We extend the classification of approximately locality-preserving unitaries (ALPUs) on spin chains recently obtained by [1] to take a local symmetry action of a finite symmetry group G into account. We classify such equivariant ALPUs up to strongly continuous deformation and allowing to `stack' with auxiliary spin chains. We find that the equivalence classes are uniquely labelled by a chiral index taking values in a discrete subset of $\mathbb{R}$ , and a cohomology index taking values in the second cohomology group $H^2(G, U(1))$ . We discuss the relation of this cohomology index to the index for one-dimensional symmetry protected topological (SPT) phases, which also takes values in $H^2(G, U(1))$ .We extend the classification of approximately locality-preserving unitaries (ALPUs) on spin chains recently obtained by [1] to take a local symmetry action of a finite symmetry group G into account. We classify such equivariant ALPUs up to strongly continuous deformation and allowing to `stack' with auxiliary spin chains. We find that the equivalence classes are uniquely labelled by a chiral index taking values in a discrete subset of $\mathbb{R}$, and a cohomology index taking values in the second cohomology group $H^2(G, U(1))$. We discuss the relation of this cohomology index to the index for one-dimensional symmetry protected topological (SPT) phases, which also takes values in $H^2(G, U(1))$.

Jun 3, 20211 citation

Enlarged Symmetries of Spin Chains and Loop Models

Hubert Saleur

Aug 8, 20071 citation

Volledigheid van de classificatie van symmetriebeschermde topologische toestanden

Bruno de Oliveira Carvalho

This thesis investigates the structure of topological phases of quantum lattice systems. We focus on the classification of unique gapped ground states and symmetric short-range entangled states in one and two spatial dimensions. Our main results establish the completeness of well-known classification indices, proving that they form complete invariants with respect to stable equivalence of states. The principal technical challenge lies in constructing the corresponding adiabatic paths that connect equivalent states, both in the presence and absence of symmetries. In the first part of this thesis, we study unique gapped ground states of one-dimensional quantum spin systems. We first show that such systems are topologically trivial in the bulk, in the sense that their ground states are short-range entangled. We then incorporate symmetries modelled by compact topological groups and classify the resulting symmetry-protected topological (SPT) phases of spin chains. In particular, we prove that the well-known cohomological classification is complete under stable equivalence: if two SPT states share the same index, then they can be connected by an adiabatic evolution generated by a local, symmetric Hamiltonian. In the second part, we investigate two-dimensional SPT phases and their relation to one-dimensional locality-preserving symmetries (LPSs). We show that stable equivalence classes of LPSs form a group that is isomorphic to the third cohomology group for a finite symmetry group G. Moreover, LPSs emerge at the boundaries of two-dimensional SPT states that are entangled by globally symmetric finite-depth quantum circuits, and we prove that such a subclass of SPT states is completely classified by stable equivalence classes of boundary LPSs.This thesis investigates the structure of topological phases of quantum lattice systems. We focus on the classification of unique gapped ground states and symmetric short-range entangled states in one and two spatial dimensions. Our main results establish the completeness of well-known classification indices, proving that they form complete invariants with respect to stable equivalence of states. The principal technical challenge lies in constructing the corresponding adiabatic paths that connect equivalent states, both in the presence and absence of symmetries. In the first part of this thesis, we study unique gapped ground states of one-dimensional quantum spin systems. We first show that such systems are topologically trivial in the bulk, in the sense that their ground states are short-range entangled. We then incorporate symmetries modelled by compact topological groups and classify the resulting symmetry-protected topological (SPT) phases of spin chains. In particular, we prove that the well-known cohomological classification is complete under stable equivalence: if two SPT states share the same index, then they can be connected by an adiabatic evolution generated by a local, symmetric Hamiltonian. In the second part, we investigate two-dimensional SPT phases and their relation to one-dimensional locality-preserving symmetries (LPSs). We show that stable equivalence classes of LPSs form a group that is isomorphic to the third cohomology group for a finite symmetry group G. Moreover, LPSs emerge at the boundaries of two-dimensional SPT states that are entangled by globally symmetric finite-depth quantum circuits, and we prove that such a subclass of SPT states is completely classified by stable equivalence classes of boundary LPSs.

May 19, 20260 citations

Spin Chains with Dynamical Lattice Supersymmetry

Christian Hagendorf

arXiv:1207.0357Journal of Statistical PhysicsFeb 1, 201331 citations

Quantum spin chains with quantum group symmetry

M. Fannes, Bruno Nachtergaele, R. F. Werner

arXiv:cond-mat/9504002Communications in Mathematical PhysicsJan 1, 199628 citations

Classification of symmetry protected states of quantum spin chains for continuous symmetry groups

Bruno de O. Carvalho, Wojciech De Roeck, Tijl Jappens

Symmetry protected states (SPT's) of quantum spin systems were studied by several authors. For one-dimensional systems (spin chains), there is an essentially complete and rigorous understanding: SPT's corresponding to finite on-site symmetry groups $G$ are classified by the second cohomology group $H^2(G,U(1))$ , as established by Kapustin et al. [J. Math. Phys. (2021)]. We extend this result to the case of compact topological symmetry groups $G$ . We also strengthen the existing results in the sense that our classification results holds within the class of spin chains with locally bounded on-site dimensions.Symmetry protected states (SPT's) of quantum spin systems were studied by several authors. For one-dimensional systems (spin chains), there is an essentially complete and rigorous understanding: SPT's corresponding to finite on-site symmetry groups $G$ are classified by the second cohomology group $H^2(G,U(1))$, as established by Kapustin et al. [J. Math. Phys. (2021)]. We extend this result to the case of compact topological symmetry groups $G$. We also strengthen the existing results in the sense that our classification results holds within the class of spin chains with locally bounded on-site dimensions.

arXiv:2409.01112Sep 2, 20240 citations

\mathbb{R}

, and a cohomology index taking values in the second cohomology group

H^2(G, U(1))

. We discuss the relation of this cohomology index to the index for one-dimensional symmetry protected topological (SPT) phases, which also takes values in

H^2(G, U(1))

G

are classified by the second cohomology group

H^2(G,U(1))

, as established by Kapustin et al. [J. Math. Phys. (2021)]. We extend this result to the case of compact topological symmetry groups

G

. We also strengthen the existing results in the sense that our classification results holds within the class of spin chains with locally bounded on-site dimensions.