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)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\ifmmode\times\else\texttimes\fi{}
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 have developed a numerical procedure to clarify the critical behavior near a quantum phase transition by analyzing a multipoint correlation function characterizing the ground state. This work presents a successful application of this procedure to the string order parameter of the
S=1 XXZ chain with uniaxial single-ion anisotropy. The finite-size string correlation function is estimated by the density-matrix renormalization-group method. We focus on the gradient of the inverse-system-size dependence of the correlation function on a logarithmic plot. This quantity shows that the finite-size scaling sensitively changes at the critical point. The behavior of the gradient with increasing system size is divergent, is stable at a finite value, or rapidly decreases to zero when the system is in the disordered phase, at the critical point, or in the ordered phase, respectively. The analysis of the finite-size string correlation functions allows precise determination of the boundary of the Haldane phase and estimation of the critical exponent of the correlation length. Our estimates of the transition point and the critical exponents, which are determined only by the ground-state quantities, are consistent with results obtained from the analysis of the energy-level structure. Our analysis requires only the correlation functions of several finite sizes under the same condition as a candidate for the long-range order. The quantity is treated in the same manner irrespective of the kind of elements which destroy the order concerned. This work will assist in the development of a method for directly observing quantum phase transitions.
String correlations are investigated in an infinite-size XXZ spin-1 chain. By using the infinite matrix product state representation, we calculate a long-range string order. In the XY phase, the string correlations decay within a relatively very large lattice distance, which makes a finite-size study difficult to verify the non-existence of the string order. Thus, in the Haldane phase, the non-vanishing string correlations in the limit of a very large distance allow to characterize the phase boundaries to the XY phase and the Neel phase, which implies that the transverse long-range string order is the order parameter for the Haldane phase. In addition, the singular behaviors of the von Neumann entropy and the fidelity per lattice site are shown to capture clearly the phase transition points that are consistent with the results from the string order. The estimated critical points including a BKT transition from the XY phase to the Haldane phase agree well with the previous results:
Δc2=0 for the XY-Haldane phase transition and
Δc3=1.185 for the Haldane-N'eel phase transition from the density renormalization group. From a finite-entanglement scaling of the von Neumann entropy with respect to the truncation dimension, the central charges are found to be
c≃1.0 at
Δc2=0 and
c≃0.5 at
Δc3=1.185, respectively, which shows that the XY-Haldane phase transition at
Δc2=0 belongs to the Heisenberg universality class, while the Haldane-Neel phase transition at
Δc2=1.185 belongs to the two-dimensional classical Ising universality class. It is also shown that, the long-range order parameters and the von Neumann entropy, as well as the fidelity per site approach, can be applied to characterize quantum phase transitions as a universal phase transition indicator for one-dimensional lattice many-body systems.
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.