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arXiv:1910.00589 · Physical review. B./Physical review. B

Anomalies and unusual stability of multicomponent Luttinger liquids in Zn×Zn spin chains

Yahya Alavirad, Maissam Barkeshli

Published Jul 30, 2021• 14 citations
arXivHTMLPDF

Abstract

We study translationally invariant spin chains where each unit cell contains an $n$-state projective representation of a ${\mathbb{Z}}_{n}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{n}$ internal symmetry, generalizing the spin-1/2 XYZ chain. Such spin chains possess a generalized Lieb-Schultz-Mattis (LSM) constraint, and we demonstrate that certain $(n\ensuremath{-}1)$-component Luttinger liquids possess the correct anomalies to satisfy these LSM constraints. For $n=3$, using both numerical and analytical approaches, we find that such spin chains with nearest-neighbor interactions appear to be gapless for a wide range of microscopic parameters and described by a two-component conformally invariant Luttinger liquid. This implies the emergence of $n\ensuremath{-}1$ conserved U(1) charges from only discrete microscopic symmetries. Remarkably, the system remains gapless for an unusually large parameter regime despite the apparent existence of symmetry-allowed relevant operators in the field theory. This suggests that either these spin chains have hidden conserved quantities not previously identified, or the parameters of the field theory are simply unusual due to frustration effects of the lattice Hamiltonian. We argue that similar features are expected to occur in (1) ${\mathbb{Z}}_{n}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{n}$ symmetric chains for $n$ odd and (2) ${\mathbb{S}}_{n}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{n}$ symmetric chains for all $n>2$. Finally, we suggest the possibility of a lower bound growing with $n$ on the minimum central charge of field theories that possess such LSM anomalies.

DOI
10.1103/physrevb.104.045151

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We report that a spin-1/2 tetrahedral Heisenberg chain realizes a gapless symmetry-protected topological (gSPT) phase characterized by the coexistence of the Tomonaga-Luttinger-liquid criticality due to chirality degrees of freedom and the symmetry-protected edge state due to spin degrees of freedom. This gSPT phase has an interesting feature that no symmetry forbids the trivial spin gap opening but a discrete symmetry, Z3×Z2T\mathbb Z_3\times\mathbb Z_2^TZ3​×Z2T​, forbids the unique gapped ground state. In the first part of the paper, we numerically show the coexistence of a critical entanglement entropy and a nontrivially degenerate entanglement spectrum based on the density-matrix renormalization group (DMRG) method.Next, we clarify that chirality degrees of freedom form the Tomonaga-Luttinger liquid while spin degrees of freedom form the spin-1 Haldane state based on a degenerate perturbation theory. Last but not least, we discuss the Lieb-Schultz-Mattis-type ingappability in the gSPT phase, using a local Z3\mathbb{Z}_3Z3​ rotation. We can thus characterize our gSPT phase as a symmetry-protected critical phase protected by the Z3\mathbb{Z}_3Z3​ on-site symmetry, the Z2T\mathbb{Z}_2^TZ2T​ time-reversal symmetry, the lattice translation symmetry, and the U(1) spin-rotation symmetry.