Irreducible many-body correlations in topologically ordered systems

Topologically ordered systems exhibit large-scale correlation in their ground states, which may be characterized by quantities such as topological entanglement entropy. We propose that the concept of irreducible many-body correlation (IMC), the correlation that cannot be implied by all local correlations, may also be used as a signature of topological order. In a topologically ordered system, we demonstrate that for a part of the system with holes, the reduced density matrix exhibits IMCs which become reducible when the holes are removed. The appearance of these IMCs then represents a key feature of topological phase. We analyze the many-body correlation structures in the ground state of the toric code model in external magnetic fields, and show that the topological phase transition is signaled by the IMCs.

Topologically ordered phases may not be characterized by any local order parameter associated with Laudau's symmetry breaking picture [1]. How to characterize this type of exotic phase is one of the biggest challenges in modern condensed matter physics. These topological phases may be characterized by many distinguished features, including: the degeneracy of ground states depends on the topology of the manifold that supports the system; the existence of anyonic elementary excitations; the existence of edge states on the open boundaries. Moreover, all these properties characterizing topological phase must be stable against local perturbations, making topologically ordered systems promising candidates for fault-tolerant quantum computing [2,3].
The topological entanglement entropy is firstly proposed to characterize the ground state with topological order by  and Levin-Wen [5], which builds a nontrivial connection between many-body physics and quantum information theory. The underlying picture of the topological entanglement entropy is to 'retrieve' a many-body correlation which cannot be built up from its parts. In general, calculating with large enough parts of the system, the entanglement entropy is successfully used to identify topological order in several microscopic models [6][7][8].
In this paper, we provide a novel perspective to signal this many-body correlation in topologically ordered ground states, building on the concept of irreducible many-body correlation (IMC) [9,10]. This approach intuitively sounds, as irreducible r-body correlation is nothing but the correlation that cannot be build up from any r 1 ( )  --body correlations [9,10]. (See appendix A for the definition of IMC.) In a topologically ordered system, we demonstrate that for a region of the lattice with holes, the reduced density operator exhibits IMCs.
To be more precise, for an n-body quantum state and any r n  , the irreducible r-party correlation (or irreducible correlation of order r) characterizes how much information is contained in the r-particle reduced density matrices (r-RDMs) but not in the r 1 ( ) --RDMs. For a state σ, denote C r ( ) ( ) s its r-party irreducible correlation. Here the reduced state for a region   Í is defined by , where  r is the state on the region , and ⧹   is the complement set of  relative to . The total correlation is then the information contained in the state beyond that in the 1-RDMs. In this sense the irreducible r-party correlations provide a natural hierarchy of correlations in the system-the sum of all the irreducible r-party correlations equals the total correlation [10].
Throughout the paper we consider lattice spin models. For any regions ,   of the lattice, with   Ì , one naturally expects that there are more correlations in  than those in , as all the particles in  are contained in . This is in general also the case for IMCs. However counter-intuitively, in topologically ordered systems, one could have is the reduced state of the region  (). The extreme case could be that C 0 r ( ) This may happen when  and  have different topology (e.g., ⧹   is a hole). In this case when we consider the correlations in region , the r-party correlation in  r must become reducible, i.e., the information in the r-RDMs of the region  is contained in the information of the r ¢-RDMs of the region , with r r ¢ < . It is worthy to note that the total correlation in the reduced state of the region  is always not less than that of the region .
It turns out that the appearance of these IMCs in a region with holes, or the validity of equation (1), represents a key feature of topologically ordered systems. This key feature, the emergence of long-range interaction on subregions, is also known for probability distributions (see for example corollary 7 of [11]). We will analyze these IMC structures in the ground state of the toric code model in an external magnetic field. In addition to demonstrating the appearance of these irreducible correlations and the nontrivial phenomena of equation (1) in these systems, we show that the topological phase transition is also identified by the creation of these IMCs.

Results
The Toric code mode-We start with the toric code model [2], which is a spin- When the periodic boundary condition is considered (i.e., a torus), the model shows typical features of topological order such as ground state degeneracy. And the degenerate ground state space is a quantum errorcorrecting code with macroscopic distance (i.e., the distance grows with system size). As the star and plaquette operators commute the ground state space is given by the stabilizer formalism of quantum code where the loop C pv (C ph ) is the set of spins on one vertical (horizontal) line of the lattice, and C sv (C sh ) is the corresponding one on the dual lattice, which are demonstrated in figure 1. Hence the ground state degeneracy is 4.
IMCs in toric code model-We start from considering the IMC for any ground state G | ñ of the toric code model shown in figure 2. For the convenience of the following discussion, we use dashed red lines to divide the system into six parts, denoted as 1  , 2  , 3  , R I 4 , II 4  , and 5  respectively. The state  r is denoted as the RDM of G | ñ for the region .
Recall that for any n-qubit state σ, the IMC C S S , where i s is 1-RDM of the ith particle. To analyze the structure of IMCs for any ground state G | ñ of the toric code model, a basic tool is theorem 2 in [10], which gives the analytical results of IMCs for all the stabilizer states. It states that the degree of r-party irreducible correlation in an n-qubit stabilizer state is equal to the number of independent elements with length r in the state's stabilizer group when we choose the independent elements with less length as possible.
Since the 1-RDMs of G | ñ are maximally mixed, the total correlation of G | ñ is C G n T (| ) ñ = . Furthermore, for any r L 4 h  < , the maximally mixed state M r supported on the ground-state space has the maximum entropy among all states with the same r-RDMs as those of G | ñ. Therefore we have and the IMCs of all the other orders are 0, see a proof of equation (6) in appendix A.1. This implies that there are 2 bits of IMCs of macroscopic order in any ground state of toric code model on a torus. This then raises an interesting question: where do these 2 bits of macroscopic correlations come from? To answer this question, we examine in detail the correlations in the local reduced states of the system, which will demonstrate the essential feature of topological order as given in equation (1).
Notice that the RDM of G | ñ for a singly connected region of the lattice, e.g., the regions R 1 and R R , is independent of the choice of the ground states. Furthermore, the RDM is a stabilizer state, whose  generator elements may all have the shortest length 4, which implies that the reduced state has and only has irreducible 4-party correlations according to theorem 2 in [10]. Therefore the total correlations in the reduced state for a singly connected region, e.g., 1 However, if the region contains a hole, e.g., the region 2  , then situation could be dramatically different. Because the reduced state 2  r has two generator elements, one is j j z s  , where j takes over the sites in the loop marked by 14 rectangles, the other is k k x s  , where k takes over the sites in the loop marked by 10 crosses. According to theorem 2 in [10], we have It is worthy to emphasize that equations (8) and (9) can be regarded as a typical example of equation (1) when we take Intuitively, these IMCs in 2  are contributed by the correlations in the reduced state of the hole ( 1  ), which are reduced correlations in the region with the hole ( 1 2   È ). In general, the reduced states for any region in the lattice topologically equivalent to 2  exhibit IMCs of macroscopic order, which become reducible for the region including the hole. And the existence of these IMCs of macroscopic order in regions with holes is an essential feature of topological order.
When the hole becomes larger and finally encounters the boundary, e.g., the hole 1  (4) and (5). It is worthy to note that the region is not topologically equivalent to 2  , and the reduced state for the region 4  depends on the ground state we choose. Characterizing topological phase transition-Based on the discussions above, it is natural to use the IMCs of orders proportional to the boundary length in a region with holes to signal the topologically ordered phase. As a typical example, we consider the toric model in an external magnetic field along the n  direction [12], with the Our calculation is based on the numerical exact diagonalization method: to calculate the largest magnitude eigenvalues, we use the Lanczos algorithm, which is realized in matlab as the function eigs(). Our system consists of 24 spins on a 4 3 lattice, and the irreducible 6-particle correlation r is not a topological invariant, but it does reflect the power of creating higher order correlations from lower order correlations.
In the thermodynamic limit, we expect that C 0 ¹ for a topological phase, while it is zero for a nontopological phase. This implies that a kind of discontinuity appears in the maximal entropy interference, which is discussed recently, see [13] and references therein. However, the numerical value of correlation might not be topological invariant if calculating with finite size systems. Nevertheless, even for the system of this small size, the rate change of correlation already clearly signals the phase transition. In this sense we suggest to use the maximal changing rate of C 5 ( ) ) with respect to h as an indicator of the phase transition point: Based on the numerical algorithms proposed in [14,15], we obtain the results of C  in [18,19]. It is worthy to note that since the toric code model only contains x s and z s terms, the direction of the external magnetic field is symmetric with respect to e x  and e z  , but is asymmetric for e x  and e z  .
From the numerical results we know that the ground state of region 2  , i.e. 2  r contains only 1 bit C 6 ( ) without external field h, which can also be obtained analytically as given in appendix. While h is infinite, there are no correlations of order greater than 1. The distribution of correlation among the different order correlation of 2  r under different h is showed in figure 5. When h h  < , the total correlation largely stems from from C 6 ( ) . However, the lower order correlation C 2 ( ) contributes to the main part of the total correlation when h h  > .
General correlation structure in topological order-Although we discussed the toric code model, a similar correlation structure should also be valid for topologically ordered systems in general. In the thermodynamic limit, the ground states are degenerate and any ground state exhibits the following correlation structure: where r 0 is a positive integer independent of the system size L h . Furthermore, the value of IMCs of macroscopic order C G L h (| ) ( )  ñ is a topological invariant. In a finite system, the ground state space is generally unique, which does not exhibit any IMCs of macroscopic order i.e., C G 0 r (| ) ( ) ñ = for r r 0 > . However, there are always IMCs of higher order in the reduced states of G | ñ, which is manifested in a region with holes, e.g. 2  in figure 2, as the IMCs are of order proportional When the region 2  is large enough (typically larger than the correlation length), it then contains the IMCs

Discussion
The relation with topological entanglement entropy-The construction of topological entanglement entropy by Levin-Wen [5], denoted by LW  may be regarded as an approximation for obtaining the IMCs of macroscopic order in a large enough region with a hole, e.g., the region 2  . To calculate LW  , they divided the region into three parts A, B, and C as demonstrated in figure 6, where A and C are far apart so they there should be no correlation between them. It is worth pointing out that here we study correlations among A, B, and C, but not those among single spins as before.
The Levin-Wen topological entropy is then given by the total correlation  and 3  [20], which implies the area law. Notice that for the topological phase, the IMCs of macroscopic order in the region 2  will decrease the correlations between 2  and 3  , thus decreases the entanglement entropy compared to the area law, which then gives the topological entanglement entropy.
It is worth mentioning that [21] proved that topological entanglement entropy is equivalent to IMC if the state has zero-correlation length. Hence it gives a strict mathematical reason for the similarities between the topological entanglement entropy and IMC found in our numerical results of appendix B.
Summary-In sum, we use the concept of IMCs to analyze the correlation structure in the ground states of the toric code model. Based on the analysis, we suggest that the appearance of IMCs of macroscopic order in a region with holes represents an essential feature of topological order. For the toric code model in an external magnetic field, we also demonstrate that the power to create IMCs of higher orders for a region  with holes, from IMCs of lower order for a region   É , signals the topological transition phase transition. Our calculation uses a relatively small system, which clearly indicates the transition. Our concept has intimate relations with the idea of topologically entanglement entropy and may be applied to study other systems with topologically order, by calculation with relatively small system size. Our work may shed light on a better understanding of the general many-body correlation structure of a quantum state in topologically ordered phase.

Acknowledgments
We would like to thank Z-F Ji for pointing out the relationship between Levin-Wen entanglement entropy and the strong subadditivity of quantum entropy. We also thank X Chen, X-G Wen, and C-P Sun for helpful discussions. YL and DLZ are supported by NSF of China under Grant No. 11175247 and 11475254, and

Appendix A. Irreducible multiparty correlations
In this section, we will first briefly review the definition of irreducible multiparty correlations. Then two typical examples, the toric code model without magnetic field, are given to demonstrate how to get the analytical results of irreducible multiparty correlations. In addition, the latter example explain how the higher order correlations are generated from lower order correlations. In fact, the simplest configuration (only seven spins whose correlation structure to be analyzed) for the toric code model suffices to demonstrate this essential feature of topologically ordered phase. The concept of irreducible multiparty correlation is introduced [9, 10] to classify the total correlations in a multi-party quantum state into bipartite correlations, tripartite correlations, etc.
The definition of irreducible multiparty correlations for an n-party quantum state , where a | |is the cardinality of the set a. Second, since the set k  is convex, there is a unique state with maximal von Neumann entropy in k  : where the von Neumann entropy of a state τ is defined as Then the irreducible k-party correlation ( k n 2   ) for the n-party state n [ ] r is defined as r is the single particle reduced state of the ith party. This implies that the irreducible k-party correlations are a complete classification of the total correlation in an n-party state.
Notice that the analytical results on irreducible multiparty correlations of all the stabilizer states are given in theorem 2 in [10]. Since the ground state and its reduced states for the toric code model without magnetic field are stabilizer states, we can obtain the distribution of irreducible multi-party correlations in these states by directly applying the above-mentioned theorem.
Here we present two examples to demonstrate how to get the analytical results of the IMCs for the toric code model without magnetic field.
state G r has the same r-RDMs ( r L 1 h  < ) with the state M r . Notice that the state M r is a stabilizer state with generators from independent elements from A B , Since the number of independent elements of the stabilizer is n 2 -, and the lengths of the elements are 4, theorem 2 in [10] implies that it has only n 2 bits of irreducible 4-party correlation. Therefore the state G r has the same irreducible r-party correlations ( r L 2 h  < ), which completes the proof of equation (6).
A.2. Example 2: Irreducible multiparty correlations in regions of G | ñ In this subsection, we will try to explain why the concept of IMCs plays a key role in characterizing topological orders using the case as shown in figure 3. That is, why we choose C 6 ( ) in our numerical analysis of the topological phase transitions.
We consider the model given by equation (10) with the magnetic field h = 0 in the lattice shown in figure 3. To give the details of our calculations, we redraw figures 3-7(a), where the spin labels are given for regions 1  and 2  . Notice that the region 2  has a hole, i.e., the region 1  (the spin 0). Our main results for this example are given as follows. Let us denote the density matrix of one of the ground states as ρ. Then Obviously this is a typical example of equation (1), i.e., the 6-party correlation is generated from the 4-party correlations, which is the essential feature of the topological order phase as suggested by us. Now we come to the calculation details. The generator of the stabilizer reduced state 1 2   r È may be taken as The transition line is also determined by equation (13) and labeled by the black lines in figure 8. The phase diagram is similar with the previous one in [18]. r in the above two cases, which are shown in figures 11 and 12. When the magnetic field is along the y direction, we observe that E LW shows almost the same behavior as C 5

( )
 , which is consistent with the argument we present in the article. However, when the magnetic field is along the x direction, the behaviors between E LW and C 5 ( )  show obvious differences, particularly in the smaller size of the system. The relation between these two quantities will be studied further in the next section.
Appendix C. When will irreducible correlation coincide with topological entropy The numerical results demonstrated above implies that in many cases the irreducible multiparty correlation and the topological entanglement entropy proposed by Levin-Wen are very similar, which motivates us to ask when will the irreducible correlations coincide with topological entropy.      One can verify that the right hand side is indeed a quantum state whose reduced density matrix on BC is exactly BC r . But it is not always the case that the reduced density matrix on AB is also AB r . In fact, the following condition is necessary and sufficient for the equality of E IC and E LW : .
C . 8 Ä may be neither sufficient nor necessary. Notice that the above discussion is valid for all the three-party states of finite dimension. It is certainly correct to the configuration in figure 5 for the Levin-Wen topological entropy.