Correlations, spectral gap and entanglement in harmonic quantum systems on generic lattices

We investigate the relationship between the gap between the energy of the ground state and the first excited state and the decay of correlation functions in harmonic lattice systems. We prove that in gapped systems, the exponential decay of correlations follows for both the ground state and thermal states. Considering the converse direction, we show that an energy gap can follow from algebraic decay and always does for exponential decay. The underlying lattices are described as general graphs of not necessarily integer dimension, including translationally invariant instances of cubic lattices as special cases. Any local quadratic couplings in position and momentum coordinates are allowed for, leading to quasi-free (Gaussian) ground states. We make use of methods of deriving bounds to matrix functions of banded matrices corresponding to local interactions on general graphs. Finally, we give an explicit entanglement-area relationship in terms of the energy gap for arbitrary, not necessarily contiguous regions on lattices characterized by general graphs.


Introduction
On physical grounds, one expects that correlations in ground states of gapped many-body systems-quantum systems for which the energy of the lowest excitation is strictly larger than the ground state energy-decay exponentially. This implication of the energy gap to the exponential decay of two-point equal-time correlation functions is a well-established observation on noncritical quantum many-body systems [1]- [9]. Yet, surprisingly perhaps, rigorous proofs of this implication for spin models in higher dimensional cubic or general lattices have been found only very recently. Notably, [6] reexamines this question for finite-dimensional constituents, which has been generalized to the case of general lattices defined by graphs that may have any non-integer dimension in [4,7].
In this paper, we rigorously reconsider this question for a class of systems on general lattices the constituents of which are infinite-dimensional quantum systems: the quasi-free case of harmonic systems on general lattices. This harmonic case is particularly transparent, as the entire discussion of state properties can be done in terms of the second moments of the states. This case has notably taken a central role in recent discussions of questions of entanglement scaling in many-body systems, as a class of natural physical systems where sophisticated questions on the scaling of entanglement-in form of, e.g., entanglement-area relationships [10]- [12] and other issues [13]- [17]-are relatively accessible.
The general lattices are not assumed to be necessarily translationally invariant, and the underlying graphs may have any spatial dimension. Harmonic systems, needless to say, have been considered many times before, in particular the case of nearest-neighbour interaction [18]. Here, we investigate in a very comprehensive manner the implications of a gap to the twopoint correlation functions and the converse direction for all harmonic lattice systems with local Hamiltonians reflecting general couplings in position as well as in momentum, both for the ground 3 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT state and Gibbs states (nonzero temperature). Notably, these findings enable us to formulate an entanglement-area law for regions of arbitrary shape within the considered context of harmonic systems on general graphs. Such harmonic systems model discrete versions of Klein-Gordon fields, vibrational modes of crystal lattices or of ions in a trap, or serve as approximations to not strictly harmonic systems. In contrast to, for example [4], we will not rely on using variants of Lieb-Robinson [19] bounds on the group velocity to prove the validity of our bounds. Instead we will make use of generalizations of the methods introduced in [20] to the case of general lattices.
• We briefly introduce the notion of quasi-free states as ground or thermal states of harmonic Hamiltonians on generic lattices described by graphs, with coupling in position and momentum coordinates. • For local harmonic Hamiltonians on generic lattices we prove that whenever the system is gapped, exponential decay of the correlations in the ground state with the canonical coordinates follows (theorem 1). • Conversely, for systems (not necessarily locally) coupled in position, we demonstrate that for sufficiently fast decay of the correlations, the existence of a spectral gap can be deduced. This includes sufficiently strongly algebraically decaying correlations (theorems 2 and 3). • These findings give rise to a proven equivalence of exponentially decaying couplings and the existence of a spectral gap for systems coupled in position on generic lattices (corollary 1). • We prove that for gapped harmonic lattice systems on generic lattices in Gibbs states (thermal states), exponential decay of the correlations follows (theorem 4). • We give a connection [11,12,21] to entanglement properties in many-body systems, in this instance of harmonic lattice systems. We show that similarly to [12], the entropy of a subsystem of a gapped lattice system is bounded from above by a quantity linear in the boundary area of the distinguished subsystem, which may be of arbitrary shape (theorem 5). This generalizes the area theorem of [11,12] to systems on general lattices. • We finally discuss several examples in detail, such as instances of disordered systems or thermal states of rotating wave Hamiltonians. We also investigate a case of non-local algebraically decaying interactions. Note that simultaneously, similar findings were published in [22], where in contrast to this paper, emphasis was put on ground state properties of translationally invariant harmonic systems on cubic lattices, but where in turn, non-local interactions and critical cases were considered in much more detail.

Harmonic systems on general lattices
We consider quadratic bosonic Hamiltonians on generic lattices. This means that the lattice is characterized by general graphs [23]. The set L of vertices with cardinality |L| is associated with the set of sites in the lattice, each of which corresponding to a bosonic degree of freedom. The simple graph G = (L, E) is identified with the lattice, where E is the edge set or, equivalently, the adjacency matrix characterizing a neighbourhood relation between physical systems. In a cubic lattice, say, G will represent just this general lattice, regardless of the interaction, which may be finite-ranged beyond nearest or next-to-nearest neighbour interactions. We will consider paths in the sense of standard graph theory, as sequence of vertices connecting a start and an end vertex, each of which are connected by an edge. For two vertices i, j ∈ L, the integer dist(i, j) denotes the graph-theoretical distance, so the length of the shortest path connecting i and j [23], with respect to the edge set E.
(i) The Hamiltonian is assumed to be of the form where V x , V p ∈ R |L|×|L| are positive matrices, withx T = (x 1 , . . . ,x |L| ) andp T = (p 1 , . . . ,p |L| ) being the canonical coordinates satisfying the canonical commutation relations (CCRs). Hence, we allow for any general coupling in position and momentum coordinates (but no coupling involving both simultaneously). We write (V x,p ) i,j , Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT i, j = 1, . . . , |L|, to label the element of the matrices V x,p that belongs to the coupling between two vertices i, j, respectively. For simplicity of notation, these couplings in position and momentum can be subsumed into the tuple C = (G, V x , V p ). Whenever two sites i, j ∈ L are coupled in position, the element (V x ) i,j will be nonzero, and similarly for the momentum coordinates. The range of the interaction is taken with respect to the metric dist(·, ·), such that V x and V p inherit a neighbourhood relation.
Note that V x and V p can in turn be conceived as adjacency matrices of a weighted graph with the same vertex set L, but a different edge set. The above description is similar to the assessment of the correlation function for generic spin systems in [4,7]. Note that such bosonic harmonic lattice systems resemble to some extend the concept of graph states for spin or qubit systems in the sense of [24,25], compare also [12,14,16]. The dimension of the underlying lattice G can take any positive value in the following sense: we may define a sphere S r (i) for some i ∈ L, centred at site i with integer radius r as Then there exists a smallest d > 0 of the lattice, notably not necessarily integer [7], such that for all r ∈ N, for some c > 0. This number d is taken as the dimension of the lattice. Note that for cubic lattices this dimension coincides with the natural underlying spatial dimension, and the above expression is the same for open and periodic boundary conditions. Similarly, we can define a ball centred at site i ∈ L with radius r as of a ball with radius r in the graph theoretical sense. For a consideration of the dimension of general graphs and the discussion of self-similarity in this context, see, e.g., [26]. For our purposes, the dimension will only enter the bounds through the volume of a ball of some radius. Concerning the locality, we assume the following: (ii) most Hamiltonians that we consider are local, corresponding to a finite-ranged interaction, if not otherwise specified. This means that there exists an m ∈ N, such that for all i ∈ L for all j ∈ L for which dist(i, j) > m/2. 6 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Note that this in turn means that the Hamiltonian is of the form as in equation (1) and can be written asĤ X ⊂ L being subsets the diameter of which satisfy It will turn out to be convenient to collect the |L| conjugate pairs of canonical coordinates in a vectorr T = (x 1 , . . . ,x |L| ,p 1 , . . . ,p |L| ), the entries of which satisfying the CCR, giving rise to a symplectic scalar product [27]. The Hamiltonian can now be brought into diagonal form by means of a symplectic transformation, so by means of linear transformations S ∈ Sp(2|L|, R) preserving the symplectic form [27], defined by the skew-symmetric symplectic matrix and withr = Sr , the Hamiltonian finally takes the form The ground state energy is thus given by The energy of the first excited state is given by where we denote by λ min (A) the minimum eigenvalue of a matrix A. The corresponding ground state of such a Hamiltonian is a quasi-free (Gaussian) state [27], completely characterized by the first and second moments. Quasi-free bosonic or Gaussian state means that the characteristic 7 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT function-expectation value of the Weyl displacement operator-is a Gaussian function in state space. Equivalently, a state is Gaussian if its Wigner function is a Gaussian function in state space. The second moments can be embodied in the covariance matrix with respect to the ground state Here, first moments vanish as the ground state is the vacuum. We find and in the original coordinates we have This expression simplifies significantly in case of commuting V x and V p . Often, V p = 1 1, in which case see [10]. The language chosen here is the one used also in the assessment of entanglement properties of harmonic chains [10,13] or more general harmonic systems on lattices [11,12,14,15,27,28], mildly generalized to coupling in both position and momentum and to general lattices.

Exponential decay of correlations in ground states
The first result links the spectral gap of the Hamiltonian to the exponentially decaying correlation functions of the system. Note that the bound on the right-hand side depends only on the dimension of the underlying graph, the gap, the parameter characterizing the range of the interactions, and bounds to the coupling strength.

Theorem 1 (exponentially decaying correlation functions). Consider a Hamiltonian on a general lattice of dimension d with a coupling
where where · denotes the operator norm and E = 2λ is the energy difference between the first excited state and the ground state.
Note that if the energy is expressed in units of V x V p , then the correlation length depends only on the gap and the range of the interactions.

Spectral gap from algebraically decaying correlation functions
In this subsection, we consider an instance of the converse direction, compare [2]: we assume that the system has algebraically decaying correlation functions in the position coordinates and assume that we have a-not necessarily local-coupling in these coordinates and V p = 1 1, then we can conclude that the system must be gapped if the decay is sufficiently strong. This is made more rigorous in the subsequent theorem. (3). If the ground states satisfy

Theorem 2 (spectral gap from algebraic decay). Consider a sequence of couplings
for all i, j ∈ L (n) and for all n, then the energy difference between the first excited and the ground state satisfies where ζ is the Riemann zeta function.
Note that, if in addition to the assumptions of the above theorem the coupling V (n) is (i) of finite range m for all n, and (ii) the momentum correlations decay as in equations (24) or alternatively, (iii) the coupling has a finite coupling strength, sup V (n) < ∞, 4 it immediately follows from theorem 1 that in fact the correlations decay exponentially. This is an interesting observation, showing that in a sense the class of harmonic lattice systems is not rich enough to-roughly speaking-show all 'types of decays' of the correlations.

Equivalence of spectral gap and exponentially decaying correlation functions
For exponentially decaying correlations we can always conclude that the system is gapped. This statement can be formulated as follows.
Theorem 3 (spectral gap from exponential decay). Let K 0 and ξ 0, and consider a sequence of couplings for all i, j ∈ L (n) and all n, then the energy difference between the first excited and the ground state satisfies where Li 1−d is the polylogarithm of degree 1 − d.
Together with theorem 1 this establishes for locally coupled systems the following equivalence.

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Corollary 1 (equivalence of spectral gap and exponentially decaying correlations).
Consider the sequence of couplings C (n) = (G (n) , V (n) , 1 Then the following statements are equivalent.
1. There exist constants K, ξ > 0 such that both correlations with respect to the ground state satisfy for all i, j ∈ L (n) and all n.

The energy difference between the first excited state and the ground state satisfies
inf E (n) =: E > 0 (29) and the coupling is of finite strength, sup V (n) < ∞.
This result establishes that in local harmonic systems on generic lattices, being non-critical in the sense of being gapped is equivalent with finding exponentially decaying correlation functions.

Exponential decay in Gibbs states
In this subsection, we consider thermal Gibbs states [27] corresponding to some temperature T > 0, Such states are again quasi-free (Gaussian) states and can thus also be uniquely characterized by their covariance matrices, appropriately modified for this case of nonzero temperature. For a study of two-point correlations in Fermi-systems at nonzero temperature see also [5]. In the diagonal basis (13), it takes the form and in original coordinates we have In the following, we will now assume that matrices V x and V p commute. This yields a simplified covariance matrix Furthermore, we require the following assumption on the lattice. This assumption is very similar to the ones in [4], and is satisfied for a large class of natural lattice systems.
Assumption 1 (Lattice structure). Consider a Hamiltonian on a general lattice G = (L, E) with couplings C = (G, V x , V p ) as in theorem 1. Then, it is assumed that there exist constants l 0 0 and ν > 0 such that for all i, j ∈ L and for For example, for a cubic lattice in d dimensions, we have that L = [1, . . . , n] ×d , and dist(i, j) = d δ=1 |i δ − j δ |. Therefore, we arrive at i.e., in this cubic case, we have ν = µ/2. Under the previous assumption, one can arrive at the subsequent statement on two-point correlations in Gibbs states. Note that it is not merely trivially true that the correlations are shorter-ranged in thermal states: there are examples of Hamiltonians where Gibbs states at higher temperatures have longer-ranged correlations than at zero temperature, see example 2.
Theorem 4 (correlations in systems at finite temperature). Consider a finite-ranged Hamiltonian corresponding to C = (G, V x , V p ), [V x , V p ] = 0 , on a general lattice G = (L, E) with assumptions as in theorem 1 and equipped with assumption 1. Then, for dist(i, j) m, the Gibbs state with respect to some temperature T satisfies Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT This theorem ends the list of main statements on the relation between the system being gapped and the decay of the two-point correlation functions with respect to the ground state and thermal states. In the next section we will study the implications of the above findings on the entanglement scaling of distinguished regions of general lattices

Entanglement scaling in ground states of harmonic systems on general lattices
The above statements on the decay of correlation functions have immediate implications on the scaling of entanglement. More specifically, in a lattice system we may distinguish a certain part I ⊂ L and ask for the degree of entanglement between the degrees of freedom of this region and the rest of the lattice. This question is that of the scaling of the geometric entropy of this region. This question goes back to seminal numerical investigations in [29,30] (see also [31]), suggesting a linear relationship between the geometric entropy and the boundary area of a distinguished region. In turn, in [11,12] a rigorous relationship between the boundary area and the entropy of the region in cubic harmonic lattice systems has first been analytically established.
In fact, in the light of the above findings, one may infer the validity of such an 'area theorem' for general lattices, not only for cubic lattices, exploiting exactly the same methods of proof. Defining the surface area of a distinguished region I of the whole lattice as see figure 2, one arrives at a bound to the von-Neumann entropy, of the reduced ground state, I = tr L\I [ ], with respect to the distinguished region.

Theorem 5 (entanglement-area law for general lattices). Consider a general lattice G = (L, E) of dimension d equipped with a coupling (G, V, 1 1). If the coupling is of finite range m, the entropy of entanglement satisfies
where Li 1−2d is the polylogarithm of degree 1 − 2d, E the energy gap above the ground state, s(I) the surface area of I, c as defined in equation (3), and In this form, the contribution of a geometrical factor, as well as one originating from the correlation length, is very transparent. Note that, using theorem 4 and the bounds derived in [12], one arrives at a similar bound for the distillable entanglement of thermal states. Indeed, this theorem covers the entanglement-area relationship in all generality for gapped harmonic models on generic lattices. In particular, for cubic regions with volume L d in d dimensions this means that the geometric entropy is bounded by expressions linear in L d−1 . For critical fermionic quasi-free systems, in turn, there are instances where one finds a behaviour of L d−1 log L for the geometric entropy [32,33]. In contrast, there is numerical evidence that for bosonic harmonic systems, even in the critical case, the validity of the entanglement-area 12 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT relationship of L d−1 is preserved [34,35]. To strictly prove or refute this relationship in this critical case constitutes one of the intriguing open problems in the field.

Proofs
This section will contain the proofs of the statements made before. We will make extended use of the bandedness of the interaction matrix in the sense of the metric dist(·, ·). The main ingredient will be polynomial approximations to matrix functions of matrices. We generalize a statement of [20] to the case of general lattices. This extends the generalization put forth in [12] for general cubic lattices. Note that for statements of the type following, random walk methods would similarly be suitable. In particular, for the case of nearest-neighbour interaction, this analysis has been done, see, e.g., [18] for a comprehensive introduction.
In the following, we will need the subsequent lemma which is concerned with the range of matrices in the sense of dist(·, ·). Note that the matrix power is taken in the sense of the ordinary matrix power.

Lemma 1 (range of matrix powers). Let the |L| × |L| matrix A be a coupling matrix on a general lattice G = (L, E). If
Proof. This can be proven by induction over n. For n = 1 we have A n = A is of range nm. Now suppose A n has range nm, i.e., we have (A n ) i,j = 0 for dist(i, j) > nm/2. Now, Suppose dist(i, j) > m(n + 1)/2. It follows that for dist(i, k) > nm/2 we have (A n ) i,k = 0 as A n is of range nm. For dist(i, k) nm/2 we have i.e., dist(k, j) > m/2, i.e., A k,j = 0. In turn, for dist(k, j) m/2 we have i.e., dist(i, k) > nm/2, i.e., (A n ) i,k = 0 as A n is of range nm. Finally, we hence arrive at (A (n+1) ) i,j = 0 for dist(i, j) > m(n + 1)/2, which concludes the proof.

Proof of theorem 1
From the covariance matrix γ, equation (19), we have

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We can now compute the matrix (V 1/2 x V p V 1/2 x ) −1/2 and similarly (V 1/2 p V x V 1/2 p ) −1/2 using the power-series expansion of the square root, which is valid for |x| < 1, where the coefficients a k are bounded by a k 1. Now, i.e., and similarly Note that clearly, (V 1/2 x V p V 1/2 x ) −1/2 is a positive matrix, whereas W p,x is in general not a symmetric matrix (and the matrix power is the k-fold concatenation of matrix multiplication). Thus, the correlation functions take the form Assuming V x and V p to be of range m/2, i.e., (V x ) i,j = (V p ) i,j = 0 for dist(i, j) > m/2, we have that V x V p and V p V x are of range m. Note that for a matrix to be of finite range in the sense of the above lemma it is not required that it is symmetric. Hence, we can conclude that (V x V p ) k and (V p V x ) k are of range km. Thus, For the diagonal terms we have |(W p,x ) i,i | W p,x = V x V p 1/2 /( E/2) and for the offdiagonal terms we find The same bound holds for the entries of W x,p , i.e., for all i, j ∈ L we have For the correlation function x ixj this yields Now dist(i, j) dist(i, k) + dist(k, j) m/2 + dist(k, j), and thus we find for dist(i, j) m For | p ipj | one can argue in the same manner. Noting that the sum on the right-hand side is just |B m/2 (i)|, we finally have for dist(i, j) m which concludes the proof of theorem 1.

Proof of theorem 2
For each n, the energy gap is given by The operator norm is bounded from above by the norm corresponding to the maximum sum of elements of any row of the matrix, and hence we may write and hence and therefore where ζ is the Riemann zeta function, and c (n) > 0 is defined as in equation (3). Thus, we can conclude that independent of n.

Proof of theorem 3
For each n, the energy gap is given by equation (67). Analogous to the previous proof, we have that where Li 1−d is the polylogarithm of degree 1 − d. As the above bound holds for all n, we can conclude that

Proof of theorem 4
In the finite-temperature case, the covariance matrix γ(T ) is given by We will now proceed by demonstrating the exponential decay of the entries of the matrices G and (V x V p ) −1/2 = (V 1/2 x V p V 1/2 x ) −1/2 (V x and V p are assumed to be commuting) and will then apply the assumption on the lattice to finally arrive at theorem 2. To show the exponential decay, we will now prove a generalization of a theorem of [20] to matrices reflecting general lattices. This latter work is concerned with the exponential decay of entries of matrix functions of banded matrices. The proof will be very similar to the one in [20], only that the notion of a distance is different: the distance from the main diagonal of a matrix versus dist(·, ·) in the graph theoretical sense. So quite surprisingly, the ideas of [20] carry over to this more general case with little modifications. Note that a mere naive embedding of the potential matrices into banded matrices would be insufficient to find the above conclusion, as for every ordering of physical systems, the range of the banded matrices could not be kept finite. We nevertheless state the full proof here for completeness. We denote by the minimal and maximal eigenvalues of a real symmetric matrix V of finite range. Let a function f : C → C be such that f • ψ is analytic in the interior of an ellipse 5 ε χ , χ > 1, with focii in −1 and 1 and continuous on ε χ . Furthermore, suppose (f • ψ)(z) ∈ R for z ∈ R. Here We can now state the generalization of [20].
Theorem 6 (exponential decay of entries of matrix functions). Let V = (V i,j ) be a positive real symmetric matrix of finite range m, V i,j = 0 for dist(i, j) > m/2, let f : C → C be such that it fulfils the above assumptions. Then there exist constants K and q, 0 K, 0 q < 1 such that where Proof. For a function g : C → C analytic in the interior of the ellipse ε χ , χ > 1, continuous on ε χ , and with g(z) ∈ R for real z, one has inf max where P k denotes the set of all polynomials with real coefficients and degree less than or equal to k. This result is due to Bernstein (see [36] for a proof) and applies to the function f • ψ. As V Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT is assumed to be of range m, we have that p(ψ −1 (V )) is of range km for all polynomials p ∈ P k . Thus, for dist(i, j) > km/2 we have For given i = j we now choose k such that i.e., we have k < 2 dist(i, j)/m k + 1, which yields where the last equation follows from the fact that the spectrum of ψ −1 (V ) is contained in the interval [ − 1, 1]. Applying Bernstein's theorem, we know that there exists a sequence of polynomials p (n) of degree k that satisfy where the last inequality follows from the choice of k and χ > 1. Along with the fact that (f(V )) i,i f(V ) this concludes the proof.
We will now apply theorem 3 to the function f defined as reflecting the matrix function that is needed in order to evaluate the second moments of a Gibbs state, and the matrix V x V p in order to prove theorem 2. In the notation of theorem 3 we have and therefore f • ψ is analytic for all z ∈ C with (z) < (a + b)/(b − a). So for χ = b/ (b − a) = α + β the function f fulfils the requirements of theorem 3 and |(f • ψ)| attains its maximum at z = −α. Thus, Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT To summarize, we have To bound the entries of (V x V p ) −1/2 we could also apply theorem 3, however, for this special function, the bound can be given in a more straightforward manner. To this end, consider again the power-series expansion of the square root, equation (52). Denote by a and b the minimal and maximal eigenvalues of V = V x V p , respectively. As matrices V x and V p are assumed to be of range m, V is of range 2m. Thus, for given i = j choosing k = dist(i, j)/m − 1, i.e., k < dist(i, j)/m k + 1, yields For diagonal terms, we have |(V −1/2 ) i,i | V −1/2 = 1/( E/2), i.e., for all i, j we have and thus, together with the assumption on the lattice, The last step is now analogous to the proof in the zero temperature case, equation (64). We finally arrive at for dist(i, j) m, which concludes the proof.
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Discussion and examples
In this section, we discuss a few special cases to exemplify the previous results. It is important to note that the previous results hold true in case of general lattices, beyond structures with a very high translational symmetry.
Example 1 (disordered one-dimensional system). This example is to highlight that besides locality, no assumptions on the coupling are required. In particular, we allow for random coupling, as for example in the following sense. Consider a sequence of one-dimensional systems on a one-dimensional chain with periodic boundary conditions, such that G (n) = (L (n) , E (n) ) is characterized by L (n) = [1, . . . , n] and the adjacency matrix with entries E (n) i,j = δ i,j+1 + δ i,j−1 + δ i,n δ j,1 + δ j,n δ i,1 , i,j = 1, . . . , n.
From Gershgorin's theorem we then know that E = 2λ 1/2 min (V (n) x ) 2.Also, V (n) 5 for all n. Hence, theorem 1 can be applied, and we find exponentially decaying correlation functions, even in this disordered system. Note that this example can also be generalized to d-dimensional general lattices, as one can straightforwardly find upper and lower bounds to the spectral values of V using the same arguments as above.
Subsequently, we merely sketch the argument leading to the statement that the correlation length is nonzero for T > 0, but it should be clear how this can be made a rigorous statement. For high temperature we can approximate exp(2λ k (V )/T ) − 1 ≈ 2λ k (V )/T , and thus x ixj = p ipj ≈ δ i,j + T((V (n) ) −1 ) i,j .
conditions, the geometric entropy, so the von-Neumann entropy of the reduction, is bounded from above by a quantity that is linear in the boundary area of the distinguished region. Equivalently, this quantifies the degree of entanglement of the region with respect to the rest of the lattice. This result on a connection between the surface area and the degree of entanglement can hence also be identified on general graphs, hence establishing an rigorous entanglement-area relationship in general finite-ranged gapped harmonic lattice systems.