Finite-Temperature Phase Diagram and Algebraic Paramagnetic ...

Finite-Temperature Phase Diagram and Algebraic Paramagnetic Liquid in the Spin-1/2 Kagomé Heisenberg Antiferromagnet Xi Chen,1 Shi-Ju Ran,2 Tao Liu,3 Cheng Peng,1 Yi-Zhen Huang,1 and Gang Su1, 4, ∗ 1

arXiv:1711.01001v1 [cond-mat.str-el] 3 Nov 2017

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School of Physical Sciences, University of Chinese Academy of Sciences, P. O. Box 4588, Beijing 100049, China ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain 3 School of Science, Hunan University of Technology, 412007 Zhuzhou 412007, China 4 Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China 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 spin-1/2 kagomé Heisenberg antiferromagnet (KHA). Its ground state is shown to be a gapless U(1) QSL. By studying the behaviors of specific heat and magnetic susceptibility, we propose a finite-temperature phase diagram for the spin-1/2 KHA in a magnetic field. It is unveiled that there exists an algebraic paramagnetic liquid (APL) that possesses both paramagnetic and algebraic behaviors inherited from the QSL. The APL is induced under the interplay between quantum fluctuations from geometrical frustration and thermal fluctuations. Our work suggests that even when the QSL at zero temperature is frozen into a solid state by a magnetic field, proper thermal fluctuations could “melt” the system into a liquid-like state. We speculate that an APL phase could generally emerge after proper thermal fluctuations intervene in a gapless QSL. PACS numbers: 75.10.Jm, 75.10.Kt, 75.60.Ej, 05.10.Cc

Quantum spin liquid (QSL) is an exotic state in which the interactions between spins fail to order the system at temperature even down to zero. Since it was firstly suggested by Anderson [1] forty years ago for a possible ground state of the triangular Heisenberg antiferromagnet, QSL has received much attention in condensed matter physics owing to its likely connection with high temperature superconductor and topological phases in quantum magnetism. The search for a QSL has been attempted over past decades [2]. As a magnetic system with low spin, low dimensionality and strong frustration is usually thought to favor a QSL [1], the spin-1/2 Heisenberg antiferromagnet on kagomé lattice is widely viewed as the most promising candidate. Nonetheless, after decades of intensive investigations, a mass of works reveal that, both theoretically and experimentally, the spin-1/2 kagomé Heisenberg antiferromagnet (KHA) should be a spin liquid (SL) [3], but the nature of its ground state, say, a gapless or gapped SL, has still no consensus and is currently actively debated. A number of theoretical studies on the spin-1/2 KHA by density matrix renormalization group (DMRG) [4–8], symmetric tensor network state (TNS) [9], etc., tend to give a gapped Z2 SL. However, a few inconsistencies exist as well. For instance, the four-fold topological degeneracy and spinon excitations in a Z2 SL have not yet been observed. In addition, a chiral spin liquid (CSL) was also proposed to this model by adding second and third nearest neighbor spin interactions [10, 11]. It is hard to understand from a Z2 SL to CSL with altering the couplings, because such a transition is expected to be continuous [12–14]. On the other hand, there are other works by e.g. variational Monte Carlo [15–19], and various TNS based algorithms [20, 21], favoring a gapless QSL. In particular, a recent large-scale DMRG simulation [22] finds indications for a gapless Dirac spin liquid. It is obvious that these studies make this issue quite subtle. The mineral Herbertsmithite ZnCu3 (OH)6Cl2 [25] is

widely considered as an ideal model compound of the spin1/2 KHA. Earlier inelastic neutron scattering and Raman spectroscopy [26, 27] as well as thermodynamic measurements [25, 28, 29] on Herbersmithite appear to support a gapless QSL, but a recent NMR study shows a finite spin gap around 0.03 ∼ 0.07J [30], in favor of a gapped Z2 SL. The incompatibility in experiments also remains unsolved, albeit the Dzyaloshinskii-Moriya interaction is believed to play a role in physics of Herbertsmithite [29, 31–33]. Besides the controversial results of the ground state, the lack of the finite-temperature simulations requires more accurate and reliable works on the KHA. There are only a few investigations by, e.g., the high-temperature series expansion [23, 34], projected-wave-function techniques [15, 19], and Hamsde Raedt method [24]. These approaches are severely limited by finite sizes or the temperature regime of interest. The finite-temperature simulations on KHA with tensor network (TN) methods [42–47], which possess unprecedented advantages on capturing many-body features and have achieved great successes on the ground states, are still missing. Here we utilize the TN-based numerical methods with two distinct optimization algorithms to intensively study both the ground-state and thermodynamic properties of the spin-1/2 KHA with high accuracy. We find no zero-magnetization plateau in the ground state, and algebraic behaviors of magnetic susceptibility and specific heat at low temperatures. All these results are in accordance with the characteristics of the U(1) Dirac spin liquid [15]. By studying the behaviors of the specific heat and susceptibility in magnetic fields, the finitetemperature phase diagram is firstly proposed for the spin-1/2 KHA, where we discover an algebraic paramagnetic liquid (APL) phase. The APL inherits the algebraic properties from the ground-state QSL, but exhibits paramagnetism in the absence of a magnetic field. We speculate that the APL could be

2 a novel state of matter that generally emerges after introducing proper thermal fluctuations to a gapless QSL. The phase diagram also reveals an interesting phenomenon that even when the QSL at zero temperature is frozen by the magnetic field to a solid state, proper thermal fluctuations could “melt” the system into an APL state. Model and Methods.— Let us consider the spin-1/2 Heisenberg antiferromagnet on kagomé lattice in a magnetic field. The Hamiltonian reads X X H=J Si · S j − h S iz , (1)

i

where J is the coupling constant, < i j > stands for the summation over nearest neighbors, Si is the spin operator on the ith site, and h is the magnetic field along the z direction. We assume gµB = 1 (g the Landé factor, µB the Bohr magneton) for simplicity. We adopt the tensor network (TN) algorithms to study this system numerically. To contract the tensor networks of the ground state and the partition function, two optimization algorithms, cluster update [36–38] and full update [39–41] schemes, are employed. In the cluster update algorithm, we choose the unit cell formed by a few tensors that preserve the symmetry of the state and use the Bethe approximation to simulate the environment outside the cluster; in the full update algorithm, we contract the whole TN. At finite temperatures, we invoke the tensor product density operator (TPDO) [42–47]. Refer to Supplemental Material for the details of calculating algorithms. In ground state calculations, the Trotter step τ is taken from 10−1 to 10−5 , and in simulations of thermodynamic properties, τ is fixed to be 10−2 . In cluster and full update schemes, we choose a unit cell of nine spins that comprises of one hexagon and six triangles. The benchmark on the ground state energy is given in the Supplemental Material. Gapless U(1) quantum spin liquid in the ground state.— The gapless U(1) QSL in the ground state is supported by the behaviors of ground-state magnetization, and of the lowtemperature specific heat and susceptibility. Fig. 1 presents the ground-state magnetization per site m, defined by m = P (1/N) i hS iz i (with N the total number of lattice sites) as a function of the magnetic field h. It can be seen that m depends linearly on h at small magnetic fields, i.e., m ∼ hα with α = 1, and vanishes as h → 0. These results suggest a U(1) Dirac spin liquid, consistent with the projected wave function study [15] and a recent simulation by DMRG on infinite long cylinders [22]. Note that previous DMRG calculations on cylinders [4–8, 54] claimed a gapped Z2 spin liquid with a small gap on this model. There was also a recent work based on the tensor entanglement renormalization group with symmetry by a simple update scheme giving a gapped ground state [9]. However, the four-fold topological degeneracy on a torus in a gapped Z2 spin liquid was not observed, thereby casting doubt on the gapped spin liquid. In addition, we calculate the overall magnetic curve shown in the lower inset of Fig. 1. One may see that there are four plateau states occurring at m/m s = 1/9, 1/3, 5/9, 7/9, with

FIG. 1. (Color online) The magnetization per site m versus magnetic field h at low fields. The magnetization vanishes linearly when h tends to zero. Lower inset presents the overall magnetic curve, where the magnetization plateaus at m/ms = 1/9, 1/3, 5/9, 7/9 (with ms the saturated magnetization per site) are clearly observed, but zero magnetization plateau is absent. Upper inset gives the spin configuration of the 1/3-magnetization plateau state, in which a singlet hexagon valence bond state (bold green hexagon) forms with the outer three spins almost fully polarized. These results are calculated by the cluster update algorithm with Dc = 10.

m s the saturated magnetization per site, consistent with the DMRG calculations [54]. The plateaus can be written in a compact form of m/m s = (1 − 2ℓ/9), ℓ = 1, 2, 3, 4. For the m/m s = 1/9 plateau state, we observe a ferrimagnetic order as shown in the upper inset of Fig. 1. The six spins on hexagon of the unit cell displayed are partially polarized with directions all pointing down while the outer three spins (each of the six spins is shared by two unit cells) are also partially polarized but in the opposite direction. It is interesting to note that the m/m s = 1/9 plateau was observed in a previous DMRG calculation on cylinders, where this 1/9-plateau was claimed to be a Z3 spin liquid [54]. It is also worth mentioning that for the m/m s = 1/3 plateau state, the six spins on the hexagon of the unit cell form a singlet hexagon valence bond state with vanishing magnetic moment on each site, while the outer three spins are almost fully polarized. This is in contrast to an upup-down spin configuration observed on an infinite Husimi lattice [20]. Besides the absence of zero magnetization plateau, more solid evidences of the U(1) QSL are obtained from the algebraic behaviors of the susceptibility χ = ∂m/∂h and specific heat C = ∂U/∂T (with U the internal energy) at low temperatures. Fig. 2 (a) shows the temperature-dependence of χ(T ) in zero magnetic field (h = 0). At high temperatures, our results coincide with the fifteenth-order HTE calculation [34, 35] under the parameter Pade [7,8]. For h = 0, χ(T ) can be nicely fitted by a polynomial χ(T ) = 0.13277 + 0.02096(T/J) + 1.20337(T/J)2 + O(T 3 ) at low temperatures (inset of Fig. 2 (a)). When T → 0, χ(T ) approaches linearly to a constant

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FIG. 2. (Color online) The susceptibility χ (a) and specific heat C (b) as a function of temperature T for the spin-1/2 kagomé Heisenberg antiferromagnet in the absence of magnetic field. Here the cluster update algorithm with bond dimension Dc = 17 is adopted. χ (blue open circle in (a)) and C(T ) (blue open square in (b)) are compared with those of the fifteenth-order high temperature series expansion (HTE) (red dash line in (a)) and the sixteenth-order HTE (red dash line in (b)) under parameters Pade[7,8], respectively. As shown in the insets, both χ and C exhibit algebraic behaviors at low temperatures.

0.13277, which is consistent with the ground state susceptibility χ(0) = 0.1298 given by the slope of the magnetic curve at small h (Fig. 1). This temperature dependent behavior of the susceptibility resembles the result from the modified spin-wave theory [56], where the linear T -dependence of χ at T → 0 is obtained for the spin-1/2 Heisenberg antiferromagnet on square lattice, which has gapless excitations (Goldstone modes) because it has an antiferromagnetic long-range ordered ground state. However, we do not find any long-range order in the spin-1/2 KHA, suggesting that the spin excitations in this system could be essentially different from the square Heisenberg antiferromagnet. For the specific heat C, a peak appears in the absence of magnetic field as demonstrated in Fig. 2 (b). The C at high temperatures obtained by TN scheme also coincides with the results from the sixteenth-order HTE with the parameter Pade [7,8] [34, 35]. At low temperatures, C(T ) can be well fitted by a polynomial function C(T ) = 88.991(T/J)2 − 1991.199(T/J)5/2 + O(T 3 ) (inset of Fig. 2 (b)). It suggests, when T → 0, we have C(T ) ∼ T 2 , which is another evidence for the gapless U(1) QSL [53].

FIG. 3. (Color online) The phase diagram of the spin-1/2 KHA in the plane of temperature and magnetic field, which consists of the phases including gapless quantum spin liquid (QSL), algebraic paramagnetic liquid, field-induced ordered phase, crossover phase, and conventional paramagnetic state. The upper (red circles) and lower (black squares) phase boundaries are determined from the highand low-temperature peaks of the specific heat in different magnetic fields, respectively. The dash line is obtained by fitting the upper boundary for large fields.

Finite-temperature phase diagram and algebraic paramagnetic liquid.— The phase diagram of the spin-1/2 KHA in the plane of temperature and magnetic field is given in Fig. 3. The phase boundaries denoted by red circles and black squares are determined by the positions of the peaks of the temperature dependent specific heat [see also Fig. 4 (b)]. We observe that there are no phase transitions among different phases, but only crossovers. For the boundary at relatively higher temperatures (denoted by T p2 , corresponding to the high-temperature peaks of C(T )), we see two different behaviors when h changes. For 1.5 < h < 2.5, T p2 increases linearly with increasing the magnetic field h, which is consistent with the crossover temperaP tures in a noninteracting spin model Hˆ = h i Sˆ iz . It suggests that the crossover occurring at the peak of C(T ) is dominated by the high-temperature trivial paramagnetic states. Note that near the saturated fields, such a linear behavior slightly deviates owing to the occurrence of independent magnons. For h < 1.5, T p2 changes very little with h. The peak of C(T ) in this region is dominated by the low-energy physics of the QSL emerging from frustrations. The insensitivity of T p2 to h is consistent with the fact that the ground-state QSL has no magnetic momentum to gain or loss energy when h changes. Moreover, a universal scaling equation for the specific heat, C(1 + h/J)α = q0 + b1 (1 + h/J)β(1 − TT∗ ), is found in the vicinity of an isosbestic point T ∗ /J = 0.25126. Refer to Supplemental Material for details. This implies that the phase below this upper boundary should be dominated by the low-energy physics of the QSL, which gives rise to a new phase dubbed as the APL. Above this boundary, we have the high-temperature

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dependence of the susceptibility χ for different h. It is clear that at high temperatures, it obeys the Curie-Weiss law [χ−1 ∝ (T/J + 1.4)], and at low temperatures, the behavior of χ is similar to that of h = 0. The paramagnetism is unveiled by the linearly field-dependent magnetization, as shown in the inset of Fig. 4 (a). Meanwhile, the low-temperature behavior of C in the APL phase is also similar to that of h = 0 [see Fig. 4 (b)]. These algebraic properties indicate that the APL is a liquid-like state that essentially differs from the conventional paramagnetic phase.

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FIG. 4. (Color online) The temperature dependences of the susceptibility χ(T ) (a) and the specific heat C(T ) (b) for the spin-1/2 kagomé Heisenberg antiferromagnet in various magnetic fields (h). At high temperatures, the susceptibility obeys the Curie-Weiss paramagnetic law. When temperature goes down to the APL region, the magnetization depends linearly on h [inset of (a)]. The positions of round peaks of C(T ) almost appear at the same temperature for small h, but grow linearly with h when h/J ≤ 1.5. In the APL phase, C(T ) shows an algebraic behavior [a logarithmic plot in the inset of (b)].

trivial paramagnetic phase. The boundary indicated by the black squares are given by the positions (denoted by T p1 ) of the low-temperature peak of C(T ) that appears for h > 1.8 approximately [see also Fig. 4 (b)]. Compared with the positions of the magnetization plateaus in the ground state, this boundary separates the field-induced magnetic ordered phase and a crossover phase. In the field-induced ordered phase (T < T p1 ), our calculations show that the magnetization per site does not vanish, characterized by ferrimagnetic features. When T > T p1 , the thermal fluctuations tend to destroy the ordering and drive the system into a canted (crossover) phase, where the magnetization decreases when T rises, and finally vanishes for T > T p2 . There should be another boundary that separates the APL and crossover phases, since they exhibit different physics. One reasonable choice should be the peak of the specific heat contributed by the noninteracting magnetic states. However, this peak cannot be observed directly for h < 1.5 since the dominant contribution of C(T ) is from the QSL. For these reasons, we extrapolate the boundary according to the linear relation to T = 0 (dash line). The APL is a phase that possesses both paramagnetic and the QSL-like algebraic properties. Fig. 4 (a) shows the T -

Our work poses an interesting phenomenon. Starting from h = 0 and T = 0, where the strong quantum fluctuations permit a disordered QSL, the system can be frozen into a solidlike state by increasing the magnetic field to, e.g., h = 1. The reason is that a larger field suppresses the frustration, reducing the quantum fluctuations. Then by properly increasing temperature to, e.g., T = 0.5, the system is melted by thermal fluctuations, and enters into the APL state that possesses nontrivial QSL-like properties. Thus, the APL is the consequence of interplay between quantum and thermal fluctuations. We speculate that the APL phase could generally emerge after introducing proper thermal fluctuations to a gapless QSL. Conclusion.— By utilizing state-of-the-art tenser network algorithms, we show that the ground state of the spin-1/2 KHA is a gapless U(1) QSL, which is evidenced by the facts such as the absence of zero-magnetization plateau, as well as the algebraic behaviors of susceptibility and specific heat at low temperatures. By studying the effects of thermal fluctuations on the QSL, we propose a phase diagram in the h-T plane, where five phases including QSL, APL, field-induced ordered state, canted (crossover) phase, and high-temperature trivial paramagnetic state are identified. The APL phase is unveiled emerging from both quantum and thermal fluctuations, which possesses paramagnetic and the QSL-like algebraic properties. It is interesting to note that our phase diagram looks similar to the 17 O NMR measurements in Herbertsmithite in a magnetic field [29]. As the thermodynamic properties of the spin-1/2 KHA in a magnetic field are still sparse, our findings would stimulate more experimental explorations, and shed deeper insight on the novel states of matter. The authors acknowledge Wei Li and Xin Yan for useful discussions. This work was supported in part by the MOST of China (Grants No. 2013CB933401), the NSFC (Grant No. 14474279), and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB07010100). SJR was supported by ERC AdG OSYRIS (ERC-2013-AdG Grant No. 339106), the Spanish MINECO grants FOQUS (FIS2013-46768-P), FISICATEAMO (FIS2016-79508P), and “Severo Ochoa” Programme (SEV-2015-0522), Catalan AGAUR SGR 874, Fundació Cellex, EU FETPRO QUIC, CERCA Programme / Generalitat de Catalunya and Fundació Catalunya - La Pedrera · Ignacio Cirac Program Chair.

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