Resonating valence bond states in the PEPS formalism

We study resonating valence bond (RVB) states in the projected entangled pair states (PEPS) formalism. Based on symmetries in the PEPS description, we establish relations between the toric code state, the orthogonal dimer state, and the $\mathrm{SU}\left(2\right)$ singlet RVB state on the kagome lattice: We prove the equivalence of toric code and dimer state, and devise an interpolation between the dimer state and the RVB state. This interpolation corresponds to a continuous path in Hamiltonian space, proving that the RVB state is the fourfold degenerate ground state of a local Hamiltonian on the (finite) kagome lattice. We investigate this interpolation using numerical PEPS methods, studying the decay of correlation functions, the change of overlap, and the entanglement spectrum, none of which exhibits signs of a phase transition.

Figure 1

(a) Oriented kagome lattice. Spins are associated with vertices. (b) Dimer covering of the kagome lattice. The lattice is completely covered with dimers (marked blue), i.e., disjoint pairs of adjacent vertices.

Figure 2

PEPS construction of the RVB state: Place $|\epsilon \rangle$ states (green), Eq. (1), and then apply the map $\mathcal{P}$ (red), Eq. (2), as indicated. To obtain the dimer state, replace $\mathcal{P}$ with ${\mathcal{P}}_{\perp }$, Eq. (3).

Figure 3

Construction of the toric code as a tensor network. (a) The toric code is defined on a square lattice with two types of plaquettes. By taking $2\times 2$ blocks over $A$-type plaquettes, a particularly convenient tensor network description can be found. (b) Tensor for the toric code; depicted are the nonzero entries. The four auxiliary indices are in the $X$ basis, while the physical sites are in the $Z$ basis.

Figure 4

Tensor network notation for the RVB state. (a) Tensors are denoted as boxes with one leg per index. Connecting legs correspond to summing over that index; this panel, e.g., shows ${\sum }_{\beta }{A}_{\alpha \beta }^i{B}_{\beta \gamma }^j$. (b) Tensor network description of the RVB state; the triangular link is shorthand for the tensor $E$ implicitly defined in Eq. (8); see panel (c). Depending on whether we choose $P$ as in Eq. (9) or replace it with $P_{\perp }$, Eq. (10), we obtain a tensor network for either the RVB or the dimer state; in the latter case, the physical indices $i_k$ form double indices $(i_k,j_k)$.

Figure 5

${\mathbb{Z}}_2$-invariant tensor network. (a) A tensor is called ${\mathbb{Z}}_2$ invariant if it is invariant under some $\overline{Z}$ applied to all virtual indices, where $\{1,\overline{Z}\}$ forms a representation of ${\mathbb{Z}}_2$. (b) ${\mathbb{Z}}_2$ invariance is stable under concatenation of tensors.

Figure 6

Transformation of the dimer state into the toric code. (a) $P_{\perp }$ can be rewritten as a tensor network with bond dimension 2. (b) Network obtained by replacing $P_{\perp }$ in Fig. 4b with the tensor network of panel (a). The tensor network can be grouped into triangular tensors which contain only $A$ or $B$ tensors, and which are connected by bonds of dimension 2.

Figure 7

Nonzero orthogonal virtual states and the corresponding orthogonal physical states for the dimer state tensors (the red lines denotes singlets). The $A$ ($B$) type tensors are supported on the odd (even) parity subspace.

Figure 8

Definition of the tensor $P_{✡}$. (a) $P_{✡}$ is obtained by blocking 12 tensors $P$ with 6 triangular “bond tensors” $E$; it maps 6 virtual qutrits to 12 physical qubits. (b) Effective depiction of the tensor $P_{✡}$.

Figure 9

Complete covering of the kagome lattice with stars, Fig. 8. The stars tensor $P_{✡}$ (or projectors ${\mathcal{P}}_{✡}$), together with triangular tensors $E$ (or $|\varepsilon \rangle$ bonds), describe the RVB state (or its orthogonal version, if we choose $P_{\perp ,✡}$ instead). By acting with ${\mathcal{O}}_{✡}$ on each of the stars, we can reversibly convert between the RVB and the orthogonal RVB.

Figure 10

(a) Lattice used for the numerical calculations. The red dots denote $P\left(\theta \right)$ tensors. (b) By blocking three spins, we transform this to a square lattice. (c) Resulting square lattice. For OBC, we set the bonds at the boundary to $|2\rangle$ as shown; for CBC, we consider a vertical cylinder where left and right indices are connected. (d) Tensor representation of the scalar product $\langle \Psi \left(\theta \right)|\Psi \left({\theta }^{\prime }\right)\rangle$.

Figure 11

Two-point correlations as a function of the distance for different values of $\theta$. The plot shows data for an infinite system where in the PEPS contraction (Ref. 14) the bond dimension has been truncated at $D=40$ and $D=60$, respectively.

Figure 12

Gap of the transfer operator as a function of $\theta$ along the RVB-dimer interpolation for $N_h=4,\cdots ,10$ and PBC. As the transfer operator is parity preserving (changing) for $N_h$ even (odd), it has two (almost) degenerate largest eigenvalues corresponding to different parity sectors (for $N_h$ even, there is an exponentially small splitting in $N_h$); the plot shows the ratio of the third eigenvalue and the average of the two maximal eigenvalues. Note that this quantity upper bounds the decay of any correlation function. The data for $N_h=\infty$ has been obtained by fitting the data for $N_h=6,\cdots ,10$ with $a+b\exp (-c{N}_h)$. The inset shows the fit for $\theta =0,0.1,0.2,0.3,0.5$ (from bottom to top, the curves have been shifted vertically in the plot); note that the two rightmost data points have not been used for the fit.

Figure 13

The overlap change ${\widehat{F}}_{N_h}$, Eq. (14), as a function of $\theta$ for both open boundary conditions ($N_h=4,6,8$) and periodic boundary conditions ($N_h=4,6,8,10$). The solid black and red lines give the extrapolated data for $N_h=\infty$ for open and periodic boundaries. The inset shows a zoom of the $N_h=\infty$ data for $\theta ⩽0.18$, where the black (red) crosses correspond to open (periodic) boundaries. The details of the scaling analysis are given in Fig. 14.

Figure 14

Finite size scaling analysis for ${\widehat{F}}_{N_h}$. The plot shows OBC and PBC data for ${\widehat{F}}_{N_h}$. Due to the absence of boundary effects, the PBC data converge very quickly to the $N_h=\infty$ value once $N_h$ is beyond the correlation length. We have fitted the PBC data for $N_h=6,8,10$ with $a+b\exp (-c{N}_h)$. The OBC data, on the other hand, have a $1/N_h$ leading term from the boundary; we have fitted the data for $N_h=4,6,8$ with $a+b/N_h$. Note that the correction for the data points for OBC would most likely be concave, suggesting that the true $N_h=\infty$ value is closer to the PBC data.

Figure 15

Entanglement spectrum of an infinite (vertical) cylinder for decreasing $\theta$, and perimeter $N_h=8$. The spectrum is shown as a function of the momentum $K$ along the one-dimensional (horizontal) edge. The eigenstates are also labeled according to their spin quantum numbers ($\frac{1}{2}\oplus 0$ representation) according to the symbols in the legend. Panel (a) shows the integer spin sector of the spectrum, panel (b) the half-integer sector; the two sectors are not coupled by the entanglement Hamiltonian. Note that for both sectors, the same energy scale has been chosen (i.e., trace-normalizing the boundary states in both sectors), corresponding to a local entanglement Hamiltonian (Ref. 24).

Figure 16

Gap between the lowest $S=1$ and $S=0$ state (left) and $S=\frac{1}{2}$ and $S=0$ state (right) of the entanglement Hamiltonian for the dimer-RVB interpolation for $N_h=6,8$.

Figure 17

PEPS representation for the RVB. Place maximally entangled “bonds” $|01\rangle -|10\rangle +|22\rangle$ along the edges of the lattice (blue), and subsequently apply the linear map ("PEPS projector") P4 (red circle), Eq. (A2), which maps the four qutrits at each vertex (encircled) to a spin-$\frac{1}{2}$ degree of freedom. Alternatively, we can obtain the (orthogonal) dimer state by replacing ${\mathcal{P}}_4$ with ${\mathcal{P}}_{4,\perp }$, Eq. (A3): ${\mathcal{P}}_{4,\perp }$ ensures that only one qutrit per vertex holds a singlet, but keeps the full Hilbert space, thus ensuring local orthogonality of different dimer configurations.

Figure 18

Simplified PEPS construction for the RVB and dimer state. (a) Start from the representation Fig. 17 and rewrite ${\mathcal{P}}_4$ in two layers, ${\mathcal{P}}_4=\mathcal{P}(\mathcal{Q}\otimes \mathcal{Q})$, Eq. (A4). The $\mathcal{Q}$'s act as indicated by the green circles, and make sure at most one singlet degree of freedom is kept; they thus return one qutrit each. ${\mathcal{P}}_2$ makes sure exactly one singlet is kept, which can come from either of the two $\mathcal{Q}$'s. (b) The simplified PEPS construction: Applying three $\mathcal{Q}$'s to three bonds around a loop results in a state $|\epsilon \rangle$ which holds at most one singlet, Eq. (A5), and which is placed across every triangle. Subsequently, $\mathcal{P}$ is applied to pick one of the singlets at each vertex.

Figure 19

(a) Tensor network and Hamiltonian for the toric code on the kagome lattice. Two types of terms ensure the parity constraints over the triangular and hexagonal plaquettes, while the vertex term enforces an equal weight superposition of all even parity configurations. Note that the $-Z^{\otimes 3}$ acting on the triangles can be understood as enforcing each individual tensor to be in the correct subspace. (b) Hamiltonian terms obtained by transforming each tensor unitarily into a projector onto the symmetric subspace. $Z^{\otimes 3}$ still enforces that the tensor is in the correct subspace; further, there is a $Z^{\otimes 2}$ at each vertex enforcing equality of the adjacent qubits—translated to the dimer state, this will enforce that there is exactly one singlet adjacent to each vertex. Finally, the $X^{\otimes 12}$ term around the hexagon makes sure that all such configurations appear with equal weight—it will make the dimer configurations resonate in the orthogonal RVB state. (Note that this model can also be understood as a toric code model on a “decorated” hexagonal lattice with two sites per edge and a vertex between, and $X$ stabilizers around plaquettes and $Z$ stabilizers across vertices.)

Figure 20

Resonance moves in the dimer state. (a) Resonance move implemented by a single $\Xi$, Eq. (C1): $\Xi$ acts on the lower two qutrits only and flips the two configurations singlet/no singlet, and the two configurations singlet left/singlet right. (b) Applying ${\Xi }^{\otimes 6}$ around the hexagon gives a resonance between pairs of dimer configurations around the hexagon (marked red and green, respectively). Note that any such pair of dimer configuration can be described by a closed loop of even length on the star; cf. Ref. 8.

Figure 21

Graphical proof that ${\mathbb{Z}}_2$ injectivity is stable as growing a region. We illustrate it in the case of growing from one to two partially overlapping stars. Our aim is to prove the existence of a left inverse on the invariant subspace on the two stars, using the existence of the left inverse on a single star. Since we are considering the tensor network representation of the RVB state made out of tensors $P$—with physical index—and triangular links $E$ without physical index, given a region, all one needs is to clarify whether the outgoing auxiliary indices come from a $P$ tensor or from an $E$ tensor. In our case, we start with two overlapping stars with outgoing indices of type $P$ and divide them into three regions, the middle one (bow tie) with all outgoing indices of type $P$ and the rest. By gathering indices together, we have panel (a), where the circle represents the bow tie and each square one of the other two regions. The crucial property is that the circle together with any of the two squares is ${\mathbb{Z}}_2$ injective. We proceed as follows. We apply the inverse to the circle and left square, getting panel (b), where $\overline{Z}$ now means $\otimes \overline{Z}$. The next step, illustrated in (c), is to build a new circle tensor attached to the right square tensor, and passing the ${\overline{Z}}^r$ through using the symmetry of the circle tensor. Now we invert the remaining two tensors, getting (d). By tracing the middle (extra) line we obtain ${\delta }_{rs}$ and therefore finish the argument since we have constructed the desired inverse.

Figure 22

Graphical proof of the intersection property for the case $h_1+h_2$ where $h_1$ acts on the 3 stars colored in blue-red in (a) and $h_2$ acts on those colored in red-green. It is easy to see that by separating the common part (2 red stars) and the rest, with the convention that the common part has all outgoing indices of type $P$, $\ker h_1\otimes {\mathcal{H}}_{\mathrm{green}}$ is given by panel (b), where $N$ is an arbitrary tensor, whereas ${\mathcal{H}}_{\mathrm{blue}}\otimes \ker h_2$ is given by panel (c), where $M$ is arbitrary. Here ${\mathcal{H}}_{\mathrm{green}}$ and ${\mathcal{H}}_{\mathrm{blue}}$ represent the Hilbert spaces associated with the physical indices in the green and blue regions, respectively. As before, we have gathered together all virtual (as well as the physical) indices within each type of tensor. Given a state $|\psi \rangle$ in the intersection of both kernels, there exist $N$ and $M$ such that $|\psi \rangle$ can be expressed both in the form (b) and (c). In the equality (b)=(c) we invert the circle tensor, obtaining the equality given in panel (d). After that, we apply an auxiliary extra circle tensor, then the inverse of the red-green tensors (3 stars), which is also ${\mathbb{Z}}_2$ injective by the previous section, and then trace the extra virtual system. Since one can assume without loss of generality that both $M$ and $N$ have the symmetry $\otimes \overline{Z}$, we can also eliminate the sums. After all that we arrive at panel (e) for some tensor $L$. Substituting (e) in (c) gives the desired form for $|\psi \rangle$ and hence the result.

Figure 23

The kagome lattice with periodic boundary conditions can be decomposed as in (b), where all outgoing indices on the black regions are of type $P$, whereas all outgoing indices of the blue regions are of type $E$. All tensors in (b) have an additional physical index “going out of the paper.” A possible way to obtain such a decomposition is illustrated in (a). The black regions are ${\mathbb{Z}}_2$ injective; the same happens if one takes two black regions with the intermediate red or green one or if one takes the four black regions with the intermediate red, green, and blue ones. The blue region is the one corresponding to the cross points. Note that the black and the blue only interact through a red or green region, exactly as required in (b).

Figure 24

Any state which is in the ground state of the parent Hamiltonian when we close the boundaries in a periodic way [where we close the boundaries as indicated in Fig. 23a] can be represented graphically in any of the forms given in the figure. As in Fig. 23, each of the tensors, including the “boundary conditions,” have a physical index going “out of the paper.”

Figure 25

The ground space of the Hamiltonian with periodic boundary conditions has dimension four and is spanned by the states $|{\psi }_{r,s}\rangle$ in the figure. As above, $\overline{Z}$ means $\otimes \overline{Z}$.

Figure 26

Intermediate step in the proof that the ground space of the Hamiltonian is spanned by the vectors given in Fig. 25 above.

Figure 27

The figure illustrates the first step in the reduction to show that the parent Hamiltonian on two overlapping stars suffices to give the RVB state as a unique ground state—up to the string operators given in Fig. 25 which induce a 4-fold degeneracy. The left (right) upper part of (b) represents the kernel of the Hamiltonian grown from 2-star regions which acts on the stars with a dot (cross). The circles represent the stars with the same color in (a), with all outgoing indices of type $P$. The squares represent the rest of (a), having outgoing indices both of type $P$ (those connecting with the boundary) and type $E$ (those connecting with the circles). As in the previous figures, all bonds connecting the same objects have been gathered together in the same bond. Also, all tensors have an additional physical index which “goes out of the paper.” The boundary conditions in the upper part of (b) have also a physical index corresponding to the part of (a) with the same color. The intersection of these two kernels, which is the kernel of $H_6={\sum }_{x,\circ }{H}_2$, is the sum of the two subspaces depicted in the lower part of (b). Needless to say that the boundaries of the lower part of (b) do not have physical index.

Figure 28

The figure illustrates the second step in the reduction to a 2-star Hamiltonian. (a) represents the space $\mathcal{S}$, coming from the lower-right part of Fig. 27b, each state of which we want to penalize by the 2-star term marked by squares in Fig. 27a. All outgoing indices are of type $P$. (b) illustrates the idea of “completing the path”: for each set of free sites—marked by a circle—there exist 2 disjoint dimer coverings (in green and violet, respectively) compatible with the choice of free sites, which in addition form a Hamiltonian cycle in the remaining graph.