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arXiv:2207.02485

Gapless chirality liquid with symmetry-protected edge spins

Shunsuke C. Furuya, Katsuhiro Morita

Published Jul 6, 2022• 0 citations
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Abstract

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, $\mathbb Z_3\times\mathbb Z_2^T$, 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 $\mathbb{Z}_3$ rotation. We can thus characterize our gSPT phase as a symmetry-protected critical phase protected by the $\mathbb{Z}_3$ on-site symmetry, the $\mathbb{Z}_2^T$ time-reversal symmetry, the lattice translation symmetry, and the U(1) spin-rotation symmetry.

DOI
10.48550/arxiv.2207.02485

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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.
We study a three-leg spin-1/2 ladder with geometrically frustrated interleg interactions. We call this model an anisotropic triangular-strip (ATS) model. We numerically and field-theoretically show that its ground state belongs to a gapless symmetry-protected topological (SPT) phase. The numerical approach is based on density-matrix renormalization group analyses of the entanglement entropy and the entanglement spectrum. Whereas the entanglement entropy exhibits a critical behavior, the entanglement spectrum is nontrivially degenerate. These entanglement properties imply that the ground state is a gapless topological phase. We investigate the ATS model using a quantum field theory to support the numerical findings. When the frustrated interchain interaction is deemed a perturbation acting on the three spin chains, the frustrated interchain interaction almost isolates the second chain from the other two chains. However, at the same time, the second chain mediates a ferromagnetic interaction between the first and third chains. Therefore, the ground state of the ATS model is a gapless Tomonaga-Luttinger liquid weakly coupled to a spin-1 Haldane chain with irrelevant interactions. Last but not least, we show that the gapless SPT phase of the ATS model is a symmetry-protected critical phase. We point out that the symmetry protection of criticality is essential in characterization of the gapless SPT phase.
Many quantum phases are described by the Landau theory, which is based on the spontaneous symmetry breaking.However, the symmetry-protected topological (SPT) phases have been discovered as new quantum phases that cannot be described by the Landau theory.One of the examples is the Haldane phase in the one-dimensional spin-1 Heisenberg chain.This concept of the SPT phase is developed for gapped states.However, many researchers are attempting to extend this concept to gapless states.Such as SPT phase in a gapless system is called a gapless SPT phase.To calculate the ground state of a spin system, we perform the numerical calculation based on the density-matrix renormalization group (DMRG) method.All the quantum spin states on a lattice is written in the form of a matrix product state (MPS).The DMRG method is one of the variational methods by which we optimize the MPS.In addition, an MPS characterizes SPT phases.An entanglement spectrum is defined using singular values of an MPS and becomes even-fold degenerate when the state is in the Haldane state.We can also calculate an entanglement entropy that characterizes the gapless system.In this thesis, we investigate a three-leg spin-1/2 ladder with geometrically frustrated interleg interactions.We call this model an anisotropic triangular-strip (ATS) model.According to the quantum field theory and renormalization group theory, when the rung coupling is sufficiently small, the frustrated interchain interaction almost isolates the second chain in the middle from the other two chains.In addition, at the same time, the first and third chains effectively mediate the ferromagnetic interaction.As a result, the ground state of the ATS model is a gapless Tomonaga-Luttinger liquid weakly coupled to a spin-1 Haldane chain with irrelevant interactions.By calculating the entanglement spectrum and the entanglement entropy, we show that the ground state of the ATS model with weak rung coupling is the gapless SPT state.Moreover, the ground state of the ATS model with sufficiently large rung coupling becomes ferrimagnetic state.We also investigate the other phases of the ATS model by the DMRG method.Through this study, we found the gapless SPT state which is not trivially decoupled into the gapless system and the nontrivial gapped SPT system in the ATS model, and we determine the phase diagram of the ATS model.This doctoral thesis would not have been completed without the support of many people.I am grateful to Shunsuke C. Furuya for proposing the basic concept of this thesis and supporting my investigation.He also gives me much fruitful advice especially when I got stuck in my research.I also would like to thank Atsushi Ueda especially for teaching me how to use the supercomputer in ISSP.Thanks to Yasuhiro Tada for teaching me great advice which is valuable to my study and for polishing this thesis.This study is performed with their cooperation, and some results are published in the journal Physical Review B.I am grateful to my supervisor, Prof. Masaki Oshikawa, for supporting many things about my research.He introduced the gapless SPT phase to me.I had meaningful discussions with him and was given much fruitful advice for completing this doctoral thesis.Oshikawa Group members gave me great and meaningful advice when I presented in the group seminars
Using density-matrix renormalization-group calculations for infinite cylinders, we elucidate the properties of the spin-liquid phase of the spin-$\frac{1}{2}\phantom{\rule{4pt}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the triangular lattice. We find four distinct ground states characteristic of a nonchiral, Z2{Z}_{2}Z2​ topologically ordered state with vison and spinon excitations. We shed light on the interplay of topological ordering and global symmetries in the model by detecting fractionalization of time-reversal and space-group dihedral symmetries in the anyonic sectors, which leads to the coexistence of symmetry protected and intrinsic topological order. The anyonic sectors, and information on the particle statistics, can be characterized by degeneracy patterns and symmetries of the entanglement spectrum. We demonstrate the ground states on finite-width cylinders are short-range correlated and gapped; however, some features in the entanglement spectrum suggest that the system develops gapless spinonlike edge excitations in the large-width limit.
We study symmetric gapped ground states of one-dimensional spin SSS chains with a continuous exponential symmetry U(1)⋊Z2U(1)\rtimes\mathbb Z_2U(1)⋊Z2​ with integer base qqq, where the U(1) charge has a modulated unit of charge qL−jq^{L-j}qL−j decaying exponentially with site jjj. The simplest model is a quantum breakdown model that can be viewed as a qqq-deformation of the XXZ Haldane spin chain. We show that the topology of their symmetric ground states crucially depends on SSS and qqq. For integer SSS, odd qqq allows a nontrivial Z2\mathbb Z_2Z2​ Haldane-type symmetry-protected topological (SPT) phase with protected edge modes, while even qqq allows no nontrivial SPT phase. For half-odd integer SSS, odd qqq prohibits any unique symmetric gapped ground state due to a Lieb-Schultz-Mattis obstruction, whereas even qqq enforces a novel symmetric gapped phase which we name as the ``half-SPT" phase. Unconventionally, the half-SPT phase only has a symmetry protected edge mode on one edge, and has half of the string orders of an SPT phase. Our results uncover a new type of symmetry-protected gapped phases beyond existing classification schemes.
Two-dimensional topological paramagnets protected by Z3\mathbb{Z}_3Z3​ symmetry: Properties of the boundary Hamiltonian
We systematically study gapless edge modes corresponding to \mathbb{Z}_3 ℤ 3 symmetry-protected topological (SPT) phases of two-dimensional three-state Potts paramagnets on a triangular lattice. First, we derive microscopic lattice models for the gapless edge and, using the density-matrix renormalization group (DMRG) approach, investigate the finite-size scaling of the low-lying excitation spectrum and the entanglement entropy. Based on the obtained results, we identify the universality class of the critical edge, namely the corresponding conformal field theory and the central charge. Finally, we discuss the inherent symmetries of the edge models and the emergent winding number symmetry. As a result, one-dimensional chains with this symmetry form a model that supports gapless excitations due to its tricritical symmetry. Numerically, we show that low-energy states in the continuous limit of the edge model can be described by conformal field theory (CFT) with central charge c=1 c = 1 , given by the coset SU_k(3)/SU_k(2) S U k ( 3 ) / S U k ( 2 ) CFT at level k=1.
We introduce a systematic construction of a gapless symmetry-protected topological phase in one dimension by ``decorating'' the domain walls of Luttinger liquids. The resulting strongly interacting phases provide a concrete example of a gapless symmetry-protected topological (gSPT) phase with robust symmetry-protected edge modes. Using boundary conformal field theory arguments, we show that while the bulks of such gSPT phases are identical to conventional Luttinger liquids, their boundary critical behavior is controlled by a different, strongly coupled renormalization group fixed point. Our results are checked against extensive density matrix renormalization group calculations.