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# Quantum critical phenomena in a spin-1/2 frustrated square lattice with
spatial anisotropy

H. Yamaguchi1, Y. Iwasaki2, Y. Kono3, T. Okubo4, S. Miyamoto1, Y. Hosokoshi1,
A. Matsuo5, T. Sakakibara5, T. Kida6, and M. Hagiwara6 1Department of Physical
Science, Osaka Prefecture University, Osaka 599-8531, Japan
2Center for Advanced High Magnetic Field Science (AHMF), Graduate School of
Science, Osaka University, Osaka 560-0043, Japan
3Department of Physics, Chuo University, Tokyo 112-8551, Japan
4Department of Physics, the University of Tokyo, Tokyo 113-0033, Japan
5Institute for Solid State Physics, the University of Tokyo, Chiba 277-8581,
Japan
6Center for Advanced High Magnetic Field Science (AHMF), Graduate School of
Science, Osaka University, Osaka 560-0043, Japan

###### Abstract

We present a model compound with a spin-1/2 spatially anisotropic frustrated
square lattice, in which three antiferromagnetic interactions and one
ferromagnetic interaction are competing. We observe an unconventional gradual
increase in the low-temperature magnetization curve reminiscent of the quantum
critical behavior between gapped and gapless phases. In addition, the specific
heat and electron spin resonance signals indicate one-dimensional
characteristics. These results demonstrate quantum critical behavior
associated with one-dimensionalization caused by frustrated interactions in
the spin-1/2 spatially anisotropic square lattice.

###### pacs:

75.10.Jm,

††preprint: APS/123-QED

One current key area in condensed matter physics is the search for quantum
phenomena associated with magnetic frustration, in which neighboring spins
interact via competing exchange interactions that cannot be satisfied
simultaneously. The archetypal example of frustrated magnets is a triangular
lattice antiferromagnet. The quantum spin liquid picture on the spin-1/2
triangular lattice proposed by Anderson in 1973 has stimulated extensive
research on ground states caused by frustration RVB . However, subsequent
numerical studies have established that its ground state has an
antiferromagnetic (AF) order 120_1 ; 120_2 . A square lattice can also be made
to exhibit frustration via the coexistence of AF and ferromagnetic
interactions. Such a system has been theoretically studied in association with
Josephson-junction arrays FFSL_1 ; FFSL_2 ; FFSL_3 ; however, there have been
no reports of ideal model compounds. Recently, our spin arrangement design
using organic radicals enabled the formation of a spin-1/2 frustrated square
lattice with spatial anisotropy. Such compounds demonstrate that the strong
quantum fluctuations and frustrated plaquettes stabilize the quantum states
TCNQ ; PF6 . The spin model focused on in the present study is one example of
such a spin-1/2 spatially anisotropic frustrated square lattice, where three
AF interactions and one ferromagnetic interaction form a plaquette, yielding
spatial anisotropy.

Frustrated systems have a delicate energy balance; accordingly, additional
factors can easily change their ground states. One such factor is spatial
anisotropy, which is equivalent to lattice distortion. The spin-1/2 Heisenberg
triangular lattice case has been extensively studied. An anisotropic
triangular lattice can be viewed as 1D chains with frustrated interactions
between them. The frustration effectively decouples the chains, creating a 1D
chain system. As a consequence, a gapless disordered state analogous to a
Tomonaga-Luttinger liquid (TLL) in a 1D system becomes stable in a spin-1/2
anisotropic triangular lattice Cs2CuCl4_exp1 ; Cs2CuCl4_exp2 ; spin1/2_1 ;
spin1/2_2 ; spin1/2_3 ; spin1/2_4 . Here we show that a similar frustration-
induced dimensional reduction also can occur in certain spin-1/2 anisotropic
square lattices as we describe below.

In this letter, we present a model compound with a spin-1/2 spatially
anisotropic frustrated square lattice. We successfully synthesized single
crystals of the verdazyl-based salt [$m$-MePy-V-($p$-F)2]SbF6. Our molecular
orbital (MO) calculations indicate the formation of a spatially anisotropic
square lattice composed of three AF interactions and one ferromagnetic
interaction causing frustration. The low-temperature magnetization curve
exhibits an unconventional gradual increase reminiscent of the quantum
critical behavior between gapped and gapless phases. The specific heat and the
electron spin resonance (ESR) signals indicate gapless 1D characteristics.
These results demonstrate the quantum critical behavior between gapped
1D-based disordered and two-dimensional (2D) magnetic ordered phases.

We prepared $m$-MePy-V-($p$-F)2 using a conventional procedure gosei  and
synthesized [$m$-MePy-V-($p$-F)2]SbF6 using a reported procedure for salts
with similar chemical structures PF6 ; 3Dhoneycomb . Recrystallization using
acetonitrile yielded dark-brown [$m$-MePy-V-($p$-F)2]SbF6 crystals. The
crystal structure was determined from intensity data obtained using a Rigaku
AFC-8R Mercury CCD RA-Micro7 diffractometer with a Japan Thermal Engineering
XR-HR10K. High-field magnetization measurements were made using a
nondestructive pulse magnet under pulsed magnetic fields of up to
approximately 50 T. The magnetization curve at 80 mK was measured using a
capacitive Faraday magnetometer in a dilution refrigerator. The magnetic
susceptibility was measured using a commercial SQUID magnetometer (MPMS-XL,
Quantum Design). The experimental result was corrected considering the
diamagnetic contributions calculated via Pascal’s method. The specific heat
was measured using a commercial calorimeter (PPMS, Quantum Design) through a
thermal relaxation method down to approximately 0.7 K. The ESR measurements
were performed using a vector network analyzer (ABmm), superconducting magnet
(Oxford Instruments), and a laboratory-built cylindrical cavity. Considering
the isotropic nature of organic radical systems, all the experiments were
performed using small randomly oriented single crystals.

Figure 1(a) shows the molecular structure of [$m$-MePy-V-($p$-F)2]SbF6, which
has an $S$ = 1/2 within a molecule. No indications of structural phase
transitions have been observed down to 25 K supple1 . Because this study
focuses on low-temperature magnetic properties, we examined the low-
temperature crystallographic data at 25 K. The crystallographic parameters at
25 K are as follows: triclinic, space group $P\bar{1}$, $a$ = 7.134(4)
$\rm{\AA}$, $b$ = 11.302(6) $\rm{\AA}$, $c$ = 13.902(7) $\rm{\AA}$, $\alpha$ =
68.83(2) ∘, $\beta$ = 82.81(3) ∘, $\gamma$ = 85.55(3) ∘, $V$ = 1036.4
$\rm{\AA}^{3}$, and $Z$ = 2. We performed MO calculations MOcal  to evaluate
the exchange interaction between the $S$ = 1/2 spins on the radicals and found
four types of dominant interactions supple1 , as shown in Fig. 1(b). They are
identified as $J_{1}/k_{\rm{B}}$ = $54.3$ K, $J_{2}/k_{\rm{B}}$ = 21.1 K,
$J_{3}/k_{\rm{B}}$ = $-20.8$ K, and $J_{4}/k_{\rm{B}}$ = 18.0 K, which are
defined in the Heisenberg spin Hamiltonian given by
$\mathcal{H}=J_{n}{\sum_{\langle
i,j\rangle}}\textbf{{{S}}}_{i}{\cdot}\textbf{{{S}}}_{j}$, where $\sum_{\langle
i,j\rangle}$ denotes the sum over the neighboring spin pairs. As a
consequence, $S$ = 1/2 spins form a spatially anisotropic square lattice in
the $ab$ plane, in which three AF interactions and one ferromagnetic
interaction cause a frustration, as shown in Fig. 1(c). The SbF6 anions act as
spacers between the 2D planes and enhance the 2D characteristics of the
present model supple1 .

Figure 2(a) shows the temperature dependence of the magnetic susceptibility (
$\chi$ = $M/H$) at 0.1 T. Above 150 K, it follows the Curie-Weiss law, and the
Weiss temperature is estimated to be $\theta_{\rm{W}}$ = $-15.2$(5) K.
Considering a mean-field approximation with $\theta_{\rm{W}}$, we obtain
($J_{4}$+$J_{2}$+$J_{3}$+$J_{4}$)/$k_{\rm{B}}$ $\sim$ 61 K, which is close to
evaluated value using the MO calculation. A broad peak appears at
approximately 30 K, which indicates an AF short-range order in the 2D lattice.
The upturn below approximately 10 K is caused by slight paramagnetic
impurities associated with lattice defects; this is commonly observed in
verdazyl-based compounds PRL ; Zn ; TCNQ_JPSJ . Assuming conventional
paramagnetic behavior $C_{\rm{imp}}/T$ and gapless excitations, where
$C_{\rm{imp}}$ is the Curie constant of the impurities, the paramagnetic
impurities are evaluated as accounting for approximately 2.7 $\%$ of all
spins, which is close to the percentage evaluated in a verdazyl-based salt
with a similar molecular structure 3Dhoneycomb .

The magnetization curve exhibits an unusually convex behavior up to
approximately 20 T, as shown in Fig. 2(b). In higher fields, the magnetization
increases almost linearly. Note that this convex behavior becomes more
prominent at a lower temperature of 80 mK, which cannot be explained by the
contributions of the paramagnetic impurities. A slight paramagnetic
contribution is found in the low-field region of the magnetization at 80 mK,
in which paramagnetic impurities are almost fully-polarized above 0.5 T, as
shown in Fig. 2(b) The trend of the gradual increase is completely different
from the concave shape of the magnetization curve observed in general 2D
quantum spin systems a235Cl3V ; tanaka ; kagome . The overall magnetization
behavior is reminiscent of the quantum critical behavior with a square root
dependence in the region between gapped singlet and gapless magnetic phases.
In 1D gapped spin systems, the energy gap disappears when a magnetic field
applied, and a quantum phase transition to the TLL phase appears accompanied
by a singular square root behavior of the magnetization curve TCNQ_JPSJ ; kono
; okunishi1 ; okunishi2 .

The temperature dependence of the specific heat $C_{\rm{p}}$ is shown in Fig.
3(a). There is no peak showing a phase transition to a magnetic order up to 14
T. In the low-temperature regions below approximately 2 K, where magnetic
contributions are expected to be dominant, as observed in other verdazyl-based
compounds, $T$-linear behavior ($C_{\rm{p}}/T$ = const.) is observed, as shown
in Fig. 3(b). This $T$-linear behavior is robust against an applied magnetic
field and appears even at 14 T. Considering the contributions of the low-
energy excitations to the specific heat, this $T$-linear behavior indicates
the existence of gapless linear dispersions in a 1D AF spin system. Such 1D
characteristics in the dispersion relation are observed in $S$ = 1/2 spatially
anisotropic triangular lattices and are confirmed to originate from one-
dimensionalization caused by the frustration effect Cs2CuCl4_exp1 ;
Cs2CuCl4_exp2 . Accordingly, the observed 1D characteristics in the present
spatially anisotropic square lattice are also expected to originate from one-
dimensionalization owing to frustrated interactions. Figure 3(c) shows the
entropy $S_{\rm{p}}$ obtained via the integration of $C_{\rm{p}}/T$. As shown
in the following ESR results, the short-range correlation develops remarkably
below approximately 15 K. Therefore, we evaluated the magnetic entropy
$S_{\rm{m}}$ by subtracting the lattice contribution such that $S_{\rm{m}}$
approaches a constant value above approximately 15 K. We assumed the lattice
contribution of the specific heat given by $\alpha$$T^{3}$, which corresponds
to Debye’s $T^{3}$ law for low-temperature regions and has been confirmed to
be effective for verdazyl-based compounds. We obtained $S_{\rm{m}}$ with an
asymptotic behavior toward 15 K using $\alpha$ = 0.013, as shown in Fig. 3(c).
The value of $\alpha$ is consistent with those used for other verdazyl-based
compounds Zn ; a235Cl3V ; pBrV ; b26Cl2V ; 3Br4FV . Considering that the
entropy change below 15 K is close to the total magnetic entropy of 5.76
($R$ln2), the magnetic entropy associated with the present $S$ = 1/2 system is
almost entirely consumed in the experimental low-temperature regions.

Figure 4(a) shows the temperature dependence of the ESR absorption spectra at
34.13 GHz. Sharp resonance signals characteristic of organic radical systems
are observed, and the changes in the resonance spectra become noticeable with
decreasing temperature. The resonance fields are converted into $g$ $\sim$ 2.0
at all temperatures, which confirms the isotropic nature of the present
system. The temperature dependences of the resonance shift
$\Delta$$H_{\rm{res}}$ and the absorption linewidth $\Delta$$H_{\rm{1/2}}$ are
shown in Figs. 4(b) and 4(c), respectively. The resonance shift
$\Delta$$H_{\rm{res}}$ changes dramatically below approximately 15 K, which
indicates that the change in the internal field accompanied by the development
of short-range AF correlations becomes remarkable below 15 K. Correspondingly,
the absorption linewidth $\Delta$$H_{\rm{1/2}}$ exhibits a broadening with
decreasing temperature. We fitted the ESR linewidth assuming
$\Delta$$H_{\rm{1/2}}$ $\propto$ $T^{-p}$ and obtained $p$ = 0.90(6). The low-
temperature parts of $\Delta$$H_{\rm{1/2}}$ in weakly coupled 1D spin chains
are described by a $T^{-1}$ broadening furuya ; furuya_rei1 ; furuya_rei2 .
Considering that the evaluated $p$ is close to 1.0, the development of
correlations in the present system has 1D characteristics.

Here, we consider the ground state of the present system. The experimental
results exhibited neither an energy gap nor 2D characteristics, while gapless
1D characteristics were observed. Furthermore, the nonmonotonic increase in
the magnetization curve suggests that the ground state is in the vicinity of a
quantum critical point between gapped and gapless phases. Considering the
observed 1D characteristics, we assumed two types of possible 1D correlations:
a $J_{1}$-$J_{2}$ AF alternating chain and a $J_{1}$-$J_{3}$ AF-ferromagnetic
alternating chain. In either case, a singlet ground state formed by the
stronger AF interaction $J_{1}$ is separated from a low-lying triplet by an
energy gap, which supports the expected quantum critical region near the
gapped phase. Furthermore, in the later case, the ground state is considered
to be equivalent to a Haldane state in the spin-1 AF chain. Assuming such a
situation, this system can be mapped to a spin-1 spatially anisotropic
triangular lattice, where the spins coupled by the ferromagnetic $J_{3}$ form
an effective spin-1, as shown in Fig. 1(c). In spin-1 triangular lattice, a
dimensional reduction associated with the spatial anisotropy has been
suggested in analogy with the spin-1/2 case spin1_5 ; spin1_7 ; spin1_6 ;
spin1_1 . The phase boundaries have been evaluated in several numerical
studies as a function of $J^{{}^{\prime}}/J$, where 1D chains are formed
through $J$ and coupled via the frustrated zigzag path $J^{{}^{\prime}}$. Even
though the critical $J^{{}^{\prime}}/J$ is still controversial, the phase
boundary is predicted to be within the range of 0.3 ${\textless}$
$J^{{}^{\prime}}/J$ ${\textless}$ 0.8 spin1_5 ; spin1_7 ; spin1_6 ; spin1_1 .
In the present mapped triangular lattice, considering that $J_{2}$ and $J_{4}$
are close values, the ratio $J_{2}/J_{1}$ corresponds to $J^{{}^{\prime}}/J$.
The values of the interactions evaluated from the MO calculations give
$J_{2}/J_{1}$ $\sim$ 0.4, which is indeed within the range of the predicted
quantum critical point. Even though it is not clear in which case the actual
1D correlation is, the present work demonstrates the quantum critical behavior
between the 1D gapped and 2D gapless magnetic phases, which indicates one-
dimensionalization caused by the frustrated interactions in the spin-1/2
spatially anisotropic square lattice

In summary, we succeeded in synthesizing single crystals of the verdazyl-based
salt [$m$-MePy-V-($p$-F)2]SbF6. The MO calculations indicated that the
$m$-MePy-V-($p$-F)2 molecules formed an $S$ = 1/2 spatially anisotropic
frustrated square lattice, in which three AF interactions and one
ferromagnetic interaction are competing. The low-temperature magnetization
curve exhibited an unconventional gradual increase reminiscent of the quantum
critical behavior between gapped and gapless phases. Furthermore, the specific
heat and the ESR signals indicated gapless 1D characteristics. These
properties can be understood as the quantum critical behavior between the
gapped 1D-based disordered and gapless 2D magnetic ordered phases. Therefore,
we demonstrated that the model compound of the $S$=1/2 spatially anisotropic
frustrated square lattice, [$m$-MePy-V-($p$-F)2]SbF6, realizes quantum
critical behavior associated with one-dimensionalization caused by frustrated
interactions. The present material provides insights into the ground states
realized in frustrated square lattices, which have not been fully explored,
and will stimulate further studies on quantum phenomena associated with
magnetic frustration in condensed matter physics.

Figure 1: (color online) (c) Molecular structure of [$m$-MePy-V-($p$-F)2]SbF6.
(d) Crystal structure forming the spin model in the $ab$ plane. Hydrogen atoms
are omitted for clarity. (c) Spin-1/2 spatially anisotropic frustrated square
lattice composed of three AF interactions, $J_{1}$, $J_{2}$, and $J_{4}$, and
one ferromagnetic interaction $J_{3}$ in [$m$-MePy-V-($p$-F)2]SbF6. The mapped
model corresponds to a spin-1 spatially anisotropic triangular lattice
composed of the spins dimerized by $J_{3}$. Figure 2: (color online) (a)
Temperature dependence of the magnetic susceptibility ($\chi$ = $M/H$) of
[$m$-MePy-V-($p$-F)2]SbF6 at 0.1 T . (b) Magnetization curves of
[$m$-MePy-V-($p$-F)2]SbF6 at 80 mK and 1.3 K. The arrow shows the contribution
of the paramagnetic impurities. Figure 3: (color online) (a) Temperature
dependence of the specific heat $C_{\rm{p}}$ of [$m$-MePy-V-($p$-F)2]SbF6 at 0
and 14 T. (c) Low-temperature region of $C_{\rm{p}}/T$. (b) Entropy
$S_{\rm{p}}$ obtained via the integration of $C_{\rm{p}}/T$ at zero-field.
$S_{\rm{m}}$ is the magnetic entropy evaluated by subtracting the lattice
contribution.  Figure 4: (color online) (a) Temperature dependence of the ESR
absorption spectra of [$m$-MePy-V-($p$-F)2]SbF6 at 34.13 GHz. Temperature
dependence of the (b) resonance field shift and (c) linewidth (full width at
half maximum) obtained from the ESR spectra. The solid line indicates the
$T^{-p}$ fit.

###### Acknowledgements.

We thank S. C. Furuya for valuable discussion. This research was partly
supported by Grant for Basic Science Research Projects from KAKENHI (No.
19J01004 and No. 19H04550), the Murata Science Foundation, Kyoto Technoscience
Center, SEI Group CSR Foundation, and Nanotechnology Platform Program
(Molecule and Material Synthesis) of the Ministry of Education, Culture,
Sports, Science and Technology (MEXT), Japan. A part of this work was
performed at the Center for Advanced High Magnetic Field Science in Osaka
University under the interuniversity cooperative research program of the
joint-research program of ISSP, the University of Tokyo.

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