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.
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
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, 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 rotation. We can thus characterize our gSPT phase as a symmetry-protected critical phase protected by the Z3 on-site symmetry, the Z2T time-reversal symmetry, the lattice translation symmetry, and the U(1) spin-rotation symmetry.
We investigate the ground-state phase diagram of a frustrated spin-1/2 Heisenberg ladder in the transverse magnetic field with an anisotropic inter-rung exchange coupling α. This bond-anisotropic parameter continuously interpolates among the Ising-like limit α=0, the isotropic point α=1, and the XY-dominated regime α≫1. Using density-matrix renormalization group calculations within a matrix-product-state framework, we analyze bipartite entanglement, magnetization, and spin correlations to characterize the emergent quantum phases. We identify six distinct ground-state phases, including rung-singlet, Haldane-like, Tomonaga-Luttinger liquid, canted Ising-ordered, XY-polarized, and ordered ferromagnetic states. While magnetization and local correlations provide a first insight into phase classifications, the finite-size scaling of the entanglement measures offers a more sensitive and unified diagnostics of both gapped and gapless regimes. In the gapless regime, we extract the central charge c≃1, confirming Tomonaga-Luttinger liquid behavior, while at the transition between the canted Ising and ferromagnetic phases, we observe c≃1/2, consistent with the Ising universality. Finally, we find that both the Haldane-like behavior and the extended critical Tomonaga-Luttinger liquid regime are strongly confined to the vicinity of the isotropic point α=1.
We study the interplay between the Kitaev and Ising interactions on a ladder geometry. We show that the ground state of the Kitaev ladder is a symmetry-protected topological (SPT) phase, which is protected by a ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry. The nature of the SPT phase is confirmed by degeneracy of the entanglement spectrum. Nonlocal order parameters that indirectly measure phase factors (inequivalent projective representations of the symmetries) of the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry explicitly show the protection of the SPT phase under the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry. We derive the effective theory to describe the topological phase transition on the ladder geometry, which is given by a transverse field Ising model with/without next-nearest-neighbor coupling based on the primary Ising configurations. The ladder has three phases, namely, the Kitaev SPT, symmetry-broken ferro/antiferromagnetic order, and classical spin liquid. The nonzero quantum critical point and its corresponding central charge are provided by the effective theory, which are in full agreement with the numerical results, i.e., the divergence of entanglement entropy at the critical point and change of the entanglement spectrum degeneracy. The central charge of the critical points are either c=1 or c=2, with the magnetization and correlation exponents being 1/4 and 1/2, respectively. The transition from the classical spin-liquid phase of the frustrated Ising ladder to the Kitaev SPT phase is mediated by a floating phase, which shows strong finite entanglement scaling.
We systematically study gapless edge modes corresponding to 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, given by the coset SUk(3)/SUk(2) CFT at level k=1.
Two-dimensional topological paramagnets protected by Z3 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.