Correlations in Quantum Spin Systems from the Boundary Effect

We introduce the boundary effect on the ground state as an attribute of general local spin systems that restricts the correlations in the ground state. To this end, we introduce what we call a boundary effect function, which characterises not only the boundary effect, but also the thermodynamic limit of the ground state. We prove various aspects of the boundary effect function to unfold its relationship to other attributes of the system such as a finite spectral gap above the ground state, two-point correlation functions, and entanglement entropies. In particular, it is proven that an exponentially decaying boundary effect function implies the exponential clustering of two-point correlation functions in arbitrary spatial dimension, the entanglement area law in one dimension, and the logarithmically corrected area law in higher dimension. It is also proven that gapped local spin systems with nondegenerate ground states ordinarily fall into that class. In one dimension, the area law can also result from a moderately decaying boundary effect function, in which case the system is thermodynamically gapless.

One of the prominent approaches to quantum manybody theory is to explore universal features of typical classes of many-body systems, thereby guiding our intuition into more specific problems. Commonly studied in that context are spin systems with finite-range interaction. The localness of the interaction is then manifested as various dynamical and static features. A quintessential example is the Lieb-Robinson bound, which demonstrates that local interaction manifests itself as a locality in the emergent dynamics [1].
Regarding the static features, various aspects of correlation that ground states exhibit are of primary concern. In particular, along with the condition of local interaction, the existence of a finite spectral gap above the ground state highly restricts the correlation that the ground state can accommodate. In such systems, arbitrary two-point correlation functions for the ground state decay exponentially with distance, called the exponential clustering theorem [2,3]. Moreover, such ground states can accommodate only a restricted amount of entanglement. When the entanglement between a subregion and the rest, called the entanglement entropy, scales at most as the boundary size of the region, such a state is said to obey the area law for entanglement entropy [4]. It turns out that in one-dimensional gapped spin systems with local interaction, nondegenerate ground states obey the area law [5]. This contrasts with the case of random states, which exhibit an entanglement entropy proportional to the volume of the region [6]. Conceptually, this implies that the ground states obeying the area law occupy only an extremely small portion of the Hilbert space, which again suggests that we do not actually need too many parameters to describe them. The area law is thus of crucial importance both conceptually and practically, e.g., in the area of the simulation of quantum many-body systems [7] and the Hamiltonian complexity theory [8]. For this reason, a general proof of the area law in more than one dimension has been awaited for quite a while, but is still lacking [4,5,[9][10][11][12][13][14][15][16].
Apparently, all the attributes mentioned abovelocal interaction, spectral gap, spatial dimension, Lieb-Robinson bound, exponential clustering, area law, etc.are deeply related to each other and inseparable. How-ever, our understanding of them and their relationship is far from being satisfactory. Owing to the inherent complexity, the ordinary task is rather to figure out various aspects of their relationship. For example, although one can infer various features assuming the existence of a spectral gap, it is extremely hard to know whether a given Hamiltonian is gapped. In this paper, we bring in the thermodynamic limit, in a specific sense we clarify later, as another attribute of many-body systems that strongly dictates the ground state correlations. This allows us to offer a sensible and intuitive picture on how a local nature emerges in the ground states and how the gap plays a role in there. In order to address the thermodynamic limit, we consider a sequence of Hamiltonians with increasing number of spins and characterize the convergence towards the thermodynamic limit in terms of what we call a boundary effect function. When the boundary effect function decays exponentially, meaning the system approaches the thermodynamic limit exponentially fast, the ground state obeys the exponential clustering in any spatial dimension, the area law for entanglement entropy in one dimension, and the logarithmically-corrected area law in higher dimensions. In particular, we prove that nondegenerate ground states of gapped frustration-free Hamiltonians have an exponentially decaying boundary effect function and hence exhibit all the aforementioned correlations. A Hamiltonian is called frustration-free when its ground state is also the ground state of every individual term in the Hamiltonian. Frustration-free Hamiltonians are intensively studied as they account for a large class of many-body systems. In particular, they underpin a number of paradigmatic models, such as the AKLT model [17] and the Kitaev model [18], and also the Hamiltonian complexity theory [8]. Furthermore, a gapped local Hamiltonian can be transformed to a frustration-free one [19]. Our result would thus serve as a significant step towards a more general proof of the area law in high dimensions.
Let us first clarify the problem we deal with in this paper. Suppose an N -particle Hamiltonian H N . The problem at hand, in its most general form, would be concerning the ground state |Ψ 0 N of general H N . However, ordinary many-body problems we face are in fact somewhat restricted. Let us recall our empirical wisdom. How do we define a certain many-body system? We usually specify its constituent particles, their mutual interactions, the external potential, etc. In other words, what we specify is rather a rule to construct a Hamiltonian and thus for any N , the corresponding H N is accordingly given. Moreover, N is usually irrelevant as far as it is sufficiently large; because N changes, we do not call it a different matter. In other words, for some large M < N , there exist a sequence of Hamiltonians {H M , H M+1 , ..., H N } that share the same characteristics. In this paper, we take this as our key assumption. For example, when we say a certain system is gapped, we mean that there exist such a sequence of gapped Hamiltonians. This is also related to the existence of a thermodynamic limit. In most systems, when N is large, any local observables become intensive quantities independent of N . Going a bit further, suppose the thermodynamic limit also exists at the zero temperature, i.e., for the ground state. If we take a subregion R A and obtain the reduced density matrix Hereafter, we take this as the definition of the existence of a thermodynamic limit for the ground state. Of course, this definition contains some ambiguities in that there are many different ways to increase the system size. In our discussion, it will be clear from the context.
Let us set out the notation and assumptions in more detail. We consider a system of N finite-dimensional spins on a D-dimensional lattice. The graph distance ℓ G (s, s ′ ) is defined as the number of edges in the shortest path connecting sites s and s ′ . For analytical simplicity, we make two assumptions on the lattice. First, there is a constant a 0 such that ℓ E (s, s ′ ) ≤ a 0 ℓ G (s, s ′ ) for any sites s and s ′ , where ℓ E (s, s ′ ) is the Euclidean distance. Second, one can take a unit volume (δl) D such that the number of sites in a unit volume is bounded by n 0 (δl) D for some constant n 0 . Note that these two properties are very general. For D ≥ 2, the shape of the system may affect the bulk properties (e.g., imagine a fractal shape). To avoid such unnecessary complication, we assume the system has a sufficiently simple shape. For the same reason, we choose the shape of a region in the proof of the area law to be a D-dimensional sphere. Generalization to the case of other simple-shaped regions is straightforward and thus omitted. We denote by B k s the set of sites s ′ such that ℓ G (s, s ′ ) < k and U(B k s ) the set of unitary operators supported on B k s . As we consider only finiterange interactions, the n-body Hamiltonian can be written as H n = n s=1 h s without loss of generality, where h s is supported on B k0 s with k 0 being a constant bounding the range of interaction. H n is assumed to have a nondegenerate ground state |Ψ 0 n for any n. For an operator A, we use both the trace norm A 1 = Tr|A| and the operator norm A ∞ that is the largest eigenvalue of |A|. For a vector |ψ , |ψ denotes the Euclidean norm ψ|ψ . We let J = sup s h s ∞ .
To begin with, recall that the locality as implied by the Lieb-Robinson bound is a dynamical manifestation of local interaction. That being so, what would be an analogous static manifestation? Consider a local Hamiltonian H with a ground state |Ψ 0 and another Hamiltonian differing only locally as s0 , it is reasonable to define for r ≥ r 0 an optimal uni- is a non-increasing function and lim r→∞ µ s0 (r) = 0. Intuitively speaking, a local change to the Hamiltonian is likely to change the ground state only locally. However, this intuition fails to hold because, unlike the case of the Lieb-Robinson bound, there is no particular length or time scale involved to bound the distance up to which information can reach. Nonetheless, this reasoning gives us an important clue. If somehow µ s0 (r) is nonzero for r ≤ r m for some constant r m and has an exponentially decreasing tail outside, we may argue that the ground state possesses some kind of local nature. In this case, we can call U ∞ s0 a quasi-local unitary transformation as it can be approximated by a local unitary transformation.
In the same context, consider a mapping from |Ψ 0 n−1 to |Ψ 0 n . This mapping effectively attaches a spin on the boundary, making a local change to the Hamiltonian. For this, we first need to introduce a Hamiltoniañ H n−1 = H n−1 + S n , where S n is a single-spin operator on site n with a nondegenerate ground state |0 n , so as to match the Hilbert space dimension. The ground state is then |Ψ 0 n−1 = |Ψ 0 n−1 |0 n , which can now be transformed unitarily to |Ψ 0 n . Note that H n −H n−1 is generally supported on B 2k0 n , not on B k0 n , because the boundary terms, if exist, should change around the site n. As above, we can again define the optimal U r n ∈ U(B r n ) minimizing |Ψ 0 n − U r n |Ψ 0 n−1 . As µ n (r) is a nonincreasing function, there exists a non-increasing function µ(r) = sup n µ n (r), which satisfies µ n (r) ≤ µ(r) for any n. For convenience, let us call µ(r) a boundary effect function. When the boundary effect function has an exponentially decreasing tail, we will call the mapping |Ψ 0 M → |Ψ 0 M+1 → · · · → |Ψ 0 N a quasi-local extension of the state in the sense that the size is enlarged by a series of quasi-local unitary transformations.
Before proceeding, it is remarkable that the boundary effect function characterizes the convergence towards the thermodynamic limit. For example, consider a onedimensional chain of N spins and let region R A be the sites from 1 to M . If a thermodynamic limit exists, one can take a Cauchy sequence such that lim N →∞ η A , which easily follows from Uhlmann's theorem and the inequalities between the trace distance and the fidelity [20]. The boundary effect function thus characterizes how fast the system approaches a thermodynamic limit as N is increased. From a different perspective, it also characterizes how far the boundary effect penetrates into the bulk. For example, if µ(r) = 0 for r > r m and N > M +r m , the region R A cannot recognize the existence of a boundary and ρ A N becomes independent of N . Note that the same argument also holds for higher dimensional systems, although in that case the boundary effect function gives only an upper bound on η A N . Our next step is to show that the boundary effect function also characterizes the ground state correlations. Let us first consider a one-dimensional chain of N spins. To see the exponential clustering, we divide the chain into three regions: R A is from site 1 to M − 1, R B is site M , and R C is from site M + 1 to N . We start from a system in which there is no spin in R B and there is no interaction between R A and R C as if they were two independent systems. The ground state of this system can then be written as |Ψ 0 A |Ψ 0 C . We then add a spin in R B to recover our desired system. The final state can then be written as where Q 1 and Q 2 are arbitrary single-spin operators supported on sites M ± r 0 , respective. Our strategy is to approxi- We use two inequalities. First, for any states |ψ and |φ , | ψ|A|ψ − φ|A|φ | = |Tr[(|ψ ψ| − |φ φ|)A]| ≤ |ψ ψ| − |φ φ| 1 A ∞ [21]. Second, |ψ ψ| − |φ φ| 1 ≤ 2 inf θ |ψ − e iθ |φ , which follows from the inequality between the trace distance and the fidelity. From these, we have |f (Q 1 , . The boundary effect function thus bounds the two-point correlation function. Note that if the ground state is quasi-locally extended, it obeys the exponential clustering. As for the area law, we divide the chain differently: R A is from site 1 to M, R B is from M + 1 to M + m, and R C is from M + m + 1 to N , where m > 0 will be chosen later. We are interested in S(ρ A N ). The starting point is the ground state |Ψ 0 AB = |Ψ 0 M+m . Note that Suppose we extend the system by filling up the spins in R C . Let ρ n = |Ψ 0 n Ψ 0 |. From the variant of Fannes' inequality [22], If the boundary effect function decays exponentially as µ(r) ≤ µ 0 e −κr , S(ρ A N ) ≤ m + c 0 me −κm + c 1 e −κm/2 for some constants c 0 and c 1 . By arbitrarily choosing m, S(ρ A N ) is bounded by a constant, i.e., the area law is obeyed.
The same logic applies for D ≥ 2. Here, we take a near-continuum limit for analytical simplicity. To see the exponential clustering, we take the Cartesian coordinates (x 1 , · · · , x D ) and choose the regions such that R B is a (D − 1)-dimensional plane with thickness l defined by l/2 ≤ x 1 ≤ l/2 and R A (R C ) is the region x 1 < l/2 (x 1 > l/2). We again bound the two-point correlation function f (Q 1 , Q 2 ), where Q 1 and Q 2 are, respectively, supported on the lattice sites at r ± = (±r 0 , 0, · · · , 0). Suppose there are M spins in R A + R C . We start from the state |Ψ 0 M = |Ψ 0 A |Ψ 0 C and extend it to |Ψ 0 N = |Ψ 0 ABC . For brevity, let |Ψ 0 n = |Ψ 0 n |0 n+1 · · · |0 N .
, which also decreases exponentially if µ(r) does so. See the Appendix for the proof.
As for the area law, we choose the regions such that R A , R B , and R C are defined byr ≤ r 0 , r 0 <r ≤ r 0 + l, andr > r 0 + l, respectively, wherer is the Euclidean distance from the origin and l ≪ r 0 will be chosen later. Suppose there are M , m, and N − M − m spins in R A , R B , and R C , respectively. We again start from |Ψ 0 AB and extend it by filling up the spins in R C in such a way that the shape of the system is kept almost the same but only the size is gradually increased. In this way, the integral is simplified.
). In the same way as above, The remaining question is when does the boundary effect function decay exponentially? For gapped Hamiltonians, we can obtain an intuitive picture as follows. Consider an extension of |Ψ 0 n−1 to |Ψ 0 n . Suppose the Hamiltonian has a finite spectral gap larger than ∆ between the ground state and the first excited state and furthermore H(λ(t)) = (1−λ)H n−1 +λH n is also gapped for 0 ≤ λ(t) ≤ 1. Let ∆ a be the lower bound of the spectral gap of H(λ). We can then consider an adiabatic passage of the ground state from |Ψ 0 n−1 to |Ψ 0 n . Note that H(λ) = H n + (λ − 1)K n , where K n = H n −H n−1 . As the adiabatic passage can be performed in a finite time of O(1/∆ a ) and K n is local, owing to the Lieb-Robinson bound, the local change can affect the system only locally, causing the boundary effect function to decay exponentially. This idea is formulated in the Appendix.
The above argument is valid only when H(λ) remains gapped during the adiabatic process. In fact, this requirement is mitigated because we are free to add an arbitrary term acting locally around the site n to the Hamiltonian to restrain the gap from being accidentally closed. Given that |Ψ 0 n−1 and |Ψ 0 n are essentially the same kind of states and hence there is no quantum phase transition, only a small portion of the Hamiltonian is changed during the process, and the extra term can be freely adjusted, it is highly likely that such an adiabatic path always exists. It is indeed the case for frustrationfree Hamiltonians. Proving this is our final task. Let µ 0 = µ(r 0 ) < 1 for some constant r 0 . There always exists such r 0 unless µ(r) = 1 for all r, which is unrealistic. Let n † so that |Ψ ′ 0 n−1 = U r0 n |Ψ 0 n−1 . It suffices to prove that |Ψ ′ 0 n−1 can be adiabatically trans- , where I n is the identity operator. This Hamiltonian retains all the essential properties of H(λ) and is readily diagonalizable. It is easy to see that for any state |φ orthogonal to |Ψ ′ 0 n−1 , a 0| φ| H ′ (λ) |φ |0 a > ∆ and for |ψ orthogonal to |Ψ 0 n , a 1| ψ| H ′ (λ) |ψ |1 a > ∆. Consequently, |G 1 = |Ψ ′ 0 n−1 |0 a and |G 2 = |Ψ 0 n |1 a constitute the two lowest energy eigenstates with energies λ∆ and (1 − λ)∆, respectively, while all the other eigenstates have energy larger than ∆. The two levels cross only at λ = 1/2. This degeneracy can be lifted by adding an extra term that couples |G 1 and |G 2 . It thus suffices to add a term proportional to f (λ)I n ⊗ σ X a , where f (λ) is a smooth function with f (0) = f (1) = 0 and f (1/2) = 1, and σ X a = |0 a 1| + |1 a 0|. As | G 1 | I n ⊗ σ X a |G 2 | > 1 − µ 2 0 > 0, the degeneracy should be lifted, making H ′ (λ) gapped.
Summing up, the convergence towards the thermodynamic limit has been characterized by a boundary effect function. In particular, for gapped frustrationfree Hamiltonians with a nondegenerate ground state, the boundary effect function decays exponentially, which implies that the ground state is quasi-locally extended, which again implies that it obeys the exponential clustering and the area law up to a logarithmic correction. We note that the logarithmic correction disappears if the state is extended by strictly local operations, which corresponds to the case wherein the boundary effect function has a cut-off. For example, cluster states are locally extended states [23] and thus obey the area law without correction.
The author thanks M.-J. Hwang, H.-J. Kim, and S.-W. Ji for discussions.
Appendix A: Proof of the exponential clustering for D ≥ 2 when µ(r) ≤ µ0e −κr From the inequality it follows that For r ≫ l,