PEPS as ground states: degeneracy and topology

PEPS as ground states: degeneracy and topology

Norbert Schuch, Ignacio Cirac, and David Pérez-García
Abstract

We introduce a framework for characterizing Matrix Product States (MPS) and Projected Entangled Pair States (PEPS) in terms of symmetries. This allows us to understand how PEPS appear as ground states of local Hamiltonians with finitely degenerate ground states and to characterize the ground state subspace. Subsequently, we apply our framework to show how the topological properties of these ground states can be explained solely from the symmetry: We prove that ground states are locally indistinguishable and can be transformed into each other by acting on a restricted region, we explain the origin of the topological entropy, and we discuss how to renormalize these states based on their symmetries. Finally, we show how the anyonic character of excitations can be understood as a consequence of the underlying symmetries.

California Institute of Technology, Institute for Quantum Information,

MC 305-16, Pasadena CA 91125, USA

Max-Planck-Institut für Quantenoptik,

Hans-Kopfermann-Str. 1, D-85748 Garching, Germany

Dpto. Analisis Matematico and IMI,

Universidad Complutense de Madrid, 28040 Madrid, Spain

1 Introduction

What are the entanglement properties of quantum many-body states which characterize ground states of Hamiltonians with local interactions? The answer seems to be “an area law”: the bipartite entanglement between any region and its complement grows as the area separating them – and not as their volume, as is the case for a random state (see [1] for a recent review). Moreover, particular corrections to this scaling law are linked with critical points (logarithmic corrections) or topological order (additive corrections). A rigorous general proof of the area law, however, could up to now only be given for the case of one-dimensional systems [2], where an area law has been proven for all systems with an energy gap above the ground state, whereas the currently strongest result for two dimensions [3, 4] requires a hypothesis on the eigenvalue distribution of the Hamiltonian. Surprisingly, there is a completely general proof in arbitrary dimensions if instead, we consider the corresponding quantity for thermal states [5], and similar links to topological order persist [6].

The area law can be taken as a guideline for designing classes of quantum states which allow to faithfully approximate ground states of local Hamiltonians. There are several of these classes in the literature: Matrix Product States (MPS) [7] and Projected Entangled Pair States (PEPS) [8] are most directly motivated by the area law, but there are other approaches such as MERA (the Multi-Scale Entanglement Renormalization Ansatz) [9] which e.g. is based on the scale invariance of critical systems; all these classes are summarized under the name of Tensor Network or Tensor Product States. Though the main motivation to introduce them was numerical – they consitute variational ansatzes over which one minimizes the energy of a target Hamiltonians and thus obtains an approximate description of the ground state – they have turned out to be powerful tools for characterizing the role of entanglement in quantum many body systems, and thus helped to improve our understanding of their physics.

In this paper, we are going to present a theoretical framework which allows us to understand how MPS and PEPS appear as ground states of local Hamiltonians, and to characterize the properties of their ground state subspace. This encompasses previously known results for MPS and particular instances of PEPS, while simultaneously giving rise to a range of new phenomena, in particular topological effects. Our work is motivated by the contrast between the rather complete understanding in one and the rather sparse picture in two dimensions, and we will review what is known in the following. We will thereby focus on analytical results, and refer the reader interested in numerical aspects to [10].

1.1 Matrix Product States

Matrix Product States (MPS) [7] form a family of one-dimensional quantum states whose description is inherently local, in the sense that the degree to which two spins can be correlated is related to their distance. The total amount of correlations across any cut is controlled by a parameter called the bond dimension, such that increasing the bond dimension allows to grow the set of states described. MPS have a long history, which was renewed in 1992 when two apparently independent papers appeared: In [11], Fannes, Nachtergaele, and Werner generalized the AKLT construction of [12] by introducing the so-called Finitely Correlated States, which in retrospect can be interpreted as MPS defined on an infinite chain; in fact, this work layed the basis for our understanding of MPS and introduced many techniques which later proved useful in characterizing MPS [13]. The other was [14], where White introduced the Density Matrix Renormalization Group (DMRG) algorithm, which can now be understood as a variational algorithm over the set of MPS. In [15], MPS were explained from a quantum information point of view by distributing “virtual” maximally entangled pairs between adjacent sites which can only be partially accessed by acting on the physical system. This entanglement-based perspective has since then fostered a wide variety of results.

I. The complexity of simulating 1D systems

Motivated by the extreme success of DMRG, people investigated how hard or easy the problem of approximating the ground state of a 1D local Hamiltonian (or simply its energy) was. The history of this problem is full of interesting positive and negative results. A number of them was devoted to prove that every ground state of a gapped 1D local Hamiltonian can be approximated by an MPS [16, 17]; this was finally proven by Hastings [2], justifying the use of MPS as the appropriate representation of the state of one-dimensional spin systems. Very recently, also in the positive, it was shown that dynamical programming could be used to find the best approximation to the ground state of a one-dimensional system within the set of MPS with fixed bond dimension in a provably efficient way [18, 19]. On the other hand, in the negative it could be shown that finding the ground state energy of Hamiltonians whose ground states are MPS with a bond dimension polynomial in the system size is NP-hard [20]; this is based on a previous result of Aharonov et al. [21] proving that finding the ground state energy of 1D Hamiltonians is QMA-complete (the quantum version of NP-complete).

Ii. Hidden orders, symmetries and entanglement in spin chains

As we have seen, MPS provide the right description for one-dimensional quantum spin chains. Therefore, and given their simple structure, one can employ MPS to improve our understanding of the physics of one-dimensional systems. One field in which significant insight could be gained was the characterization of symmetries in terms of entanglement. First, the relation between string order parameters and localizable entanglement was explained in [22, 23]. In [24] (see also [25]), global symmetries in generic MPS have been characterized, and related to the existence of string order parameters, thus explaining many of the properties of string order, for instance its fragility [26]. This characterization of global symmetries was generalized to arbitrary MPS in [27], where it was used to shed light on the Hamiltonian-free nature of the Lieb-Schultz-Mattis theorem as well as to find new –invariant two-body Hamiltonians with MPS ground states, beyond the AKLT and Majumdar-Ghosh models. Other examples of MPS with global symmetries were already provided in [11, 28, 29]. Recently, also reflection symmetry has been investigated, showing how it provides topological protection of some MPS such as the odd-spin AKLT model, as opposed to the even-spin case [30].

MPS have also been extremely useful in understanding the scaling of entanglement in quantum spin chains, where special attention has been devoted to the case of quantum phase transitions. In [31], MPS were used to give examples of phase transitions with unexpected properties, namely analytic ground state energy and finite entanglement entropy of an infinite half-chain; the entanglement properties of these examples were further analyzed in [32, 33]. In [34], MPS theory was used to compute how the geometric entanglement with respect to large blocks diverges logarithmically with the correlation length near a critical point, and thus takes the role of an order parameter (see also [33]).

Apart from that, MPS theory has been used to decompose global operations (such as cloning or the creation of an entangled state) into a sequence of local operations [35, 36], to characterize renormalization group transformations and their fixed points in 1D [37], to understand which quantum circuits can be simulated classically [38], or even to propose new numerical methods to solve differential equations [39] or to compress images [40].

But what is it that makes MPS so useful in deriving all these results?

Iii. The structure of MPS

The main reason seems to be that MPS, despite being able to faithfully represent the states of one-dimensional systems, have a simple and well-understood structure, which makes them quite easy to deal with. For instance, as shown in [38], they naturally reflect the Schmidt decomposition at any cut across the chain, which makes dealing with their entanglement properties particularly easy. If one moreover restricts to the physically relevant case of translational invariant states, it turns out that one can fully characterize the set of all translationally invariant MPS by bringing them into a canonical form [11, 13]; in fact this canonical form constitutes one of the main ingredients in many of the results mentioned above.

In the canonical form, the matrices characterizing the MPS obtain a block diagonal form, and the properties of the state can be simply read off the structure of these blocks. In particular, for the case of one block (termed the injective case), it can be shown that the MPS arises as the unique ground state of a so-called “parent” Hamiltonian with local interactions, which moreover is frustration free. For the non-injective case where one has several blocks, the number of blocks determines the degeneracy of the parent Hamiltonian, and the ground state subspace is spanned by the injective MPS described by the individual blocks [13]. Beyond that, the injective case has other nice properties, such as an exponential decay of correlations. For the case of an infinite chain, all these properties were proven in [11], together with the fact that the parent Hamiltonian of an injective MPS has an energy gap above the ground state. The block structure of the canonical form is also useful beyond the relation of Hamiltonians and ground states, and e.g. allows to read off the type of the RG fixed point of a given 1D system.

1.2 Peps

Projected Entangled Pair States (PEPS) constitute the natural generalization of MPS to two and higher dimensions, motivated by the quantum information perspective on MPS which views them as arising from virtual entangled pairs between nearest neighbors [41, 8]. Though there has not yet been a complete formal proof that PEPS approximate efficiently all ground states of gapped local Hamiltonians, this could be proven under a (realistic) assumption on the spectral density in the low-energy regime [42], showing that PEPS are the appropriate class to describe a large variety of two-dimensional systems. However, as compared to MPS, PEPS are much harder to deal with: For instance, computing expectation values of local observables, which would be the key ingredient in any variational algorithm such as DMRG, is a #P-complete problem, and thus in particular NP-hard [43]. This poses an obstacle to numerical methods, and different ideas to overcome this problem have been proposed (we refer again to [10] for numerical issues). Fortunately, the bad news comes with good ones: The increase in complexity allows to find a much larger variety of different interesting behavior within PEPS as compared to MPS; for instance, in [44] it is shown that there exist PEPS with a power-law decay of two-point correlation functions, something which cannot be achieved for MPS.

I. Many examples

There have been identified various different classes of PEPS which exhibit rich properties. To start with, in [45] it has been shown how the interpretation of the 2D cluster state as a PEPS can be used to understand measurement based quantum computation [46] – a way of performing a quantum computation solely by measurements – by viewing it as a way to carry out the computation as teleportation-based computation on the virtal maximally entangled states underlying the PEPS description. This motivated the search for different models for measurement based quantum computation, and indeed models with very different properties have been subsequently proposed [47, 48, 49].

Another category of examples has been found with regard to topological models, where it was realized that many topologically ordered states have a PEPS representation with a small bond dimension: The case of Kitaev’s toric code was observed in [44], and this was later generalized to all string net models in [50]. Yet, despite their ability to describe these states, up to now PEPS did not help much in understanding the topological behavior of these states, which is one of the things we will assess in this work.

Ii. Few general results

As we have seen, the class of PEPS is rich enough to incorporate states with a variety of different behaviors. However, given the complexity of topological systems or of (measurement based) quantum computation, is this class still simple enough to prove useful as a tool to improve our understanding of 2D systems? That is, can PEPS help to uncover new effects and relations in nature, or to give a better understanding of the mechanisms behind quantum effects in two dimensions?

Judging from the experience with one-dimensional systems, in order to do so it would be highly desirable to have an understanding of the structure of PEPS comparable to the one obtained using the canonical form in one dimension. As it turns out, in the case of “injective” PEPS, several 1D results can be transferred; in particular, injective PEPS appear as unique ground states of their parent Hamiltonian [51]. Also, in [52] it is shown how global symmetries can be characterized in injective PEPS, which helped to understand the mechanism behind the 2D version of the Lieb-Schultz-Mattis theorem [53], to define an appropriate analogue for string orders in 2D [24], and to improve the PEPS-based algorithms used to simulate 2D systems with symmetries [54]. While these results illustrate that PEPS are a useful tool to understand properties of quantum states which appear as unique ground states of local Hamiltonians, it is also true that some of the most interesting physics in two dimensions takes place in systems which do not have unique ground states, but rather lowly degenerate ones, such as systems with symmetry broken phases or states with topological order.

Yet, in order to be able to fully apply the toolbox of PEPS to the understanding of these systems, it would be crucial to have a mathematical characterization of the structure of PEPS, providing a framework similar to the one which proved so useful for one-dimensional systems: Is there a canonical form for PEPS which allows to easily determine their properties, and how can it be found? How do PEPS appear as ground states of local “parent” Hamiltonians, and what is the ground state degeneracy? What is the structure of the ground state subspace, and how do these states relate to the PEPS under consideration?

1.3 Content of the paper

In this paper, we introduce a framework for characterizing both MPS and PEPS in terms of symmetries of the underlying tensors. This classification allows us to rederive all results known for one-dimensional systems, while it can be equally applied to the characterization of two- and higher-dimensional PEPS states, answering the aforementioned questions: Using the characterization based on symmetries, we prove how PEPS appear as ground states of local Hamiltonians with finitely degenerate ground states, and how these states can be obtained as variants of the original state. Subsequently, we demonstrate the power of our framework by using it to explain in a simple and coherent way the topological properties of these ground states: We prove that these states are locally indistinguishable and can be transformed into each other by acting on a restricted region, we explain the origin of the topological entropy, and we discuss how to renormalize these states based on their symmetries. We also discuss the excitations of these Hamiltonians, and demonstrate how to understand their anyonic statistics as a consequence of their symmetries. Thus, the characterization of PEPS in terms of symmetries provides a powerful framework which allows to explain a large range of their properties in a coherent and natural way.

The material in this paper is structured as follows: In Sec. 2, we introduce MPS and PEPS and discuss their basic properties. In Sec. 3, we review the proof for the “injective” case, in which the MPS appears as a unique ground state of the parent Hamiltonian. In Sec. 4, we show how general MPS can be classified in terms of symmetries, and use this description to generalize the results of the preceding section; we also discuss how our results relate to the ones obtained using the original canonical form of [13]. In Sec. 5, we generalize the symmetry-based classification of MPS to the case of PEPS, where we derive the parent Hamiltonian and characterize the structure of its ground state space. In Sec. 6, we consider a more restricted class of PEPS with symmetries, for which we derive a variety of results concerning the structure of the ground state space and the parent Hamiltonian, such as topological entropy, local indistinguishability, renormalization transformations, or the fact that the parent Hamiltonian commutes. We conclude the section by characterizing the excitations of the system and explaining how their anyonic statistics emerges from the symmetries of the PEPS. Finally, in Sec. 7, we discuss examples which illustrates the applicability of our classification, before we conclude in Sec. 8.

2 MPS and PEPS

We start by defining Matrix Product States, which describe the state of a one-dimensional chain of -level systems of length .

Definition 2.1.

A state is called a Matrix Product State (MPS) if it can be written as

(2.1)

Here, (the space of matrices over ), and is called the bond dimension.

Note that we restrict to the translational invariant setting with only one tensor , whereas a general MPS can be defined with a different tensor at each site. However, for each MPS describing a translationally invariant state a translationally invariant description (2.1) can be constructed; note, however, that the bond dimension can increase with .

In a more graphical representation, can be written as a three-index tensor,

Here, refers to the “physical” index [characterizing the actual state in (2.1)], and to the “virtual” index, which is only used to construct the MPS (2.1) and does not appear in the final state. For clarity, we assign arrows to the virtual indices, pointing from bras to kets . Connecting the legs of two tensors (called a “bond”) denotes contraction,

Thus, we can write the MPS (2.1) as

Having this picture in mind, we can immediately define a two-dimensional generalization:

Definition 2.2.

A state is called a Projected Entangled Pairs State (PEPS) of bond dimension if it can be written as

(2.2)

with , , a five-index tensor, where the index denoted corresponds to the physical system and “goes out of the paper”.

Note that in the definition (2.2) we have implicitly defined the direction of the arrows, and thus the assignments of kets and bras in :

Let us now introduce two simplifications to the MPS tensors, which will allow us to characterize all MPS based on their non-local properties.

First, we define the “projector” – the map which creates the physical system from the virtual layer.

Definition 2.3.

For an MPS given by a tensor ,

(2.3)

is the map which maps the virtual system to the physical one. The definition extends directly to PEPS.

The importance of the map lies in the fact that it tells us how virtual and the physical space of the MPS are connected. On the one hand, it tells us which physical states can be created by acting on the virtual degrees of freedom, and on the other hand, it tells us which virtual configurations – which in turn enforce physical configurations on the surrounding sites – can be realized by acting on the physical system. Thus, studying the properties of will allow us to infer properties of the MPS.

Let us now see which simplifications we can make in the analysis of , given that we are only interested in the non-local properties of the MPS (PEPS). Define , and . Then, can be inverted on and , respectively:

Note that characterizes the local support of the MPS: Projecting sites on any basis state leaves us with (where ), and thus, the single-site reduced operator is supported on . Thus, we can restrict the MPS to the subspace , i.e., each local system can be mapped to a –dimensional system by a local isometry, i.e., without changing any non-local properties. This implies that we can w.l.o.g. restrict our analysis to MPS with the following property.

Observation 2.4.

Any MPS/PEPS can be characterized (up to local isometries) by a tensor for which has a right inverse (denoted by in the diagram) such that:

(2.4)

where is the orthogonal projector on . W.l.o.g., we will assume a description with this property from now on.

The purpose of this paper is to characterize MPS and PEPS by looking at the structure of the subspace , and especially at its symmetries. As it turns out, it is sufficient to consider two cases: First, , the space of all linear operators on , and second, the case where is an arbitrary -algebra, i.e., a linear space of matrices closed under multiplication and hermitian conjugation.

In order to see why we only need to consider these two cases, take an MPS with tensor , and group its sites into super-blocks of sites each. This results in a new MPS with tensor

(2.5)

The map for this new tensor goes from a to a –dimensional space: Thus, for , we have that , which means that the map will typically be injective, and thus .111See [13] for a discussion of how to understand “typical” in this context. These MPS are known as injective, and they appear as unique ground states of their associated parent Hamiltonians.

The second case – being a -algebra – arises e.g. if the are block diagonal matrices, but within each block span the whole space of linear operators. As shown in [13], this does in fact cover the case of a general MPS, as any MPS can be brought into this form. While these states are no longer unique ground states of local Hamiltonians, their parent Hamiltonians have a finite ground state degeneracy, and the ground states are described by the individual blocks of the .

In the following, we will first review the situation of injective MPS, and show how to prove that they are unique ground states of local Hamiltonians. We will then turn towards the case where is a general -algebra, which we translate into a condition on the symmetry of under unitaries. While in one dimension this reproduces the results previously derived using the block structure of the matrices  [11, 13], it will enable us to generalize these results to the two-dimensional scenario, where we will find states exhibiting topological order and anyonic excitations, all of which can be understand purely in terms of symmetries.

3 MPS: the injective case

3.1 Definition and basic properties

We start by analyzing the injective case in which . According to Observation 2.4, we can choose such that is invertible. This leads us to the following formal definition of injective.

Definition 3.1 (Injectivity).

A tensor is called injective if has a left inverse

(3.1)

(The corresponding MPS will also be termed injective.) Intuitively, injectivity means that we can achieve any action on the virtual indices by acting on the physical spins.

Lemma 3.2 (Stability under concatenation).

The injectivity property (3.1) is stable under concatenation of tensors: If and are injective, then the tensor obtained by concatenating and is also injective, since

Note that we generally omit normalization constants in the diagrams (the contraction of the loop is ).

3.2 Parent Hamiltonians

Let us now see how injective MPS give rise to parent Hamiltonians, to which they are unique ground states. To this end, note that the two-particle reduced operator of the MPS is given by

with the projector onto the two-particle state obtained by projecting sites to the basis state ,

with . Thus, is supported on the subspace

(3.2)

Moreover, the injectivity of implies that spans the space of all matrices, and thus, has actually full rank on . Analogously to (3.2), one can define a sequence of subspaces

which by the same arguments exactly support the -body reduced operator .

The idea for obtaining a parent Hamiltonian is now as follows: Define a two-body Hamiltonian which has as its ground state subspace, and let the parent Hamiltonian be the sum of these local terms. The proof consists of two parts: First, we show that for a chain of length with open boundaries, the ground state subspace is (i.e., optimal), and second, when closing the boundaries, the only state remaining is . This is formalized in the following two theorems.

Theorem 3.3 (Intersection property).

Let and be injective tensors. Then,

(3.3)

with .

Proof.

It is clear that the right side is contained in the intersection, since for any we can choose

(3.4)

Conversely, for any in the l.h.s. of (3.3), there exist , such that

Applying the left inverse of and to the and index, we find that

i.e., is of the form (3.4), and thus is contained in the r.h.s. of (3.3). ∎

Theorem 3.4 (Closure property).

For injective and ,

(3.5)
Proof.

As in the proof of Theorem 3.3, the r.h.s. is trivially contained in the intersection by choosing . Conversely, by taking an arbitrary element in the intersection and applying the left inverses of and , we find that

which proves (3.5). ∎

Let us now put Theorems 3.3 and 3.4 together to show that the MPS arises as the unique frustration free ground state of a local Hamiltonian.

Theorem 3.5 (Parent Hamiltonians).

Let be injective, and as in Eq. (3.2). Define

as the orthogonal projector on the subspace orthogonal to on sites and (modulo ). Then,

has as its unique and frustration free ground state.

Proof.

For , define and . Rewriting

(and similarly for and ), Theorem 3.3 implies that , and thus by induction

(3.6)

i.e., the subspace supporting the length chain is given by the intersection of the two-body supports .

Now let , and . As the are projectors, the null space of is given by the intersection (3.6), i.e., by

Correspondingly, the null space of is

and thus by the closure property, Theorem 3.4, is the unique zero-energy (i.e., frustration free) ground state of . ∎

Note that all these results hold equally for injective PEPS [51]. We will discuss the case of PEPS in detail for the non-injective scenario, which includes the injective one as a special case. Note also that the results of this section do not rely on the translational invariance of the PEPS – the central Theorems 3.3 and 3.4 hold for any pair , of injective tensors.

4 MPS: the -injective case

4.1 Definition and basic properties

In the following, we will consider the case where is not injective, but where nevertheless

(4.1)

forms a –algebra.

In the following, we will characterize the structure of , and thus of , in terms of symmetries. Using Observation 2.4 – that has a left inverse on – together with an appropriate characterization of will allow us to base proofs on this left inverse, similar to the injective case.

Theorem 4.1.

For any -algebra there exists a finite group and a unitary representation such that is the commutant of , i.e.,

(4.2)
Proof.

Any -algebra can be decomposed as

(4.3)

where denotes unitary equivalence, . Now choose a finite group and a representation with irreducible representations of dimensions and multiplicity , respectively,222 This can be achieved, e.g., by choosing finite groups which have a -dimensional irreducible representation , and letting , and . In particular, the choice of and is not unique.

Now Schur’s lemma implies Eq. (4.2). ∎

Note that conversely, any unitary representation has a -algebra as its commutant (4.2), since . Thus, the characterization in terms of unitaries is equivalent to the characterization in terms of , the span of the . This motivates the following definition.

Definition 4.2.

Let be a unitary representation of a finite group . We say that an MPS tensor is –injective if
i) , and
ii) the map [Eq. 2.3] has a left inverse on the subspace

(4.4)

of –invariant matrices,

(4.5)

In graphical notation, we will denote unitary representations as circles labelled , or simply when unambigous. Note that the arrow now point towards the ket on which to apply . Condition i) of Definition 4.2 then reads

Note that the group is more important than its representation (though we will require certain properties at some point), and in fact, a different representation can be attached to each bond.

Lemma 4.3.

The orthogonal projector onto the –invariant subspace (4.4) is given by

Here, is the cardinality of the group . Thus, condition (4.5) corresponds to

(We generally omit normalization in diagrams.)

Proof.

Since

is a projection. As it leaves invariant,

(i.e., the image is contained in ), and it is hermitian, it is the orthogonal projector on . ∎

We will now show the analogue of Lemma 3.2: –injectivity is stable under concatenation of tensors. To this end, we will use the following identity.

Lemma 4.4.

For any unitary representation of a finite group,

with

(4.6)

where are dimensions and multiplicites of the irreducible representations of , and is diagonal in the basis in which . In a formula,

(4.7)
Proof.

The group orthogonality theorem implies

(4.8)

Thus,

For a restricted class of representations, we can make a stronger statement.

Definition 4.5 (Semi-regular representations).

A unitary representation of a finite group is called semi-regular if contains all irreducible representations of .

Lemma 4.6 (Linear independence of semi-regular representations).

Let be a semi-regular representation of a group . Then,

with as of (4.6). Note that for the regular representation, .

Proof.

This follows as is the character of the regular representation, which is . ∎

We are now ready to prove the analogue of Lemma 3.2 for the –injective case.

Lemma 4.7 (Stability under concatenation).

Let and be –injective tensors. Then is also –injective with left inverse

(4.9)

Here, is defined as in Lemma 4.4.

Proof.

–invariance of follows from

(4.10)

Moreover, (4.9) is the left inverse of on the –invariant subspace since

from Lemma 4.4. ∎

4.2 Parent Hamiltonians

Let us now proceed to the relation of –injective MPS and parent Hamiltonians. The construction is exactly analogous to the injective case: We define

and prove the analogues of Theorem 3.3 (Intersection Property) and Theorem 3.4 (Closure Property), but now for –injective MPS. While the intersection property will be the same as in the injective case, the closure will give rise to a subspace whose dimension equals the number of conjugacy classes of , for a properly chosen group .

Theorem 4.8 (Intersection property).

Let , be –injective. Then,

(4.11)
Proof.

Using the –invariance of and , we can infer that we can restrict , , and to also be invariant in the virtual indices. For instance, for any ,

this is, we can replace by

the same holds true for and . We will always assume symmetrized tensors from now on.

We are now ready to prove (4.11). First, the r.h.s. is clearly contained in the l.h.s., by choosing , . On the other hand, each element in the intersection can be simultaneously characterized by a pair , of tensors, and by applying the inverse maps we find that

This shows that (and equally ) are of the form required by Eq. (4.11), and thus proves the theorem. ∎

Theorem 4.9 (Closure property).

For –injective and ,

(4.12)

Before proving the theorem, let us give an intuition why the closure can be done using any (and nothing else). To this end, regard as the -fold blocking of ’s. The closures are exactly the operators which commute with , and therefore, they can be moved to any position in the chain. Thus, no local block of ’s needs to hold the closing , i.e., the state looks the same locally independent of the closure.

Proof.

We may again assume that and are –invariant. It is clear that the r.h.s. is contained in the l.h.s., by moving to the relevant link and setting or , respectively. On the other hand, any element in the intersection can be written using some and , for which it holds that

(In the first step, we have used that is -invariant.) Thus, any element of the intersection is of the form of the r.h.s. of (4.12), with , . ∎

Let us now formally define translational invariant MPS with an operator in the closure, as they appear in the above theorem.

Definition 4.10.

For an MPS tensor and , we define

(4.13)

to be the MPS given by with closure 333Note that one can use a different closure to reduce dramatically the bond dimension. One example is the W-state, whose bond dimension grows to infinity with L with the standard closure [13, 55], but has bond dimension 2 with a different closure..

Theorem 4.11 (Parent Hamiltonians).

Let be –injective, as in (3.2), and let

Then,

has a subspace of frustration free ground states spanned by the MPS with -closed boundaries.

Proof.

The proof is exactly the same as for Theorem 3.5, except that is is now based on Theorems 4.8 and 4.9; the latter leading to the degeneracy of the ground state subspace. ∎

Theorem 4.12 (Structure of ground state subspace).

Let be the irreducible representations of , of dimension . Then, the ground state subspace of Theorem 4.11 is -fold degenerate, and it is spanned by the MPS , where

(4.14)

is the projector onto the subspace supporting the irreducible representation in [proven in (4.8)].

Moreover, if is a semi-regular representation, is equal to the number of conjugacy classes of , and the subspace is spanned by the linearly independent states , where for each conjugacy class , one representative is chosen.

Note that the first part of the theorem corresponds to the known form of the different ground states, corresponding to the block structure of the ’s [11, 13], while the second part is the new symmetry-based classification of ground states which can be extended to the two-dimensional scenario.

Proof.

First, the –invariance of the implies that

i.e., all closures from the same conjugacy class are equivalent. Let , , be all irreducible representations of . Since the characters form an orthonormal set for the space of class functions (i.e. the functions which are constant over conjugacy classes), the ground state space is equally spanned by

Note that for (i.e. if does not contain ).

Conversely, linear independence of , can be seen as follows: For any class function , and with the -fold blocking of , we have that

(4.15)

Thus,

which proves linear independence. (Note that we had to use that is contained in , otherwise .)

The second statement follows from the fact that is constant on conjugacy classes, and that the number of conjugacy classes equals the number of irreducible representations of . ∎

5 Two dimensions: –injective PEPS

5.1 Definition and basic properties

Having understood the one-dimensional case of –injective MPS, let us now turn towards two dimensions. We will introduce some new conventions for the diagrams (as in principle, we need a third dimension), which we will explain right after the definition of –injectivity.

Definition 5.1.

Let be a semi-regular representation of a finite group . We call a PEPS tensor –injective if
i) It is invariant under on the virtual level,

(5.1)

ii) There exists a left inverse to such that , the projector on the -invariant subspace:

(5.2)

Let us briefly explain the differences in notation: We will try to avoid three-dimensional plots as far as possible. PEPS tensors are generally depicted “from the top”: The four legs in (5.1) are the virtual indices, and a black dot denotes the physical index (for the inverse, it is in the lower right instead of the upper left corner). As we use the let inverse to make the virtual subspace accessible via the physical indices, we will depict only the situation after applying the left inverse whenever possible, as depicted on the very right of (5.2). In order to distinguish the “original” virtual level and the one after the application of the left-inverse, we shade the latter gray. (This corresponds to the lower and upper layer, respectively, in the 1D case.)

Note that there is no need to choose the same representation for the horizontal and vertical direction. In fact, as mentioned earlier representations are assigned to links, and every link can carry its own representation – all that matters is that the two tensors acting on a link act with the same representation. It is this possibility of changing the representations which enables us to prove that –injectivity is is preserved under blocking.

Lemma 5.2 (Stability under concatenation).

Let and be –injective tensors. Then,

(5.3)

(with blocked up, down, and physical indices) is also –injective with left inverse

(5.4)
Proof.

First, note that for any two semi-regular representations and , is again a semi-regular representation. Then, it is clear that is also –invariant, as the action of the on the inner link cancels. That (5.4) is left-inverse to follows using , Lemma 4.6:

(5.5)

Observation 5.3.

Note that –injectivity is also preserved when contracting legs of an already connected block, e.g. the up leg of with the down leg of in (5.3): The resulting tensor is clearly again –invariant, and the left-inverse is obtained by contracting the corresponding legs of (5.4) with any any operator with nonzero trace (e.g., ): The group elements attached to the two legs cancel out, as they belong to the same tensor.

5.2 Parent Hamiltonians

The idea to construct parent Hamiltonians is essentially the same as in one dimension. We define the local Hamiltonian as minus the projector on the span of a block (the smallest block which allows for an overlapping tiling of the lattice), and study how the ground state subspace behaves when growing the block. For simplicity, we first grow the block in one direction until we reach a lattice, and then in the other direction until we have the full lattice with open boundaries. Finally, we study what happens when we close the boundaries. While the growing will work essentially exactly as in 1D, closing the boundaries will give a richer structure.

Theorem 5.4 (Intersection property).

Let , be –injective. Then,

Here, we have chosen to shade the inside of the “boundary condition” tensors , , and .

Proof.

The proof is exactly analogous to the one-dimensional case, Theorem 4.8. The r.h.s. is contained in the l.h.s., as any can be written as both and . Conversely, any element in the intersection can be written as for some and ; as in the one-dimensional case, we can assume both to be –invariant. To recover , we apply the left inverse (5.4) to the left two physical modes of and obtain

which implies that the state is of the form . ∎

Theorem 5.5 (Closure property).

For –injective , , , and ,

(5.6)

Here, the sum runs over all pairs such that .

Note that if , , , and arise from blocking the original tensor, this corresponds to a closure with two unitaries and at the horizontal and vertical closure, respectively.

Proof.

The proof again follows closely the proof for one dimension. First, it is clear that the r.h.s. in contained in the intersection by choosing , , , and appropriately. To show that every element in the intersection is of the form , we consider an element of the intersecion, which can be written as with boundaries and , respectively. As before, we can assume the boundary conditions to be –invariant. To recover , we apply the left-inverse