Violation of area-law scaling for the entanglement entropy in spin 1/2 chains

Entanglement entropy obeys area law scaling for typical physical quantum systems. This may naively be argued to follow from locality of interactions. We show that this is not the case by constructing an explicit simple spin chain Hamiltonian with nearest neighbor interactions that presents an entanglement volume scaling law. This non-translational model is contrived to have couplings that force the accumulation of singlet bonds across the half chain. Our result is complementary to the known relation between non-translational invariant, nearest neighbor interacting Hamiltonians and QMA complete problems.


I. INTRODUCTION
Ground states of relevant physical Hamiltonians carry quantum correlations which decrease with distance. For instance, a two-point correlation function is expected to fall off exponentially with the separation of points in the presence of a mass gap or algebraically at critical points. This amount of quantum correlations is in correspondence with an area law scaling of entanglement entropy. To be precise, the entanglement entropy, defined by where ρ A is the density matrix of the region of space A considered, scales as the boundary of A.
In contradistinction to the above situation, random quantum states are known to carry volume law entropy [1]. Therefore, typical Hamiltonians produce grounds states which are not generic. Indeed, relevant physics corresponds to a small corner of the total Hilbert space of a quantum system. This observation is crucial to understand recent efforts to simulate quantum states with tensor networks [2][3][4][5][6][7][8]. Such approximations are able to accommodate area law scaling for the entropy.
It is then important to understand precisely what are the properties that a Hamiltonian must obey so as to produce a ground state which only displays area law entanglement. A first heuristic approach suggests that entanglement decreases at large distances because interactions are local. That is, a local degree of freedom interacts with its neighbor and gets entangled with it. This second degree of freedom interacts in turn with a further one. This sequence of interactions would eventually entangle far separated degrees of freedom, though the strength of interactions would only manage to get the standard correlations we find in Nature. On the other hand, it is unclear whether interactions could be contrived to achieved larger entangled states. The role of translational invariance is then critical.
Some results for one-dimensional systems are wellestablished. In one dimension, if a system obeys local interactions and it is gapped, area-law always emerges [9]. On the other hand, if the system is at a critical point, and therefore gapless, a logarithmic divergence is encountered. This logarithmic scaling of the entanglement entropy is explained by conformal field theory [10][11][12][13][14].
In Refs. [15,16], infinite translational invariant Fermionic systems of any spatial dimension with arbitrary interactions are considered. For such systems, it is shown that the entropy of a finite region typically scales with the area of the boundary times a logarithmic correction.
Although there has recently been further progress on this topic in higher dimensions [17][18][19], the necessary and sufficient conditions for an area-law have not been defined yet. An explicit example of a system where area-law is violated is presented in Ref. [20]. It is shown that a one dimensional non-translational invariant system composed of 12-level local quantum particles with nearest neighbor interactions presents a ground state that carries a volume law scaling of entanglement. In particular, it is proven that the problem of approximating the ground state energy of such system is QMA-complete. This precise example shows that a quantum computer could not simulate any one dimensional system, and, moreover, that there exist one-dimensional systems which take an exponential time to relax to their ground states at any temperature, making them candidates for being one-dimensional spin glasses.
The issue addressed in this work is to study how simple can be a quantum system to give a highly entangled ground state. In particular, we show that a simple spin 1/2 model with nearest neighbors interactions with a suitable fine tunning of its coupling constants can have a ground state with a volume law scaling for the entanglement entropy. Our proposal is based on the translational symmetry breaking, hence, this makes that the area-law violation can not be maintained for any bipartition of the system. Nevertheless, it will be shown that the average of the entanglement entropy over all the possible positions of the block fulfills a volume-law. Our results are presented in the following way. We first review the real space renormalization group technique which brings the fundamental intuition on how to build an area law violating Hamiltonian. We then turn to solve the proposed Hamiltonian using its exact diagonalization, where the final step can be taken both in perturbation theory or numerically. We also illustrate the real space renormalization idea in an Appendix.

II. REAL SPACE RENORMALIZATION GROUP
A. Introduction to real space Renormalization Group approach Real-space Renormalization Group (RG) approach was introduced in Ref. [21] generalizing the works presented in Ref. [22]. It is a method suited for finding the effective low energy Hamiltonian and the ground state of random spin chains. The couplings have to satisfy the hypothesis of strong disorder, i. e. the logarithm of its probability distribution is wide. Under such conditions, the ground state of the system can be very well approximated by a product state of singlets whose spins are arbitrarily distant.
Let us review the real-space RG method for the inhomogeneous XX model case First, we find the strongest bond J i ≫ J i+1 , J i−1 and diagonalize it independently of the rest of the chain. According to the previous Hamiltonian, this leads to a singlet between spins i and i + 1 (see appendix A). Therefore, the ground state at zeroth order in perturbation theory respect the couplings J i−1 and J i+1 is where ) is a singlet state between the spins i and i + 1, and |ψ x<i and |ψ x>i correspond to the states of the rest of the system.
In order to compute corrections to the ground state at higher orders, we use perturbation theory as it is shown in appendix A. This leads to an effective interaction between the distant spins i − 1 and i + 2 with an effective couplingJ In summary, real space RG integrates out two spins and reduces the Hamiltonian energy scale. Notice that this new effective low energy Hamiltonian couples the spins i − 1 and i + 2, therefore, it has non-local interactions as seen from the original Hamiltonian. Iterating this procedure for a XX model with random couplings, it is seen that the ground state can be described by a random singlet phase, i. e. each spin forms a singlet pair with another one (see Fig. 1a). Most pairs involve nearby spins, but some of them produce long distance correlations. Figure 1: Diagram of a random singlet phase (a) and the concentric singlet phase (b). Each spin forms a singlet pair with another spin indicated by the bond lines.
In Ref. [23], real-space RG was used to show that, for random spin chains where the ground state is a random singlet phase, the entanglement entropy scales logarithmically at the critical point as in the homogeneous case. That is, wherec = c ln 2 is an effective central charge proportional the central charge for the same model but without disorder c. This analytical result has later been check numerically in Refs. [24][25][26].

B. Area-law violation for the entanglement entropy
Let us now tune the couplings J i of our XX model in such a way that the entanglement entropy of the ground state of the system scales with the volume of the block of spins. An easy way of achieving this is to generate a ground state with a concentric singlet phase as it is shown in Fig. 1b. We see that the system is in a product state of distant singlets between the positions N/2 − (i − 1) and N/2 + i for 1 ≤ i ≤ N/2. It is trivial to see that the entanglement entropy of this configuration would scale with the size of the block, since it merely corresponds to the number of bonds cut by the bipartition (see Fig. 2a).
Let us note that the entanglement entropy for concentric blocks would be 0 as it is shown in Fig. 2b. As translational invariance of the system is broken, the entanglement entropy of a block not only depends on the size of it but also in its position.
In order to measure how entangled is a state for nontranslationally invariant systems, it is useful to introduce the average entanglement entropy over all the possible positions of the block, that is where S L (i) is the entanglement entropy of the block of size L from the i-th spin to the (i + L)-th one. According to the previous definition, the average en- Figure 2: Diagram of the entanglement entropy scaling for the concentric singlet phase. The entanglement entropy grows maximally if we take blocks at one extreme (a) and is zero if the blocks are centered at the middle of the chain. This is an explicit example that in the non-translationally invariant systems the entanglement entropy depends on the position of the block.
tanglement entropy of the concentric singlet phase reads Although for the concentric singlet phase the average entropy losses its linear behavior for large blocks, L ∼ 1 − 1 √ 3 N , it always fulfills the conditionS L ≥ 1 2 L. Thus, the concentric singlet phase represents a simple and explicit example of area-law violation of scaling of the entanglement.
The aim of our work is to tune the coupling constants J i of the XX model, such that, the concentric singlet phase becomes the ground state of the system, and, in this way, to obtain an explicit example of a Hamiltonian with nearest neighbor interactions of spins that violate the area-law scaling of entanglement.
Due to the symmetry of the state that we pretend to generate, let us consider a XX chain of N spins where the central coupling between spins N/2 and N/2 + 1 is J 0 and the rest of them are chosen as follows where 1 ≤ i ≤ N 2 − 1 and the coupling J N/2±i connects the spins N/2 ± i and N/2 ± i + 1.
We are going to use real space renormalization group ideas in order to see at which values we have to tune the coupling constants, such that, the concentric singlet phase becomes the ground state of the system. If J 0 ≫ J 1 , in the low-energy limit, an effective interaction between the spins N/2−1 and N/2+2 appears. We label this effective coupling asJ 1 and, according to Eq. (4), it Then, ifJ 1 ≫ J 2 , the effective low-energy Hamiltonian will have an effective bond between the spins N/2−2 and N/2 + 3. We would like to proceed in this way in order to generate iteratively the concentric singlet phase.
we expect that the ground state of the system is the concentric singlet phase. Specifically, if we impose that J i = ǫJ i−1 for any i, such that it is always possible to apply Eq. (10), we see that the couplings J i must decay very rapidly In general, we are going to study chains with couplings that decay where α(i) is a function that is monotonically increasing. If α(i) ∼ i 2 , we would have a Gaussian decaying. Next, we want to solve the XX model with the coupling constants defined in Eq. (12), and study how the entanglement entropy scales depending on the kind of decay law for the couplings.

III. SOLUTION OF A SPIN MODEL AND ITS ENTANGLEMENT ENTROPY
Let us consider a finite spin chain with nearest neighbor couplings J x i , J y i and an arbitrary transverse magnetic field λ i in each spin. This system is described by the Hamiltonian: where L is the size of the system and σ x,z i are Paulimatrices at site i. The XX model presented before is a particular case of this Hamiltonian (13) for J x i = J y i and λ i = 0 ∀i.

A. Jordan-Wigner transformation
The essential technique in the solution of H is the well-known mapping to spinless fermions by means of the Jordan-Wigner transformation. First, we express the spin operators σ x,y,z i in terms of fermion creation (anni- Doing this, H can be rewritten in a quadratic form in fermion operators where the matrices A and B are defined by with 1 ≤ i, j ≤ N .

B. Bogoliubov transformation
In a second step, the Hamiltonian is diagonalized using a Bogoliubov transformation where the Φ k and Ψ k are real and normalized vectors: The fermionic excitation energies, Λ k , and the components of the vectors, Φ k and Ψ k , are obtained from the solution of the following equations: It is easy to transform them into an eigenvalue problem, from where Λ k , Φ k and Ψ k can be determined.

C. Ground State
In Eqs. (19) and (18), we realize that transforming Φ k into −Φ k (or Ψ k into −Ψ k ), Λ k is changed to −Λ k . This allows us to restrict ourselves to the sector corresponding to Λ k ≥ 0, k = 1, 2, . . . , N . Thus, considering Eq. (17) and the fact that all Λ k are positive, the ground state is a state |GS which verifies, In practice, what we do to restrict ourselves to the sector of positive Λ k is to determine Φ k and Λ k by solving Eq. (20), and calculate

D. Computation of Von Neumann entropy
Following Refs. [11,27,28], the reduced density matrix ρ L = tr N −L |GS GS| of the ground state of a block of L sites in a system of free fermions can be written as where κ is a normalization constant and H a free fermion Hamiltonian. Let us very briefly justify why the density matrix must have this structure. First, notice that the Hamiltonian defined by Eq. (14) has Slater determinants as eigenstates. Thus, according to Wick theorem, any correlation function of the ground state (or any other eigenstate) can be expressed in terms of correlators of couples of creation and annihilation operators. For instance, If all these indices belong to a subsystem of L sites, the reduced density matrix ρ L must reproduce the expectation values of the correlation functions, i. e.
This is only possible if ρ L is the exponential of an operator H which also contains creation and annihilation processes, i. e.
We can diagonalize this Hamiltonian H by means of another Bogoliubov transformation where v k (i) and u k (i) are real and normalized. Then, the Hamiltonian reads where ξ † k and ξ k are the creation and annihilation operators of some fermionic modes. In terms of these modes, the density matrix ρ L is uncorrelated and can simply be expressed as In the previous equation, the new parameters ν k have been introduced in order to ensure the normalization of ρ k , tr (ρ k ) = 1. This way of expressingρ k will be useful next. Thus, the entanglement entropy of the density matrix ρ L is merely the sum of binary entropies is the binary Shannon entropy. In order to determine the spectrum ofρ k , let us consider the correlation matrix, Notice that the matrix G can be computed using the Φ k and Ψ k vectors, where the correlations η † k η q = δ kq and η k η q = 0 have been considered.
In the subspace of L spins, G is completely determined by the reduced density matrix. To avoid any confusion, let us define T ≡ G(1 : L, 1 : L) as the L × L upper-left sub-matrix of the correlation matrix G. Then, T can be expressed in terms of the expected values ξ † k ξ q , where the i and j indices run from 1 to L. This equation leads to the relations, that can be translated to the eigenvalue problem Once the ν q variables are computed, we can determine the entanglement entropy by means of Eq. (31).

E. Summary of the calculation
To sum up, let us enumerate the steps that we have to follow in order to calculate the entanglement entropy of a block L.

IV. EXPANSION OF THE ENTANGLEMENT ENTROPY
We would like to tune the coupling constants of the Hamiltonian (13), such that the scaling of the entanglement entropy of its ground state violates the area law. The entanglement entropy only depends on the variables ν k . Then, we can separate the Shannon entropy of the probabilities 1±ν k where , is a positive function. Thus, the entanglement entropy reads Notice that the scaling of the entropy only depends on the sum L k=1 h(ν k ). More concretely, we can define the parameter, that describes the asymptotic behavior of the scaling of the entropy for large blocks: • β = 0: maximal entanglement, • β < 1: volume-law, • β = 1: sub-volume-law.
Let us focus on the case β ∼ 0. Let us analyze if it is possible to design a spin chain with nearest neighbor interactions whose ground state is maximally entangled. First, we realize that β is strictly zero if and only if all the variables ν k = 0. Thus, if we want to consider small deviations of the maximally entangled case, we can assume that ν k ∼ 0 and expand β in series of ν k , Considering Eq. (38), we can express β in terms of the matrix-elements of T , Let us notice that to fulfill this condition requires that the average of the matrix-elements of T tend to zero for large L, If we assume a smooth behavior for the matrix-elements of T , according to Eq. (43), they must decay faster than the inverse square root function, such that β = 0.
In conclusion, in order that the entanglement entropy scales close to the maximal way, the matrix-elements of T matrix have to be very close (or decay rapidly) to zero. If this is the case, the entanglement entropy can be simplified to where ||T || F is the Frobenius norm of T , defined by ||T || F = tr (T T T ).
Let us now study if it is possible to tune the coupling constants of a spin chain in order that ||T || F is strictly (or close to) zero. The possibility of having a null T is discarded because it cannot be achieved with nearest neighbor interactions models. Despite this, there is a wide freedom to tune the coupling constants such that the matrix-elements of T fulfill condition (43). This arbitrariness makes very difficult to specify the shape of the distribution of coupling constants in order that area-law is violated. With this aim, we can exploit the idea of real space Renormalization Group presented before.

V. NUMERICAL RESULTS
We can follow the steps described in Sec. III E in order to calculate the entanglement entropy of the XX chain  This XX model is characterized by having the strongest bond in the middle of the chain, J 0 , while the value of the rest of bonds J n decrease rapidly with the distance n to the central one. In particular, we have studied two different kinds of decay for the coupling constants J n : (i) Gaussian decay, J n = e −n 2 , and (ii) exponential decay J n = e −n . Let us notice that due to the rapidly decaying of the coupling constants and the finite precision of the computer, we can only consider small systems.
In Fig. 4(a), the entanglement entropy is plotted for the Gaussian case. As we expected scales linearly with the size of the block L with a slope practically equal to one. Notice that although the slope is 1 for large blocks, the entanglement is not the maximal due to the nonlinear behavior of the entropy for the smaller ones. This can be better understood analyzing if the approximation of the previous section given by Eq. (46) is fulfilled. With this aim, the square Frobenius norm ||T || 2 F and the sum S(L)+||T || 2 F are also plotted. In fact, we observe that the sum S(L) + ||T || 2 F coincides with the maximal entropy, as Eq. (46) suggests.
The same plot can be realized for an exponential decay of the coupling constants, see Fig. 4(b). In this case, although the entanglement entropy also scales linearly, its slope is less than one. Thus, we observe a volume law, but the entropy is not maximal. Therefore, Eq. (46) is not fulfilled in this case. We can, actually, see that the Frobenius norm ||T || F increases linearly with L instead of saturating to a small value.
We have repeated the same computations for the same kind of decays but other basis. The same behaviors for the Gaussian and the exponential cases have been obtained. For the Gaussian case, a faster decay implies a saturation to a smaller value for ||T || F , that is, a closer situation to the maximal entropy. For the exponential case, ||T || F continues increasing linearly but with a smaller slope.

VI. CONCLUSIONS
We have constructed a one dimensional system composed by spin-1 2 particles with nearest neighbor interactions with a entanglement entropy of the ground state that scales with the volume of the size of the block.
This results further confirms that violations of area law scaling for entanglement entropy are possible for local interacting Hamiltonians. Furthermore, such a behavior is found possible for spin 1/2 degrees of freedom. The price to be paid for violating area law scaling is to break the translational symmetry of the system. Indeed, in Refs. [29,30], it is shown that, although translationinvariant one dimensional states give rise to arbitrary fast sublinear entropy growth, they cannot support a linear scaling.
Let us also recall that two-local Hamiltonian problems have been proven to provide QMA-complete problems.
To be more precise, the problem of finding out whether the ground state energy of a two-local Hamiltonian is larger than a or smaller than b, where |a − b| > O(1/n) is QMA-complete. We may further argue that an efficient classical simulation of such a problem is likely to be impossible, otherwise we could solve any NP-complete problem by just simulating Quantum Mechanics on a classical computer. The obstruction to obtain faithful simulations of quantum mechanical systems is in turn related to the amount of entanglement found in the system. Thus, exponentially large entanglement should be found in some one-dimensional quantum systems. Our results is somehow completing this idea. Even spin 1/2 chains can pro-duce highly entangled states if couplings are adequately tuned.
At zero order in λ we recover the trivial non-perturbed Schrödinger equation. If we collect terms of order λ, we get In order to calculate the first correction to the energy, we project the previous equation (A7) to the degenerate ground state subspace where V m,m ′ ≡ m|V |m ′ is the projection of the interaction to this subspace. In our particular case, the matrixelements V m,m ′ = 0 for all m and m ′ , hence, E