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

arXiv:1107.2911 · Physical Review B

Intermediate Haldane phase in spin-2 quantum chains with uniaxial anisotropy

Hong-Hao Tu, Román Orús

Published Oct 12, 2011• 22 citations
arXivHTMLPDF

Abstract

We provide evidence of an intermediate Haldane phase in a spin-2 quantum chain. By combining effective field theory and numerical approaches, we show that the phase diagram of the proposed model includes SO(5) Haldane, intermediate Haldane, and large-$D$ phases. We determine the characteristic properties of these phases, including edge states, string order parameters, and degeneracies in the entanglement spectrum. The symmetries responsible for the degeneracy patterns observed in the entanglement spectrum are also discussed.

DOI
10.1103/physrevb.84.140407

Related papers

A novel intermediate phase in S=2 antiferromagnetic chains with uniaxial anisotropy

Masaki Oshikawa, Masaru Yamanaka, Seiji Miyashita

We study ground state properties of the $S=2$ quantum antiferromagnetic chain with a uniaxially anisotropic Hamiltonian: $ H = \sum_{j} [S_{j} \cdot S_{j+1} + D (S^{z}_{j})^2 ] $ by a Monte Carlo calculation. While it has been reported that a string order parameter vanishes exponentially at the isotropic Heisenberg point ($D=0$), we found it tends to remain finite in the thermodynamic limit around $D=1.2$. This implies that the model for $S=2$ has a novel intermediate phase between the Haldane phase and the large-$D$ phase, as was anticipated from the earlier argument based on the Valence-Bond-Solid picture.

arXiv:cond-mat/9507098•Jul 24, 1995•0 citations

Haldane, Large-D, and Intermediate-DStates in anS= 2 Quantum Spin Chain with On-Site andXXZAnisotropies

Takashi Tonegawa, Kiyomi Okamoto, Hiroki Nakano +3 more

Using mainly numerical methods, we investigate the ground-state phase diagram of the S=2 quantum spin chain described by $H = \sum_j (S_j^x S_{j+1}^x + S_j^y S_{j+1}^y + ΔS_j^z S_{j+1}^z) + D \sum_j (S_j^z)^2$, where $Δ$ denotes the $XXZ$ anisotropy parameter of the nearest-neighbor interactions and $D$ the on-site anisotropy parameter. We restrict ourselves to the case with $Δ\ge 0$ and $D \ge 0$ for simplicity. Each of the phase boundary lines is determined by the level spectroscopy or the phenomenological renormalization analysis of numerical results of exact-diagonalization calculations. The resulting phase diagram on the $Δ$-$D$ plane consists of four phases; the XY 1 phase, the Haldane/large-$D$ phase, the intermediate-$D$ phase and the Néel phase. The remarkable natures of the phase diagram are: (1) the Haldane state and the large-$D$ state belong to the same phase; (2) there exists the intermediate-$D$ phase which was predicted by Oshikawa in 1992; (3) the shape of the phase diagram on the $Δ$-$D$ plane is different from that believed so far. We note that this is the first report of the observation of the intermediate-$D$ phase.

arXiv:1011.6568•Journal of the Physical Society of Japan•Apr 12, 2011•49 citations

All spin-1 topological phases in a single spin-2 chain

Augustine Kshetrimayum, Hong-Hao Tu, Román Orús

Here we study the emergence of different symmetry-protected topological (SPT) phases in a spin-2 quantum chain. We consider a Heisenberg-like model with bilinear, biquadratic, bicubic, and biquartic nearest-neighbor interactions, as well as uniaxial anisotropy. We show that this model contains four different effective spin-1 SPT phases, corresponding to different representations of the $({\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2})+T$ symmetry group, where ${\mathbb{Z}}_{2}$ is some $\ensuremath{\pi}$ rotation in the spin internal space and $T$ is time reversal. One of these phases is equivalent to the usual spin-1 Haldane phase, while the other three are different but also typical of spin-1 systems. The model also exhibits an $SO(5)$-Haldane phase. Moreover, we also find that the transitions between the different effective spin-1 SPT phases are continuous and can be described by a $c=2$ conformal field theory. At such transitions, indirect evidence suggests a possible effective field theory of four massless Majorana fermions. The results are obtained by approximating the ground state of the system in the thermodynamic limit using matrix product states via the infinite time-evolving block decimation method, as well as by effective field theory considerations. Our findings show that different large effective spin-1 SPT phases separated by continuous quantum phase transitions can be stabilized in a simple quantum spin chain.

arXiv:1412.3370•Physical Review B•May 19, 2015•14 citations

Ground-state phase diagram of the spin-1/2 ferromagnetic-antiferromagnetic alternating Heisenberg chain with anisotropy

Kazuo Hida

The ground state of the ferromagnetic-antiferromagnetic alternating spin-1/2 Heisenberg chain is studied with Ising-type anisotropy on the ferromagnetic bond. This model tends to the spin-1 antiferromagnetic Heisenberg chain with on-site anisotropy in the limit of strong ferromagnetic bond. It is found that the Haldane-gap phase, the large-D phase, and the two types of N\'eel phases appear as the ground state of this model. The duality relations between these four phases are proved for several limiting cases. The string order parameter characterizing the large-D phase is introduced based on these duality relations. The ground-state phase diagram is determined by the numerical diagonalization of the finite-size system.

Physical review. B, Condensed matter•Oct 1, 1992•81 citations

Ground-state phase diagram of S = 2 quantum spin chain with the XXZ and on-site anisotropies

Kiyomi Okamoto, Takashi Tonegawa, Hiroki Nakano +3 more

By use of mainly numerical methods, we investigate the ground-state phase diagram of an S = 2 quantum spin chain with the XXZ and on-site anisotropies described by = Σ j ( S j x S j +1 x + S y j S y j +1 + Δ S z j S z j +1 + D Σ j ( S z j ) 2 , where Δ denotes the XXZ anisotropy parameter of the nearest-neighbor interactions and D the on-site anisotropy parameter. In the phase diagram with Δ > 0 and D > 0 there appear the XY phase, the Haldane/large- D (LD) phase, the intermediate- D (ID) phase, and the Néel phase. Among them the ID phase was proposed by Oshikawa in 1992 and has been considered to be absent since the DMRG studies in the latter half of 1990's. And also, the Haldane state and the LD state belong to the same phase, which should be named Haldane/LD phase.

Journal of Physics Conference Series•Jul 20, 2011•11 citations

How to distinguish the Haldane/Large-Dstate and the intermediate-Dstate in anS= 2 quantum spin chain with theXXZand on-site anisotropies

Kiyomi Okamoto, Takashi Tonegawa, Hiroki Nakano +3 more

We numerically investigate the ground-state phase diagram of an S = 2 quantum spin chain with the XXZ and on-site anisotropies described by = Σ j ( S j x S j + 1 x + S j y S j + 1 y + Δ S j z S j+1 z ) + D Σ j ( S j z ) 2 , where Δ denotes the XXZ anisotropy parameter of the nearest-neighbor interactions and D the on-site anisotropy parameter. We restrict ourselves to the Δ > 0 and D > 0 case for simplicity. Our main purpose is to obtain the definite conclusion whether there exists or not the intermediate- D (ID) phase, which was proposed by Oshikawa in 1992 and has been believed to be absent since the DMRG studies in the latter half of 1990's. In the phase diagram with Δ > 0 and D > 0 there appear the XY state, the Haldane state, the ID state, the large- D (LD) state and the Néel state. In the analysis of the numerical data it is important to distinguish three gapped states; the Haldane state, the ID state and the LD state. We give a physical and intuitive explanation for our level spectroscopy method how to distinguish these three phases.

arXiv:1101.2799•Journal of Physics Conference Series•Sep 28, 2011•18 citations
We provide evidence of an intermediate Haldane phase in a spin-2 quantum chain. By combining effective field theory and numerical approaches, we show that the phase diagram of the proposed model includes SO(5) Haldane, intermediate Haldane, and large-DDD phases. We determine the characteristic properties of these phases, including edge states, string order parameters, and degeneracies in the entanglement spectrum. The symmetries responsible for the degeneracy patterns observed in the entanglement spectrum are also discussed.
We study ground state properties of the S=2S=2S=2 quantum antiferromagnetic chain with a uniaxially anisotropic Hamiltonian: $ H = \sum_{j} [S_{j} \cdot S_{j+1} + D (S^{z}_{j})^2 ] $ by a Monte Carlo calculation. While it has been reported that a string order parameter vanishes exponentially at the isotropic Heisenberg point (D=0D=0D=0), we found it tends to remain finite in the thermodynamic limit around D=1.2D=1.2D=1.2. This implies that the model for S=2S=2S=2 has a novel intermediate phase between the Haldane phase and the large-DDD phase, as was anticipated from the earlier argument based on the Valence-Bond-Solid picture.
Using mainly numerical methods, we investigate the ground-state phase diagram of the S=2 quantum spin chain described by H=∑j(SjxSj+1x+SjySj+1y+ΔSjzSj+1z)+D∑j(Sjz)2H = \sum_j (S_j^x S_{j+1}^x + S_j^y S_{j+1}^y + ΔS_j^z S_{j+1}^z) + D \sum_j (S_j^z)^2H=∑j​(Sjx​Sj+1x​+Sjy​Sj+1y​+ΔSjz​Sj+1z​)+D∑j​(Sjz​)2, where ΔΔΔ denotes the XXZXXZXXZ anisotropy parameter of the nearest-neighbor interactions and DDD the on-site anisotropy parameter. We restrict ourselves to the case with Δ≥0Δ\ge 0Δ≥0 and D≥0D \ge 0D≥0 for simplicity. Each of the phase boundary lines is determined by the level spectroscopy or the phenomenological renormalization analysis of numerical results of exact-diagonalization calculations. The resulting phase diagram on the ΔΔΔ-DDD plane consists of four phases; the XY 1 phase, the Haldane/large-DDD phase, the intermediate-DDD phase and the Néel phase. The remarkable natures of the phase diagram are: (1) the Haldane state and the large-DDD state belong to the same phase; (2) there exists the intermediate-DDD phase which was predicted by Oshikawa in 1992; (3) the shape of the phase diagram on the ΔΔΔ-DDD plane is different from that believed so far. We note that this is the first report of the observation of the intermediate-DDD phase.
Here we study the emergence of different symmetry-protected topological (SPT) phases in a spin-2 quantum chain. We consider a Heisenberg-like model with bilinear, biquadratic, bicubic, and biquartic nearest-neighbor interactions, as well as uniaxial anisotropy. We show that this model contains four different effective spin-1 SPT phases, corresponding to different representations of the $({\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2})+T$ symmetry group, where Z2{\mathbb{Z}}_{2}Z2​ is some $\ensuremath{\pi}$ rotation in the spin internal space and TTT is time reversal. One of these phases is equivalent to the usual spin-1 Haldane phase, while the other three are different but also typical of spin-1 systems. The model also exhibits an SO(5)SO(5)SO(5)-Haldane phase. Moreover, we also find that the transitions between the different effective spin-1 SPT phases are continuous and can be described by a c=2c=2c=2 conformal field theory. At such transitions, indirect evidence suggests a possible effective field theory of four massless Majorana fermions. The results are obtained by approximating the ground state of the system in the thermodynamic limit using matrix product states via the infinite time-evolving block decimation method, as well as by effective field theory considerations. Our findings show that different large effective spin-1 SPT phases separated by continuous quantum phase transitions can be stabilized in a simple quantum spin chain.
The ground state of the ferromagnetic-antiferromagnetic alternating spin-1/2 Heisenberg chain is studied with Ising-type anisotropy on the ferromagnetic bond. This model tends to the spin-1 antiferromagnetic Heisenberg chain with on-site anisotropy in the limit of strong ferromagnetic bond. It is found that the Haldane-gap phase, the large-D phase, and the two types of N\'eel phases appear as the ground state of this model. The duality relations between these four phases are proved for several limiting cases. The string order parameter characterizing the large-D phase is introduced based on these duality relations. The ground-state phase diagram is determined by the numerical diagonalization of the finite-size system.
By use of mainly numerical methods, we investigate the ground-state phase diagram of an S = 2 quantum spin chain with the XXZ and on-site anisotropies described by = Σ j ( S j x S j +1 x + S y j S y j +1 + Δ S z j S z j +1 + D Σ j ( S z j ) 2 , where Δ denotes the XXZ anisotropy parameter of the nearest-neighbor interactions and D the on-site anisotropy parameter. In the phase diagram with Δ > 0 and D > 0 there appear the XY phase, the Haldane/large- D (LD) phase, the intermediate- D (ID) phase, and the Néel phase. Among them the ID phase was proposed by Oshikawa in 1992 and has been considered to be absent since the DMRG studies in the latter half of 1990's. And also, the Haldane state and the LD state belong to the same phase, which should be named Haldane/LD phase.
We numerically investigate the ground-state phase diagram of an S = 2 quantum spin chain with the XXZ and on-site anisotropies described by = Σ j ( S j x S j + 1 x + S j y S j + 1 y + Δ S j z S j+1 z ) + D Σ j ( S j z ) 2 , where Δ denotes the XXZ anisotropy parameter of the nearest-neighbor interactions and D the on-site anisotropy parameter. We restrict ourselves to the Δ > 0 and D > 0 case for simplicity. Our main purpose is to obtain the definite conclusion whether there exists or not the intermediate- D (ID) phase, which was proposed by Oshikawa in 1992 and has been believed to be absent since the DMRG studies in the latter half of 1990's. In the phase diagram with Δ > 0 and D > 0 there appear the XY state, the Haldane state, the ID state, the large- D (LD) state and the Néel state. In the analysis of the numerical data it is important to distinguish three gapped states; the Haldane state, the ID state and the LD state. We give a physical and intuitive explanation for our level spectroscopy method how to distinguish these three phases.