We report measurements of the specific heat and magnetization of single crystal samples of the spin-1/2 kagome compound ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$ (herbertsmithite), a promising quantum spin-liquid candidate, in high magnetic fields and at low temperatures. The magnetization was measured up to $μ_{0}H$ = 55 T at $T$ = 0.4 K, showing a saturation of the weakly interacting impurity moments in fields above $\sim10$ T. The specific heat was measured down to $T < 0.4$ K in magnetic fields up to 18 T, revealing $T$-linear and $T$-squared contributions. The $T$-linear contribution is surprisingly large and indicates the presence of gapless excitations in large applied fields. These results further highlight the unusual excitation spectrum of the spin liquid ground state of herbertsmithite.
Strongly correlated Fermi systems are among the most intriguing and fundamental systems in physics, whose realization in some compounds is still to be discovered. We show that herbertsmithite ZnCu_3(OH)_6Cl_2 can be viewed as a strongly correlated Fermi system whose low temperature thermodynamic in magnetic fields is defined by a quantum critical spin liquid. Our calculations of its thermodynamic properties are in good agreement with recent experimental facts and allow us to reveal their scaling behavior which strongly resembles that observed in HF metals and 2D 3He.
We report on temperature dependence of the infrared reflectivity spectra of a single crystalline herbertsmithite in two polarizations --- parallel and perpendicular to the kagome plane of Cu atoms. We observe anomalous broadening of the low frequency phonons possibly caused by fluctuations in the exotic dynamical magnetic order of the spin liquid.
Based on experimental data and our theoretical analysis, we provide a strategy for unambiguous establishing of gapless quantum spin liquid state (QSL) in herbertsmithite and other materials. To clarify the nature of QSL, we recommend measurements of heat transport, low-energy inelastic neutron scattering and optical conductivity under the application of external magnetic field at low temperatures. We also suggest that artificially introduced inhomogeneity into $\rm ZnCu_{3}(OH)_6Cl_2$ can stabilize QSL, and serves as a test elucidating the contribution coming from impurities. We predict the re…
We consider the polarization function P^{exp}_Z(t) measured by the muon-spin relaxation (muSR) technique for the SrCr8Ga4O19 and ZnCu3(OH)6Cl2 spin-liquid systems. We show the functional form of P^{exp}_Z(t) to imply that, in the temperature range of order 0.1 K, the spectral-weight function F(ω) of the magnetic correlations scales with 1/|ω|^{1 -x} (0 < x < 1) in the energy range of one microelectronvolt, i.e. \hbar ω~ 1 μeV. We derive the parameters involved in F(ω) from fits to available experimental data. Inelastic neutron scattering data probing F(ω) in the millielectronvolt energy range …
Recently Han et al. have provided an analysis of the observed behavior of $\rm ZnCu_{3}(OH)_6Cl_2$ Herbertsmithite based on a separation of the contributions to its thermodynamic properties due to impurities from those due to the kagome lattice. The authors developed an impurity model to account for the experimental data and claimed that it is compatible with the presence of a small spin gap in the kagome layers. We argue that the model they advocate is problematic, conflicting with the intrinsic properties of $\rm ZnCu_{3}(OH)_6Cl_2$ as observed and explained in recent experimental and theore…
Spin liquids are exotic phases of quantum matter challenging Landau's paradigm of symmetry-breaking phase transitions. Despite strong exchange interactions, spins do not order or freeze down to zero temperature. While well-established for 1D quantum antiferromagnets, in higher dimension where quantum fluctuations are less acute, realizing and understanding such states represent major issues, both theoretical and experimental. In this respect the simplest nearest-neighbor Heisenberg antiferromagnet Hamiltonian on the highly frustrated kagome lattice has proven to be a fascinating and inspiring …
Herbertsmithite, ZnCu3(OH)6Cl2, is the prototype candidate for a spin liquid behavior on a geometrically perfect kagome lattice. Its discovery and the absence of any evidence for spin-freezing down to the lowest probed temperature to-date enable one to explore the properties of kagome-related physics in an unprecedented temperature range. We review its properties and discuss some open issues. A tentative comparison to models is also performed.
Quantum spin liquid (QSL) is a novel state of matter with exotic excitations and was theoretically predicted to be realized most possibly in an S=1/2 kagome antiferromagnet. Experimentally searching for the candidate materials is a big challenge in condensed matter physics and only two such candidates were reported so far. Here we report the successful synthesis of a new spin-1/2 kagome antiferromagnet ZnCu3(OH)6SO4. No magnetic ordering is observed down to 50 mK, despite a moderately high Weiss temperature of θW ~ -79 K. It strongly suggests that the material is a new QSL candidate. Most inte…
We report thermodynamic measurements of the S=1/2 kagome lattice antiferromagnet ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$, a promising candidate system with a spin-liquid ground state. Using single crystal samples, the magnetic susceptibility both perpendicular and parallel to the kagome plane has been measured. A small, temperature-dependent anisotropy has been observed, where $χ_{z}/ χ_{p} > 1$ at high temperatures and $χ_{z}/ χ_{p} < 1$ at low temperatures. Fits of the high-temperature data to a Curie-Weiss model also reveal an anisotropy. By comparing with theoretical calculations, the presence of a s…
We analyze recent heat capacity measurements in herbertsmithite $\rm ZnCu_3(OH)_6Cl_2$ single crystal samples subjected to strong magnetic fields. We show that the temperature dependence of specific heat $C_{mag}$ formed by quantum spin liquid at different magnetic fields $B$ resembles the electronic heat capacity $C_{el}$ of the HF metal $\rm YbRh_2Si_2$. We demonstrate that the spinon effective mass $M^*_{mag}\propto C_{mag}/T$ exhibits a scaling behavior like that of $C_{el}/T$. We also show that the recent measurements of $C_{mag}$ are compatible with those obtained on powder samples. Thes…
We find that the best RVB state of the spin-$\frac{1}{2}$ Kagome antiferromagnetic Heisenberg model(spin-$\frac{1}{2}$ KAFH) is described by a $Z_{2}$ gapped mean field ansatz, which hosts a mean field spinon dispersion very different from that of the widely studied $U(1)$ Dirac spin liquid state. However, we find that the physical spin fluctuation spectrum calculated from the Gutzwiller projected RPA(GRPA) theory above such an RVB state is actually gapless and is almost identical to that above the $U(1)$ Dirac spin liquid state. We find that such a peculiar behavior can be attributed to the u…
We argue that neutron spectra in a short-range ordered Valence Bond state is dominated by two-spinon states localized in small spatial regions such as a pinwheel. These excitations lead to angle averaged dynamic structure factor that is spread over a wide frequency range up to about $2.5 J$, whereas its wavevector dependence at all frequencies remains very close to that of isolated dimers. These results are in excellent agreement with recent Neutron scattering data in the Herbertsmithite materials ZnCu$_3$(OH)$_6$Cl$_2$.
We analyze measurements of $C_{\rm mag}/T$, specific heat $C_{\rm mag}$ divided by temperature $T$, of the recently observed ultra spin liquid. The measurements are carried out in magnetic fields on the triangular lattice compound $\rm Lu_3Cu_2Sb_3O_{14}$. We show that the obtained heat capacity $C_{\rm mag}/T$ formed by ultra spin liquid as a function of temperature $T$ versus magnetic field $B$ behaves very similar to the electronic specific heat $C_{el}/T$ of the heavy fermion (HF) metal $\rm YbRh_2Si_2$ and that of the quantum magnet $\rm ZnCu_3(OH)_6Cl_2$. We further demonstrate that the …
The emergent behavior of spin liquids that are born out of geometrical frustration makes them an intriguing state of matter. We show that in the quantum kagome antiferromagnet ZnCu$_3$(OH)$_6$SO$_4$ several different correlated, yet fluctuating states exist. By combining complementary local-probe techniques with neutron scattering, we discover a crossover from a critical regime into a gapless spin-liquid phase with decreasing temperature. An additional unconventional instability of the latter phase leads to a second, distinct spin-liquid state that is stabilized at the lowest temperatures. We …
Arguably the most favorable situation for spins to enter the long-sought quantum spin liquid (QSL) state is when they sit on a kagome lattice. No consensus has been reached in theory regarding the true ground state of this promising platform. The experimental efforts, relying mostly on one archetypal material ZnCu$_3$(OH)$_6$Cl$_2$, have also led to diverse possibilities. Apart from subtle interactions in the Hamiltonian, there is the additional degree of complexity associated with disorder in the real material ZnCu$_3$(OH)$_6$Cl$_2$ that haunts most experimental probes. Here we resort to heat…
Neutron scattering measurements have been performed on deuterated powder samples of claringbullite and Zn-doped claringbullite (Zn$_x$Cu$_{4-x}$(OD)$_{6}$FCl). At low temperatures, claringbullite Cu$_4$(OD)$_{6}$FCl forms a distorted pyrochlore lattice with long-range magnetic order and spin-wave-like magnetic excitations. Partial Zn doping leads to the nominal ZnCu$_3$(OD)$_{6}$FCl compound, a geometrically frustrated spin-1/2 kagome antiferromagnet that shows no transition to magnetic order down to 1.5 K. The magnetic excitations form a gapless continuum, a signature of fractional excitation…
In our review we focus on the quantum spin liquid, defining the thermodynamic, transport and relaxation properties of geometrically frustrated magnets (insulators) represented by herbertsmithite $\rm ZnCu_{3}(OH)_6Cl_2$.
Recent neutron scattering experiments on the spin-1/2 kagome lattice antiferromagnet ZnCu3(OH)6Cl2 (Herbertsmithite) provide the first evidence of fractionalized excitations in a quantum spin liquid state in two spatial dimensions. In contrast to existing theoretical models of spin liquids, the measured dynamic structure factor reveals an excitation continuum which is remarkably flat as a function of frequency and has almost no momentum dependence along several high-symmetry directions. Here we show that many experimentally observed features can be explained by the presence of topological viso…
Herbertsmithite, ZnCu$_3$(OH)$_6$Cl$_2$, is a two dimensional kagomé lattice realization of a spin liquid, with evidence for fractionalized excitations and a gapped ground state. Such a quantum spin liquid has been proposed to underlie high temperature superconductivity and is predicted to produce a wealth of new states, including a Dirac metal at $1/3$rd electron doping. Here we report the topochemical synthesis of electron-doped ZnLi$_x$Cu$_3$(OH)$_6$Cl$_2$ from $x$ = 0 to $x$ = 1.8 ($3/5$th per Cu$^{2+}$). Contrary to expectations, no metallicity or superconductivity is induced. Instead, we…
We report 17O NMR measurements in the S=1/2 Cu2+ kagome antiferromagnet Herbertsmithite ZnCu3(OH)6Cl2 down to 45mK in magnetic fields ranging from 2T to 12T. While Herbertsmithite displays a gapless spin-liquid behavior in zero field, we uncover an instability toward a spin-solid phase at sub-kelvin temperature induced by an applied magnetic field. The latter phase shows largely suppressed moments $\lesssim 0.1\muB$ and gapped excitations. The H-T phase diagram suggests the existence of a quantum critical point at the small but finite magnetic field mu0 Hc=1.55(25)T. We discuss this finding in…
We address the problem of a lightly doped spin-liquid through a large-scale density-matrix renormalization group (DMRG) study of the $t$-$J$ model on a Kagome lattice with a small but non-zero concentration, $δ$, of doped holes. It is now widely accepted that the undoped ($δ=0$) spin 1/2 Heisenberg antiferromagnet has a spin-liquid groundstate. Theoretical arguments have been presented that light doping of such a spin-liquid could give rise to a high temperature superconductor or an exotic topological Fermi liquid metal (FL$^\ast$). Instead, we infer that the doped holes form an insulating cha…
The ground state of the quantum kagome antiferromagnet Zn-brochantite, ZnCu$_3$(OH)$_6$SO$_4$, which is one of only a few known spin-liquid (SL) realizations in two or three dimensions, has been described as a gapless SL with a spinon Fermi surface. Employing nuclear magnetic resonance in a broad magnetic-field range down to millikelvin temperatures, we show that in applied magnetic fields this enigmatic state is intrinsically unstable against a SL with a full or a partial gap. A similar instability of the gapless Fermi-surface SL was previously encountered in an organic triangular-lattice ant…
Strongly correlated Fermi systems are among the most intriguing and fundamental systems in physics. We show that the herbertsmithite ZnCu3(OH)6Cl2 can be viewed as a new type of strongly correlated electrical insulator that possesses properties of heavy-fermion metals with one exception: it resists the flow of electric charge. We demonstrate that herbertsmithite's low temperature properties are defined by a strongly correlated quantum spin liquid made with such hypothetic particles as fermionic spinons which carry spin 1/2 and no charge. Our calculations of its thermodynamic and relaxation pro…
New physics can emerge in magnetic materials where quantum fluctuations are enhanced due to reduced dimensionality and strong frustration. One long sought example is the resonating-valence-bond (RVB) state, where atomic magnetic moments are strongly correlated but do not order or freeze even in the limit of T -> 0. The RVB ground state does not break conventional symmetries, such as lattice translation or spin-rotation. The realization of such a quantum spin liquid in two-dimensions would represent a new state of matter. It is believed that spin liquid physics plays a role in the phenomenon of…
Quantum spin liquids attract great interest due to their exceptional magnetic properties characterized by the absence of long-range order down to low temperatures despite the strong magnetic interaction. Commonly, these compounds are strongly correlated electron systems, and their electrodynamic response is governed by the Mott gap in the excitation spectrum. Here we summarize and discuss the optical properties of several two-dimensional quantum spin liquid candidates. First we consider the inorganic material Herbertsmithite ZnCu$_3$(OH)$_6$Cl$_2$ and related compounds, which crystallize in a …
The kagome Heisenberg antiferromagnet formed by frustrated spins arranged in a lattice of corner-sharing triangles is a prime candidate for hosting a quantum spin liquid (QSL) ground state consisting of entangled spin singlets. But the existence of various competing states makes a convincing theoretical prediction of the QSL ground state difficult, calling for experimental clues from model materials. The kagome lattice materials Zn-barlowite ZnCu$_{3}$(OD)$_{6}$FBr and herbertsmithite ZnCu$_{3}$(OD)$_{6}$Cl$_2$ do not exhibit long range order, and they are considered the best realizations of t…
Low energy inelastic neutron scattering on single crystals of the kagome spin liquid compound ZnCu3(OD)6Cl2 (Herbertsmithite) reveals antiferromagnetic correlations between impurity spins for energy transfers E < 0.8 meV (~J/20). The momentum dependence differs significantly from higher energy scattering which arises from the intrinsic kagome spins. The low energy fluctuations are characterized by diffuse scattering near wavevectors (1 0 0) and (0 0 3/2), which is consistent with antiferromagnetic correlations between pairs of nearest neighbor Cu impurities on adjacent triangular (Zn) interlay…
Impurities, which are inherently present in any real material, may play an important role in the magnetism of frustrated spin systems with spin-liquid ground states. We address the impurity issue in quantum kagome antiferromagnets by investigating ZnCu$_3$(OH)$_6$SO$_4$ (Zn-brochantite) by means of muon spin spectroscopy. We show that muons couple to the impurity magnetism, originating from Cu-Zn intersite disorder, and that the impurities are highly correlated with the kagome spins, allowing us to probe the intrinsic kagome physics via a Kondo-like effect. The low-temperature plateau in local…
The spin-1/2 kagomé Heisenberg antiferromagnet is generally accepted as one of the most promising two-dimensional models to realize a quantum spin liquid state. Previous experimental efforts were almost exclusively on only one archetypal material, the herbertsmithite ZnCu$_3$(OH)$_6$Cl$_2$, which unfortunately suffers from the notorious orphan spins problem caused by magnetic disorders. Here we turn to YCu$_3$(OH)$_{6.5}$Br$_{2.5}$, recently recognized as another host of a globally undistorted kagomé Cu$^{2+}$ lattice free from the orphan spins, thus a more feasible system for studying the int…
We present a Raman spectroscopic investigation of the Herbertsmithite ZnCu3(OH)6Cl2, the first realization of a Heisenberg s=1/2 antiferromagnet on a perfect kagome lattice. The magnetic excitation spectrum of this compound is dominated by two components, a high temperature quasi elastic signal and a low temperature, broad maximum. The latter has a linear low energy slope and extends to high energy. We have investigated the temperature dependence and symmetry properties of both signals. Our data agree with previous calculations and point to a spin liquid ground state.
Despite tremendous investigations, a quantum spin liquid state realized in spin-1/2 kagome Heisenberg antiferromagnet remains largely elusive. In herbertsmithite ZnCu$_3$(OH)$_6$Cl$_2$, a quantum spin liquid candidate on the perfect kagome lattice, precisely characterizing the intrinsic physics of the kagome layers is extremely challenging due to the presence of interlayer Cu/Zn antisite disorder within its crystal structure. Here we measured the specific heat and thermal conductivity of single crystal herbertsmithite in magnetic fields with high resolution. Our results are highlighted by the …
We present a theory of the dynamic magnetic susceptibility of quantum spin liquid. The obtained results are in good agreement with experimental facts collected on herbertsmithite ZnCu3(OH)6Cl2 and on heavy-fermion metals, and allow us to predict a new scaling in magnetic fields in the dynamic susceptibility. Under the application of strong magnetic fields quantum spin liquid becomes completely polarized. We show that this polarization can be viewed as a manifestation of gapped excitations when investigating the spin-lattice relaxation rate.
This article is an introductory review of the physics of quantum spin liquid (QSL) states. Quantum magnetism is a rapidly evolving field, and recent developments reveal that the ground states and low-energy physics of frustrated spin systems may develop many exotic behaviors once we leave the regime of semi-classical approaches. The purpose of this article is to introduce these developments. The article begins by explaining how semi-classical approaches fail once quantum mechanics become important and then describes the alternative approaches for addressing the problem. We discuss mainly spin …
We followed the Shastry--Shraiman formulation of Raman scattering in Hubbard systems and considered the Raman intensity profile in the spin-1/2 "perfect" kagome lattice herbertsmithite ZnCu_3(OH)_6Cl_2, assuming the ground state is well-described by the U(1) Dirac spin-liquid state. In the derivation of the Raman T-matrix, we found that the spin chirality term appears in the A_{2g} channel in the kagome lattice at the t^4/(ω_i-U)^3 order, but (contrary to the claims by Shastry and Shraiman) vanishes in the square lattice to that order. In the ensuing calculations on the spin-1/2 kagome lattice…
The kagome Heisenberg antiferromagnet is a leading candidate in the search for a spin system with a quantum spin-liquid ground state. The nature of its ground state remains a matter of great debate. We conducted 17-O single crystal NMR measurements of the S=1/2 kagome lattice in herbertsmithite ZnCu$_3$(OH)$_6$Cl$_2$, which is known to exhibit a spinon continuum in the spin excitation spectrum. We demonstrate that the intrinsic local spin susceptibility $χ_{kagome}$ deduced from the 17-O NMR frequency shift asymptotes to zero below temperature T ~ 0.03 J, where J ~ 200 K is the Cu-Cu super-exc…
A quantum spin liquid (QSL) is a state of matter in which magnetic spins interact strongly, but quantum fluctuations inhibit long-range magnetic order even at zero temperature. A QSL has been predicted to have a host of exotic properties, including fractionalized excitations and long-range quantum entanglement. Despite the numerous theoretical studies, experimental realization of a QSL has proved to be challenging due to the lack of candidate materials. The triangular organic salts EtMe3Sb[Pd(dmit)2]2 and κ-(BEDT-TTF)2Cu2(CN)3, and kagome ZnCu3(OH)6Cl2 (Herbertsmithite) have recently emerged a…
The spin-1/2 kagome lattice antiferromagnet herbertsmithite, ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$, is a candidate material for a quantum spin liquid ground state. We show that the magnetic response of this material displays an unusual scaling relation in both the bulk ac susceptibility and the low energy dynamic susceptibility as measured by inelastic neutron scattering. The quantity $χT^α$ with $α\simeq 0.66$ can be expressed as a universal function of $H/T$ or $ω/T$. This scaling is discussed in relation to similar behavior seen in systems influenced by disorder or by the proximity to a quantum crit…
The spin-1/2 Heisenberg model on the kagome lattice, which is closely realized in layered Mott insulators such as ZnCu$_3$(OH)$_6$Cl$_2$, is one of the oldest and most enigmatic spin-1/2 lattice model. While the numerical evidence has accumulated in favor of a quantum spin liquid, the debate is still open as to whether it is a $Z_2$ spin liquid with very short-range correlations (some kind of Resonating Valence Bond spin liquid), or an algebraic spin-liquid with power-law correlations. To address this issue, we have pushed the program started by Rokhsar and Kivelson in their derivation of the …
The field of frustrated magnetism has been enriched significantly by the discovery of various kagome lattice compounds. These materials exhibit a great variety of macroscopic behaviours ranging from magnetic orders to quantum spin liquids. Using large-scale exact diagonalization, we construct the phase diagram of the $S=1/2$ $J_1$-$J_2$ kagome Heisenberg model with $z$-axis Dzyaloshinskii-Moriya interaction $D_z$. We show that this model can systematically account for many of the experimentally observed phases. Small $J_2$ and $D_z$ can stabilize respectively a gapped and a gapless spin liquid…
Barlowite, Cu$_{4}$(OH)$_{6}$FBr, has attracted much attention as the parent compound of a new series of quantum spin liquid candidates, Zn$_{x}$Cu$_{4-x}$(OH)$_{6}$FBr. While it is known to undergo a magnetic phase transition to a long-range ordered state at $T_{N} = 15$ K, there is still no consensus over either its nuclear or magnetic structures. Here, we use comprehensive powder neutron diffraction studies on deuterated samples of barlowite to demonstrate that the only space group consistent with the observed nuclear and magnetic diffraction at low-temperatures is the orthorhombic $Pnma$ s…
To capture the high-field magnetization process of herbertsmithite (ZnCu3(OH)6Cl2), Faraday rotation (FR) measurements were carried out on a single crystal in magnetic fields of up to 190 T. The magnetization data evaluated from the FR angle exhibited a saturation behavior above 150 T at low temperatures, which was attributed to the 1/3 magnetization plateau. The overall behavior of the magnetization process was reproduced by theoretical models based on the nearest-neighbor Heisenberg model. This suggests that herbertsmithite is a proximate kagome antiferromagnet hosting an ideal quantum spin …
Quantum spin liquids (QSLs) represent a novel state of matter in which quantum fluctuations prevent conventional magnetic order from being established, and the spins remain disordered even at zero temperature. There have been many theoretical developments proposing various QSL states. On the other hand, experimental movement was relatively slow largely due to limitations on the candidate materials and difficulties in the measurements. In recent years, the experimental progress has been accelerated. In this topical review, we give a brief summary of experiments on the QSL candidates under magne…
In frustrated magnets the Dzyaloshinsky-Moriya interaction arising from spin-orbit coupling can induce a magnetic long-range order. Here, we report a theoretical prediction of thermal Hall effect in frustrated kagomé magnets such as KCr$_3$(OH)$_6$(SO4)$_2$ and KFe$_3$(OH)$_{6}$(SO$_{4}$)$_2$. The thermal Hall effects in these materials are induced by scalar spin chirality as opposed to DMI in previous studies. The scalar spin chirality originates from magnetic-field-induced chiral spin configuration due to non-coplanar spin textures, but in general it can be spontaneously developed as a macro…
Several heavy-fermion metals display a quantum phase transition from an antiferromagnetic metal to a heavy Fermi liquid. In some materials, however, recent experiments seem to find that the heavy Fermi liquid phase can be directly tuned into a non-Fermi liquid phase without apparent magnetic order. We analyze a candidate state for this scenario where the local moment system forms a spin liquid with gapless fermionic excitations. We discuss the thermal conductivity and spin susceptibility of this fractionalized state both in two and, in particular, three spatial dimensions for different tempera…
A quantum spin liquid (QSL) is a state of matter where unpaired electrons' spins in a solid are quantum entangled, but do not show magnetic order in the zero-temperature limit. Because such a state may be important to the microscopic origin of high-transition temperature superconductivity and useful for quantum computation, the experimental realization of QSL is a long-sought goal in condensed matter physics. Although neutron scattering experiments on the two-dimensional (2D) spin-1/2 kagome-lattice ZnCu3(OD)6Cl2 and effective spin-1/2 triangular lattice YbMgGaO4 have found evidence for a cont…
Quantum fluctuations from frustration can trigger quantum spin liquids (QSLs) at zero temperature. However, it is unclear how thermal fluctuations affect a QSL. We employ state-of-the-art tensor network-based methods to explore the ground state and thermodynamic properties of the spin-1/2 kagome Heisenberg antiferromagnet (KHA). Its ground state is shown to be consistent with a gapless QSL by observing the absence of zero-magnetization plateau as well as the algebraic behaviors of susceptibility and specific heat at low temperatures, respectively. We show that there exists an \textit{algebraic…
