# Quantum Heisenberg model on a sawtooth-chain lattice:
Rotation-invariant Green’s function method

Taras Hutak  Institute for Condensed Matter Physics, National Academy of
Sciences of Ukraine, Svientsitskii Street 1, 79011 L’viv, Ukraine  Taras
Krokhmalskii  Institute for Condensed Matter Physics, National Academy of
Sciences of Ukraine, Svientsitskii Street 1, 79011 L’viv, Ukraine  Oleg
Derzhko  Institute for Condensed Matter Physics, National Academy of Sciences
of Ukraine, Svientsitskii Street 1, 79011 L’viv, Ukraine  Johannes Richter
Institut für Physik, Otto-von-Guericke-Universität Magdeburg, P.O. Box 4120,
39016 Magdeburg, Germany

(April 27, 2022)

###### Abstract

We apply the rotation-invariant Green’s function method (RGM) to study the
spin $S=1/2$ Heisenberg model on a one-dimensional sawtooth lattice, which has
two nonequivalent sites in the unit cell. We check the RGM predictions for
observable quantities by comparison with the exact-diagonalization and finite-
temperature-Lanczos calculations. We discuss the thermodynamic and dynamic
properties of this model in relation to the mineral atacamite Cu2Cl(OH)3
complementing the RGM outcomes by results of other approaches.

sawtooth-chain lattice, quantum Heisenberg spin model, Green’s function method

###### pacs:

75.10.Jm

##  I Introduction

The finite-temperature properties of frustrated quantum spin systems are
interesting for a broad community of solid-state researchers. On the other
hand, the theoretical description of these quantum many-body systems is
challenging. Moreover, the powerful quantum Monte Carlo approach suffers from
the infamous sign problem and exact solutions are notoriously rare.

Among the methods used to study interacting quantum spin systems the double-
time Green’s function method Tyablikov1967 ; Zubarev1971 ; Gasser2001 ;
Rudoy2011  is a well-established approach, see, e.g., Kondo1972 ;
Shimahara1991 ; Barabanov1992 ; Winterfeldt1997 ; Yu2000 ; Siurakshina2001 ;
Bernhard2002 ; Junger2004 ; Froebrich2006 ; Schmalfuss2006 ; Haertel2008 ;
Antsygina2008 ; Miheyenkov2013 ; Vladimirov2015 ; Mueller2017 ; Mikheenkov2018
; Sun2018 ; Mueller2018 ; Mueller2019 ; Wieser2019 ; Savchenkov2021 . There
are some attractive features of this method. It is a universal and
straightforward way to investigate quantum lattice spin systems which yields
on an equal footing both the thermodynamic and dynamic properties at zero and
nonzero temperatures. Since the Green’s functions are obtained after some
decoupling procedure in the equations of motion, the approximation is not
well-controlled and a direct comparison with the results of other approximate
methods is desirable, see, e.g., Refs. Junger2004 ; Schmalfuss2006 ;
Haertel2008 ; Mueller2017 ; Mueller2018 ; Mueller2019 .

A prominent feature of many frustrated quantum spin systems is the absence of
magnetic long-range order. Within the double-time Green’s function method,
this can be accounted by adopting the so-called Kondo-Yamaji decoupling in
equations of motion Kondo1972 . Such a double-time Green’s function approach
is known as the rotation-invariant Green’s function method (RGM): It
preservers the rotational symmetry in the spin space. There are quite a lot
studies of the frustrated quantum spin systems within the RGM approach, see,
e.g., Refs. Shimahara1991 ; Winterfeldt1997 ; Siurakshina2001 ; Schmalfuss2006
; Haertel2008 ; Miheyenkov2013 ; Mikheenkov2018  and references therein. The
studied systems include in particular the quantum spin-$S$ Heisenberg
antiferromagnet on the paradigmatic kagome and pyrochlore lattices Yu2000 ;
Bernhard2002 ; Mueller2017 ; Mueller2018 ; Mueller2019 . However, as far as we
know, the RGM approach has never been applied to the quantum Heisenberg model
on the lattices with nonequivalent sites in the unit cell, such as, for
example, the sawtooth chain in one dimension Schulenburg2002 ; Tonegawa2004 ;
Zhitomirsky2005 ; Richter2008 ; Dmitriev2016 ; Yamaguchi2020 ;
Metavitsiadis2020 ; Richter2020  or the square-kagome lattice in two
dimensions Siddharthan2001 ; Tomczak1996 ; Richter2009 ; Rousochatzakis2013 ;
Nakano2013 ; Hasegawa2018 ; Lugan2019 ; McClarty2020 ; Mizoguchi2021 ;
Astrakhantsev2021 ; Richter2022 . Both quantum spin models, on the sawtooth-
chain lattice and on the square-kagome lattice, are widely used as a play-
ground to study frustrated quantum magnetism. On the other hand, there are
several solid-state realizations of these models, see Refs. Kikuchi2011 ;
Baniodeh2018 ; Heinze2018 ; Inosov2018 ; Gnezdilov2019 ; Heinze2021 ; Nawa2021
; Richter2020  (see also Zhang2015 ) and Refs. Fujihala2020 ; Yakubovich2021 ;
Liu2022 .

In the present paper we focus on the quantum $S=1/2$ Heisenberg model on the
sawtooth-chain lattice bearing in mind a three-fold aim. First, to elaborate
the RGM approach for a lattice with nonequivalent sites in the unit cell. Less
lattice symmetry has some important consequences, e.g., the moment and
frequency matrices become complex, the frequency matrix loses Hermiticity, and
more vertex parameters which improve the Kondo-Yamaji decoupling may be
introduced. We intend to test the RGM approach by comparison with other
techniques in one dimension. Second, such a study paves the way to the square-
kagome $S=1/2$ Heisenberg antiferromagnet which becomes of renewed interest
now because of recent experimental studies Fujihala2020 . The square-kagome
lattice has two sets of nonequivalent sites in the unit cell too. And third,
there is real compound for which the sawtooth-chain $S=1/2$ Heisenberg
antiferromagnet is of relevance, that is, the natural mineral atacamite
Cu2Cl(OH)3 for which some measurements are available Heinze2018 ; Heinze2021 .

The rest of the paper is organized as follows. In Sec. II we introduce the
spin model and illustrate that various thermodynamic and dynamic quantities
can be expressed in terms of the double-time Green’s functions (2.9)
constructed on the spin operators (II). In Sec. III we adopt the Kondo-Yamaji
approximation (III) to obtain the set of equations for the Green’s functions
(3.9) with the solution given in Eq. (III). We derive a set of self-consistent
equations for determining correlators and vertex parameters, Eqs. (III) –
(III). In Sec. IV and Appendix we show how to solve numerically the obtained
equations for correlators and vertex parameters (IV). We compare the RGM
predictions for correlators with exact diagonalizations, Fig. 2. In Sec. V we
report a set of observable thermodynamic and dynamic quantities as they follow
from the RGM approach and compare these results with exact diagonalization
(ED) data (thermodynamics and static correlations) and finite-temperature
Lanczos method (FTLM) data (thermodynamics). Here, we consider the set of
Hamiltonian parameters which is relevant for atacamite Cu2Cl(OH)3. Finally, we
summarize our study and present an outlook for future researches in Sec. VI.

##  II Model and quantities of interest

We consider a quantum $S=1/2$ Heisenberg model on a sawtooth-chain lattice of
$N=2{\cal N}$ sites or of ${\cal N}$ two-site cells, see Fig. 1. The lattice
sites are unambiguously given by two integer numbers $j=1,\ldots,{\cal N}$ and
$\alpha=1,2$: The first one determines the unit cell whereas the second one
specifies the site in the cell. The Hamiltonian of the model reads

$\displaystyle H\\!=\\!\sum_{j=1}^{\cal{N}}\\!\left[J_{1}{\bf
S}_{j,1}\\!\cdot\\!{\bf S}_{j+1,1}\\!+\\!J_{2}\left({\bf
S}_{j,1}\\!\cdot\\!{\bf S}_{j,2}\\!+\\!{\bf S}_{j,2}\\!\cdot\\!{\bf
S}_{j+1,1}\right)\right],\,\,\,$ (2.1)

see Fig. 1, and periodic boundary conditions are imposed for convenience. Here
the spin-1/2 operators satisfy the relations ($\hbar=1$):
$S^{+}S^{z}=-S^{+}/2$, $S^{z}S^{+}=S^{+}/2$, $S^{-}S^{z}=S^{-}/2$,
$S^{z}S^{-}=-S^{-}/2$, $S^{+}S^{-}=1/2+S^{z}$, $S^{-}S^{+}=1/2-S^{z}$ etc., if
they are attached to the same site, but they commute if they are attached to
different sites.

Figure 1: The sawtooth-chain lattice, see Eq. (2.1). We also illustrate by
white lines the correlators $c_{10}$, $c_{01}$, $c_{20}$, $c_{11}$, and
$c_{02}$ (from left to right) to be introduced within the RGM calculations,
see Eqs. (II) and (3.7).

The introduced sawtooth-chain Heisenberg model (2.1) has the continuous SU(2)
symmetry which cannot be spontaneously broken at any temperature $T\geq 0$ and
the methods exploited to calculate its properties should respect this feature.
However, for a special relation between exchange couplings, $J_{1}=J_{2}$, a
discrete symmetry may emerge which can be broken in the ground state only
Monti1991 ; Kubo1993 .

It is worth noting that our model with $J_{1}\simeq 3.294$, $J_{2}=1$ is
appropriate for atacamite Cu2Cl(OH)3 for temperatures above $T_{\rm
N}/J_{2}\simeq 0.087$ (for atacamite $J_{1}=336$ K, $J_{2}=102$ K and a three-
dimensional ordering shows up below a magnetic transition at $T_{\rm N}=8.9$ K
Heinze2021 ).

Let us briefly discuss the physical quantities of interest. The internal
energy (per cell) of the model follows immediately from Eq. (2.1):

$\displaystyle e(T)=\frac{3}{2}J_{1}c_{10}+3J_{2}c_{01},$ $\displaystyle
c_{10}=\langle S_{j,1}^{-}S_{j+1,1}^{+}\rangle,\;\;\;c_{01}=\langle
S^{-}_{j,2}S^{+}_{j+1,1}\rangle$ (2.2)

[we have used the lattice symmetry and the relations $\langle
S^{x}_{A}S^{x}_{B}\rangle=\langle S^{y}_{A}S^{y}_{B}\rangle=\langle
S^{z}_{A}S^{z}_{B}\rangle$, and $2\langle S^{z}_{A}S_{B}^{z}\rangle=(\langle
S^{+}_{A}S_{B}^{-}\rangle+\langle S^{-}_{A}S_{B}^{+}\rangle)/2=\langle
S^{+}_{A}S_{B}^{-}\rangle=\langle S^{-}_{A}S_{B}^{+}\rangle$]. Here the angle
brackets denote the thermodynamic average $\langle(\ldots)\rangle={\rm
Tr}[\exp(-H/T)(\ldots)]/{\rm Tr}\exp(-H/T)$ ($k_{\rm B}=1$) and the
thermodynamic limit $N\to\infty$ is implied. By differentiating Eq. (II) we
get the specific heat $c(T)=\partial e(T)/\partial T$,

$\displaystyle c(T)=\frac{3}{2}J_{1}\frac{\partial c_{10}}{\partial
T}+3J_{2}\frac{\partial c_{01}}{\partial T},$ (2.3)

which in turn yields the entropy $s(T)=\int_{0}^{T}{\rm d}{\sf T}c({\sf
T})/{\sf T}$,

$\displaystyle s(T)=s(\infty)+\frac{e(T)}{T}-\int\limits_{T}^{\infty}{\rm
d}{\sf T}\frac{e({\sf T})}{{\sf T}^{2}}$ (2.4)

with $e(T)$ given in Eq. (II). The corresponding equations for the uniform
susceptibility will be given below.

Let us turn to dynamic quantities. The dynamic spin susceptibility
$\chi^{zz}_{\bf q}(\omega)=\chi^{+-}_{\bf q}(\omega)/2$ describes the magnetic
linear response to an infinitesimally small space- and time-varying magnetic
field with the wave vector ${\bf q}$ and the frequency $\omega$. Assuming that
${\bf q}$ is directed along the chain and setting the distance between the
sites $j,1$ and $j+1,1$ to unity, see Fig. 1, we arrive at

$\displaystyle\chi^{zz}_{q}(\omega)\\!=\\!\frac{\chi^{+-}_{q11}\\!(\omega)\\!+\\!{\rm
e}^{{\rm i}\frac{q}{2}}\\!\chi^{+-}_{q12}\\!(\omega)\\!+\\!{\rm e}^{-{\rm
i}\frac{q}{2}}\\!\chi^{+-}_{q21}\\!(\omega)\\!+\\!\chi^{+-}_{q22}\\!(\omega)}{4}\\!,$
$\displaystyle\chi^{+-}_{q\alpha\beta}(\omega)=\int\limits_{-\infty}^{\infty}{\rm
d}t{\rm e}^{{\rm i}\omega t}\chi^{+-}_{q\alpha\beta}(t),$
$\displaystyle\chi^{+-}_{q\alpha\beta}(t)={\rm
i}\theta(t)\left\langle\left[S_{q\alpha}^{+}(t),S_{q\beta}^{-}\right]\right\rangle,$
$\displaystyle S_{q\alpha}^{\pm}=\frac{1}{\sqrt{{\cal N}}}\sum_{j=1}^{\cal
N}{\rm e}^{\mp{\rm i}qj}S_{j,\alpha}^{\pm},\,\,\,\,\,$ (2.5)

where $\theta(t)$ is the Heaviside step function and $O(t)=\exp({\rm
i}Ht)O\exp(-{\rm i}Ht)$ stands for the operator $O$ in the Heisenberg
representation. It is worth noting the spatial phase shifts in the formula for
$\chi^{zz}_{q}(\omega)$ which arise owing to the position of the site $2$ with
respect to the site $1$ in the unit cell of the sawtooth-chain lattice, see
Fig. 1.

Similarly, the dynamic spin structure factor $S^{zz}_{\bf
q}(\omega)=S^{+-}_{\bf q}(\omega)/2$, which can be measured in scattering
experiments, for the wave vector transfer ${\bf q}$ and the energy transfer
$\omega$, for ${\bf q}$ directed along the chain becomes

$\displaystyle
S^{zz}_{q}(\omega)\\!=\\!\frac{S^{+-}_{q11}\\!(\omega)\\!+\\!{\rm e}^{{\rm
i}\frac{q}{2}}\\!S^{+-}_{q12}\\!(\omega)\\!+\\!{\rm e}^{-{\rm
i}\frac{q}{2}}\\!S^{+-}_{q21}\\!(\omega)\\!+\\!S^{+-}_{q22}\\!(\omega)}{4}\\!,$
$\displaystyle
S^{+-}_{q\alpha\beta}\\!(\omega)=\int\limits_{-\infty}^{\infty}{\rm d}t{\rm
e}^{{\rm i}\omega t}\langle S_{q\alpha}^{+}(t)S_{q\beta}^{-}\rangle\,\,\,\,\,$
(2.6)

with $S_{q\alpha}^{\pm}$ defined in Eq. (II). Moreover, the dynamic structure
factor $S^{zz}_{q}(\omega)$ yields the static structure factor $S_{q}$ via the
formula

$\displaystyle\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}{\rm d}\omega
S_{q}^{zz}(\omega)=S^{zz}_{q}=\frac{1}{3}S_{q}.$ (2.7)

Obviously, Eqs. (2.7), (II), (II) agree with the definition of the static
structure factor

$\displaystyle S_{\bf q}=\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{N}{\rm e}^{-{\rm
i}{\bf q}\cdot({\bf R}_{i}-{\bf R}_{j})}\langle{\bf S}_{i}\cdot{\bf
S}_{j}\rangle,$ (2.8)

if ${\bf q}$ is directed along the chain and the component of ${\bf R}_{i}$
along the chain is either $i$ for the site with $\alpha=1$ or $i+1/2$ for the
site with $\alpha=2$, see Fig. 1. $S_{q}$ is also accessible in scattering
experiments. Moreover, $S_{q=0}$ gives the (initial) uniform susceptibility
$\chi_{0}^{zz}=\beta S_{q=0}/3$.

These formulas illustrate why it is important to find the Green’s functions
Zubarev1971

$\displaystyle G_{q\alpha\beta}(\omega)\equiv
G^{+-}_{q\alpha\beta}(\omega)=-\chi^{+-}_{q\alpha\beta}(\omega)$ (2.9)

constructed with the operators $S_{q\alpha}^{\pm}$, see Eq. (II). On the one
hand, the Green’s functions give straightforwardly the dynamic susceptibility
$\chi_{q}^{zz}(\omega)$ (II). On the other hand, they give the time-dependent
correlation functions after applying the spectral theorem Zubarev1971

$\displaystyle\langle S_{q\beta}^{-}S_{q\alpha}^{+}(t)\rangle=\frac{{\rm
i}}{2\pi}\lim_{\epsilon\to+0}\int\limits_{-\infty}^{\infty}{\rm
d}\omega\frac{{\rm e}^{-{\rm i}\omega t}}{{\rm e}^{\frac{\omega}{T}}-1}$
$\displaystyle\times\left(G_{q\alpha\beta}(\omega+{\rm
i}\epsilon)-G_{q\alpha\beta}(\omega-{\rm i}\epsilon)\right).$ (2.10)

Inserting $\langle S_{q\alpha}^{+}(t)S_{q\beta}^{-}\rangle$ (see Ref.
Zubarev1971 ) into (II) gives

$\displaystyle S_{q\alpha\beta}^{+-}(\omega)={\rm
i}\lim_{\epsilon\to+0}\frac{G_{q\alpha\beta}(\omega+{\rm
i}\epsilon)-G_{q\alpha\beta}(\omega-{\rm i}\epsilon)}{1-{\rm
e}^{-\frac{\omega}{T}}}$ (2.11)

yielding the dynamic structure factor $S^{zz}_{q}(\omega)$ (II) and the static
structure factor $S_{q}$ (2.7). Setting $t=0$ in Eq. (II) we get the equal-
time correlation functions $\langle S_{q\beta}^{-}S_{q\alpha}^{+}\rangle$,
which according to Eq. (II) can be written as

$\displaystyle\langle S_{q\beta}^{-}S_{q\alpha}^{+}\rangle=\sum_{l=0}^{{\cal
N}-1}{\rm e}^{-{\rm i}ql}\langle S_{j,\beta}^{-}S_{j+l,\alpha}^{+}\rangle,$
(2.12)

and therefore

$\displaystyle\langle S_{j,\beta}^{-}S_{j+l,\alpha}^{+}\rangle=\frac{1}{{\cal
N}}\sum_{q}{\rm e}^{{\rm i}ql}\langle S_{q\beta}^{-}S_{q\alpha}^{+}\rangle.$
(2.13)

Here we have to replace $\sum_{q}(\ldots)/{\cal N}\to\int_{-\pi}^{\pi}{\rm
d}q(\ldots)/(2\pi)$ in the thermodynamic limit. In particular, setting
$\alpha=\beta=1$ and $l=1$ we get $c_{10}$ and setting $\alpha=1$, $\beta=2$,
and $l=1$ we get $c_{01}$ which enter Eq. (II) for the internal energy.

These are generally known relations from the double-time Green’s function
tool-box fitted to the model at hand; more details can be found in Refs.
Tyablikov1967 ; Zubarev1971 ; Gasser2001 . Our next task is to find the
introduced Green’s functions $G_{q\alpha\beta}(\omega)$, see Eqs. (2.9) and
(II).

##  III RGM equations

To determine the Green’s functions we write down the equations of motion for
them. Differentiating the definition of $G_{q\alpha\beta}(t)$ gives

$\displaystyle\frac{{\rm{d}}G_{q\alpha\beta}(t)}{{\rm{d}}t}\\!=\\!-2{\rm{i}}\delta(t)\delta_{\alpha\beta}\langle
S^{z}\rangle\\!-\\!{\rm{i}}\theta(t)\\!\left\langle\\!\left[\\!\frac{{\rm{d}}S^{+}_{q\alpha}}{{\rm{d}}t}\\!(t),S^{-}_{q\beta}\\!\right]\\!\right\rangle\\!.$
(3.1)

Since we seek for rotation-invariant Green’s functions (i.e., $\langle
S^{z}\rangle=0$), the first term in Eq. (3.1) drops out. We need to take the
derivative with respect to time $t$ once more arriving at

$\displaystyle\frac{{\rm{d}}^{2}G_{q\alpha\beta}(t)}{{\rm{d}}t^{2}}=-{\rm{i}}\delta(t)\left\langle\\!\left[\frac{{\rm{d}}S^{+}_{q\alpha}}{{\rm{d}}t},S^{-}_{q\beta}\right]\\!\right\rangle$
$\displaystyle-{\rm{i}}\theta(t)\left\langle\\!\left[\frac{{\rm{d}}^{2}S^{+}_{q\alpha}}{{\rm{d}}t^{2}}(t),S^{-}_{q\beta}\right]\\!\right\rangle.$
(3.2)

Let us introduce the moment matrix:

$\displaystyle
M_{q\alpha\beta}={\rm{i}}\left\langle\\!\left[\frac{{\rm{d}}S^{+}_{q\alpha}}{{\rm{d}}t},S^{-}_{q\beta}\right]\\!\right\rangle$
$\displaystyle=\frac{1}{\sqrt{\cal{N}}}\sum_{j=1}^{\cal{N}}{\rm{e}}^{-{\rm{i}}qj}\left\langle\\!\left[\left[S_{j,\alpha}^{+},H\right],S^{-}_{q\beta}\right]\\!\right\rangle.$
(3.3)

By straightforward calculations in the second line of Eq. (III) we immediately
obtain

$\displaystyle M_{q11}=-4J_{1}c_{10}\left(1-\cos q\right)-4J_{2}c_{01},$
$\displaystyle M_{q12}={\sf M}_{q12}\left(1+{\rm e}^{-{\rm
i}q}\right)=\left(M_{q21}\right)^{*},\;\;\;{\sf M}_{q12}=2J_{2}c_{01},$
$\displaystyle M_{q22}=-4J_{2}c_{01}\,\,\,\,\,$ (3.4)

with $c_{10}$ and $c_{01}$ defined in Eq. (II).

The second term in the right-hand side of Eq. (III) can be rewritten as
follows:

$\displaystyle{\rm{i}}\theta(t)\left\langle\\!\left[\frac{{\rm{d}}^{2}S^{+}_{q\alpha}}{{\rm{d}}t^{2}}(t),S^{-}_{q\beta}\right]\\!\right\rangle$
$\displaystyle=-\\!{\rm{i}}\theta(t)\frac{1}{\sqrt{\cal N}}\sum_{j=1}^{\cal
N}{\rm e}^{-{\rm
i}qj}\left\langle\\!\left[\left[\left[S^{+}_{j,\alpha},H\right],H\right](t),S^{-}_{q\beta}\right]\\!\right\rangle$
$\displaystyle\stackrel{{\scriptstyle\text{Kondo-Yamaji
decoupling}}}{{\approx}}\sum_{\gamma=1}^{2}F_{q\alpha\gamma}G_{q\gamma\beta}(t).$
(3.5)

Here $F_{q\alpha\beta}$ are the elements of the frequency matrix. To arrive at
the last line of Eq. (III) we follow the strategy of J. Kondo and K. Yamaji
Kondo1972  and utilize the Kondo-Yamaji decoupling:

$\displaystyle
S_{A}^{-}S_{B}^{+}S_{C}^{+}\to\tilde{\alpha}_{AB}S_{C}^{+}+\tilde{\alpha}_{AC}S_{B}^{+},$
$\displaystyle
S_{A}^{z}S_{B}^{z}S_{C}^{+}\to\frac{1}{2}\tilde{\alpha}_{AB}S_{C}^{+},$ (3.6)

where $A$, $B$, and $C$ denote different lattice sites,
$\tilde{\alpha}_{AB}=\alpha_{AB}c_{AB}$, $c_{AB}\equiv\langle
S^{-}_{A}S^{+}_{B}\rangle=\langle S^{+}_{A}S^{-}_{B}\rangle=2\langle
S^{z}_{A}S^{z}_{B}\rangle$, and $\alpha_{AB}$ are the vertex parameters
introduced to improve the decoupling of higher-order correlators. After all,
$c_{AB}$ and $\alpha_{AB}$ must be determined self-consistently. Besides the
first-neighbor correlations $c_{10}$ and $c_{01}$ (II), we introduce the
second-neighbor correlations

$\displaystyle c_{20}\\!=\\!\langle
S^{-}_{i,1}S^{+}_{i+2,1}\rangle\\!,c_{11}\\!=\\!\langle
S^{-}_{i,2}S^{+}_{i+2,1}\rangle\\!,c_{02}\\!=\\!\langle
S^{-}_{i,2}S^{+}_{i+1,2}\rangle\\!,\,\,\,\,\,$ (3.7)

see Fig. 1. After tedious but straightforward calculations in the second line
of Eq. (III) with accounting Eq. (III) we find the elements of the frequency
matrix $F_{q\alpha\beta}$,

$\displaystyle
F_{q11}\\!=\\!J_{1}^{2}\\!\left(\\!1-2\tilde{\alpha}_{10}\\!+\\!2\tilde{\alpha}_{20}\\!\right)\\!+\\!J_{2}^{2}\\!\left(\\!1\\!+\\!2\tilde{\alpha}_{02}\\!\right)+4J_{1}\\!J_{2}\\!\left(\\!\tilde{\alpha}_{01}\\!+\\!\tilde{\alpha}_{11}\\!\right)$
$\displaystyle\\!+\\!\left[\\!-J_{1}^{2}\\!\left(\\!1\\!+\\!2\tilde{\alpha}_{10}\\!+\\!2\tilde{\alpha}_{20}\\!\right)\\!+\\!2J_{2}^{2}\\!\tilde{\alpha}_{01}\\!-\\!2J_{1}\\!J_{2}\\!\left(\\!3\tilde{\alpha}_{01}\\!+\\!\tilde{\alpha}_{11}\\!\right)\\!\right]\cos\\!q$
$\displaystyle\\!+\\!4J_{1}^{2}\\!\tilde{\alpha}_{10}\cos^{2}\\!q,$
$\displaystyle F_{q12}={\sf F}_{q12}\left(1+{\rm e}^{-{\rm i}q}\right),$
$\displaystyle{\sf
F}_{\\!q12}\\!\\!=\\!\\!-J_{2}^{2}\\!\left(\\!\\!\frac{1}{2}\\!+\\!\tilde{\alpha}_{10}\\!+\\!\tilde{\alpha}_{01}\\!\\!\right)\\!-\\!2J_{1}\\!J_{2}\tilde{\alpha}_{10}\\!+\\!2J_{1}\\!J_{2}\tilde{\alpha}_{10}\cos\\!q,$
$\displaystyle F_{q21}={\sf F}_{q21}\left(1+{\rm e}^{{\rm i}q}\right),$
$\displaystyle{\sf
F}_{\\!q21}\\!\\!=\\!\\!-\\!J_{2}^{2}\\!\left(\\!\\!\frac{1}{2}\\!+\\!\tilde{\alpha}_{01}\\!+\\!\tilde{\alpha}_{02}\\!\\!\right)\\!\\!-\\!\\!J_{1}\\!J_{2}\\!\left(\\!\tilde{\alpha}_{01}\\!\\!+\\!\\!\tilde{\alpha}_{11}\\!\right)\\!+\\!2J_{1}\\!J_{2}\tilde{\alpha}_{01}\cos\\!q,$
$\displaystyle
F_{q22}=J_{2}^{2}\left(1+2\tilde{\alpha}_{10}+2\tilde{\alpha}_{01}\cos
q\right).\,\,\,\,\,$ (3.8)

As a result, Eq. (III) becomes a closed set of equations for the Green’s
functions and after utilizing the representation for the $\delta$-function
$2\pi\delta(t)=\int_{-\infty}^{\infty}{\rm d}\omega{\rm e}^{-{\rm i}\omega
t}$, we arrive at the $2\times 2$ set of equations:

$\displaystyle\sum_{\gamma=1}^{2}\left(\omega^{2}\delta_{\alpha\gamma}-F_{q\alpha\gamma}\right)G_{q\gamma\beta}(\omega)=M_{q\alpha\beta}.$
(3.9)

Importantly, in all previous RGM studies the moment matrix and the frequency
matrix were real and symmetric Mueller2017 ; Mueller2018 . Both matrices (III)
and (III) are complex. Moreover, while the moment matrix is Hermitian, the
frequency matrix is not. This is a direct result of the two nonequivalent
sites in the unit cell.

Inverting the matrix $\omega^{2}{\bf I}-{\bf F}_{q}$ one gets ${\bf
G}_{q}(\omega)$. The final result reads:

$\displaystyle G_{q\alpha\beta}(\omega)=\frac{{\sf
A}_{q\alpha\beta}(f_{+})}{\omega^{2}-f_{+}}-\frac{{\sf
A}_{q\alpha\beta}(f_{-})}{\omega^{2}-f_{-}},$ $\displaystyle{\sf
A}_{q11}(\omega^{2})=\frac{\left(\omega^{2}-F_{q22}\right)M_{q11}+F_{q12}M_{q21}}{f_{+}-f_{-}},$
$\displaystyle{\sf
A}_{q12}(\omega^{2})=\frac{\left(\omega^{2}-F_{q22}\right)M_{q12}+F_{q12}M_{q22}}{f_{+}-f_{-}},$
$\displaystyle{\sf
A}_{q21}(\omega^{2})=\frac{F_{q21}M_{q11}+\left(\omega^{2}-F_{q11}\right)M_{q21}}{f_{+}-f_{-}},$
$\displaystyle{\sf
A}_{q22}(\omega^{2})=\frac{F_{q21}M_{q12}+\left(\omega^{2}-F_{q11}\right)M_{q22}}{f_{+}-f_{-}},$
$\displaystyle
f_{\pm}=\frac{F_{q11}\\!+\\!F_{q22}}{2}\pm\sqrt{\left(\\!\frac{F_{q11}\\!-\\!F_{q22}}{2}\\!\right)^{\\!2}\\!+\\!F_{q12}F_{q21}}.$
(3.10)

Here $f_{\pm}$ stand for the eigenvalues of the frequency matrix ${\bf
F}_{q}$. We search for non-negative eigenvalues $f_{\pm}\geq 0$ for all
$-\pi\leq q<\pi$. The solution (III) of the set of equations (3.9) for the
Green’s functions is not the final result yet, since
$G_{q\alpha\beta}(\omega)$ (III) contain five unknown correlation functions
and vertex parameters which must be determined self-consistently.

To proceed further, we calculate the correlation function $\langle
S_{q\beta}^{-}S_{q\alpha}^{+}\rangle$ (II) doing standard manipulations.
Namely, we rewrite the Green’s function (III) in the form

$\displaystyle G_{q\alpha\beta}(\omega)\\!=\\!\frac{{\sf
A}_{q\alpha\beta}(f_{+})}{2\sqrt{f_{+}}}\\!\left(\\!\frac{1}{\omega\\!-\\!\sqrt{f_{+}}}\\!-\\!\frac{1}{\omega\\!+\\!\sqrt{f_{+}}}\\!\right)$
$\displaystyle-\frac{{\sf
A}_{q\alpha\beta}(f_{-})}{2\sqrt{f_{-}}}\\!\left(\\!\frac{1}{\omega\\!-\\!\sqrt{f_{-}}}\\!-\\!\frac{1}{\omega\\!+\\!\sqrt{f_{-}}}\\!\right),$
(3.11)

use the Sokhotski-Plemelj theorem to calculate

$\displaystyle\lim_{\epsilon\to+0}\left(G_{q\alpha\beta}(\omega+{\rm
i}\epsilon)-G_{q\alpha\beta}(\omega-{\rm i}\epsilon)\right)$
$\displaystyle\\!=\\!-\frac{{\rm i}\pi{\sf
A}_{q\alpha\beta}(f_{+})}{\sqrt{f_{+}}}\left(\\!\delta(\omega\\!-\\!\sqrt{f_{+}})\\!-\\!\delta(\omega\\!+\\!\sqrt{f_{+}})\\!\right)$
$\displaystyle+\frac{{\rm i}\pi{\sf
A}_{q\alpha\beta}(f_{-})}{\sqrt{f_{-}}}\left(\\!\delta(\omega\\!-\\!\sqrt{f_{-}})\\!-\\!\delta(\omega\\!+\\!\sqrt{f_{-}})\\!\right),$
(3.12)

and finally obtain

$\displaystyle\langle S_{q\beta}^{-}S_{q\alpha}^{+}\rangle\\!=\\!\frac{{\sf
A}_{q\alpha\beta}(f_{+})}{2\sqrt{f_{+}}}\\!\coth\\!\frac{\sqrt{f_{+}}}{2T}\\!-\\!\frac{{\sf
A}_{q\alpha\beta}(f_{-})}{2\sqrt{f_{-}}}\\!\coth\\!\frac{\sqrt{f_{-}}}{2T}.$

Thus, we are in position to calculate correlation functions, see Eq. (2.13),
and in particular the ones that were introduced while deriving the closed-set
of equations of motion (3.9). That is,

$\displaystyle c_{10}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}{\rm d}q{\rm
e}^{{\rm i}q}\langle S_{q1}^{-}S^{+}_{q1}\rangle,$ $\displaystyle
c_{01}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}{\rm d}q{\rm e}^{{\rm i}q}\langle
S_{q2}^{-}S^{+}_{q1}\rangle,$ $\displaystyle
c_{20}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}{\rm d}q{\rm e}^{2{\rm
i}q}\langle S_{q1}^{-}S^{+}_{q1}\rangle,$ $\displaystyle
c_{11}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}{\rm d}q{\rm e}^{2{\rm
i}q}\langle S_{q2}^{-}S^{+}_{q1}\rangle,$ $\displaystyle
c_{02}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}{\rm d}q{\rm e}^{{\rm i}q}\langle
S_{q2}^{-}S^{+}_{q2}\rangle.$ (3.14)

Moreover, we have two more equations $\langle S_{j,1}^{z}\rangle=0$ and
$\langle S_{j,2}^{z}\rangle=0$ (sum rules), which read:

$\displaystyle\langle
S^{-}_{j,1}\\!S^{+}_{j,1}\rangle=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}{\rm
d}q\langle S_{q1}^{-}S^{+}_{q1}\rangle=\frac{1}{2},$ $\displaystyle\langle
S^{-}_{j,2}\\!S^{+}_{j,2}\rangle=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}{\rm
d}q\langle S_{q2}^{-}S^{+}_{q2}\rangle=\frac{1}{2}.$ (3.15)

Note that we have two sum rules (III) since there are two nonequivalent sites
in the unit cell, whereas in previous RGM studies with one site in the unit
cell only one sum rule could be exploited.

We adopt a generalization of the minimal RGM scheme setting

$\displaystyle\tilde{\alpha}_{10}=\alpha_{1}c_{10},\;\;\;\tilde{\alpha}_{20}=\alpha_{1}c_{20},$
$\displaystyle\tilde{\alpha}_{01}=\alpha_{2}c_{01},\;\;\;\tilde{\alpha}_{11}=\alpha_{2}c_{11},\;\;\;\tilde{\alpha}_{02}=\alpha_{2}c_{02},$
(3.16)

i.e., the correlators which contain only the site $1$ are improved by the
vertex parameter $\alpha_{1}$, whereas the correlators which contain the site
$2$ are improved by the vertex parameter $\alpha_{2}$, see Fig. 1. Two vertex
parameters go hand in hand with two sum rules for two nonequivalent sites in
the unit cell (III). We note in passing that the described scheme reproduces
the results of the Kondo-Yamaji paper Kondo1972  in the limit
$J_{1}/J_{2}\to\infty$, i.e., for the $S=1/2$ Heisenberg chain.

To summarize this section, we face a set of seven coupled equations (III),
(III), and (III) for the determination of the correlators $c_{10}$, $c_{01}$,
$c_{20}$, $c_{11}$, $c_{02}$ and two vertex parameters $\alpha_{1}$,
$\alpha_{2}$. In the next section (and in the Appendix) we discuss how to
solve these self-consistent nonlinear equations numerically. It is worth
reminding that in the previous RGM studies of frustrated quantum spin systems
there was a smaller number of unknown quantities and equations to determine
them. For instance, for the $S=1/2$ pyrochlore-lattice Heisenberg
antiferromagnet one deals with three correlation functions and one vertex
parameter to be found from a set of four equations Mueller2019 . This means
that now we are facing a more challenging computational task.

##  IV Solution of the self-consistent equations

Let us multiply Eqs. (III), (III) by $\alpha_{2}$ and introduce
$\rho=\alpha_{2}/\alpha_{1}$ to rewrite the self-consistent equations in the
following form:

$\displaystyle\rho\tilde{\alpha}_{10}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}{\rm
d}q{\rm e}^{{\rm i}q}\langle
S_{q1}^{-}S^{+}_{q1}\rangle_{\tilde{\alpha},\rho},$
$\displaystyle\tilde{\alpha}_{01}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}{\rm
d}q{\rm e}^{{\rm i}q}\langle
S_{q2}^{-}S^{+}_{q1}\rangle_{\tilde{\alpha},\rho},$
$\displaystyle\rho\tilde{\alpha}_{20}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}{\rm
d}q{\rm e}^{2{\rm i}q}\langle
S_{q1}^{-}S^{+}_{q1}\rangle_{\tilde{\alpha},\rho},$
$\displaystyle\tilde{\alpha}_{11}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}{\rm
d}q{\rm e}^{2{\rm i}q}\langle
S_{q2}^{-}S^{+}_{q1}\rangle_{\tilde{\alpha},\rho},$
$\displaystyle\tilde{\alpha}_{02}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}{\rm
d}q{\rm e}^{{\rm i}q}\langle
S_{q2}^{-}S^{+}_{q2}\rangle_{\tilde{\alpha},\rho},$
$\displaystyle\frac{\alpha_{2}}{2}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}{\rm
d}q\langle S_{q1}^{-}S^{+}_{q1}\rangle_{\tilde{\alpha},\rho},$
$\displaystyle\frac{\alpha_{2}}{2}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}{\rm
d}q\langle S_{q2}^{-}S^{+}_{q2}\rangle_{\tilde{\alpha},\rho}.$ (4.1)

Here $\langle S_{q\beta}^{-}S^{+}_{q\alpha}\rangle_{\tilde{\alpha},\rho}$ are
given in Eqs. (III), (III), (III), and (III), however, with
$\tilde{M}_{q\alpha\beta}\equiv\alpha_{2}M_{q\alpha\beta}$ instead of
$M_{q\alpha\beta}$. In turn, $\tilde{M}_{q\alpha\beta}$ is given by Eq. (III)
in which $c_{10}$ and $c_{01}$ are replaced by $\rho\tilde{\alpha}_{10}$ and
$\tilde{\alpha}_{01}$, respectively. We may set for convenience $J_{2}=1$.

After solving the first five equations in Eq. (IV) and the sixth equation

$\displaystyle\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}{\rm d}q\langle
S_{q1}^{-}S^{+}_{q1}\rangle_{\tilde{\alpha},\rho}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}{\rm
d}q\langle S_{q2}^{-}S^{+}_{q2}\rangle_{\tilde{\alpha},\rho}$ (4.2)

we get $\tilde{\alpha}_{10}$, $\tilde{\alpha}_{01}$, $\tilde{\alpha}_{20}$,
$\tilde{\alpha}_{11}$, $\tilde{\alpha}_{02}$, and $\rho$. Then we use one of
the last two equations in Eq. (IV) to calculate $\alpha_{2}$ and determine
$\alpha_{1}=\alpha_{2}/\rho$. After determining the five correlation functions
$c_{10}$, $c_{01}$, $c_{20}$, $c_{11}$, and $c_{02}$ and the two vertex
parameters $\alpha_{1}$ and $\alpha_{2}$ we will have the Green’s functions
$G_{q\alpha\beta}$ (2.9), (II) within the Kondo-Yamaji approximation, see Eqs.
(III), (III), and (III).

The crucial point in the numerical solution of the set of self-consistent
equations is to figure out among its solutions the relevant one. Our strategy
is to solve Eq. (IV) starting from the high-temperature limit $T\to\infty$ and
gradually decrease the temperature $T$ to zero, see, e.g., Refs. Mueller2017 ;
Mueller2019 . In the high-temperature limit, the first non-vanishing terms in
the high-temperature series can be easily obtained (see, e.g., Ref.
Lohmann2014 ):

$\displaystyle
c_{10}\\!\\!=\\!-\\!\frac{J_{1}}{8T}\\!\\!+\\!\frac{-\\!J_{1}^{2}\\!+\\!J_{2}^{2}}{32T^{2}}\\!+\\!\ldots,c_{01}\\!\\!=\\!-\\!\frac{J_{2}}{8T}\\!\\!+\\!\frac{J_{1}J_{2}\\!-\\!J_{2}^{2}}{32T^{2}}\\!+\\!\ldots,$
$\displaystyle
c_{20}\\!\\!=\\!\frac{J_{1}^{2}}{32T^{2}}\\!+\\!\ldots,c_{11}\\!\\!=\\!\frac{J_{1}J_{2}}{32T^{2}}\\!+\\!\ldots,c_{02}\\!\\!=\\!\frac{J_{2}^{2}}{32T^{2}}\\!+\\!\ldots.\,\,\,\,\,$
(4.3)

In this limit, i.e., up to order $\beta^{2}$, we have $c_{20}=2c_{10}^{2}$,
$c_{11}=2c_{10}c_{01}$, $c_{02}=2c_{01}^{2}$. We may compliment these first
non-vanishing terms in the high-temperature series for correlators by the
obvious high-temperature series for the vertex parameters,
$\alpha_{1}=1+\ldots$ and $\alpha_{2}=1+\ldots\,$. In contrast to simpler
cases like in Refs. Kondo1972 ; Haertel2008 ; Menchyshyn2014 , we were not
able to derive the high-temperature asymptotes from the self-consistent
equations. However, we have numerical evidence that solutions of the equations
(IV) coincide with the high-temperature asymptotes given in Eq. (IV). Further
details on the numerical solution of the self-consistent equations (IV) can be
found in the Appendix.

Figure 2: RGM solutions for correlators $c_{10}(T)$, $c_{01}(T)$, $c_{20}(T)$,
$c_{11}(T)$, $c_{02}(T)$ versus ED data for $J_{1}=3.294$ and $J_{2}=1$. Gray
curves correspond to high-temperature asymptotes (IV). Dotted, dash-dotted,
and dashed lines correspond to ED data for $N=12$, 16, and 20, respectively.

Temperature dependences of $\tilde{\alpha}_{10}$, $\tilde{\alpha}_{01}$,
$\tilde{\alpha}_{20}$, $\tilde{\alpha}_{11}$, $\tilde{\alpha}_{02}$ and of
$\rho$, $\alpha_{1}$, $\alpha_{2}$ as well as an illustration of the accuracy
of the performed calculations for the specific set of parameters
$J_{1}=3.294$, $J_{2}=1$ relevant for atacamite are reported in Appendix (Fig.
8). In Fig. 2 we report the RGM predictions for correlators $c_{10}$,
$c_{01}$, $c_{20}$, $c_{11}$, and $c_{02}$ and compare them to ED data for
finite chains of $N=12,16,20$ sites (for the same set of parameters). While
for high temperatures both data coincide and agree with asymptotes (IV), some
discrepancy between the $N\to\infty$ RGM and finite-$N$ ED data is seen at low
temperatures. Note, that the low-temperature ED data move towards the RGM data
as $N$ increases (cf. $N=12$, $N=16$ and $N=20$ data).

The results for $c_{10}$, $c_{01}$, $c_{20}$, $c_{11}$, $c_{02}$,
$\alpha_{1}$, and $\alpha_{2}$ illustrate the temperature dependencies of the
moment matrix (III) and the frequency matrix (III), and therefore the
temperature dependence of the Green’s functions (III) and of the dynamic
susceptibility (2.9), (II). While the moment matrix vanishes at high
temperatures, the frequency matrix remains finite having the following
eigenvalues:

$\displaystyle
f_{\pm}=J_{1}^{2}\sin^{2}\frac{q}{2}+J_{2}^{2}\pm\sqrt{J_{1}^{4}\sin^{4}\frac{q}{2}+J_{2}^{4}\cos^{2}\frac{q}{2}}$
(4.4)

(these are the eigenvalues for $T=\infty$). Evidently, $\sqrt{f_{-}}$
corresponds to acoustic excitations having the velocity
$\sqrt{J_{1}^{2}+J_{2}^{2}}/2$ whereas $\sqrt{f_{+}}$ corresponds to optical
excitations having the lowest energy $\sqrt{2J_{2}^{2}}$ at $q=0$. As the
temperature decreases, $c_{10},\ldots,c_{02}$ and
$\tilde{\alpha}_{10},\ldots,\tilde{\alpha}_{02}$ become nonzero controlling
temperature dependencies of the moment and frequency matrices. Although both
vertex parameters $\alpha_{1}$ and $\alpha_{2}$ are not far from unity, their
ratio has a nonmonotonous temperature dependence, see the middle panel of Fig.
8 in the Appendix.

In summary, we have completed the finding of the Green’s functions
$G_{q\alpha\beta}(\omega)$ (2.9), (II): Their form is fixed in Eq. (III) and
all parameters in the moment matrix (III) and the frequency matrix (III) are
determined now (Fig. 2 and the upper panel of Fig. 8 in Appendix). We are now
in the position to discuss thermodynamics as well as the static and dynamic
correlations of the spin model at hand.

##  V Thermodynamic and dynamic properties

Figure 3: RGM results for thermodynamic quantities per site: (upper panel) the
internal energy $e(T)$, (middle panel) the specific heat $c(T)$, and (lower
panel) the entropy $s(T)$; $J_{1}=3.294$, $J_{2}=1$. We also show ED data
($N=16,20$) and FTLM data ($N=24,28,32$; $R=50$ and $N=36$; $R=20$) FTLM . The
ground-state energy for the $S=1/2$ antiferromagnetic Heisenberg chain
($J_{1}=3.294$, $J_{2}=0$) is shown in the upper panel by the orange triangle
whereas the specific heat for the case $J_{1}=3.294$, $J_{2}=0$ is shown in
the middle panel by the orange line. Figure 4: RGM results for
$\chi^{zz}_{q}(0)$ at various temperatures; $J_{1}=3.294$, $J_{2}=1$.

Figure 5: RGM results for $S_{q}^{zz}(\omega)$ for three temperatures
$T=0.01$, $0.1$, $1$ and $J_{1}=3.294$, $J_{2}=1$. White lines show
$\sqrt{f_{\pm}}$ ($f_{\pm}$ are the eigenvalues of the frequency matrix). In
the upper panel ($T=0.01$) we also show the lower boundary of the two-spinon
continuum for the $S=1/2$ antiferromagnetic Heisenberg chain ($J_{1}=3.294$,
$J_{2}=0$, orange line), see discussion in the text. Figure 6: RGM results
(solid) and ED data (crosses for $N=16$ and squares for $N=20$) for $S_{q}$ at
various temperatures; $J_{1}=3.294$, $J_{2}=1$.

In this section we discuss the thermodynamic and dynamic quantities of a
$S=1/2$ antiferromagnetic Heisenberg sawtooth-chain model as they follow from
the RGM calculations and compare them to the outcomes of other approaches, see
Figs. 3, 4, 5, and 6.

First we note that the system at hand with $J_{1}=3.294$, $J_{2}=1$ may be
viewed as a simple $S=1/2$ antiferromagnetic Heisenberg chain with
$J_{1}=3.294$ perturbed by an extra coupling with the strength $J_{2}\approx
0.3J_{1}$ running along the zig-zag path, see Fig. 1. Let us check how many
traces of the $S=1/2$ antiferromagnetic Heisenberg chain with $J_{1}=3.294$
($J_{2}=0$), which is amenable to a rigorous study via the famous Bethe
ansatz, can be seen in the obtained data for the set $J_{1}=3.294$, $J_{2}=1$.
The famous ground-state energy value $e_{0}/J=1/4-\ln 2$ Hulthen1938 ;
Karbach1998  corresponds to $e_{0}\approx-0.730$ and it is shown by the orange
triangle in the upper panel of Fig. 3. For $J_{2}=1$, the RGM result is
$e_{0}\approx-0.673$ whereas $e_{0}$ from finite-system numerics lies in the
region $-0.776\ldots-0.759$, see the inset in the upper panel of Fig. 3.
Moreover, for the $S=1/2$ antiferromagnetic Heisenberg chain, the most
important two-spinon excitations form a continuum with the following lower and
upper boundaries: $\epsilon_{l}(q)=(\pi J/2)|\sin q|$ and $\epsilon_{u}(q)=\pi
J\left|\sin(q/2)\right|$ Cloizeaux1962 ; Karbach1998 . Thus, the two-spinon
excitations account for about 73% of the total intensity in
$S_{q}^{zz}(\omega)$ in the ground state Karbach1997 . Moreover,
$S_{q}^{zz}(\omega)$ diverges at the lower boundary $\epsilon_{l}(q)$
Karbach1997 , see Fig. 2 of Ref. Klauser2011 . The two-spinon excitation lower
boundary for the $S=1/2$ antiferromagnetic Heisenberg chain with $J_{1}=3.294$
is shown by the orange line in the upper panel of Fig. 5 which corresponds to
$T=0.01$. In the case when $J_{2}=1$, the high-energy branch $\sqrt{f_{+}}$,
which manifests itself in the RGM result for $S_{q}^{zz}(\omega)$ at $T=0.01$,
is close to the two-spinon excitation lower boundary $\approx 5.174|\sin q|$
(orange line) and, in addition, the low-energy branch $\sqrt{f_{-}}$ shows up.
Furthermore, the specific heat of the $S=1/2$ antiferromagnetic Heisenberg
chain with $J_{1}=3.294$ (see Ref. Kluemper2000 ) corresponds to the orange
line in the middle panel of Fig. 3: Its high-temperature features are hardly
changed when $J_{2}=1$ but the low-temperature features are obviously
completely different. Finally, the entropy in the case $J_{1}=3.294$,
$J_{2}=0$ consists of the entropy of the $S=1/2$ antiferromagnetic Heisenberg
chain with $J_{1}=3.294$ and the entropy $(N/2)\ln 2$ of $N/2={\cal N}$ spins
$S=1/2$ sitting in sites $j,2$, $j=1,\ldots,{\cal N}$ and not entering the
Hamiltonian, see Fig. 1. That implies the residual ground-state entropy
$s(T=0)=(\ln 2)/2\approx 0.347$. A remnant of this is a plateau-like behavior
of $s(T)$ in the region $0.1\ldots 0.5$ seen in all reported data for
$J_{2}=1$ in the lower panel of Fig. 3. Similar arguments are applicable to
the uniform susceptibility: In the limit $J_{2}=0$, the uniform susceptibility
is a sum of the $\propto 1/T$ contribution from the $S=1/2$ spins at sites
$j,2$ and the uniform susceptibility of the antiferromagnetic Heisenberg chain
with $J_{1}=3.294$ consisting of the $S=1/2$ spins at sites $j,1$. If
$J_{2}=1$, such a picture remains valid for high enough temperatures but fails
as the temperature becomes sufficiently low.

Now let us comment further on the thermodynamics of the $S=1/2$
antiferromagnetic Heisenberg sawtooth-chain model with $J_{1}=3.294$ and
$J_{2}=1$. The existence of a double-peak structure in $c(T)$ indicating two
energy scales is obvious, see the middle panel of Fig. 3, where the position
and the height of the additional low-temperature maximum is noticeably
affected by the system size (see the ED and FTLM data for different $N$).
Moreover, the RGM data obviously deviate from the ED and FTLM data at low
temperatures. However, the RGM predicts correctly the very existence of the
additional low-temperature maximum in the specific heat $c(T)$. That is in
accordance with previous RGM studies, see Refs. Junger2004 ; Haertel2008 .
Thus we may conclude, that the RGM scheme has some predictive power also at
low temperatures which is important in cases where no reliable data from
alternative approaches are available.

The RGM results for the entropy $s(T)$ obtained from the internal energy
according to Eq. (2.4) does not vanish as the temperature decreases but
approaches a finite value about $\approx 0.408$, see the lower panel in Fig.
3. In other words, the RGM loses about 60% of entropy in the case at hand.
Clearly, the sum rule like $\int_{0}^{\infty}{\rm d}{\sf T}c({\sf T})/{\sf
T}=s(\infty)$ implies some restriction for correlators which can be hardly
satisfied within the RGM approach. Another sum rule, $\int_{0}^{\infty}{\rm
d}{\sf T}c({\sf T})=-e(0)$, is also beyond control within the RGM
calculations. Again, the RGM predicts only qualitatively a plateau-like shape
of $s(T)$ at $T=0.1\ldots 0.2$.

In Fig. 4 we report $\chi_{q}^{zz}(0)$ for several temperatures $T=0.01\ldots
5$. $\chi_{q}^{zz}(0)$ is finite and small for all $-\pi\leq q<\pi$ even at
very low temperatures in accordance with the absence of a phase transition in
the system at hand.

Let us comment on the dynamic structure factor (II). To obtain the data
reported in Fig. 5 we replaced the $\delta$-functions $\delta(x)$ in Eq. (III)
by the Lorentzian functions $\epsilon/[\pi(x^{2}+\epsilon^{2})]$ with the half
width at half maximum $\epsilon=0.01$. $S_{q}^{zz}(\omega)$ shows two
excitation branches in accordance with two sites in the unit cell. The higher-
frequency one is related to the two-spinon continuum lower boundary of the
$S=1/2$ antiferromagnetic Heisenberg chain with $J_{1}=3.294$, see the
discussion above, and the lower-frequency one is controlled by the smaller
energy scale $J_{2}=1$. These excitation branches should be detectable in
inelastic neutron scattering experiments on atacamite above the magnetic
transition temperature $T_{\rm N}/J_{2}\simeq 0.087$.

In Fig. 6 we report the static structure factor $S_{q}$, cf. Eqs. (2.7) and
(2.8), for several temperatures $T=0.01\ldots 5$. $S_{q}$ approaches $3/4$ in
the high-temperature limit, see the red curve for $T=5$ in Fig. 6 (and also
the results for $T=10$ and $T=100$ in Fig. 11 in Appendix), as it should. From
Fig. 6 one can notice a slow approaching of the RGM and ED data with growing
temperature. This is conditioned by a too large RGM value of $c_{02}$ at high
temperatures, see the discussion in Appendix. The static structure factor
should satisfy the sum rule: $[2/(3\pi)]\int_{-\pi}^{\pi}{\rm d}qS_{q}=1$, see
Eq. (2.8). The left-hand side of this equation for the RGM outcome deviates
from 1: It is slightly above 60% of 1 as $T$ varies from 0.001 to 0.1 and then
it starts to approach 1 exceeding 95% of 1 at $T=10$.

Figure 7: Specific heat per site $c$ and uniform susceptibility per site
$\chi_{0}^{zz}$ versus temperature $T$ for the $S=1/2$ Heisenberg sawtooth
chain with $J_{1}=3.294$, $J_{2}=1$: RGM [$\chi_{0}^{zz}=\chi_{0}^{zz}(0)$],
ED for $N=16,20$ [$\chi_{0}^{zz}=S_{q=0}/(3T)$ is obtained by inserting
computed correlators into Eq. (2.8)], FTLM for $N=24,28,32,36$, Padé
approximant $[5,5]$ of high-temperature series Lohmann2014 , and experiments
for atacamite Heinze2018  (crosses), Heinze2021  (squares). We use logarithmic
temperature scale to make transparent both the low- and the high-temperature
features.

Let us compare various theoretical predictions to experimental data for
atacamite: Temperature dependences of $c$, $s$, and $\chi_{0}^{zz}$ were
reported in Refs. Heinze2021 ; Heinze2018 . In Fig. 7 we show besides the
experimental data (symbols) and RGM results also the ED and FTLM data as well
as the Padé approximant $[5,5]$ of high-temperature series Lohmann2014 .
($\chi_{0}^{zz}$ Heinze2018  is bound to the value of Padé approximant $[5,5]$
at $T=2.945$.) As can be seen from Fig. 7, the RGM approach can describe
experiments only qualitatively. Furthermore, the finite-size lattice
calculations explain pretty well the temperature profile of the specific heat
$c$ (upper panel) but fail to reproduce the temperature dependence of the
uniform susceptibility $\chi_{0}^{zz}$ (lower panel). Thus, the RGM
underestimates the experimental value of $\chi_{0}^{zz}$ at $T=0.1$ by about
39%, whereas the ED (Padé approximant $[5,5]$) overestimates it by about 100%
(45%). It should be underlined that from the theoretical side we deal with a
pure one-dimensional-system susceptibility whereas in experiments three-
dimensional interactions may be relevant Scalapino1975 ; Schulz1996 .

##  VI Summary

In conclusion, we have used the RGM approach to investigate the properties of
a $S=1/2$ antiferromagnetic Heisenberg sawtooth-chain model. Exact statements
are rather scarce for frustrated quantum spin models and the analytical or
numerical methods available to study these systems involve approximations.
Therefore, it is valuable to obtain results by different techniques. Although
various aspects of the $S=1/2$ antiferromagnetic Heisenberg sawtooth chain
have been studied by many authors using various approximate techniques, a
consistent analysis of the thermodynamic and dynamic properties still remains
an interesting issue.

The sawtooth-chain lattice geometry corresponds to two nonequivalent sites in
the unit cell. We have elaborated the RGM approach for the calculation of the
thermodynamic and dynamic quantities of the $S=1/2$ Heisenberg model on such a
lattice. A unit cell with nonequivalent sites has less symmetry. The frequency
matrix is complex and non-Hermitian. Nevertheless, its eigenvalues are real
(but not necessarily nonnegative) and the application of a minimal RGM scheme
(i.e., with the number of vertex parameters equals to the number of
nonequivalent sites in the unit cell) gives reasonable results. Although the
basic RGM equations (III) – (III) are valid for any set of parameters, we
discuss in detail how to derive relevant solutions for $J_{2}=1$ and $J_{1}$
slightly larger than $J_{2}$ and illustrate that the RGM approach gives
reasonable results up to very low temperatures. In particular, the RGM is
applicable for atacamite Cu2Cl(OH)3 with $J_{2}=1$, $J_{1}=3.294$ Heinze2018 ;
Heinze2021 . As it was mentioned in Sec. V, the RGM results violate three sum
rules: Two for the specific heat $c(T)$ and one for the static structure
factor $S_{q}$. It might be interesting to check whether these sum rules (with
$e_{0}$ as an extra input parameter) may be used to introduce more vertex
parameters this way improving the minimal RGM scheme results. We hope the
gained experience would be useful for other RGM studies in the case of
nonequivalent sites in the unit cell, in particular, for the square-kagome
$S=1/2$ Heisenberg antiferromagnet compounds Fujihala2020 ; Yakubovich2021 ;
Liu2022 .

## Acknowledgements

We are grateful to Dieter Ihle, Patrick Müller, and Jürgen Schnack for
comments and suggestions. We thank Taras Verkholyak and Stanislav Pidhorskyi
for useful discussions.

## Appendix: Numerical solution of the self-consistent equations

Figure 8: Numerical solution of Eq. (IV) using (Appendix: Numerical solution
of the self-consistent equations) (solid) and (A.5) (dashed), see the text of
appendix: (upper panel) $\tilde{\alpha}_{ij}$; (middle panel) $\rho$,
$\alpha_{1}$, $\alpha_{2}$; (lower panel) achieved values of the objective
functions. Figure 9: Values of correlators $|c_{ij}|$ as they follow by
numerical solution of Eq. (IV) using (Appendix: Numerical solution of the
self-consistent equations) (solid) and (A.5) (dashed). Gray straight lines
correspond to high-temperature asymptotes.

Figure 10: RGM results for thermodynamic quantities per site, (upper panel)
$e(T)$, (middle panel) $c(T)$, (lower panel) $s(T)$, cf. Fig. 3, as they
follow by numerical solution of Eq. (IV) using (Appendix: Numerical solution
of the self-consistent equations) (solid) and (A.5) (dashed). Figure 11: RGM
results for $S_{q}$ at various temperatures, cf. Fig. 6, as they follow by
numerical solution of Eq. (IV) using (Appendix: Numerical solution of the
self-consistent equations) (solid) and (A.5) (dashed).

In this appendix, we discuss how to solve numerically the self-consistent
equations reported in Sec. IV. To this end, we consider a six-dimensional
space of values $\xi_{1}\equiv\tilde{\alpha}_{10}$,
$\xi_{2}\equiv\tilde{\alpha}_{01}$, $\xi_{3}\equiv\tilde{\alpha}_{20}$,
$\xi_{4}\equiv\tilde{\alpha}_{11}$, $\xi_{5}\equiv\tilde{\alpha}_{02}$, and
$\xi_{6}\equiv\rho$, see Eq. (IV). We introduce the (nonnegative) objective
function Menchyshyn2014

$\displaystyle\mathfrak{F}(\xi_{1}\\!,\\!\ldots\\!,\\!\xi_{6})\\!=\\!\\!\left[\\!\xi_{1}\xi_{6}\\!-\\!\Xi_{1}(\\!\xi_{1}\\!,\\!\ldots\\!,\\!\xi_{6}\\!)\\!\right]^{2}\\!+\\!\left[\\!\xi_{2}\\!-\\!\Xi_{2}(\\!\xi_{1}\\!,\\!\ldots\\!,\\!\xi_{6}\\!)\\!\right]^{2}$
$\displaystyle\\!+\\!\left[\\!\xi_{3}\xi_{6}\\!-\\!\Xi_{3}(\\!\xi_{1}\\!,\\!\ldots\\!,\\!\xi_{6}\\!)\\!\right]^{2}\\!+\\!\left[\\!\xi_{4}\\!-\\!\Xi_{4}(\\!\xi_{1}\\!,\\!\ldots\\!,\\!\xi_{6}\\!)\\!\right]^{2}$
$\displaystyle\\!+\\!\left[\\!\xi_{5}\\!-\\!\Xi_{5}(\\!\xi_{1}\\!,\\!\ldots\\!,\\!\xi_{6}\\!)\\!\right]^{2}\\!+\\!\left[\Xi_{6}(\xi_{1}\\!,\\!\ldots\\!,\\!\xi_{6})\right]^{2}\\!\geq\\!0\,\,\,\,\,$
(A.1)

with

$\displaystyle\Xi_{1}(\xi_{1}\\!,\\!\ldots\\!,\\!\xi_{6})\\!\\!=\\!\\!\frac{1}{2\pi}\\!\int\limits_{-\pi}^{\pi}\\!\\!\\!{\rm
d}q{\rm e}^{{\rm i}q}\langle
S_{q1}^{-}S_{q1}^{+}\rangle_{\tilde{\alpha},\rho},$ $\displaystyle\vdots$
$\displaystyle\Xi_{5}(\xi_{1}\\!,\\!\ldots\\!,\\!\xi_{6})\\!\\!=\\!\\!\frac{1}{2\pi}\\!\int\limits_{-\pi}^{\pi}\\!\\!\\!{\rm
d}q{\rm e}^{{\rm i}q}\langle
S_{q2}^{-}S_{q2}^{+}\rangle_{\tilde{\alpha},\rho},$
$\displaystyle\Xi_{6}(\xi_{1}\\!,\\!\ldots\\!,\\!\xi_{6})\\!\\!=\\!\\!\frac{1}{2\pi}\\!\\!\int\limits_{-\pi}^{\pi}\\!\\!\\!{\rm
d}q\langle
S_{q1}^{-}S_{q1}^{+}\rangle_{\tilde{\alpha},\rho}\\!\\!-\\!\\!\frac{1}{2\pi}\\!\\!\int\limits_{-\pi}^{\pi}\\!\\!\\!{\rm
d}q\langle S_{q2}^{-}S_{q2}^{+}\rangle_{\tilde{\alpha},\rho},\,\,\,\,\,$ (A.2)

see Eqs. (IV), (4.2). The objective function
$\mathfrak{F}(\xi_{1},\ldots,\xi_{6})$ can be evaluated according to Eqs.
(Appendix: Numerical solution of the self-consistent equations) and (Appendix:
Numerical solution of the self-consistent equations) for any point in the six-
dimensional space $(\xi_{1},\ldots,\xi_{6})$ presuming that
$\Xi_{i}(\xi_{1},\ldots,\xi_{6})$, $i=1,\ldots,6$ exist. Obviously,
$\mathfrak{F}$ defined in Eq. (Appendix: Numerical solution of the self-
consistent equations) vanishes at the point $(\xi_{1}^{*},\ldots,\xi_{6}^{*})$
which corresponds to the solution of Eq. (IV), i.e.,
$\mathfrak{F}(\xi_{1}^{*},\ldots,\xi_{6}^{*})=0$. Importantly, the set of
equations at hand [Eq. (IV)] does not have a unique solution, i.e., we are
interested in the physical one only, which should be discriminated from
irrelevant ones.

We begin with a sufficiently high temperature $T$ (e.g., $T=50$ or $T=100$),
when the correlation functions can be calculated using the asymptotic high-
temperature values $c_{10}=-J_{1}/(8T)$, $c_{01}=-J_{2}/(8T)$,
$c_{20}=J_{1}^{2}/(32T^{2})$, $c_{11}=J_{1}J_{2}/(32T^{2})$,
$c_{20}=J_{2}^{2}/(32T^{2})$ and $\rho=1$; these initial values are denoted as
$\xi_{1}^{(0)},\ldots,\xi_{6}^{(0)}$ and
$\mathfrak{F}(\xi_{1}^{(0)},\ldots,\xi_{6}^{(0)})\neq 0$. However, if
$(\xi_{1}^{(0)},\ldots,\xi_{6}^{(0)})$ is quite close to
$(\xi_{1}^{*},\ldots,\xi_{6}^{*})$ the objective function
$\mathfrak{F}(\xi_{1},\ldots,\xi_{6})$ has (approximately) the form of a
paraboloid in the seven-dimensional space around
$(\xi_{1}^{(0)},\ldots,\xi_{6}^{(0)})$, i.e.,

$\displaystyle\mathfrak{F}(\xi_{1},\ldots,\xi_{6})\\!$ $\displaystyle\approx$
$\displaystyle\\!{\cal F}(\xi_{1},\ldots,\xi_{6});$ $\displaystyle{\cal
F}(\xi_{1},\ldots,\xi_{6})\\!$ $\displaystyle=$
$\displaystyle\\!\sum_{i=1}^{6}\sum_{j=i}^{6}C_{ij}\left(\xi_{i}-\xi_{i}^{*}\right)\left(\xi_{j}-\xi_{j}^{*}\right)\\!.$
(A.3)

Considering close points with
$\xi_{1}^{\pm}\\!=\\!\xi_{1}^{(0)}\\!\pm\\!\delta\xi_{1}/p$, …,
$\xi_{6}^{\pm}\\!=\\!\xi_{6}^{(0)}\\!\pm\\!\delta\xi_{6}/p$, where
$\delta\xi_{1},\ldots,\delta\xi_{5}$ are (small) differences of the asymptotic
high-temperature values at $T$ and, e.g., at $1.01T$, $\delta\xi_{6}$ is,
e.g., $0.001$, whereas $p$ is, e.g., $600$, and using ${\cal
F}(\xi_{1},\ldots,\xi_{6})\approx\mathfrak{F}(\xi_{1},\ldots,\xi_{6})$, we
determine the coefficients $C_{ij}$ in Eq. (Appendix: Numerical solution of
the self-consistent equations). Furthermore, we obtain the prediction for
$\xi_{1}^{*},\ldots,\xi_{6}^{*}$ from Eq. (Appendix: Numerical solution of the
self-consistent equations). The objective function $\mathfrak{F}$, in general,
does not vanish at the determined point $(\xi_{1}^{*},\ldots,\xi_{6}^{*})$
(since $\mathfrak{F}$ and ${\cal F}$, in general, are only approximately
equal). We declare this point as the initial one, i.e.,
$\xi_{1}^{*}\to\xi_{1}^{(0)},\ldots,\xi_{6}^{*}\to\xi_{6}^{(0)}$, and repeat
calculations. While seeking for the new coefficients $C_{ij}$ and the new
prediction for $\xi_{1}^{*},\ldots,\xi_{6}^{*}$ we decrease $\delta\xi_{1}/p$,
…, $\delta\xi_{6}/p$ by factor 2. We repeat calculations (e.g., 10 times),
evaluating the value of the objective function $\mathfrak{F}$ at this
temperature $T$, see the solid line in the lower panel of Fig. 8, and its
small values allow us to conclude that we have found the solution of Eq. (IV)
at the temperature $T$.

Next step is to decrease the temperature: $T\to T-\Delta T$; $\Delta T$ varies
from $0.01$ to $0.000\,01$, see below. We do not use the asymptotic high-
temperature values any more. Instead, we use the determined values
$\xi_{1}^{*},\ldots,\xi_{6}^{*}$ at the temperature $T$ as the initial values
$\xi_{1}^{(0)},\ldots,\xi_{6}^{(0)}$ for the lower temperature $T-\Delta T$.
Furthermore, $\delta\xi_{i}$ now are the differences of $\xi_{i}^{(0)}$ at
$T-\Delta T$ and $\xi_{i}^{(0)}$ at $T$. This way we proceed approaching
extremely low temperatures; simultaneously we observe the objective function
$\mathfrak{F}$ which should be small enough. As can be seen in the lower panel
of Fig. 8, $\mathfrak{F}$ is as small as $10^{-40}\ldots 10^{-50}$ (solid
curve) and thus evidences that we have found the solution of Eq. (IV).

Few comments on the explained scheme of numerical solution of the self-
consistent equations are in order here. First, the described procedure, which
is based on the assumption about a paraboloid for the objective function
(Appendix: Numerical solution of the self-consistent equations), requires a
reasonable amount of time on personal computer that is obviously an advantage
in comparison with the numerical solution described in Ref. Menchyshyn2014
(seeking for the minimum of the objective function within a cuboid).

Second, at certain low temperature the described scheme may fail. What is the
reason for that? We observed that it may occur because $f_{-}$ [see Eq. (III)]
becomes negative at the points of the six-dimensional space which are used to
determine the coefficients $C_{ij}$ in Eq. (Appendix: Numerical solution of
the self-consistent equations). Sometimes, this obstacle can be overcome by a
change of the specific parameter values employed in the described scheme. Here
we have arrived at the third comment, which regards a jump behavior of the
solid curve in the lower panel of Fig. 8. The jumps are related to a change of
the step $\Delta T$: If the described scheme fails at some temperature, one
may decrease $\Delta T$ or increase $p$ etc. and this may allow to proceed
further decreasing the temperature. For example, we set $\Delta T=0.01$ at
high temperatures, but $\Delta T=0.001$ while approaching $T=1$, $\Delta
T=0.000\,1$ for $T=1\ldots 0.1$ and $\Delta T=0.000\,01$ below $T=0.1$.

Fourth, it is worth noting that the second and the fourth equations (IV) may
be replaced by the physically equivalent ones

$\displaystyle\tilde{\alpha}_{01}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}{\rm
d}q\langle S_{q1}^{-}S^{+}_{q2}\rangle_{\tilde{\alpha},\rho},$
$\displaystyle\tilde{\alpha}_{11}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}{\rm
d}q{\rm e}^{{\rm i}q}\langle
S_{q1}^{-}S^{+}_{q2}\rangle_{\tilde{\alpha},\rho},$ (A.4)

respectively. This simply corresponds to another possible choice of
$c_{01}=\langle S_{j,1}^{-}S_{j,2}^{+}\rangle$ and $c_{11}=\langle
S_{j,1}^{-}S_{j+1,2}^{+}\rangle$, see Fig. 1 and Eqs. (II) and (3.7). In these
cases, however, the explained numerical scheme for the resulting set of self-
consistent equations fails at higher temperatures and therefore they were
discarded.

Finally, for the set $J_{1}=3.294$, $J_{2}=1$ we have detected the following
problem. At high temperatures $\tilde{\alpha}_{02}$ is the smallest quantity
and can be hardly controlled by the objective function (Appendix: Numerical
solution of the self-consistent equations). Therefore $c_{02}$ does not follow
(IV) in the temperature range where other four correlators, $c_{10}$,
$c_{01}$, $c_{20}$, and $c_{11}$, are quite close their high-temperature
asymptotes (IV). However, the correct high-temperature behavior of all
correlators is inherent in the self-consistent equations (IV): Fixing $c_{02}$
by the relation $c_{02}=2c_{01}^{2}$ which holds at high temperatures and
utilizing the described numerical scheme now in a five-dimensional space we
achieve the values of the objective function as small as $10^{-20}\ldots
10^{-30}$.

Yet another comment is worth mentioning. In the beginning of appendix we
introduce a six-dimensional space defining $\xi_{1},\ldots,\xi_{6}$ and then
apply the explained numerical scheme. However, another choice of the
coordinates which describe the points of a six-dimensional space is also
possible. More specifically, we may consider another six-dimensional space of
values, $\xi_{1}\equiv\rho\tilde{\alpha}_{10}$,
$\xi_{2}\equiv\tilde{\alpha}_{01}$, $\xi_{3}\equiv\rho\tilde{\alpha}_{20}$,
$\xi_{4}\equiv\tilde{\alpha}_{11}$, $\xi_{5}\equiv\tilde{\alpha}_{02}$, and
$\xi_{6}\equiv\rho$, and use the following objective functions:

$\displaystyle\mathfrak{F}(\xi_{1}\\!,\\!\ldots\\!,\\!\xi_{6})\\!=\\!\sum_{i=1}^{5}\\!\left[\\!\xi_{i}\\!-\\!\Xi_{i}(\\!\xi_{1}\\!,\\!\ldots\\!,\\!\xi_{6}\\!)\\!\right]^{2}\\!+\\!\left[\Xi_{6}(\xi_{1}\\!,\\!\ldots\\!,\\!\xi_{6})\right]^{2}\\!\\!,\,\,\,$
(A.5)

where $\Xi_{i}(\xi_{i},\ldots,\xi_{6})$, $i=1,\ldots,6$ are defined in Eq.
(Appendix: Numerical solution of the self-consistent equations), see Eqs.
(IV), (4.2). As can be seen in Figs. 8 and 9, dashed curves, in such a case
the high-temperature asymptote for $c_{02}$ is reproduced better and this
correlator has smaller values. However, the values of the objective function
are much larger (dashed curve in the lower panel of Fig. 8). Thermodynamic
quantities along with the static structure factor as they follow by numerical
solution of Eq. (IV) using (Appendix: Numerical solution of the self-
consistent equations) and (A.5) are compared in Figs. 10 and 11 (solid versus
dashed curves). Although the results are different in detail (and correspond
to a big difference of the objective function values) they look qualitatively
quite similar.

Summarizing this appendix, we emphasize that a numerical solution of the self-
consistent equations emerging after the Kondo-Yamaji approximation is an
important ingredient of the RGM approach. While in the previous studies this
issue has not been discussed in great detail, in the case of nonequivalent
sites in the unit cell, when the number of equations increases, a controlled
solving of this set of equations is vitally necessary to make possible a
successful application of the RGM approach.

## References

  * (1) S. V. Tyablikov, Methods in the Quantum Theory of Magnetism (Plenum Press (Plenum Publishing Corporation), New York, 1967).
  * (2) D. N. Zubarev, Nonequilibrium Statistical Thermodynamics (Consultants Bureau, New York, 1974).
  * (3) W. Gasser, E. Heiner, and K. Elk, Greensche Funktionen in Festkörper- und Vielteilchenphysik (Wiley-VCH, Berlin, 2001).
  * (4) Yu. G. Rudoy, Theoretical and Mathematical Physics 168, 1318 (2011).
  * (5) J. Kondo and K. Yamaji, Prog. Theor. Phys. 47, 807 (1972).
  * (6) H. Shimahara and S. Takada, J. Phys. Soc. Jpn. 60, 2394 (1991).
  * (7) A. F. Barabanov and O. A. Starykh, J. Phys. Soc. Jpn. 61, 704 (1992).
  * (8) S. Winterfeldt and D. Ihle, Phys. Rev. B 56, 5535 (1997).
  * (9) W. Yu and S. Feng, Eur. Phys. J. B 13, 265 (2000).
  * (10) L. Siurakshina, D. Ihle, and R. Hayn, Phys. Rev. B 64, 104406 (2001).
  * (11) B. H. Bernhard, B. Canals, and C. Lacroix, Phys. Rev. B 66, 104424 (2002).
  * (12) I. Junger, D. Ihle, J. Richter, and A. Klümper, Phys. Rev. B 70, 104419 (2004).
  * (13) P. Fröbrich and P. J. Kuntz, Physics Reports 432, 223 (2006).
  * (14) D. Schmalfuß, R. Darradi, J. Richter, J. Schulenburg, and D. Ihle, Phys. Rev. Lett. 97, 157201 (2006).
  * (15) T. N. Antsygina, M. I. Poltavskaya, I. I. Poltavsky, and K. A. Chishko, Phys. Rev. B 77, 024407 (2008).
  * (16) M. Härtel, J. Richter, D. Ihle, and S.-L. Drechsler, Phys. Rev. B 78, 174412 (2008).
  * (17) A. V. Miheyenkov, A. V. Shvartsberg, and A. F. Barabanov, JETP Letters 98, 156 (2013).
  * (18) A. A. Vladimirov, D. Ihle, and N. M. Plakida, Eur. Phys. J. B 88, 148 (2015).
  * (19) P. Müller, A. Lohmann, J. Richter, O. Menchyshyn, and O. Derzhko, Phys. Rev. B 96, 174419 (2017); T. Hutak, P. Müller, J. Richter, T. Krokhmalskii, and O. Derzhko, Condensed Matter Physics 21, 33705 (2018).
  * (20) A. V. Mikheenkov, V. E. Valiulin, A. V. Shvartsberg, and A. F. Barabanov, J. Exp. Theor. Phys. 126, 404 (2018).
  * (21) N.-N. Sun and H.-Y. Wang, J. Magn. Magn. Mater. 454, 176 (2018).
  * (22) P. Müller, A. Zander, and J. Richter, Phys. Rev. B 98, 024414 (2018).
  * (23) P. Müller, A. Lohmann, J. Richter, and O. Derzhko, Phys. Rev. B 100, 024424 (2019).
  * (24) R. Wieser, J. Phys.: Condens. Matter 31, 325801 (2019).
  * (25) P. S. Savchenkov and A. F. Barabanov, J. Magn. Magn. Mater. 521, 167505 (2021).
  * (26) J. Schulenburg, A. Honecker, J. Schnack, J. Richter, and H.-J. Schmidt, Phys. Rev. Lett. 88, 167207 (2002).
  * (27) T. Tonegawa and M. Kaburagi, J. Magn. Magn. Mater. 272-276, 898 (2004).
  * (28) M. E. Zhitomirsky and H. Tsunetsugu, Prog. Theor. Phys. Suppl. 160, 361 (2005).
  * (29) J. Richter, O. Derzhko, and A. Honecker, Int. J. Mod. Phys. B 22, 4418 (2008).
  * (30) D. V. Dmitriev and V. Ya. Krivnov, J. Phys.: Condens. Matter 28, 506002 (2016).
  * (31) T. Yamaguchi, S.-L. Drechsler, Y. Ohta, and S. Nishimoto, Phys. Rev. B 101, 104407 (2020).
  * (32) A. Metavitsiadis, C. Psaroudaki, and W. Brenig, Phys. Rev. B 101, 235143 (2020).
  * (33) J. Richter, J. Schulenburg, D. V. Dmitriev, V. Ya. Krivnov, and J. Schnack, Condensed Matter Physics 23, 43710 (2020).
  * (34) R. Siddharthan and A. Georges, Phys. Rev. B 65, 014417 (2001).
  * (35) P. Tomczak and J. Richter, Phys. Rev. B 54, 9004 (1996).
  * (36) J. Richter, J. Schulenburg, P. Tomczak, and D. Schmalfuß, Condensed Matter Physics 12, 507 (2009).
  * (37) H. Nakano and T. Sakai, J. Phys. Soc. Jpn. 82, 083709 (2013).
  * (38) I. Rousochatzakis, R. Moessner, and J. van den Brink, Phys. Rev. B 88, 195109 (2013).
  * (39) Y. Hasegawa, H. Nakano, and T. Sakai, Phys. Rev. B 98, 014404 (2018).
  * (40) T. Lugan, L. D. Jaubert, and A. Ralko, Phys. Rev. Research 1, 033147 (2019).
  * (41) P. A. McClarty, M. Haque, A. Sen, and J. Richter, Phys. Rev. B 102, 224303 (2020).
  * (42) T. Mizoguchi, Y. Kuno, and Y. Hatsugai, Phys. Rev. B 104, 035161 (2021).
  * (43) N. Astrakhantsev, F. Ferrari, N. Niggemann, T. Müller, A. Chauhan, A. Kshetrimayum, P. Ghosh, N. Regnault, R. Thomale, J. Reuther, T. Neupert, and Y. Iqbal, Phys. Rev. B 104, L220408 (2021).
  * (44) J. Richter, O. Derzhko, and J. Schnack, Phys. Rev. B 105, 144427 (2022).
  * (45) H. Kikuchi, Y. Fujii, D. Takahashi, M. Azuma, Y. Shimakawa, T. Taniguchi, A. Matsuo, and K. Kindo, Journal of Physics: Conference Series 320, 012045 (2011).
  * (46) A. Baniodeh, N. Magnani, Y. Lan, G. Buth, C. E. Anson, J. Richter, M. Affronte, J. Schnack, and A. K. Powell, npj Quantum Mater. 3, 10 (2018).
  * (47) L. Heinze, R. Beltran-Rodriguez, G. Bastien, A. U. B. Wolter, M. Reehuis, J.-U. Hoffmann, K. C. Rule, and S. Süllow, Physica B 536, 377 (2018).
  * (48) D. S. Inosov, Adv. Phys. 67, 149 (2018).
  * (49) V. P. Gnezdilov, Yu. G. Pashkevich, V. S. Kurnosov, O. V. Zhuravlev, D. Wulferding, P. Lemmens, D. Menzel, E. S. Kozlyakova, A. Yu. Akhrorov, E. S. Kuznetsova, P. S. Berdonosov, V. A. Dolgikh, O. S. Volkova, and A. N. Vasiliev, Phys. Rev. B 99, 064413 (2019).
  * (50) L. Heinze, H. O. Jeschke, I. I. Mazin, A. Metavitsiadis, M. Reehuis, R. Feyerherm, J.-U. Hoffmann, M. Bartkowiak, O. Prokhnenko, A. U. B. Wolter, X. Ding, V. S. Zapf, C. Corvalán Moya, F. Weickert, M. Jaime, K. C. Rule, D. Menzel, R. Valenti, W. Brenig, and S. Süllow, Phys. Rev. Lett. 126, 207201 (2021).
  * (51) K. Nawa, M. Avdeev, P. Berdonosov, A. Sobolev, I. Presniakov, A. Aslandukova, E. Kozlyakova, A. Vasiliev, I. Shchetinin, and T. J. Sato, Scientific Reports 11, 24049 (2021).
  * (52) T. Zhang and G.-B. Jo, Scientific Reports 5, 16044 (2015).
  * (53) M. Fujihala, K. Morita, R. Mole, S. Mitsuda, T. Tohyama, S.-i. Yano, D. Yu, S. Sota, T. Kuwai, A. Koda, H. Okabe, H. Lee, S. Itoh, T. Hawai, T. Masuda, H. Sagayama, A. Matsuo, K. Kindo, S. Ohira-Kawamura, and K. Nakajima, Nature Communications 11, 3429 (2020).
  * (54) O. V. Yakubovich, L. V. Shvanskaya, G. V. Kiriukhina, A. S. Volkov, O. V. Dimitrova, and A. N. Vasiliev, Inorganic Chemistry 60, 11450 (2021).
  * (55) B. Liu, Z. Zeng, A. Xu, Y. Sun, O. Yakubovich, L. Shvanskaya, S. Li, and A. Vasiliev, Low-temperature specific-heat studies on two square-kagome antiferromagnets, arXiv:2203.03804 (2022).
  * (56) F. Monti and A. Sütö, Phys. Lett. A 156, 197 (1991).
  * (57) K. Kubo, Phys. Rev. B 48, 10552 (1993); T. Nakamura and K. Kubo, Phys. Rev. B 53, 6393 (1996); D. Sen, B. S. Shastry, R. E. Walstedt, and R. Cava, Phys. Rev. B 53, 6401 (1996).
  * (58) A. Lohmann, H.-J. Schmidt, and J. Richter, Phys. Rev. B 89, 014415 (2014).
  * (59) O. Menchyshyn, T. Krokhmalskii, and O. Derzhko, Simple-cubic-lattice spin-1/2 Heisenberg model within Green-function method, preprint Inst. Cond. Matt. Phys., Nat. Acad. Sci. of Ukraine, ICMP-14-01E (L’viv, 2014).
  * (60) Within the FTLM scheme the sum over an orthonormal basis in the partition function is replaced by a much smaller sum over $R$ random vectors, see J. Jaklič and P. Prelovšek, Phys. Rev. B 49, 5065(R) (1994) and also J. Schnack, J. Richter, and R. Steinigeweg, Phys. Rev. Research 2, 013186 (2020).
  * (61) L. Hulthén, Ark. Mat., Astron. Fys. 26 A 11, 1 (1938).
  * (62) M. Karbach and G. Müller, Computers in Physics 11, 36 (1997); M. Karbach, K. Hu, and G. Müller, Computers in Physics 12, 565 (1998); M. Karbach, K. Hu, and G. Müller, Introduction to the Bethe Ansatz III, arXiv:cond-mat/0008018 (2000).
  * (63) J. des Cloizeaux and J. J. Pearson, Phys. Rev. 128, 2131 (1962).
  * (64) M. Karbach, G. Müller, A. H. Bougourzi, A. Fledderjohann, and K.-H. Mütter, Phys. Rev. B 55, 12510 (1997).
  * (65) A. Klauser, J. Mossel, J.-S. Caux, and J. van den Brink, Phys. Rev. Lett. 106, 157205 (2011).
  * (66) A. Klümper and D. C. Johnston, Phys. Rev. Lett. 84, 4701 (2000); D. C. Johnston, R. K. Kremer, M. Troyer, X. Wang, A. Klümper, S. L. Bud’ko, A. F. Panchula, and P. C. Canfield, Phys. Rev. B 61, 9558 (2000).
  * (67) D. J. Scalapino, Y. Imry, and P. Pincus, Phys. Rev. B 11, 2042 (1975).
  * (68) H. J. Schulz, Phys. Rev. Lett. 77, 2790 (1996).