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Physical review. B, Condensed matter

Numerical study of the hidden antiferromagnetic order in the Haldane phase

Yasuhiro Hatsugai, Mahito Kohmoto

Published Dec 1, 1991• 60 citations
Publisher

Abstract

We investigate the string correlation functions proposed by den Nijs and Rommelse for S=1 Heisenberg antiferromagnets in one dimension. The Hamiltonian with D${\mathrm{\ensuremath{\Sigma}}}_{\mathit{i}}$(${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$${)}^{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 ${\mathit{Z}}_{2}$\ifmmode\times\else\texttimes\fi{}${\mathit{Z}}_{2}$ 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.

DOI
10.1103/physrevb.44.11789

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Journal of Physics Condensed Matter•Aug 5, 1991•6 citations
We investigate the string correlation functions proposed by den Nijs and Rommelse for S=1 Heisenberg antiferromagnets in one dimension. The Hamiltonian with D\ensuremathΣi{\mathrm{\ensuremath{\Sigma}}}_{\mathit{i}}\ensuremathΣi​({\mathit{S}}_{\mathit{i}}^{\mathit{z}}$${)}^{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 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.
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=0Δ_{c2} = 0Δc2​=0 for the XY-Haldane phase transition and Δc3=1.185Δ_{c3} = 1.185Δc3​=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≃1.0c \simeq 1.0c≃1.0 at Δc2=0Δ_{c2} = 0Δc2​=0 and c≃0.5c \simeq 0.5c≃0.5 at Δc3=1.185Δ_{c3} = 1.185Δc3​=1.185, respectively, which shows that the XY-Haldane phase transition at Δc2=0Δ_{c2} = 0Δc2​=0 belongs to the Heisenberg universality class, while the Haldane-Neel phase transition at Δc2=1.185Δ_{c2} = 1.185Δc2​=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.
We have investigated Haldane's conjecture for the S=2 antiferromagnetic quantum spin chain with nearest-neighbor exchange J. Using a density matrix renormalization group algorithm for chains up to L=350 spins, we find in the thermodynamic limit a finite gap of \ensuremath{\Delta}=0.085(5)J and a finite spin-spin correlation length \ensuremath{\xi}=49(1) lattice spacings. We have confirmed the gap value by a zero-temperature quantum Monte Carlo study. We show that the ground state has a hidden topological order that is revealed in a nonlocal string correlation function which saturates to a nonzero value in the thermodynamic limit. We investigate the behavior of the spin-2 chain under an easy-plane anisotropy D(Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​)2{)}^{2})2, D\ensuremath{\gtrsim}0, and find that the Haldane and the large-D phase are separated by an XY phase. The string correlation function vanishes only in the D\ensuremath{\rightarrow}\ensuremath{\infty} limit and does not distinguish between the Haldane phase and the perturbative large-D phase. An analysis of the transition mechanism and of the S=2 phase diagram in the presence of easy-plane and exchange anisotropy, markedly different from the S=1 phase diagram, allow us to conjecture how the classical limit is reached from increasing integer spins. \textcopyright{} 1996 The American Physical Society.
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.
The author studies the S=1 antiferromagnetic Heisenberg chain with uniaxial anisotropy, which exhibits the Haldane gap phenomenon. He argues that, when a magnetic field in a direction transverse to the anisotropy axis is applied, a sharp phase transition takes place at a critical value of the field. The transition is characterized by vanishing of the hidden antiferromagnetic order which can be measured by the string order parameter of den Nijs and Rommelse. The existence of such a transition distinguishes a Haldane gap system from other disordered quantum spin systems.