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:2008.12062 · Journal of the Physical Society of Japan

Universality Class around the SU(3) Symmetric Point of the Dimer–Trimer Spin-1 Chain

Tohru Mashiko, Shunji Moriya, Kiyohide Nomura

Published Jan 18, 2021• 4 citations
arXivHTMLPDF

Abstract

We study critical phenomena of the SU(3) symmetric spin-1 chains when adding the SU(3) asymmetric term. To investigate such system, we numerically diagonalize the Dimer-Trimer (DT) model Hamiltonian around the SU(3) symmetric point, named the pure trimer (PT) point. We analyze our numerical results with the conformal field theory (CFT). First of all, we discover soft modes at the wave number q = 0 and q = 2π/3 for the PT point, and then the system is critical. Secondly, we find that the system at the PT point belongs to the CFT with the central charge c = 2 and the scaling dimension x = 2/3. Finally, by investigating the eigenvalues of the Hamiltonian in the vicinity of the PT point, we find that there is a phase transition at the PT point from a massive phase to a massless phase. From these numerical results, the phase transition at the PT point belongs to the Berezinskii-Kosterlitz-Thouless (BKT)-like universality class that is explained by the level-1 SU(3) Wess-Zumino-Witten (SU(3) 1 WZW) model.

DOI
10.7566/jpsj.90.024005

Related papers

Universality Class around the SU(3) Symmetric Point of the Dimer-Trimer Spin-1 Chain

Tohru Mashiko, マシコ, トオル, Shunji Moriya +4 more

We study critical phenomena of an SU(3) symmetric spin-1 chain when adding an SU(3) asymmetric term. To investigate such phenomena, we numerically diagonalize the dimer–trimer (DT) model Hamiltonian around the SU(3) symmetric point, named the pure trimer (PT) point. We analyze our numerical results on the basis of the conformal field theory (CFT). First of all, we discover that soft modes appear at the wave number q = 0 and ±2π/3 for the PT point, and then the system is critical. Secondly, we find that the system at the PT point can be described by the CFT with the central charge c = 2 and the scaling dimension x = 2/3. Finally, by investigating the eigenvalues of the Hamiltonian in the vicinity of the PT point, we find that there is a phase transition at the PT point from a massive phase to a massless phase. From these numerical results, the phase transition at the PT point belongs to the Berezinskii–Kosterlitz–Thouless (BKT)- like universality class that is explained by the level-1 SU(3) Wess–Zumino–Witten [SU(3)1 WZW] model.

•Jan 18, 2021•4 citations

Phase transition of an SU(3) symmetric spin-1 chain

Tohru Mashiko, K. Nomura

We investigate a phase transition in an SU(3) symmetric spin-1 chain. To study this transition, we numerically diagonalize an SU(3) symmetric Hamiltonian combining the Uimin-Lai-Sutherland (ULS) model Hamiltonian with the Hamiltonian of the exact trimer ground state (${\mathbb{Z}}_{3}$ symmetry breaking). Our numerical results are discussed on the basis of the conformal field theory (CFT) and the renormalization group (RG). We then show the phase transition between a trimer liquid phase (massless, no long-range order) and a trimer phase (spontaneously ${\mathbb{Z}}_{3}$ symmetry breaking). Next, we find the central charge $c=2$ in the massless phase, and $c<2$ in the ${\mathbb{Z}}_{3}$ symmetry broken phase. From these results, we identify the universality class of the critical trimer liquid phase, as well as the phase boundary between these two phases.

arXiv:2108.01882•Physical review. B./Physical review. B•Oct 4, 2021•2 citations

Critical phenomena around the SU(3) symmetric tricritical point of a spin-1 chain

Tohru Mashiko, K. Nomura

We investigate critical phenomena of a spin-1 chain in the vicinity of the SU(3) symmetric critical point, which we already specified in a previous study [Mashiko and Nomura, Phys. Rev. B 104, 155405 (2021)]. We numerically diagonalize a Hamiltonian combining the bilinear-biquadratic Hamiltonian with the trimer Hamiltonian. We then discuss the numerical results based on the conformal field theory and the renormalization group. As a result, we first verify that the critical point found in our previous study is the tricritical point among the Haldane phase, the trimer phase, and the the trimer-liquid (TL) phase. Second, with regard to the TL-trimer transition and the TL-Haldane transition, we find that the critical phenomena around this tricritical point belong to the Berezinskii-Kosterlitz-Thouless-like universality class. Third, we find the boundary between the Haldane phase and the trimer phase, which is illustrated by the massive self-dual sine-Gordon model.

arXiv:2207.10857•Physical review. B./Physical review. B•Mar 7, 2023•1 citation

Dimerization and exotic criticality in spin-S chains

Natalia Chepiga

In this thesis we have studied the emergence of spontaneously dimerized phases in frustrated spin-S chains, with emphasis on the nature of the critical lines between the dimerized and non-dimerized phases. The main numerical method used in this thesis is the Density Matrix Renormalization Group (DMRG). The DMRG algorithm is a relatively old and well established method for the investigation of the ground-state. In this thesis, we show how to use this algorithm to calculate the excitation spectra of one-dimensional critical systems, known in the context of conformal field theory as conformal towers of states. We have demonstrated that the method works very well for two simple minimal models (the transverse-field Ising model and the three-state Potts model), and we have used it systematically to identify the universality classes and the underlying conformal field theories of various one-dimensional spin systems. It has been known for a long time that the transition to a spontaneously dimerized phase in a spin-1 chain can be either continuous, in the Wess-Zumino-Witten (WZW) SU(2) level 2 universality class, or first order. By combining a careful numerical investigation with a conformal field theory analysis, we were able to detect in a frustrated spin-1 chain with competing next-nearest-neighbor and three-site interactions the presence of yet another type of continuous phase transition that belongs to the Ising universality class. In contrast to the WZW SU(2) level 2 critical line, at which the singlet-triplet gap closes, the Ising transition occurs entirely in the singlet sector, while the singlet-triplet gap remains open. The use of the standard DMRG approach, along the lines mentioned above, has allowed us to provide explicit numerical evidence for the presence of a conformal tower of singlets inside the spin gap. Moreover, according to field theory, a WZW SU(2) level k critical line can turn into a first order transition due to the presence of a marginal operator in the WZW SU(2) level k model. A careful investigation of the conformal towers along the critical lines has allowed us to find the precise location of this point in both S=1 and S=3/2 chains. We have also shown that the nature of the continuous dimerization transitions is related to the topological properties of the corresponding phases, and that the phase diagrams of various frustrated spin chains can be effectively extracted by looking at the local topological order parameter - the degeneracy of the lowest state in the entanglement spectrum. When coupled with the conformal field theory of open systems, DMRG appears to be an extremely powerful tool to characterize not only the phase diagram and the ground-state correlations of quantum one-dimensional systems, but also the excitation spectrum and the conformal structure along critical lines.

•Jan 1, 2017•3 citations

Dimerization transitions in spin-1 chains

Natalia Chepiga, Ian K. Affleck, Frédéric Mila

We study spontaneous dimerization transitions in a Heisenberg spin-1 chain with additional next-nearest-neighbor and three-site interactions using extensive numerical simulations and a conformal field-theory analysis. We show that the transition can be second order in the Wess-Zumino-Witten (WZW) $\mathrm{SU}{(2)}_{2}$ or Ising universality class, or first order. We argue that these features are generic because of a marginal operator in the WZW $\mathrm{SU}{(2)}_{2}$ model and because of two topologically distinct nondimerized phases with or without edge states. We also provide explicit numerical evidence of conformal towers of singlets inside the spin gap at the Ising transition. Implications for other models are briefly discussed.

arXiv:1603.01395•Physical review. B./Physical review. B•Jun 20, 2016•44 citations

Quantum Critical Spin-2 Chain with Emergent SU(3) Symmetry

Pochung Chen, Zhi-Long Xue, Ian P. McCulloch +3 more

We study the quantum critical phase of an SU(2) symmetric spin-2 chain obtained from spin-2 bosons in a one-dimensional lattice. We obtain the scaling of the finite-size energies and entanglement entropy by exact diagonalization and density-matrix renormalization group methods. From the numerical results of the energy spectra, central charge, and scaling dimension we identify the conformal field theory describing the whole critical phase to be the SU(3)_{1} Wess-Zumino-Witten model. We find that, while the Hamiltonian is only SU(2) invariant, in this critical phase there is an emergent SU(3) symmetry in the thermodynamic limit.

arXiv:1402.5713•Physical Review Letters•Apr 9, 2015•24 citations
We study critical phenomena of the SU(3) symmetric spin-1 chains when adding the SU(3) asymmetric term. To investigate such system, we numerically diagonalize the Dimer-Trimer (DT) model Hamiltonian around the SU(3) symmetric point, named the pure trimer (PT) point. We analyze our numerical results with the conformal field theory (CFT). First of all, we discover soft modes at the wave number q = 0 and q = 2π/3 for the PT point, and then the system is critical. Secondly, we find that the system at the PT point belongs to the CFT with the central charge c = 2 and the scaling dimension x = 2/3. Finally, by investigating the eigenvalues of the Hamiltonian in the vicinity of the PT point, we find that there is a phase transition at the PT point from a massive phase to a massless phase. From these numerical results, the phase transition at the PT point belongs to the Berezinskii-Kosterlitz-Thouless (BKT)-like universality class that is explained by the level-1 SU(3) Wess-Zumino-Witten (SU(3) 1 WZW) model.
We study critical phenomena of an SU(3) symmetric spin-1 chain when adding an SU(3) asymmetric term. To investigate such phenomena, we numerically diagonalize the dimer–trimer (DT) model Hamiltonian around the SU(3) symmetric point, named the pure trimer (PT) point. We analyze our numerical results on the basis of the conformal field theory (CFT). First of all, we discover that soft modes appear at the wave number q = 0 and ±2π/3 for the PT point, and then the system is critical. Secondly, we find that the system at the PT point can be described by the CFT with the central charge c = 2 and the scaling dimension x = 2/3. Finally, by investigating the eigenvalues of the Hamiltonian in the vicinity of the PT point, we find that there is a phase transition at the PT point from a massive phase to a massless phase. From these numerical results, the phase transition at the PT point belongs to the Berezinskii–Kosterlitz–Thouless (BKT)- like universality class that is explained by the level-1 SU(3) Wess–Zumino–Witten [SU(3)1 WZW] model.
We investigate a phase transition in an SU(3) symmetric spin-1 chain. To study this transition, we numerically diagonalize an SU(3) symmetric Hamiltonian combining the Uimin-Lai-Sutherland (ULS) model Hamiltonian with the Hamiltonian of the exact trimer ground state (Z3{\mathbb{Z}}_{3}Z3​ symmetry breaking). Our numerical results are discussed on the basis of the conformal field theory (CFT) and the renormalization group (RG). We then show the phase transition between a trimer liquid phase (massless, no long-range order) and a trimer phase (spontaneously Z3{\mathbb{Z}}_{3}Z3​ symmetry breaking). Next, we find the central charge c=2c=2c=2 in the massless phase, and c<2c<2c<2 in the Z3{\mathbb{Z}}_{3}Z3​ symmetry broken phase. From these results, we identify the universality class of the critical trimer liquid phase, as well as the phase boundary between these two phases.
We investigate critical phenomena of a spin-1 chain in the vicinity of the SU(3) symmetric critical point, which we already specified in a previous study [Mashiko and Nomura, Phys. Rev. B 104, 155405 (2021)]. We numerically diagonalize a Hamiltonian combining the bilinear-biquadratic Hamiltonian with the trimer Hamiltonian. We then discuss the numerical results based on the conformal field theory and the renormalization group. As a result, we first verify that the critical point found in our previous study is the tricritical point among the Haldane phase, the trimer phase, and the the trimer-liquid (TL) phase. Second, with regard to the TL-trimer transition and the TL-Haldane transition, we find that the critical phenomena around this tricritical point belong to the Berezinskii-Kosterlitz-Thouless-like universality class. Third, we find the boundary between the Haldane phase and the trimer phase, which is illustrated by the massive self-dual sine-Gordon model.
In this thesis we have studied the emergence of spontaneously dimerized phases in frustrated spin-S chains, with emphasis on the nature of the critical lines between the dimerized and non-dimerized phases. The main numerical method used in this thesis is the Density Matrix Renormalization Group (DMRG). The DMRG algorithm is a relatively old and well established method for the investigation of the ground-state. In this thesis, we show how to use this algorithm to calculate the excitation spectra of one-dimensional critical systems, known in the context of conformal field theory as conformal towers of states. We have demonstrated that the method works very well for two simple minimal models (the transverse-field Ising model and the three-state Potts model), and we have used it systematically to identify the universality classes and the underlying conformal field theories of various one-dimensional spin systems. It has been known for a long time that the transition to a spontaneously dimerized phase in a spin-1 chain can be either continuous, in the Wess-Zumino-Witten (WZW) SU(2) level 2 universality class, or first order. By combining a careful numerical investigation with a conformal field theory analysis, we were able to detect in a frustrated spin-1 chain with competing next-nearest-neighbor and three-site interactions the presence of yet another type of continuous phase transition that belongs to the Ising universality class. In contrast to the WZW SU(2) level 2 critical line, at which the singlet-triplet gap closes, the Ising transition occurs entirely in the singlet sector, while the singlet-triplet gap remains open. The use of the standard DMRG approach, along the lines mentioned above, has allowed us to provide explicit numerical evidence for the presence of a conformal tower of singlets inside the spin gap. Moreover, according to field theory, a WZW SU(2) level k critical line can turn into a first order transition due to the presence of a marginal operator in the WZW SU(2) level k model. A careful investigation of the conformal towers along the critical lines has allowed us to find the precise location of this point in both S=1 and S=3/2 chains. We have also shown that the nature of the continuous dimerization transitions is related to the topological properties of the corresponding phases, and that the phase diagrams of various frustrated spin chains can be effectively extracted by looking at the local topological order parameter - the degeneracy of the lowest state in the entanglement spectrum. When coupled with the conformal field theory of open systems, DMRG appears to be an extremely powerful tool to characterize not only the phase diagram and the ground-state correlations of quantum one-dimensional systems, but also the excitation spectrum and the conformal structure along critical lines.
We study spontaneous dimerization transitions in a Heisenberg spin-1 chain with additional next-nearest-neighbor and three-site interactions using extensive numerical simulations and a conformal field-theory analysis. We show that the transition can be second order in the Wess-Zumino-Witten (WZW) SU(2)2\mathrm{SU}{(2)}_{2}SU(2)2​ or Ising universality class, or first order. We argue that these features are generic because of a marginal operator in the WZW SU(2)2\mathrm{SU}{(2)}_{2}SU(2)2​ model and because of two topologically distinct nondimerized phases with or without edge states. We also provide explicit numerical evidence of conformal towers of singlets inside the spin gap at the Ising transition. Implications for other models are briefly discussed.
We study the quantum critical phase of an SU(2) symmetric spin-2 chain obtained from spin-2 bosons in a one-dimensional lattice. We obtain the scaling of the finite-size energies and entanglement entropy by exact diagonalization and density-matrix renormalization group methods. From the numerical results of the energy spectra, central charge, and scaling dimension we identify the conformal field theory describing the whole critical phase to be the SU(3)_{1} Wess-Zumino-Witten model. We find that, while the Hamiltonian is only SU(2) invariant, in this critical phase there is an emergent SU(3) symmetry in the thermodynamic limit.