# Thermodynamics of the $S=1/2$ hyperkagome-lattice Heisenberg antiferromagnet

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  Professor Ivan
Vakarchuk Department for Theoretical Physics, Ivan Franko National University
of L’viv, Drahomanov Street 12, 79005 L’viv, Ukraine  Fakultät für Physik,
Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany  Jürgen
Schnack  Fakultät für Physik, Universität Bielefeld, Postfach 100131, 33501
Bielefeld, Germany  Johannes Richter  Institut für Physik, Otto-von-Guericke-
Universität Magdeburg, P.O. Box 4120, 39016 Magdeburg, Germany  Max-Planck-
Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden,
Germany

(August 31, 2024)

###### Abstract

The $S=1/2$ hyperkagome-lattice Heisenberg antiferromagnet, which for instance
is related to the experimentally accessible spinel oxide Na4Ir3O8, allows to
study the interplay of geometrical frustration and quantum as well as thermal
fluctuations in three dimensions. We use 16 terms of a high-temperature series
expansion complemented by the entropy-method interpolation to examine the
specific heat and the uniform susceptibility of the $S=1/2$ hyperkagome-
lattice Heisenberg antiferromagnet. We obtain thermodynamic quantities for the
two possible scenarios of either a gapless or a gapped energy spectrum. We
have found that the specific heat $c$ exhibits, besides the high-temperature
peak around $T\approx 0.669$, a low-temperature one at $T\approx 0.021\ldots
0.033$. The functional form of the uniform susceptibility $\chi$ below about
$T=0.5$ depends strongly on whether the energy spectrum is gapless or gapped.
The value of the ground-state energy can be estimated to
$e_{0}\approx-0.440\ldots-0.435$. In addition to the entropy-method
interpolation we use the finite-temperature Lanczos method to calculate $c$
and $\chi$ for finite lattices of $N=24$ and $36$ sites. A combined view on
both methods leads us to favour a gapless scenario since then the maximum of
the susceptibility agrees better between both methods.

##  I Introduction

Frustrated quantum spin systems are a subject of intense ongoing research in
the field of magnetism [1, 2, 3, 4]. Geometric frustration and quantum
fluctuations may evade any ground-state ordering even in three dimensions.
Among several famous examples, the $S=1/2$ pyrochlore-lattice Heisenberg
antiferromagnet has attracted much attention, being for decades a candidate
for the realization of a spin-liquid state in three dimensions [5]. After
intense numerical studies, a lattice symmetry breaking in the ground state has
been revealed [6, 7, 8, 9].

A closely related example is the $S=1/2$ hyperkagome-lattice Heisenberg
antiferromagnet. Inspired by experiments on the spinel oxide Na4Ir3O8 [10], in
which low spin $d^{5}$ Ir4+ ions reside on the vertices of a hyperkagome
lattice, several theoretical studies for the classical ($S\to\infty$) and
quantum ($S=1/2$) Heisenberg antiferromagnet on such a lattice have been
performed [11, 12, 13, 14, 15, 16, 17, 18, 19]. The main focus of these
studies is at ground-state properties of the $S=1/2$ hyperkagome-lattice
Heisenberg antiferromagnet. For the ground state of this model a gapped
quantum spin liquid with topological order [12] and a gapless quantum spin
liquid with spinon Fermi surfaces [15] were proposed by Lawler et al.. In
contrast, Bergholtz et al. [16] proposed a valence bond crystal with a 72 site
unit cell as the ground state of this model; this implies a spin gap with a
huge number of singlet excitations below the lowest triplet state and thus a
power law for the specific heat and a vanishing susceptibility for vanishing
temperature.

Less attention has been paid to the finite-temperature properties of the
$S=1/2$ hyperkagome-lattice Heisenberg antiferromagnet [15, 16, 17, 19]. It
was argued that $c(T)\propto T^{2}$ at low $T$ [15] (similar to what is
observed for Na4Ir3O8 [10]) and that $\chi(T)$ has a non-zero value at $T=0$
and almost no temperature dependence as $T\to 0$ [19] (again in agreement with
experimental data for Na4Ir3O8 [10]). In addition, high-temperature series
expansions for $c$ and $\chi$ were developed and compared with the
experimental data for Na4Ir3O8 [17].

On the experimental side, apart from the mentioned iridate compound Na4Ir3O8
[10], there are other solid-state realizations of the hyperkagome-lattice
Heisenberg anitiferromagnet, see, e.g., Refs. [20, 21, 22]. Note, however,
that the $5d$-based transition-metal oxides, such as Na4Ir3O8, are known for
having a large spin-orbit coupling so that the Heisenberg Hamiltonian
apparently should be accomplished by other terms relevant for such materials
[23].

In the present paper, we consider the thermodynamics of the $S=1/2$
hyperkagome-lattice Heisenberg antiferromagnet. The toolbox to tackle
thermodynamics of frustrated quantum spin systems is rather scarce. Quantum
Monte Carlo suffers from the sign problem [24], exact diagonalization or
finite-temperature Lanczos methods are restricted to too small lattices [25,
26, 27], the density-matrix renormalization group technique requires a mapping
via a “snake” path to a one-dimensional system [28]. Moreover, the
pseudofermion functional renormalization group approach focuses on the wave-
vector-dependent susceptibility [19], whereas one more universal method, the
rotation-invariant Green’s function method [29, 30, 31, 32, 33, 34], has not
been applied to the $S=1/2$ hyperkagome-lattice Heisenberg antiferromagnet so
far.

In our study, we use the high-temperature series expansions to the order of
$\beta^{16}$ ($\beta=1/T$) provided by Singh and Oitmaa in Ref. [17]. Singh
and Oitmaa used the high-temperature series to compute various thermodynamic
properties down to a temperature 111Temperatures are given as multiples of the
exchange interaction. $T=0.25$ thus means $k_{\rm B}T/J=0.25$. of about
$T\approx 0.25$ [17]. In this paper, we combine the series expansion with the
entropy method using knowledge about the low-energy spectrum of the spin model
in order to obtain thermodynamic functions for the whole temperature range.
The entropy-method interpolation of high-temperature series expansions was
suggested by Bernu and Misguish [36] and later used in several studies [37,
38, 39, 40, 41, 42, 43, 44]. Besides the temperature dependence for the
specific heat $c(T)$ and the uniform susceptibility $\chi(T)$ for the $S=1/2$
hyperkagome-lattice Heisenberg antiferromagnet, we also obtain a prediction
for the ground-state energy of the model $e_{0}$, which provides self-
consistency of the entropy-method calculations. Our entropy-method
calculations are accompanied by finite-temperature Lanczos calculations for
finite lattices up to 36 sites.

The remainder of this paper is organized as follows. In Section II we
introduce the model whereas in Sections III and IV we briefly explain the
methods to be used for obtaining the thermodynamic quantities. Then, in
Section V, we report our results for the ground-state energy $e_{0}$, the
specific heat $c(T)$, and the uniform susceptibility $\chi(T)$. Finally, we
summarize our findings in Section VI.

##  II Model

Figure 1: The hyperkagome lattice. Besides the 12 sites from the same unit
cell ($1,\ldots,12$ stand for $\bm{r}_{1},\ldots,\bm{r}_{12}$), we show 13
more sites of the nearby unit cells (we use, e.g., $11-x+y+z$ for
$\bm{r}_{11}-\bm{e}_{x}+\bm{e}_{y}+\bm{e}_{z}$ to lighten notations).
Moreover, we show here 28 bonds (black lines); 15 bonds connect the sites from
the same unit cell and 9 bonds connect the sites of the neighboring cells. The
remaining 4 bonds, which connect the sites $1+y$ and $12+y$, $8-x$ and $9-x$,
$9-x$ and $11-x$, $11-x+y+z$ and $12+y$, are shown for better clarity. We also
display the underlying pyrochlore lattice.

The hyperkagome lattice has been described in several papers. It can be viewed
as a three-dimensional network of corner-sharing triangles with 12 sites in a
cubic unit cell. It also can be viewed as a 1/4 depleted pyrochlore lattice,
meaning that three out of the four sites in every tetrahedron are occupied by
spins. As a result, each spin of the three-dimensional hyperkagome lattice has
only four nearest neighbors just as for the two-dimensional kagome lattice.
There are several different conventions regarding the coordinates of lattice
sites (see, e.g., Refs. [23, 45, 46, 47]). According to Fig. 1, we define the
sites on the hyperkagome lattice sites by
$\bm{R}_{\bm{n}\alpha}=\bm{R}_{\bm{n}}+\bm{r}_{\alpha}$. Here,
$\bm{R}_{\bm{n}}=n_{x}\bm{e}_{x}+n_{y}\bm{e}_{y}+n_{z}\bm{e}_{z}$, where
$n_{x},n_{y},n_{z}$ are integers and $\bm{e}_{x}=(1,0,0)$,
$\bm{e}_{y}=(0,1,0)$, $\bm{e}_{z}=(0,0,1)$, generates a simple cubic lattice.
Moreover, the origins of the 12 equivalent sites in the unit cell may be
defined by $\bm{r}_{\alpha}$, $\alpha=1,\dots,12$ with
$\bm{r}_{1}=(3/8,1/8,5/8)$, $\bm{r}_{2}=(5/8,7/8,5/8)$,
$\bm{r}_{3}=(3/8,7/8,3/8)$, $\bm{r}_{4}=(5/8,3/8,1/8)$,
$\bm{r}_{5}=(3/8,3/8,7/8)$, $\bm{r}_{6}=(5/8,5/8,7/8)$,
$\bm{r}_{7}=(1/8,5/8,3/8)$, $\bm{r}_{8}=(7/8,5/8,5/8)$,
$\bm{r}_{9}=(7/8,3/8,3/8)$, $\bm{r}_{10}=(1/8,7/8,1/8)$,
$\bm{r}_{11}=(7/8,1/8,1/8)$, $\bm{r}_{12}=(1/8,1/8,7/8)$. In Fig. 1, we denote
$\bm{r}_{1},\dots,\bm{r}_{12}$ by $1,\ldots,12$. Moreover, we display there 13
sites of the nearby unit cells by $11-x+y+z$, $11-x+z$, $8-x$ and so on,
where, e.g., $11-x+y+z$ means $\bm{r}_{11}-\bm{e}_{x}+\bm{e}_{y}+\bm{e}_{z}$,
and so on.

The isotropic Heisenberg Hamiltonian of the model is given by

$H=J\sum_{\langle\bm{m}\alpha;\bm{n}\beta\rangle}\bm{S}_{\bm{m}\alpha}\cdot\bm{S}_{\bm{n}\beta}\
.$ (1)

Here, we set the antiferromagnetic interaction $J=1$, the sum in the equation
runs over the nearest-neighbor bonds of the hyperkagome lattice, and
$\bm{S}_{\bm{m}\alpha}$ represents the $S=1/2$ spin-vector operator at the
lattice site $\bm{R}_{\bm{m}\alpha}$. Expanding the sum in Eq. (1) for fixed
$\bm{m}$, one gets 24 bonds, that is, 15 bonds connecting the sites within the
unit cell with the same cell index $\bm{m}$ and 9 bonds connecting the sites
of the unit cell $\bm{m}$ with the sites of the neighboring unit cells
$\bm{m}-\bm{e}_{x}$, $\bm{m}-\bm{e}_{y}$, $\bm{m}-\bm{e}_{z}$,
$\bm{m}-\bm{e}_{x}+\bm{e}_{y}$, $\bm{m}+\bm{e}_{x}-\bm{e}_{z}$, and
$\bm{m}+\bm{e}_{y}-\bm{e}_{z}$, see Fig. 1. The remaining 4 bonds in Fig. 1,
i.e., the ones which connect the sites $1+y$ and $12+y$, $8-x$ and $9-x$,
$9-x$ and $11-x$, $11-x+y+z$ and $12+y$ (cf. the bonds connecting the sites
$1$ and $12$, $8$ and $9$, $9$ and $11$, $11-x+z$ and $12$), are shown here
for the sake of clarity.

In what follows, we compare the properties of model (1) on the hyperkagome
lattice to the properties of the two-dimensional kagome-lattice Heisenberg
antiferromagnet, which features also four nearest neighbors, as well as of the
three-dimensional pyrochlore-lattice Heisenberg antiferromagnet, which could
be considered the “mother” crystal structure, featuring six nearest neighbors
for each spin.

##  III Numerics for finite-size lattices

Previously, exact-diagonalization and finite-temperature Lanczos methods were
used for the $S=1/2$ pyrochlore-lattice Heisenberg antiferromagnet [48, 41].
At first glance, the hyperkagome-lattice case is simpler, since 1/4 of the
pyrochlore sites are empty, however, the unit cell now contains 12 sites
(instead of 4 sites for the pyrochlore) and the finite-system sizes may be
$N=12,\,24,\,36,\,48$, and so on. The exact diagonalization for one unit cell
with $N=12$ was reported in Ref. [15]. A chain-like arrangement of two
($N=24$) and three ($N=36$) unit cells is shown in Appendix A. Periodic
boundary conditions are applied in all cases, also for $N=12$ in order to
guarantee the correct number of nearest neighbors.

Figure 2: Finite-lattice results for (top) the specific heat and (bottom) the
uniform susceptibility. Exact-diagonalization ($N=12$) and finite-temperature
Lanczos method ($N=24$, $R=200$ and $N=36$, $R=20$) data. Details of
calculations are given in Appendix A.

We report exact-diagonalization ($N=12$) and finite-temperature Lanczos
($N=24,\,36$) data in Fig. 2. The results for the specific heat (top) and the
uniform susceptibility (bottom) for $N=24$ and $N=36$ differ from each other
below about $T\approx 0.2$. Surprisingly, the high-temperature peak of the
specific heat does not show any finite-size scaling; it is already provided by
the calculations for one unit cell ($N=12$). On these grounds, we thus
speculate that the curve of the specific heat at temperatures of the high-
temperature peak and above represents the thermodynamic limit. The position of
the low-temperature peak, on the other hand, does depend on the size; it is at
$T\approx 0.101$ for $N=12$, at $T\approx 0.069$ for $N=24$, and at $T\approx
0.055$ for $N=36$. Moreover, the height decreases notably with growing $N$.

The maximum of $\chi(T)$ has a mild dependence on system size; it occurs at
$T\approx 0.204$ for $N=12$, at $T\approx 0.168$ for $N=24$, and at $T\approx
0.158$ for $N=36$. Moreover, the height remains practically unchanged. This
behavior can be traced back to the size of the singlet-triplet gap for these
systems. Its value is $\Delta_{\rm s{-}t}\approx 0.383,0.216,0.136$ for
$N=12,24,36$, respectively. The whole behavior is reminiscent of that of the
kagome-lattice antiferromagnet [49].

In contrast, the results for the $S=1/2$ pyrochlore-lattice Heisenberg
antiferromagnet of $N=32$ sites [41] reflect the thermodynamic limit only for
$T>0.7$, well above the temperature of the high-temperature peak of $c(T)$.
Therefore, the finite-lattice results for the hyperkagome case allow a
reliable discussion of thermodynamic properties for much lower temperatures
down to $T\approx 0.2$. We provide more details in Sec. V on results.

##  IV Entropy method

As have been mentioned above, in our study we use the high-temperature series
expansion up to 16th order, which was reported in Ref. [17], see series
expansion coefficients for $\ln Z$ and the uniform structure factor in Table I
of Ref. [17]. On the other hand, we employ the Magdeburg HTE code developed
mainly by Lohmann [50, 51] (which is freely available at http://www.uni-
magdeburg.de/jschulen/HTE/) in an extended version up to 13th order to check
the series of the specific heat and the static uniform susceptibility. Our
results coincide with those of Ref. [17].

The high-temperature series expansion may be improved by simple Padé
approximants $[u,d](T)=P_{u}(\beta)/Q_{d}(\beta)$, where $P_{u}(\beta)$ and
$Q_{d}(\beta)$ are polynomials of order $u$ and $d$, $u+d\leq 16$, and the
series expansion of $[u,d](T)$ coincides with the high-temperature series of
$c$ or $\chi$ up to $16$th order with respect to $\beta=1/T$. Comparing close
to diagonal Padé approximants
$[5,5],\,[5,6],\,[6,6],\,[6,7],\,[7,7],\,[7,8],\,[8,8]$ (not shown here), we
conclude that they start to deviate notably one from another below $T\approx
0.5$ and thus can reproduce the high-temperature peak of $c(T)$ at $T\approx
0.669$, but not any of the specific features of $\chi(T)$ since $\chi(T)$
increases monotonously to temperatures well below $T=0.5$ and also has got its
maximum below $T=0.5$.

In order to study the thermodynamic behavior at lower temperatures we use the
entropy-method interpolation scheme introduced by Bernu et al. [36, 37, 38]
and further used in several studies [39, 40, 41, 42, 43, 44]. Within the
entropy method one interpolates the entropy (per site) $s$ as a function of
the mean (internal) energy (per site) $e$, $s(e)$. As $e$ approaches its
maximal value $e_{\infty}=E(T\rightarrow\infty)/N=\text{tr}(H)/N=0$, the
entropy is known from high-temperature series expansion, $s(e)=\ln
2+\sum_{i>1}a_{i}e^{i}$ (i.e., the coefficients $a_{2},\ldots,a_{16}$ are
known, see Ref. [36]). As $e$ approaches its minimal (ground-state) value
$e_{0}$, the entropy behaves as $s(e)\propto(e-e_{0})^{\alpha/(1+\alpha)}$ if
$c(T)=AT^{\alpha}$ for $T\to 0$ (gapless low-energy excitations) or as
$s(e)\propto-[(e-e_{0})/\Delta](\ln[\Delta(e-e_{0})]-1)$ if $c(T)\propto
e^{-\Delta/T}/T^{2}$ for $T\to 0$ (gapped low-energy excitations). Next, we
interpolate, instead of $s(e)$, an auxiliary function $G(e)$, different for
the two types of low-energy excitations, which immediately gives $s(e)$. For
the gapless case we have

$\displaystyle
G(e){=}\frac{\left[s(e)\right]^{\frac{1+\alpha}{\alpha}}}{e-e_{0}}\to G_{\rm
app}(e){=}\frac{\left(\ln
2\right)^{\frac{\alpha}{1+\alpha}}}{-e_{0}}\frac{P_{u}(e)}{Q_{d}(e)}\ ;$
$\displaystyle s_{\rm app}(e)=\left[\left(e-e_{0}\right)G_{\rm
app}(e)\right]^{\frac{\alpha}{1+\alpha}}\ .$ (2)

And for the gapped case we have

$\displaystyle
G(e){=}\left(e-e_{0}\right)\left[\frac{s(e)}{e-e_{0}}\right]^{\prime}\to
G_{\rm app}(e){=}\frac{\ln 2}{e_{0}}\frac{P_{u}(e)}{Q_{d}(e)}\ ;$
$\displaystyle\frac{s_{\rm app}(e)}{e-e_{0}}=\frac{\ln
2}{-e_{0}}-\int_{e}^{0}{\rm d}\xi\frac{G_{\rm app}(\xi)}{\xi-e_{0}}\ .$ (3)

Here, $P_{u}(e)$ and $Q_{d}(e)$ are the polynomials of order $u$ and $d$,
$u+d\leq 16$, and the series expansion of the quotient
$[u,d](e)=P_{u}(e)/Q_{d}(e)$ coincides with the Maclaurin series of $G(e)$
known up to $16$th order. Moreover, the prime denotes the derivative with
respect to $e$. Knowing the dependence $s(e)$, we obtain the desired
temperature dependence of the specific heat $c(T)$ in the parametric form:
$T=1/s^{\prime}(e)$ and $c=-[s^{\prime}(e)]^{2}/s^{\prime\prime}(e)$. Finally,
we can calculate the prefactor $A$, $A_{\rm
app}=[\alpha^{1+\alpha}/(1+\alpha)^{\alpha}][G_{\rm app}(e_{0})]^{\alpha}$,
for the gapless case and the energy gap $\Delta$, $\Delta_{\rm app}=-1/G_{\rm
app}(e_{0})$, for the gapped case. In the presence of a (small) external
magnetic field $h$ one gets the entropy $s_{\rm app}(e,h)$ which yields the
uniform susceptibility $\chi$ via the relations:
$m=[1/s^{\prime}(e,h)]\partial s(e,h)/\partial h$, $\chi=m/h$ ($h\to 0$).

Thus, to obtain the thermodynamic quantities within the framework of the
entropy method one needs, besides the high-temperature series for $c$ and
$\chi$, to know i) the ground-state energy $e_{0}$, ii) how $c(T)$ vanishes as
$T\to 0$, and iii) the value of $\chi_{0}\equiv\chi(T=0)$ in the case of
gapless low-energy excitations. Even if the precise value of $e_{0}$ is not
available and both gapless and gapped excitations are acceptable, one can
proceed as in Ref. [40]. First, one has to assume some reasonable value
$e_{0}$ (being prepared to explore systematically a certain region of
$e_{0}$). Second, one has to assume the exponent $\alpha$ in the case of a
gapless spectrum or one has to assume that the spectrum is gapped. Then, for
the assumed $e_{0}$ and gapless/gapped energy spectrum one has to calculate
within the entropy method the specific heat $c(T)$ using all $n_{\rm P}$
available Padé approximants $[u,d](e)$. There are $n+1$ Padé approximants
based on the series up to $n$th order; we discard from the very beginning four
Padé approximants $[n,0],\,[n-1,1],\,[1,n-1],\,[0,n]$ so that $n_{\rm P}=n-3$.
Next, one has to examine the “closeness” of all $n_{\rm P}$ profiles $c(T)$ by
examining them thoroughly from some (high enough, but not too high)
temperature $T_{\rm i}$ down to some (sufficiently low) temperature $T_{\rm
f}$ with temperature steps $\Delta T$. If the absolute value of the difference
of a certain $c$ from the average value (arithmetic mean) $\overline{c}$ at a
running temperature $T$ ($T_{\rm f}\leq T\leq T_{\rm i}$) is less than some
bound, e.g., 0.001, this $c$ belongs to the set of “coinciding” Padé
approximants. In the opposite case, this Padé approximant is discarded and not
considered for lower temperatures. According to Refs. [40, 43], a large number
of coinciding curves $n_{\rm cP}$, or more precisely a large value of
$p=n_{\rm cP}/n_{\rm P}$, provides evidence that the assumptions made about
$e_{0}$ and the low-energy excitations are self-consistent.

Figure 3: The ratio of the number of “coinciding” entropy-method Padé
approximants $n_{\text{cP}}$ to the number of all considered entropy-method
Padé approximants $n_{\rm P}$, $p=n_{\text{cP}}/n_{\rm P}$, based on the
series of 15th (thin green) and 16th (thick red) orders as a function of the
chosen value of $e_{0}$. Here $T_{\rm i}=0.5$, $\Delta T=0.025$, $T_{\rm
f}=0.1$, see the main text. We consider both assumptions, gapless (solid) and
gapped (dot-dashed) low-energy excitations.

In Fig. 3 we illustrate such an analysis based on $n_{\rm P}$ Padé
approximants following from the 15th (thin green) and 16th order (thick red)
in Eqs. (IV) or (IV) for the specific heat $c(T)$ under the assumption of a
gapless spectrum with $\alpha=2$ (solid) or a gapped spectrum (dot-dashed).
Here we set $T_{\rm i}=0.5$, $\Delta T=0.025$, $T_{\rm f}=0.1$, see Appendix
B. If $e_{0}$ is taken in the range $-0.440\,2\ldots-0.437\,9$ assuming a
gapless spectrum, i.e., $c(T)=AT^{2}$ as $T\to 0$, and the analysis is based
on 16th order ($n_{\rm P}=13$), we find that $n_{\text{cP}}=6$ and $p\approx
0.46$. Moreover, for the prefactor $A$ we get $A=493\ldots 727$. If $e_{0}$ is
taken in the range $-0.438\,1\ldots-0.435\,3$ assuming a gapped spectrum and
the analysis is based on 16th order, we find $p=4/13\approx 0.31$. Moreover,
the energy gap is $\Delta=0.025\ldots 0.018$. All these findings are
visualized by the thick red curves in Fig. 3. Slightly different values of
$e_{0}$ which provide maximal values of $p$ follow from the analysis based on
15th order, see the thin green curves in Fig. 3. Namely, for the gapless
spectrum with $e_{0}=-0.441\,5\ldots-0.438\,5$ we have $p=5/12\approx 0.42$,
$A=377\ldots 563$; for the gapped spectrum with
$e_{0}=-0.438\,5\ldots-0.437\,2$ we have $p=6/12=0.5$, $\Delta=0.027\ldots
0.024$.

Following the strategy of Refs. [40, 43], we may conclude that the entropy-
method prediction for the ground-state energy $e_{0}$ is
$-0.440\,2\ldots-0.437\,9$ (gapless excitations) or $-0.438\,1\ldots-0.435\,3$
(gapped excitations). By combination of both cases we arrive at $e_{0}=-0.440$
…$-0.435$. In what follows we use this missing input parameter $e_{0}$ for the
entropy method, considering both assumptions about $c(T)$ as $T\to 0$. We note
in passing that the uniform susceptibility $\chi(T)$ is less convenient for
seeking a large value of $p=n_{\text{cP}}/n_{\rm P}$, since it requires the
additional parameter $\chi_{0}$ if the spectrum is gapless.

##  V Results

###  V.1 Ground-state energy $e_{0}$

Table 1: Ground-state energy $e_{0}$ obtained by different authors $N{=}12$ [15] | ${-}0.454$
---|---
$N{=}12/24/36$ (present paper) | ${-}0.453\,74/{-}0.446\,33/{-}0.445\,10$
QSL [15] | ${-}0.424$
VBC [16] | ${-}0.430\,115$
EM (present paper) | ${-}0.440\ldots{-}0.435$

We begin with the discussion of the ground-state energy of the $S=1/2$
hyperkagome-lattice Heisenberg antiferromagnet. Various proposals about the
nature of the ground state, i.e., spin liquids or valence-bond crystals, yield
for $e_{0}$ the values $-0.430\ldots-0.424$, see Table 1. Exact
diagonalizations for $N=12,24,36$ yield $-0.453\,74$, $-0.446\,33$,
$-0.445\,10$, see Table 1, that, apparently, are overestimated values of the
thermodynamically large systems. As explained above, to provide consistency of
the entropy-method calculations, we have to assume for $e_{0}$ the values
$-0.440\ldots-0.435$. Yet another plausible simple approach to determine
$e_{0}$ from the high-temperature series expansion [6] is discussed in
Appendix C; it yields $e_{0}$ about $-0.448$. The determination of $e_{0}$
based on the high-temperature series expansion seems to be rather formal,
since it does not use any specific picture for the ground state. However, the
experience from other models, including exactly solvable ones and precisely
examined numerically ones, gives hints that it may yield quite reasonable
predictions [40, 43].

It is worth noting that the ground-state energy for the kagome lattice is
quite close: $-0.438\,6(5)$ [52, 53], $-0.438\,7\ldots$ [54] (i.e., about
$-0.219$ per bond), but for the pyrochlore lattice it is rather different:
$-0.490(6)$ [6], $-0.483\,1(1)$ [7], $-0.489\ldots$ [9] (i.e., about $-0.163$
per bond).

###  V.2 Thermodynamic properties

Figure 4: Entropy-method results, obtained as explained in Sec. IV, for (top)
the specific heat and (bottom) the uniform susceptibility of the hyperkagome-
lattice $S=1/2$ Heisenberg antiferromagnet. Blue curves correspond to the
gapless spectrum $c=AT^{2}$ and red ones to the gapped spectrum
$c\\!\propto\\!e^{-\Delta/T}/T^{2}$. The shaded area (light blue and light
red) represents the region of $e_{0}$ where $p$ has a maximum (see Fig. 3). We
also show $N=36$ data (yellow) and two simple Padé approximants $[7,7]$ and
$[8,8]$ for $c(T)$ ($T\geq 0.35$) and for $\chi(T)$ ($T\geq 0.25$) and color
in gray the region between them. The simple Padé approximants almost coincide
with the entropy-method curves especially in the top panel. In the case of
gapless excitations, we set $\chi_{0}=0.13$.

We pass to finite-temperature properties of the $S=1/2$ hyperkagome-lattice
Heisenberg antiferromagnet. In Fig. 4 we report the temperature dependence of
the specific heat and the uniform susceptibility obtained by the entropy-
method. The ground-state energy is determined from the analysis of $c(T)$ as
was explained in Sec. IV. Both possibilities, the gapless spectrum with
$\alpha=2$ and the gapped spectrum, were considered, see blue and red curves,
respectively. For the gapless excitations we set $\chi_{0}=0.13$ inspired by
experimental data [10] and other theoretical papers [17, 19].

The specific heat $c(T)$ besides the high-temperature peak at $T\approx 0.669$
has an additional low-temperature one at about $T\approx 0.033$ (gapless
excitations) or $T\approx 0.021$ (gapped excitations); the height of the low-
temperature peak is about two times smaller than the height of the main peak.
These features, at least at intermediate temperatures and above, are quite
similar to what is known for the kagome-lattice (and also the square-kagome-
lattice) case (a peak at $T=0.67$, a shoulder of two times smaller height at
$T=0.1\ldots 0.25$ [49, 55]), but differ from those for the pyrochlore-lattice
case, where only one maximum in $c(T)$, but no additional low-temperature
feature (peak/shoulder) was found [41, 56]. Concerning high-temperature series
expansions for $c$, they coincide for the hyperkagome and kagome lattices up
to $\beta^{5}$, but differ for the pyrochlore lattice already in terms
proportional to $\beta^{2}$.

The uniform susceptibility $\chi(T)$ behaves identically at $T$ above about
$0.5$ for gapless and gapped excitations. For lower temperatures, $\chi(T)$
has a maximum at $T\approx 0.118$ (gapless excitations) or $T\approx 0.309$
(gapped excitations) and approaches either $\chi_{0}$ (the former case) or
zero (the latter case) as the temperature goes to zero. This resembles the
maximum of $\chi(T)$ for the finite-size kagome lattices [49] and for the
infinite kagome lattice analysed by the entropy method [38]. In contrast, for
the pyrochlore lattice we have several scenarios none of which can be excluded
to date [57, 41, 56]. Concerning high-temperature series expansions for
$\chi$, they coincide for the hyperkagome- and kagome-lattice cases up to
$\beta^{6}$ but differ for the pyrochlore-lattice case already in terms
proportional to $\beta^{3}$.

A general message that can be taken from Fig. 4 is that the entropy-method and
finite-system numerics data (and even simple Padé approximants for $\chi$)
favour the assumption of a gapless spectrum. This conclusion agrees with
experiments for Na4Ir3O8.

Figure 5: Comparison with the kagome- and pyrochlore-lattice cases for (top)
the specific heat and (bottom) the uniform susceptibility. Finite-temperature
Lanczos method data for $N=36$ (hyperkagome lattice, see Fig. 2, and kagome
lattice [49]) and $N=32$ (pyrochlore lattice [41]). The extra low-temperature
peak of $c(T)$ at $T\approx 0.117$ for the pyrochlore lattice (top panel,
black curve) is a finite-size effect and is not present for $N\to\infty$ [41].

As has been mentioned above, the hyperkagome-lattice Heisenberg
antiferromagnet has similar properties with the kagome-lattice one, but
different to the pyrochlore-lattice one. In Fig. 5 we compare the finite-
temperature Lanczos method data for $c(T)$ and $\chi(T)$ for the hyperkagome-
lattice case with the kagome- and pyrochlore-lattice cases. We have to remark
here that the energy scale is different for the pyrochlore (each site has six
neighbors) and the kagome or hyperkagome (each site has four neighbors) and
one may rescale $T\to T/z$ and $\chi\to\chi z$ so that different lattices,
with $z=4$ and $z=6$, can be compared, but the conclusions below remain
unchanged. Namely, Fig. 5 illustrates a good agreement above about $T=0.25$
for the specific heat (top panel) and even for all temperatures for the
uniform susceptibility (bottom panel) for the hyperkagome- and kagome-lattice
Heisenberg antiferromagnets. In contrast, the pyrochlore-lattice Heisenberg
antiferromagnet shows different temperature profiles $c(T)$ and $\chi(T)$
(also after rescaling). Thus, we may conclude that the three-dimensional
hyperkagome lattice is closer to highly frustrated two-dimensional lattices
(kagome, square-kagome) than to the three-dimensional pyrochlore lattice.
However, it is worth noting the difference: For the kagome lattice the low-
temperature peak of $c(T)$ moves to higher temperatures with increasing $N$
[49], opposite to what is observed for the hyperkagome lattice (recall the top
panel of Fig. 2). Thus, for the kagome lattice one yields a low-temperature
shoulder of the main peak in the thermodynamic limit [58].

Before closing this section, let us comment on the relevance for experiments.
Previous theoretical papers [15, 17, 19] compare $c(T)$ or $c(T)/T$ and
$\chi(T)$ or $1/\chi(T)$ to available experimental data for the $S=1/2$
hyperkagome antiferromagnet Na4Ir3O8 [10]. These comparisons exhibit
noticeable discreapancies roughly below $J/2$ ($J$ is about 300 K for
Na4Ir3O8) and even at higher temperatures for the specific heat. The authors
attributed this disagreement to an incomplete subtraction of nonmagnetic
contribution to the experimentally measured $c(T)$ [17] and an insufficiency
of the model (1) for description of the $S=1/2$ hyperkagome antiferromagnet
Na4Ir3O8 [16, 23].

##  VI Summary

In the present paper, we consider the $S=1/2$ hyperkagome-lattice Heisenberg
antiferromagnet. Using finite-lattice calculations and high-temperature series
expansion up to 16th order [17] complemented by plausible assumptions about
low-temperature properties we have obtained the temperature dependences for
the specific heat and the uniform susceptibility. Our main findings are as
follows: We observe a two-peak profile for $c(T)$, and we do not see any
difference on whether the excitations are gapless or gapped for $\chi(T)$
above $T\approx 0.5$. As a byproduct, we obtain the ground state energy
$e_{0}$, which provides self-consistency of the entropy-method calculations.
We have found that the thermodynamics of the three-dimensional hyperkagome-
lattice Heisenberg antiferromagnet is quite similar to the two-dimensional
kagome-lattice one, but differs from that of the pyrochlore lattice.

Future work on thermodynamics may be related to application of universal and
specific tools to tackle the problem. Evidently, the $S=1/2$ hyperkagome-
lattice Heisenberg antiferromagnet can be studied by the rotation-invariant
Green’s function method for obtaining approximate thermodynamic and dynamic
quantities on an equal footing. Similar studies for the quantum kagome-lattice
and pyrochlore-lattice Heisenberg antiferomagnets were reported in Refs. [33,
34]. On the other hand, the hyperkagome-lattice Heisenberg antiferromagnet
represents a flat-band system, since the one-magnon energy spectrum has
dispersionless (flat) bands. The flat-band states may be relevant at high
fields and low temperatures and their dominant contribution to thermodynamics
can be elaborated by special methods of flat-band systems, see Refs. [59, 60].

## Acknowledgements

T. H. was supported by the fellowship of the President of Ukraine for young
scholars and by the Projects of research works of young scientists of the
National Academy of Sciences of Ukraine (project # 29-04/18-2023, Frustrated
quantum magnets at finite temperatures). O. D. thanks J. Strečka for the kind
hospitality at the MECO48 conference (Stará Lesná, Slovakia, May 22-26, 2023).
O. D. acknowledges the kind hospitality of the University of Bielefeld in
October-December of 2023 (supported by Erasmus+ and DFG). This work was
supported by the Deutsche Forschungsgemeinschaft (DFG SCHN 615/28-1 and RI
615/25-1).

##  Appendix A Finite-lattice calculations

Figure 6: Finite-size hyperkagome lattices with (left) $N=24$ and (right)
$N=36$ for finite-temperature Lanczos method studies.

In this appendix we present more details about finite-lattice calculations.
For exact diagonalizations, we use the 12-site unit cell with periodic
boundary conditions. Note that the open 12-site unit cell has sites with only
two neighbors, as, e.g., site 1, since the bonds between this site and sites
$2$ and $3$ are absent for open boundary conditions. Periodic boundary
conditions appear after superposing the sites $1$ and $1+y$, $4$ and $4+z$,
$8$ and $8-x$, $9$ and $9-x$, $10$ and $10+z$, $11$ and $11-x+y+z$, $12$ and
$12+y$ in Fig. 1. In Fig. 6 we show the lattices of 24 and 36 sites (two and
three unit cells arranged in a row along the $x$ axis) which are used for
studies with the finite-temperature Lanczos method. In Fig. 6, the sites
$1,\ldots,12$ of the first unit cell (cf. Fig. 1) are numbered as
$13,\ldots,24$ and $25,\ldots,36$ in the second and third unit cell,
respectively. Periodic boundary conditions are implied in $x$, $y$, and $z$
directions, i.e., the gray disks with identical numbers in Fig. 6 coincide.

In our finite-lattice numerics we use the conservation of the $z$-component of
the total spin as well as lattice symmetries, i.e., the Hilbert space
${\cal{H}}$ splits into orthogonal subspaces ${\cal{H}}(\gamma)$ with $\gamma$
labeling the respective symmetry. Exact and complete diagonalizations are
performed for $N=12$ using Schulenburg’s publicly available package spinpack
[61, 62].

Within the finite-temperature Lanczos method, the sum over an orthonormal
basis in the partition function $Z$ is replaced in a Monte-Carlo fashion by a
much smaller sum over $R$ random vectors where each random vector is employed
for a trace estimation [25],

$\displaystyle
Z\approx\sum_{\gamma=1}^{\Gamma}\frac{\dim({\cal{H}}(\gamma))}{R}\sum_{\nu=1}^{R}\sum_{n=1}^{N_{\rm
L}}$ $\displaystyle\exp\left(-\frac{\epsilon_{n}^{(\nu)}}{T}\right)$ (A1)
$\displaystyle\times\left|\langle n(\nu)|\nu\rangle\right|^{2}\ ,$

where $|\nu\rangle$ labels random vectors for each symmetry-related orthogonal
subspace ${\cal{H}}(\gamma)$. The exponential of the Hamiltonian $H$ in Eq.
(A1) is approximated by its spectral representation in a Krylov space spanned
by the $N_{\rm L}$ Lanczos vectors starting from the respective random vector
$|\nu\rangle$, where $|n(\nu)\rangle$ is the $n$th eigenvector of $H$ in this
Krylov space with the energy $\epsilon_{n}^{(\nu)}$. In the present study we
take $R=200$ for $N=24$ and $R=20$ for $N=36$, cf. Fig. 2.

##  Appendix B More details of the entropy-method calculations

In this appendix we provide some technical details about the entropy-method
calculations which are outlined in Sec. IV. Our approach is similar to the one
of Refs. [40, 43].

First we construct for a reference a bundle of close to diagonal Padé
approximants $[u,d](T)$ of the high-temperature series for $c(T)$ and select
those Padé approximants which are “indistinguishable” unless the temperature
becomes lower than, say, $T=0.5$. Clearly, the entropy-method profiles $c(T)$
should coincide with the bundle of simple Padé approximants above this
temperature. Next, we take the initial temperature $T_{\rm i}=0.5$ at which
the constructed simple Padé approximants begin to deviate one from another and
start to lower the temperature by $\Delta T=0.025$ making 16 steps to reach
the final temperature $T_{\rm f}=0.1$. We assume a certain value of $e_{0}$
(and will scan the ground-state energies $e_{0}$ with the step $\Delta
e_{0}=0.000\,1$), and consider both scenarios for the low-energy spectrum
(gapped or gapless) separately. This way, we obtain a set of entropy-method
curves $c(T)$ according to all considered $n_{\rm P}$ Padé approximants
$[u,d](e)$ in Eqs. (IV) or (IV). The entropy-method curves $c(T)$ coincide
with the bundle of simple Padé approximants for $T\geq T_{\rm i}$ with the
absolute accuracy below, e.g., 0.001. Now, for each temperature $T_{\rm
i},T_{\rm i}-\Delta T,\ldots,T_{\rm f}$, we find those $[u,d](e)$ that yield
$c(T)$ which deviates from the average value (arithmetic mean)
$\overline{c(T)}$ by an absolute value less than 0.001. We count the number of
such Padé approximants $n_{\text{cP}}$ and calculate $p=n_{\text{cP}}/n_{\rm
P}$.

In Sec. IV we mention that the choice of $T_{\rm i}$ is important: We took
$T_{\rm i}=0.5$ which is below the temperature of the high-temperature peak of
$c(T)$ at $T\approx 0.669$. We also like to remark that achieving large values
of the ratio $p=n_{\text{cP}}/n_{\rm P}$ should be considered with caution.
For example, if taking $T_{\rm i}=1$ (and $\Delta T=0.05$, $T_{\rm f}=0.2$),
we obtain values of $p$ quite close to 1 (above 0.69 and up to 0.92). However,
by comparing the related curves of $c(T)$ and $\chi(T)$ with simple Padé
approximants one notices a larger discrepancies at $T=0.5$ than in Fig. 4.
Therefore, since simple Padé approximants converge well above $T\approx 0.5$
we conclude that the choice of $T_{\rm i}=0.5$ is preferable.

##  Appendix C Estimation of $e_{0}$ from high-temperature series expansion

The high-temperature series expansion can be used for the estimation of the
ground-state energy of the $S=1/2$ hyperkagome-lattice Heisenberg
antiferromagnet following the lines of the Supplementary Material of Ref. [6].
Below we repeat these arguments for the $S=1/2$ hyperkagome-lattice Heisenberg
antiferromagnet.

Figure 7: Estimate of the ground-state energy $e_{0}$ from the high-
temperature series expansion up to 16th order (black curve). We also plot
simple Padé approximants for the internal energy $e=(-\partial\ln
Z/\partial\beta)/N$ and show the results for $e_{0}$ of Refs. [15, 16] along
with the entropy-method and finite $N$ data for $e_{0}$, see Table 1.

Since $c(T)=\partial e(T)/\partial T$, the ground-state energy $e_{0}$ is
given by $e_{0}=-\int_{0}^{\infty}{\rm d}Tc(T)$. Moreover, its upper bound is
given by $e_{0}^{*}=-\int_{T^{*}}^{\infty}{\rm d}Tc(T)$, where $T^{*}>0$ is a
temperature above which simple Padé approximants $[u,d](T)$ of the high-
temperature series expansion of $c(T)$ show a reliable convergence. Replacing
$c(T)$ by $[8,8](T)$ and assuming $T^{*}=0.35$ we obtain for the upper bound
$e_{0}^{*}\approx-0.401$, see Fig. 7.

Furthermore, consider the $e(T)$ as it is given by the $[7,8](T)$ Padé
approximant and fit $e(T)$ by a quadratic polynomial $e_{0}+e_{1}T+e_{2}T^{2}$
in the range between $T=0.4$ and $T=0.5$ or between $T=0.4$ and $T=0.55$. This
way we get an estimate for the ground-state energy $e_{0}\approx-0.448$, see
Fig. 7. If the taken range is between $T=0.35$ and $T=0.55$, we arrive at only
about 0.1% lower energy $e_{0}$. Just this latter quadratic fit is shown by
the black curve in Fig. 7. In Fig. 7 we also compare different predictions for
$e_{0}$.

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