Detection of symmetry-protected topological phases in one dimension

A topological phase is a phase of matter which cannot be characterized by a local order parameter. It has been shown that gapped symmetric phases in one-dimensional (1D) systems can be completely characterized using tools related to projective representations of the symmetry groups. We explain two ways to detect these symmetry protected topological phases in 1D. First, we give a numerical approach for directly extracting the projective representations from a matrix-product state representation. Second, we derive nonlocal order parameters for time-reversal and inversion symmetry, and discuss a generalized string order for local symmetries for which the regular string-order parameter cannot be applied. We furthermore point out that the nonlocal order parameter for these topological phases is actually related to topological surfaces.

Figure 1

(a) Diagrammatic representation of an iMPS formed by the tensors $\Gamma$ and $\Lambda$. The horizontal line represents the bond indices $1,\cdots ,\chi$ and the vertical lines the physical indices $1,\cdots ,d$. (b) Condition for the MPS to be in the canonical form [i.e., the transfer matrices Eqs. (2) and (3) have the identity as eigenvectors with eigenvalue 1].

Figure 2

(a) Transformation of an iMPS which is invariant under an internal symmetry operation $\Sigma$. Here $\tilde{\Gamma }$ can be $\Gamma$, ${\Gamma }^{*}$, or ${\Gamma }^T$. (b) Eigenvalue equation $T^{\Sigma }X=\eta X$ for the generalized transfer matrix. The upper part corresponds to the transformed wave function and the lower part to the original one. We find $\left|\eta \right|=$1 if and only if the state is symmetric under this transformation. (c) Overlap of Schmidt states $|\alpha R\rangle$ with their symmetry-transformed partners. If the chain is assumed to be very long, the overlap can be expressed in terms of the eigenvector $X$ corresponding to the largest-magnitude eigenvector $\left|\eta \right|=1$ of the generalized transfer matrix (filled gray circles). The right boundary yields an overall phase factor which we ignore here (see text for details).

Figure 3

Phase diagram of a spin-1 Heisenberg Hamiltonian in the presence of a single-ion anisotropy $D$ and a transverse magnetic field in the $x$ direction with magnitude $B$.

Figure 4

Different phases of Hamiltonian (13) are distinguished by ${\mathcal{O}}_{Z_2\times Z_2}$ and ${\mathcal{O}}_{\mathcal{I}}$ (defined in the text). These quantities are equal to zero if the symmetry is broken and $\pm 1$ distinguishes the trivial and nontrivial phases.

Figure 5

Diagrammatic derivation of the string order $\mathcal{S}$ for a wave function which is symmetric under an internal transformation $\Sigma$ and represented by an MPS in canonical form: (a) String order involving a segment of transformed sites terminated by operators $O^A$ and $O^B$. (b) The matrices ${\Gamma }_j$ transform according to Eq. (7) and all matrices $U$ and $U^{†}$ vanish except the ones at the edges. (c) Using the properties of the transfer matrices (defined in the text), the expectation value can be simplified for long segments.

Figure 6

Diagrammatic derivation of the string order ${\mathcal{S}}_{\mathcal{I}}$ for a wave function which is inversion symmetric and represented by an iMPS in canonical form: (a) Overlap of a wave function with the wave function for an infinite chain in which a segment of $n$ sites has been inverted. (b) The overlap can be untwisted by reversing the segment using the unitaries $U_{\mathcal{I}}$. (c) For large $L$ and $n$, the expression can be simplified by keeping only the largest-magnitude eigenvector of the transfer matrix $T$, yielding ${\mathcal{S}}_{\mathcal{I}}$.

Figure 7

The nonlocal order parameter ${\mathcal{S}}_{\mathcal{I}}$ distinguishing the Haldane ($D=0.0,0.5$) and large-D ($D=1.5$) phases in the presence of inversion symmetry.

Figure 8

Diagrammatic derivation of the string order ${\mathcal{S}}_{\mathrm{TR}}$ for a wave function which is time-reversal symmetric and represented by an MPS in canonical form: (a) Representation of the wave function $d^{n/2}\langle R_{1n}|{\mathrm{Swap}}_{n+1,2n}|{\psi }_2\rangle$ containing three domain walls, where $\Sigma =\exp \left(i\pi S^y\right)/\sqrt{d}$ and $d$ is the physical dimension. The $\Sigma$ may be replaced by the operator $U_{\mathrm{TR}}$ and the complex conjugation of the segment in the upper row. (b) Representation of the wave function $d^{n/2}\langle {\psi }_2|R_{1n}\rangle$. (c) Contraction of the two wave functions over open indices with the same letter ($a$–$h$) and keeping only the largest-magnitude eigenvector of the transfer matrix $T$ yields for larger $m$,$n$ the order parameter ${\mathcal{S}}_{\mathrm{TR}}$. The dashed red lines indicate the positions of the domain walls.

Figure 9

An illustration of a nonlocal order parameter for measuring the phases of local symmetries directly. A segment is chosen from the chains and divided into three consecutive sections. The symmetry operations are applied to the middle sections. The left and right sections are permuted such that the end points with the same number were connected to each other before applying the permutation. The order parameter is then obtained by calculating the overlap with the four original replicas of the state. The labels $\Gamma$ and $\Lambda$ have been left out so that the figure does not become too busy; furthermore, there are additional domain walls at the other ends of the segments $L$ and $R$ which are not shown.

Figure 10

(a) String-order parameters are equivalent to path integrals on topological surfaces. This is derived by piecing different regions together. (b) The ground-state wave function is represented by a partition function on a half plane with bonds sticking off the lower edge. (c),(d) A simple string operator $\prod a_i$ sandwiched between two copies of the wave function is represented by a plane with a branch cut in it. The torus in (a) is derived by a more complicated gluing procedure.

Figure 11

The first step in the gluing. Applying the symmetries to the four copies of the wave functions and then summing over the middle section produces these four patches. Applying the permutation operators and summing over the remaining spins will glue these together into a single surface: the rule for gluing is described by the black and the white dots, which are supposed to be matched with one another. In this figure, the orientation of the arrow represents whether the symmetry is supposed to be $a$ or $a^{-1}$.

Figure 12

Middle steps in the gluing. (a)–(d) Distorting the planar regions of the previous step into triangles. (e) Pasting the triangles together to form a square. The last gluings cause the square to be rolled into a torus.

Figure 13

How one proves that the branch cuts can be moved around. The partition function has a microscopic representation as a grid of contracted tensors (each site of the lattice corresponds to a tensor $T$), and the branch cut is represented by matrices on the bonds of the tensors. The symmetry of each tensor implies that the branch cut can hop over a single site, and iterating this allows it to be transferred anywhere.