Journal of Physics Condensed Matter

Haldane gap antiferromagnets in a transverse magnetic field

Hal Tasaki

Published Aug 5, 1991• 6 citations

Abstract

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.

DOI: 10.1088/0953-8984/3/31/010

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Journal of Physics Condensed Matter

Haldane gap antiferromagnets in a transverse magnetic field

Hal Tasaki

Published Aug 5, 1991• 6 citations

Publisher

Abstract

DOI: 10.1088/0953-8984/3/31/010

Related papers

Transverse Antiferromagnetism of a Magnetized Haldane-Gap System

Tôru Sakai, Minoru Takahashi

A 1D S =1 Heisenberg antiferromagnet in a magnetic field H (// z -axis) at T =0 is studied by numerical diagonalizations. It is found that a phase transition exists at H c1 (=Δ), where Δ is the Haldane gap, and the transverse staggered susceptibility χ st x x diverges and the spin correlation is asymptotically < S 0 x S r x >∼(-1) r r -η between H c1 and H c2 (=4). If the system is quasi-1D, even small interchain interactions can make canting Néel order within a mean field approximation for interchain interactions. This is consistent with a recent NMR measurement for NENP at low temperature.A 1D S =1 Heisenberg antiferromagnet in a magnetic field H (// z -axis) at T =0 is studied by numerical diagonalizations. It is found that a phase transition exists at H c1 (=Δ), where Δ is the Haldane gap, and the transverse staggered susceptibility χ st x x diverges and the spin correlation is asymptotically < S 0 x S r x >∼(-1) r r -η between H c1 and H c2 (=4). If the system is quasi-1D, even small interchain interactions can make canting Néel order within a mean field approximation for interchain interactions. This is consistent with a recent NMR measurement for NENP at low temperature.

Journal of the Physical Society of JapanDec 15, 19901 citation

AnisotropicS=1 Antiferromagnetic Heisenberg Chain in a Magnetic Field

Tôru Sakai, Minoru Takahashi

We study the S =1 antiferromagnetic Heisenberg chain with the anisotropies E ∑ j [( S j x ) 2 -( S j y ) 2 ] and D ∑ j ( S j x ) 2 in a magnetic field H along z -axis, using numerical diagonalization up to N =14 and the finite-size scaling and phenomenological renormalization. We give the phase diagrams in the H E and H D planes repectively at T =0. It is found that a magnetic field breaking the axial symmetry of the Hamiltonian brings about canted Néel order for H c1 < H < H c2 , if the system has the Haldane gap. The two transitions at H c1 and H c2 are revealed to belong to the same universality class as the two-dimensional Ising model. In addition it is found that even if the system is in the large- D phase ( D > D c ≈1), the same transitions occur in the region where the value of D is about 1.0∼1.5.We study the S =1 antiferromagnetic Heisenberg chain with the anisotropies E ∑ j [( S j x ) 2 -( S j y ) 2 ] and D ∑ j ( S j x ) 2 in a magnetic field H along z -axis, using numerical diagonalization up to N =14 and the finite-size scaling and phenomenological renormalization. We give the phase diagrams in the H E and H D planes repectively at T =0. It is found that a magnetic field breaking the axial symmetry of the Hamiltonian brings about canted Néel order for H c1 < H < H c2 , if the system has the Haldane gap. The two transitions at H c1 and H c2 are revealed to belong to the same universality class as the two-dimensional Ising model. In addition it is found that even if the system is in the large- D phase ( D > D c ≈1), the same transitions occur in the region where the value of D is about 1.0∼1.5.

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Crossover between the Haldane-gap phase and the dimer phase in the spin-1/2 alternating Heisenberg chain

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S=2 antiferromagnetic quantum spin chain

Ulrich Schollwöck, O. Golinelli, Th. Jolicœur

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( ${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$ ${)}^{2}$ , 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.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(${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$${)}^{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.

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Numerical study of the hidden antiferromagnetic order in the Haldane phase

Yasuhiro Hatsugai, Mahito Kohmoto

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.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.

Physical review. B, Condensed matterDec 1, 199160 citations

{\mathit{S}}_{\mathit{i}}^{\mathit{z}}

{)}^{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.

{\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.