Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Chinese Physics B

Commensurate and incommensurate Haldane phases for a spin-1 bilinear–biquadratic model

Yan-Wei Dai, Ai-Min Chen, Xijing Liu, Yaoheng Su

Published Dec 16, 2022• 2 citations
Publisher

Abstract

Commensurate and incommensurate Haldane phases for a spin-1 bilinear-biquadratic model are investigated using an infinite matrix product state algorithm. The bipartite entanglement entropy can detect a transition point between the two phases. In both phases, the entanglement spectrum shows double degeneracy. We calculate the nonlocal order parameter of the bond-centered inversion in both phases, which rapidly approaches a saturation value of −1 as the segment length increases. The nonlocal order parameter of the bond-centered inversion with a saturation value −1 and the nonzero value string order indicate that the Haldane phase is a symmetry-protected topological phase. To distinguish the commensurate and incommensurate Haldane phases, the transversal spin correlation and corresponding momentum distribution of the structure factor are analyzed. As a result, the transversal spin correlations exhibit different decay forms in both phases.

DOI
10.1088/1674-1056/acac0c

Related papers

Quantum entanglement and non-local string order in spin-1, bond-alternating Heisenberg model

Yaoheng Su, Chen Ai-Min, Chunlan Wang +1 more

Quantum entanglement and non-local string order are investigated in the one-dimensional spin-1 bond-alternating Heisenberg model for quantum phase transitions. Employing the infinite matrix product state representation, ground state wavefunctions are numerically obtained by using the infinite time evolving block decimation approach in the infinite lattice system. From a bipartite entanglement measure, i.e., von Neumann entropy, a quantum phase transition point corresponding to the singular point of the von Neumann entropy between the Haldane phase and the dimerised phase is found to be δc=0.26. In addition, based on the infinite matrix product state algorithm, the non-local string order is calculated. The result shows that the non-local string order is able to characterize the topological phase transitions.

•Mar 30, 2017•0 citations

Bulk-edge correspondence in the Haldane phase of the bilinear-biquadratic spin-1 Hamiltonian

Sudipto Singha Roy, Silvia N. Santalla, Javier Rodríguez-Laguna +1 more

Abstract The Haldane phase is the prototype of symmetry protected topological (SPT) phases of spin chain systems. It can be protected by several symmetries having in common the degeneracy of the entanglement spectrum (ES). Here we explore in depth this degeneracy for the spin-1 Affleck–Kennedy–Lieb–Tasaki and bilinear-biquadratic Hamiltonians and show the emergence of a bulk-edge correspondence that relates the low energy properties of the entanglement Hamiltonian of a periodic chain and that of the physical Hamiltonian of an open chain. We find that the ES can be described in terms of two spins-1/2 behaving as the effective spins at the end of the open chain. In the case of non-contiguous partitions, we find that the entanglement Hamiltonian is given by the spin-1/2 Heisenberg Hamiltonian, which suggests a relationship between SPT phases and conformal field theory. We finally investigate the string order parameter and the relation with the bulk-edge correspondence.

arXiv:2011.05452•Journal of Statistical Mechanics Theory and Experiment•May 1, 2021•1 citation

Quantum phase transitions in the spin-1/2 X X Z model with staggered γ interaction

Xia Jin, Dongchang Liu, Yu Shi +2 more

The one-dimensional spin-1/2 XXZ model with the limited case of the staggered Gamma interaction which includes the staggered DM and KSEA interactions is investigated by utilizing the infinite-time evolving block decimation method in the infinite matrix product state representation. It is shown that the staggered DM interaction does not lead to any new phase but expands the XY phase. Contrasted to the DM interaction, the region of the ferromagnetic phase expands into the nematic-ferromagnetic and dimer-Haldane phases with the deconfined quantum phase transition happening in the case of KSEA interaction. We observe a pronounced effect in the entanglement entropy and Schmidt gap, the odd-bond and even-bond von Neumann entropies exhibit different values, and the value of the Schmidt gap on the even bond becomes zero in the dimer-Haldane phase. This implies that quantum entanglement and the Schmidt gap can distinguish the two degenerate ground states caused by the existence of dimerized interaction. The zero value of the Schmidt gap works as a sign of the topological phase, which is also evidenced by the string correlation. From the perspective of magnetism, the existence of chiral order and nematic order suggests the helical magnetic structure induced by Gamma interaction. By comparing the DM and KSEA interactions, it is found that DM interaction does not change ferromagnetism, whereas the KSEA interaction inhibits ferromagnetism and destroys the product state in the ferromagnetic phase. Furthermore, the behaviors of the spin structure factor for the transversal spin correlation can distinguish different phases. From an appropriate fitting exponent of the spin correlation oscillating decay, commensurate or incommensurate phases can be identified.

Physical review. E•Jan 6, 2025•10 citations

Angular-time evolution and edge-spin dynamics in the Haldane phase of the S=1 bilinear-biquadratic chain

Takuma Kaise, Kouichi Okunishi

We investigate the angular-time evolution -- a parameter-time evolution generated by the entanglement Hamiltonian -- for the bipartitioned ground state of the S=1 bilinear-biquadratic chain under the open boundary condition with the up edge spin. Using a matrix-product-state representation of the ground-state wavefunction, we calculate the angular-time spin correlation functions $\langle S_n^{α}(τ)S_{n'}^{α}(0)\rangle$ in the Haldane phase, and extract its dominant oscillation mode attributed to the nearly two-fold-degenerate entanglement spectrum associated with the $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry. We also compute the effective edge-spin dynamics under a uniform magnetic field applied to the system part and numerically verify its correspondence to the dominant angular-time mode by precisely comparing the subsystem-size dependence of their amplitudes.

cond-mat.str-elCross-listedarXiv:2606.23117•Jun 23, 2026•0 citations

Symmetry-protected topological phases beyond groups: The q -deformed bilinear-biquadratic spin chain

Thomas Quella

We study the phase diagram of the ${\text{SO}}_{q}(3)$ quantum group invariant spin-1 bilinear-biquadratic spin chain for real values of $q>1$. Numerical computations suggest that the chain has at least three clearly distinguished phases: A chiral analog of the Haldane phase, a dimerized phase, and a ferromagnetic phase. In contrast, the counterpart of the extended critical region that is known to exist for $q=1$ remains elusive. Our results show that the Haldane phase fails to exhibit a twofold degeneracy in the entanglement spectrum but that the degeneracy is restored upon a suitable $q$ deformation of the entanglement Hamiltonian, which can be interpreted as a Zeeman field. The structure of the phase diagram is confirmed through analytical calculations in the extreme anisotropic limit $q\ensuremath{\rightarrow}\ensuremath{\infty}$. Our results suggest that symmetries of the form ${\mathcal{U}}_{q}\left[su(2)\right]$ for distinct choices of $q$ should be interpreted as one single family instead of separate symmetries when defining SPT phases, leading naturally to the notion of a $q$-SPT phase.

arXiv:2011.12679•Physical review. B./Physical review. B•Feb 2, 2021•9 citations

Emergent Haldane phase in the S=1 bilinear-biquadratic Heisenberg model on the square lattice

Ido Niesen, Philippe Corboz

Infinite projected entangled pair states simulations of the $S=1$ bilinear-biquadratic Heisenberg model on the square lattice reveal an emergent Haldane phase in between the previously predicted antiferromagnetic and three-sublattice ${120}^{\ensuremath{\circ}}$ magnetically ordered phases. This intermediate phase preserves SU(2) spin and translational symmetry but breaks lattice rotational symmetry, and it can be adiabatically connected to the Haldane phase of decoupled $S=1$ chains. Our results contradict previous studies which found a direct transition between the two magnetically ordered states.

arXiv:1701.05142•Physical review. B./Physical review. B•May 22, 2017•44 citations
Commensurate and incommensurate Haldane phases for a spin-1 bilinear-biquadratic model are investigated using an infinite matrix product state algorithm. The bipartite entanglement entropy can detect a transition point between the two phases. In both phases, the entanglement spectrum shows double degeneracy. We calculate the nonlocal order parameter of the bond-centered inversion in both phases, which rapidly approaches a saturation value of −1 as the segment length increases. The nonlocal order parameter of the bond-centered inversion with a saturation value −1 and the nonzero value string order indicate that the Haldane phase is a symmetry-protected topological phase. To distinguish the commensurate and incommensurate Haldane phases, the transversal spin correlation and corresponding momentum distribution of the structure factor are analyzed. As a result, the transversal spin correlations exhibit different decay forms in both phases.
Quantum entanglement and non-local string order are investigated in the one-dimensional spin-1 bond-alternating Heisenberg model for quantum phase transitions. Employing the infinite matrix product state representation, ground state wavefunctions are numerically obtained by using the infinite time evolving block decimation approach in the infinite lattice system. From a bipartite entanglement measure, i.e., von Neumann entropy, a quantum phase transition point corresponding to the singular point of the von Neumann entropy between the Haldane phase and the dimerised phase is found to be δc=0.26. In addition, based on the infinite matrix product state algorithm, the non-local string order is calculated. The result shows that the non-local string order is able to characterize the topological phase transitions.
Abstract The Haldane phase is the prototype of symmetry protected topological (SPT) phases of spin chain systems. It can be protected by several symmetries having in common the degeneracy of the entanglement spectrum (ES). Here we explore in depth this degeneracy for the spin-1 Affleck–Kennedy–Lieb–Tasaki and bilinear-biquadratic Hamiltonians and show the emergence of a bulk-edge correspondence that relates the low energy properties of the entanglement Hamiltonian of a periodic chain and that of the physical Hamiltonian of an open chain. We find that the ES can be described in terms of two spins-1/2 behaving as the effective spins at the end of the open chain. In the case of non-contiguous partitions, we find that the entanglement Hamiltonian is given by the spin-1/2 Heisenberg Hamiltonian, which suggests a relationship between SPT phases and conformal field theory. We finally investigate the string order parameter and the relation with the bulk-edge correspondence.
The one-dimensional spin-1/2 XXZ model with the limited case of the staggered Gamma interaction which includes the staggered DM and KSEA interactions is investigated by utilizing the infinite-time evolving block decimation method in the infinite matrix product state representation. It is shown that the staggered DM interaction does not lead to any new phase but expands the XY phase. Contrasted to the DM interaction, the region of the ferromagnetic phase expands into the nematic-ferromagnetic and dimer-Haldane phases with the deconfined quantum phase transition happening in the case of KSEA interaction. We observe a pronounced effect in the entanglement entropy and Schmidt gap, the odd-bond and even-bond von Neumann entropies exhibit different values, and the value of the Schmidt gap on the even bond becomes zero in the dimer-Haldane phase. This implies that quantum entanglement and the Schmidt gap can distinguish the two degenerate ground states caused by the existence of dimerized interaction. The zero value of the Schmidt gap works as a sign of the topological phase, which is also evidenced by the string correlation. From the perspective of magnetism, the existence of chiral order and nematic order suggests the helical magnetic structure induced by Gamma interaction. By comparing the DM and KSEA interactions, it is found that DM interaction does not change ferromagnetism, whereas the KSEA interaction inhibits ferromagnetism and destroys the product state in the ferromagnetic phase. Furthermore, the behaviors of the spin structure factor for the transversal spin correlation can distinguish different phases. From an appropriate fitting exponent of the spin correlation oscillating decay, commensurate or incommensurate phases can be identified.
We investigate the angular-time evolution -- a parameter-time evolution generated by the entanglement Hamiltonian -- for the bipartitioned ground state of the S=1 bilinear-biquadratic chain under the open boundary condition with the up edge spin. Using a matrix-product-state representation of the ground-state wavefunction, we calculate the angular-time spin correlation functions ⟨Snα(τ)Sn′α(0)⟩\langle S_n^{α}(τ)S_{n'}^{α}(0)\rangle⟨Snα​(τ)Sn′α​(0)⟩ in the Haldane phase, and extract its dominant oscillation mode attributed to the nearly two-fold-degenerate entanglement spectrum associated with the Z2×Z2\mathbb{Z}_2 \times \mathbb{Z}_2Z2​×Z2​ symmetry. We also compute the effective edge-spin dynamics under a uniform magnetic field applied to the system part and numerically verify its correspondence to the dominant angular-time mode by precisely comparing the subsystem-size dependence of their amplitudes.
We study the phase diagram of the SOq(3){\text{SO}}_{q}(3)SOq​(3) quantum group invariant spin-1 bilinear-biquadratic spin chain for real values of q>1q>1q>1. Numerical computations suggest that the chain has at least three clearly distinguished phases: A chiral analog of the Haldane phase, a dimerized phase, and a ferromagnetic phase. In contrast, the counterpart of the extended critical region that is known to exist for q=1q=1q=1 remains elusive. Our results show that the Haldane phase fails to exhibit a twofold degeneracy in the entanglement spectrum but that the degeneracy is restored upon a suitable qqq deformation of the entanglement Hamiltonian, which can be interpreted as a Zeeman field. The structure of the phase diagram is confirmed through analytical calculations in the extreme anisotropic limit $q\ensuremath{\rightarrow}\ensuremath{\infty}$. Our results suggest that symmetries of the form Uq[su(2)]{\mathcal{U}}_{q}\left[su(2)\right]Uq​[su(2)] for distinct choices of qqq should be interpreted as one single family instead of separate symmetries when defining SPT phases, leading naturally to the notion of a qqq-SPT phase.
Infinite projected entangled pair states simulations of the S=1S=1S=1 bilinear-biquadratic Heisenberg model on the square lattice reveal an emergent Haldane phase in between the previously predicted antiferromagnetic and three-sublattice ${120}^{\ensuremath{\circ}}$ magnetically ordered phases. This intermediate phase preserves SU(2) spin and translational symmetry but breaks lattice rotational symmetry, and it can be adiabatically connected to the Haldane phase of decoupled S=1S=1S=1 chains. Our results contradict previous studies which found a direct transition between the two magnetically ordered states.