Exact diagonalization of Heisenberg $SU(N)$ chains in the fully symmetric and antisymmetric representations

Abstract

Motivated by recent experimental progress in the context of ultra-cold multi-color fermionic atoms in optical lattices, we have developed a method to exactly diagonalize the Heisenberg $SU(N)$ Hamiltonian with several particles per site living in a fully symmetric or antisymmetric representation of $SU(N)$. The method, based on the use of standard Young tableaux, takes advantage of the full $SU(N)$ symmetry, allowing one to work directly in each irreducible representations of the global $SU(N)$ group. Since the $SU(N)$ singlet sector is often much smaller than the full Hilbert space, this enables one to reach much larger system sizes than with conventional exact diagonalizations. The method is applied to the study of Heisenberg chains in the symmetric representation with two and three particles per site up to $N=10$ and up to 20 sites. For the length scales accessible to this approach, all systems except the Haldane chain ($SU(2)$ with two particles per site) appear to be gapless, and the central charge and scaling dimensions extracted from the results are consistent with a critical behaviour in the $SU(N)$ level $k$ Wess-Zumino-Witten universality class, where $k$ is the number of particles per site. These results point to the existence of a cross-over between this universality class and the asymptotic low-energy behavior with a gapped spectrum or a critical behavior in the $SU(N)$ level $1$ WZW universality class.

16 pages; 9 figures

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Cited by in corpus (28)

• One-Dimensional Symmetry Protected Topological Phases and their Transitions
• Anomaly and global inconsistency matching: $\theta$-angles, $SU(3)/U(1{)}^2$ nonlinear sigma model, $SU(3)$ chains and its generalizations
• Generalization of the Haldane conjecture to SU(3) chains
• Solving correlation clustering with QAOA and a Rydberg qudit system: a full-stack approach
• DMRG simulations of SU(N) Heisenberg chains using standard Young tableaux: fundamental representation and comparison with finite-size Bethe ansatz
• Combined use of translational and spin-rotational invariance for spin systems
• Linear Flavor-Wave Theory for Fully Antisymmetric $\mathrm{SU}(N)$ Irreducible Representations
• Novel families of $\mathrm{SU}(N)$ AKLT states with arbitrary self-conjugate edge states
• Exact diagonalization of $\mathrm{SU}(N)$ Heisenberg and AKLT chains using the full $\mathrm{SU}(N)$ symmetry
• Variational Quantum Eigensolver for SU($N$) Fermions
• Exact ordering of energy levels for one-dimensional interacting Fermi gases with $SU(n)$ symmetry
• Engineering infinite-range SU($n$) interactions with spin-orbit-coupled fermions in an optical lattice
• Haldane Gap of the Three-Box Symmetric $\mathrm{SU}(3)$ Chain
• Suppression and Control of Pre-thermalization in Multi-component Fermi Gases Following a Quantum Quench
• Hidden Bethe states in a partially integrable model
• Symmetry-protected topological phases in the SU(N) Heisenberg spin chain: a Majorana-fermion approach
• Thermodynamic signature of the $\mathrm{SU}(4)$ spin-orbital liquid and symmetry fractionalization from the Lieb-Schultz-Mattis theorem
• Symmetries and Correlations in Strongly Interacting One-dimensional Quantum Gases
• Critical exponents of nonlinear sigma model on Grassmann manifold $U(N)/U(m)U(N-m)$ by $1/N$ expansion
• Even-odd effects in the $J_1-J_2$ SU($N$) Heisenberg spin chain
• Transition from band insulator to Mott insulator and formation of local moment in half-filled ionic SU($N$) Hubbard model
• State selective cooling of $\mathrm{SU}(N)$ Fermi-gases
• Bridging conformal field theory and parton approaches to SU(n)_k chiral spin liquids
• Ground State Phase Diagram of $\text{SU}(3)$ $t$-$J$ Chain
• Density Matrix Renormalization Group simulations of the SU(N) Fermi-Hubbard chain implementing the full SU(N) symmetry via Semi-Standard Young Tableaux and Unitary Group Subduction Coefficients
• Exact quantization of topological order parameter in SU($N$) spin models, $N$-ality transformation and ingappabilities
• Bosonic realization of SU(3) chiral Haldane phases
• Edge states and universality class of the critical two-box symmetric SU(3) chain