,j\leq-1\,\,\text{and}\,\,i,j\geq
1.$ (23b)

Utilising Eqs. (21a)-(21d), (23a)-(23b) and (22), we obtain the magnetic
susceptibility of the vacancy:

$\displaystyle\chi(T)-\chi_{0}(T)$ $\displaystyle\approx
2\beta\left(\frac{e^{-2\beta J_{2}}-1}{1+e^{-2\beta J_{2}}}-\frac{1-e^{-\beta
E_{D}}}{1+e^{-\beta E_{D}}}\right)$
$\displaystyle\quad+4\beta\left[\frac{e^{-2\beta J_{2}}}{1+e^{-2\beta
J_{2}}}\,\frac{1-e^{2\beta J_{1}+2\beta J_{2}}}{1+e^{2\beta J_{1}+2\beta
J_{2}}}-\frac{1}{1+e^{-\beta E_{D}}}\,\frac{1-e^{-\beta E_{D}}}{1+e^{-\beta
E_{D}}}\right]\left(1+e^{\beta E_{D}}\right)$
$\displaystyle\quad+\frac{\beta}{2}\left[\frac{e^{-2\beta
J_{2}}-1}{1+e^{-2\beta J_{2}}}\left(\frac{1-e^{2\beta J_{1}+2\beta
J_{2}}}{1+e^{2\beta J_{1}+2\beta J_{2}}}\right)^{2}-\left(\frac{1-e^{-\beta
E_{D}}}{1+e^{-\beta E_{D}}}\right)^{3}\right]\left(1+e^{\beta
E_{D}}\right)^{2}$ (24)

At low temperatures, $T\ll 2J_{2},\,2|J_{1}|-4J_{2}$

$\chi(T)-\chi_{0}(T)=-\beta e^{-4\beta J_{1}-8\beta J_{2}}+\beta e^{-4\beta
J_{1}-10\beta J_{2}}+o\left(\beta e^{-4\beta J_{1}-10\beta J_{2}}\right)$ (25)

The presence of a spin vacancy has an exponentially large effect on magnetic
susceptibility. The leading order effect is equivalent to reducing the length
of the chain by twice the correlation length $b(T)\approx 2\xi\approx
e^{-2\beta J_{1}-4\beta J_{2}}$.

##  Appendix D Chains with ferromagnetic NN and NNN interactions, $J_{1}<0$,
$J_{2}<0$

In this section, we consider the case of an Ising chain with ferromagnetic NN
and NNN interactions, $J_{1}<0$, $J_{2}<0$.

In the ground state, all spins are parallel to each other, while excited
states correspond to sequences of ferromagnetic domain walls. Low-temperature
spin correlators are given by

$\langle\sigma_{i}\sigma_{j}\rangle_{0}\approx\left(\frac{1-e^{-\beta
E_{D}}}{1+e^{-\beta E_{D}}}\right)^{|j-i|}.$ (26)

where

$E_{D}=-2J_{1}-4J_{2}$ (27)

is the energy of the domain wall.

The magnetic susceptibility of a vacancy-free chain is given by

$\chi_{0}(T)=\beta\sum_{i,j}\langle\sigma_{i}\sigma_{j}\rangle_{0}\approx\beta
Ne^{-2\beta J_{1}-4\beta J_{2}}$ (28)

and is exponentially large at low temperatures.

In the presence of a vacancy, all spins in the chain are still parallel. A
domain wall at the location of the vacancy, i.e. an excitation corresponding
to antiparallel spins at sites $-1$ and $1$, has an energy of $2|J_{2}|$. A
domain-wall excitation between sites $1$ and $2$ or between sites $-1$ and
$-2$ has an energy of $-2J_{1}-2J_{2}$. The energy of domain walls at the
other locations is unchanged by the vacancy, $E_{D}=-2J_{1}-4J_{2}$. Low-
temperature correlators of the spins in the chain are given by

$\displaystyle\langle\sigma_{-1}\sigma_{1}\rangle$
$\displaystyle=\frac{1-e^{2\beta J_{2}}}{1+e^{2\beta J_{2}}},$ (29a)
$\displaystyle\langle\sigma_{1}\sigma_{j}\rangle$
$\displaystyle\approx\frac{1-e^{2\beta J_{1}+2\beta J_{2}}}{1+e^{2\beta
J_{1}+2\beta J_{2}}}\left(\frac{1-e^{-\beta E_{D}}}{1+e^{-\beta
E_{D}}}\right)^{j-2},$ (29b)
$\displaystyle\langle\sigma_{-1}\sigma_{j}\rangle$
$\displaystyle\approx\frac{1-e^{2\beta J_{2}}}{1+e^{2\beta
J_{2}}}\,\frac{1-e^{2\beta J_{1}+2\beta J_{2}}}{1+e^{2\beta J_{1}+2\beta
J_{2}}}\left(\frac{1-e^{-\beta E_{D}}}{1+e^{-\beta E_{D}}}\right)^{j-2},$
(29c) $\displaystyle\langle\sigma_{i}\sigma_{j}\rangle$
$\displaystyle\approx\frac{1-e^{2\beta J_{2}}}{1+e^{2\beta
J_{2}}}\left(\frac{1-e^{2\beta J_{1}+2\beta J_{2}}}{1+e^{2\beta J_{1}+2\beta
J_{2}}}\right)^{2}\left(\frac{1-e^{-\beta E_{D}}}{1+e^{-\beta
E_{D}}}\right)^{j-i-4},$ (29d)

for $i\leq-2$, $j\geq 2$. For $i,j\leq-2$ and $i,j\geq 2$,
$\langle\sigma_{i}\sigma_{j}\rangle$, the correlators are unchanged by the
presence of the vacancy and are given by (26). All the correlators satisfy
symmetry relation
$\langle\sigma_{i}\sigma_{j}\rangle=\langle\sigma_{-i}\sigma_{-j}\rangle$. The
correlators (29a)-(29d) are identical to those in the case of
antiferromagnetic NNN interactions (21a)-(21d).

The magnetic susceptibility of the vacancy is given by

$\displaystyle\chi(T)-\chi_{0}(T)$
$\displaystyle=2\beta\left(\langle\sigma_{-1}\sigma_{1}\rangle-\langle\sigma_{-1}\sigma_{1}\rangle_{0}\right)+4\beta\sum_{j=2}^{\infty}\left(\langle\sigma_{1}\sigma_{j}\rangle-\langle\sigma_{1}\sigma_{j}\rangle_{0}\right)$
$\displaystyle\quad+4\beta\sum_{j=2}^{\infty}\left(\langle\sigma_{-1}\sigma_{j}\rangle-\langle\sigma_{-1}\sigma_{j}\rangle_{0}\right)+2\beta\sum_{i=-2}^{-\infty}\sum_{j=2}^{\infty}\left(\langle\sigma_{i}\sigma_{j}\rangle-\langle\sigma_{i}\sigma_{j}\rangle_{0}\right)$
$\displaystyle\approx 2\beta\left(\frac{1-e^{2\beta J_{2}}}{1+e^{2\beta
J_{2}}}-\frac{1-e^{-\beta E_{D}}}{1+e^{-\beta E_{D}}}\right)$
$\displaystyle\quad+4\beta\left[\frac{1}{1+e^{2\beta
J_{2}}}\,\frac{1-e^{2\beta J_{1}+2\beta J_{2}}}{1+e^{2\beta J_{1}+2\beta
J_{2}}}-\frac{1}{1+e^{-\beta E_{D}}}\,\frac{1-e^{-\beta E_{D}}}{1+e^{-\beta
E_{D}}}\right]\left(1+e^{\beta E_{D}}\right)$
$\displaystyle\quad+\frac{\beta}{2}\left[\frac{1-e^{2\beta J_{2}}}{1+e^{2\beta
J_{2}}}\left(\frac{1-e^{2\beta J_{1}+2\beta J_{2}}}{1+e^{2\beta J_{1}+2\beta
J_{2}}}\right)^{2}-\left(\frac{1-e^{-\beta E_{D}}}{1+e^{-\beta
E_{D}}}\right)^{3}\right]\left(1+e^{\beta E_{D}}\right)^{2}$ (30)

At low temperatures, $T\ll 2|J_{1,2}|$,

$\chi(T)-\chi_{0}(T)=-\beta e^{-4\beta J_{1}-6\beta J_{2}}-2\beta e^{-2\beta
J_{1}-6\beta J_{2}}+o\left(\beta e^{-2\beta J_{1}-6\beta J_{2}}\right)$ (31)

The susceptibility of vacancy is exponentially large. The leading order effect
is equivalent to reducing the length of the chain by $b(T)\approx e^{-2\beta
J_{1}-2\beta J_{2}}$.

##  Appendix E Magnetic susceptibilities for finite antiferromagnetic chains

Figure 4: The ground state of Ising chains with antiferromagnetic NN
interactions and weaker NNN interactions. (a) A chain with two free ends. (b)
A chain with periodic boundary conditions.

This section provides details of the calculation of the magnetic
susceptibility of a finite chain with antiferromagnetic NN interactions and
weaker NNN interactions ($J_{1}>0$, $|J_{2}|<J_{1}/2$). We assume the length
of the chain $N=2n$ to significantly exceed the correlation length $\xi$. In
what follows, we label the spins as shown in Fig. 4a.

The energies of domain walls are modified near the free ends. The energies of
the domain walls between sites $1$ and $2$ and between sites $-1$ and $-2$ are
given by $2J_{1}-2J_{2}$.

In the limit of a long chain, spin correlators are given by

$\displaystyle\langle\sigma_{i}\sigma_{j}\rangle$
$\displaystyle=0\quad\text{for}\quad-n\leq i\leq-1,\,\,1\leq j\leq n;$ (32a)
$\displaystyle\langle\sigma_{1}\sigma_{j}\rangle$
$\displaystyle\approx\frac{e^{-2\beta J_{1}+2\beta J_{2}}-1}{1+e^{-2\beta
J_{1}+2\beta J_{2}}}\left(\frac{e^{-\beta E_{D}}-1}{1+e^{-\beta
E_{D}}}\right)^{j-2}\quad\text{for}\quad 2\leq j\leq n,$ (32b)

where $E_{D}=2J_{1}-4J_{2}$ is the energy of a domain wall in the bulk of the
chain. For $i,j\leq-1$ and $i,j\geq 1$, $\langle\sigma_{i}\sigma_{j}\rangle$,
the correlators match those in a chain with periodic boundary conditions (see
Fig. 4b) and are given by Eq. (3). All the correlators satisfy symmetry
relation
$\langle\sigma_{i}\sigma_{j}\rangle=\langle\sigma_{-i}\sigma_{-j}\rangle$.

The contribution of the free ends to the susceptibility is given by the
difference of the susceptibilities of a chain with open boundary conditions
and that with periodic boundary conditions:

$\displaystyle\chi_{\text{open}}-\chi_{0}$
$\displaystyle=4\beta\sum_{j=2}^{n}\left(\langle\sigma_{1}\sigma_{j}\rangle-\langle\sigma_{1}\sigma_{j}\rangle_{0}\right)+2\beta\sum_{i=-1}^{-n}\sum_{j=1}^{n}\left(\langle\sigma_{i}\sigma_{j}\rangle-\langle\sigma_{i}\sigma_{j}\rangle_{0}\right)$
$\displaystyle\approx 2\beta\left(\frac{e^{-2\beta J_{1}+2\beta
J_{2}}-1}{1+e^{-2\beta J_{1}+2\beta J_{2}}}-\frac{e^{-\beta
E_{D}}-1}{1+e^{-\beta E_{D}}}\right)\left(1+e^{-\beta
E_{D}}\right)-\frac{\beta}{2}\frac{e^{-\beta E_{D}}-1}{1+e^{-\beta
E_{D}}}\left(1+e^{-\beta E_{D}}\right)^{2}$ (33)

At low temperatures, $T\ll 2J_{1}-2J_{2},\,2J_{1}-4J_{2}$,

$\chi_{\text{open}}-\chi_{0}=\frac{\beta}{2}+o\left(\beta e^{-2\beta
J_{1}+2\beta J_{2}},\beta e^{-\beta E_{D}}\right).$ (34)

According to the definition of quasipins in Eq. (1), the leading order
contribution corresponds to a quasispin $\sqrt{\left\langle
S^{2}\right\rangle}=\frac{1}{2}$ from each of the free ends.