# Bounded Entanglement Entropy in the Quantum Ising Model

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## Abstract

A rigorous proof is presented of the boundedness of the entanglement entropy of a block of spins for the ground state of the one-dimensional quantum Ising model with sufficiently strong transverse field. This is proved by a refinement of the stochastic geometric arguments in the earlier work by Grimmett et al. (J Stat Phys 131:305–339, 2008). The proof utilises a transformation to a model of classical probability called the continuum random-cluster model. Our method of proof is fairly robust, and applies also to certain disordered systems.

## Keywords

Quantum Ising model Entanglement Entropy Area law Random-cluster model## Mathematics Subject Classification

82B20 60K35## 1 The Quantum Ising Model and Entanglement

The purpose of this note is to give a rigorous proof of the area law for entanglement entropy in the quantum Ising model in one dimension. This is achieved by an elaboration of the stochastic geometrical approach of [21]. We prove the boundedness of entanglement entropy of a block of spins of size \(L+1\) in the ground state of the model with sufficiently strong transverse field, uniformly in *L*. The current paper is presented as a development of the earlier work [21] by the same authors, to which the reader is referred for details of the background and basic theory.

*x*, corresponding to the eigenvalues \(\pm 1\). The other two Pauli operators with respect to this basis are represented by the matrices

*x*) of the eigenstates of \(\sigma ^{(3)}_x\). In the following, \(|\phi \rangle \) denotes a vector and \(\langle \phi |\) its adjoint. As a notational convenience, we shall represent sub-intervals of \({{\mathbb {Z}}}\) as real intervals, writing for example \(\Delta _m=[-m, m+L]\).

The Hamiltonian \(H_m\) has a unique pure ground state \(|\psi _m\rangle \) defined at zero temperature (as \(\beta \rightarrow \infty \)) as the eigenvector corresponding to the lowest eigenvalue of \(H_m\). This ground state \(|\psi _m\rangle \) depends only on the ratio \(\theta =\lambda /\delta \). We work here with a free boundary condition on \(\Delta _m\), but we note that the same methods are valid with a periodic (or wired) boundary condition, in which \(\Delta _m\) is embedded on a circle.

*L*] and \(\Delta _m\setminus [0, L]\) and by considering the entropy of the

*reduced density operator*

*L*]. By the spectral theorem for normal matrices [10], this operator may be diagonalised and has real, non-negative eigenvalues, which we denote in decreasing order by \(\lambda _j^{\downarrow }(\rho _m^L)\).

### Definition 1.1

*entanglement (entropy)*of the interval [0,

*L*] relative to its complement \(\Delta _m \setminus [0, L]\) is given by

Here are our two main theorems.

### Theorem 1.2

### Remark 1.3

The value \(\theta =2\) is critical for the quantum Ising model in one dimension, and therefore the condition \(\theta <2\) is sharp for \(\gamma >0\) in (1.5). See the discussion following [13, Thm 7.1].

### Theorem 1.4

Weaker versions of Theorems 1.2 and 1.4 were proved in [21, Thms 2.2, 2.8], namely that (1.5) holds subject to a power factor of the form \(L^\alpha \), and (1.7) holds with \(c_1\) replaced by \(C_1+C_2\log L\) (and subject to a slightly stronger assumption on \(\gamma \)). As noted in Remark 1.3, Theorem 1.2 is a further strengthening of [21, Thm 2.2] in that (1.5) holds for \(\theta <2\), rather then just \(\theta <1\). Stronger versions of these two theorems may be proved similarly, with the interactions \(\lambda \) and field intensities \(\delta \) varying with position while satisfying a suitable condition. A formal statement for the disordered case appears at Theorem 1.5.

There is a considerable and growing literature in the physics journals concerning entanglement entropy in one and more dimensions. For example, paper [17] is an extensive review of area laws. The relationship between entanglement entropy and the spectral gap has been explored in [4, 5], and polynomial-time algorithms for simulating the ground state are studied in [6]. Related works include studies of the XY spin chain [1], oscillator systems [7], the XXZ spin chain [8], and free fermions [25]. The connection between correlations and the area-law is explored in [14].

We make next some remarks about the proofs of the above two theorems. The basic approach of these mathematically rigorous proofs is via the stochastic geometric representation of Aizenman, Klein, Nachtergaele, and Newman [2, 3, 23]. Geometric techniques have proved of enormous value in studying both classical systems (including Ising and Potts models, see for example [19]), and quantum systems (see [11, 12, 13, 15, 18, 26]).

The proofs of Theorems 1.2, 1.4 and the forthcoming Theorem 1.5 have much in common with those of [21, Thms 2.2, 2.8] subject to certain improvements in the probabilistic estimates. The general approach and many details are the same as in the earlier paper, and indeed there is some limited overlap of text. We make frequent reference here to [21], and will highlight where the current proofs differ, while omitting arguments that may be taken directly from [21]. In particular, the reader is referred to [21, Sects. 4, 5] for details of the percolation representation of the ground state, and of the associated continuum random-cluster model. In Sect. 2, we review the relationship between the reduced density operator and the random-cluster model, and we state the fundamental inequalities of Theorem 2.5 and Lemma 2.6. Once the last two results have been proved, Theorems 1.2 and 1.4 follow as in [21]: the first as in the proof of [21, Thm 2.2], and the second as in that of [21, Thm 2.8] (see the notes for the latter included in Sect. 5).

### Theorem 1.5

The situation is more complicated when \(\varvec{\lambda }\), \(\varvec{\delta }\) are random but do not a.s. satisfy (1.9) with \(\lambda /\delta <2\).

### Remark 1.6

The authors acknowledge Massimo Campanino’s announcement in a lecture on 12 June 2019 of his perturbative proof with Michele Gianfelice of a version of Theorem 1.2 for sufficiently small \(\theta \), using cluster expansions. That announcement stimulated the authors of the current work.

## 2 Estimates via the Continuum Random-Cluster Model

We write \({{\mathbb {R}}}\) for the reals and \({{\mathbb {Z}}}\) for the integers. The *continuum percolation model* on \({{\mathbb {Z}}}\times {{\mathbb {R}}}\) is constructed as in [20, 21]. For \(x\in {{\mathbb {Z}}}\), let \(D_x\) be a Poisson process of points in \(\{x\}\times {{\mathbb {R}}}\) with intensity \(\delta \); the processes \(\{D_x: x\in {{\mathbb {Z}}}\}\) are independent, and the points in the \(D_x\) are termed ‘deaths’. The lines \(\{x\}\times {{\mathbb {R}}}\) are called ‘time lines’.

For \(x\in {{\mathbb {Z}}}\), let \(B_x\) be a Poisson process of points in \(\{x+\frac{1}{2}\}\times {{\mathbb {R}}}\) with intensity \(\lambda \); the processes \(\{B_x: x\in {{\mathbb {Z}}}\}\) are independent of each other and of the \(D_y\). For \(x\in {{\mathbb {Z}}}\) and each \((x+\frac{1}{2},t)\in B_x\), we draw a unit line-segment in \({{\mathbb {R}}}^2\) with endpoints (*x*, *t*) and \((x+1,t)\), and we refer to this as a ‘bridge’ joining its two endpoints. For \((x,s), (y,t) \in {{\mathbb {Z}}}\times {{\mathbb {R}}}\), we write \((x,s)\leftrightarrow (y,t)\) if there exists a path \(\pi \) in \({{\mathbb {R}}}^2\) with endpoints (*x*, *s*), (*y*, *t*) such that: \(\pi \) comprises sub-intervals of \({{\mathbb {Z}}}\times {{\mathbb {R}}}\) containing no deaths, together possibly with bridges. For \(\Lambda ,\Delta \subseteq {{\mathbb {Z}}}\times {{\mathbb {R}}}\), we write \(\Lambda \leftrightarrow \Delta \) if there exist \(a\in \Lambda \) and \(b\in \Delta \) such that \(a\leftrightarrow b\). Let \({{\mathbb {P}}}_{\Lambda ,\lambda ,\delta }\) denote the associated probability measure when restricted to the set \(\Lambda \), and write \(\theta =\lambda /\delta \).

Let \({{\mathbb {P}}}_{\lambda ,\delta }\) be the corresponding measure on the whole space \({{\mathbb {Z}}}\times {{\mathbb {R}}}\), and recall from [9, Thm 1.12] that the value \(\theta =1\) is the critical point of the continuum percolation model.

The *continuum random-cluster model* on \({{\mathbb {Z}}}\times {{\mathbb {R}}}\) is defined as follows. Let \(a,b\in {{\mathbb {Z}}}\), \(s,t\in {{\mathbb {R}}}\) satisfy \(a \le b\) and \(s \le t\), and write \(\Lambda =[a,b]\times [s,t]\) for the box \(\{a,a+1,\ldots ,b\} \times [s,t]\). Its boundary \(\partial \Lambda \) is the set of all points \((x,y)\in \Lambda \) such that: either \(x\in \{a,b\}\), or \(y\in \{s,t\}\), or both.

As sample space we take the set \(\Omega _\Lambda \) comprising all finite subsets (of \(\Lambda \)) of deaths and bridges, and we assume that no death is the endpoint of any bridge. For \(\omega \in \Omega _\Lambda \), we write \(B(\omega )\) and \(D(\omega )\) for the sets of bridges and deaths, respectively, of \(\omega \).

The *top/bottom periodic boundary condition* is imposed on \(\Lambda \): for \(x\in [a,b]\), we identify the two points (*x*, *s*) and (*x*, *t*). The remaining boundary of \(\Lambda \), denoted \(\partial ^{\mathrm h}\Lambda \), is the set of points of the form \((x,u)\in \Lambda \) with \(x\in \{a,b\}\) and \(u\in [s,t]\).

*Z*is the appropriate partition function. As at [21, eqn (5.3)],

We make a note concerning exponential decay which will be important later. The critical point of the infinite-volume (\(q=2\)) continuum random-cluster model on \({{\mathbb {Z}}}\times {{\mathbb {R}}}\) with parameters \(\lambda \), \(\delta \) is given by \(\theta _{\mathrm {c}}=2\) where \(\theta =\lambda /\delta \) (see [13, Thm 7.1]). Furthermore, as in [19, Thm 5.33(b)], there is a unique infinite-volume weak limit, denoted \(\phi _{\lambda ,\delta }\), when \(\theta <2\). In particular (as in the discussion of [13]) there is exponential decay of connectivity when \(\theta <2\). Let \(\Lambda _m=[-m,m]^2 \subseteq {{\mathbb {Z}}}\times {{\mathbb {R}}}\), with boundary \(\partial \Lambda _m\).

### Theorem 2.1

Henceforth the function \(\gamma \) denotes that of Theorem 2.1. (The function \(\gamma \) in Theorems 1.2, 1.4 is derived from that of Theorem 2.1.) By stochastic domination, (2.3) holds with \(\phi _{\lambda ,\delta }\) replaced by \({{\mathbb {P}}}_{\Lambda ,\lambda ,\delta ,2}\) for general boxes \(\Lambda \).

It is explained in [21] that a random-cluster configuration \(\omega \) gives rise, by a cluster-labelling process, to an Ising configuration on \(\Lambda \), which serves (see [2]) as a two-dimensional representation of the quantum Ising model of (1.2). We shall use \(\overline{\phi }_{m,\beta }\) and \(\phi _{m,\beta }\) to denote the respective couplings of the continuum random-cluster measures and the corresponding (Ising) spin-configurations, and \(\overline{\phi }_{m,\beta }^\eta \), \(\phi _{m,\beta }^\eta \) for the measures with spin-configuration \(\eta \) on \(\partial ^{\mathrm h}\Lambda _{m,\beta }\).

### Remark 2.2

Theorem 2.1 is an important component of the estimates that follow. At the time of the writing of [21], the result was known only when \(\theta <1\), and the corresponding exponential-decay theorem [21, Thm 6.7] was proved by stochastic comparison with continuum percolation (see (2.2)). More recent progress of [13] has allowed its extension to the \(q=2\) continuum random-cluster model directly. In order to apply it in the current work, a minor extension of the ratio weak-mixing theorem [21, Thm 7.1] is needed, namely that the mixing theorem holds with \(\overline{\phi }\) taken to be the random-cluster measure on \(\Lambda \) with *free boundary conditions*. The proof is unchanged.

### Remark 2.3

In the proofs that follow, it would be convenient to have a stronger version of (2.3) with \(\phi _{\lambda ,\delta }\) replaced by the finite-volume random-cluster measure on \(\Lambda _{m,\beta }\) with wired boundary condition on \(\partial ^{\mathrm h}\Lambda _{m,\beta }\) and periodic top/bottom boundary condition. It may be possible to derive such an inequality as in [16], but we do not pursue that option here.

### Remark 2.4

Let \(\Omega _{m,\beta }\) be the sample space of the continuum random-cluster model on \(\Lambda _{m,\beta }\), and \(\Sigma _{m,\beta }\) the set of admissible allocations of spins to the clusters of configurations, as in [21, Sect. 5]. For \(\sigma \in \Sigma _{m,\beta }\) and \(x\in S_L\), write \(\sigma _x^\pm \) for the spin-state of \(x^\pm \). Let \(\Sigma _L=\{-1,+1\}^{L+1}\) be the set of spin-configurations of the vectors \(\{x^+: x\in S_L\}\) and \(\{x^-: x\in S_L\}\), and write \(\sigma ^+_L= (\sigma _x^+: x\in S_L)\) and \(\sigma ^-_L= (\sigma _x^-: x\in S_L)\).

Here is the main estimate of this section, of which Theorem 1.2 is an immediate corollary with adapted values of the constants. It differs from [22, Thm 6.5] in the removal of a factor of order \(L^\alpha \), and the replacement of the condition \(\theta <1\) by the weaker assumption \(\theta <2\).

### Theorem 2.5

In the proof of Theorem 2.5, we make use of the following two lemmas (corresponding, respectively, to [21, Lemmas 6.8, 6.9]), which are proved in Sect. 3 using the method of ratio weak-mixing.

### Lemma 2.6

*K*is such that \(R_K \le \tfrac{1}{2}\).

In the second lemma we allow a general spin boundary condition on \(\partial ^{\mathrm h}\Lambda _{m,\beta }\).

### Lemma 2.7

### Proof of Theorem 2.5

Let \(\theta <2\), and let \(\gamma \) be as in Theorem 2.1. It suffices to prove (2.7) with \(\phi _m\) (respectively, \(\phi _n\)) replaced by \(\overline{\phi }_{m,\beta }\) (respectively, \(\overline{\phi }_{n,\beta }\)), and \(a_m\) (respectively, \(a_n\)) replaced by \(a_{m,\beta }\) (respectively, \(a_{n,\beta }\)). Having done so, we let \(\beta \rightarrow \infty \) to obtain (2.7) by Remark 2.4.

*A*, \(C_1\), \(R_K\) be as in Lemma 2.6, and let \(L\ge 3\) and \(1\le K<\frac{1}{2} L\) be such that

*L*are covered in (2.7) by adjusting

*C*.

*D*be the set of all pairs \((\omega _1,\omega _2)\in \Omega _{n,\beta }\times \Omega _{n,\beta }\) such that: \(\omega _2\) contains no path joining \(\partial B\) to \(\partial ^{\mathrm h}\Lambda _{m,\beta }\), where

Having constructed the measure \(\nu \) accordingly, we may now allocate spins to the clusters of \(\omega _1\) and \(\omega _2\) in the manner described in [21, Sect. 5]. This may be done in such a way that, on the event *D*, the spin-configurations associated with \(\omega _1\) and \(\omega _2\) within *B* are identical. We write \(\sigma _1\) (respectively, \(\sigma _2\)) for the spin-configuration on the clusters of \(\omega _1\) (respectively, \(\omega _2)\), and \(\sigma _{i,L}^\pm \) for the spins of \(\sigma _i\) on the slit \(S_L\).

*D*, and \(1_E\) is the indicator function of

*E*.

*D*, we have that \(\sigma _{1,L}^\pm = \sigma _{2,L}^\pm \), so that

*B*with free boundary conditions, namely,

*C*and \(M_1\) above such that

*B*but nothing further about \(\omega _1\) within

*B*. Combining this with (2.20)–(2.22), we find that, for \(r \ge M_3(\lambda ,\delta )\) and some \(C_3=C_3(\lambda ,\delta )\),

Finally, we remark that *C* and *M* depend on both \(\lambda \) and \(\delta \). The left side of (2.7) is invariant under re-scalings of the time-axes, that is, under the transformations \((\lambda ,\delta ) \mapsto (\lambda \eta , \delta \eta )\) for \(\eta \in (0,\infty )\). We may therefore work with the new values \(\lambda '=\theta \), \(\delta '=1\), with appropriate constants \(\alpha (\theta ,1)\), \(C(\theta ,1)\), \(M(\theta ,1)\). \(\square \)

## 3 Proofs of Lemmas 2.6 and 2.7

Let \(\Lambda \) be a box in \({{\mathbb {Z}}}\times {{\mathbb {R}}}\) (we shall later consider a box \(\Lambda \) with a slit \(S_L\), for which the same definitions and results are valid). A *path*\(\pi \) of \(\Lambda \) is an alternating sequence of disjoint intervals (contained in \(\Lambda \)) and unit line-segments of the form \([z_0,z_1]\), \(b_{12}\), \([z_2,z_3]\), \(b_{34}\), \(\ldots \), \(b_{2k-1,2k}\), \([z_{2k},z_{2k+1}]\), where: each pair \(z_{2i}\), \(z_{2i+1}\) is on the same time line of \(\Lambda \), and \(b_{2i-1,2i}\) is a unit line-segment with endpoints \(z_{2i-1}\) and \(z_{2i}\), perpendicular to the time-lines. The path \(\pi \) is said to join \(z_0\) and \(z_{2k+1}\). The *length* of \(\pi \) is its one-dimensional Lebesgue measure. A *circuit**D* of \(\Lambda \) is a path except inasmuch as \(z_0=z_{2k+1}\). A set *D* is called *linear* if it is a disjoint union of paths and/or circuits. Let \(\Delta \), \(\Gamma \) be disjoint subsets of \(\Lambda \). The linear set *D* is said to *separate*\(\Delta \) and \(\Gamma \) if every path of \(\Lambda \) from \(\Delta \) to \(\Gamma \) passes through *D*, and *D* is minimal with this property in that no strict subset of *D* has the property.

Let \(\omega \in \Omega _\Lambda \). An *open path*\(\pi \) of \(\omega \) is a path of \(\Lambda \) such that, in the notation above, the intervals \([z_{2i},z_{2i+1}]\) contain no death of \(\omega \), and the line-segments \(b_{2i-1,2i}\) are bridges of \(\omega \).

Let \(\Gamma \) be a measurable subset and \(\Delta \) a finite subset of \(\Lambda \) such that \(\Delta \cap \Gamma =\varnothing \). We shall make use of the ‘ratio weak-mixing property’ of the spin-configurations in \(\Delta \) and \(\Gamma \) that is stated and proved in [21, Thm 7.1]; note Remark 2.2.

*K*be an integer satisfying \(1\le K < \frac{1}{2} L\), and let

### Lemma 3.1

### Proof

*D*is a linear subset of \(\Lambda _{m,\beta }\) that separates \(\Delta \) and \(\Gamma \). Let \(t_1\), \(t_2\),

*t*be as in [21, Thm 7.1], namely,

We now prove Lemmas 2.6 and 2.7 .

### Proof of Lemma 2.6

*x*is a death, so that

*x*satisfying \(0\le x <K\) and \(L-K < x \le L\), and

### Proof of Lemma 2.7

*k*is an integer. (If either

*m*is small or

*k*is non-integral, the constant

*C*may be adjusted accordingly.) Let \(D_0\) be the circuit illustrated in Fig. 2, comprising a path in the upper half-plane from \((-k,0)\) to \((L+k,0)\) together with its reflection in the

*x*-axis. Let \(D=D_0 \cap \Lambda _{m,\beta }\). Thus, \(D=D_0\) in the case \(\beta =\beta _2\) of the figure. In the case \(\beta =\beta _1\),

*D*comprises two disjoint paths of \(\Lambda _{m,\beta }\). In each case,

*D*separates \(\Delta \) and \(\Sigma \).

*t*be as in (3.2). By the ratio weak-mixing theorem [21, Thm 7.1] and Remark 2.2,

## 4 Quenched Disorder

The parameters \(\lambda \) and \(\delta \) have so far been assumed constant. The situation is more complicated in the disordered case, when either they vary deterministically, or they are random. The arguments of this paper may be applied in both cases, and the outcomes are summarised in this section. Let the Hamiltonian (1.2) be replaced by (1.8), and write \(\varvec{\lambda }=(\lambda _{x,x+1}: x \in {{\mathbb {Z}}})\) and \(\varvec{\delta }=(\delta _x: x \in {{\mathbb {Z}}})\).

Consider now the situation in which (1.9) does not hold with probability one. Suppose that the \(\lambda _{x,x+1}\), \(x\in {{\mathbb {Z}}}\), are independent, identically distributed random variables, and similarly the \(\delta _x\), \(x\in {{\mathbb {Z}}}\), and assume that the vectors \(\varvec{\lambda }\) and \(\varvec{\delta }\) are independent. We write *P* for the corresponding probability measure, viewed as the measure governing the ‘random environment’.

A quenched area law might assert something along the following lines: subject to suitable conditions, there exists a random variable *Z* which is *P*-a.s. finite such that \(S(\rho _m^L)<Z\) for all appropriate *m*, *L*. Such a uniform upper bound will not generally exist, owing to the fluctuations in the system as \(L \rightarrow \infty \). In the absence of an assumption of the type of (1.9), there may exist sub-domains of \({{\mathbb {Z}}}\) where the environment is not propitious for such a bound.

Partial progress may be made using the methods of [21, Sect. 8], but this is too incomplete for inclusion here.

## 5 Proof of Theorem 1.4

Since this proof is very close to that of [21, Thm 2.12], we include only details that are directly relevant to the strengthened claims of the current theorem, namely the removal of the logarithmic term of [21] and the weakened assumption on \(\gamma \).

*C*and \(\gamma >2\ln 2\) be as in Theorem 1.2, and choose an integer \(K =K(\theta )\ge 2\) such that

*Q*satisfying \(Q\le 1\), we have

## Notes

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