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:2305.03072 · Physical review. B./Physical review. B

G2 integrable point characterization via isotropic spin-3 chains

Chengshu Li, Victor L. Quito, Dirk Schuricht, Pedro L. S. Lopes

Published Oct 16, 2023• 2 citations
arXivHTMLPDF

Abstract

We investigate the physical properties of ${G}_{2}$-symmetric integrable chains with local degrees of freedom in the fundamental representation; given the typical connection between integrability and critical points, we test the model's properties against a hypothesis of conformal-invariant long-distance behavior. Leveraging an embedding between the ${G}_{2}$ exceptional Lie algebra and $\mathrm{SU}(2)$-symmetric chains with local spin-3 representations, we perform numerical analyses via exact diagonalization (ED) targeted at specific spin sectors, as well as via non-Abelian density-matrix renormalization group . A basic study of the momentum-resolved ED spectrum suggests the low-energy system is effectively described by a ${({G}_{2})}_{1}$ Wess-Zumino-Witten (WZW) theory, but we find challenges in further numerical characterization of conformal data. The study and control of the phenomenology of this model may have implications for the development of accessible models for Fibonacci anyons.

DOI
10.1103/physrevb.108.165123

Related papers

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

The spectrum properties of an integrable G2 invariant vertex model

Marcio Jose Martins

This paper is concerned with the study of properties of the exact solution of the fundamental integrable G2 vertex model. The model R-matrix and respective spin chain are presented in terms of the basis generators of the G2 Lie algebra. This formulation permits us to relate the number of the Bethe roots of the respective Bethe equations with the eigenvalues of the U(1) conserved charges from the Cartan subalgebra of G2. The Bethe equations are solved by a peculiar string structure which combines complex three-strings with real roots allowing us to determine the bulk properties in the thermodynamic limit. We argue that G2 spin chain is gapless but the low-lying excitations have two different speeds of sound and the underlying continuum limit is therefore not strictly Lorentz invariant. We have investigated the finite-size corrections to the ground state energy and proposed that the critical properties of the system should be governed by the product of two c=1 conformal field theories. By combining numerical and analytical methods we have computed the bulk free-energy of the G2 vertex model. We found that there are three regimes in the spectral parameter in which the free-energy is limited and continuous. There exists however at least two sharp corner points in which the bulk-free energy is not differentiable.

arXiv:2301.05963•Nuclear Physics B•Feb 25, 2023•3 citations

The case of SU(3) criticality in spin-2 chains

Chengshu Li, Victor L. Quito, E. Miranda +3 more

It was proposed by Chen et al., [Phys. Rev. Lett. 114, 145301 (2015)] that spin-2 chains display an extended critical phase with enhanced $\mathrm{SU}(3)$ symmetry. This hypothesis is highly unexpected for a spin-2 system and, as we argue, would imply an unconventional mechanism for symmetry emergence. Yet, the absence of convenient critical points for renormalization group perturbative expansions, allied with the usual difficulty in the convergence of numerical methods in critical or small-gapped phases, renders the verification of this hypothetical $\mathrm{SU}(3)$-symmetric phase a non-trivial matter. By tracing parallels with the well-understood phase diagram of spin-1 chains and searching for signatures robust against finite-size effects, we draw criticism on the existence of this phase. We perform non-Abelian density matrix renormalization group studies of the multipolar static correlation function, energy spectrum scaling, single-mode approximation, and entanglement spectrum to shed light on the problem. We determine that the hypothetical $\mathrm{SU}(3)$ spin-2 phase is, in fact, dominated by ferro-octupolar correlations and also observe a lack of Luttinger-liquid-like behavior in correlation functions that suggests that is perhaps not critical. We further construct an infinite family of spin-$S$ systems with similar ferro-octupolar-dominated quasi-$\mathrm{SU}(3)$-like phenomenology; curiously, we note that the spin-3 version of the problem is located in a subspace of exact ${\mathrm{G}}_{2}$ symmetry, making this a point of interest for search of Fibonacci topological properties in magnetic systems.

arXiv:2108.10329•Physical review. B./Physical review. B•Feb 23, 2022•4 citations

DMRG studies of critical SU(N) spin chains

M. Führinger, Stephan Rachel, Ronny Thomale +2 more

Abstract The DMRG method is applied to integrable models of antiferromagnetic spin chains for fundamental and higher representations of SU(2), SU(3), and SU(4). From the low energy spectrum and the entanglement entropy, we compute the central charge and the primary field scaling dimensions. These parameters allow us to identify uniquely the Wess–Zumino–Witten models capturing the low energy sectors of the models we consider.

Annalen der Physik•Nov 20, 2008•13 citations

DMRG studies of critical SU(N) spin chains

M. Führinger, Stephan Rachel, Ronny Thomale +2 more

The DMRG method is applied to integrable models of antiferromagnetic spin chains for fundamental and higher representations of SU(2), SU(3), and SU(4). From the low energy spectrum and the entanglement entropy, we compute the central charge and the primary field scaling dimensions. These parameters allow us to identify uniquely the Wess-Zumino-Witten models capturing the low energy sectors of the models we consider.

arXiv:0806.2563•Annalen der Physik•Oct 20, 2008•58 citations

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
We investigate the physical properties of G2{G}_{2}G2​-symmetric integrable chains with local degrees of freedom in the fundamental representation; given the typical connection between integrability and critical points, we test the model's properties against a hypothesis of conformal-invariant long-distance behavior. Leveraging an embedding between the G2{G}_{2}G2​ exceptional Lie algebra and SU(2)\mathrm{SU}(2)SU(2)-symmetric chains with local spin-3 representations, we perform numerical analyses via exact diagonalization (ED) targeted at specific spin sectors, as well as via non-Abelian density-matrix renormalization group . A basic study of the momentum-resolved ED spectrum suggests the low-energy system is effectively described by a (G2)1{({G}_{2})}_{1}(G2​)1​ Wess-Zumino-Witten (WZW) theory, but we find challenges in further numerical characterization of conformal data. The study and control of the phenomenology of this model may have implications for the development of accessible models for Fibonacci anyons.
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.
This paper is concerned with the study of properties of the exact solution of the fundamental integrable G2 vertex model. The model R-matrix and respective spin chain are presented in terms of the basis generators of the G2 Lie algebra. This formulation permits us to relate the number of the Bethe roots of the respective Bethe equations with the eigenvalues of the U(1) conserved charges from the Cartan subalgebra of G2. The Bethe equations are solved by a peculiar string structure which combines complex three-strings with real roots allowing us to determine the bulk properties in the thermodynamic limit. We argue that G2 spin chain is gapless but the low-lying excitations have two different speeds of sound and the underlying continuum limit is therefore not strictly Lorentz invariant. We have investigated the finite-size corrections to the ground state energy and proposed that the critical properties of the system should be governed by the product of two c=1 conformal field theories. By combining numerical and analytical methods we have computed the bulk free-energy of the G2 vertex model. We found that there are three regimes in the spectral parameter in which the free-energy is limited and continuous. There exists however at least two sharp corner points in which the bulk-free energy is not differentiable.
It was proposed by Chen et al., [Phys. Rev. Lett. 114, 145301 (2015)] that spin-2 chains display an extended critical phase with enhanced SU(3)\mathrm{SU}(3)SU(3) symmetry. This hypothesis is highly unexpected for a spin-2 system and, as we argue, would imply an unconventional mechanism for symmetry emergence. Yet, the absence of convenient critical points for renormalization group perturbative expansions, allied with the usual difficulty in the convergence of numerical methods in critical or small-gapped phases, renders the verification of this hypothetical SU(3)\mathrm{SU}(3)SU(3)-symmetric phase a non-trivial matter. By tracing parallels with the well-understood phase diagram of spin-1 chains and searching for signatures robust against finite-size effects, we draw criticism on the existence of this phase. We perform non-Abelian density matrix renormalization group studies of the multipolar static correlation function, energy spectrum scaling, single-mode approximation, and entanglement spectrum to shed light on the problem. We determine that the hypothetical SU(3)\mathrm{SU}(3)SU(3) spin-2 phase is, in fact, dominated by ferro-octupolar correlations and also observe a lack of Luttinger-liquid-like behavior in correlation functions that suggests that is perhaps not critical. We further construct an infinite family of spin-SSS systems with similar ferro-octupolar-dominated quasi-SU(3)\mathrm{SU}(3)SU(3)-like phenomenology; curiously, we note that the spin-3 version of the problem is located in a subspace of exact G2{\mathrm{G}}_{2}G2​ symmetry, making this a point of interest for search of Fibonacci topological properties in magnetic systems.
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.