We show that the ground-state expectation value of twisting operator is a topological order parameter for U(1)- and Z_{N}-symmetric symmetry-protected topological (SPT) phases in one-dimensional "spin" systems-it is quantized in the thermodynamic limit and can be used to identify different SPT phases and to diagnose phase transitions among them. We prove that this (nonlocal) order parameter must take values in Nth roots of unity, and its value can be changed by a generalized lattice translation acting as an N-ality transformation connecting distinct phases. This result also implies the Lieb-Schultz-Mattis (LSM) ingappability for SU(N) spins if we further impose a general translation symmetry. Furthermore, our exact result for the order parameter of SPT phases can predict a large number of LSM ingappabilities by the general lattice translation. We also apply the N-ality property to provide an efficient way to construct possible multicritical phase transitions starting from a single Hamiltonian with a unique gapped ground state.
We show that the ground-state expectation value of twisting operator is a topological order parameter for U(1)- and Z_{N}-symmetric symmetry-protected topological (SPT) phases in one-dimensional "spin" systems-it is quantized in the thermodynamic limit and can be used to identify different SPT phases and to diagnose phase transitions among them. We prove that this (nonlocal) order parameter must take values in Nth roots of unity, and its value can be changed by a generalized lattice translation acting as an N-ality transformation connecting distinct phases. This result also implies the Lieb-Schultz-Mattis (LSM) ingappability for SU(N) spins if we further impose a general translation symmetry. Furthermore, our exact result for the order parameter of SPT phases can predict a large number of LSM ingappabilities by the general lattice translation. We also apply the N-ality property to provide an efficient way to construct possible multicritical phase transitions starting from a single Hamiltonian with a unique gapped ground state.
We investigate the extension of pure-state symmetry protected topological phases to mixed-state regime with a strong U(1) and a weak Z2 symmetries in one-dimensional spin systems by the concept of quantum channels. We propose a corresponding topological phase order parameter for short-range entangled mixed states by showing that it is quantized and its distinct values can be realized by concrete spin systems with disorders, sharply signaling phase transitions among them. We also give a model-independent way to generate two distinct phases by various types of translation and reflection transformations. These results on the short-range entangled mixed states further enable us to generalize the conventional Lieb-Schultz-Mattis theorem to mixed states, even without the concept of spectral gaps and lattice Hamiltonians.
We propose and prove a family of generalized Lieb-Schultz-Mattis~(LSM) theorems for symmetry protected topological~(SPT) phases on boson/spin models in any dimensions. The ``conventional'' LSM theorem, applicable to e.g. any translation invariant system with an odd number of spin-1/2 particles per unit cell, forbids a symmetric short-range-entangled ground state in such a system. Here we focus on systems with no LSM anomaly, where global/crystalline symmetries and fractional spins within the unit cell ensure that any symmetric SRE ground state must be a non-trivial SPT phase with anomalous boundary excitations. Depending on models, they can be either strong or ``higher-order'' crystalline SPT phases, characterized by non-trivial surface/hinge/corner states. Furthermore, given the symmetry group and the spatial assignment of fractional spins, we are able to determine all possible SPT phases for a symmetric ground state, using the real space construction for SPT phases based on the spectral sequence of cohomology theory. We provide examples in one, two and three spatial dimensions, and discuss possible physical realization of these SPT phases based on condensation of topological excitations in fractionalized phases.
Although symmetry-protected topological phases (SPTs) can be distinguished by their entanglement properties, it has been unclear how to extract this information directly from expectation values beyond the 1D case. Here, we close this gap and propose a pair of nonlocal order parameters that can detect and distinguish all bosonic SPTs in 2D protected by internal discrete, Abelian unitary symmetries. The desired topological invariants are extracted by these quantities by effectively simulating the SPT path integral on topologically non-trivial spacetime manifolds. Our order parameters are defined in terms of expectation values of partial symmetry and permutation operations acting on fixed numbers of replicas of the system in finite spatial regions. These expectation values correspond to symmetry-twisted versions of multipartite entanglement quantities known as multi-entropies. We show explicitly that our two order parameters detect symmetry-protected four-party and six-party entanglement, respectively, and we constrain possible "spurious" contributions. We analytically test our proposal in fixed-point lattice models. Our results suggest multipartite entanglement to be a defining feature of SPTs; indeed, we expect our methods to generalize to fermionic and higher-dimensional systems.
We study (1 + 1)-dimensional SU(N ) spin systems in the presence of global SU(N ) rotation and lattice translation symmetries.Knowing the mixed anomaly of the two symmetries at low energy, we identify, by anomaly matching argument, a topological index for spin model-the total number of Young-tableaux boxes of spins per unit cell modulo N -characterizing the "ingappability" of the system.A nontrivial index implies either a ground-state degeneracy in a gapped phase, which can be thought of as a field-theory version of the Lieb-Schultz-Mattis (LSM) theorem, or a restriction of the possible universality classes in a critical phase, regarded as the symmetry-protected critical phases.As an example of the latter case, we show only a class of SU(N ) Wess-Zumino-Witten theories can be realized in the low-energy limit of the given lattice model in the presence of the symmetries.Similar constraints also apply when a higher global symmetry emerges in the model with a lower symmetry.We also predict a general constraint on the structure factor which is measurable in experiments.To generalize to higher dimensions, we introduce a new boundary condition which renders the flux-insertion argument for the LSM-type theorems in two or higher dimensional lattices free from the specific choice of system sizes.It also enables a formulation of the LSM-type theorems in arbitrary dimensions in terms of the anomaly in field theories of bulk correspondences.Moreover, by our new boundary condition, we also show that the higher-form U(1) LSM theorem and U(1) and SU(N ) LSM theorems in arbitrary higher dimensions have unified form in the bulk viewpoint.At last, we also discuss the discrete-symmetry LSM-type ingappabilities in the lattice language and we find that projective symmetry representations per unit cell implies that the system cannot be gapped with a unique translation-symmetric ground state.I would like to express my sincere gratitude to all those having helped me during my Master and Doctoral courses at the University of Tokyo.First of all, I would like to thank my parents who have supported me financially and encouraged me to do whatever I am keen on in the past 28 years.Also, I am grateful to every one that has taught me great knowledges on life and study at various steps.More specifically, I have to appreciate the guidance and kind supports by my supervisor, Professor Masaki Oshikawa who has given me patient guidance on study and kind financial supports.Moreover, I would thank Prof. Masahiro Sato for his cooperation in our paper, "Theory of electron spin resonance in one-dimensional topological insulators with spinorbit couplings: Detection of edge states".I am also grateful for Dr. Chang-Tse Hsieh for his collaboration on our work "Anomaly Matching and Symmetry-Protected Critical Phases in SU (N ) Spin Systems in 1+1 Dimensions", on which this Thesis is largely based, and Dr. Yoshiki Fukusumi on the work "Bosonization in a background U (1) gauge field".I would also thank Prof. Yuji Tachikawa and Prof. David Tong for patient and kind help on resolving my confusion during study.I would also like to acknowledge with much appreciation the help from our group members.
The equivalence of topological indices for U(1)⋊Z2-symmetric quantum spin chains
Symmetry-protected topological (SPT) phases in quantum spin chains are distinguished by topological indices. Ogata's operator-algebraic indices apply to a broad range of symmetry classes without requiring continuous symmetry, whereas the elementary twist index of Tasaki requires U(1) symmetry. We prove their equivalence in the common setting of integer-spin chains with on-site U(1)⋊Z2 symmetry and a symmetric locally-unique gapped ground state, without assuming translation invariance or a matrix product representation. A charge-fluctuation estimate yields a strong-limit construction of half-chain rotations with their phases fixed. An exact group commutator identity then identifies the Ogata index with the limit of local twist expectations and gives an explicit error bound. This provides a computable formula for the index when a lower bound on the gap and the relevant ground-state expectations are available. Combining the equivalence with a recent boundary-field gap theorem of OpenAI and our earlier twist-index calculation, we obtain a spin-rotation-invariant locally-unique gapped ground state of the infinite S=1 antiferromagnetic Heisenberg chain with both indices equal to −1, without assuming infinite-volume uniqueness.
The author formulated the problem and proposed the basic strategy, while ChatGPT Astra made substantial contributions to the development of the proof.
We review the Lieb–Schultz–Mattis theorem and its variants, which are no-go theorems that state that a quantum many-body system with certain conditions cannot have a locally-unique gapped ground state. We restrict ourselves to one-dimensional quantum spin systems and discuss both the generalized Lieb–Schultz–Mattis theorem for models with U(1) symmetry and the extended Lieb–Schultz–Mattis theorem for models with discrete symmetry. We also discuss the implication of the same arguments to systems on the infinite cylinder, both with the periodic boundary conditions and with the spiral boundary conditions. For models with U(1) symmetry, we here present a rearranged version of the original proof of Lieb, Schultz, and Mattis based on the twist operator. As the title suggests, we take a modern topological point of view and prove the generalized Lieb–Schultz–Mattis theorem by making use of a topological index (which coincides with the filling factor). By a topological index, we mean an index that characterizes a locally-unique gapped ground state and is invariant under continuous (or smooth) modification of the ground state. For models with discrete symmetry, we describe the basic idea of the most general proof based on the topological index introduced in the context of symmetry-protected topological phases. We start from background materials such as the classification of projective representations of the symmetry group. We also review the notion that we call a locally-unique gapped ground state of a quantum spin system on an infinite lattice and present basic theorems. This notion turns out to be natural and useful from the physicists’ point of view. We have tried to make the present chapter readable and almost self-contained. We only assume basic knowledge about quantum spin systems.