Home

Search

Collections

Journals

About

Contact us

My IOPscience

Cluster mean-field theory study of J1−J2 Heisenberg model on a square lattice

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 115601 (http://iopscience.iop.org/0953-8984/26/11/115601) View the table of contents for this issue, or go to the journal homepage for more

Download details: IP Address: 129.130.252.222 This content was downloaded on 11/07/2014 at 15:48

Please note that terms and conditions apply.

Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 115601 (7pp)

doi:10.1088/0953-8984/26/11/115601

Cluster mean-field theory study of J1–J2 Heisenberg model on a square lattice Yong-Zhi Ren1 , Ning-Hua Tong1 and Xin-Cheng Xie2 1

Department of Physics, Renmin University of China, 100872 Beijing, People’s Republic of China International Center for Quantum Materials and School of Physics, Peking University, Beijing 100871, People’s Republic of China 2

E-mail: [email protected] Received 31 August 2013, revised 14 January 2014 Accepted for publication 21 January 2014 Published 3 March 2014

Abstract

We study the spin-1/2 J1 –J2 Heisenberg model on a square lattice using the cluster mean-field theory. We find a rapid convergence of phase boundaries with increasing cluster size. By extrapolating the cluster size L to infinity, we obtain accurate phase boundaries J2c1 ≈ 0.42 (between the N´eel antiferromagnetic phase and non-magnetic phase), and J2c2 ≈ 0.59 (between non-magnetic phase and the collinear antiferromagnetic phase). Our results support the second-order phase transition at J2c1 and the first-order one at J2c2 . For the spin-anisotropic J1 –J2 model, we present its finite temperature phase diagram and demonstrate that the non-magnetic state is unstable towards the first-order phase transition under intermediate spin anisotropy. Keywords: J1 –J2 Heisenberg model, quantum phase transition, cluster mean-field theory (Some figures may appear in colour only in the online journal)

where Si is the spin 12 operator on site i, and J1 and J2 are the nearest neighbor and the next-nearest-neighbor coupling coefficients, respectively. In the following, we set J1 = 1 as the unit of energy. For the next-nearest-neighbor coupling J2 , we confine ourself to the AFM case J2 > 0. This model has been the subject of numerous studies in the past two decades, using various methods including exact diagonalization (ED) [6–11], series expansion [12–16], coupled cluster [17, 18], spin wave approximation [3, 19], Green’s function method [20], density-matrix renormalization group (DMRG) [21, 22], matrix-product or tensor-network based algorithms [23–26], high temperature expansion [27], resonating valence bond approaches [28–32], bond operator formalism [33, 34], mean-field theories [12, 35, 36], and field theoretical methods [37–39]. It has been established that in the regime 0 < J2 /J1 . 0.4, the ground state of J1 –J2 model is an AFM phase with N´eel order. In J2 /J1 & 0.6, an AFM phase with collinear LRO is stable, due to the dominance of the next-nearest-neighbor coupling J2 . One of the most controversial regimes is the intermediate regime 0.4 . J2 /J1 . 0.6 where the ground state is non-magnetic

1. Introduction

It was suggested by Anderson [1] that low spin, low spatial dimension, and high frustration are the three main factors which favor the melting of magnetic long-range order (LRO) and lead to exotic spin liquid ground state. Such a state was closely related to the appearance of high temperature superconductivity in Cu-based oxides upon doping [2]. The spin-1/2 J1 –J2 Heisenberg model in a two-dimensional square lattice is such a model that bears all the three factors, hence its ground state is a promising candidate for the exotic spin liquid state [3]. Besides the interest for spin liquid, this model in the large J2 /J1 regime is relevant to materials such as Li2 VOSiO4 [4, 27], and the S > 1/2 version is relevant to the parent material of iron-based high temperature superconductors [5]. The Hamiltonian of the antiferromagnetic (AFM) J1 –J2 model reads X X Hˆ = J1 Si · S j + J2 Si · S j , (1) hi, ji 0953-8984/14/115601+07$33.00

hhi, jii 1

c 2014 IOP Publishing Ltd

Printed in the UK

J. Phys.: Condens. Matter 26 (2014) 115601

Y-Z Ren et al

and hence the SU (2) symmetry is not broken. The nature of this intermediate non-magnetic ground state is still a much debated issue. The possible candidates of this ground state, as proposed by various authors, include dimerized valence bond solid (VBS) which breaks both the translation and the rotation symmetries of the lattice [8, 12–14], the plaquette VBS which breaks only the translation symmetry [33, 38, 36], the nematic spin liquid which breaks only the rotational symmetry [39], and the gapped [21, 26, 29] or gapless [31] spin liquid which conserves all the symmetries of the lattice. The difficulty of this issue lies in that there is no unbiased and accurate method to study the ground state of the J1 –J2 model in the thermodynamical limit. Most of the numerical studies heavily rely on the extrapolation of the finite size results to the thermodynamical limit. In cases where there is little guidance from analytical knowledge, this practice may have uncertainties [40, 24], as demonstrated by a recent study on the J–Q model [41]. Besides the nature of the non-magnetic state, there are other important issues under various physical contexts. Previous studies show that the AFM N´eel phase transits into the non-magnetic state at J2 /J1 ≈ 0.4 through a continuous quantum phase transition. If the intermediate region actually possesses a VBS order, this transition is an abnormal one, as a continuous transition between two phases without the group– subgroup symmetries violates the conventional ‘Landau rule’. A ‘deconfined’ quantum critical point was proposed to exist between the N´eel and the VBS states [42]. For the parameter regime J2 /J1 & 0.6, this model also invoked much interest since lots of real materials are related to this parameter regime, such as the La–O–Cu–As iron-based superconductors [43, 44] and Li2 VOSiO4 [4, 27]. Another interesting issue in this parameter regime is the possible finite temperature symmetry breaking. For this model, although the spin SU (2) symmetry cannot be broken spontaneously at finite temperature due to the Mermin–Wagner theorem [45], symmetry breaking of the lattice C4 symmetry could occur below a finite T < Tc [37, 46, 47]. However, there is also a different opinion on this issue [15]. The effect of spin anisotropy in the J1 –J2 model is also an interesting issue, given that the anisotropy is quite common in real materials. Theoretical studies on this issue are rare [48, 49]. In this paper, we focus on the phase boundary of the J1 –J2 model and attempt to present accurate critical values J2c1 and J2c2 . We use the cluster mean-field theory (CMFT), which is the cluster extension of the Weiss mean-field theory [50, 51]. We obtained the N´eel AFM phase, the collinear AFM phase, and the non-magnetic phase. Using the reshaping method for plotting multiple-valued curves [52], we studied the fine structure of the first-order phase transition between the non-magnetic phase and the collinear AFM phases, including the stable, meta-stable and unstable phases. This information is important when the system is under external influence but has often been neglected in previous studies. The critical values J2c1 and J2c2 are found to converge very fast with increasing cluster size, allowing us to obtain an accurate estimation of them. We also study the effect of spin anisotropy,

producing the finite temperature phase diagram and analyzing the stability of the non-magnetic phase under the spin anisotropy. The rest of this paper is organized as follows. In section 2, we introduce the CMFT and the method we used to obtain the fine structure of the first-order phase transition. In section 3, we first present the zero temperature results, including the phase diagram and magnetic susceptibility. Then, the effect of the spin anisotropy is considered and the phase diagram at finite temperature is given. 2. Method

The simplest mean-field theory for spin systems is Weiss’s single-site mean-field theory [50]. In this theory, the influence of surrounding spins to a central spin is approximated by an effective static field, which is then determined self-consistently. The Weiss mean-field theory thus neglects the spatial fluctuations and often overestimates the stability of LRO. Based on a similar idea, Bethe–Peierls–Weiss (BPW) [54–56] and Oguchi [57] improved the approximation by mapping the lattice model into clusters subjected to self-consistently determined effective fields. Interactions inside a cluster are treated exactly while interactions between clusters are approximated by mean fields. Since the short-range spatial fluctuations inside a cluster are taken into account, the results are expected to improve as cluster size increases. In this work, we study the J1 –J2 model on a square lattice using the cluster extension of Weiss mean-field theory. Although being simple, this theory produces surprisingly accurate boundaries between various phases, as compared to results from the more sophisticated methods. We first divide the lattice into identical clusters of L sites. To separate the spin couplings inside a cluster from those between clusters, the Hamiltonian of J1 –J2 model is rewritten as   X X X  J1 Hˆ = Si,cn · S j,cn + J2 Si,cn · S j,cn  cn

hi ji

hhi jii

 +

X c n 6 =c m

 J1

 X hi ji

Si,cn · S j,cm + J2

X

Si,cn · S j,cm  .

hhi jii

(2) The operator Si,cn denotes the spin operator on the ith site in the cluster cn . The first term in equation (2) represents the Hamiltonian of decoupled clusters, while the second one represents interactions between clusters. We make the standard mean-field approximation for the interactions between two spins belonging to different clusters cn 6 = cm , z z z Si,cn · S j,cm ≈ Si,c hS zj,cm i + hSi,c iS zj,cm − hSi,c ihS zj,cm i. n n n (3)

Here, the z-axis is chosen as the quantization axis. This approximation breaks both spin SU (2) symmetry and spatial translation symmetry of the original Hamiltonian. Substituting it into the second term of equation (2) and neglecting a constant, we obtain the cluster-decoupled mean-field 2

J. Phys.: Condens. Matter 26 (2014) 115601

Y-Z Ren et al

or nonzero at the same time, indicting the appearance or disappearance of the magnetic LRO. We use an iterative method to solve the mean-field equations. For a given set of effective fields h i , we use the Lanczos method (for T = 0) or full ED method (for T > 0) to calculate the magnetization m i which are fed back to equation (5). This process iterates until all the m i converge. For a given J2 and T , the calculation starts from an initial set of m i , which we usually get from the self-consistent solution of a slightly deviated parameter J2 (or T ). Thus we can scan the parameter space from small J2 (or T ) to larger values, or vice versa. It turns out that the set of mean-field equations has more than one solution, stabilized respectively by scanning from left to right or from right to left along the J2 (or T ) axis. For those multiple solutions at a fixed (J2 , T ), we compare their energies (T = 0) or free energies (T > 0) to determine the physical solution of this system. After the solutions of m i (i = 1, 2, . . . , L) are obtained, its LRO can be identified easily from the magnetization pattern. Near J2 ≈ 0.6, naive scanning of J2 produces a discontinuous m–J2 curve: m jumps from 0 to a finite value or vice versa (m is the magnetization of a center site of the cluster). We suppose that this is the numerical instability due to the multiple-valued relation of m–J2 . If such structure does exist, ordinary calculation can only produce one branch of solution and neglect the others, leading to a jump at some J2 where the relative stabilities of two solutions invert. To overcome this problem, we use the ‘stretching trick’ proposed in the study of first-order phase transitions in correlated electron systems [52]. If the mean-field solution m = F(J2 ) is a continuous curve in the m–J2 plane but has a S- or Z -shaped turn, the new equation m = F(J2 –V |m|) will produce a single-valued m–J2 curve, given a proper selection of V > 0. Pictorially this single-valued curve is obtained by ‘stretching’ the original curve. We can then solve this modified equation first and recover the original solutions by plotting m versus J2 –V |m|.

Figure 1. (a) The square lattice is divided into 2 × 2 clusters (solid

lines). The interactions between different clusters are denoted by dot–dashed lines (J1 ) and dashed lines (J2 ). (b) Upper: picture of N´eel AFM order, dominated by the nearest antiferromagnetic interaction J1 (solid lines). Lower: picture of collinear AFM order, dominated by the next-nearest antiferromagnetic interaction J2 (dashed lines).

Hamiltonian, X Hˆ mf = Hˆ cn cn

Hˆ cn = J1

X

Si,cn · S j,cn + J2

hi ji

+

L X

X

Si,cn · S j,cn

hhi jii

z h i Si,c . n

(4)

i=1

Here h i is the effective static field felt by the spin Si,cn . It is a linear combination of hS zj,cm i, the magnetization of boundary site j on the neighboring cluster cm . Figure 1 shows an example of 2 × 2 clusters and their couplings between each other. We use the spatial translation z symmetry of clusters to ensure hSi,c i = hSiz i = m i . For a n cluster with L sites, m i (i = 1, 2, . . . , L) are our magnetic order parameters that can characterize different magnetic orders. In this paper, we do not consider the possibility of LRO in the intermediate non-magnetic regime, as it is still an open issue how to incorporate the non-magnetic order parameters into the CMFT. With this notation, the effective field h i reads X X h i = J1 m δ + J2 m δ0 . (5) δ

3. Results and discussions 3.1. Zero temperature

δ0

In this work, we use the rectangular clusters of size L = L x × L y . To avoid odd number of spins in a cluster, we use even L x and L y . The total number of spins L is confined as L ≤ 16 due to the exponential increase of computational cost with L. We choose 2 × 2 and 4 × 4 clusters for qualitative study, and use L y = 2 and L x = 2, 4, 6, 8 for quantitative size dependence analysis. In figure 2, we show |m| versus J2 for three successively larger clusters. |m| is measured on the center site of the cluster. For all the clusters we used, the N´eel order is stable for small J2 regime. As J2 increases, |m| decreases and vanishes continuously at a critical value J2c1 ≈ 0.41–0.42, which indicates a second-order transition to a non-magnetic phase. As J2 increases above J2c2 ≈ 0.6–0.7, |m| jumps from zero to a finite value, with a collinear magnetic pattern. In both N´eel and collinear phases, m decreases with increasing L, showing that more and more quantum fluctuations are taken into account by using large clusters, and hence the increasing

δ0

Here δ, ∈ [1, L] denote the nearest neighbor site and the next-nearest neighbor site in the neighboring clusters of site i, respectively. The CMFT equations are completed by solving m i from a central cluster Hamiltonian Hˆ c in equation (4). In the limit of single-site cluster L = 1, the above approximation recovers the Weiss mean-field theory. As the cluster size increases, longer and longer range correlations contained in the cluster are treated exactly. Therefore, the results are expected to become exact as L tends to infinity. To solve the CMFT equations, we use open boundary conditions for the cluster. The L magnetization values m i (i = 1, 2, . . . , L) are solved independently without symmetry constraints. Due to the lack of translation symmetry within the cluster, |m i | has a weak site-dependence, being smaller on the center of the cluster, and larger on the edge and even larger at the corner. The qualitative behavior of magnetization on different sites are exactly the same, i.e. they will be zero 3

J. Phys.: Condens. Matter 26 (2014) 115601

Y-Z Ren et al

Figure 2. Magnetization m versus J2 obtained using various cluster

geometries. The pattern of LROs are marked in the figure. PM denotes paramagnetic. Here m is the magnetization of a spin at the center of the cluster.

quality of our results. The exact value m = 0.307 [53] for J2 = 0 is only asymptotically approached in the L = ∞ limit. It is interesting to observe that from L = 4 to L = 16, the |m|–J2 curve always vanishes continuously and the critical point J2c1 does not change much. Although it is difficult to numerically distinguish a second-order phase transition from a weak first-order one, the rapid convergence of results with L supports the second-order nature of the transition at J2c1 . Taking the L = 16 result as our estimation for the thermodynamical limit, we obtain J2c1 ≈ 0.42. Compared to other methods such as the ED [8, 9], series expansion [12, 14] and DMRG [21], CMFT is surprisingly accurate and simple in producing the ground state phase boundaries. In figure 3(a), we take a closer look at the fine structure of the |m|–J2 curve near J2c2 , where the transition between the non-magnetic phase and collinear AFM phase occurs. It is obtained by the ‘stretching trick’ mentioned above. In order to see the systematic cluster size dependence, we fix L x = 2 and increase L y from 2 to 8. We always obtain continuous curves with S-shaped structures which contain the stable, meta-stable, and the unstable phases and are generic features of the first-order phase transition. The width of the coexistence region W decreases as L y increases. As shown in the inset of figure 3(a), W is found to scale with 1/L y as W ∝ αeβ/L y for the calculated cluster size. Fitting of the data gives α = 0.038 and β = 0.42. α = 0.038 > 0 means that the first-order phase transition still exists even if we use a cluster L x = 2, L y = ∞. This seems to be a strong support to the first-order phase transition between non-magnetic phase and collinear AFM phase in the thermodynamical limit. For a more convincing conclusion, one should extrapolate L x and L y to infinity simultaneously. However, due to the rapid increase of the numerical cost, this is not done in our present study. In figure 3(b), the ground state energy per site versus J2 is plotted for the N´eel AFM, non-magnetic, and the collinear AFM phases. We show the result obtained using 2 × 2 cluster for demonstration purpose. As J2 increases up to 0.42

Figure 3. (a) Magnetization |m| of a center spin versus J2 near the first-order phase transition for L y = 2 and L x = 2, 4, 6, and 8. Inset: the width of coexistence region versus 1/L x . (b) Ground state energy per site versus J2 in different phases, obtained using L x = L y = 2 cluster. The arrows mark the second-order

N´eel-to-non-magnetic transition (arrow a), the first-order non-magnetic-to-collinear transitions (arrow b), and the meta-stable second-order non-magnetic-to-collinear transition (arrow c). Symbols are data and the dashed lines are guides for the eyes. The solid line is the result of the hierarchical mean-field approach using a 2 × 2 cluster in [36].

(marked by arrow ‘a’), the energy of N´eel AFM continuously approaches that of the non-magnetic phase from below, consistent with the scenario of a second-order transition. The transition between the non-magnetic phase and the collinear AFM phase occurs at the energy crossing point marked by the arrow ‘b’ in figure 3(b), which we denote as J2c2 . In the coexistence region, a third collinear AFM solution has the highest energy. It corresponds to the unstable solution with negative |m|–J2 slope in figure 3(a). In this first-order transition, a continuous transition does exist at the meta-stable level, between collinear AFM and non-magnetic phases (marked by arrow ‘c’). This scenario is common in first-order phase transitions described by mean-field equations, as disclosed by the dynamical mean-field theory study for the correlated electron systems [52]. Extrapolating L y to infinity, we get J2c2 ≈ 0.59, which should be very close to the exact value in the thermodynamical limit. This value agrees quite well with the more sophisticated calculations such as DMRG [21] (see table 1) and resonating valence bond trial wavefunction method [32]. It is noted that our energy curve agree quantitatively with the result from the hierarchical mean-field approach (HMFA) on a 2 × 2 cluster [36] (solid lines in figure 3(b)). Although HMFA is based on the sophisticated Schwinger boson representation and mean-field approximation, the quantitative agreement makes us believe that the HMFA is equivalent to the cluster mean-field method that we have used here, at least for the case of the 2 × 2 4

J. Phys.: Condens. Matter 26 (2014) 115601

Y-Z Ren et al

Table 1. Comparison of J2c1 and J2c2 from various works. The

methods are abbreviated as ED (exact diagonalization), SE (series expansion), DMRG (density-matrix renormalization group), HMFT (hierarchical mean-field theory), VMC (variational Monte Carlo), and CMFT (cluster mean-field theory). Reference [9] Met. ED

[12] [21] [18] [36] [31] This work SE DMRG CC HMFT VMC CMFT

J2c1

0.35 0.41 0.41

0.44 0.42

0.45

0.42

J2c2

0.66 0.64 0.62

0.59 0.66

0.6

0.59

cluster. The critical values of J2 have been obtained in many works, using different methods with varied sophistication. In table 1, we summarize some of the previous results and compare them with ours. Note that a similar CMFT study on the J1 –J2 model was carried out in [12], but the cluster size effect was not analyzed systematically. A central issue in the study of the J1 –J2 model concerns the properties of the intermediate non-magnetic phase. The key open question is whether it is a spin liquid or a state which spontaneously breaks some symmetry of the original model, such as the lattice translation and/or rotation symmetry. Since in CMFT the translation symmetry of the original lattice is broken by hand, we cannot answer this question directly. In the non-magnetic phase, the effective fields of CMFT become zero and Hmf describes uncorrelated clusters. Then CMFT is equivalent to ED on a cluster with open boundary condition. Previously, ED studies have been carried out for clusters up to 40 sites with periodic boundary condition [7–9]. Various candidate ground states were proposed based on these ED calculations, including the dimerized state and the chiral state [8], and the twisted magnetic LRO [7]. However, due to the severe finite size effect, it is difficult to draw definite conclusion from these studies. It is the same situation for our study here on clusters up to 16 sites. In contrast to the periodic boundary condition used in previous ED studies, here the open boundary condition will induce nonzero VBS order parameter in small clusters. For an example, the operator of plaquette order parameter reads [58]

Figure 4. Zero temperature N´eel susceptibility χn (squares on the guiding line) and collinear susceptibility χc (dots on the guiding line) as functions of J2 . The data are obtained by numerical

derivation with the applied field h = 0.01.

We also investigate the N´eel as well as the collinear magnetic susceptibility at zero temperature. These susceptibilities are defined as

h→0+

hTr[e−β( Hˆ −h Mα ) ]

.

(7)

Here, the N´eel susceptibility χn and collinear susceptibility χc are defined using staggered magnetization Mn and Mc , respectively. For the 2 × 2 cluster shown in figure 1, Mn = S1z − S2z + S3z − S4z and Mc = S1z + S2z − S3z − S4z . We apply a small staggered field h and evaluate χn and χm using numerical derivation. The results obtained are shown in figure 4. The continuously diverging behavior of χn at J2 ≈ 0.42 confirms the continuous transition from N´eel AFM phase to non-magnetic phase. In contrast, near the collinear transition J2c2 , an abrupt jump of χc is observed, being consistent with a first-order phase transition. Note that both χn and χc are much larger in the non-magnetic regime than in their corresponding long-range ordered regime. This shows that the intermediate non-magnetic ground state is rich of short-range spin fluctuations at various momentums, and different types of spin correlation compete strongly with each other. This leads to the notorious difficulty in the study of the non-magnetic state. The mean-field approximation usedPin our study introL z duces a symmetry breaking term H 0 = i=1 h i Si,c , which n breaks the SU (2) symmetry of the original Hamiltonian. For CMFT calculation using a finite cluster, this term effectively suppresses the quantum fluctuation and tends to exaggerate the stability of LRO in the ground state. As a result, the obtained |m| is larger than the exact value (as checked in the J2 = 0 case). The region of the magnetic LRO is enlarged and the non-magnetic region suppressed. Here, to phenomenologically study the effects of enhancing or reducing quantum fluctuations, we introduce artificial fluctuations by multiplying a tunable factor λ to the mean-field term H 0 . The total Hamiltonian becomes Heff = Hcn + λH 0 . λ < 1 enhances the fluctuation of Heff , and it mimics the effects of larger cluster or smaller

Q αβγ δ = 2[(Sα · Sβ )(Sγ · Sδ ) + (Sα · Sδ )(Sβ · Sγ ) − (Sα · Sγ )(Sβ · Sδ )] + 12 Sα · Sβ + Sγ · Sδ  + Sα · Sδ + Sβ · Sγ + Sα · Sγ + Sβ · Sδ + 14 .

Tr[e−β( H −h Mα ) Mα ] ˆ

χα = lim

(6)

Here α, β, γ , δ denote the four sites of a plaquette clockwise. At J2 = 0.5, the plaquette order parameter is evaluated on a 2 × 2 cluster as Q αβγ δ ≈ 0.988, very close to its saturated value 1.0. Evaluating Q αβγ δ on a larger cluster also gives nonzero result. However, this is the boundary effect of the cluster and does not support a true VBS state. It is an interesting open question how to incorporate the order parameter of various VBS state into the mean-field approximation. If such a mean-field theory does exist, considering that it tends to exaggerate the LRO, a negative result about the existence of VBS may rule out the possibility of VBS in the intermediate parameter regime. 5

J. Phys.: Condens. Matter 26 (2014) 115601

Y-Z Ren et al

Figure 6. Phase diagram of the anisotropic J1 –J2 model (δ = 0.5) in the T –J2 plane, obtained using 2 × 2 cluster mean-field theory.

Figure 5. Phase diagram on the λ–J2 plane obtained using 2 × 2

cluster. Squares on the solid line show the second-order transition between N´eel AFM and non-magnetic phases. Dots and diamonds on the solid lines represent the phase coexistence boundary of non-magnetic phase and the collinear phase. The triangles on the dashed line is the actual first-order phase transition line. The lines are a guide for the eye.

Squares on the eye guiding line is the second-order N´eel-to-paramagnetic phase transition. Solid dots represent the boundaries of the coexistence of N´eel (low T ) or paramagnetic phase (high T ) and the collinear AFM phase. The solid dot at (J2c = 0.653, Tc = 0.4) is the critical point above which the first-order transition changes into a second-order line (diamonds on the solid line).

spin. λ > 1 reduces the fluctuation of Heff and it mimics the effects of anisotropy or larger spin. Figure 5 shows a phase diagram in the λ–J2 plane. For larger λ, the LRO region is enlarged and the non-magnetic region shrinks. At λ = 1.4, the non-magnetic region diminishes, leading to a direct first-order transition between N´eel phase and collinear phase. At this point the phase diagram resembles that of the J1 –J2 Ising model where quantum fluctuation disappears. For smaller λ, the non-magnetic region enlarges and for sufficiently small λ, the LRO regime will disappear. This phase diagram resembles the phase diagram of anisotropic Heisenberg model [48]. Note that this artificial fluctuation does not influence the width of the coexistence region, showing that the first-order phase transition at J2c2 is robust against quantum fluctuations.

isotropic J1 –J2 model. For δ = 1, it becomes the Ising model. In figure 6, we show the phase diagram in the T –J2 plane for an intermediate spin anisotropy δ = 0.5, obtained using the 2 × 2 cluster. It is seen that for such a intermediate anisotropy, the non-magnetic ground state disappears from the phase diagram. At T = 0, as J2 increases, the ground state changes from N´eel AFM into collinear AFM through a first-order phase transition, accompanied by a coexistence regime in 0.53 < J2 < 0.77. Upon increasing temperature, the N´eel AFM in the small J2 regime and the collinear AFM in the large J2 regime extend up to a finite critical temperature where they disappear continuously. Although the 2 × 2 CMFT tends to overestimate the transition temperature, the finite critical temperature is qualitatively reasonable considering that the Mermin–Wagner theorem does not apply in this case. Meanwhile a precise Monte Carlo calculation shows that even small anisotropy will significantly enhance transition temperature in the nonfrustrated antiferromagnetic model [58]. For the intermediate J2 value, the coexistence region shrinks as T increases. Above the critical point (J2c = 0.653, Tc = 0.4), the first-order phase transition changes into the second-order one. In a small temperature window just below Tc , the coexistence occurs between the PM phase and the collinear AFM phase. The whole phase diagram is similar to that of the anisotropic J1 –J2 model studied in [48] using the effective field theory. Our results show that strong quantum fluctuations are crucial for the existence of the non-magnetic phase in the J1 –J2 Heisenberg model. It is noted that for the isotropic J1 –J2 model, a finite temperature phase transition in regime J2 > J2c2 may exist to break the C4 rotation symmetry of the lattice, according to Chandra et al [37, 46, 47]. However, what we obtained in figure 6 has nothing to do with this transition. It would be interesting to develop our CMFT for further study of this novel Ising transition. We leave this issue for the future.

3.2. Effect of anisotropic coupling

In this section, we study the effect of spin-anisotropic coupling which breaks the SU (2) symmetry of the original J1 –J2 Heisenberg Hamiltonian. It is expected that the spin anisotropy strongly suppresses the quantum fluctuations. Studying the effect of spin anisotropy can help us understand the role of quantum fluctuations in stabilizing the non-magnetic ground state. Also, the anisotropy will induce a finite temperature phase transition even in two-dimensions. A similar study of the spin anisotropy has been carried out for the J1x x z –J2 model [48]. In the following, we study the spin-anisotropy effect using the following Hamiltonian: X X Hˆ ani = (1 − δ)J1 Si · S j + (1 − δ)J2 Si · S j hi, ji

+ δ J1

X hi, ji

Siz Szj + δ J2

hhi, jii

X

Siz Szj .

(8)

hhi, jii

Here, the parameter δ is used to tune the strength of the spin anisotropy in this Hamiltonian. For δ = 0, Hani reduces to the 6

J. Phys.: Condens. Matter 26 (2014) 115601

Y-Z Ren et al

4. Summary

[18] Darradi R et al 2008 Phys. Rev. B 78 214415 [19] Dotsenko A V and Sushkov O P 1994 Phys. Rev. B 50 13821 [20] Siurakshina L, Ihle D and Hayn R 2001 Phys. Rev. B 64 104406 [21] Jiang H C, Yao H and Balents L 2012 Phys. Rev. B 86 024424 [22] Gong S S, Zhu W, Sheng D N, Motrunich O I and Fisher M P A 2013 (arXiv:1311.5962) [23] Murg V, Verstraete F and Cirac J I 2009 Phys. Rev. B 79 195119 [24] Yu J F and Kao Y J 2012 Phys. Rev. B 85 094407 [25] Furukawa S, Sato M, Onoda S and Furusaki A 2012 Phys. Rev. B 86 094417 [26] Wang L, Gu Z C, Verstraete F and Wen X G 2011 (arXiv:1112.3331) [27] Misguich G, Bernu B and Pierre L 2003 Phys. Rev. B 68 113409 [28] Capriotti L, Becca F, Parola A and Sorella S 2001 Phys. Rev. Lett. 87 097201 [29] Li T, Becca F, Hu W and Sorella S 2012 Phys. Rev. B 86 075111 [30] Beach K S D 2009 Phys. Rev. B 79 224431 [31] Hu W J, Becca F, Parola A and Sorella S 2013 Phys. Rev. B 88 060402 [32] Zhang X and Beach K S D 2013 Phys. Rev. B 87 094420 [33] Zhitomirsky M E and Ueda K 1996 Phys. Rev. B 54 9007 [34] Ueda H T and Totsuka K 2007 Phys. Rev. B 76 214428 [35] Mila F, Poilblanc D and Bruder C 1991 Phys. Rev. B 43 7891 [36] Isaev L, Ortiz G and Dukelsky J 2009 Phys. Rev. B 79 024409 [37] Chandra P, Coleman P and Larkin A I 1990 Phys. Rev. Lett. 64 88 [38] Takano K, Kito Y, Ono Y and Sano K 2003 Phys. Rev. Lett. 91 197202 [39] Lante V and Parola A 2006 Phys. Rev. B 73 094427 [40] Mambrini M, L¨auchli A, Poilblanc D and Mila F 2006 Phys. Rev. B 74 14442 [41] Sandvik A W 2012 Phys. Rev. B 85 134407 [42] Senthil T et al 2004 Science 303 1490 [43] Dai J H, Si Q, Zhu J X and Abrahams E 2009 Proc. Natl Acad. Sci. USA 106 4118 [44] Bruning E M et al 2008 Phys. Rev. Lett. 101 117206 [45] Mermin N D and Wagner H 1966 Phys. Rev. Lett. 17 1133 [46] Weber C et al 2003 Phys. Rev. Lett. 91 177202 [47] Capriotti L, Fubini A, Roscilde T and Tognetti V 2004 Phys. Rev. Lett. 92 157202 [48] Viana J R and de Sousa J R 2007 Phys. Rev. B 75 052403 [49] Wang H Y 2012 Phys. Rev. B 86 144411 [50] Weiss P 1907 J. Phys. Radium 6 661 [51] For a recent development, see Yamamoto D 2009 Phys. Rev. B 79 144427 [52] Tong N H and Pu F C 2000 Phys. Rev. B 62 9425 Tong N H, Shen S Q and Pu F C 2001 Phys. Rev. B 64 235109 Tong N H, Shen S Q and Bulla R 2004 Phys. Rev. B 70 085118 [53] Sandvik A W 1997 Phys. Rev. B 56 11678 Buonaura M C and Sorella S 1998 Phys. Rev. B 57 11446 [54] Bethe H A 1953 Proc. R. Soc. A 150 552 [55] Peierls R E 1936 Proc. Camb. Phil. Soc. 32 477 [56] Weiss P R 1948 Phys. Rev. 74 1493 [57] Oguchi T 1955 Prog. Theor. Phys. 13 148 [58] Fouet J B, Mambrini M, Sindzingre P and Lhuillier C 2003 Phys. Rev. B 67 054411

In summary, we use the cluster mean-field theory to study the J1 –J2 Heisenberg model on a square lattice. For small, intermediate, and large J2 /J1 regime, we obtain the N´eel AFM phase, the non-magnetic phase, and the collinear AFM phase, respectively. The N´eel-to-non-magnetic transition is found to be of second order, and the non-magnetic-to-collinear transition is of first order. The respective critical values J2c1 and J2c2 are found to converge rapidly with increasing L. From the largest 4 × 4 cluster we obtain J2c1 ≈ 0.42, which is very close to the results of the 2 × 2 cluster 0.41. Extrapolating the cluster size to infinity, we obtain J2c2 ≈ 0.59. Both J2c1 and J2c2 agree with the previous results very well. We also investigate the finite temperature phase diagram of the spin- anisotropic J1 –J2 model. It is demonstrated that with sufficiently strong Ising coupling, the non-magnetic ground state is unstable towards a first-order phase transition. Above the critical temperature Tc , the first-order phase transition changes into a second-order transition. Our results show that the cluster mean-field theory is not only a very useful tool for studying classical phase transitions [51], but can also give surprisingly accurate ground state phase boundaries for the frustrated quantum magnet. Acknowledgments

This work is supported by National Program on Key Basic Research Project (973 Program) under Grant No. 2009CB29100, 2012CB821402, and 2012CB921704, and by the NSFC under Grant No. 91221302 and 11074302. References [1] Anderson P W 1987 Science 235 1196 [2] Lee P A, Nagaosa N and Wen X G 2006 Phys. Mod. Phys. 78 17 [3] Chandra P and Doucot B 1988 Phys. Rev. B 38 9335 [4] Melzi R et al 2000 Phys. Rev. Lett. 85 1318 Melzi R et al 2001 Phys. Rev. B 64 024409 [5] Yildirim T 2008 Phys. Rev. Lett. 101 057010 Ma F, Lu Z Y and Xiang T 2008 Phys. Rev. B 78 224517 [6] Figueirido F et al Phys. Rev. B 41 4619 [7] Dagotto E and Moreo A 1989 Phys. Rev. Lett. 63 2148 [8] Schulz H J and Ziman T A L 1992 Europhys. Lett. 18 355 [9] Richter J and Schulenburg J 2010 Eur. Phys. J. B 73 117 [10] Capriotti L and Sorella S 2000 Phys. Rev. Lett. 84 3173 [11] Mambrini M et al 2006 Phys. Rev. B 74 144422 [12] Gelfand M P, Singh R R P and Huse D A 1989 Phys. Rev. B 40 10801 [13] Gelfand M P 1990 Phys. Rev. B 42 8206 [14] Singh R R P, Weihong Z, Hamer C J and Oitmaa J 1999 Phys. Rev. B 60 7278 [15] Singh R R P et al 2003 Phys. Rev. Lett. 91 017201 [16] Sirker J, Weihong Z, Sushkov O P and Oitmaa J 2006 Phys. Rev. B 73 184420 [17] Schmalfuß D et al 2006 Phys. Rev. Lett. 97 157201

7

Cluster mean-field theory study of J1-J2 Heisenberg model on a square lattice.

We study the spin-1/2 J1-J2 Heisenberg model on a square lattice using the cluster mean-field theory. We find a rapid convergence of phase boundaries ...
429KB Sizes 2 Downloads 3 Views