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 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 four-fold 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.
Resonating valence bond (RVB) states have been introduced by Anderson anderson:rvb () as a wavefunction in the context of high-temperature superconductivity and have since then received significant attention. RVB wavefunctions are obtained as the superposition of all nearest-neighbor (or otherwise constrained) singlet coverings on a given lattice, and they have been studied as an ansatz capturing the behavior of frustrated quantum spin systems, in particular Heisenberg antiferromagnets on frustrated lattices. misguich:AFM-review () They are believed to describe so-called topological spin liquids, i.e., exotic phases with degenerate ground states which however do not break rotational or translational symmetry, despite the presence of strong antiferromagnetic interactions. Particular interest, both theoretically and experimentally, has been devoted to frustrated magnets on the kagome lattice, as those systems are realized in actual materials and form candidates for the first experimental observation of a topological spin liquid. mendels:kagome (); meng:spinliquid-2d-diracfermions (); mezzacapo:honeycomb-j1j2 (); yan:heisenberg-kagome (); depenbrock:kagome-heisenberg-dmrg ()
Unfortunately, models with potential RVB ground states are typically difficult to study. One the one hand, this is due to presence of frustration. Yet, the bigger issue seems to be that different singlet configurations are not orthogonal, which up to now has e.g. hindered a full understanding of how RVB states appear as ground states of local Hamiltonians. seidel:kagome () In order to better understand the RVB phase, simplified so-called dimer models have been introduced, rokhsar:dimer-models () where the system is re-defined on the Hilbert space spanned by all dimer coverings of the lattice, making different dimer configurations orthogonal by definition. Dimer models have subsequently been studied for various lattices, and it has in particular been found that for certain geometries, such as for the triangular moessner:dimer-triangular () or the kagome lattice, misguich:dimer-kagome () such states appear as ground states of local Hamiltonians with topological ground state degeneracy (four-fold on the torus), forming a topological spin liquid. Unfortunately, these findings cannot be mapped back to the true RVB state due to the different underlying Hilbert spaces which are not related by a simple mapping, and thus, the way in which the corresponding RVB states can describe ground states of local Hamiltonians is only partially understood (although very remarkable progress in this direction has been made during the last years cano:rvb-hamiltonians (); seidel:kagome ()).
More recently, Projected Entangled Pair States (PEPS) have been introduced as a tool to study quantum many-body wavefunctions. verstraete:mbc-peps (); verstraete:2D-dmrg () PEPS give a description of quantum many-body states based on their entanglement structure in terms of local tensors, and form a framework which allows to both analytically understand these wavefuctions, and to study their properties numerically. In particular, there is a clear way to understand how PEPS arise as ground states of local Hamiltonians, especially for systems with unique ground states perez-garcia:parent-ham-2d () and with topological order, schuch:peps-sym () based on the symmetry properties of the underlying tensor representation. Numerically, they form a tool for computing quantities such as correlation functions or overlaps efficiently with very high accuracy, using transfer operators of matrix product form. verstraete:2D-dmrg ()
In this paper, we perform a systematic study of RVB and dimer states in the PEPS formalism, focusing on the kagome lattice. We introduce closely related PEPS representations for the RVB wavefunction verstraete:comp-power-of-peps () and a version of the dimer state in which different dimer configurations are locally orthogonal. These representations allow us to derive a reversible local mapping between the RVB and the dimer state, as well as to prove local unitary equivalence of the dimer state and Kitaev’s toric code.kitaev:toriccode () This yields a Hamiltonian for our dimer model, and subsequently a Hamiltonian for the RVB state: This is, we can for the first time prove that the RVB state on the kagome lattice is the ground state of a local Hamiltonian with topological ground state degeneracy (four-fold on the torus) for any finite lattice. The PEPS formalism further allows us to construct an interpolation between RVB and dimer state where the Hamiltonian changes smoothly along the path. Using techniques for numerical PEPS calculations, verstraete:2D-dmrg () we try to assess whether the RVB state is in the same phase as the dimer state (which is equivalent to the toric code and thus a topological spin liquid) by looking for signatures of a phase transition along the dimer–RVB interpolation. We have considered the behavior of correlation functions, the rate at which the ground state changes in terms of the wave function overlap,zanardi:overlap-criterion () and the entanglement spectrum of the system, and have found that all of these quantities behave smoothly and show no sign of a phase transition.
Let us describe the organization of the paper. For the sake of conciseness, alternative definitions, proofs, etc. have been moved to appendices. In Sec. II, we introduce the RVB and dimer states and their PEPS representations, as well as the formulation using tensor networks. Alternative PEPS representations are discussed in Appendix A. In Sec. III, we discuss the relations between the toric code, the dimer state, and the RVB state, using the symmetry of the underlying tensors. Based on these findings, we show how to construct a smooth interpolation between the dimer state and the RVB state in Sec. IV, and subsequently study its properties numerically. In Sec. V, we use the relation between the toric code, the dimer state, and the RVB state to construct parent Hamiltonians for the dimer state and the RVB, as well as a smooth path of Hamiltonians for the interpolation between them. Appendix C provides a simpler parent Hamiltonian for the dimer state, based on a more direct mapping to the toric code, and Appendix D shows how to directly derive a (much more compact) parent Hamiltonian for the RVB state by generalizing the techniques developed in Ref. schuch:peps-sym, for PEPS with symmetries to the symmetries found in the RVB.
ii.1 The RVB and orthogonal RVB state
We start by introducing dimer states and resonating valence bond (RVB) states. We focus on the kagome lattice, Fig. 1a, both for its relevance and for clarity of the presentation; but our techniques generalize to other lattices (see conclusions). A dimer is a pair of vertices connected by an edge. A dimer covering is a complete covering of the lattice with dimers, Fig. 1b. We can associate orthogonal quantum states with each dimer covering . Then, the dimer state is given by the equal weight superposition , where the sum runs over all dimer coverings . Usually, it is favorable to ensure orthogonality of different states and locally; we will introduce such a version of the dimer state which is particularly suited for our purposes in Sec. II.3.
Let us now turn towards the resonating valence bond (RVB) state. We first associate to each vertex of the lattice a spin- particle, or qubit, with basis states and . Then, for each dimer covering we define a state which is a tensor product of singlets (we omit normalization throughout) between the pairs of spins in each dimer in the covering, where the singlets are oriented according to the arrows in Fig. 1a. The resonating valence bond (RVB) state is then defined as the equal weight superposition over all dimer coverings.
ii.2 PEPS representation of the RVB state
We will now give a description of the RVB state in terms of Projected Entangled Pair States (PEPS).verstraete:mbc-peps () PEPS are states which can be described by first placing “virtual” entangled states between the sites of the system, and subsequently applying linear maps at each site to obtain the physical system. A PEPS representation of RVB states has first been given in Ref. verstraete:comp-power-of-peps, ; a detailed discussion how it is related to our description can be found in Appendix A.
To obtain a PEPS description of the RVB state, we first place -qutrit states
inside each triangle of the kagome lattice, as depicted in Fig. 2. Here, is the completely antisymmetric tensor with , and , , and are oriented clockwise (i.e., consistent with the arrows in Fig. 1a). Second, we apply the map
at each vertex, which maps the two qutrits from the adjacent states to one qubit. It is straightforward to check that this construction exactly gives the resonating valence bond state defined above (see Appendix A for a detailed discussion).
ii.3 PEPS representation of dimer state
In a similar way as for the RVB state, we can also obtain a PEPS representation of the dimer state. To this end, we enlarge our local Hilbert space and replace the map in Fig. 2 by the map
It is straightforward to check that the resulting state is an equal weight superposition of all dimer coverings, and that different dimer configurations are orthogonal, since the position of the dimers can be unambiguously inferred from the location of the ’s. Note that while we write the physical space as a subspace of a –dimensional space, in fact only a four-dimensional subspace is used. A more detailed discussion of the construction can again be found in Appendix A.
As one might expect, it is possible to obtain the RVB state from the dimer state by coherently discarding the information about where the singlet is located and only keeping the singlet subspace . This is most easily seen by noting that
and thus, applying to each site of the dimer state yields the RVB state.
Note that our representation of the dimer state differs from the formulations typically used in the literature in two points: Firstly, while dimer models have been usually studied on the space spanned by valid dimer configurations, our formulation allows to locally check whether a dimer is present on a given link. This allows us to enforce the restriction to the space of valid dimer configurations using a local Hamiltonian, and ultimately enables us to map our dimer model to the toric code by local unitaries. Secondly, unlike mappings where the presence or absence of a dimer is represented by a spin, nayak:dimer-models () our mapping uses actual singlets built of two entangled particles. This allows us to locally interpolate between the dimer and the RVB model, and gives rise to a gapped entanglement spectrum, which is in contrast to the gapless entanglement spectra which had been observed previously for topologically ordered systems.
ii.4 Gauge transformation
The PEPS representation of the RVB and dimer state are built using unnormalized singlets, as the weight of and in (1) is the same. Since the number of singlets in all dimer configurations is the same, this is not an issue; however, in some situations (cf. Sec. III.2) it will be more convenient to normalize the singlets, i.e., replace (1) by
Note that , where . Since on the other hand, (and thus also for , as well as the interpolation defined in Sec. IV), we find that these two PEPS do not only represent the same total state, but are in fact related by a local gauge transformation, allowing to directly relate all properties of their PEPS description.
ii.5 The toric code
where the sums run over all and type plaquettes and , respectively, and the tensor products of Paulis act on the qubits adjacent to each plaquette.
The toric code state can be written as a PEPS verstraete:comp-power-of-peps () with bond dimension . A particularly useful representation is obtained by blocking the qubits in blocks around type plaquettes, as indicated in Fig. 3a and letting the PEPS projector be schuch:peps-sym ()
for each of these blocks. Here, the sums are modulo , and the ordering of the indices is illustrated in Fig. 3b; the virtual qubits are expressed in the eigenbasis , .
ii.6 Formulation using tensor networks
PEPS can also be expressed in terms of tensor networks; while both descriptions are mathematically equivalent we will utilize either of them when more convenient.
A tensor network state is a state
where can be expressed efficiently by a tensor network, i.e., as the sum over of product of tensors with indices and , where each of the tensors only has a few indices. Tensor networks are often expressed graphically, with tensors denoted by boxes with legs corresponding to the indices, where connecting legs corresponds to contracting over the corresponding index, cf. Fig. 4a.
It is straightforward to see that the simplified PEPS representation of the RVB state can be expressed using two types tensors, and , defined such that
to obtain a PEPS representation of the dimer state.
In this section, we will show how one can transform reversibly between the toric code, the dimer state, and the RVB state. To this end, we will start by recalling the concept of -injectivity of PEPS schuch:peps-sym () and subsequently apply it it to prove equivalence of the toric code and the dimer state, and to devise a mapping which transforms between the dimer and the RVB state.
We start by introducing –injectivity, which has been introduced (as –injectivity, with any finite group) in Ref. schuch:peps-sym, as a tool to characterize PEPS with topological properties. Let be a unitary such that forms a representation of , i.e., . A –invariant tensor network consists of tensors which are invariant under the symmetry action on all virtual indices simulateously, as depicted in Fig. 5a (we restrict to equal representations on all indices). This property is stable under blocking, i.e., two (or more) –invariant tensors together are still –invariant, see Fig. 5b. A –invariant tensor is called –injective if the corresponding PEPS projector is injective on the invariant subspace (the subspace belonging to the trivial representation), i.e., if it has a left inverse such that is the projection onto the invariant subspace. Differently speaking, this means that there exists a linear map which we can apply to the physical system to obtain direct access to the virtual system, up to a projection onto the symmetric subspace . –injectivity is stable under concatenation: When composing two –injective tensors, we obtain another –injective tensor. schuch:peps-sym ()
The key point to note is that any two –injective tensor networks with identical geometry and symmetry representation, but different tensors, are locally equivalent: There exists a map acting on the physical system of individual tensors which transforms the networks into another. This follows since both PEPS projectors, and , have left inverses which yield the projector onto the symmetric subspace, , and thus, , and vice versa.
An important special case are –isometric tensors where is a partial isometry, : Any two –isometric tensors and can be transformed into each other by a unitary acting on the physical system. Again, –isometry is a property which is stable under concatenation. schuch:peps-sym ()
iii.2 Toric code and dimer state
In the following, we will show that the Toric Code state and the orthogonal dimer state are both built of –isometric tensors, which implies that they can be transformed into each other by local unitaries. (This relation has been observed previously by mapping the presence or absence of dimers to a spin degree of freedom.nayak:dimer-models ()) In this section, we will consider the representation , with .
For the toric code tensor Eq. (6), it is straightforward to check that it is invariant under , and acts isometrically on the invariant subspace, i.e., is –isometric. In order to see the same for the dimer state, we first rewrite , Eq. (10), as
where , and . Using this decomposition, we replace each tensor in Fig. 4b with a pair of tensors and connected by a two-dimensional bond, cf. Fig. 6a, oriented as in Fig. 6b. This orientation allows to block the tensors into triangles, labelled and , containing only and type tensors, cf. Fig. 6b; the triangles are connected by bonds of dimension two.
The bonds are associated to vertices; a bond state () indicates that the dimer lies inside the () triangle (similar to the “arrow representation” of Ref. elser:rvb-arrow-representation, ). As each triangle contains either no or one dimer, the () tensors are odd (even) under symmetry, and moreover isometric on the (anti)symmetric subspace if we use the gauge of Eq. (4); the possible mappings from bond to physical configurations are given in Fig. 7.
We now group one and one tensor to obtain square tensors, which are antisymmetric and isometric; grouping two of these square tensors finally yields a –isometric tensor with symmetry representations and , respectively. Blocking two tensors of the toric code also yields a –isometric tensor with the very same symmetry, and thus, the dimer state and the toric code can be converted into one another by local unitaries.
iii.3 RVB and dimer state
Let us now show that we can reversibly (though not unitarily) transform the dimer state to the RVB state. In principle, the forward direction is obvious, as ; however, this transformation cannot be inverted. In order to construct an invertible transformation, we will show that both states can be understood as being constructed from –injective tensors connected by bonds: Following the reasoning of Sec. III.1, we can reversibly transform between the two –injective tensors and thus between the RVB state and the dimer state.
For this section, we will use the PEPS representation Fig. 4 of both the RVB and the dimer state. The representation of the symmetry is given by
It is straightforward to see that both , , and the bond tensor are anti-symmetric under the action of the symmetry on all virtual indices. Let us now form a new tensor by blocking all the tensors covering one star, cf. Fig. 8. We call the new tensors and , respectively (with corresponding PEPS projector and ). Clearly, we can tile the lattice with tensors connected by bonds, see Fig. 9. Due to stability of –invariance, one finds that and are –invariant. On the other hand, it can be checked analytically using a computer algebra system that is indeed invertible on the invariant subspace of , i.e., is –injective. This immediately implies –injectivity of (as ).
We can now use the argument of Sec. III.1 to construct an invertible mapping: –injectivity of implies the existence of such that is the projector onto the –invariant subspace on the virtual qutrits along the boundary of the star. If we now define , it follows that , i.e., it (locally) converts the RVB into the dimer state. [Note that is only well-defined on the range of , ; we choose to vanish on its orthogonal complement.] Analogously, we can define such that [again such that it vanishes on ], and it is easily verified that and , i.e., is the pseudoinverse of .
Applying the maps or its inverse to all stars in the tiling Fig. 9, we can now reversibly convert between the dimer and the RVB state, and .
iv.1 Interpolation between dimer model and RVB
Let us now show how to continuously deform the dimer state into the RVB state; as we will show in Sec. V.3, this interpolation does correspond to a continuous change of the associated local parent Hamiltonian. To this end, we define a family of PEPS projectors
with a two-qubit physical space. Clearly, , and , both up to local isometries. By placing in the tensor network Fig. 2, we therefore obtain a smooth interpolation between the RVB and the dimer state. Clearly, we can accordingly define and subsequently , which is a continuous invertible map from to . Note that while it might seem that the interpolation (12) results in a discontinuity as , the discussion about the equivalence of RVB and dimer state in the preceding section shows that this discontinuity disappears when considering whole stars.
This interpolation can be understood as a way to make different dimer configurations more and more orthogonal, analogous to what is achieved by decorating the lattice in Ref. raman:rvb-decorated-lattice, : Overlapping two different dimer configurations gives rise to a loop pattern, and the absolute value of their overlap is , with the number of loops (including length- ones) and their total length (which equals the number of lattice sites). Here, for the RVB and for the dimer state, and Eq. (12) corresponds to an interpolation .
iv.2 Numerical results
We have numerically studied the interpolation (12) between the dimer state (which is a topological spin liquid) and the RVB state to determine whether they are in the same phase. We use the tensor network representation of Fig. 4, where the interpolating path is characterized by tensors corresponding to of Eq. (12). For the numerical calculations, it is particularly convenient to block three spins, i.e., tensors, together with two tensors, as indicated by the dashed squares in Fig. 10a; this allows us to rewrite the system on a square lattice with three spins per site (Fig. 10b). We perform simulations both for open boundary conditions and for cylindrical boundary conditions. For open boundary conditions (OBC), we set all open indices in the tensor network to (i.e., there are no outgoing singlets at the edges), as indicated in Fig. 10c. The case of cylindrical boundary conditions is obtained by putting the system on a vertical cylinder, i.e., connecting the left and right indices, while the upper and lower outgoing indices are still set to . We will denote by and the number of sites in each row and column of the square lattice, respectively. We will generally consider the case where for finite . In this case, cylindrical boundary conditions give the same results as periodic boundary conditions as long as topological degeneracies are appropriately taken into account, and we will therefore refer to them as periodic boundary conditions (PBC) in the following.
iv.2.1 Correlation functions and gap of the transfer operator
Firstly, we have studied the decay of correlation functions along the dimer–RVB interpolation. Correlation functions can be computed for lattices of arbitrary size using standard methods for the contraction of PEPS.verstraete:2D-dmrg () We have computed various correlation functions, all of which show an exponential decay with a slope which changes smoothly with , and do not exhibit any sign of a phase transition; as an example, Fig. 11 shows the connected dimer-dimer correlation function between two operators , where each dimer is supported on one square tensor of Fig. 10b and the two squares are in the same row, as a function of the distance between the squares, for different values of , evaluated on an infinite system. (This data has been obtained by row-wise contraction using infinite Matrix Product States and iTEBD, and then evaluating the correlation function using the fixed point tensor.)
The decay of any correlation function is bounded by the gap of the transfer operator, i.e., one row in Fig. 10d. We have used exact diagonalization to determine the gap of the transfer operator for PBC. The result for , as well as the extrapolated data for , is shown in Fig. 12. Again, the gap of the transfer operator stays finite and converges rapidly for growing . Note that the fact that the ground state is a PEPS throughout the interpolation (and thus satisfies an area law) does not preclude critical behavior. verstraete:comp-power-of-peps (); barthel:peps-mera ()
In order to detect phase transitions which are not reflected in correlation functions, we have also studied the rate at which the state changes as we change , as quantified by the overlap (or fidelity). zanardi:overlap-criterion () Concretely, let be the state interpolating between and , as defined in the preceding section. We define the change of overlap
and are interested in the behavior of the limit
which describes how quickly the state changes in the thermodynamic limit.
We have computed this quantity in two different ways. In the first approach, we use that for sufficiently small we can exactly compute for arbitrary values of and by exact row-wise contraction of the tensor network Fig. 10d, and use this to extrapolate the limit
to arbitrary accuracy for ; this data is then used in a finite size scaling analysis to infer . We apply this approach to the OBC case. The second approach makes use of the fact that can be expressed as an average over correlation functions, as we show in Appendix B. Again, this average over correlation functions can be computed exactly for any value of , and (to reach , we use that total bond dimension of the transfer operator of the tensor, and thus the overall bond dimension in the tensor network in Fig. 10d, can be reduced from to ), which again allows us to determine for , and subsequently apply a finite size scaling analysis. We apply this approach to the PBC case.
The resulting data is shown in Fig. 13, together with extrapolated data for from the finite size scaling. (See Fig. 14 for a discussion of the finite size scaling analysis.) The extrapolated curves for open and periodic boundary conditions agree very well. The extrapolated value for seems to become essentially constant for , which might suggest a non-analytic behavior of ; however, zooming into this region (inset of Fig. 13) shows clearly that decreases again for , showing no evidence for the presence of a phase transition. (See Fig. 14 for a discussion of the difference observed between the extrapolated OBC and PBC data.)
iv.2.3 Entanglement spectrum
Finally, we have also studied the behavior of the entanglement spectrum along the interpolation for an infinite cylinder, using the techniques described in Refs. cirac:peps-boundaries, ; poilblanc:rvb-boundaries, (in particular, using exact contraction). In Fig. 15 we give the entanglement spectra for for the integer and half-integer spin sector (the boundary Hamiltonian does not couple the two sectors). While at first, it seems that the lowest lying and states cross with the ground state at some small , a scaling analysis (Fig. 16) suggests that the gap of the entanglement Hamiltonian is more likely to vanish at ; also, note that there is no prior reason to assume that a vanishing gap in the entanglement spectrum implies any critical behavior in the actual system, as the boundary state is a thermal state of the entanglement Hamiltonian and therefore will have finite correlations even for a critical entanglement Hamiltonian.cirac:peps-boundaries ()
Let us also point out an interesting aspect of the entanglement spectrum: While the system under consideration is topological (rigorously provable in an environment of the dimer point bravyi:tqo-long ()), the entanglement Hamiltonian has a unique ground state, i.e., it is gapped, which shows that the connection between topologically ordered systems and gapless entanglement spectra might be less clear than generally believed. Note that the reason that the entanglement spectrum is gapped (unlike for the toric code, the RK model, misguich:dimer-kagome () or the mapping of Ref. nayak:dimer-models, ) is due to the fact that our representation of the dimer state involves singlets (i.e., an entangled state of two spins) rather than single spins indicating the presence or absence of a dimer, and that we split these singlets when computing the entanglement spectrum, just as one would do for the RVB state itself.
v.1 Hamiltonian for the dimer state
As has been proven in Ref. schuch:peps-sym, , every –injective PEPS on a square lattice appears as the ground state of a local Hamiltonian, acting on at most elementary cells, which has a fixed degeneracy determined by the symmetry group. In particular, this result can be directly applied to the dimer state to infer that it is the ground state of a local Hamiltonian with four-fold degeneracy. Alternatively, we can obtain a Hamiltonian for the dimer state by using its local isometric equivalence to the toric code: The same transformation will also transform the toric code Hamiltonian into a local Hamiltonian for the dimer state. Note, however, that we need to add extra terms to the dimer Hamiltonian which ensure that the state is constrained to its local support, as the isometry is only defined as a map between the local support of the toric code and of the dimer state, respectively.
In Appendix C, we explicitly derive a Hamiltonian for the dimer state using the duality with the toric code Hamiltonian. It has three types of terms (all of them projectors), acting on vertices, triangles, and hexagons, respectively:
acts on individual vertices. It ensures that there is exactly one singlet per vertex.
acts inside the triangles (as used in the ––blocking). It makes sure that each triangle holds zero or one singlet.
acts on the six vertices adjacent to each hexagon. It makes sure that all singlet configurations around the corresponding star appear with equal weight.
(The exact form of these three terms, together with their derivation, can be found in Appendix C.)
How does this Hamiltonian compare to the Rokhsar-Kivelson (RK) Hamiltonian for dimer models?rokhsar:dimer-models (); moessner:dimer-triangular (); misguich:dimer-kagome () The resonance terms of our Hamiltonian correspond to those of RK Hamiltonian; in both cases, they ensure that the ground state is the even weight superposition of all dimer configurations. On the other hand, our Hamiltonian has two additional types of terms ( and ) which ensure that the system is in a valid dimer configuration; these terms are not present in the conventional dimer models as they are defined right away on the subspace of valid dimer configurations. While this makes our Hamltonian more complicated, it is outweighed by the fact that we have a completely local description of our system; in particular, it is this locality of the description which allows us to interpolate between the dimer state and the RVB state locally. Beyond that, the particular way in which we make the dimer configurations locally orthogonal allows us to define the resonance moves on a single hexagon (cf. Appendix C), as compared to the star required in the usual RK Hamiltonian.misguich:dimer-kagome ()
v.2 Hamiltonian for the RVB state
The Hamiltonian for the dimer state is frustration free, this is, all Hamiltonian terms—which we have chosen to be projectors—annihilate the dimer state, . We can make use of this fact to directly obtain a corresponding frustration free Hamiltonian for : Define
where for each , the product only involves the stars in the covering Fig. 9 which overlap with . Then, is positive semi-definite, and . Moreover, for each star we add a term to the Hamiltonian which projects onto the orthogonal complement of , and thus restricts the ground state (=zero-energy) subspace of the overall Hamiltonian to . [Note that conversely, those terms in which restrict the ground state space to give in Eq. (15).] As is a bijection between and , it follows that there is a one-to-one correspondence between ground states of and of ; in particular, the constructed RVB parent Hamiltonian has a topology-dependent degeneracy of the ground space (four-fold on the torus), which is obtained from the ground space of the dimer Hamiltonian by applying .
The Hamiltonian constructed this way is rather large: In particular, those triangle terms which lie at the intersection of three stars give rise to terms acting on three stars in Fig. 9. In order to obtain simpler terms, one can choose to adapt the framework for parent Hamiltonians of –injective PEPS derived in Ref. schuch:peps-sym, to the RVB state, which results in a parent Hamiltonian acting on two overlapping stars, i.e., spins; we give the full derivation in Appendix D.
v.3 Hamiltonian for the dimer-RVB interpolation
The relation between the dimer and RVB Hamiltonian extends to the whole interpolating path introduced in Sec. IV.1, and gives rise to a continuous path of local Hamiltonians interpolating between the dimer model and the RVB state: In order to ensure continuity, we first replace the dimer Hamiltonian by an equivalent Hamiltonian . Here, is the projection of onto , where the tensor product goes over all stars supporting and we omit vanishing , and is the projector onto on star , where the sum ranges over all stars. As is a parent Hamiltonian, the restriction to is ensured by the terms in locally; thus, has the same ground state space as with a spectal gap above, and we can smoothly interpolate between and without changing the ground state space or closing the gap. For , it is now straightforward to construct a continuous interpolating path , where , and projects onto for star .
In this paper, we have applied the PEPS formalism to the study of the Resonating Valence Bond states and dimer models. We have discussed PEPS representations of the RVB and the dimer model and studied their structure and relation. In particular, we have given a local unitary mapping between the dimer model and the toric code; furthermore, by defining the dimer state with locally orthogonal dimers, we were able to devise a local reversible mapping between the dimer state and the RVB state which allowed us to prove that the RVB state is the four-fold degenerate ground state of a local parent Hamiltonian for any finite lattice. Subsequently, this allowed us to devise a smooth interpolation between the dimer state and the RVB state and the corresponding Hamiltonians. We have studied this interpolating path numerically, considering the behavior of correlation functions, the rate at which the ground state changes, and the entanglement spectrum, and have found that all of these quantities behave smoothly and show no sign of a phase transition.
Our results make heavy use of the formalism of PEPS and their associated parent Hamiltonians, and in particular of –injectivity and –isometry, which we generalize in order to assess the RVB state on the kagome lattice. Similarly, the PEPS representation of the dimer–RVB interpolation allowed for efficient numerical simulations, enabling us to study the phase of the RVB state. We believe that these techniques will be of further use in the study of related systems. In particular, the results can be generalized to other lattices: Firstly, the PEPS description of RVB states applies to arbitrary graphs. All lattices with the “linear independence property” chayes:linindep (); seidel:kagome (); wildeboer:linear-independence () are –injective, and whenever the linearly independent blocks allow cover the lattice up to disconnected patches (such as in Fig. 9), this allows to interpolate between the RVB and the corresponding dimer state. (This is the case, for instance, for the square-octagon or the star lattice in Ref. wildeboer:linear-independence (), whereas for the hexagonal lattice it appears that no such covering can be found.) Further, if the dimer model can be expressed in terms of –injective tensors, this allows to conclude that it is equivalent to the toric code. (This is the case e.g. for the star lattice, but not for the square-octagon lattice.) Also note that our findings are not restricted to spin- singlets as dimers, but can be generalized to higher dimensional singlets or any other state, as long as –injectivity can be established.
We acknowledge helpful comments by S. Kivelson and A. Seidel. We wish to thank the Perimeter Institute for Theoretical Physics in Waterloo, Canada, and the Centro de Ciencias Pedro Pascual in Benasque, Spain, where parts of this work were carried out, for their hospitality. NS acknowledges support by the Alexander von Humboldt foundation, the Caltech Institute for Quantum Information and Matter (an NSF Physics Frontiers Center with support of the Gordon and Betty Moore Foundation) and the NSF Grant No. PHY-0803371. DP acknowledges support by the “Agence Nationale de la Recherche” under grant No. ANR 2010 BLANC 0406-0, and CALMIP (Toulouse) for super-computer ressources under Project P1231. JIC acknowledges the EU project QUEVADIS, the DFG Forschergruppe 635, and Caixa Manresa. DP-G acknowledges QUEVADIS and Spanish grants QUITEMAD and MTM2011-26912.
Appendix A Different PEPS representations for the RVB and dimer state
In this appendix, we give an overview of different PEPS representations of the RVB and dimer state and how they are related. We will start from the representation introduced in Ref. verstraete:comp-power-of-peps, , and subsequently show how to derive from it the representation used in this paper.
a.1 PEPS representation of the RVB
along all edges of the lattice (observing its orientation); this associates four three-level systems (qutrits) with each vertex. Then, we apply the following map to the qutrits at each vertex, which maps them to one physical qubit:
Here, in the second formulation the sum runs over the four virtual systems , and act on the virtual system , and acts on all virtual systems but .
What is the intuition underlying this construction? The Hilbert space holding the bond state, , Eq. (16), should be understood as a direct sum [with a corresponding representation ]. The two-dimensional subspace, , provides the spin- degree of freedom which holds the singlets used to construct the RVB state, while the third level, , is used as a tag indicating that there is no singlet along that edge, synchronizing this choice for both ends. In order to understand what happens when we apply to the bonds , let us rewrite as
where is a projector acting within the virtual space as
with the projector onto the subspace on site . Thus, projects one virtual qutrit onto the “singlet space” , while the others are projected onto the subspace. Due to the form of , it follows that both sides of each bond have to be projected onto the same subspace, i.e., either the singlet is selected, or both ends of the bond need to be in the state. From these two observations, it follows that by applying to the bonds , we obtain configurations with singlets along the edges, such that exactly one singlet is incident to each vertex, while all other edges are in the “unused” state . This way, all possible dimer configurations are orthogonal—we can infer the location of the dimers by measuring whether we are in the subspace or in the state. Moreover, the final state has an equal weight of all dimer configurations, as and appear with the same relative weight everywhere. Thus, it follows that the state obtained by applying the maps to the bonds is a realization of the (orthogonal) dimer state.
It is now straightforward to see that applying to this dimer state gives the actual (non-orthogonal) RVB state: keeps the singlet degree of freedom at every vertex, but it (coherently) erases the information about the edge the singlet is associated to; as the phases between different dimer locations are chosen to be , we indeed obtain an equal weight superposition of all dimer coverings with singlets, i.e., the RVB state.
a.2 Simplified PEPS representation
Here, is defined just as , but acting only on two virtual systems,
[cf. Eq. (2)], and is defined as
where is the completely antisymmetric tensor with , where , , and are oriented clockwise. Thus, we obtain a PEPS construction for the RVB state where we place tripartite bond states across triangles, and apply to each adjacent pair of qutrits, as shown in Fig. 18b.
As already explained in Sec. II.3, this PEPS representation of the RVB state can again be understood as arising from a PEPS representation of the dimer state, with . Note that different dimer coverings are again locally orthogonal, although in a different fashion—they become orthogonal on triangles. Note that this dimer state can be obtained from the one discussed in the previous section A.1 using that , where : While removes some degrees of freedom, different dimer configurations remain orthogonal on triangles.
Appendix B Change of overlap and correlation functions
In this appendix, we show how to compute the change of the overlap of the state along the dimer–RVB interpolation from two-point correlation functions; it can be seen as a Hamiltonian-free PEPS version of the result that the change of the overlap can be expressed as an integral over imaginary time correlation functions. campos-venuti:overlap-correlationfunction () While we will illustrate the derivation for the particular case of the dimer–RVB interpolation, it will equally apply to most other PEPS interpolations.
The interpolating path is given by
with as in Eq. (12), and where is the dimer state. can be split as
We want to compute the change of the normalized overlap
As the first-order change to can be at most a (unphysical) phase change, we are interested in the second order in . We write , and from Eqs. (21) and (23), it follows that can be expanded in orders of .
For notational convenience, let us define
Note that since both and in (21) are real, is real, and thus is real; we will use this in the following derivation. Using the above definitions (where is first order and second order in ) we have (neglecting third and higher order terms)
where the sum is over all possible positions of the . (As only appears as in the overlap, we can neglect the second order term.) It follows that
where we have used the notation
On the other hand, to second order
and thus, the second-order term in the expansion
can be expressed as an average over connected correlation functions:
where , and is the number of sites. This derivation shows that for PEPS interpolations, we can infer the change of the overlap from two-point correlators; this argument can be extended to relate the -th order expansion coefficient of the overlap to -point correlators.
For systems with finite correlation length , the sum in Eq. (24) is proportional to , and thus, . Therefore, the only way can diverge when the correlation functions decay exponentially is if . Note, however, that for , , so whether diverges at depends on the rate at which the latter vanishes.
Note that Eq. (24) does not apply when . For that case, one can easily check that and thus . On the other hand, is the norm of the PEPS where has been replaced by at a single site (summed over all sites), which immediately shows how to compute .
Appendix C A simple parent Hamiltonian for the orthogonal RVB
In this appendix, we show how to directly derive a parent Hamiltonian for the dimer state (in the representation of Sec. II.3) by relating it to the toric code on the same lattice.
Let us start by considering the toric code model on a kagome lattice, where the spins sit on the edges of the lattice, as illustrated in Fig. 19a. The toric code Hamiltonian has two types of terms: First, it has terms acting on the four qubits across each vertex, as
and second, it has terms acting across plaquettes—triangles and hexagons—as
where and are the Pauli matrices. For future purposes, we have introduced different labels for the two types of triangles: refers to an upward pointing tensor (sketched blue in Fig. 19a), and thus to a downwards triangle, and vice versa. In analogy to the square lattice toric code, Eq. (6), we can write the toric code state with triangular –invariant tensors
The states on the virtual level are again in the eigenbasis, which makes them invariant under . The tensors are marked blue in Fig. 19a, with the virtual indices associated to vertices of the kagome lattice. can be transformed by a unitary acting on the physical system to the projector onto the invariant subspace,
where the physical qubits are now associated to vertices, cf. Fig. 19b. It can be easily checked that the action of the Hamiltonian terms in the transformed basis is as follows (illustrated in Fig. 19b):
enforces that the two qubits per vertex are in the same state (namely equal to the value of the virtual index);
acts on the qubits around each hexagon and ensures that all bond configuration appear with the same probability; finally, in the original representation, enforced that the qubits at each tensor are in the proper subspace, it thus has to be replaced by the corresponding term for the transformed tensor,
The three terms are depicted in Fig. 19b.
Let us now consider what happens if we change all –type tensors (upwards pointing triangles) to projectors onto the antisymmetric subspace,
while the –type tensors (downwards pointing triangles) remain of the form
Clearly, this gives rise to a different triangular Hamiltonian for all –type tensors,
which enforces odd parity, while we keep for the –type tensors. All other Hamiltonian terms remain unchanged, since we still need to ensure that the qubits on both sides of the vertex are equal (), and that all configurations appear in an equal weight superposition ().
The projectors and can now be transformed to the and tensors for the dimer state by local isometries, which allows us to explicitly construct its parent Hamiltonian: First, the terms and enfore that the sites at each tensor are in the correct subspace; they are thus mapped to a local term which have the four states of Fig. 7 as ground states. The vertex term is transformed to