Hidden order and dynamics in supersymmetric valence-bond solid states: Super-matrix product state formalism

Supersymmetric valence-bond solid models are extensions of the VBS model, a paradigmatic model of “solvable” gapped quantum antiferromagnets, to the case with doped fermionic holes. In this paper, we present a detailed analysis of physical properties of the models. For systematic studies, a supersymmetric version of the matrix-product formalism is developed. On 1D chains, we exactly evaluate the hole-doping behavior of various physical quantities, such as the spin or the charge excitation spectrum, the superconducting order parameter. A generalized hidden order is proposed, and the corresponding string nonlocal order parameter is also calculated. The behavior of the string-order parameter is discussed in the light of the entanglement spectrum.

Figure 1

The type I sVBS is a superposed state of hole-pair doped VBS states. With finite hole-doping parameter $r$, all of the hole-pair doped VBS states are superposed to form the sVBS state, and the sVBS state exhibits the superconducting property. At $r=0$, the sVBS state is reduced to the original VBS state (depicted as the first chain), while $r\to \infty$, the sVBS state is reduced to the MG dimer state (depicted as the last two chains).

Figure 2

The type II sVBS states are expressed as superposition of the hole-pair doped VBS states. Unlike the type I sVBS states, the spinless sites, depicted by the large white circles with double holes, appear. The MG states are realized in the “middle” of the sequence. The original VBS state and the hole-VBS state are respectively realized in the first and last lines.

Figure 3

Plot of absolute values of the five different eigenvalues of $T$. The largest eigenvalue is always unique and nondegenerate.

Figure 4

Plot of $\langle S_j^z\rangle$ ($r=0.3$) for various (left) edge states “hole,” “$\uparrow$,” and “$\downarrow$” (with the right edge state fixed to $s_{\mathrm{R}}=\uparrow$). The system is nonmagnetic in the bulk and magnetic moments exist only around the edges of the chain.

Figure 5

Action of fermion operator on the MPS. (a) Due to the fermionic anticommutation relation, extra factors ${(-1)}^F$ appear in the $A$ matrices on the left of site $j$. Accordingly, a new transfer matrix (c) is necessary as well as the standard one (b) when we calculate expectation values containing fermionic operators.

Figure 6

Plot of ${\mathcal{O}}_{\mathrm{sc}}=\langle {\Delta }_j\rangle$, the hole density $\langle n_{\mathrm{hole}}\left(j\right)\rangle =\langle f_j^{†}{f}_j\rangle$, and the hole-number fluctuation $\langle f_j^{†}{f}_j\rangle -{\langle f_j^{†}{f}_j\rangle }^2$ as a function of $r$. Here the bulk values are plotted. Inset: Profile of the hole density ($r=0.5$) for a finite system ($L=20$) with different left edge states ($\uparrow$, $\downarrow$, and “hole”). Only the left edge state is changed with the right one fixed to $s_{\mathrm{R}}=\uparrow$. The hole density approaches exponentially to the bulk value as we move away from the edge.

Figure 7

Height plot of typical spin configurations in spin-1 VBS state (a) and $M=1$ sVBS state (b). Note that heights are confined within a region of width 1. Although a simple “diluted” Néel picture does not hold because of the presence of hole pairs, still we can find string order when hole pairs are grouped together in (b).

Figure 8

String correlation function in the bulk $C_{\mathrm{string}}(\infty ;n)$ for various values of $r$: (i) $r=0$ (top; pure spin AKLT), (ii) $r=5.0$ (middle), and (iii) $r=20.0$ (bottom). Note that for the pure spin AKLT model ($r=0$), the string correlation function is constant $4/9$. For $r\ne 0$, the string correlation functions exponentially approach the limiting values shown by dashed lines.

Figure 9

The infinite-distance limit of the string correlation function ${\mathcal{O}}_{\mathrm{string}}^{\infty }={\lim }_{n\to \infty }{C}_{\mathrm{string}}(\infty ;n)$ for several values of $M$ plotted as a function of $r$. Note that ${\mathcal{O}}_{\mathrm{string}}^{\infty }(r=0)=0$ for even $M$ corresponding to the vanishing of string-order parameter for even $S$.

Figure 10

Action of local spin operator $S^z$ (a) and fermionic generator $K_1$ (b) onto the sVBS state. The local operators $S_j^a$ ($a=x,y,z$) and $K_{1,2}$ respectively create a triplet bond and a spinon-hole pair (crackion) on either of the two adjacent bonds $(j-1,j)$ and $(j,j+1)$.

Figure 11

The spin excitation (triplon) spectrum ${\omega }_{\mathrm{SMA}}^{\mathrm{s}}\left(k\right)$ obtained by single-mode approximation (SMA). At $r=0$, it reduces to the well-known dispersion ${\omega }_{\mathrm{SMA}}\left(k\right)=10(5+3\cos k)/27$ of the spin-1 VBS model. When $r\nearrow \infty$ (Majumdar-Ghosh limit), dispersion becomes flat.

Figure 12

The excitation spectrum ${\omega }_{\mathrm{SMA}}^{\mathrm{h}}\left(k\right)$ of a spinon-hole pair obtained by single-mode approximation [Eq. (87)]. This spinon-hole pair state is created by fermionic generator $K_1$ except at $r=0$, where the transition matrix elements of $K_1$ from the ground state vanish.

Figure 13

Plot of absolute values $|{\lambda }_i|$ of the seven different eigenvalues of $G$. Since $|{\lambda }_i|$ are symmetric with respect to $r\mapsto -r$, only the $r>0$ part is shown.

Figure 14

Plot of local magnetization profile $\langle S_j^z\rangle$ for different (left) edge states (with right edge fixed). Inset: Spin correlation length ${\xi }_{\mathrm{spin}}\left(r\right)$ as a function of $r$. It monotonically decreases as $r$ is increased and approaches to zero as ${\xi }_{\mathrm{spin}}\sim 1/\log (1+r^2)$.

Figure 15

Plot of ${\mathcal{O}}_{\mathrm{sc}}=\langle {\Delta }_j\rangle$, the hole density $\langle n_{\mathrm{hole}}\left(j\right)\rangle =\langle f_j^{†}{f}_j\rangle$, and the hole-number fluctuation $\langle f_j^{†}{f}_j\rangle -{\langle f_j^{†}{f}_j\rangle }^2$ as a function of $r$. Here the bulk values are plotted. Inset: Profile of the hole density [$r=0.5$ or $r^2/(1+r^2)=0.2$] for a finite system ($L=20$). Only the left edge state is changed with the right one fixed to $s_{\mathrm{R}}=\uparrow$. The hole density approaches exponentially to the bulk value as we move away from the edge.

Figure 16

Plot of hole correlation $C_{\mathrm{SC}}^{fg}\left(n\right)=(a_j{b}_{j+n}-b_j{a}_{j+n})(f_j^{†}{g}_{j+n}^{†}+g_j^{†}{f}_{j+n}^{†})$ for various $r$. Due to the form of the wave function, the hole correlation identically vanishes when the distance $n$ is even. Inset: Correlation length ${\xi }_{\mathrm{SC}}\left(r\right)$ of the hole correlation.

Figure 17

The infinite-distance limit of the string correlation function $\langle S_x^z\exp \left[i\pi {\sum }_{j=x}^{x+n-1}{S}_j^z\right]S_{x+n}^z\rangle$ ($n\nearrow \infty$) as a function of $r$. The value of string correlation smoothly decreases from the AKLT value $4/9$ to 0 (no spins left).