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Thermodynamic Properties of Kagome Lattice in ZnCu_3(OH)_6Cl_2 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.

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 ZnCu3(OH)6Cl2 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 3 He. An explanation of the rich behavior of strongly correlated Fermi systems still continues to be among the main problems of the condensed matter physics. One of the most interesting and puzzling issues in the research of strongly correlated Fermi systems is the non-Fermi liquid (NFL) behavior detected in their thermodynamic properties. Under the application of external fields, e.g. magnetic field B, the system can be driven to a Landau Fermi liquid behavior (LFL). Such a behavior was observed in quite different objects such as heavy-fermion (HF) metals [1,2] and two-dimensional 3 He [2][3][4][5] Recently the herbertsmithite ZnCu 3 (OH) 6 Cl 2 has been exposed as a S = 1/2 kagome antiferromagnet [6] and new experimental investigations have revealed its unusual behavior [7][8][9]. Because of the electrostatic environment, Cu 2+ is expected to occupy the distorted octahedral kagome sites. Magnetic kagome planes Cu 2+ S = 1/2 are separated by nonmagnetic Zn 2+ layers. Observations have found no evidence of long range magnetic order or spin freezing down to temperature of 50 mK, indicating that ZnCu 3 (OH) 6 Cl 2 is the best model found of quantum kagome lattice [7][8][9][10]. The specific heat C, arising from the Cu spin system, at T < 1 K appears to be governed by a power law with an exponent which is less than or equal to 1. At the lowest explored temperature, namely over the temperature range 106 < T < 400 mK, C follows a linear law temperature dependence, C ∝ T , and for temperatures of a few Kelvin and higher, the specific heat becomes C(T ) ∝ T 3 and is dominated by the lattice contribution [7][8][9]. At low temperatures T ≤ 1, the strong magnetic field dependence of the specific heat C suggests that C is predominately magnetic in origin [7][8][9]. There are a number of papers suggesting that the S = 1/2 model on the kagome lattice can be viewed as the gapless critical spin liquid [7][8][9][13][14][15][16]. These facts allow us to test both the NFL and LFL behavior of ZnCu 3 (OH) 6 Cl 2 and to show that a Fermi quantum spin liquid formed in the herbertsmithite determines its low temperature thermodynamic properties.
Contrary to the C ∝ T behavior [7,8], the observed spin liquids contribute a T 2 specific heat which in the model is not sensitive to an applied magnetic field [15,16]. Moreover the magnetic susceptibility χ(T ) of ZnCu 3 (OH) 6 Cl 2 shown in Fig. 1 displays an unusual behavior [9]. At B ≥ 3 T, χ(T ) has a maximum χ max (T ) at some temperature T max (B). The maximum χ max (T ) decreases as magnetic field B grows, while T max (B) shifts to higher T reaching 15 K at B = 14 T. At B ≤ 1 T as seen from Fig. 1, χ(T ) ∝ T −α with α = 2/3. The calculated exponent [2,17] is in good agreement with the experimental value α = 2/3 ≃ 0.66 [9]. The observed behavior of χ strongly resembles that in HF metals and is associated with their proximity to a quantum critical point (QCP) [2,17,18]. As a result, we safely assume that a quantum deconfined Fermi critical spin liquid with essentially gapless excitations formed by neutral fermions is realized in ZnCu 3 (OH) 6 Cl 2 and located very near QCP [7]. Thus, ZnCu 3 (OH) 6 Cl 2 turns out to be located at its QCP without tuning this substance to QCP using a control parameter such as magnetic field, pressure, or chemical composition. This observation is in sharp contrast to the common practice applied to tune HF metals to their QCP's. A simple kagome lattice may have a dispersionless topologically protected branch of the spectrum with zero excitation energy that is the flat band [11,12]. In that case a fermion condensation quantum phase transition (FCQPT) [2] can be considered as QCP of the ZnCu 3 (OH) 6 Cl 2 quantum spin liquid.
In this letter we uncover the quantum spin liquid phase and its QCP in the herbertsmithite ZnCu 3 (OH) 6 Cl 2 and explain its low temperature thermodynamic in magnetic fields. For the first time we calculate the susceptibility χ, magnetization M and specific heat C as functions of temperature T versus magnetic field B. Our calculations are in good agreement with the experimental facts and allow us to reveal their scaling behavior which strongly resembles that observed in HF metals and 2D 3 He.
To study the low temperature thermodynamic and scaling behavior, we use the model of homogeneous heavy-fermion liquid [2]. This model permits to avoid complications associated with the crystalline anisotropy of solids. We propose that the critical quantum spin liquid is composed of fermions. These fermions with zero charge and spin σ = 1/2 occupy the corresponding Fermi sphere with the Fermi momentum p F . The ground state energy E(n) is given by the Landau functional depending on the quasiparticle distribution function n σ (p), where p is the momentum. Near FCQPT point, the effective mass M * is governed by the Landau equation [2,19] 1 Here we have rewritten the quasiparticle distribution function as n σ (p, T, B) ≡ n σ (p, T = 0, B = 0) + δn σ (p, T, B). The Landau amplitude F is completely defined by the fact that the system has to be at QCP of FCQPT [2,17,21]. The sole role of the Landau amplitude is to bring the system to FCQPT point, where Fermi surface alters its topology so that the effective mass acquires temperature and field dependence [2,17,18]. At this point, the term 1/M * vanishes and Eq. (1) becomes homogeneous. It can then be solved analytically [2,17]. At B = 0, the effective mass strongly depends on T demonstrating the NFL behavior [2,17] At finite T , the application of magnetic field B drives the system the to LFL region with At finite B and T near FCQPT, the solutions of Eq. (1) M * (B, T ) can be well approximated by a simple universal interpolating function. The interpolation occurs between the LFL (M * (T ) ∝ const) and NFL (M * (T ) ∝ T −2/3 ) regions [2,17]. It is convenient to introduce the normalized effective mass M * N and the normalized temperature T N dividing the effective mass M * by its maximal values, M * M , and temperature T by T max at which the maximum occurs. Equation (1) allows us to calculate the thermodynamic properties for the normalized suscep- We note that our calculations of M * N based on Eq. (1) do not contain any free fitting parameters. The normalized effective mass M * N = M * /M * M as a function of the normalized temperature y = T N = T /T max is given by [2,17] Here c 0 = (1 + c 2 )/(1 + c 1 ), c 1 and c 2 are fitting parameters, approximating the Landau amplitude. Magnetic field B enters Eq. (1) only in the combination µ B B/k B T , making k B T max ≃ µ B B where k B is the Boltzmann constant and µ b is the Bohr magneton [2,17]. Thus, in the presence of magnetic fields the variable y becomes The variables T and B enter Eq. We are now in a position to construct the schematic phase diagram of ZnCu 3 (OH) 6 Cl 2 . The phase diagram is reported in Fig. 2. At T = 0 and B = 0 the system is located at QCP of FCQPT without tuning. Both magnetic field B and temperature T play the role of the control parameters, shifting the system from its QCP and driving it from the NFL to LFL regions as shown by the vertical and horizontal arrows. At fixed temperatures the increase of B drives the system along the horizontal arrow from the NFL region to LFL one. On the contrary, at fixed magnetic field and increasing temperatures the system transits along the vertical arrow from the LFL region to NFL one. The inset to Fig. 2 demonstrates the behavior of normalized effective mass M * N versus normalized temperature T N following from Eq. (4). It is seen that the temperature region T N ∼ 1 represents a transition region between the LFL behavior with almost constant effective mass and the NFL one, having the T −2/3 dependence. It is seen from Eqs. (4) and (5) Fig. 3. It is clearly seen that the data collected on both ZnCu 3 (OH) 6 Cl 2 and YbRh 2 Si 2 collapse into the same curve, obeying the scaling behav-ior. Consistent with the phase diagram displayed in Fig.  2, at growing temperatures (y ≃ 1) the LFL behavior first converts into the transition one and then disrupts into the NFL behavior. This demonstrates that the spin liquid of ZnCu 3 (OH) 6 Cl 2 is close to QCP and behaves as the HF liquid of YbRh 2 Si 2 . It is seen, that the lowtemperature ends (T N ≤ 0.5) of the curves do not merge and their values decrease as B grows representing the full spin polarization of the HF band at the highest reached magnetic fields [21]. Indeed, at low T N , χ N at B = 14 T is close to (C/T ) N at B = 18 T, while our calculations shown by the solid curve is close to both functions.  Fig. 4. Here, T k is the temperature at which the magnetization demonstrates the kink, while the system enters the transition region shown in Fig. 2 [2]. The normalized entropy S N (y) = S(T /T inf )/S(T inf ) is obtained from measurements of the entropy S on 2D 3 He [3]. Here T inf is the temperature at which the system enters the transition region and S possesses its inflection point, clearly seen in the data (see the Supporting Online Material for [3], Fig. S8, A). It is seen from Eq. (4) and from the inset to Fig. 2, that at y < 1, S N = M N ∝ y, and at y > 1, S N = M N ∝ y 1/3 . This behavior produces the kink and makes the scaled data merge into a single curve in terms of the variable y [2]. Our calculations are in good agreement with the measurements. In Fig. 5, panel A, the solid squares denote temperatures T max (B) at which the maxima of χ(T ) occur versus magnetic field B. In the panel B, the corresponding values of the maxima χ max (B) are shown by the solid diamonds versus B. It is seen that the agreement between the theory and experiment is good in the entire magnetic field domain. Our calculations of the specific heat C(B, T ) are shown in Fig. 6. For T of a few Kelvin and higher, the lattice contribution to the specific heat is the most significant contribution. However, this contribution diminishes at low temperatures, and at T ≤ 1 K C is predominately formed by the spin liquid [7]. It is seen from Fig. 6, that in the LFL region at k B T µ B B, C(B, T ) ∝ M * T ∝ B −2/3 T , and field B completely defines the M * (B) behavior given by Eq. (3). Clearly, our calculations are in good agreement with the measurements when the system demonstrates the LFL behavior. Indeed, at T = 1 K the system exhibits the LFL behavior at B ≥ 2 T, while at T = 0.5 K the LFL behavior is observed even at lower values of B, namely B ≥ 1 T.
In summary, we have shown that the kagome lattice of ZnCu 3 (OH) 6 Cl 2 can be viewed as a strongly correlated Fermi system whose thermodynamic is defined by the quantum critical spin liquid located at FCQPT. Our calculations of the thermodynamic properties are in good agreement with the experimental facts and their scaling behavior coincides with that observed in HF metals and 2D 3 He. We have also demonstrated that ZnCu 3 (OH) 6 Cl 2 exhibits the LFL, NFL and the transition behavior as HF metals and 2D 3 He do.
We grateful to V.A. Khodel and V.A. Stephanovich for valuable discussions. This work was supported by U.S. DOE, Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research, AFOSR and the RFBR # 09-02-00056.