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Coupled-cluster calculations for the ground- and excited-states of the spin-half XXZ model

The coupled-cluster method is applied to the spin-1/2 antiferromagnetic XXZ model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, sublattice magnetization, and the critical anisotropy when comparing with the approximation including the two-body correlations alone. We also obtain the full excitation spectrum which is in good agreement with the quantum Monte Carlo results and the high-order spin-wave theory.


Introduction
The coupled-cluster method (CCM) is one of the most precise microscopic formulations of quantum many-body theories [1][2][3][4][5][6][7][8][9]. There is a large number of successful applications of CCM to a wide range of physical and chemical systems. In particular, the applications of CCM to spin systems on discrete spatial lattices have produced one of the most accurate results [10][11][12][13][14][15][16][17][18][19][20]. Several approximation schemes have been developed for the application of the CCM to the spin lattice systems. Two such successful schemes are the socalled SUBn scheme in which all correlations of any range for up to n spins are retained and the localised LSUBm scheme in which m or fewer adjacent spin sites over all distinct locales on the lattice are retained. Other highorder localized approximation schemes such as DSUBm [19] and LPSUBm [20] have also been employed. Up to now, most recent studies have presented results for the high-order calculations mainly based on the LSUBm scheme in which the long-range order correlations are ignored [13][14][15][16][17][18][19][20]. In this paper we present results for the ground and excitation states for an antiferromagnetic square lattice by combining the SUB2 and LSUB4 approximation schemes (SUB2+LSUB4). Due to inclusion of the two-body long-range correlations, we are able to obtain improved results for the ground-state properties, including the critical value of the anisotropy, as well as the full excitation spectrum which is difficult to calculate by using the localised approximation scheme alone.
The spin-1/2 antiferromagnetic XXZ Heisenberg Hamiltonian in terms of spin rasing s + and lowering s − operators is given by, where ∆ is the anisotropy and the sum on i, j runs over all the nearest neighbor pairs once. The isotropic Heisenberg model is given by ∆ = 1. Classically, the ground-state of Eq. (1) is ferromagnetic, with all spins aligned along z-axis for all lattice when ∆ ≤ −1; for |∆| ≤ 1 it is antiferromagnetic for all bipartite lattice with all spins are aligned along some arbitrary direction in the xy−plane; for ∆ ≥ 1 it is antiferromagnetic with spins aligned along (±) directions of the z-axis. The classical Néel ground state with all up-spins on one sublattice and all down-spins on the other is chosen to be the model state in our CCM calculation. In this article, as before, we use index i to label sites of the down-spin sublattice and index j for the up-spin sublattice. It is useful to introduce a transformation for the local spin axes of one sublattice. This is achieved by rotating all up-spins by 180 • around the y−axis and hence every spin of the system points down in the Néel model state with s z = −1/2. This transformation is given by for all j-sublattice operators, s ∓ = s x ∓ is y → −s ± and s z → −s z . The Hamiltonian of Eq. (1) after the rotation is rewritten as, The ket and bra ground states of the CCM are given in terms of correlation operators S andS a respectively, where the model state |Φ is the rotated Néel state as mentioned earlier with all the spins pointing down, C † I and C I are the so-called configurational creation and destruction operators respectively with the nominal index I labeling the multi-spin raising and lowering operators as, with the ket-and bra-state correlation coefficients S I andS I to be determined variationally as shown below. We note that the bra-state Ψ | and the ket-state |Ψ are not manifestly hermitian conjugate to one another. The normalization conditions Ψ |Ψ ≡ Φ|Ψ ≡ Φ|Φ ≡ 1 is satisfied by construction. The ground-state Schrödinger equation, H|Ψ = E g |Ψ , can now be written as,Ĥ |Φ = E g |Φ , where the similarity-transformed HamiltonianĤ can be written in terms of a series of nested commutations as, The expectation value of an arbitrary operator O can be written as, The correlation coefficients {S I ,S I } are determined variationally by the following equations, In the followings we will consider a specific approximation, namely the SUB2+LSUB4 scheme as defined earlier, by a similar truncation in both S andS.
2 Ground-state energy for the SUB2+LSUB4 approximation scheme As mentioned in Introduction, the SUB2 approximation retains two-spin-flip configurations of all orders. In the SUB4 scheme, additional 4-spin correlations are also included. We hence write the SUB4 ket-state operators as, where b i,j and g i 1 ,i 2 ;j 1 ,j 2 are the two-spin-flip and four-spin-flip correlation coefficients respectively. The full SUB4 scheme equations were obtained before [11], but they are difficult to solve. Here we consider the SUB2+LSUB4 scheme which retains ten local configurations as shown in Fig. 1, in additional to the other two-body high-order coefficients of the SUB2 scheme. As described in general by Eq. (10), the SUB4 approximation consists of two sets of equations, the two-spin-flip and four-spin-flip equations. The two-spin-flip equations are given by, from which we obtain the subset of the SUB2+LSUB4 approximation as, where ρ is the nearest-neighbor index vector with four possible values for a square lattice, ρ 0 is any one of them, G α with α = 1, 2, 3 are defined as, and ρ 3 are 2D vectors containing ρ with ρ 3a =(3ρ x ,0), and ρ 3b =(2ρ x , ρ y ). The four-spin-flip equations are similarly given by, from which we obtain the following four coupled equations, These nonlinear equations for the SUB2+LSUB4 scheme are solved firstly by Fourier transformation of Eq. (14) and then by iteration method for Eqs. (17)- (20). In particular, Eq. (14) becomes after Fourier transformation, which is easily solved with the physical solution, where the constant K, and the function E(q) are given by respectively, and where γ(q), γ 3a (q) and γ 3b (q) are defined respectively by, γ 3b (q) = 1 2 (cos 2q x + cos q y ), with the constants k 2 1 , k 2 2 and k 2 3 defined by, In any approximation scheme of CCM, the ground-state energy for the Hamiltonian of Eq. (2) is always given by [11], where z is the coordination number. In Fig. 2 and Table 1, we present numerical results for the ground-state energy as a function of the anisotropy parameter ∆ in our SUB2+LSUB4 scheme, together with those of the SUB2, SUB2+g a 4 and LSUB4 schemes obtained earlier [11] for comparison. As can be seen, the SUB2+LSUB4 results are lower than any of the other schemes. Furthermore, the critical value of the anisotropy ∆ c =0.847 beyond which the solution of Eq. (22) becomes imaginary, is also improved and closer to the expected value of 1 than that of the SUB2 scheme (0.798) or that of the SUB2+g a 4 scheme (0.818). In the high-order LSUBm scheme [16], the critical values are obtained as ∆ c = 0.763 and 0.843 for m = 6 and 8 respectively, and ∆ c = 1 after extrapolation of m = ∞ is made. The corresponding value of ∆ c in the localized schemes are 0.637 in DSUB10 [19] and 0.766 in LPSUB5 [20]. The physics of this critical point was discussed in details in Ref. [11].  Figure 2: The ground state-energy per spin as a function of ∆ for spin-1/2 XXZ model in the full SUB2, SUB2+g a 4 and SUB2+LSUB4 schemes. The critical terminating points for each scheme are also indicated.

Staggered Magnetization
The staggered magnetization for a general spin quantum number s can be defined as, where l runs over all the lattice sites for our rotated Hamiltonian of Eq. (2).
In the SUB2+LSUB4 scheme we obtain, where two-body and four-body bra-state coefficientsb r andg 4 are determined by the second variational Eqs. (11). We solve these equations for the brastate in similar fashion as for the ket-state, namely by Fourier transformation for the two-body coefficients and by iteration methods for the four-body coefficients. We leave the details to Appendix and show the results in Fig. 3. We find that at the critical ∆ c , M c = 0.649 in our SUB2+LSUB4 scheme, compared with M c = 0.663 in the SUB2+g a 4 scheme and M c = 0.682 in the SUB2 obtained earlier [11]. Our SUB2+LSUB4 result is in good agreement with M = 0.6138 of the 3rd-order spin-wave results [21], M = 0.614 of the series expansion calculations [22], M = 0.615 of the quantum Monte Carlo calculations [23] at ∆ c = 1. The highe-order LSUBm scheme with m=8 produces M = 0.705 at ∆ = 1 before extrapolation and M = 0.616 after an extrapolation has been made [16]. The corresponding values of M at ∆ = 1 are 0.712 in DSUB11 scheme [19] and 0.708 in LPSUB6 scheme [20].

Spin-wave excitation spectra
The excited state in CCM is given by applying an excitation operator X e to the ket-state wave function, where X e in general is written in terms of the configurational creation operators C + I only as, with the excitation coefficient χ e I . From the Schrödinger equation H|Ψ e = E e |Ψ e , it is straightforward to derive the following equation for the excitation coefficient, where ε e ≡ E e − E g is the excitation energy. Here, we consider the spin-wave excitations by including only a single spin-flip operator, C † I s + i , similar to the SUB2 scheme as before [11]. After Fourier transform we obtain the energy spectrum in this linear approximation as, where K and E(q) are as defined before in Eqs. (23) and (24), and z is the coordination number. We present the excitation gap, ε(q) at q = 0, as a function of ∆ in Fig. 4. As can be seen from the figure, the energy gap in the SUB2+LSUB4 scheme is smaller than that of the SUB2 and SUB2+g a 4 schemes, implying that the energy gap is reduced in the higher-order approximations. For all these three schemes, the energy gap disappears at their corresponding critical anisotropy ∆ c . It is interesting to compare our results for the energy gap with that of the high-order LSUBm scheme [16]. At ∆ = 1 our SUB2+LSUB4 gap value is ε(0) = 1.05 while the LSUB4 and LSUB8 values are much lower at ε(0) = 0.851 and 0.473 respectively. By employing an extrapolation, the LSUBm scheme produces an energy gap close to zero, corresponding to the SUB2+LSUB4 result at the critical ∆ c . The much lower energy gap values away from the critical region by the higherorder LSUBm scheme are clearly due to the inclusion of the higher-order correlations in the excitation operators whereas we only include the linear excitation operators in our calculations as given by Eq. (33) with C † I s + i . However, our SUB2+LSUB4 scheme has an advantage of capable of producing the full energy spectra due to inclusion of the long-range two-body correlations as discussed below.
In Fig. 5, we present our SUB2+LSUB4 results for the spin-wave energy spectrum of Eq. (35) at ∆ c together with that of the SUB2 results [11], and at ∆ = 1, the results of the linear spin-wave theory (LSWT), the series expansion calculations (SE) [24], and quantum Monte Carlo calculations (QMC) [25]. The spin-wave velocity correction factor to the linear spin-wave theory in our SUB2+LSUB4 scheme is given by K c = 1.23, in good agreement with 1.18±0.02 from the series expansion and 1.21±0.03 from the quantum Monte Carlo calculations.  Figure 4: The excitation energy gap ε(0) for the 2D spin-1/2 XXZ Heisenberg model as a function of ∆, for the full SUB2, SUB2+g a 4 and SUB2+LSUB4 schemes. The two gap values at ∆ = 1 are given by the LSUB4 scheme (•) and LSUB8 scheme ( ) of Ref. [16] where the high-order excitation correlations are included as discussed in the text.  Figure 5: The spin-wave excitation spectra for the 2D spin-1/2 XXZ Heisenberg model at ∆ c for the CCM (SUB2 and SUB2+LSUB4) results, and at ∆ = 1 for the linear spin-wave theory (LSWT), the series expansion (SE) [24], and quantum Monte Carlo calculations [25]. The energy spectra in (a) are for q x = q y and those in (b) are for q y = 0.

Summary and Conclusion
In summary, we have obtained here numerical results for the ground-state energy, sublattice magnetization, and excitation energy for the spin-half squarelattice antiferromagnetic XXZ model using the SUB2+ LSUB4 scheme of CCM. We find that our results for the ground-state properties in general are improved when compared with those obtained by the SUB2 or LSUB4 scheme alone. In particular, due to inclusion of the two-body long-range-order correlations, the SUB2+LSUB4 scheme is capable of producing improved results around the critical regions of the anisotropy, the excitation gaps at q = 0, and the full spin-wave energy spectra. Good agreement for the spin-wave spectra is found with the high-order series expansion and the quantum Monte Carlo calculations. This is contrast to the recent state-of-the-art calculations of the LSUBm scheme using computer algebra, where good results of the critical properties have been obtained after an extrapolation in the limit m → ∞ is made [16][17][18][19][20]. Away from the critical points, the long-range correlations are less important and the high-order LSUBm clearly provides better numerical results due to inclusion of the high-order local correlations. We believe that the different approximation schemes in CCM complement each other for a more complete description of the physics of the spin-lattice Hamiltonian model, and in particular the SUB2+LSUBm scheme as presented here has the advantage of producing the full excitation energy spectrum. Further improvement for the excitation energies away from the critical points can be obtained by including the higher-order local correlations in the excitations operator X e as demonstrated in the LSUBm scheme of Ref. [16]. It will be interesting to apply our SUB2+LSUBm scheme to other models such as the spin-1/2 XY model.
Using the solution for Γ(q) of Eq. (22) with the definition for E(q) in Eq. (24), the physical solution of Eq. (49) for the bra-state is, where the constant D is defined as, The value of D can be determined self-consistently as follows. We first rewrite Eq. (48) as an integral in Fourier space as, The bra-state coefficientb r is obtained by inverse Fourier transformation of Γ(q),b r = 1 π 2 π 0 e −ir.q Dγ(q) + 2A(q) 4KE(q) dq, and in particular,b 1 is given by, Combining Eqs. (51),(52) and (54), we obtain the following expression for D, where the constant c is given by, with the integral I defined as, Using the above self-consistency equations forb 1 ,b a 3 ,b b 3 , D and Ξ and by iteration method, we obtain the numerical values forg a 4 ,g b 4 ,g c 4 andg d 4 of the four-body bra-state coefficients. The staggered magnetization is then calculated by using Eq. (31).