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Journal of the Physical Society of Japan

String Correlation of Quantum Antiferromagnetic Spin Chains withS=1 and 2

Yasuhiro Hatsugai

Published Nov 15, 1992• 25 citations
Publisher

Abstract

Antiferromagnetic Heisenberg spin chains for S =1 and S =2 are treated numerically. We investigate a generalized θ -string correlation function O θ ( j )=< S z 1 exp ( i θ ∑ j -1 k =1 S z k ) S j z > which is a generalization of the string correlation function of den Nijs-Rommelse. When the value of θ lies in an appropriate region, the phase of O θ ( j ) is fixed and O θ ( j ) behaves smoothly as a function of j. This means alternative sign changes of the usual spin-spin correlation O θ =0 ( j ) are well canceled in O θ ( j ) for such values of θ . The cancellation is best at θ =π and π /2 for the S=1 and 2 cases, respectively. The generalized string which appears in the θ -string correlation function plays a role similar to that of a fictitious gauge potential in the definition of the hidden order parameter of the fractional quantized Hall state by Girvin and MacDonald.

DOI
10.1143/jpsj.61.3856

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Antiferromagnetic Heisenberg spin chains for S =1 and S =2 are treated numerically. We investigate a generalized θ -string correlation function O θ ( j )=< S z 1 exp ( i θ ∑ j -1 k =1 S z k ) S j z > which is a generalization of the string correlation function of den Nijs-Rommelse. When the value of θ lies in an appropriate region, the phase of O θ ( j ) is fixed and O θ ( j ) behaves smoothly as a function of j. This means alternative sign changes of the usual spin-spin correlation O θ =0 ( j ) are well canceled in O θ ( j ) for such values of θ . The cancellation is best at θ =π and π /2 for the S=1 and 2 cases, respectively. The generalized string which appears in the θ -string correlation function plays a role similar to that of a fictitious gauge potential in the definition of the hidden order parameter of the fractional quantized Hall state by Girvin and MacDonald.
We investigate the string correlation functions proposed by den Nijs and Rommelse for S=1 Heisenberg antiferromagnets in one dimension. The Hamiltonian with D${\mathrm{\ensuremath{\Sigma}}}_{\mathit{i}}$(Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​)2{)}^{2})2 term is diagonalized by the Lanczos method to obtain the ground state. We calculate both the usual spin-spin correlation functions and the string correlation functions not only in the z direction (quantized direction) but also in the x direction to investigate the Z2{\mathit{Z}}_{2}Z2​\ifmmode\times\else\texttimes\fi{}Z2{\mathit{Z}}_{2}Z2​ symmetry breaking recently proposed by Kennedy and Tasaki. We find that the long-range string correlation, which is argued to exist in the Haldane disordered phase, in fact, exists at the Heisenberg point D=0 by a finite-size analysis. We can show explicitly that the string correlation in the x direction signifies the difference between the Haldane phase and the N\'eel phase, which appears for the D0, \ensuremath{\Vert}D\ensuremath{\Vert}\ensuremath{\gtrsim}1 case. In the large-D (D\ensuremath{\gtrsim}1) phase, all spin-spin correlations are of short range as expected. There is a significant enhancement in the usual and string correlations in the x direction at the boundary between the Haldane phase and the large-D phase.
We demonstrate the existence of nonlocal topological (string) order in half-integer-spin antiferromagnetic Heisenberg chains on macroscopic scale on the basis of analytical scaling analysis and density matrix renormalization group calculations. Strong numerical evidence leads to a conjecture that chains with S = (2m-1)/2 and m (m = integers) belong to the same topological class defined by the topological angle theta/pi = 1/m that plays a role similar to the fictitious gauge field in the fractional quantum Hall effect.
We study the string correlation functions proposed by den Nijs and Rommelse for the S=1 XXZ Hamiltonian by an exact diagonalization method. Using a finite size analysis, we estimate values of the string order parameters in the thermodynamic limit. Using the extrapolated values of the string order parameter we investigate the phase boundary between the two expected phases of the model, namely, the massive Haldane phase and the x-y--like massless phase. In the massless phase the exponent of the string correlation functions on the periodic system seems to be given by dimensions, which normally occur in the operator content of the chain when antiperiodic boundary condition is imposed.
The S=2 antiferromagnetic Heisenberg open chain is analyzed with the real-space numerical renormalization-group method improved by White. The low-ly ing spectrum and the one-point function are evaluated to show the degeneracy and the boundary effect on the ground state. Generalized string correlation funct ions were calculated for θ=π and θ=π 2. The θ=π 2 string correlation is found to indicate the long-range hidden order of the ground state. These results are quite consistent with the S=2 VBS argument.
The phase-dependent generalized string order introduced by Oshikawa in the VBS (valence bond solid)-type states of the integer spin chain is extended to the spin-1/2 J - J ′ alternating Heisenberg chain with antiferromagnetic J . The numerical diagonalization and the bosonization analysis show that the generalized string order remains finite in the ground state of the present model for -∞< J ′ < J , taking the maximum at J ′ =0. The characteristic feature of the phase dependence of the generalized string order also coincides with that of the spin-1 VBS state. This reconfirms the notion that the Haldane state of the spin-1 chain and the dimerized state of the alternating chain belong to a single phase characterized by the local singlet-type correlation.
The ground-state properties of the spin-1/2 Heisenberg chain, which has two exchange couplings (J>J') alternatingly, are studied. This model interpolates the spin-1 antiferromagnetic Heisenberg chain (AFHC) (J'\ensuremath{\rightarrow}-\ensuremath{\infty}, J>0), the dimerized spin-1/2 AFHC (J'\ensuremath{\simeq}J>0), and the uniform spin-1/2 AFHC (J'=J>0) continuously. The string order introduced by den Nijs--Rommelse and Tasaki changes continuously through the Haldane phase and the dimer phase to vanish at J=J'. It is at a maximum for J'=0. This implies that the string order is related to the local singlet correlation in terms of the spin-1/2 model. The lowest excitation can be regarded as the renormalized triplet wave. The reversal of the anisotropy effect on the excitation spectrum of the spin-1 AFHC is also explained in a intuitive way based on this picture.