The spin- - Heisenberg antiferromagnet on the square lattice: Exact diagonalization for spins
We present numerical exact results for the ground state and the low-lying excitations for the spin- - Heisenberg antiferromagnet on finite square lattices of up to sites. Using finite-size extrapolation we determine the ground-state energy, the magnetic order parameters, the spin gap, the uniform susceptibility, as well as the spin-wave velocity and the spin stiffness as functions of the frustration parameter . In agreement with the generally excepted scenario we find semiclassical magnetically ordered phases for and separated by a gapful quantum paramagnetic phase. We estimate and .
75.10.Jm Quantized spin models
75.45.+j Macroscopic quantum phenomena in magnetic systems
The spin- Heisenberg antiferromagnet (HAFM) with nearest-neighbor (NN) and frustrating next-nearest-neighbor (NNN) coupling (- model) on the square lattice has attracted a great deal of interest during the last twenty years (see, e.g., Refs. [1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 9, 12, 13, 16, 14, 17, 15, 18, 19, 20, 21, 22, 23, 24, 26, 25, 33, 29, 27, 30, 28, 31, 32, 34] and references therein). The corresponding Hamiltonian reads
where the sums over and run over all NN and NNN pairs, respectively, counting each bond once. In what follows we set . The synthesis of layered magnetic materials [36, 37] which can be described by the spin- - model has stimulated a renewed interest in this model. Another new promising perspective is also opened by the recently discovered layered Fe-based superconducting materials  which may have a magnetic phase that can be described by a - model with spin quantum number [39, 40, 41].
The - model is a canonical model to study quantum phase transitions between semiclassical magnetically ordered ground state (GS) phases and novel non-magnetic quantum phases. Moreover, the - model might be also a candidate for a deconfined critical point separating a semiclassical magnetic phase from non-magnetic quantum phase [20, 30, 31, 42, 43].
Due to frustration highly efficient quantum Monte-Carlo codes , such as the stochastic series expansion suffer from the minus sign problem. Therefore, many other approximate methods, e.g., the Green’s function method [15, 22], the series expansion [13, 18, 16, 20, 31], the Schwinger boson approach , the coupled cluster method [12, 22, 29, 30], variational techniques [17, 33], the path integral quantization , the cluster effective-field theory [24, 27], the hierarchical mean-field approach , the projected entangled pair states method  as well as the stochastic state selection method  were used to find the GS phases of the model. One of the most powerful methods is the Lanczos exact diagonalization method which yields numerically exact results for finite lattices. It has been succesfully applied to the - model in various papers, see, e.g., Refs. [2, 3, 4, 6, 7, 8, 9, 11, 14, 21]. In particular, the paper of Schulz and coworkers  presenting for the first time results for lattices up to sites has set a benchmark for such studies and provides detailed information on the model. In addition, these numerical exact results are often used to test new approximate methods, see, e.g., Refs. [35, 26, 30].
Due to the progress in the computer hardware and also to the increase in the efficiency in programming the Lanczos algorithm, very recently the GS and low-lying excitations of the unfrustrated (i.e., ) spin- HAFM  and of a spin- Heisenberg model with ring exchange  have been calculated for a square lattice with sites. The largest two-dimensional quantum spin model for which the GS has been calculated so far is the spin- HAFM on the star lattice with sites . Note, however, that for higher sectors of which are relevant for finite magnetic fields much larger system sizes can be considered, see, e.g., Ref. .
As a result of numerous investigations it seems to be clear now that the - model exhibits two magnetically long-range ordered phases at small and at large separated by an intermediate quantum paramagnetic phase without magnetic long-range order (LRO) in the parameter region , where and . The GS at low exhibits semiclassical Néel magnetic LRO with the magnetic wave vector . The GS at large shows so-called collinear magnetic LRO with the magnetic wave vectors or . These two collinear states are characterized by a parallel spin orientation of nearest neighbors in vertical (horizontal) direction and an antiparallel spin orientation of nearest neighbors in horizontal (vertical) direction.
The nature of the transition between the Néel and the quantum paramagnetic phase as well as the properties of the quantum paramagnetic phase and the precise values of the transition points are still under discussion [1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 16, 14, 17, 15, 18, 19, 20, 21, 22, 23, 24, 26, 25, 33, 29, 27, 30, 28, 31].
In this paper we present new results for the GS and the low-lying excitations of the - model for a finite lattice of sites shown as in Fig. 1. These results are presented in detail, on the one hand, as benchmark data for approximate methods. On the other hand, we combine our new results with the known results  (which we have recalculated) to improve the finite-size extrapolation. For all the finite lattices periodic boundary conditions were imposed.
2 Results of Lanczos exact diagonalization for a finite square lattice of sites
Although Schulz and coworkers performed the Lanczos diagonalization for the spin- - HAFM on the finite square lattice of sites more then 10 years ago the corresponding Lanczos diagonalization for the finite lattice of sites is still a challenging problem. The number of basis states on the square lattice in the GS symmetry sector is for compared to for . The GS belongs to a translational symmetry with .
The most important quantities in order to analyze semiclassical GS magnetic ordering are appropriate order parameters corresponding to the classical Néel and collinear LRO. Following Schulz et al.  we use here the -dependent susceptibilities (square of order parameters) defined as
For the magnetic wave vector the quantity is the relevant order parameter for the Néel ordered GS phase present at small , whereas with magnetic wave vectors or corresponds to the collinear magnetic LRO present at large .
In table 1 we give the singlet GS energy, , the energy of the first triplet excitation , the energy of the first singlet excitation , as well as the -dependent susceptibilities , . The low-lying energies are also displayed in Fig. 2 where more data points are included than shown in table 1. It is obvious that the first excited singlet state becomes very close to the singlet GS near . Since both states belong to the same lattice symmetry, a level crossing is forbidden due to the von Neumann-Wigner theorem . The numerical data shown in the right inset of Fig. 2 clearly give a numerical check of this general theorem. The energies of the low-lying states show the typical frustration induced maximum known from previous ED calculations for smaller lattices, see, e.g., Refs. [2, 4, 6]. Since we have data for the singlet GS and the first triplet excitation, we can also give results for the spin gap , see Fig. 3. Note that the spin gap was not presented in the paper of Schulz et al.. Hence we compare in Fig. 3 spin gap data for , , and . It is obvious that with increasing the gap starts to grow at . It reaches a maximum at about and becomes again small for . This behavior yields an indication of a gapful quantum paramagnetic GS phase around . However, it is also obvious that there is no monotonous finite-size behavior in that parameter region, i.e., a reliable finite-size extrapolation of the spin gap around is not possible,
Next we consider the spin-spin correlation functions . There are altogether 11 different correlations functions given in table 2 for the same data points as in table 1. Graphically the variation of some selected correlation functions are shown inf Fig. 4. For small the spin-spin correlations are quite strong and according to the Néel order we have for connecting sites of sublattices and but for connecting sites within a sublattice. Increasing the frustration leads to a weakening of the spin-spin correlation. This weakening is particularly strong for larger separations. For strong frustration around all correlation functions except for are very small which is an indication for a magnetically disordered phase. Beyond the increasing strength of correlation functions for lattice vectors connecting sites within the same sublattice (or ) indicates the emerging collinear LRO. The corresponding order parameters defined in Eq. (2) and listed in table 1 are shown in Fig. 5. The suppression of magnetic order around is again obvious. Although there is no level crossing (see the discussion above) we observe in Figs. 4 and 5 a sharp change of the magnetic quantities near , i.e., at that point where both singlet levels become very close to each other (see Fig. 2, right inset). This behavior might be a hint on a first-order transition between the phase with collinear LRO and the magnetically disordered phase, see also, e.g., Refs. [4, 13, 16, 22, 30].
Another interesting point is the breakdown of the Marshall-Peierls sign rule (MPSR) at . Writing the GS as , where is an Ising basis state of typical form , the MPSR determines the sign of the coefficients . The MPSR has been proved exactly for bipartite lattices and arbitrary site spins by Lieb, Schultz and Mattis . As pointed out in several papers the knowledge of the sign of the is of great importance in different numerical methods, e.g. for the construction of variational wave functions [7, 17, 52, 33], in quantum Monte-Carlo methods (which suffer from the sign problem in frustrated systems ) and also in the density matrix renormalization group method, where the application of the MPSR has substantially improved the method in a frustrated spin system .
For the - model on the square lattice the violation of the MPSR was considered as one signal of destruction of magnetic LRO [7, 8, 11, 12, 17, 33]. It was found that the MPSR (proved exactly only for ) survives till about . The exact Lanczos results for the ground state presented in Refs. [7, 8] and  were, however, restricted to systems of up to sites. Here we present data for , , and in Fig. 6. From Fig. 6 it is obvious that the MPSR is valid in very good approximation even till . The MPSR starts to be significantly violated beyond , where the Néel LRO breaks down.
3 Finite-size extrapolation
We use now our new data for together with the known data for to perform a finite-size extrapolation. Note that we do not include the data for the lattice with sites. This lattice has an extra symmetry because it is equivalent to a hypercube in 4 dimensions. It was argued that, therefore, the lattice exhibits an anomalous behavior . The finite-size extrapolation rules for the two-dimensional HAFM are well known [55, 56, 57, 58, 59]. The extrapolation of the magnetic order parameters yields an estimate of the transition points between the semiclassically ordered phases (Néel and collinear) and the magnetically disordered quantum phase. In addition, since the scaling behavior is related to the low-energy degrees of freedom of the model, via the extrapolation procedure one can extract the dependence of other quantities such as the spin-wave velocity and the spin stiffness . It has been demonstrated that Lanczos data for finite lattices of up to N=36 sites can provide quite accurate data for these quantities in the case of unfrustrated lattices [4, 58, 60].
In the seminal paper of Schulz et al.  the finite-size extrapolation procedure as well as some particular problems of the extrapolation appearing for the - model were discussed in great detail. We follow here the lines of that paper, but do not repeat a detailed discussion of the extrapolation scheme.
where is the spin-wave velocity and is the spin stiffness. The corresponding order parameter for the collinear LRO at large is defined as  . The finite-size behavior of is
The results for and are shown in Fig. 7. As expected we find a magnetically disordered GS phase around . The transition points are determined to and , i.e., the range of the magnetically disordered GS phase obtained here is slightly larger than predicted, e.g., from series expansion [13, 16, 18, 20] or coupled cluster approach . For the unfrustrated square lattice HAFM () we obtain which is in good agreement with corresponding results obtained by other methods, e.g., 3rd order spin-wave theory () , quantum Monte Carlo ()  or coupled cluster method () .
The finite-size behavior of the GS energy is given by
The extrapolated energy is shown in Fig. 8. Because of an irregular finite-size behavior near the maximum in (see inset in Fig. 8) the extrapolation becomes unreliable around (this parameter region is indicated by the dotted line in the main panel of Fig. 8). Nevertheless, one can speculate that the kink in the extrapolated GS energy near might be another hint for a first-order transition at between the semiclassical collinear phase and the quantum paramagnetic phase. The extrapolated GS energy, , for is again in very good agreement with corresponding results obtained by other methods, e.g., 3rd order spin-wave theory () , quantum Monte Carlo ()  or coupled cluster method () .
We can also perform a finite-size extrapolation of the gap to the first triplet excitation (spin gap) . The corresponding formula is
The results for are shown in Fig. 9. In the magnetically ordered Néel and collinear phases gapless Goldstone modes exist and consequently the spin gap should vanish. The finite values of the extrapolated gap for and give a hint of the limits of precision of the finite-size extrapolation of the spin gap. Nevertheless, it becomes obvious that there is a significant increase of in the quantum paramagnetic phase . Note that for larger the energy scale of the system becomes proportional to that explains the increase in with for . A finite spin gap for would be consistent with the findings of many papers which suggest that the quantum paramagnetic phase exhibits valence-bond order, see, e.g., Refs.  and  and references therein. Note, however, that our finding is in contrast to recent DMRG studies on frustrated odd-leg ladders  where arguments for a gapless spectrum of the - model on the square lattice have been given.
Next we consider the uniform susceptibility which is related to the spin gap by . For the finite-size extrapolation of the uniform susceptibility we use 
The results for and the extrapolated value are shown in Fig. 10. In a gapful quantum paramagnetic phase, i.e., for , should vanish. Indeed, we observe a vanishing of around . However, the parameter region where is zero is clearly smaller than that where the order parameters and vanish. At we obtain which is in good agreement with data obtained by other methods, e.g., quantum Monte Carlo () , series expansion () , third-order spin-wave theory () , or coupled cluster method () .
So far we have considered quantities which can be calculated directly for each finite lattice and for which the corresponding value for is the leading term in a finite-size extrapolation formula, see Eqs. (3), (4), (5), (6), and (7). A test of the accuracy the finite-size extrapolation is given by the comparison with best available data for the unfrustrated limit (), see above. In addition to the direct calculation of certain magnetic quantities the finite size-extrapolation allows also an indirect determination of the spin-wave velocity and the spin stiffness which enter the finite size-extrapolation formulas (3) and (5) as prefactors of the leading finite-size corrections. First we can determine the spin-wave velocity via Eq. (5). Using this result and also we find then the stiffness via Eq. (3). Since is proportional to , it vanishes at the same point . Alternatively, we can use the hydrodynamic relation , see, e.g., Refs.  and , to determine the stiffness . We show the spin-wave velocity and both values of the stiffness in Fig. 11. Although, obtained via Eq. (3) and obtained via the hydrodynamic relation are quite different the qualitative behavior of both is similar to that obtained by direct calculation of , e.g., using Schwinger boson  or coupled cluster approach . The spin-wave velocity also decreases with growing . However, remains finite at the magnetic-nonmagnetic transition at as predicted from the non-linear sigma model . At we get , and . These values are significantly lower than corresponding values obtained by quantum Monte Carlo method , 3rd order spin-wave theory  or coupled cluster method [61, 67]. Hence we may argue that the indirect determination of magnetic quantities via the prefactors of the leading finite-size corrections is less accurate.
In this paper we have presented detailed information on the GS and the low-lying excitations for the spin- - HAFM on a finite square lattice of sites. These data for the GS energy, the lowest triplet and singlet excitations, the spin-spin correlation functions and the magnetic order parameters may serve as benchmarks for approximate methods.
Including data for finite lattices of sites we have performed finite-size extrapolations of several magnetic quantities. Using the well-known extrapolation formulas for the GS energy, the order parameters, the spin gap, and the uniform susceptibility we have determined these quantities for . To estimate the accuracy of the extrapolated quantities we have compared them with best available results for the unfrustrated limit , where the high-order spin-wave theory and the quantum Monte Carlo method work well. At the extrapolated GS energy is in excellent agreement with spin-wave theory and quantum Monte Carlo results. The deviation of the Néel order parameter obtained by the finite-size extrapolation from the spin-wave and quantum Monte Carlo results is only about 1%.
Besides these quantities which have been determined directly for each finite lattice, in addition, we have used the formulas of finite-size extrapolation to determine the spin-wave velocity as well as the spin stiffness which appear in the prefactors of the leading finite-size corrections of the GS energy and the Néel order parameter. This ’indirect way’ to determine spin-wave velocity and stiffness, however, yields only qualitative agreement with known results.
From the extrapolated magnetic order parameters
we find the transition point from the magnetically ordered Néel phase
to the gapful quantum paramagnetic phase
to and transition point between the magnetically ordered
collinear phase and the quantum paramagnetic phase
Acknowledgment: This work was supported by the DFG (Ri615/16-1).
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