Entanglement renormalization and symmetry fractionalization

TL;DR

This paper investigates how symmetry fractionalization manifests differently in matrix product states and MERA representations of 1D quantum ground states, revealing that symmetry fractionalization does not occur in gapped MERA but may in critical cases.

Contribution

It demonstrates that symmetry does not fractionalize in MERA for gapped ground states, contrasting with MPS, and explores potential fractionalization in critical states, informing tensor network algorithms.

Findings

Symmetry does not fractionalize in gapped MERA.

Symmetry can fractionalize in critical MERA.

Results support using symmetric tensors in MERA algorithms.

Abstract

It is well known that the matrix product state (MPS) description of a gapped ground state with a global on-site symmetry can exhibit "symmetry fractionalization". Namely, even though the symmetry acts as a linear representation on the physical degrees of freedom, the MPS matrices---which act on some virtual degrees of freedom---can transform under a projective representation. This was instrumental in classifying gapped symmetry protected phases that manifest in one dimensional quantum many-body systems. Here we consider the multi-scale entanglement renormalization ansatz (MERA) description of 1D ground states that have global on-site symmetries. We show that, in contrast to the MPS, the symmetry does not fractionalize in the MERA description if the ground state is gapped, assuming that the MERA preserves the symmetry at all length scales. However, it is still possible that the symmetry can fractionalize in the MERA if the ground state is critical, which may be relevant for characterizing critical symmetry protected phases. Our results also motivate the presumed use of symmetric tensors to implement global on-site symmetries in MERA algorithms.

Figures

Equations

$$\vbox{\hbox{\includegraphics[width=99.58464pt]{symmps}}}.$$

$$O_g^{\prime }\equiv W^{†}{O}_{g}W,\forall g\in \mathcal{G}.$$

$$\vbox{\hbox{\includegraphics[width=99.58464pt]{merasymconstraint}}},$$

$$\begin{gathered} \text{parbox}80.00012ptMERAmadeof \\ symmetrictensors\Rightarrow \text{parbox}150.00023ptsymmetryremainsglobal\& \\ on-siteundercoarse-graining \end{gathered}$$

$$\begin{gathered} \text{parbox}125.00018ptsymmetryremainsglobal\& \\ on-siteundercoarse-graining\Rightarrow \text{parbox}80.00012ptMERAtensors \\ aresymmetric. \end{gathered}$$

$$|{\Psi }^{-}⟩=(V_{\varphi }\otimes V_{\varphi })|{\Psi }^{-}⟩,$$

$$|\Psi ⟩=\sum _{i_1{i}_2\dots i_L}\text{mbox}Tr(A_{i_1}{A}_{i_2}\dots A_{i_L})|i_1⟩\otimes |i_2⟩\dots \otimes |i_L⟩,$$

$$A_j^{\times s}\equiv A_{i_k}{A}_{i_{k+1}}\dots A_{i_{k+s}},$$

$$\vbox{\hbox{\includegraphics[width=199.16928pt]{A_decompose}}}.$$

$$\vbox{\hbox{\includegraphics[width=71.13188pt]{A_unitary}}}.$$

$$\vbox{\hbox{\includegraphics[width=78.24507pt]{A_svd}}}.$$

$$\vbox{\hbox{\includegraphics[width=99.58464pt]{A_densitymatrix}}},$$

$$\vbox{\hbox{\includegraphics[width=71.13188pt]{A_proj}}}.$$

Full text

I Introduction

Characterizing symmetries in tensor network states has recently played an instrumental role in the classification of gapped quantum phases of matter. For example, on a one dimensional lattice, ground states of gapped local Hamiltonians can be efficiently described as matrix product states (MPS), whose probability amplitudes are obtained by contracting a tensor network such as the one illustrated in Fig. 1(a) mps1 ; mps2 . Consider an MPS description of a 1D gapped ground state with a global on-site symmetry $\mathcal{G}$, such that the symmetry acts as a linear representation on each site, see Fig. 1(b). If the MPS is also normalmps3 (see also Appendix A) then the MPS tensor $B$ fulfills, up to certain gauge transformationsgauge ,

[TABLE]

Here, even though the symmetry acts linearly on the physical sites, the matrices $Y_{g}$ can form a projective representation of the symmetryMPSSym ; MPSSym2 , which fulfills the group product only up to a phase as $Y_{g}Y_{h}=e^{i\omega(g,h)}Y_{gh}$.cocycle This is sometimes referred to as ‘symmetry fractionalization’ in the MPS. (If the MPS is also in a certain canonical formmps3 , the matrices $Y_{g}$ are unitary.) If two (normal) symmetric MPSs carry inequivalentprojrep projective ‘bond representations’ $Y_{g}$ and $Y^{\prime}_{g}$ respectively, then they belong to different quantum phases protected by the same symmetry $\mathcal{G}$.CohmologyClassification ; CohmologyClassification1 ; CohmologyClassification2 ; phaseDetection ; phaseDetection1 Thus, the MPS description of ground states characterizes different quantum phases in one dimension.

Notice that Eq. (1) is also a local constraint—i.e., fulfilled by individual tensors—that results from imposing a global symmetry on the total MPS tensor network. Besides setting the stage for possible symmetry fractionalization, this local realization of the global symmetry is also convenient in MPS simulations, where one may be interested in protecting the symmetry against numerical errors. Tensor $B$, which fulfills Eq. (1), is an example of a symmetric tensor, which loosely speaking is a tensor that commutes with the symmetrysymTensorsProj . It turns out that symmetric tensors have a sparse structure, which is determined by the representation theory of the symmetrysymmetricTensors . The sparse structure is often exploited to reduce computational costs in MPS simulations, while also protecting the symmetry.MPSSymImplement

In this paper, we characterize symmetries in another prominent and efficient tensor network description of 1D ground states—the multi-scale entanglement renormalization ansatzER ; MERA (MERA). Unlike the MPS, the MERA is also suitable for describing critical ground statescriticalMERA . Inspired by the MPS, global on-site symmetries have also been implemented in the MERA by making judicious use of symmetric tensors. However, there is no formal proof that a global on-site symmetry necessarily implies that the MERA tensors must be symmetric. That is, there is no MERA analog of Eq. (1).

Despite this, the use of symmetric tensors has played a central role in several MERA applications, e.g., (i) targetting specific symmetry sectors and reducing computational costs in MERA simulations while exactly protecting the symmetry against numerical errorssymmera , (ii) determining non-local scaling and topological defect operators of conformal field theories that have a global symmetrynonlocalMERA , (iii) building exact MERA descriptions of RG fixed points in 1D symmetry protected phasessymProtectedMERA , and (iv) realizing the bulk gauging of a global boundary symmetrysymProtectedMERA ; holospin in a holographic interpretation of the MERASwingle .

In this paper, we first show that (under reasonable assumptions) global on-site symmetries in the MERA also necessarily lead to symmetric tensors (again, up to gauge transformations). More precisely, we show that if a MERA has a global on-site symmetry and the symmetry is protected at all length scales (as described in the next section), then one can always make its tensors symmetric by introducing suitable gauge transformationsoptimality .

This, for one, motivates the presumed use of symmetric tensors for implementing global on-site symmetries in MERA algorithms e.g. in the context of applications (i)-(iv) listed above. Second, it prompts the question: Can the symmetry fractionalize in the MERA description of a 1D ground state with a global on-site symmetry? (Since symmetric tensors are a prerequisite for symmetry fractionalization in the MPS.) We will argue that, in contrast to the MPS, the answer is negative for a gapped ground state. However, our argument does not apply to critical ground states. Thus, in the absence of any other restrictions, it is possible that the symmetry can still fractionalize in the MERA description of symmetric critical ground states. We suggest the possibility that inequivalentprojrep symmetry-fractionalized MERA states belong to different critical symmetry protected phases, for example, see Refs. symProtectedCritical, ; symProtectedCritical1, ; AnomaliesMERA, .

II MERA and symmetric tensors

From the outset it is apparent that the MERA tensor network is quite different from the MPS. (In particular, this means that the proof of Eq. (1), which relied on the specific structure of the MPS tensor network, cannot be applied directly to the case of the MERA.) For example, while the MPS is a 1D tensor network, the MERA extends in two dimensions.

The extra dimension in the MERA can be understood as a length scale, since the MERA is generated by a real space renormalization group (RG) transformation, known as entanglement renormalization.ER This RG transformation acts on the lattice by removing entanglement between blocks of sites before coarse-graining them. It can be described by a tensor network composed from isometric tensors $u$ and $w$ which represent the disentangling and coarse-graining components respectively, see Fig. 2(a,b). For convenience, we will contract tensors $u$ and $w$ to obtain tensor $A$, as shown in Fig. 2(c). The resulting tensor network, comprised of copies of $A$, is a (translation-invariant) matrix product operator—the operator analog of an MPS—which implements a linear map $W$ from a fine-grained lattice $\mathcal{L}$ to coarse-grained lattice $\mathcal{L}^{\prime}$, see Fig. 2(d).

Entanglement renormalization is evidently capable of generating RG flows with proper fixed points (in the thermodynamic limit) in 1D—both in gappedsymProtectedMERA ; symProtectedMERA1 and criticalcriticalMERA systems, and also in 2D quantum systems with topological ordertopoMERA .

The MERA is a striped tensor network that is generated by composing several entanglement renormalization transformations, as illustrated in Fig. 2(e). The MERA description of a ground state can be interpreted as the RG flow of the ground state: discarding bottom stripes of the MERA yields a description of the ground state on a sequence of increasingly coarse-grained lattices $\mathcal{L}\rightarrow\mathcal{L}^{\prime}\rightarrow\mathcal{L}^{\prime\prime}\cdots$.

We are now ready to address the following question: If a MERA describes a state with a global onsite symmetry $\mathcal{G}$ then is tensor $A$ (in each strip) symmetric? To proceed, consider the symmetry operators $O^{\prime}_{g}$ on the coarse-grained lattice $\mathcal{L}^{\prime}$, given by

[TABLE]

Note that without suitably constraining the individual $A$ tensors, or the transformation $W$ as a whole, the coarse-grained operators $O^{\prime}_{g}$ are not necessarily on-site. However, in the rest of the paper, we will assume that the coarse-grained symmetry operators $O^{\prime}_{g}$, in fact, remain on-site (see Fig. 3(a)) at all length scales. That is, the RG flow preserves the global and on-site character of the symmetry, at each step along the flow.

This assumption is consistent with the t’Hooft anomaly matching condition on the lattice, see e.g. Ref. AnomaliesMERA, . A global symmetry that acts in an on-site way can be regarded as having a trivial t’Hooft anomaly, since such a global on-site symmetry can always be gauged. If anomalies are preserved along the RG flow, a trivial anomaly (that is, an on-site action) must remain trivial (on-site action) along the RG flow. In two dimensions, however, anomaly matching may be more subtle in the presence of topological order. Thus, our assumption is reasonable at least in one dimension.

Therefore, in this paper, we will regard the MERA description of a 1D symmetric ground state legitimate only if it is generated by an RG flow that preserves the global and on-site character of the symmetry.

Next, we observe that if tensor $A$ is symmetric symcontract and transforms under the symmetry as

[TABLE]

where $X_{g}$ is unitary, then the coarse-grained operators $O^{\prime}_{g}$ are, in fact, on-site. This is demonstrated in Fig. 3. Note that in Eq. (3) we will require that $U_{g}$ and $U^{\prime}_{g}$ are linear representations of $\mathcal{G}$ but $X_{g}$ is allowed to be a projective representation of $\mathcal{G}$. ($U_{g}$ is required to be linear to allow for the possibility of symmetry fractionalizaton. Symmetric disentanglers and isometries then ensure that $U^{\prime}_{g},U^{\prime\prime}_{g},\ldots$ are also linear.)

So we have the following implication:

[TABLE]

However, it is not apparent that symmetric tensors are also necessary for this implication. It is possible that a global on-site symmetry could generally be preserved by enforcing some global constraints which are satisfied, say, by the total tensor network without requiring the individual tensors to be symmetric. But we show next that the implication (4), in fact, also holds in reverse, and therefore a global on-site symmetry can always be implemented in the MERA by means of symmetric tensors (assuming that the global and on-site character of the symmetry is preserved at all length scales).

II.1 Implication (4) also holds in reverse

To prove the reverse of (4), let us begin with our main working assumption, namely, the coarse-grained symmetry operators $O^{\prime}_{g}$ remain on-site. This is depicted again in Fig. 4(a), and then re-arranged as shown in Fig. 4(b,c). Next, we vectorize the MPO $W$ as a (translation-invariant) matrix product state (MPS) $|W\rangle$. Graphically, this corresponds to bending some indices of the MPO, see Fig. 4(d). Equation Fig. 4(c) implies that the MPS $|W\rangle$ has a global on-site symmetry $\mathcal{G}$, see Fig. 4(e). We could now recall Eq. (1) and conclude that tensor $A$ must be symmetric.

However, Eq. (1) holds only for a symmetric MPS that is normal. An MPS $B$ is said to be in the canonical form if the dominant eigenvalue of the matrix $E_{B}\equiv\sum_{i}B_{i}\otimes B^{*}_{i}$ is equal to 1, and the left (right) dominant eigenvector is the identity while the right (left) dominant eigenvector is a positive semi-definite matrix. If the dominant eigenvalue of $E_{B}$ is unique, then the MPS is also said to be normal.

MPS $|W\rangle$ is, in fact, both in the canonical form and normal. This is thanks to the isometric constraints Fig. 2(b), which are fulfilled by the MERA tensors, as shown in Fig. 5. (MPS $|W\rangle$ is actually injective, a property stronger than normality, see Appendix A).

Thus, applying Eq. (1) to the MPS $|W\rangle$, it follows that tensor $A$ must be symmetric as depicted in Fig. 4(e). By unvectorizing (bending back an index) Fig. 4(e), we obtain Eq. (3). Thus, implication (4) also holds in reverse. We remark that the symmetric tensor $A$ can also, under reasonable assumptions, be decomposed into a symmetric disentangler $u$ and a symmetric isometry $w$, as shown in Appendix B.

If the global on-site symmetry is preserved at all length scales, we can apply the above argument iteratively to all strips of the MERA. Thus, it follows that if a MERA has a global on-site symmetry and preserves the symmetry at all length scales, then its tensors must necessarily be symmetric (up to gauge transformations). That is,

[TABLE]

Next, we turn to the question of symmetry fractionalization in the MERA.

III Gapped ground states

In this section, we show that the symmetry does not fractionalize in the MERA representation of a 1D gapped symmetric ground state. Our strategy will be to translate an MPS description (possibly fractionalized) of a symmetry protected ground state to a MERA description, which we will examine for possible symmetry fractionalization. Recall, that the MERA describes the RG flow of a ground state. Therefore, to build the MERA description, we will coarse-grain the MPS (by means of entanglement renormalization) until a fixed point is reached.

III.1 Symmetry does not fractionalize along the RG flow

First, let us consider a single coarse-graining step. Let MPS $B$ describe a ground state with a global on-site symmetry $\mathcal{G}$, and let MPS $B^{\prime}$ denote the coarse-grained version obtained by means of entanglement renormalization. For convenience, we will assume that MPS $B$ satisfies Property 1 stated in Appendix A, which also implies that it is normal and thus exhibits the local symmetry Eq. (1). (1D symmetry protected ground states always admit an MPS description that satisfies Property 1.mps3 )

The coarse-graining proceeds as the following sequence of elementary operations, see Fig. 6: (i) block, for example, all odd pairs of sites and contract together the MPS tensors corresponding to each pair, (ii) apply on-site unitaries (the disentanglers) on the blocked MPS, (iii) decompose the resulting MPS back to the original lattice, (iv) block all even pairs of sites and contract together the MPS tensors corresponding to each pair, and finally (v) apply on-site isometries, which project to the support of the reduced density matrix of each blocked site.

We can determine how the resulting coarse-grained MPS tensor $B^{\prime}$ transforms under the action of the symmetry, by tracking the action of the symmetry through this sequence of operations (as indicated in Fig. 6). We find:

[TABLE]

Notice that the representation $X_{g}$, which appears in the coarse-graining MPO via Eq. (3), has been transferred to the coarse-grained MPS $B^{\prime}$.

If the coarse-grained MPS $B^{\prime}$ belongs to the same phase as MPS $B$, and is normal, it’s bond representation $Y_{g}\otimes X^{\dagger}_{g}$ must belong to the same equivalence class as $Y_{g}$, that is, the bond representation of the input MPS $B$. Futhermore, this must be true if the input MPS $B$ belongs to any symmetry protected phase, since the coarse-graining did not assume a specific phase to which MPS $B$ belongs. This is possible if, and only if, the representation $X_{g}$ is linear.projrep Thus, the symmetry is not fractionalized in the coarse-graining MPO—which constitutes a strip of the MERA.

Below we show that MPS $B^{\prime}$ indeed satisfies the necessary criteria for the above argument: it belongs to the same phase as MPS $B$, and is normal. (Normality ensures that the bond representation can be trusted to correspond to the phase of the MPS.) To this end, we argue that each of the elementary operations (i)-(v), which implement the coarse-graining, preserve the symmetry, phase, and normality of the MPS.

Symmetry is preserved.— Since the MPS tensors, the disentanglers, and isometries are all symmetric, the result of all the contraction steps is also a symmetric tensor symmetricTensors , which ensures that the symmetry is preserved. The decomposition step, however, requires a more careful consideration. It turns out that a symmetric tensor can always be decomposed into symmetric tensors, if the decomposition is carried out blockwise, as explained in Fig. 6.

Phase is preseved.— Broadly speaking, two quantum many-body states are in the same phase if they can be connected by a finite depth circuit of finite range interactions; such states are expected to have the same large length scale properties.CohmologyClassification ; CohmologyClassification1 ; CohmologyClassification2 The disentanglers are local unitary transformations, and therefore keep the state in the phase by definition. The isometries are composed of local unitary transformations followed by a projection to the support of the local density matrix. Such a projection also preserves the ground state, and thus the phase of the state. (More generally, the disentanglers can also be isometries instead of unitaries, in which case they are also chosen to project to the support of the local reduced density matrices, an operation which preserves the phase.) The blocking and decomposition steps do not involve any truncation of the Hilbert space. Thus, they preserve all the information in the quantum state, including the phase it belongs to.

Normality is preserved.— Follows from known properties of normal matrix product states, which are reviewed in Appendix A.

Thanks to these properties, we can deduce that the representation $X_{g}$ is linear as argued previously. Futhermore, iterating the above RG procedure, generates an RG flow where the resulting representation $X_{g}$ remains linear all along the RG flow.

III.2 Symmetry also does not fractionalize at the RG fixed points

The RG flow described in the previous section is also consistent with the expected RG fixed-point wavefunctions in a 1D symmetry protected phase. A representative RG fixed-point wavefunction $|\Psi^{\mbox{\tiny fixed}}_{\phi}\rangle$ in a symmetry protected phase that is labelledprojrep by $\phi\in H^{2}(\mathcal{G},U(1))$ is given byCohmologyClassification ; CohmologyClassification1 ; CohmologyClassification2 :

[TABLE]

where $|\Psi^{-}\rangle\equiv\includegraphics[width=14.22636pt]{singlet}$ denotes a singlet under the action of a suitable projective representation $V_{\phi}$ of $\mathcal{G}$, that is,

[TABLE]

and denotes a site of the lattice, which transforms as the linear representation $(V_{\phi}\otimes V_{\phi})$. By ‘suitable’ we mean $V_{\phi}$ is the smallest irreducible projective representation in the equivalence class labelled by $\phi\in H^{2}(\mathcal{G},U(1))$.

As described in Ref. symProtectedMERA, , the wavefunctions $|\Psi^{\mbox{\tiny fixed}}_{\phi}\rangle$ can be seen as fixed points of the coarse-graining transformation composed of the symmetric tensors:

[TABLE]

where denotes the Identity in the irreducible representation $V_{\phi}$ and $\langle\Psi^{-}|\equiv\includegraphics[width=14.22636pt]{singletbra}$. Notice that each index of both the tensors here correspond to double lines, and thus carries the linear representation $(V_{\phi}\otimes V_{\phi})$.

However, as also described in Ref. symProtectedMERA, , if symmetry fractionalization is allowed then the wavefunctions $\{|\Psi^{\mbox{\tiny fixed}}_{\phi}\rangle\}_{\phi}$ are no longer fixed points, since they can all be coarse-grained to a product state by means of the tensors:

[TABLE]

This means that entanglement renormalization reproduces the expected fixed point $|\Psi^{\mbox{\tiny fixed}}_{\phi}\rangle$ in each symmetry protected phase, provided the symmetry does not fractionalize. We have shown that this is in fact the case, and therefore also confirm the observations presented in Ref. symProtectedMERA, .

The MERA description of a ground state belonging to phase $\phi$ consists of the RG flow to the fixed-point state $|\Psi^{\mbox{\tiny fixed}}_{\phi}\rangle$. We have seen that the symmetry does not fractionalize during the RG flow or at the fixed point. Thus we conclude that the symmetry does not fractionalize in the MERA description of a gapped symmetry protected ground state.

We remark that symmetry fractionalization in the MPS has been exploited to devise practical schemes to detect symmetry protected phases in MPS ground state simulations: either by computing non-local order parametersphaseDetection1 (topological invariants) or by making use of symmetric tensors to directly detect the equivalence class of the bond representation for the statesymMPSDetection . In the MERA, one must instead examine the fixed point of the RG flow to detect the phase, see Refs. symProtectedMERA, ; symProtectedMERA1, .

IV Critical ground states

In the previous section, we argued that the symmetry does not fractionalize in a MERA description of a gapped symmetry protected ground state. The argument relied on the fact that a gapped ground state admits a faithful MPS description. On the other hand, a critical ground state cannot be faithfully described as a MPS; therefore, our argument does not carry over to critical states.

However, we expect that the RG flow at a critical point also preserves the global and on-site character of the symmetry—thus, the local symmetry constraint Eq. (3) still holds. This means that the MERA description of a critical ground is also composed of symmetric tensors (up to gauge transformations), Eq. (5). Therefore, it is at least possible that the symmetry can fractionalize in the MERA description of a critical symmetry protected ground state. (Since symmetric tensors were a prerequisite for symmetry fractionalization in the MPS.)

A possible argument against symmetry fractionalization in critical systems could be that perhaps symmetry fractionalization results from, and is intimately tied to, the short-range entanglement structure that is characteristic of gapped ground states. However, below we illustrate a counter-example.

Consider the MERA composed from copies of the following tensors:

[TABLE]

These are almost the same tensors as the ones in Eq. (10), except that we have swapped the top indices in the $w_{\phi}$ tensor. The symmetry is clearly still fractionalized on some of the indices.

However, in contrast to the MERA from Eq. (10), the average entanglement entropy of a block of $\ell$ sites in the MERA from Eq. (11) grows as log $\ell$, as illustrated in Fig. 7. This is also the characteristic scaling of entanglement entropy in 1D critical ground states. Note, however, that the state represented by this MERA (Eq. (11)) cannot be the ground state of a critical system, since, for example, it is not translation-invariant. One can speculate that it may be possible to construct a fractionalized critical ground state by taking superpositions of translations of such a MERA. Nevertheless, this example illustrates that a MERA can exhibit both symmetry fractionalization and long range entanglement characteristic of 1D critical systems.

If the symmetry can fractionalize at a critical point, however, there are at least a couple of potentially interesting questions one can ask. First, analogous to the MPS, can symmetry fractionalization in the MERA be used to characterize critical symmetry protected phases which were, for example, introduced recently in Refs. symProtectedCritical, ; symProtectedCritical1, ; AnomaliesMERA, . Second, how does symmetry fractionalization in the MERA interplay with the emergent conformal symmetry at the critical fixed point. And third, what role, if any, does symmetry fractionalization play in holographic interpretations of the MERA.

V Summary

We first showed that a MERA with a global on-site symmetry necessarily consists of symmetric tensors (up to gauge transformations), provided we assume that the symmetry remains global and on-site along the RG flow. This motivates the existing use of symmetric tensors for implementing global on-site symmetries in MERA algorithms, and also sets the stage for exploring symmetry fractionalization in the MERA. Subsequently, we argued that the symmetry does not fractionalize in MERA descriptions of 1D gapped symmetry protected ground states. However, without imposing any other constraints on the RG flow, symmetry fractionalization can still occur in MERA descriptions of 1D critical symmetry protected ground states, which could potentially lead to interesting applications.

Acknowledgements.— SS thanks Frank Verstraete for useful discussions and acknowledges the support provided by the Alexander von Humboldt Foundation and the Federal Ministry for Education and Research through the Sofja Kovalevskaja Award. This research was funded in part by the Australian Research Council Centre of Excellence for Engineered Quantum Systems (Project number CE170100009)

Appendix A Normal Matrix Product States

In this appendix, we briefly review the definition and some standard properties of normal matrix product states, which we used in this paper. The following material has been taken mostly from Refs. MPSSym, ; MPSSym2, ; mps3, .

A translation-invariant MPS $|\Psi\rangle$ of a lattice $\mathcal{L}$ with $L$ sites is composed from a set of matrices $\{A_{i}\}_{i=1}^{d}$ as

[TABLE]

where $i_{k}$ labels an orthonormal basis on site $k$. The matrices $\{A_{i}\}$ can be glued together to obtain a 3-index tensor $A$, such that fixing a particular value of the index $i$ selects a matrix $A_{i}$ from the tensor. We succintly refer to $|\Psi\rangle$ as MPS $A$. We will also represent the MPS graphically as shown below:

[TABLE]

Each circle represents a copy of tensor $A$. Each open index $i_{k}$ labels an orthonormal basis $|i_{k}\rangle$ on site $k$ of $\mathcal{L}$. The probability amplitude for a given configuration $|i_{1}\rangle\otimes|i_{2}\rangle\ldots\otimes|i_{L}\rangle$ is obtained by fixing the open indices to the corresponding values and performing the trace of the product of the resulting selection of matrices.

Definition 1. (Injectivity) Let us define a matrix $M_{A}$ as

[TABLE]

whose rows are indexed by the pair $(a,b)$ and columns are indexed by $i$. MPS $A$ is said to be injective if the matrix $M_{A}$ has a (pseudo-)inverse $M^{-1}_{A}$, $M_{A}M^{-1}_{A}=I$. This can also be expressed directly in terms of $A$ as: there exists a 3-index tensor $A^{-1}$ such that

[TABLE]

Definition 2. (Normality) Consider another lattice $\mathcal{L}^{\times s}$ that is obtained by blocking together $s$ sites of $\mathcal{L}$. State $|\Psi\rangle$ can be described on the lattice $\mathcal{L}^{\times s}$ by an MPS whose matrices $\{A^{\times s}_{j}\}$ are given by

[TABLE]

$j\equiv(i_{k},i_{k+1},\ldots,i_{k+s})$ labels an orthonormal basis $|j\rangle\equiv|i_{k}\rangle\otimes|i_{k+1}\rangle\ldots\otimes|i_{k+s}\rangle$ on a site of $\mathcal{L}^{\times s}$. For example, for $s=2$ we have

[TABLE]

MPS $A$ is called normal if there exists an $s$ such that the MPS $A^{\times s}$ is injective.

Definition 3. (Canonical form) Define the map $E_{A}(\circ)=\sum_{i}A_{i}(\circ)A^{\dagger}_{i}$, depicted as

[TABLE]

where $\dagger$ denotes the Hermitian adjoint. MPS $A$ is said to be in the canonical form if (i) the spectral radius of the map $E_{A}$ is equal to one, and (ii) $E_{A}$ has an eigenvalue equal to its spectral radius (i.e. =1) and (iii) the left (right) dominant eigenvector is the identity while the right (left) dominant eigenvector is a positive semi-definite matrix.

Property 1. (A sufficient condition for normality.) If MPS $A$ (i) is in the canonical form and (ii) the largest eigenvalue of $E_{A}(X)$ is equal to 1 and non-degenerate, then it is normal.

See e.g. Proposition II.1. in Ref. MPSSym2, .

In the remainder, we will assume that MPS $A$ satisfies Property 1 (and is thus normal).

Property 2. (Blocking) MPS $A^{\times s}$ is also normal.

Proof. Since $E_{A^{\times s}}=(E_{A})^{s}$, the spectral properties of the map $E_{A^{\times s}}$ satisfy the conditions for normality demanded in Property 1.

Property 3. (Equivalence) If a normal MPS $A$ is equal (up to a phase) to another normal MPS $A^{\prime}$ then there exists an invertible matrix $X$ such that $A^{\prime}_{i}=XA_{i}X^{-1},\forall i$, depicted as

[TABLE]

Proved as Theorem 7 in Ref. mps3, .

Property 4. (Decomposition) Consider MPS $A^{\times 2}$ obtained from MPS $A$ by blocking two sites, Eq. (16). Let us perform an arbitrary decomposition $A^{\times 2}_{ij}\equiv P_{i}Q_{j}$ and obtain a new MPS ${A^{\prime}}^{\times 2}_{ij}\equiv Q_{i}P_{j}$,

[TABLE]

MPS ${A^{\prime}}^{\times 2}$ is also normal.

Proof. Clearly, MPS $A^{\times 2}$ and MPS ${A^{\prime}}^{\times 2}$ both represent the same state. (Since the trace in Eq. (12) remains unaffected by the decomposition considered here.) Thus, the two MPSs are equivalent and related according to Eq. (18), that is, there exists an invertible matrix $Y$ such that

[TABLE]

As a result, the map $E_{{A^{\prime}}^{\times 2}}$ is related to the map $E_{{A}^{\times 2}}$ as

[TABLE]

Thus, both maps have the same spectrum. We know that the dominant left and right eigenvectors of $E_{{A}^{\times 2}}$ are positive-semidefinite (in fact, one of the eigenvectors is the identity). Since the left and right eigenvectors of $E_{{A^{\prime}}^{\times 2}}$ are related to those of $E_{{A}^{\times 2}}$ by conjugation under $Y(.)Y^{\dagger}$ [$Y^{-1}(.)(Y^{-1})^{\dagger}$], the dominant left and right eigenvectors of $E_{{A^{\prime}}^{\times 2}}$ are also positive-semidefinite (since positive-definiteness is preserved under any conjugation of the form $Z(.)Z^{\dagger}$).

Property 5. (On-site unitary) Consider MPS $A^{\prime}$ obtained from MPS $A$ by acting with unitary $U$ on each site of $\mathcal{L}$. We have $A^{\prime}_{i}=\sum_{j}U_{ij}A_{j}$, depicted as

[TABLE]

MPS $A^{\prime}$ is also normal.

Proof. Follows from the fact that the maps $E_{A}$ and $E_{A^{\prime}}$ [Eq. (17))] are equal.

Property 6. Let $M_{A}=usv$ denote the singular value decomposition of the matrix $M_{A}$ of Eq. (13) where $u^{\dagger}u=\mathbb{I},vv^{\dagger}=\mathbb{I},$ and $s$ is a diagonal matrix with positive entries (depicted below).

[TABLE]

Then the range of the matrix $v$ contains the support of the one-site density matrix obtained from MPS $A$.

Proof. The one-site density matrix $\rho$ is given by

[TABLE]

where $L,R$ are the dominant left and right eigenvectors of $E_{A}$ respectively. The equality is obtained by replacing tensor $A$ (and $A^{\dagger}$) by the singular value decomposition Eq. (23). Consequently, $v\rho v^{\dagger}$ has the same spectrum as $\rho$ (since $vv^{\dagger}=I$). Thus, $v$ preserves the support of $\rho$.

Property 7. (On-site projector) Consider MPS $A^{\prime}$ obtained from MPS $A$ by acting with the isometry $v$ of Eq. (23) on each site of $\mathcal{L}$. (The isometry $v$ projects to the support of the one-site density matrix, Property 6). We have $A^{\prime}_{i}=\sum_{j}v_{ij}A_{j}$, depicted below

[TABLE]

MPS $A^{\prime}$ is also normal.

Proof. Follows from the fact that the maps $E_{A}$ and $E_{A^{\prime}}$ [Eq. (17)] are equal, since

[TABLE]

where we used $vv^{\dagger}=\mathbb{I}$.

Appendix B Symmetric disentanglers and isometries from tensor $A$

Consider a symmetric tensor $A$ that fulfills Eq. (3), which we recall below:

[TABLE]

Recall that tensor $A$ is obtained by contracting a disentangler $u$ and an isometry $w$ as

[TABLE]

Below we show tensors $u$ and $w$ can always be chosen to be symmetric (under reasonable assumptions).

Let us contract a disentanger and an isometry differently to obtain another tensor $A^{\prime}$ as

[TABLE]

where, in comparison to Eq. (28), we have shifted the relative positions of $u$ and $w$. We can now repeat the argument in the main text to prove that tensor $A^{\prime}$ is also symmetric. (Since the argument did not depend on how the MERA tensors were paired and combined to form a matrix product operator.) Thus, we also have

[TABLE]

Next, define tensor $Q$ obtained by contracting $A^{\prime}$ and $A^{\dagger}$ as

[TABLE]

Tensor $Q$ is symmetric since it was obtained by contracting symmetric tensors.symmetricTensors Since $Q=w^{\dagger}\otimes w$, $w$ must be symmetric fulfilling:

[TABLE]

In order to show that $u$ is also symmetric, we have to assume $w$ has a pseudo-inverse $w^{-1}$ (as shown below)

[TABLE]

It can be easily shown that if a pseudo-inverse exists, it must be symmetric. (Tensor $w$ is block-diagonal when expressed in the symmetry basis for each index. A symmetric pseudo-inverse of $w$ is obtained by replacing each block with its pseudo-inverse.) If $w^{-1}$ exists then we can act with it on both sides of Eq. (28) and obtain

[TABLE]

Since $A$ and $w^{-1}$ are symmetric, $u$ must also be symmetric fulfilling:

[TABLE]