Entanglement spectrum and boundary theories with projected entangled-pair states

In many physical scenarios, close relations between the bulk properties of quantum systems and theories associated with their boundaries have been observed. In this work, we provide an exact duality mapping between the bulk of a quantum spin system and its boundary using projected entangled-pair states. This duality associates to every region a Hamiltonian on its boundary, in such a way that the entanglement spectrum of the bulk corresponds to the excitation spectrum of the boundary Hamiltonian. We study various specific models: a deformed AKLT model [I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys. Rev. Lett. 59, 799 (1987)], an Ising-type model [F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac, Phys. Rev. Lett. 96, 220601 (2006)], and Kitaev’s toric code [A. Kitaev, Ann. Phys. 303, 2 (2003)], both in finite ladders and in infinite square lattices. In the second case, some of those models display quantum phase transitions. We find that a gapped bulk phase with local order corresponds to a boundary Hamiltonian with local interactions, whereas critical behavior in the bulk is reflected on a diverging interaction length of the boundary Hamiltonian. Furthermore, topologically ordered states yield nonlocal Hamiltonians. Because our duality also associates a boundary operator to any operator in the bulk, it in fact provides a full holographic framework for the study of quantum many-body systems via their boundary.

Figure 1

(top) We consider an $N_v\times N_h$ spin lattice in a cylindrical geometry. The PEPS is obtained by replacing each lattice site with a tensor $A$ and contracting the virtual indices $\alpha$ and $\beta$ along the horizontal and vertical directions. (bottom) We cut the lattice into two pieces, left and right. The virtual indices $\alpha$ of the tensors $A$ along the cut are shown. The state $|{\Psi }_L\rangle$ acts on the spins ($i_{k,n}$) as well as on the virtual spins along the cut.

Figure 2

(a) Ribbon made of two ($N_h=2$) coupled periodic $S=1/2$ Heisenberg chains (two-leg ladder). (b) Ground state of a two-leg $S=3/2$ AKLT ladder. Each site is split into three spins-1/2 (red dots). Nearest-neighbor spins-1/2 are paired up into singlet valence bonds. (c) PEPS representation for $S=3/2$ and $S=2$ sites of AKLT wave functions in the valence bond (singlet) picture (for connection to the “maximally entangled picture”; see text). Open squares indicate the $r_{{\alpha }_1,{\alpha }_2,{\alpha }_3}^m$ and $A_{{\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4}^m$ tensors defined in the text and open circles correspond the to $2\times 2$ matrix $[0,1;-1,0]$.

Figure 3

(a) Four-leg ($N_h=4$) AKLT ladder on a cylinder partitioned (dotted green line) into two halves. (b) Schematic representation of the density matrix ${\sigma }_b^2$ of a four-leg ($N_h=4$, $\ell =2$) AKLT ladder. After being “cut” the two halves are “glued” together (physical indices are contracted).

Figure 4

Entanglement spectra of $H_b$ (with respect to the ground-state energy ${\xi }_0$) vs total momenta $K$ in the chain (vertical) direction. (a) Two-leg ($14\times 2$) quantum Heisenberg ladder, (b) two-leg ($16\times 2$) AKLT ladder, and (c) eight-leg ($16\times 8$) AKLT ladder. The eigenvalues are labeled according to their total spin quantum number using different symbols (according to the legend on the graph).

Figure 5

(a) Amplitudes $A_r$ of the (isotropic) spin-spin couplings up to distance $r=7$ of the effective boundary Hamiltonian of a quantum Heisenberg two-leg ladder in the Haldane and rung singlet phases vs $\theta$. (b) Ratio of the same amplitudes normalized to the nearest-neighbor coupling ($r=1$). Computations are carried out on $12\times 2$ (open symbols) and $14\times 2$ (closed symbols) systems. Note that when $\theta \to \pi /2$ (decoupled rung singlets), $A_r/A_0\to 0$ for $r⩾1$ and all the weights of the reduced density matrix become equal to $2^{-N_v}$ ($A_0=\ln 2$).

Figure 6

Two-leg quantum Heisenberg ladders: ratio of the amplitudes $\left|A_r\right|$ by the nearest-neighbor amplitude $A_1$ plotted using a logarithmic scale as a function of $r$ for different values of $\theta$. (a) Antiferromagnetic and (b) ferromagnetic leg couplings (the rung couplings are antiferromagnetic in both cases).

Figure 7

AKLT ladders: (a) Ratio $\left|A_r\right|/A_1$ plotted using a logarithmic scale as a function of $r$. Results are approximation free for finite $N_h$ while the $N_h\to \infty$ limit is obtained by finite size scaling [see Fig. 8b]. (b) Comparison with $\sqrt{d_{r+1}/d_2}$ (solid symbols) computed (see text) on a two-leg and infinitely long ($N_h=\infty$) cylinder.

Figure 8

(a) Finite size scaling of the amplitudes $A_r$ for a two-leg AKLT ladder vs $1/N_v$ ($N_v=14$, 16, 18, 20). (b) Finite size scaling of the amplitudes $A_r$ for $N_h$-leg AKLT ladders vs $1/N_h$ at fixed $N_v=16$ (open symbols) or $N_v=18$ (solid and $+$ symbols). (c) Von Neumann (VN) entropy per unit length [normalized by $\ln (2)$] vs $1/N_h$ at fixed $N_v=16$.

Figure 9

AKLT model with finite “nematic” field $\Delta$: (a) Relative amplitude $\sqrt{d_{r+1}/d_2}$ in a two-leg ladder plotted using a logarithmic scale as a function of $r$ and (b) the same for an infinitely long cylinder ($N_h=\infty$). From bottom to top, $\Delta$ is incremented from $0$ to ${\Delta }_{\max }$ by constant steps. (c) Effective temperature ${\beta }_{\mathrm{eff}}$ [see Eq. (27)] vs $\Delta$ for the two cases reported in (a) and (b). All results are obtained for $N_v=16$ ($D_c=50$, and 100 iterations such that the tensors $C$ already converge).

Figure 10

(a) Inverse correlation length ${\xi }^{-1}$ vs $\Delta$ for both an AKLT two-leg ladder and an infinitely long cylinder ($N_h=\infty$). These data correspond to the infinite circumference limit, i.e., $N_v=\infty$. The arrow marks the phase transition in the infinitely long cylinder. Comparison with the inverse of the emerging length scale ${\xi }_b$ is obtained by fitting the decay of the coefficients of $H_b$ plotted in Fig. 9a as $\sqrt{d_{r+1}/d_2}~\exp (-r/{\xi }_b)$. (b) Truncation error in the $N_h\to \infty$ procedure. The results are compared with those obtained with $D_c=150$, and 100 iterations have always been used.

Figure 11

Entanglement spectrum of a $16\times 2$ Ising PEPS ladder vs momentum along the ladder leg direction. Comparison between (a) $\theta =0.2$ and (b) $\theta =0.5$ using the same energy scale. (c) Zoom in of the low-energy part of (b).

Figure 12

Ising PEPS: Relative amplitude $\sqrt{d_n/d_1}$ in (a) a two-leg ladder and (b) an infinitely long ($N_h=\infty$) cylinder as a function of $n$ for $\theta$ varying from 0.1 to $~$0.51 with constant intervals (0.1, 0.1585, 0.2171, 0.2756, 0.3342, 0.3927, 0.4512, and 0.5098, from top to bottom). Logarithmic scales are used on the vertical axis in both (a) and (b). Inset shows the ratio of the effective transverse field $\sqrt{d_1}$ over the effective Ising nearest-neighbor coupling $\sqrt{d_2}$ vs $\theta$ for $N_h=\infty$ ($D_c=50$ and 100 iterations). All results are obtained for $N_v=12$.

Figure 13

Ising PEPS: Inverse correlation length ${\xi }^{-1}$ vs $\Delta$ for both two-leg ladder and infinitely long cylinder ($N_h=\infty$, $D_c=150$ and 100 iterations). These data correspond to the infinite circumference limit, i.e., $N_v=\infty$. The arrow marks the phase transition in the infinitely long cylinder. Comparison with the inverse of the emerging length scale ${\xi }_b^{-1}$ obtained by fitting the decay of the coefficients plotted in Fig. 12 as $\sqrt{d_n}~\exp (-n/{\xi }_b)$.

Figure 14

(a) Checkerboard decomposition in the Kitaev code. Spin-1/2 are represented by (red) dots at the vertices of the square lattice. $X$ and $Z$ operators act on the four spins of each type of (shaded and nonshaded) plaquettes. (b) PEPS representation of the Kitaev code (see text).