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:1108.2857 · Physical Review Letters

Simultaneous Dimerization and SU(4) Symmetry Breaking of 4-Color Fermions on the Square Lattice

Philippe Corboz, Andreas Martin Läuchli, Karlo Penc, Matthias Troyer, Frédéric Mila

Published Nov 15, 2011• 127 citations
arXivHTMLPDF

Abstract

Using infinite projected entangled-pair states, exact diagonalization, and flavor-wave theory, we show that the SU(4) Heisenberg model undergoes a spontaneous dimerization on the square lattice, in contrast with its SU(2) and SU(3) counterparts, which develop Néel and three-sublattice stripelike long-range order. Since the ground state of a dimer is not a singlet for SU(4) but a 6-dimensional irreducible representation, this leaves the door open for further symmetry breaking. We provide evidence that, unlike in SU(4) ladders, where dimers pair up to form singlet plaquettes, here the SU(4) symmetry is additionally broken, leading to a gapless spectrum in spite of the broken translational symmetry.

DOI
10.1103/physrevlett.107.215301

Related papers

Competing states in the SU(3) Heisenberg model on the honeycomb lattice: Plaquette valence-bond crystal versus dimerized color-ordered state

Philippe Corboz, Miklós Lajkó, Karlo Penc +2 more

Conflicting predictions have been made for the ground state of the SU(3) Heisenberg model on the honeycomb lattice: Tensor network simulations found a plaquette order [Zhao et al., Phys. Rev. B 85, 134416 (2012)], where singlets are formed on hexagons, while linear flavor-wave theory suggested a dimerized, color-ordered state [Lee and Yang, Phys. Rev. B 85, 100402 (2012)]. In this work we show that the former state is the true ground state by a systematic study with infinite projected-entangled pair states (iPEPS), for which the accuracy can be systematically controlled by the so-called bond dimension $D$. Both competing states can be reproduced with iPEPS by using different unit cell sizes. For small $D$ the dimer state has a lower variational energy than the plaquette state; however, for large $D$ it is the latter which becomes energetically favorable. The plaquette formation is also confirmed by exact diagonalizations and variational Monte Carlo studies, according to which both the dimerized and plaquette states are nonchiral flux states.

arXiv:1302.1108•Physical Review B•May 10, 2013•41 citations

Spectral Mapping Theorem between SU(N) and SU(Ns/N) Heisenberg Models and Exact Diagonalization of SU(4) and SU(7) Long-Range Ordered Phases on the Triangular Lattice

Samuel Gozel, Pierre Nataf

We first show that, for the SU(N) Heisenberg model with isotropic two-site interactions on an arbitrary lattice of Ns sites, the full spectrum in its singlet sector (when N divides Ns) is the negative of the full spectrum in its singlet sector of the SU(Ns/N) Heisenberg model defined on the same lattice. This theorem, valid for models with the fundamental irreducible representation (irrep) at each lattice site, follows naturally from a transposition operation in the basis of Standard Young Tableaux (SYTs). Second, this basis -- previously recognized as particularly convenient for exact diagonalization (ED) of SU(N) Heisenberg models -- admits an efficient, memory-saving implementation, thereby enabling large-scale diagonalizations for unprecedented system sizes Ns and numbers of colors N>3. We apply these ideas to an ED study of the SU(4) and SU(7) Heisenberg models on the triangular lattice (TL), extending the calculations up to Ns=28 sites. In particular, starting from the SU(4) case, whose nearest-neighbor ground state remains debated, we confirm that next-nearest-neighbor couplings stabilize four-sublattice long-range order. We further find that increasing the number of colors from N=4 to N=7 provides another route toward a Néel-like ground state, even without such additional couplings. N = 4 to N = 7 provides another route toward a Néel-like ground state, even without such additional couplings.

arXiv:2610.15162•Oct 10, 2026•0 citations

Discrete lattice symmetry breaking in a two-dimensional frustrated spin-1 Heisenberg model

Ji-Yao Chen, Sylvain Capponi, Didier Poilblanc

Spontaneous discrete symmetry breaking can be described in the framework of projected entangled pair states (PEPS) by linearly superposing local tensors belonging to two (or more) symmetry classes of tensors. This is illustrated in the case of a frustrated spin-1 Heisenberg model on the square lattice, which hosts a nematic spin liquid spontaneously breaking lattice $\ensuremath{\pi}/2$-rotation symmetry. A superposition of SU(2)-symmetric PEPS tensors belonging to two irreducible representations of the lattice point group is shown to capture accurately the properties of the nematic phase, as shown from a comparison to exact diagonalizations and density matrix renormalization group results.

arXiv:1803.11393•Physical review. B./Physical review. B•Jul 3, 2018•6 citations

Dimer description of the SU(4) antiferromagnet on the triangular lattice

Anna Keselman, L. Savary, Leon Balents

In systems with many local degrees of freedom, high-symmetry points in the phase diagram can provide an important starting point for the investigation of their properties throughout the phase diagram. In systems with both spin and orbital (or valley) degrees of freedom such a starting point gives rise to SU(4)-symmetric models. Here we consider SU(4)-symmetric "spin'' models, corresponding to Mott phases at half-filling, i.e. the six-dimensional representation of SU(4). This may be relevant to twisted multilayer graphene. In particular, we study the SU(4) antiferromagnetic "Heisenberg'' model on the triangular lattice, both in the classical limit and in the quantum regime. Carrying out a numerical study using the density matrix renormalization group (DMRG), we argue that the ground state is non-magnetic. We then derive a dimer expansion of the SU(4) spin model. An exact diagonalization (ED) study of the effective dimer model suggests that the ground state breaks translation invariance, forming a valence bond solid (VBS) with a 12-site unit cell. Finally, we consider the effect of SU(4)-symmetry breaking interactions due to Hund's coupling, and argue for a possible phase transition between a VBS and a magnetically ordered state.

arXiv:1911.03492•SciPost Physics•May 13, 2020•9 citations

Ground-state phase diagram of the SU ( 4 ) Heisenberg model on a plaquette lattice

Ryui Kaneko, Shimpei Goto, Ippei Danshita

We investigate the ground state of the $\mathrm{SU}(4)$ Heisenberg model on a square lattice with spatial anisotropy on each plaquette bond using the tensor-network method based on infinite projected entangled pair states. We find that the $\mathrm{SU}(4)$ singlet ground state appears in the strongly anisotropic limit, whereas N\'eel and valence-bond crystal orders coexist in the nearly isotropic limit. By examining the intermediate parameter region, we identify a phase transition between these phases. The nature of the phase transition is likely to be of first order, and the transition point is estimated to be around ${J}^{\ensuremath{'}}/J\ensuremath{\approx}0.85(5)$, where $J$ and ${J}^{\ensuremath{'}}$ are the interaction strengths of intra- and interplaquette bonds, respectively. We also calculate the anisotropy dependence of singlet correlations on a plaquette bond, which will be useful for future experiments of ultracold atoms in optical lattices.

arXiv:2407.05556•Physical Review A•Aug 27, 2024•1 citation

SU(4) topological resonating valence bond spin liquid on the square lattice

Olivier Gauthé, Sylvain Capponi, Didier Poilblanc

We generalize the construction of the spin-1/2 SU(2) resonating valence bond (RVB) state to the case of the self-conjugate $\mathbf{6}$ representation of SU(4). As for the case of SU(2) [J.-Y. Chen and D. Poilblanc, Phys. Rev. B 97, 161107(R) (2018)], we use the projected entangled pair state (PEPS) formalism to derive a simple (two-dimensional) family of generalized SU(4) RVB states on the square lattice. We show that, when longer-range SU(4)-singlet bonds are included, a local gauge symmetry is broken down from U(1) to ${\mathbb{Z}}_{2}$, leading to the emergence of a short-range spin liquid. Evidence for the topological nature of this spin liquid is provided by the investigation of the Renyi entanglement entropy of infinitely long cylinders and of the modular matrices. Relevance to microscopic models and experiments of ultracold atoms is discussed.

arXiv:1901.05905•Physical review. B./Physical review. B•Jun 26, 2019•8 citations
Using infinite projected entangled-pair states, exact diagonalization, and flavor-wave theory, we show that the SU(4) Heisenberg model undergoes a spontaneous dimerization on the square lattice, in contrast with its SU(2) and SU(3) counterparts, which develop Néel and three-sublattice stripelike long-range order. Since the ground state of a dimer is not a singlet for SU(4) but a 6-dimensional irreducible representation, this leaves the door open for further symmetry breaking. We provide evidence that, unlike in SU(4) ladders, where dimers pair up to form singlet plaquettes, here the SU(4) symmetry is additionally broken, leading to a gapless spectrum in spite of the broken translational symmetry.
Conflicting predictions have been made for the ground state of the SU(3) Heisenberg model on the honeycomb lattice: Tensor network simulations found a plaquette order [Zhao et al., Phys. Rev. B 85, 134416 (2012)], where singlets are formed on hexagons, while linear flavor-wave theory suggested a dimerized, color-ordered state [Lee and Yang, Phys. Rev. B 85, 100402 (2012)]. In this work we show that the former state is the true ground state by a systematic study with infinite projected-entangled pair states (iPEPS), for which the accuracy can be systematically controlled by the so-called bond dimension DDD. Both competing states can be reproduced with iPEPS by using different unit cell sizes. For small DDD the dimer state has a lower variational energy than the plaquette state; however, for large DDD it is the latter which becomes energetically favorable. The plaquette formation is also confirmed by exact diagonalizations and variational Monte Carlo studies, according to which both the dimerized and plaquette states are nonchiral flux states.
We first show that, for the SU(N) Heisenberg model with isotropic two-site interactions on an arbitrary lattice of Ns sites, the full spectrum in its singlet sector (when N divides Ns) is the negative of the full spectrum in its singlet sector of the SU(Ns/N) Heisenberg model defined on the same lattice. This theorem, valid for models with the fundamental irreducible representation (irrep) at each lattice site, follows naturally from a transposition operation in the basis of Standard Young Tableaux (SYTs). Second, this basis -- previously recognized as particularly convenient for exact diagonalization (ED) of SU(N) Heisenberg models -- admits an efficient, memory-saving implementation, thereby enabling large-scale diagonalizations for unprecedented system sizes Ns and numbers of colors N>3. We apply these ideas to an ED study of the SU(4) and SU(7) Heisenberg models on the triangular lattice (TL), extending the calculations up to Ns=28 sites. In particular, starting from the SU(4) case, whose nearest-neighbor ground state remains debated, we confirm that next-nearest-neighbor couplings stabilize four-sublattice long-range order. We further find that increasing the number of colors from N=4 to N=7 provides another route toward a Néel-like ground state, even without such additional couplings. N = 4 to N = 7 provides another route toward a Néel-like ground state, even without such additional couplings.
Spontaneous discrete symmetry breaking can be described in the framework of projected entangled pair states (PEPS) by linearly superposing local tensors belonging to two (or more) symmetry classes of tensors. This is illustrated in the case of a frustrated spin-1 Heisenberg model on the square lattice, which hosts a nematic spin liquid spontaneously breaking lattice $\ensuremath{\pi}/2$-rotation symmetry. A superposition of SU(2)-symmetric PEPS tensors belonging to two irreducible representations of the lattice point group is shown to capture accurately the properties of the nematic phase, as shown from a comparison to exact diagonalizations and density matrix renormalization group results.
In systems with many local degrees of freedom, high-symmetry points in the phase diagram can provide an important starting point for the investigation of their properties throughout the phase diagram. In systems with both spin and orbital (or valley) degrees of freedom such a starting point gives rise to SU(4)-symmetric models. Here we consider SU(4)-symmetric "spin'' models, corresponding to Mott phases at half-filling, i.e. the six-dimensional representation of SU(4). This may be relevant to twisted multilayer graphene. In particular, we study the SU(4) antiferromagnetic "Heisenberg'' model on the triangular lattice, both in the classical limit and in the quantum regime. Carrying out a numerical study using the density matrix renormalization group (DMRG), we argue that the ground state is non-magnetic. We then derive a dimer expansion of the SU(4) spin model. An exact diagonalization (ED) study of the effective dimer model suggests that the ground state breaks translation invariance, forming a valence bond solid (VBS) with a 12-site unit cell. Finally, we consider the effect of SU(4)-symmetry breaking interactions due to Hund's coupling, and argue for a possible phase transition between a VBS and a magnetically ordered state.
We investigate the ground state of the SU(4)\mathrm{SU}(4)SU(4) Heisenberg model on a square lattice with spatial anisotropy on each plaquette bond using the tensor-network method based on infinite projected entangled pair states. We find that the SU(4)\mathrm{SU}(4)SU(4) singlet ground state appears in the strongly anisotropic limit, whereas N\'eel and valence-bond crystal orders coexist in the nearly isotropic limit. By examining the intermediate parameter region, we identify a phase transition between these phases. The nature of the phase transition is likely to be of first order, and the transition point is estimated to be around ${J}^{\ensuremath{'}}/J\ensuremath{\approx}0.85(5)$, where JJJ and ${J}^{\ensuremath{'}}$ are the interaction strengths of intra- and interplaquette bonds, respectively. We also calculate the anisotropy dependence of singlet correlations on a plaquette bond, which will be useful for future experiments of ultracold atoms in optical lattices.
We generalize the construction of the spin-1/2 SU(2) resonating valence bond (RVB) state to the case of the self-conjugate 6\mathbf{6}6 representation of SU(4). As for the case of SU(2) [J.-Y. Chen and D. Poilblanc, Phys. Rev. B 97, 161107(R) (2018)], we use the projected entangled pair state (PEPS) formalism to derive a simple (two-dimensional) family of generalized SU(4) RVB states on the square lattice. We show that, when longer-range SU(4)-singlet bonds are included, a local gauge symmetry is broken down from U(1) to Z2{\mathbb{Z}}_{2}Z2​, leading to the emergence of a short-range spin liquid. Evidence for the topological nature of this spin liquid is provided by the investigation of the Renyi entanglement entropy of infinitely long cylinders and of the modular matrices. Relevance to microscopic models and experiments of ultracold atoms is discussed.