Symmetry-protected topological (SPT) phases are short-range entangled phases of matter with a non-local order parameter which are preserved under a local symmetry group. Here, by using unsupervised learning algorithm, namely the diffusion maps, we demonstrate that can differentiate between symmetry broken phases and topologically ordered phases, and between non-trivial topological phases in different classes. In particular, we show that the phase transitions associated with these phases can be detected in different bosonic and fermionic models in one dimension. This includes the interacting SSH model, the AKLT model and its variants, and weakly interacting fermionic models. Our approach serves as an inexpensive computational method for detecting topological phases transitions associated with SPT systems which can be also applied to experimental data obtained from quantum simulators.
Symmetry-protected topological (SPT) phases are short-range entangled phases of matter with a nonlocal order parameter which are preserved under a local symmetry group. Here, by using an unsupervised learning algorithm, namely, diffusion maps, we demonstrate that we can detect phase transitions between symmetry-broken and topologically ordered phases, and between nontrivial topological phases in different classes. In particular, we show that the phase transitions associated with these phases can be detected in different bosonic and fermionic models in one dimension. This includes the interacting Su-Schrieffer-Heeger model, the Affleck-Kennedy-Lieb-Tasaki model and its variants, and weakly interacting fermionic models. Our approach serves as an inexpensive computational method for detecting topological phase transitions associated with SPT systems which can also be applied to experimental data obtained from quantum simulators.
Recently, it was realized that quantum states of matter can be classified as long-range entangled states (i.e., with nontrivial topological order) and short-range entangled states (i.e., with trivial topological order). We can use group cohomology class
Hd(SG,R/Z) to systematically describe the SRE states with a symmetry
SG [referred as symmetry-protected trivial (SPT) or symmetry-protected topological (SPT) states] in
d-dimensional space-time. In this paper, we study the physical properties of those SPT states, such as the fractionalization of the quantum numbers of the global symmetry on some designed point defects and the appearance of fractionalized SPT states on some designed defect lines/membranes. Those physical properties are SPT invariants of the SPT states which allow us to experimentally or numerically detect those SPT states, i.e., to measure the elements in
Hd(G,R/Z) that label different SPT states. For example,
2+1-dimensional bosonic SPT states with
Zn symmetry are classified by a
Zn integer $m\ensuremath{\in}{\mathcal{H}}^{3}({Z}_{n},\mathbb{R}/\mathbb{Z})={\mathbb{Z}}_{n}$. We find that
n identical monodromy defects, in a
Zn SPT state labeled by
m, carry a total
Zn charge
2m (which is not a multiple of
n in general).
Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phase which is protected by SO(3) spin rotation symmetry. The topological insulator is another exam- ple of SPT phase which is protected by U(1) and time reversal symmetries. It has been shown that free fermion SPT phases can be systematically described by the K-theory. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: distinct d-dimensional bosonic SPT phases with on-site symmetry G (which may contain anti-unitary time reversal symmetry) can be labeled by the elements in H^{1+d}[G, U_T(1)] - the Borel (1 + d)-group-cohomology classes of G over the G-module U_T(1). The boundary excitations of the non-trivial SPT phases are gapless or degenerate. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: (G_H, G_{\Psi}, H^{1+d}[G_{\Psi}, U_T(1)], where G_H is the symmetry group of the Hamiltonian and G_{\Psi} the symmetry group of the ground states.
In this dissertation, we use numerical methods to study one dimensional symmetry protected topological (SPT) phases. We focus on the density matrix renormalization group (DMRG) methods and explore the machine learning methods. We investigated different SPT phases in the context of interactions and disorders. The application of machine learning methods reveals new insights into the topological phases. We begin by studying the Z3 parafermionic chain, the simplest generalization of the Kitaev p-wave wire. The quantum entanglement diagnostics we performed allow us to determine phase boundaries, and the nature of the phase transitions. An intervening incommensurate phase is found between the topological and trivial phases. We locate and characterize a putative tricritical point in the phase diagram where the three above mentioned phases meet at a single point. The phase diagram is predicted to contain a Lifshitz type transition which we con rm using entanglement measures. As another generalization of the Kitaev p-wave wire, we study the interacting inversion symmetric superconductor. We introduce interaction and inversion symmetry and preserve its original time-reversal, particle-hole and chiral symmetry. The symmetries indicates a Z2 classification. We study the quantum entanglement, teleportation and fractional Josephson effects of this system. The ground state of the topological phase is a condensation of four electrons instead of cooper-pairs. While there is a nonzero teleportation for cooper-pairs, the teleportation of one electron is suppressed. The inversion symmetry restricts the edge modes of the system to be cooper-pairs other than two uncorrelated electrons. It is also proved by the 2 pi periodicity in the fractional Josephson effects. At last we apply machine learning methods for classification of SPT phases when strong disorder is present. The entanglement spectrum is used as features to train the random forest model. We do the training using the data generated from a small fraction in the parameter space. The model can give high accuracy predictions to other regions in the phase space. It is even able to make correct predictions to system in a different symmetry class. A detailed analysis of the model indicates that it is able to capture the degeneracy in the entanglement spectrum.
Symmetry-protected topological (SPT) phases with gapless edge excitations have been shown to exist in strongly interacting bosonic/fermionic systems and it is highly desirable to identify practical systems realizing such phases through numerical simulation. A central question to be addressed is how to determine the SPT order in the system given the simulation result, while no local order parameter can be measured to distinguish the phases from a trivial one. In the tensor network approach to simulate strongly interacting systems, the quantum state renormalization algorithm has been shown to be effective in identifying intrinsic topological orders. Here, we show that a modified algorithm can identify SPT orders by extracting the symmetry-protected fixed point entanglement pattern in the ground-state wave function, which is essential for the existence of SPT order. The key to this approach is to add proper symmetry protection to the renormalization process. We demonstrate the effectiveness of this algorithm with examples of nontrivial SPT phases with internal symmetry in 1D and internal and translation symmetry in 2D.
Abstract Classifying and understanding phases of matter is an important task in condensed matter physics. The class of ‘conventional’ symmetry-broken phases is well understood in terms of Landau’s theory. A paradigmatic example is the Z2-symmetric Ising model with symmetric (paramagnetic) and symmetry-broken (ferromagnet) phases, which can be distinguished by measuring the magnetization as a local order parameter. Topological phases of matter, however, are less understood and no complete classification is known. These lecture notes discuss schemes that provide an understanding and characterization of certain topological phases of matter in models with local Hamiltonians. The main focus is on symmetry-protected topological (SPT) phases in one-dimensional bosonic systems. On the basis of the entanglement properties of such systems, a motivation is given for the matrix product state (MPS) representation of ground states. Using the MPS framework, it is demonstrated how SPT phases can be classified using projective representations of the symmetries.