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Bulletin of the American Physical Society

Accurate determination of the Gaussian transition in spin-1 chains with single ion Anisotropy

Bruce Normand, Shijie Hu, Xiaoqun Wang, Yu Lu

Published Feb 27, 2012• 0 citations
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Abstract

The Gaussian transition in the spin-one Heisenberg chain with single-ion anisotropy is extremely difficult to treat, both analytically and numerically. We introduce an improved DMRG procedure with strict error control, which we use to access very large systems. By considering the bulk entropy, we determine the Gaussian transition point to 4-digit accuracy, Dc/J = 0.96845(8), resolving a long-standing debate in quantum magnetism. With this value, we obtain high-precision data for the critical behavior of quantities including the ground-state energy, gap, and transverse string-order parameter, and for the critical exponent, ν = 1.472(2). Applying our improved technique at Jz = 0.5 highlights essential differences in critical behavior along the Gaussian transition line.

Related papers

Accurate determination of the Gaussian transition in spin-1 chains with single-ion anisotropy

Shijie Hu, Bruce Normand, Xiaoqun Wang +1 more

The Gaussian transition in the spin-1 Heisenberg chain with single-ion anisotropy is extremely difficult to treat, both analytically and numerically. We introduce an improved density-matrix renormalization group procedure with strict error control, which we use to access very large systems. By considering the bulk entropy, we determine the Gaussian transition point to four-digit accuracy, ${D}_{c}/J=0.968\phantom{\rule{0.16em}{0ex}}45(8)$, resolving a long-standing debate in quantum magnetism. With this value, we obtain high-precision data for the critical behavior of quantities, including the ground-state energy, gap, and transverse string-order parameter, and for the critical exponent $\ensuremath{\nu}=1.472(2)$. Applying our improved technique at ${J}_{z}=0.5$ highlights essential differences in critical behavior along the Gaussian transition line.

arXiv:1107.0229•Physical Review B•Dec 2, 2011•64 citations

Tangential finite-size scaling at the Gaussian topological transition in the quantum spin-1 anisotropic chain

Luan M. Veríssimo, Maria S. S. Pereira, M. L. Lyra

Scaling aspects of Gaussian topological phase transitions in quantum spin chains are investigated using the prototypical one-dimensional spin-1 XXZ Heisenberg model with uniaxial single-ion anisotropy $D$. This model presents a critical line separating the gapped Haldane and large-$D$ phases, with the relevant energy gap closing at the transition point. We show that a proper tangential finite-size scaling analysis is able to accurately locate the Gaussian critical line and to probe the continuously varying set of correlation length critical exponents. The specific features of the tangential scaling are highlighted, in contrast to the standard scaling holding in the Ising-like transition between the gapless antiferromagnetic N\'eel and gapped Haldane phases. Our results are compared with field-theoretic predictions and available high-accuracy data for specific points along the Gaussian line.

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Quantum phase transitions in spin-1 XXZ chains with rhombic single-ion anisotropy

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We explore numerically the inverse participation ratios in the ground state of one-dimensional spin-1 $XXZ$ chains with the rhombic single-ion anisotropy. By employing the techniques of density-matrix renormalization group, effects of the rhombic single-ion anisotropy on various information theoretical measures are investigated, such as the fidelity susceptibility, the quantum coherence, and the entanglement entropy. Their relations with the quantum phase transitions are also analyzed. The phase transitions from the $Y$-N\'eel phase to the large-${E}_{x}$ or the Haldane phase can be well characterized by the fidelity susceptibility. The second-order derivative of the ground-state energy indicates all the transitions are of second order. We also find that the quantum coherence, the entanglement entropy, the Schmidt gap, and the inverse participation ratios can be used to detect the critical points of quantum phase transitions. Results drawn from these quantum information observables agree well with each other. Finally we provide a ground-state phase diagram as functions of the exchange anisotropy $\mathrm{\ensuremath{\Delta}}$ and the rhombic single-ion anisotropy $E$.

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Abstract Quantum entanglement and correlations in the spin-1 Heisenberg chain with single-ion anisotropy are investigated using the quantum renormalization group method. Negativity and quantum discord (QD) are calculated with various anisotropy parameters △ and single-ion anisotropy parameters D . We focus on the relations between two abovementioned physical quantities and on transitions between the Néel, Haldane, and large- D phases. It is found that both negativity and QD exhibit step-like patterns in different phases as the size of the system increases. Interestingly, the single-ion anisotropy parameter D , which can be modulated using nuclear electric resonance (2020 Nature 579 205), plays an important role in tuning the quantum phase transition (QPT) of the system. Both the first partial derivative of the negativity and QD with respect to D or △ exhibit nonanalytic behavior at the phase transition points, which corresponds directly to the divergence of the correlation length. The quantum correlation critical exponents derived from negativity and QD are equal, and are the reciprocal of the correlation length exponent at each critical point. This work extends the application of quantum entanglement and correlations as tools for depicting QPTs in spin-1 systems.

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Wei Chen, Kazuo Hida, B. C. Sanctuary

One-dimensional $S=1$ $\mathrm{XXZ}$ chains with uniaxial single-ion-type anisotropy are studied by numerical exact diagonalization of finite-size systems. The numerical data are analyzed using conformal field theory, the level spectroscopy, phenomenological renormalization-group, and finite-size scaling method. We thus present a quantitatively reliable ground-state phase diagram of this model. The ground states of this model contain the Haldane phase, large-D phase, N\'eel phase, two $\mathrm{XY}$ phases, and the ferromagnetic phase. There are four different types of transitions between these phases: the Brezinskii-Kosterlitz-Thouless-type transitions, the Gaussian-type transitions, the Ising-type transitions, and the first-order transitions. The locations of these critical lines are accurately determined.

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Critical Properties of Spin-1 Antiferromagnetic Heisenberg Chains with Bond Alternation and Uniaxial Single-Ion-Type Anisotropy

Wei Chen, Kazuo Hida, B. C. Sanctuary

One dimensional S =1 antiferromagnetic Heisenberg chains with bond alternation and uniaxial single-ion-type anisotropy are studied by numerical exact diagonalization of finite size systems. We calculate the ground state phase diagram using the twisted boundary condition method for D ≥0. The ground states of this model contain the Haldane state, large- D state and the dimer state. Using conformal field theory and the level spectroscopy method, we calculate the D and δ-dependence of the Luttinger liquid paraments K and the critical exponent ν of the energy gap.

Journal of the Physical Society of Japan•Jan 1, 2000•28 citations
The Gaussian transition in the spin-one Heisenberg chain with single-ion anisotropy is extremely difficult to treat, both analytically and numerically. We introduce an improved DMRG procedure with strict error control, which we use to access very large systems. By considering the bulk entropy, we determine the Gaussian transition point to 4-digit accuracy, Dc/J = 0.96845(8), resolving a long-standing debate in quantum magnetism. With this value, we obtain high-precision data for the critical behavior of quantities including the ground-state energy, gap, and transverse string-order parameter, and for the critical exponent, ν = 1.472(2). Applying our improved technique at Jz = 0.5 highlights essential differences in critical behavior along the Gaussian transition line.
The Gaussian transition in the spin-1 Heisenberg chain with single-ion anisotropy is extremely difficult to treat, both analytically and numerically. We introduce an improved density-matrix renormalization group procedure with strict error control, which we use to access very large systems. By considering the bulk entropy, we determine the Gaussian transition point to four-digit accuracy, Dc/J=0.96845(8){D}_{c}/J=0.968\phantom{\rule{0.16em}{0ex}}45(8)Dc​/J=0.96845(8), resolving a long-standing debate in quantum magnetism. With this value, we obtain high-precision data for the critical behavior of quantities, including the ground-state energy, gap, and transverse string-order parameter, and for the critical exponent $\ensuremath{\nu}=1.472(2)$. Applying our improved technique at Jz=0.5{J}_{z}=0.5Jz​=0.5 highlights essential differences in critical behavior along the Gaussian transition line.
Scaling aspects of Gaussian topological phase transitions in quantum spin chains are investigated using the prototypical one-dimensional spin-1 XXZ Heisenberg model with uniaxial single-ion anisotropy DDD. This model presents a critical line separating the gapped Haldane and large-DDD phases, with the relevant energy gap closing at the transition point. We show that a proper tangential finite-size scaling analysis is able to accurately locate the Gaussian critical line and to probe the continuously varying set of correlation length critical exponents. The specific features of the tangential scaling are highlighted, in contrast to the standard scaling holding in the Ising-like transition between the gapless antiferromagnetic N\'eel and gapped Haldane phases. Our results are compared with field-theoretic predictions and available high-accuracy data for specific points along the Gaussian line.
We explore numerically the inverse participation ratios in the ground state of one-dimensional spin-1 XXZXXZXXZ chains with the rhombic single-ion anisotropy. By employing the techniques of density-matrix renormalization group, effects of the rhombic single-ion anisotropy on various information theoretical measures are investigated, such as the fidelity susceptibility, the quantum coherence, and the entanglement entropy. Their relations with the quantum phase transitions are also analyzed. The phase transitions from the YYY-N\'eel phase to the large-Ex{E}_{x}Ex​ or the Haldane phase can be well characterized by the fidelity susceptibility. The second-order derivative of the ground-state energy indicates all the transitions are of second order. We also find that the quantum coherence, the entanglement entropy, the Schmidt gap, and the inverse participation ratios can be used to detect the critical points of quantum phase transitions. Results drawn from these quantum information observables agree well with each other. Finally we provide a ground-state phase diagram as functions of the exchange anisotropy $\mathrm{\ensuremath{\Delta}}$ and the rhombic single-ion anisotropy EEE.
Abstract Quantum entanglement and correlations in the spin-1 Heisenberg chain with single-ion anisotropy are investigated using the quantum renormalization group method. Negativity and quantum discord (QD) are calculated with various anisotropy parameters △ and single-ion anisotropy parameters D . We focus on the relations between two abovementioned physical quantities and on transitions between the Néel, Haldane, and large- D phases. It is found that both negativity and QD exhibit step-like patterns in different phases as the size of the system increases. Interestingly, the single-ion anisotropy parameter D , which can be modulated using nuclear electric resonance (2020 Nature 579 205), plays an important role in tuning the quantum phase transition (QPT) of the system. Both the first partial derivative of the negativity and QD with respect to D or △ exhibit nonanalytic behavior at the phase transition points, which corresponds directly to the divergence of the correlation length. The quantum correlation critical exponents derived from negativity and QD are equal, and are the reciprocal of the correlation length exponent at each critical point. This work extends the application of quantum entanglement and correlations as tools for depicting QPTs in spin-1 systems.
One-dimensional S=1S=1S=1 XXZ\mathrm{XXZ}XXZ chains with uniaxial single-ion-type anisotropy are studied by numerical exact diagonalization of finite-size systems. The numerical data are analyzed using conformal field theory, the level spectroscopy, phenomenological renormalization-group, and finite-size scaling method. We thus present a quantitatively reliable ground-state phase diagram of this model. The ground states of this model contain the Haldane phase, large-D phase, N\'eel phase, two XY\mathrm{XY}XY phases, and the ferromagnetic phase. There are four different types of transitions between these phases: the Brezinskii-Kosterlitz-Thouless-type transitions, the Gaussian-type transitions, the Ising-type transitions, and the first-order transitions. The locations of these critical lines are accurately determined.
One dimensional S =1 antiferromagnetic Heisenberg chains with bond alternation and uniaxial single-ion-type anisotropy are studied by numerical exact diagonalization of finite size systems. We calculate the ground state phase diagram using the twisted boundary condition method for D ≥0. The ground states of this model contain the Haldane state, large- D state and the dimer state. Using conformal field theory and the level spectroscopy method, we calculate the D and δ-dependence of the Luttinger liquid paraments K and the critical exponent ν of the energy gap.