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Density Matrix Renormalization Group studies of quantum antiferromagnets on an anisotropic triangular strip

裕一朗 日髙

Published Mar 23, 2023• 0 citations
Publisher

Abstract

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

DOI
10.15083/0002011281

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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 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.
According to the classification using projective representations of the SO(3) group, there exist two topologically distinct gapped phases in spin-1 chains. The symmetry-protected topological (SPT) phase possesses half-integer projective representations of the SO(3) group, while the trivial phase possesses integer linear representations. In the present work, we implement non-Abelian symmetries in the density matrix renormalization group (DMRG) method, allowing us to keep track of (and also control) the virtual bond representations and to readily distinguish the SPT phase from the trivial one by evaluating the multiplet entanglement spectrum. In particular, using the entropies SI{S}^{I}SI (SH{S}^{H}SH) of integer (half-integer) representations, we can define an entanglement gap $G={S}^{I}\ensuremath{-}{S}^{H}$, which equals 1 in the SPT phase, and $\ensuremath{-}1$ in the trivial phase. As an application of our proposal, we study the spin-1 models on various one-dimensional (1D) and quasi-1D lattices, including the bilinear-biquadratic model on a single chain, and the Heisenberg model on a two-leg ladder and a three-leg tube. Among others, we confirm the existence of a SPT phase in the spin-1 tube model and reveal that the phase transition between the SPT and the trivial phase is a continuous one. The transition point is found to be critical, with conformal central charge c=3c=3c=3 determined by fits to the block entanglement entropy.
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 ααα. This bond-anisotropic parameter continuously interpolates among the Ising-like limit α=0α=0α=0, the isotropic point α=1α=1α=1, and the XY-dominated regime α≫1α\gg 1α≫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≃1c \simeq 1c≃1, confirming Tomonaga-Luttinger liquid behavior, while at the transition between the canted Ising and ferromagnetic phases, we observe c≃1/2c \simeq 1/2c≃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=1α= 1α=1.
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.
Effective field theory is a very useful technique for understanding quantum many body systems. We use this approach to study a certain class of critical quantum spin-1 chains and symmetry protected topological phases of matter in two spatial dimensions. In the first context we consider the SU(3) symmetric spin-1 chain with additional interactions. We use bosonization to demonstrate that the effective field theories describing the critical behaviour of these spin chains can be mapped to free compact boson conformal field theories (CFTs) with central charge c=2. We also describe how some predictions from field theory can be verified in numerical calculations using exact diagonalization (ED) and the density matrix renormalization group (DMRG) algorithm. In particular, the bosonization method gives a formula for the evolution of four Tomonaga-Luttinger liquid (TLL) parameters as a function of the lattice parameters. Using the analytic formulae for the scaling dimensions in terms of the TLL parameters and matching of lowest scaling dimensions, we numerically calculate these field-theoretic parameters and confirm that their evolution agrees with the prediction using bosonization. We also tackle aspects of the challenging problem of understanding interacting topological phases. In this context, we aim to understand the effects if interactions in certain classes of symmetry protected topological (SPT) phases of matter. We consider non-chiral SPT phases in two spatial dimensions protected by a discrete symmetry such as Z_K or Z_K x Z_K symmetry. We argue that modular invariance/noninvariance of the partition function of the one-dimensional edge theory can be used to diagnose whether, by adding a suitable potential, the edge theory can be gapped or not without breaking the symmetry. By taking bosonic phases described by Chern-Simons K-matrix theories and fermionic phases relevant to topological superconductors as examples, we demonstrate explicitly that when modular invariance is achieved, we can construct an interaction potential that is consistent with the symmetry and can completely gap out the edge. We also briefly discuss preliminary results of a numerical approach for simulating the 3 dimensional Landau Level problem. This includes a brief description of a potential application of variational Monte Carlo to spin-orbit coupled ab initio systems.
We reexamine the one-dimensional spin-1 XXZXXZXXZ model with on-site uniaxial single-ion anisotropy as to the appearance and characterization of the symmetry-protected topological Haldane phase. By means of large-scale density-matrix renormalization group (DMRG) calculations the central charge can be determined numerically via the von Neumann entropy, from which the ground-state phase diagram of the model can be derived with high precision. The nontrivial gapped Haldane phase shows up in between the trivial gapped even Haldane and N\'eel phases, appearing at large single-ion and spin-exchange interaction anisotropies, respectively. We furthermore carve out a characteristic degeneracy of the lowest entanglement level in the topological Haldane phase, which is determined using a conventional finite-system DMRG technique with both periodic and open boundary conditions. Defining the spin and neutral gaps in analogy to the single-particle and neutral gaps in the intimately connected extended Bose-Hubbard model, we show that the excitation gaps in the spin model qualitatively behave just as for the bosonic system. We finally compute the dynamical spin structure factor in the three different gapped phases and find significant differences in the intensity maximum which might be used to distinguish these phases experimentally.