Skeleton of matrix-product-state-solvable models connecting topological phases of matter

Models whose ground states can be written as an exact matrix-product state (MPS) provide valuable insights into phases of matter. While MPS-solvable models are typically studied as isolated points in a phase diagram, they can belong to a connected network of MPS-solvable models, which we call the MPS skeleton. As a case study where we can completely unearth this skeleton, we focus on the one-dimensional BDI class—noninteracting spinless fermions with time-reversal symmetry. This class, labeled by a topological winding number, contains the Kitaev chain and is Jordan-Wigner-dual to various symmetry-breaking and symmetry-protected topological (SPT) spin chains. We show that one can read off from the Hamiltonian whether its ground state is an MPS: defining a polynomial whose coefficients are the Hamiltonian parameters, MPS-solvability corresponds to this polynomial being a perfect square. We provide an explicit construction of the ground state MPS, its bond dimension growing exponentially with the range of the Hamiltonian. This complete characterization of the MPS skeleton in parameter space has three significant consequences: (i) any two topologically distinct phases in this class admit a path of MPS-solvable models between them, including the phase transition which obeys an area law for its entanglement entropy; (ii) we illustrate that the subset of MPS-solvable models is dense in this class by constructing a sequence of MPS-solvable models which converge to the Kitaev chain (equivalently, the quantum Ising chain in a transverse field); (iii) a subset of these MPS states can be particularly efficiently processed on a noisy intermediate-scale quantum computer.

Figure 1

MPS skeleton of the Ising-cluster model. The phase diagram of the model described by the Hamiltonian in Eq. (11); each dashed, gray line corresponds to setting one $t_{\alpha }$ to zero. The differently shaded regions show the different phases labeled by the topological invariant $\omega$. In the fermionic representation, these correspond to Kitaev chains with distinct winding numbers. In the dual spin chain formulations, these phases are the trivial paramagnet ($\omega =0$), the Ising magnet ($\omega =1$) and the symmetry-protected topological cluster phase ($\omega =2$). The solid red and blue lines show the parameterized paths along which we can find the MPS representation of the ground state, see Eqs. (10) and (12), the dashed gray lines show the lines where one parameter equals zero. See also Refs. [14, 44].

Figure 2

Circuit and MPS equivalence for the skeleton of the Ising-cluster model. (a) Circuit construction for the ground state in the example from $\omega =0$ to 2, given in Eq. (13). The circuit elements $M_n^{\left(1\right)}=1-a_1{h}_{n,1}$ are represented by blue boxes coupling two neighboring spins (black lines). The repeated unit element is highlighted in gray. (b) The repeating unit element of the circuit forms a tensor $A_{\alpha \beta }^j$, as defined in Eq. (14), which shows the equivalence of the circuit to a matrix-product state with bond dimension $\chi =2$.

Figure 3

MPS skeleton of generalized cluster models or Kitaev chains. The phase diagram of the model described by the function $f\left(z\right)$ in Eq. (20); the differently shaded regions show the different phases labeled by $\omega$. The solid blue, red and orange lines show the parameterized paths along which we can find the MPS representation of the ground state, the blue line corresponds to the case $\nu =\mu$, see Eq. (21), the red line to the case $\nu =\frac{\mu }{\mu +1}$, see Eq. (22), and the orange line to the case $\nu =\mu -1$, see Eq. (23). The red and orange lines intersect at the golden ratio $\phi =\frac{1+\sqrt{5}}{2}$.

Figure 4

Scaling along MPS-solvable path that converges to the critical Ising chain. Red dots correspond to different choices of $m\in \mathbb{N}$ ($400\le m\le 2\times {10}^5$) labeling the path of MPS-solvable models in Eq. (29), for which we calculate the energy gap and string order parameter. Both quantities converge to zero at the Ising critical point; the dashed black line gives scaling exponent $\mathrm{\Delta }\approx 1/8$ [see Eq. (34)].

Figure 5

Graphical notation for the circuit construction of the ground states. (a) Definition of graphical notation for gates of the form $(1+c{h}_{n,\alpha })$ with different values of $\alpha$. The squares indicate $\tilde{\gamma }$ Majorana operators while circles correspond to $\gamma$ operators. These gates can be written in terms of spin string operators using Eq. (3). (b) Commutation relations for the gates. These gates commute unless symbols of the same type are on the same wire. (c) Graphical notation for the unitary gates $U_n$ defined in Eq. (53), with double lines to distinguish them from the general form. When one of the end symbols lines up with that of another gate they satisfy the relation shown graphically, which is written explicitly in Eq. (60) and in Appendix pp3.

Figure 6

Example of the mapping from circuit to MPS for $d=2,p=3$ using the graphical notation in Fig. 5. The different colored gates correspond to the different layers in the state construction. The gray box indicates the repeating circuit element equivalent to an MPS tensor.

Figure 7

Example of the mapping from circuit to MPS for $d=2,p=3$ after simplifying (see Appendix pp3). Panel (a) corresponds to Fig. 6 after using the commutation relations for the gates. Panel (b) shows the repeating elements of the sequential circuit. Panel (c) shows the equivalent MPS tensor.

Figure 8

Putting a circuit into unitary form. (a) Illustration of the substitution explained in the main text that allows us to change a square to a circle when acting directly on the initial state $|\downarrow \rangle$ [see Eq. (62)]. The substitution turns the gates $P_n$ (if $p=1$) and those appearing in $M_n^{\left(k\right)}$ into a unitary gate if $a_k\in \mathbb{R}$. (b–e) The steps for repeatedly using the substitution in (a) and commutation relations to bring the circuit into unitary form. The figures shown correspond to either $p=0,d=3$ or to $p=1,d=2$, but can be applied more generally as explained in the main text. (f) Equality between MPS circuit and sequential unitary circuit. Without the unitary SPT entangler $W_{\alpha =1}$ (red), this applies to examples with $p=0,d=3$ or with $p=1,d=2$. With the SPT entangler, it applies to examples with $p=-4,d=3$ or with $p=-3,d=2$.

Figure 9

Contour integral for the Fourier transform. (a) The contour is the unit circle, integrating over this contour defines the Fourier coefficients of Eq. (A7). (b) The deformed contour gives the same integral and is snagged at poles and branch cuts (indicated by blue wavy lines).

Figure 10

Graphical representation of algebraic relations. These relations are satisfied by the unitary gates $U_n$ (represented with double lines) and the gates $M$ (represented with single lines). These are explained explicitly in the main text. Panels (a–c) are when the $\tilde{\gamma }$ operators (squares), labeled by $n$, line up. Panels (d–f) correspond to when the $\gamma$ operators (circles) line up.

Figure 11

Explicit example for Case (i) with $d=2$, $p=3$.

Figure 12

Explicit example for Case (ii) with $d=3$, $p=-6$.

Figure 13

Explicit example for Case (iii) with $d=4$, $p=-3$.