Abstract
The groundstates of the spinS antiferromagnetic chain \(H_{{\text {AF}}}\) with a projectionbased interaction and the spin1/2 XXZchain \( H_{{\text {XXZ}}}\) at anisotropy parameter \(\Delta =\cosh (\lambda ) \) share a common loop representation in terms of a twodimensional functional integral which is similar to the classical planar Qstate Potts model at \( \sqrt{Q}= 2S+1 =2\cosh (\lambda )\). The multifaceted relation is used here to directly relate the distinct forms of translation symmetry breaking which are manifested in the groundstates of these two models: dimerization for \(H_{{\text {AF}}}\) at all \(S> 1/2\), and Néel order for \( H_{{\text {XXZ}}} \) at \(\lambda >0\). The results presented include: (i) a translation to the above quantum spin systems of the results which were recently proven by Duminil–Copin–Li–Manolescu for a broad class of twodimensional randomcluster models, and (ii) a short proof of the symmetry breaking in a manner similar to the recent structural proof by Ray–Spinka of the discontinuity of the phase transition for \(Q>4\). Altogether, the quantum manifestation of the change between \(Q=4\) and \(Q>4\) is a transition from a gapless groundstate to a pair of gapped and extensively distinct groundstates.
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1 Introduction
The focus of this work is the structure of the groundstates in two families of antiferromagnetic quantum spin chains, each of which includes the spin1/2 Heisenberg antiferromagnet as a special case. In the infinite volume limit, with the exception of their common root, in both cases the systems exhibit symmetry breaking at the level of groundstates. The physics underlying the phenomenon is different. In one case it is extensive quantum frustration which causes dimerization with is expressed in spatial energy oscillations. In the other case, the Hamiltonian is frustration free and the symmetry breaking is expressed in longrange Néel order. Yet, in mathematical terms both phenomena are analyzable through a common random loop representation. Curiously, a similar loop system appears also as the auxiliary scaffolding of a classical planar Qstate Potts models for which the symmetry breaking relates to a discontinuity in the order parameter.
The models under consideration have been studied extensively, and hence the specific results we discuss may be regarded as known, at one level or another. The techniques which have been applied for the purpose include numerical works, Bethe ansatz calculations [1, 9,10,11,12, 27], and contour expansions [31]. The validity of Bethe ansatz calculations for similar systems has recently received support through a careful mathematical analysis [20]. The results presented here are based on nonperturbative structural arguments. They may be worth presenting since in the models considered such arguments allow full characterization of the conditions under which the symmetry breaking occurs, as well as other qualitative features of the model’s groundstates. The relation between the models may be of intrinsic interest. At the mathematical level it plays an essential role in the nonperturbative proof of symmetry breaking which is the main result presented here.
1.1 Antiferromagnetic \(SU(2S+1)\) Invariant Spin Chains with Projection Based Interaction
The most basic quantum object has a twodimensional complex state space, spanned by the two orthogonal vectors \( \vert +\rangle \equiv \left( \begin{matrix} 1 \\ 0 \end{matrix}\right) \) and \( \vert \rangle \equiv \left( \begin{matrix} 0 \\ 1 \end{matrix}\right) \). The selfadjoint operators on this space (which has the structure of \({\mathbb {C}}^2\)) are linear combinations of the three Paulispin matrices \( \pmb {\tau } = (\tau ^x,\tau ^y , \tau ^z) \),
Of particular interest is the triplet of spin operators \( \mathbf{S }= (S^x,S^y , S^z) \) with \(S^\alpha = \frac{1}{2} \tau ^\alpha \), \(\alpha = x,y,z\). These span the Lie algebra of the group SU(2) and satisfy the commutation relation
For higher spin systems the Hilbert spaces of states are given by \({\mathbb {C}}^{2S+1} \) in which one finds the \(2S+1\)dimensional representations of the Lie algebra commutation relations (1.2), with \(S\in \mathbb {N}\cup (\mathbb {N}\frac{1}{2}) \). A convenient basis is provided by the eigenvectors of \(S^z\), satisfying
In this terminology, the above binary spin system corresponds to \(S=1/2\), and the states \(\vert +\rangle \) and \(\vert \rangle \) are the eigenstates of \(S^z\) at values \(m=  \frac{1}{2}, + \frac{1}{2} \).
Our spin chains are arrays of 2L spins indexed by \( \Lambda _L:= \{L+1,\ldots ,L\} \). The corresponding state space is the tensor product Hilbert space \({\mathcal {H}}_L= \bigotimes _{v\in \Lambda _L} {\mathbb {C}}^{2S+1} \). The single spin operators are lifted to it by setting \( \pmb {\tau }_u := \mathbb {1} \otimes \cdots \mathbb {1} \otimes \pmb {\tau } \otimes \mathbb {1} \cdots \mathbb {1} \), which acts nontrivially only in the tensor product’s uth component.
Lifted to the two component product space, the above Dirac notation of states takes the form
Correspondingly, we shall use the following notation for operators acting in the corresponding twocomponent factor of \({\mathcal {H}}_L\)
Our discussion will focus on different extensions of the quantum Heisenberg antiferromagnetic spin model, which is an array of spins with the nearestneighbor interaction energy proportional to \(\mathbf{S }_u \cdot \mathbf{S }_{u+1} \). For \(S=1/2\) this can be alternatively written as
with \(\pmb {\tau }_u\cdot \pmb {\tau }_{u+1} = \sum _{\alpha = x,y,z} \tau _u^\alpha \ \tau _{u+1}^\alpha \) and \(P^{(0)}_{u,u+1} =\left( \vert D\rangle \langle D\vert \right) _{u,u+1} \) the orthogonal projection onto the state
in the corresponding twospin space. This state is of some interest: it is the only one which is annihilated by each component of the combined spin operator \(\mathbf{S }_u +\mathbf{S }_{u+1} \), and it also maximizes the entanglement between the two components.
The two expressions of the spin 1/2 Hamiltonian which are presented in (1.6) suggest slightly different extensions to higher values of the spin \( S \in {\mathbb {N}} /2 \). The one on which we focus here is
with \(P^{(0)}_{u,u+1} \) the rankone projection in the two spin space \( {\mathbb {C}}^{2S+1} \otimes {\mathbb {C}}^{2S+1} \) onto on the subspace which is invariant under rotations generated by \(\mathbf{S }_u+\mathbf{S }_{u+1}\), i.e., the joint kernel of \(S_u^\alpha +\mathbf{S }_{u+1}^\alpha \) (\(\alpha = x,y,z\)). For any \(S\in \mathbb {N}/2\) this operator is given by^{Footnote 1}
This model was studied by Affleck [1], Barber and Batchelor [9], Batchelor and Barber[10], Klümper [27], Aizenman and Nachtergaele [6], and more recently Nachtergaele and Ueltschi [31].
The classical analog of a quantum spinor with the state space \(\mathbb {C}^{2S+1}\) is a system whose states are described by a three component vector of length S. Under this correspondence, the classical analog of the projection to the groundstate(s) of \(H_{{\text {AF}}}^{(L)} \) is the restriction to configurations in which each pair of neighboring spins points in exactly opposite directions, adding to \({\underline{0}}\). However, unlike its classical analog, the quantum system exhibits frustration, and that leads to the dimerization phenomenon discussed next.
Each of the twospin interaction terms in (1.8) is minimized in the state in which the two spins are coherently intertwined into the unique state in which \(\mathbf{S }_u+\mathbf{S }_v=0\). Yet, a quantum spin cannot be locked into such a state with both its neighbors simultaneously. This effect, which results in the spinPeierls instability, is purely quantum as there is no such restriction for classical spins. (Classical spin models may be driven to frustration by other means, e.g., when placed on a nonbipartite graph with antiferromagnetic interactions, and also on arbitrary graphs under suitably mixed interactions. Such geometric frustration is shared by their quantum counterparts.)
The naive pairing depicted in Fig. 1 suggests that in finite volume the groundstates’ local energy density may not be homogeneous and have a bias triggered by the boundary conditions, i.e., the parity of L. Indeed, through approximations, numerical simulations, or the probabilistic representation of [6] (our preferred method), one may see that the local energy density of the corresponding finitevolume groundstates \( \langle \cdot \rangle _{L}^{(\text {gs})} \) is not homogeneous and satisfies
An interesting question is whether this bias persists in the limit \(L\rightarrow \infty \), in which case in the infinitevolume limit the system has (at least) two distinct groundstates, for which the expectation values of local observables F are given by
where the limit is interpreted in the weak sense, i.e., with F being any (fixed) local bounded operator. These are generated by products of spin operators
which are supported in some bounded set \( U \subset {\mathbb {Z}} \). In finitevolume, their (imaginary) timeevolved counterparts are given by
The corresponding truncated correlations also converge, e.g., for any fixed \( t \in {\mathbb {R}} \),
and similarly for \( \langle F_U(t) ; F_V\rangle _{{\text {odd}}} \).
The separate convergence of the limits (1.11) or (1.13) was established in [6] through probabilistic techniques which are enabled by the loop representation presented below. This representation also led to the following dichotomy.^{Footnote 2}
Proposition 1.1
(cf. Thm. 6.1 in [6]). For each value of \(S\in {\mathbb {N}}/2\) one of the following holds true:

1.
The two groundstates \(\langle \cdot \rangle _{{\text {even}}}\) and \(\langle \cdot \rangle _{{\text {odd}}}\) are distinct, each invariant under the 2step shift, each being the 1step shift of the other. Furthermore, their translation symmetry breaking is manifested in energy oscillations, namely, for every \(n\in {\mathbb {N}}\)
$$\begin{aligned} \langle P^{(0)}_{2n1,2n} \rangle _{{\mathrm{even}}}  \langle P^{(0)}_{2n,2n+1} \rangle _{{\mathrm{even}}} >0\, . \end{aligned}$$(1.14) 
2.
The even and odd groundstates coincide, and form a translation invariant groundstate \(\langle \cdot \rangle \) with slowly decaying correlations, satisfying
$$\begin{aligned} \sum _{v\in {\mathbb {Z}}} v\, \langle \pmb {S}_0 \cdot \pmb {S}_v \rangle  = \infty \, . \end{aligned}$$(1.15)
For \( S = 1/2 \) the second alternative is known to hold (cf. [2, 21] and references therein). In this case the model reduces to the quantum Heisenberg antiferromagnet.^{Footnote 3} In the converse direction, dimerization in this model was established for \(S\ge 8\) [31] through a contour expansion. The gap between these results is closed here through a structural proof that for all \(S> 1/2\) the first option holds (regardless of the parity of 2S).
Theorem 1.2
For all \(S>1/2\):

1.
the even and odd groundstates, defined by (1.11), differ. They are translates of each other, and exhibit the energy oscillation (1.14).

2.
there exist \(\xi =\xi (S) <\infty \) such that for all \(U,V\subset {\mathbb {Z}}\) with distance \( {\text {dist}}(U,V) \) and any \( t \in {\mathbb {R}} \):
$$\begin{aligned} \vert \langle F_U(t) ; F_V\rangle _{{\text {even}}} \vert \ \le \ C_{F_U} C_{F_V} \, e^{ ({\text {dist}}(U,V) +  t ) /\xi }\, , \end{aligned}$$(1.16)where \( C_{F_U} \) and \( C_{F_V} \) are invariant under spacetime translations of the observables \( F_U , F_V \).
The proof draws on the progress which was recently made in the study of the related loop models. In [20], the loop representation of the critical Qstate Potts model on the square lattice with \(Q>4\) was proved to have two distinct infinitevolume measures under which the probability of having large loops is decaying exponentially fast (see [28] for the case of large Q). The result was extended in [18, Theorem 1.4] to a slightly modified version of the loop model that will be redefined in this paper and connected to the spin chains (there, the model is not defined in terms of loops but in terms of percolation, as in Sect. 6). More recently, Ray and Spinka [32] provided an alternative proof of the nonuniqueness of the infinitevolume measures on the square lattice.
In this article, the inspiring proof of RaySpinka is extended to our context to provide a new proof of 1. We believe that this proof is more transparent and conceptual than the one in [18] and that even though the technique does not directly lead to 2., it illustrates perfectly the interplay between the quantum and classical realms. In fact, a careful analysis of the proofs in the paper of [18] shows that the argument there relies on two pillars: a theorem proving a stronger form of Proposition 1.1 (see also [19, 21] for versions on the square lattice), in which 1. of Theorem 1.2 is proved to imply 2., and an argument relying on the Bethe Ansatz showing that 1. indeed occurs. The adaptation of the RaySpinka argument enables us to prove 1. directly without using the Bethe Ansatz, so that the argument in this paper replaces half of the argument in [18], and that combined with the other half it also implies 2.
Let us finally note that under the dimerization scenario, which is now established for its full range (\(S > 1/2 \)), other physically interesting features follow:

1.
Spectral gap: As was argued already in [6, Theorem 7.1], the exponential decay of truncated correlations (1.16) in the tdirection implies a nonvanishing spectral gap in the excitation spectrum above the even and odd groundstates.

2.
Excess spin operators: When the decay of correlations is fast enough so that (1.15) does not hold, in particular under (1.16), in the even/odd states the spins are organized into tight neutral clusters. That is manifested in the tightness of the distribution of the block spins \(S^z_{[a,b]} = \sum _{u\in [a,b]} S^z_u\) (in a sense elaborated in [5]). That is equivalent to the existence of the excess spin operators \( {\widehat{S}}_u^z\) with which
$$\begin{aligned} \sum _{v=1}^u S^z_v = {\widehat{S}}_0^z{\widehat{S}}_u^z\, \end{aligned}$$(1.17)and such that \({\widehat{S}}_u^z\) commutes with the spins in \((\infty , u]\). The quantity \({\widehat{S}}_u^z\) can be interpreted as the total spin in \((u,\infty )\), and constructed as \(\lim _{\varepsilon \downarrow 0} \sum _{v > u } e^{\varepsilon  uv } S_v^z \) (in the strongresolvent sense), cf. [6, Sec. 6]. As was further discussed in [7], the excess spins play a role in the classification of the topological properties of the gapped groundstate phases.

3.
Entanglement entropy: Another general implication of the exponential decay of correlations is a socalled area law (which for chains equates to the boundedness) of the entanglement entropy of the groundstates, see [14] for details.
1.2 The \(S=1/2\) Antiferromagnetic XXZ Spin Chain
The second model discussed in this paper is the anisotropic XXZ spin1/2 chain with the Hamiltonian
acting on the Hilbert space \( {\mathcal {H}}_L = \bigotimes _{v=L+1}^L {\mathbb {C}}^{2} \). It consists of Pauli spin matrices (1.1) on \( {\mathbb {C}}^2 \). It is convenient to present the anisotropy parameter as
Throughout the paper and unless stated otherwise explicitly, we will take \( \lambda \ge 0 \) the nonnegative solution of (1.19).
The sign and the magnitude of \(\Delta >1\) favor antiferromagnetic order in the groundstate. The negative sign in front of the terms involving the x and ycomponent of the Pauli spin matrices can be flipped through the unitary transformation \( U_L = \exp \big ( i \tfrac{\pi }{4} \sum _u (1)^u \tau _u^z \big ) \). It renders the Hamiltonian in the manifestly antiferromagnetic form
The antiferromagnetic XXZ chain has been the subject of many works. Following Lieb’s work on interacting Bose gas [29], Yang and Yang gave a justification for the Bethe Ansatz solution of the groundstate in a series of papers [37, 38] in 1966. The groundstate has longrange order with two period2 states in the thermodynamic limit, each with mean magnetization of alternating direction. The corresponding Néel order parameter (\(M_{\text {N}\acute{\mathrm{e}}\text {el}}\) of (1.22)) vanishes in the limit \(\Delta \downarrow 1\). Since the exact solution is not very transparent, there has been interest in obtaining qualitative information by other means, e.g., expansions and other rigorous methods. These typically apply only for large \( \Delta \).
Our motivation for returning to the XXZ spin chain is that it emerges very naturally in the analysis of the thermal and groundstates of the model \(H_{{\text {AF}}}\). Furthermore, the relation between the two facilitates the proof of the symmetry breaking stated in Theorem 1.2. In the converse relation, this relation is used here to establish symmetry breakdown in the form of Néel order of the XXZ groundstate(s) for all \( \Delta > 1 \).
To prove the translation symmetry breaking we consider the pair of finitevolume groundstates for the Hamiltonian (1.18) with an added boundary field,^{Footnote 4} i.e.,
As a preparatory statement let us state:
Proposition 1.3
For any \(\Delta \ge 1\), in the limit \(L\rightarrow \infty \) with L even, the finitevolume groundstates of the XXZspin system with the above boundary terms converge to states \(\langle \cdot \rangle _+\) and \(\langle \cdot \rangle _\). Regardless of whether the two agree, each is a onestep shift of the other. The two states are different if and only if they exhibit Néel order, in the sense for all n:
at some \(M_{\text {N}\acute{\mathrm{e}}\text {el}} \ne 0\).
Let us emphasize that the system’s size is even regardless of the parity of L (the size being equal to 2L). The restriction in this theorem to sequences of constant parity is required for the consistency of the effect of the boundary conditions which are specified in (1.21).
Similarly to Proposition 1.1, this statement is proven here through the FKG inequality which is made applicable in a suitable loop representation. We postpone its proof to Sect. 6, next to the place where it is applied. Following is the XXZversion of the symmetry breaking statement.
Theorem 1.4
For any \( \Delta > 1 \) the construction described in Proposition 1.3 yields two different ground states of infinite XXZspin chain which differ by a one step shift and satisfy (1.22).
Theorem 1.4 is proven in Sect. 6 together with Theorem 1.2. In each case the symmetry breaking is initially established through the expectation value of a conveniently defined quasilocal observable. The conclusion is then boosted to the more easily recognizable statements presented in the theorems through the preparatory statements of Proposition 1.3 and respectively Proposition 1.1.
1.3 Seeding the GroundStates
Infinitevolume groundstates can be approached through their intrinsic properties (such as the energyminimizing criterion) or, constructively, as limits of finitevolume groundstate expectation value functionals. To establish their nonuniqueness, we shall consider different sequences of finite volume groundstates, and establish convergence of the expectation value functionals to limits which are extensively different. Equivalently, it suffices to construct a single limiting groundstate which does not have the Hamiltonian’s translation symmetry. A shift (or another symmetry operation) produces then another groundstate. We shall take that path in the discussion of both models.
The finitevolume groundstates will be constructed through limits of the form
with \(\vert \Psi _L\rangle \) a convenient seeding vector. To assure that the limiting functional corresponds to a groundstate (or the groundstate if it is unique) one needs to verity that this vector is not annihilated by the groundstate projection operator \( P_L^{(\text {gs})} \). That will be established by verifying that
Our choice of the seeding vectors is primarily guided not by the condition (1.24), which is generically satisfied, but rather by the goal of a transparent expression for the expectation value functional.
In view of the quantum frustration effect, a natural seed vector for the construction of a groundstate for the Hamiltonian \( H_{{\text {AF}}}^{(L)} \) on an even collection of spins in \(\Lambda _L = \{L+1,\ldots ,L\}\) is the dimerized state
The subscripts on the vectors indicate on which tensor component of \( {\mathcal {H}}_L \) they act. The role of the gauge transformation
expressed in the standard zbasis of the joint eigenstates of \(S^z_u \), \( u \in \Lambda _L \), is to ensure nonnegativity of the matrixelements of \( U_L^* e^{\beta H^{(L)}_{{\text {AF}}} } U_L \) in the zbasis. This will enable a probabilistic loop representation of this semigroup presented in Sect. 2. From this representation, we will also see that (1.24) is valid for the seed state \( \Psi _L = D_{L} \) at any finite L, cf. (3.10). The standard Perron–Frobenius argument is not applicable in this case.
Applying the semigroup operator \(e^{\beta H_L/2}\) to \(\vert D_{L} \rangle \), one gets the expectationvalue functional which assigns to each local observable F the value
and which converges as \( \beta \rightarrow \infty \) to a groundstate expectation \( \langle F \rangle _{L}^{(\text {gs})} \). It is the above expectationvalue functional which we study in the proof of Theorem 1.2 by probabilistic means.
To study the Néel order of the XXZHamiltonian we find it convenient to focus on the sequence of constant parity, say even L, and use as seed in (1.23) the vector
which is indexed by \( \lambda \). Using it, the state \(\langle \cdot \rangle _+\) of Proposition 1.3 is presentable as the double limit
of
For the state \(\langle \cdot \rangle _^{\text {(XXZ)}} \), we reverse the sign in front of \(\lambda \) in (1.28), and apply the operator \(H^{(L,)}_{\text {XXZ}}\).
Note that for fixed \( L \in 2 {\mathbb {N}} \), the limit \( \beta \rightarrow \infty \) in (1.29) converges to the finitevolume groundstate of \( H^{(L,+)}_{\text {XXZ}} \), which is found in the subspace
where it is unique. This follows from a standard Perron–Frobenius argument, which is enabled here by the positivity and transitivity of the semigroup on that subspace. As a consequence, the finitevolume groundstate can be construction through the limit \( \beta \rightarrow \infty \) starting from any nonnegative seed vector with \( S^z_{\text {tot}} = 0 \). The vectors \( N^{(L)}_\lambda \) with \( \lambda \in {\mathbb {R}} \) arbitrary are examples of such seed vectors and the limit (1.29) does not depend on the choice of \( \lambda \) in the seed (but still depends on \( \lambda \) through \( H^{(L,+)}_{\text {XXZ}} \).)
Next we start the detailed discussion by recalling the probabilistic loop representations of the states described above. The construction is included here mainly to keep the paper reasonably selfcontained, since it is already contained in [6].
2 Functional Integral Representation of the Thermal States
2.1 The General Construction
Thermal states of ddimensional quantum systems can always be expressed in terms of a \((d+1)\)dimensional functional integral. When the integrand can be expressed in positive terms, the result is a relation with a statisticmechanical system in dimension \(d+1\). General discussion of this theme and applications for specific purposes can be found, e.g., in [3, 6, 15, 22, 23, 35, 36]. Our aim in this section is to present this relation for the models discussed here.
As a starting point, let us note the following elementary identity, in which the power expansion of \(e^{\beta K} \), which is valid for any bounded operator K, is cast in probabilistic terms:
In the last expression, the sequence of times is presented as a random point subset \(\omega =(t_1,\ldots ,t_n) \subset [0,\beta ]\) distributed as a Poisson process on \([0,\beta ]\) with intensity measure dt. The Poisson probability distribution is denoted here by \(\rho _{[0,\beta ]}(d\omega )\). Attached to each point \(t\in \omega \) is a copy of the operator K labeled by t. The factors \(K_t\) are rearranged according to their time label, which is denoted using the time ordering operator \( {\mathcal {T}} \). The integral reproduces the familiar power series.
For operators which are given by sums of (local) terms, as in our case
with \(K_b\) indexed by the edgeset \({\mathcal {E}}(\Lambda ) \) of a graph \(\Lambda \), the identity (2.1) has the following extension
where \(\omega \) are the configurations of a Poisson point process over \({\mathcal {E}}(\Lambda ) \times [0,\beta ]\), which may be depicted as collections of rungs of a random multicolumnar ladder net whose rungs are listed as \(\{(b_j,t_j)\}\) in increasing order of t. We denote by \(\Omega _{\Lambda ,\beta } \) the space of such configurations, and by \(\rho _{\Lambda \times [0,\beta ]}(d\omega ) \) the Poisson process with intensity measure dt along the collection of vertical columns \(\cup _{b\in {\mathcal {E}}(\Lambda )} \{b\}\times [0,\beta ] \).
Given an orthonormal basis \(\{ \vert \alpha \rangle \}\) of the Hilbert space in which these operators operate, one has
where \({\widetilde{\alpha }}\) is summed over functions \({\widetilde{\alpha }} : [0,\beta ] \mapsto \{ \vert \alpha \rangle \}\) which are constant between the transition times \(0<t_1<...< t_{\omega }<\beta \), and the consistency constraint is expressed in the indicator function \(\mathbb {1}[\omega , {\widetilde{\alpha }}]\).
Applying this representation, one gets
The left side is obviously nonnegative. If a basis of vectors \(\vert \alpha \rangle \) can be found in which also the matrix elements of \(K_b\) are all nonnegative, then (2.4) yields a functional integral for the quantum partition function in which the integration is over \((\omega , {\widetilde{\alpha }})\) which resembles a “classical” statistic mechanical system in \(d+1\) dimensions (with \(\alpha (t)\) a timedependent configuration which changes at random times).
In that case one also gets a potentially useful decomposition of the thermal state:
with
The functional \( F \mapsto {\mathbb {E}}\left( F  \omega \right) \) was dubbed in [6] a quasistate. It does not possess the full positivity of a quantum state on all observables, but is a proper state on the subalgebra of observables which are diagonal in the basis in which the interaction terms \(K_b\) are all nonnegative.
A similar decomposition is valid for states \( \langle \Psi \vert e^{\beta H_\Lambda /2} F e^{\beta H_\Lambda /2} \vert \Psi \rangle \), which are seeded by vectors \( \Psi \) with nonnegative overlaps with the above base vectors. For that it is pictorially convenient to cyclically shift the time interval to \([\beta /2,\beta /2]\), and consider \(\omega \) given by the Poisson process over the set
whose law is denoted by \(\rho _{\Lambda _{L,\beta }}\).
Such nonnegative functional integral representations of quantum states are associated with Gibbs states of a classical statistic mechanical systems. Under this correspondence, nonuniqueness of the groundstates of a ddimensional quantum spin system, in the infinitevolume limit, is associated with a firstorder phase transition (at a nonzero temperature) of the corresponding \(d+1\)dimensional classical system.
2.2 A PotentialLike Extension
We shall also use an extension of the above expressions to operators of the form
with V an operator which is diagonal in the basis \(\{\vert \alpha \rangle \}\), with \(V\vert \alpha \rangle = V(\alpha ) \vert \alpha \rangle \). In a manner reminiscent of the way that potential appears in the Feynman–Kac formula, one has
as can be deduced from (2.4), e.g., using the LieTrotter product formula.
3 Loop Measures Associated with \(H_{{\text {AF}}}\)
3.1 The \(H_{{\text {AF}}}\) Seeded States
The positivity assumption does hold in the case of the two families of quantum spin chains considered here. Under the unitary (gauge) transformation \( U_L := \exp \left( i \tfrac{\pi }{2} \sum _u (1)^v S_u^z \right) \), the interaction terms of \(H_{\text {AF}}^{(L)}\) acquire positive matrix elements in the standard basis of the joint eigenstates of \((S^z_u)_{u\in \Lambda _L}\)
In this basis, the factors \(K_{b}= (2S+1) U_L^* P_{uv}^{(0)}U_L\) which appear in (2.3) reduce to constraints imposing the condition that before and after each rung the two spins at its edges add to zero. To compute the global effect of that, one may replace each rung by a pair of “infinitesimally separated” lines, and then decompose the graph into noncrossing loops, as indicated in Fig. 2.
By elementary considerations [6], it follows that for each rung configuration \(\omega \) drawn on \( \Lambda _{L,\beta }\):
where \(N_\ell (\omega )\) is the number of loops into which the set of lines decomposes when the vertical lines are turned into columns through “capping” them at \(t=\pm \beta /2\) over every other column starting with the leftmost, cf. Fig. 2. Depending on the parity of L, the capping rule thus follows the two pairings in Fig. 1.
More generally, for \( \vert \pmb {m}\rangle = \vert m_{L+1}, \ldots , m_L\rangle \in {\mathcal {H}}_L \) the orthonormal eigenfunctions of \( \{S_u^z\} _{u\in \Lambda _L} \), the matrix elements \(\langle \pmb {m}'\vert {\mathcal {T}} \left( \prod _{(b,t) \in \omega } K_{b,t}\right) \vert \pmb {m}\rangle \) are given by the sum over configurations of the function \(m: \Lambda _{L,\beta } \mapsto \{ S, S+1, \ldots , S \} \) for which m(x, t) is piecewise constant in time changing only at the encounters with the rungs of \(\omega \), subject to the constraints explained next to (3.1), and which at \(t=\pm \beta /2\) agree with \(\vert \pmb {m}\rangle \) and \(\vert \pmb {m}'\rangle \) correspondingly.
Adapting the quasistate decomposition to the above seeded states, one gets:
Proposition 3.1
(cf. Prop. 2.1 in [6]). For the expectation value (1.27) corresponding to the seed vector \( \vert D_L\rangle \) and any observable F:
where
and
Behind the complicated looking formula (3.4) is a simple rule which is particularly easy to describe for observables F which are functions of the spins \(S^z_u\). The conditional expectation conditioned on \(\omega \) is obtained by averaging the value of F over spin configurations which vary independently between the loops of \(\omega \). On each loop the spins are constrained to assume only two values, changing the sign upon each Uturn.
Following are some instructive examples:

1.
For each \(\omega \)
$$\begin{aligned} {\mathbb {E}}\left( S^x_u S^x_v  \omega \right) = {\mathbb {E}}\left( S^z_u S^z_v  \omega \right) = (1)^{uv} \, C_S\, \, \mathbb {1}[(u,0) {\mathop {\leftrightarrow }\limits ^{\omega }} (v,0)] \end{aligned}$$(3.6)where \(C_S= \sum _{m=S}^S m^2 / (2S+1)^{2}\) and the spacetime points \((u,0) {\mathop {\leftrightarrow }\limits ^{\omega }} (v,0)\) denotes the condition that (u, 0) and (v, 0) lie on the same loop of \(\omega \).

2.
For the projection operator defined by (1.9)
$$\begin{aligned} {\mathbb {E}}[(2S+1) P_{u,v}^{(0)}  \omega ]= & {} {\left\{ \begin{array}{ll} \,\,\, \,\, 1 &{} \hbox {if} \, (u,0) {\mathop {\leftrightarrow }\limits ^{\omega }} (v,0) \\ (2S+1)^{1} &{} \hbox {if not} \end{array}\right. } \nonumber \\= & {} \Big ( 1+ 2S \ \mathbb {1}[\,(u,0) {\mathop {\leftrightarrow }\limits ^{\omega }} (v,0)\, ] \Big )/ (2S+1)\,. \end{aligned}$$(3.7)
3.2 The \(H_{AF}\) Thermal Equilibrium States
The above representation has a natural extension to the thermal Gibbs states, for which the expectation value functional is given by
In this case the above construction yields a representation in terms of random loop decomposition of \(\Lambda _{L,\beta }\) constructed with the timeperiodic boundary conditions, with loops continuing directly from \(t=\pm \beta /2\). And if the quantum Hamiltonian \( H_{AF}^{(L)} \) is taken with periodic boundary conditions, then also the spacial coordinate is periodic, i.e., the loops are over a torus. Similarly as in (3.2) one gets
where \(N_\ell ^\text {per}(\omega )\) is the number of loops into which the set of lines decomposes with the timeperiodic boundary condition under which \(t=\pm \beta /2\) are identified.
With this adjustment in the assignment of loops to rung configurations, the state’s representation in terms of the loop system with the probability distribution (3.5) remains valid also in the presence of periodicity of either the temporal or spacial direction. This point should be borne in mind in the discussion which follows. In the pseudo spin representation, which is described next, a distinction will appear between the weights of winding versus contractible loops.
From (3.9) and (3.2) we also obtain the following explicit justification for (1.24):
Indeed, for fixed rung configuration \( \omega \), the loops in the denominator are constructed on the timeperiodic version of \(\Lambda _{L,\beta }\) and the loops in the numerator arise in the capped version of \(\Lambda _{L,\beta }\). Since the addition of a rung changes the number of loops by \( \pm 1 \) (depending on whether the two points were already connected by a loop or not), we have \(  N_\ell (\omega )  N_\ell ^\text {per}(\omega )  \le L \) and hence the lower bound in (3.10) follows.
4 The Loop Representation of the Anisotropic XXZModel
4.1 A modified 4edge presentation of the XXZ interaction
We shall now show that the loop measure which appeared quite naturally in the representation of the groundstates of \(H^{(L)}_{{\text {AF}}}\) plays a similar role also for the \(H^{(L)}_{{\text {XXZ}}}\) spin system. Preparing for that, we rewrite the Hamiltonian of the XXZ chain in terms of the slightly modified local interactions consisting of the sum of the following four rankone operators
(written in the braket notation of (1.5), with \(  \pm , \pm \rangle \) the eigenfunctions of \((\tau ^z_v, \tau ^z_{v+1})\)).
The action of \(K_{v,v+1}\) is depicted in Fig. 3 in terms of the four edge configurations with the weights:
In this representation of \(H^{(L)}_{\text {XXZ}}\), the local interaction terms are no longer invariant under spacial reflection, but their sum differs from the more symmetric expression (1.18) only in a boundary term—in fact the one which was included in (1.21) due to this correspondence. Furthermore, this boundary term does not appear in the operators’ periodic version
where the sum extends also to the edge connecting L and \(L+1 \equiv L+1 \). Following is the exact statement.
Lemma 4.1
For any \( L \in {\mathbb {N}} \) and \( \lambda \in {\mathbb {R}} \):
Furthermore, taken with the periodic boundary conditions the two operators agree without the boundary term:
Proof
The action of the sum of the first two edges (a. and b.) agrees with that of \(\left[ \tau ^x_{v} \tau ^x_{v+1}+ \tau ^y_{v} \tau ^y_{v+1}\right] /2 \), which represent the local x and yterms in (1.18). The local zterms in (1.18) and (4.3) agree with the action of the last two edges (c. and d.) in Fig. 3. However, their weight in (1.18) and (4.3) is \( \cosh (\lambda )\) for both edges c. and d. The fact that the summation over all edges in the nonperiodic box \( \Lambda _L \) yields the same result up to a boundary term is checked by noting that for a given spin configuration \( \pmb {\tau } \) the difference between these two cases can be expressed in terms of the number of up and downturns, \( n_\uparrow ^{(L)}(\pmb {\tau })\), \( n_\downarrow ^{(L)}(\pmb {\tau }) \), over the edges of \( \Lambda _L\):
The proof of (4.4) is completed by noting that \( n_\uparrow ^{(L)}(\pmb {\tau })  n_\downarrow ^{(L)}(\pmb {\tau }) = (\tau _L^z  \tau _{L+1}^z) /2 \). In the periodic case, this boundary term drops out. \(\square \)
4.2 A Link Between the \(H_{{\text {XXZ}}}\) and \(H_{{\text {AF}}}\) Loop Measures
Applying the general procedure to the operator \(e^{\beta H^{(L,+)}_{\text {XXZ}}/2}\) written as \(e^{\beta K^{(L)}/2}\) we obtain a representation of states in terms a functional integral over configurations \(\mathbf {\omega }= (\omega ,\tau )\) with binaryvalued functions
whose values may change only at the rungs of \(\omega \), consistently with the edges depicted in Fig. 3. The local condition implies that the allowed functions \(\tau \) are consistent with the loop structure of \(\omega \): Along each loop of \(\omega \) the function \(\tau \) is aligned with either its clockwise of counterclockwise orientation. We denote by \(\mathbb {1}[\omega ,\tau ] \) the indicator function expressing this consistency condition.
Theorem 4.2
For \( \lambda \ge 0 \), any L even and \(\beta \), the expectation value of any function of \(\tau ^z \) in the state defined in (1.29) is given by
with \(\mu _{L,\beta }\) the measure defined in (3.5) at
and the normalized expectation value
with the weights
where the product is over \( (+) \) and \( () \) oriented loops \( \ell \) of \( (\omega ,\tau ) \).
Proof
We spell the proof in the case \(+ \). Proceeding as described in Sect. 2, we get
with weights given by the product over all rungs of \( \omega \) in terms the four types \( \#(\tau ,b) \in \{ a., b., c., d.\} \) listed in (4.2) (cf. Fig. 3):
Lumping the factors by the loops of \(\omega \), for each loop which does not reach the upper and lower boundary of the box \( \Lambda _{L,\beta }\), one gets the total of \(e^{+ \lambda } \) per counterclockwise (\(+\)) and \(e^{ \lambda } \) per clockwise (−) oriented loop. In that case \({{\widetilde{W}}}_+(\omega ,\tau )\) reduces to the above defined \(W_+(\omega ,\tau ) \). Furthermore, with our choice of the seed vector \(\vert N^{(L)}_+\rangle \) that is also true of the loops which are reflected from the upper and/or the lower boundary.
Summing over the \(2^{N_\ell (\omega )}\) possible loop orientations one gets, for each \(\omega \)
Thus, the average in (4.11) is over \((\omega ,\tau )\) with the joint distribution whose marginal distribution of \(\omega \) is the normalized probability measure
with the conditional distribution of \(\tau \) conditioned on \(\omega \) stated in (4.9). \(\square \)
It may be instructive to pause here and compare the different perspectives on the above loop measure. Starting from the analysis of the two different quantum spin chains we arrive at a common system of random rung configurations \(\omega \), whose probability distribution in both models takes the form
with \(\rho _{\Lambda _{L,\beta }}(d\omega )\) a Poisson measure of intensity one. The factor \(\sqrt{Q}^{N_\ell (\omega )} \), by which the measure is tilted, appears through the summation over another degree of freedom, at which point the models differ. More explicitly, in the different systems this common factor is variably decomposed as
where the summations are over functions
The indicator functions impose the consistency condition requiring m or \(\tau \) to be consistent with the loop structure of \(\omega \), i.e., a switch of signs at each Uturn and otherwise be constant along each vertical segment.
Thus, the above system of the random oriented loop described by \(\mathbf {\omega } = (\omega ,\tau )\) can be presented in two equivalent forms:

1.
Locally: as a 4edge model of random oriented lines with the weights listed in Fig. 3.

2.
Globally: by the following two characteristics of its probability distribution \( {\widehat{\mu }}_{L,\beta ,\lambda } \):

(i)
\(\omega \) has the probability distribution \(\mu _{L,\beta }\) which is tilted relative to the Poisson process \(\rho _{\Lambda _{L,\beta }}(d\omega )\) by the factor \(\sqrt{Q}^{N_\ell (\omega )}\)

(ii)
conditioned on \(\omega \), the conditional distribution of \(\tau \) corresponds to independent assignments of orientation to the loops of \(\omega \), at probabilities \( e^{\pm \lambda }/[e^{\lambda }+e^{\lambda }]\) depending on whether the loop is anticlockwise (\(+\)) or cklockwise (−) oriented.
To emphasise the fact that the measure \( {\widehat{\mu }}_{L,\beta , \lambda } \) changes under a change of the sign of \( \lambda \in {\mathbb {R}} \), we keep track of it in the notation.

(i)
The above local to global relation is reminiscent of the Baxter et al. [13] correspondence between the Qstate Potts model and the 6vertex model, which followed the analysis of Temperley and Lieb [34].
In the context of the XXZoperator, the loop picture carries a particularly simple implication for sites at the boundary of \(\Lambda _L\), where the relation of \(\tau (u,t)\) to loop’s helicity is unambiguous. One gets, for the finite volume groundstates:
regardless of the value of L and \( \beta >0\).
4.3 The XXZHamiltonian with the Periodic Boundary Conditions
Under the joint distribution \( {\widehat{\mu }}_{L,\beta , \lambda } \) on oriented loops, the induced measure on \( \tau \)’s restriction to the line \( t = 0 \) was shown to agree with the seeded expectation value of zspins for the XXZHamiltonian with boundary term on \( \Lambda _L \).
This relation takes a simpler form for the thermal state of the XXZHamiltonian taken with periodic boundary conditions (4.3). To express that, we denote by \({\widehat{\mu }}_{L,\beta , \lambda }^\text {per}\) the similarly defined the 4edges measure on \(\Lambda _{L,\beta }\), taken with periodic boundary conditions in both space and time direction.
Theorem 4.3
The marginal distribution of \( {\widehat{\mu }}^\text {per}_{L,\beta , \lambda } \) on orientations \( \tau \) coincides with the quantum expectation of the XXZmodel’s tracial state, i.e., for any finite collection of spacetime points \( (u_j,t_j) \) which are ordered \( t_1< t_2< \cdots < t_N \):
where \( \sigma _j \in \{1,1\} \) are prescribed spin values and \( P_u(\sigma ) := 1 \left[ \tau ^z_{u} = \sigma \right] \) stands for the projection operator onto states with \( \sigma \) as the zcomponent of the spin at u.
Proof
The proof proceeds by plugging the operator \( K^{(L,\text {per})} \) from (4.5) into the loop representation (2.3) for each of the factors \( \exp [(t_jt_{j1} ) H^{(L,\text {per})}_{\text {XXZ}}]\) in the timeordered product in the numerator. The operator \( K^{(L,\text {per})} \) produces exactly the weights of the 4edges model with spatially periodic boundary conditions. The projection operators \( P_{u_j}(\sigma _j)\) inserted behind each factor \(\exp [(t_jt_{j1}) K^{(L,\text {per})}] \) fixes the spinvalue to \( \sigma _j \) at the particular instance \((u_j,t_j) \) in spacetime. Evaluating the trace in the joint eigenbasis of \( \tau ^z_u \) will enforce periodic boundary conditions of the oriented loops also in the time direction. \(\square \)
Since the rightside in (4.19) depends on \( \lambda \) only through the anisotropy parameter \( \cosh (\lambda ) \) entering the periodic XXZHamiltonian, the distribution of the pseudospins is easily seen to exhibit the following symmetry, which will play a crucial role in our proof of dimerization (Theorem 1.2).
Corollary 4.4
Under \( {\widehat{\mu }}^\text {per}_{L,\beta , \lambda } \), the marginal distribution of \( \tau \) is a symmetric function of \( \lambda \).
4.4 Further Symmetry Considerations
As a preparatory step toward the proof of Néel order, let us discuss the symmetries of the oriented loop’s distribution. We start by denoting three mappings on the space of functions \(\tau (u,t)\) which are defined by
and extend the last two to a similarly defined action on the unoriented edge configuration \(\omega \).
The following is a simple, but very helpful observation.
Theorem 4.5
For each finite L, \(\beta \), and \(\lambda \), the above joint probability distribution of \((\omega ,\tau )\) is invariant under \( R\circ F\). Furthermore, in any accumulation point of such measures (e.g., limit \(L\rightarrow \infty \) with L of a fixed parity) which is invariant under the twostep shift (\({\mathcal {S}}^2\)), the magnetization satisfies
for some \(M\in [1,1]\).
To avoid confusion let us stress that (4.21) does not yet establish the existence of Néel order. For that, one needs to show that \(M\ne 0\).
Proof
The first statement follows readily from the above characterization (i)(ii) of the measure, as under reflections the distribution of \(\omega \) is invariant, but the loop’s orientational preference is inverted.
To prove the second statement we combine the above symmetry with the assumed twostep shift invariance. These imply
The full oscillation (4.21) follows by another application of invariance under the double shift \({\mathcal {S}}^2\). \(\square \)
5 The Quantum Loops System’s Critical Percolation Structure
5.1 An FKGType Structure
The probability distribution (3.5) is reminiscent of the loop representation of the planar Qstate randomcluster models. For details on the randomcluster model itself, we refer to the monograph [25] and the lecture notes [17] (for recent developments).
As in that case, it is relevant to recognize here the presence of a selfdual A/Bpercolation model. To formulate it, we partition any rectangle \(\Lambda _{L,\beta } \subset {\mathbb {Z}} \times {\mathbb {R}} \) into a union of vertical columns of width 1 over the edges of \( \Lambda _L \), labeled alternatively as A and B,
with the column over (0, 1) marked as A. Rungs \( \omega \) are then distributed in the edge columns with respect to the probability measure \(\mu _{L,\beta }\).
These rungs serve a dual role. We interpret each as a cut in the column over which it lies and at the same time a bridge linking the two domains which are touched by its endpoints. To visualize the A and Bconnected components, also called A and Bclusters, which result from this convention it is convenient to think of each rung as having a small (infinitesimal) width and being bounded by a pair of segments, as is indicated in Fig. 4.
Thus, associated with each configuration \( \omega \) is a decomposition of \(\Lambda _{L,\beta } \) into Aclusters and Bclusters, with Aclusters bounded by Bclusters, and vice versa. In the topological sense this percolation model is self dual. Also, the probability distribution is symmetric, except possibly for asymmetry introduced by boundary conditions. As is explained below, this implies that the percolation model is at its phase transition point. The transition can be continuous, as is the case for independent percolation (\(Q=1\)), or discontinuous as in models with Q large enough. This distinction is tied in with the existence or not of symmetry breaking in the groundstates of the two quantum models discussed here.
The similarity with the randomcluster measures led [6] to introduce a partial order (\( {\prec }\)) on the space of rung configurations in which Aconnection is monotone increasing and Bconnection is monotone decreasing. More explicitly, labeling the rungs as of A or Btype: \(\omega _1\ \prec \ \omega _2\) if the Aconnections in \(\omega _1\) are all holding in \(\omega _2\). This notion is useful since the measures \(\mu _{L,\beta }\) satisfy the FortuinKasteleynGinibre (FKG) lattice condition which enables powerful monotonicity arguments. The FKG structure was used in the proof of the ANdichotomy [6] stated in Proposition 1.1. Here, we will use the following facts. First, it implies a FKG inequality stating, for every events E and F that are increasing (meaning that their indicator functions are increasing for \(\prec \)):
Another implication of the FKG lattice condition is the monotonicity in socalled boundary conditions. Here, the boundary conditions are imposed by the structure of the underlying graph \(\Lambda _{L,\beta }\), so we wish to draw a comparison with the randomcluster model. The construction with rungs at the top and bottom capping the loops implies that when L is odd (as in Fig. 4), the complement of the box \(\Lambda _{L,\beta }\) is treated as Bconnected, while when L is even it is Aconnected. Borrowing the language of the randomcluster model, we see that our capping procedure used in the construction of \(\mu _{L,\beta }\) can be understood as enforcing Bwired or Awired boundary conditions depending on the parity of L. To stress the type of the boundary condition and to draw an even more direct link to the standard theory of randomcluster models, in this section we write \(\mu _{L,\beta }^\#\) instead of \(\mu _{L,\beta }\), with \(\#=A\) if L is even, and \(\#=B\) if L is odd.
Now, consider \(L\ge \ell \) with \(\ell \) even and \(\beta \ge t\). The measure \(\mu _{\ell ,t}^A\) can be seen as the measure \(\mu _{L,\beta }^\#\) (with \(\#\) equal to A or B depending on L even or odd, or equal to \(\mathrm {per}\) if one wishes) in which we place the socalled Acutter, since either the points in \(\Lambda _{\ell ,t}\) were already Aconnected within \(\Lambda _{\ell ,t}\) or, in case their Aconnection ran through the complement of \(\Lambda _{\ell ,t}\), this will still be true due to the fact that the boundary conditions render this complement into a single Acluster. The monotonicity in boundary conditions therefore implies that for an increasing event E depending on rungs in \(\Lambda _{\ell ,t}\) only,
Likewise, if \(\ell \) is odd and one uses a Bcutter to cut out a smaller box, one gets
5.2 Results Based on the Percolation Analysis
By the monotonicity in the domain, the above FKG structure implies the convergence of the extremal measures, i.e., along increasing sequences of A or Bwired boundary conditions:
in the weak sense of convergence of probability measures on the configuration spaces of rungs on \({\mathbb {Z}} \times {\mathbb {R}} \).
To present the full resolution of the question posed by the dichotomy, we start with the following preparatory statements.
Theorem 5.1
For any \(Q\ge 1\), and regardless of whether the infinitevolume loop measures \(\mu ^A\), \(\mu ^B\) coincide:

1.
Each of these measures is supported on configurations with only closed loops, i.e., there are no infinite boundary lines.

2.
The convergence extends to that of the joint distribution of \((\omega ,\tau )\), i.e., of the ordered loop lines.

3.
The limiting measures’ conditional distribution of \(\tau \), conditioned on \(\omega \), is given by the same rule as in finite volume: at given \(\omega \) the loops are oriented independently of each other with probabilities \(e^{\pm \lambda }/[e^\lambda + e^{\lambda }] \), at \(\lambda \) satisfying \( \sqrt{Q}= e^\lambda + e^{\lambda }\), with \(()\) for clockwise and \((+)\) for counterclockwise orientation.
In case the measures coincide (\(\mu ^A = \mu ^B\)), then

4.
the limiting state is supported on configurations in which there is no infinite Aclusters or Bclusters, and instead each point is surrounded by an infinite family of nested loops;

5.
the loop measures with the periodic boundary conditions in both temporal and spacial direction \( \mu _{L,\beta }^\text {per} \) converge to the shared limit as \( L, \beta \rightarrow \infty \).
The proof of this theorem will follow standard arguments in percolation theory that must still be adapted to the current context.
5.3 Proofs
We begin with three statements that will play important roles. The first one deals with ergodic properties of \(\mu ^A\) and \(\mu ^B\). Let \({\mathcal {S}}_x\) be a translation by \(x\in {\mathbb {Z}}\times {\mathbb {R}}\). This translation induces a shift \({\mathcal {S}}_x\omega \) and \({\mathcal {S}}_xE\) of a configuration and an event. Furthermore, an event E is invariant under translations if for any \(x\in {\mathbb {Z}}\times {\mathbb {R}}\), \({\mathcal {S}}_xE=E\). A measure \(\mu \) is invariant under translations if \(\mu [{\mathcal {S}}_xE]=\mu [E]\) for any event E and any \(x\in {\mathbb {Z}}\times {\mathbb {R}}\). The measure is ergodic if any event invariant under translation has probability 0 or 1.
Lemma 5.2
The measures \(\mu ^A\) and \(\mu ^B\) are invariant under spacial translations by \( 2{\mathbb {Z}} \) and any timetranslation. They are ergodic separately with respect to each of these subgroups.
Proof
We will treat the case of \(\mu ^A\) only, as the case of \(\mu ^B\) is similar. By inclusionexclusion, it is sufficient to consider an increasing event E depending on rungs in \( \Lambda _{\ell ,t}\). Let \( L,k,\ell \in 2{\mathbb {N}} \) with \(L\ge \ell +k\) and \( \beta \ge t + s \). The comparison between boundary conditions implies that for \( x = (k,s) \),
Letting \(L,\beta \) tend to infinity implies the invariance under translations.
Any event can be approximated by events depending on rungs in \(\Lambda _{\ell ,t}\) for some \(\ell ,t\), hence the ergodicity follows from mixing, i.e., from the property that for any events E and F depending on finite sets,
Observe that again by inclusionexclusion, it is sufficient to prove the equivalent result for E and F increasing. Let us give ourselves these two increasing events E and F depending on rungs in \(\Lambda _{\ell ,t}\) only. The FKG inequality and the invariance under translations of \(\mu ^A\) imply that for sufficiently large \( x = (k,s) \) with k even:
In the other direction, fix \(\varepsilon >0\) and choose \(L=L(\varepsilon )\) and \(\beta =\beta (\varepsilon )\) so large than \(\mu ^{A}_{L,\beta }[E]\le \mu ^A[E]+\varepsilon \) and \(\mu ^{A}_{L,\beta }[F]\le \mu ^A[F]+\varepsilon \). If \(x=(k,s)\) with \(2\ell<k<L\ell \), then \(\Lambda _{\ell ,t}\) and its translate by x do not intersect. Thus, the FKG inequality enables to put a unique Acluster in the complement of \(\Lambda _{\ell ,t}\) and \({\mathcal {S}}_x\Lambda _{\ell ,t}\) disconnecting the two areas so that
The result therefore follows by taking x to infinity, and then \(\varepsilon \) to 0. \(\square \)
The second statement is the following important theorem.
Theorem 5.3
For any \( Q \ge 1 \) and \(\#\in \{ A, B \} \), one of the two following properties occur:

\(\mu ^\#[(1/2,0)\text { is Aconnected to infinity}]=0\) or

\(\mu ^\#[\exists \text { a unique infinite Acluster}]=1.\)
Such a result was first proved in [4] for Bernoulli percolation, and was later obtained by other means. For our needs we shall adapt the beautiful argument of Burton and Keane [16]. In the proof given below we give only its brief sketch, as its line of reasoning has by now been presented in many contexts (e.g., [17]).
Proof
We present the proof for \(\mu ^A\), since the proof for \(\mu ^B\) is the same. Let \(E_{\le 1}\), \(E_n\) (\(1<n<\infty \)), and \(E_{\ge 3}\) be the events that there are no more than one, exactly n, and finally at least 3 (possibly infinitely many) infinite Aclusters, respectively. A pair of different arguments will be used to show: i) \(\mu ^A[E_n] =0\) for any \(1<n<\infty \) (in particular \(n=2\)), ii) \(\mu ^A[E_{\ge 3}]=0\). This leaves: \(\mu ^A[E_{\le 1}]=1\).
Assume that \(\mu ^A[E_{n}] >0\) for some \(1<n<\infty \). Then there exist \(\ell \) and t large enough so that \(\mu ^A[F_{\ell ,t}]\ge \tfrac{1}{2}\mu ^A[E_{n}]>0\), where \(F_{\ell ,t}\) is the event that all the infinite Aclusters in \(({\mathbb {Z}}\times {\mathbb {R}})\setminus \Lambda _{\ell ,t}\) (if there are any) intersect \(\Lambda _{\ell ,t}\). The event \(F_{\ell ,t}\) is independent of the rungs in \(\Lambda _{L,\beta }\) and conditioned on it there is a positive probability of the event \(G_{\ell ,t}\) that all the boundary vertices of \(\Lambda _{L,\beta }\) are Aconnected in \(\Lambda _{L,\beta }\). Hence
However, by the translation invariance of the event \(E_{\le 1}\) and the ergodicity of the infinite volume probability distribution, \(\mu ^A[E_{\le 1}]\) can take only the values 0 or 1. Therefore (5.7) implies that \(\mu ^A[E_{\le 1}]=1\), and thus \(\mu ^A[E_n=0]\), contradicting the assumption.
To prove that \(\mu ^A[E_{\ge 3}]=0\) we consider trifurcation events, along the lines of Burton–Keane. A trifurcation event \({\mathcal {T}}^{(\ell )}_{n,m}\), of scale \(\ell \), is said to occur within the box \(B^\ell _{(n,m)} = [n\ell ,(n+1)\ell ]\times [m\ell ,(m+1)\ell ] \subset \mathbb {R}^2\) if for the given \(\omega \) there exists a point within \(B^\ell _{(n,m)}\) which is connected to infinity by three paths among which there is no connection outside the box.
Assume \(\mu ^A[E_{\ge 3}]>0\). Then for \(\ell \) large enough with positive probability (which tends to \(\mu ^A[E_{\ge 3}]\) for \(\ell \rightarrow \infty \)) the box \(B_\ell =[0,\ell ]\times [0,\ell ]\) intersects three distinct infinite clusters.
It is easy to see that for any exterior configuration (with locally finite edge set) for which this condition is met, there exists a nonempty open set of interior configurations for which there is a trifurcation within \(B_\ell \). Since the conditional probability of any nonempty open set of configurations is strictly positive (our analog of BK’s “finite energy” condition), one may conclude that for large enough \(\ell \)
with \(C_\ell >0 \) (by further inspection that extends to all \(\ell \ge 1\), but this refinement is not necessary).
By translation invariance the mean number of trifurcation events of scale \(\ell \) which occur within the finite region \( \Lambda _{L,T} = [0,L]\times [0,T]\) in translates of \(B_\ell \) by \((n,m)\in (2\ell {\mathbb {Z}})^2 \) increases in proportion to the volume:
The Burton–Keane argument is to combine this with the observation that in any configuration with N such trifurcation events within \(\Lambda _{L,T}\) there need to be at least N distinct infinite Aclusters intersecting the boundary of that set. However, this number cannot grow faster than the boundary. More explicitly
where \({\mathsf {N}}_1\) is the number of distinct Aclusters of the half space \((\infty , 0] \times {\mathbb {R}}\) reaching \([0,1]\times [0,1]\), \({\mathsf {N}}_2\) is the number of distinct Aclusters of the half space \({\mathbb {R}}\times (\infty ,0)\) reaching that set, and \({\mathbb {E}}^A()\) denotes the \(\mu ^A\) expectation value.
By elementary (local) estimates \(\mu ^A[{\mathsf {N}}_j]<\infty \), for both \(j=1,2\) (as distinct Aclusters require the crowding of separation events). Combining (5.9) with (5.10), and letting L and T tend to infinity at comparable speeds one learns that for all \(\ell < \infty \)
That is in contradiction with (5.8) (derived under the assumption that \(\mu ^A[E_{\ge 3}]>0\)), which proves our claim. \(\square \)
We shall also use the following statement.
Lemma 5.4
All Bclusters are finite \(\mu ^A\)almost surely.
By duality, \(\mu ^B\)almost surely all the Aclusters are finite almost surely.
Proof
Assume by contradiction that \(\mu ^A[\exists \text { infinite Bcluster}]=1\) and fix \(L,\beta \) so that
and the probability that the top, bottom, left and right of the boundary of \(\Lambda _{L,\beta }\) are Bconnected to infinity in the complement of \(\Lambda _{L,\beta }\) are the same (simply fix \(L_0,\beta _0\) large enough to get (5.12), and then increase \(L_0\) and/or \(\beta _0\) in order to obtain \(L,\beta \)).
Since a path from infinity to \(\Lambda _{L,\beta }\) ends up either on the top, bottom, left or right of it, the FKG inequality implies (through the squareroot trick^{Footnote 5}) that
and similarly for the right, bottom and top. Now, assume that the top and bottom are Bconnected to infinity, and the left and right (more precisely the Alines on the left and right of the respective boundaries) are Aconnected to infinity (the probability of the latter is larger than the probability that there exists a Bconnection since under \(\mu ^A\) the Aclusters dominate the Bones^{Footnote 6}). The union bound implies that this happens with probability \(1\tfrac{4}{10}>0\). Yet, the finiteenergy property also implies that conditionally on this event, the rungs in \(\Lambda _{L,\beta }\) are such that no boundary vertex of \(\Lambda _{L,\beta }\) are Aconnected using paths in \(\Lambda _{L,\beta }\), implying that there exist two infinite Aclusters with positive \(\mu ^A\)probability. But this contradicts the fact, proved in the previous statement, that there is zero or one infinite Acluster.\(\square \)
We are now in a position to prove this section’s main result.
Proof of Theorem 5.1
Property 1 is a direct consequence of the fact that \(\mu ^A\) does not possess any infinite Bcluster. It also means that when fixing a finite set, and taking L and \(\beta \) large enough, no loop intersecting the finite set reaches the boundary of \(\Lambda _{L,\beta }\) or winds around the vertical direction. By construction, we deduce that all these loops are oriented in an independent fashion described in the previous section. As a consequence, Properties 2 and 3 follow trivially. Property 4 is a direct consequence of \(\mu ^A=\mu ^B\), so that the distribution of the A and Bclusters is the same under \(\mu ^A\). In particular, there is no infinite A or Bcluster, which immediately implies Property 4. Finally, if the two measures are equal, \(\mu ^{\mathrm {per}}_{L,\beta }\) stochastically dominates \(\mu ^B_{L,\beta }\) and is stochastically dominated by \(\mu ^A_{L,\beta }\). Since these measures converge to the same measure \(\mu ^A=\mu ^B\), so does \(\mu ^\mathrm {per}_{L,\beta }\). \(\square \)
6 Proofs of Symmetry Breaking
We now have the tools for a structural proof of the different forms of symmetry breaking in the models considered here. We start with the translation symmetry breaking in the limiting distribution of the random loop measure for all \(Q>4\). This is then used to conclude dimerization in the groundstates of the \(H_{{\text {AF}}}\) spin chains with \(S>1/2\), and Néel order in the groundstates of spin 1/2 XXZchain at \(\Delta >1\). These results can be obtained through the rigorous analysis of the Bethe ansatz along the lines of DuminilCopin et al. [20], which also yields more quantitative information. However, for a shorter and somewhat more transparent proof we present an analog of the argument which was recently developed by Ray and Spinka [32] in the context of the 6vertex/Qstate randomcluster model on \(\mathbb {Z}^2\).
6.1 Translation Symmetry Breaking for the Loop Measure at \(Q>4\)
As in [24, 32] we shall make an essential use of a random height function \(h: (\mathbb {R}\setminus \mathbb {Z}) \times \mathbb {R}\mapsto \mathbb {Z}\), which in our case is assigned the configurations of \(\mathbf {\omega }= (\omega ,\tau ) \). The function is piecewise constant with discontinuities at lines supporting the loops of \(\omega \). Along the horizontal line \(t=0\) it is defined by:
More generally, the value of \(h_{ \mathbf {\omega }}(u,t)\) at any point off the loop lines of \(\omega \) is the sum of the fluxes of \(\tau \) across an arbitrary simple path from (1/2, 0) to (u, t), counted with the sign of the cross product of the direction of \( \tau \) with the curve’s tangent at the point of crossing. In this description of the height function one may restrict the attention to paths which avoid rungs, i.e., which crosses loops only at vertical boundaries. (To avoid misunderstanding, let us add that when the crossing occurs at horizontal rungs of \(\omega \) the hfunction increases by 0 or \(\pm 2\), depending on the orientations of the two loops which cross the rung and the direction of crossing.) An example of this construction is presented in Fig. 5.
As a random function, h(x, t) exhibits a surprising combination of properties:

1.
the function’s statistical distribution is simplest to present in the language of the loops, whose structure depends on all the loops of \(\omega \) (i.e., the erasure of any rung does change the loop structure).

2.
yet, for any \((\omega ,\tau )\) the values of \(h_{ \mathbf {\omega }}(u,t)\) at generic points can be read from just the pseudo spin function \(\tau \) (i.e., it does not require the full knowledge of \(\omega \)).
More explicitly, we have the following auxiliary statement. What makes it particularly astounding is the combination of the above with the third assertion listed below.
Lemma 6.1
For any \(Q>4\), let \( {\widehat{\mu }}\) be a probability measure on the systems of oriented loops over \(\mathbb {Z}\times \mathbb {R}\), described by the variables \( \mathbf {\omega }= (\omega , \tau ) \), with the properties:

1.
\( {\widehat{\mu }}\) almost surely the loop configurations corresponding to \(\omega \) consist of only finite loops.

2.
conditioned on \(\omega \) the loops are oriented independently of each other clockwise (−) or counterclockwise \((+)\), which the probabilities \(e^{\pm \lambda }/[e^{\lambda } +e^{+\lambda }]\), correspondingly,

3.
the marginal distribution of \(\tau \) variables is invariant under a global reversal of orientation (\( \lambda \mapsto \lambda \)).
Then the event
has zero measure.
Proof
The proof is by contradiction. Let \(\alpha _k = (u_k,t_k)\) be a sequence of points with \(\alpha _k \nearrow \infty \) each of which lies on a level set of \(h_{ \mathbf {\omega }}\) which includes a path winding around \((1/2,0)\). The values of \(h(\alpha _k)\) are given by the sum of loop orientations over those loops of \(\omega \) which separate \(\alpha _k\) from \((1/2,0)\). For \(Q>4\), these orientations are given by a sequence of iid \(\pm 1\) valued random variables, of the nonzero mean \(\tanh (\lambda )\). It readily follows that almost surely the following limit exists and satisfies
From this one may deduce that the probability distribution of \(\lim _{k\rightarrow \infty } h(\alpha _k) \) is:
The level sets of \(h_{ \mathbf {\omega }}\) can be determined from just \(\tau \) (i.e., ignoring \(\omega \)) and, under the assumption made here its distribution does not change under the flip \(\lambda \rightarrow \lambda \). Hence (6.4) yields a contradiction unless \({\widehat{\mu }}({\mathcal {N}})=0\). \(\square \)
From this we shall now deduce three symmetry breaking statements. The first concerned just the loop measure, but in the statement’s proof we make use of the measure’s significance for the two quantum spin models.
Theorem 6.2
For all \(Q >4\) (equivalently \(S>1/2\)) the loop measures corresponding to the even and odd groundstates \(\langle \cdot \rangle _{\text {even}} \) and \(\langle \cdot \rangle _{\text {odd}} \) of (1.11) differ.
Proof
Assume the two loop measures coincide. Then by Theorem 5.1 (4) the event \({\mathcal {N}}\) occurs with probability 1 with respect to the common probability measure.
However, by Theorem 5.1 (5), under the above assumption these measures also describe the limiting distribution of \(\omega \) corresponding to the groundstate of the periodic operator \(H^{(L,per)}_{XXZ}\) in the limit \(L\rightarrow \infty \), which for concreteness sake we take to be along \(2\mathbb {N}\) and with \(\lambda \) being the positive solution of (4.8). However, as was noted in Corollary 4.4 the periodic boundary condition state is actually an even function of \(\lambda \). It follows that the limiting state satisfied all the assumptions made in Lemma 6.1, and hence the event \({\mathcal {N}}\) is of probability zero.
The contradiction between the two implications of the above assumption implies that the measures are distinct. \(\square \)
6.2 Dimerization for \( S>1/2\)
Next, we extract from the above probabilistic statement a proof of dimerization in the quantum \(H_{AF}\) spin chain.
Proof of Theorem 1.2
1. From Theorem 6.2 we already know that for any \(S>1/2\) the loop measures associated with the states \( \mu ^A \equiv \langle \cdot \rangle _{\text {even}} \) and \(\mu ^B \equiv \langle \cdot \rangle _{\text {odd}} \) are different. Theorem 5.3 allows to identify the difference in percolation terms: in both cases there is almost surely a unique infinite connected cluster, which is of type A in one and B in the other case. More explicitly,
However, that leaves still the challenge to determine whether this difference between the two measures and, in each, between the even and odd sites, can be detected in terms of a physical observable, i.e., the expectation value of some function of the spin degrees of freedom.
The question was addressed in [6] where it is shown (see also the next section for a similar reasoning) that:

(i)
since the two limiting distributions of \(\omega \) are related by the FKG inequality, if they differ then the difference is also manifested in the more elementary connectivity probability to lie on the same loop, and in particular for even sites \(u=2n\):
$$\begin{aligned} \mu ^A\left[ (2n,0) {\leftrightarrow } (2n+1,0) ] \right] \ < \ \mu ^A\left[ (2n1,0) {\leftrightarrow } (2n,0) ] \right] . \end{aligned}$$(6.6) 
(ii)
combining (6.6) with (3.7) one learns that
$$\begin{aligned} \langle P_{2n,2n+1}^{(0)} \rangle _{\text {even}} \ < \ \langle P_{2n1,2n}^{(0)} \rangle _{\text {even}} \end{aligned}$$(6.7)with the opposite inequality for the odd state. Thus the differences in the two states are detectable and extensive.
2. Theorem 7.2 of [6] bounds the truncated quantum correlations from above in terms of the correlations
of the underlying loop measures for every \( \pmb {u} \in U \pm 1/2 \) and \( \pmb {v} \in V \pm 1/2 \). Now, the percolation model considered in Section 1.3 of [18] corresponds exactly to our model here.^{Footnote 7} The fifth bullet of Theorem. 1.5 of [18] thus implies, with the notation of our paper, that there exists \(c=c(q)>0\) such that for every \(n\ge 1\),
Since for \( \pmb {u} \in U \pm 1/2 \) and \( \pmb {v} \in V \pm 1/2 \) to be connected to each other, there must exist a path from \(\pmb {u}\) to the translate of \(\partial \Lambda _{n,n}\) by \(\pmb {u}\), where \(n=\max \{t,\pmb {u}\pmb {v}1\}\), we deduce that all the quantities in (6.8) are smaller than \(C\exp [({\text {dist}}(U,V)+t)/\xi ]\) for some small enough constant \(\xi =\xi (q)>0\). \(\square \)
6.3 Néel Order for \(\Delta =\cosh (\lambda )>1\)
Turning to the groundstates of the XXZspin system, let us recall the notation. Let \(\lambda >0\) be the positive solution of (1.19), and denote by \({\widehat{\mu }}^{A}_{+ \lambda } \) the even limit (cf. (5.5)) of the measure on the enhanced system of the variables \( (\omega ,\tau )\), in which the winding probabilities of loops of \(\omega \) are \(e^{\pm \lambda }/(e^\lambda + e^{\lambda } ) \), with \((+)\) winding corresponding to the counterclockwise and \(()\) the clockwise orientation. The corresponding state on (just) the spin variables is denoted \(\langle \tau (u,0) \rangle _+\). These are to be contrasted with \({\widehat{\mu }}^{A}_ \) and \(\langle \tau (u,0) \rangle _\). The superscript may be omitted, but it should be remembered that in both \(+\) and − cases, the limit \(L\rightarrow \infty \) is taken over the even sequence.
Proof of Theorem 1.4
As we saw, the probability distribution of \(\omega \) under the two measures \({\widehat{\mu }}^{A}_{\pm \lambda } \) agree with that of the corresponding \(H_{AF}\) system. Hence, in both cases there is positive probability \( p_\infty >0 \) that (1/2, 0) belongs to an infinite connected cluster. When that happens, the sign of \(\tau (0,0)\) coincides with the winding sign of the loop which passes through (0, 0). Thus the spin \( \tau (0,0) \) takes the values \( \pm \) with probabilities \( e^{\pm \lambda }/(e^\lambda + e^{\lambda } ) \). The same holds true for any even site \(u=2n\). Consequently, for any \(n\in \mathbb {Z}\):
Since \(p_\infty > 0\), we deduce that \({\widehat{\mu }}^{A}_+ \) and \({\widehat{\mu }}^{A}_ \) are different.
To that let us add the observation that percolation with respect to \(\omega \) corresponds to percolation along a level set of the height function which is readable from \(\tau \). Hence the observable which distinguishes the two states is in principle a functional of the physically meaningful spin function. We conclude that also the states \(\langle \cdot \rangle _+\) and \(\langle \cdot \rangle _\) are different, and hence the infinite XXZspin system has at least two different groundstates.
A remaining challenge is to simplify the distinction between the two states, as was done in step (ii) of the above proof of Theorem 1.2. For the XXZmodel that can be deduced using the last statement in Proposition 1.3. It allows to conclude from (6.10) (and the previously established fact that \(p_\infty >0\)) that for any \(u\in \mathbb {Z}\)
This also implies the nonvanishing of the Néel order parameter M of (4.21).
\(\square \)
In the last step, leading to (6.11), we invoked an “FKG boost” whose full discussion was postponed in order to streamline the presentation of the main results. Following is its proof.
Proof of Proposition 1.3
The convergence of each of the two finitevolume groundstates in the limit \(L \rightarrow \infty \), with L limited to even values, is based on the FKG monotonicity of the percolation model discussed in the previous section, which is common to the two systems discussed here.
It remains to establish that for the XXZ system at any \(\Delta \ge 1\), the states \(\langle \cdot \rangle _+ \) and \(\langle \cdot \rangle _\) are either equal or of different magnetization, satisfying (6.11). In the proof we shall again employ the FKG inequality but do so in a different setup than used above. This time it will be in the context of an Isinglike representation of the distribution of the staggered spins
In terms of these variables the groundstates of XXZsystem take the form of an annealed Gibbs equilibrium state of a ferromagnetic Ising model, and the two boundary conditions correspond to \((+)\) and correspondingly \(()\) fields applied along the “vertical” part of the boundary.
To present the groundstates in this form we start with the following preparatory steps:

(1)
Rewrite the XXZ Hamiltonian of (1.20), with the boundary term of (1.21), as the spin1/2 version of the \(H_{AF}\) operator with an added antiferromagnetic coupling of strength \(\delta = \Delta 1\). I.e., for any even L:
$$\begin{aligned} U_L H^{(L, \pm )}_{\text {XXZ}}U_L ^*= & {} \frac{1}{2} \sum _{v=L+1}^{L1}\, \left[ \pmb {\tau }_{v} \cdot \pmb {\tau }_{v+1} + \pm \delta \tau ^z_{v} \tau ^z_{v+1}  \Delta \cdot \mathbb {1}\right] \nonumber \\&+ \frac{ \sinh (\lambda ) }{2} [ \tau _{L+1}^z  \tau _L^z ] \,. \end{aligned}$$(6.13) 
(2)
For each even L and choice of the ± sign, construct the groundstate of this operator, applying the hybrid representation (2.10), with the \(\delta \) term treated as a potential (V) added to the \(\Delta =1\) operator, and starting from the seed state \(  N_0^{(L)} \rangle \) of (1.28) with \( \lambda = 0 \).
We are particularly interested in operators which arise from functionals of \( \tau \), or equivalently of \(\kappa \) (the two being related by (6.12)). Taking the limits indicated in (1.29), we get
with the Feynman–Kac type functional integral
with
where, in a slight abuse of notation, we denote by \(\mathbb {1}[\omega ,\kappa ]\) the consistency indicator function which is inherited from \(\mathbb {1}[\omega ,\tau ]\) through the correspondence (6.12), and the potential is
It is now important to note that in terms of \(\kappa \) the consistency condition translated into the constraint that \(\kappa (u,t)\) is constant along each of the loops of \(\omega \). Thus, for a given \(\omega \) the function \(\kappa (u,t)\) is fully characterized by the collection of binary variables \(\{ \kappa _\gamma \}\) indexed by the loops of \(\omega \). Furthermore, the potential \(V_\pm (t)\) is expressible as a ferromagnetic pair interaction among the values of this collection of observables.
By the general FKG property of the ferromagnetic Ising measures, for each \(\omega \) the \((+)\) and \(()\) measures \( {\mathbb {E}}_{L,\beta }^\pm \left[ \cdot  \omega \right] \) admit a monotone coupling (this is referred to as Strassen’s theorem). That is, there exists a joint probability distribution \( {\hat{\nu }}_\omega (\kappa _+,\kappa _) \), with marginals given by the two measures \( {\mathbb {E}}_{L,\beta }^\pm \left[ \cdot  \omega \right] \) and which is supported on pairs of configurations satisfying
In terms of such a coupling, for any functional \(F[\kappa ]\):
and for \(F[\kappa ] \) monotone nondecreasing the terms on the right are all nonnegative.
In studying the limit \(L,\beta \rightarrow \infty \) it is convenient to measure the distance of the two induced measures within rectangular spacetime domains of the form \(B_{K,T}=[K,K]\times [T,T]\), through the Wassersteintype metric
where \(\nu _\omega \) ranges over couplings of the two measures \( {\mathbb {E}}_{L,\beta }^\pm \left[ \cdot  \omega \right] \).
For the monotone coupling the absolute value can be dropped, in which case the integral reduces to the simple difference in expectation values of \(\kappa \), and thus
In the infinite volume limit, in which the mean value of \(\kappa (u,t)\) is translation invariant, one gets
We learn that if \(\langle \tau (0,0) \rangle _+ =0\) then for any \(K,T<\infty \) the Wasserstein distance between the restrictions of the two measures to the box \(B_{K,T}\) tends to zero. It follows that if the measures converge, as we know to be the case here, then they have a common limit.
In other words, if for some measurable functional of \(\tau \), \(\langle F \rangle _+ \ne \langle F\rangle _\), then we may conclude that also \( \langle \tau (0,0) \rangle _+ \ne \langle \tau (0,0) \rangle _ \,. \) In view of the relation between the two states the latter is equivalent to
\(\square \)
7 Postscript—Quantum Degrees of Freedom as Emergent Features
The analysis presented here provides another example where the categorical distinction between classical and quantum physics is blurred. We started with two quantum spin chains and moved on to their relation with a common random loop model. An alternative presentation could have started from the random loop model, based on the random rung configurations \(\omega \), which is of independent interest in probability theory and statistical mechanics and then proceed by recognising that this system’s features can be best understood through emergent quantum degrees of freedom.
The utility of such crossings of the quantum/classical divide has been noted before: In one direction, the thermodynamic of the planar Ising model is best explained in terms of emergent quantum degrees of freedom, among which are Bruria Kaufmann’s spinors [26] and Lieb–Mattis–Schultz fermions [33]. In the other direction one finds Feynman–Kac functional integral representations for thermal states of quantum particle system in terms of classical functional integrals, and analogous formulas for quantum spin chains, such as employed in [3, 6, 15, 22, 23, 35, 36].
Notes
The projection \(P_{u,v}^{(0)}\) can also be expressed as a polynomial of degree 2S in \(\pmb {S}_u\cdot \pmb {S}_{v}\), for instance \(P_{u,v}^{(0)} = ((\pmb {S}_u\cdot \pmb {S}_{v} )^21)/3\) for \(S=1\).
This version of the AN dichotomy is a bit more carefully crafted than in the original work, as the two options stated there need not be mutually exclusive. However, as (1.16) shows, ipsofacto they are.
Various features of the model are calculable through the Bethe ansatz, which was actually developed in that context [8]. However, even aside from the extra care which is required for rigorous results, the exact determination of the long distance asymptotic seems to require other means (cf. [2, 21, 30] and references therein).
One may expect that in case there is Néel order any antisymmetric boundary field would flip the groundstate into one of the extremal states. However the proof of that is simpler for the case the field’s magnitude is at least \(\sinh (\lambda )\).
The squareroot trick refers to the \(k=2\) case fo the observation that for every increasing events \(A_1,\ldots ,A_k\), the FKG inequality implies that \(\max \{\mu [A_i]  1\le i\le k\}\ge 1\mu [\left( A_1\cup \cdots \cup A_k\right) ^c]^{1/k}\). This inequality is an improvement on the union bound, as it shows that if the union of k increasing events has a probability close to 1, then this is also true for at least one of these events.
By symmetry the Bclusters under \(\mu ^A\) are distributed as the Aclusters under \(\mu ^B\), which by FKG is dominated by \(\mu ^A\).
One may be surprised by the fact that the Poisson point process there has intensity 1 and q depending on the column. This comes from the fact that the Radon–Nikodym derivative is expressed in [18] as q to the number of Aclusters, which can be shown, using Euler’s formula, to be expressed in terms of \(\sqrt{q}\) to the number of loops if one change the intensity of the Poisson point process to 1 in every column.
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Acknowledgements
Open Access funding provided by Projekt DEAL. We thank Ron Peled and Yinon Spinka, Edward Witten and Bruno Nachtergaele for stimulating discussions and relevant references. MA is supported in parts by the NSF Grant DMS1613296, and the Weston Visiting Professorship at the Weizmann Institute of Science. HDC was supported by the ERC CriBLaM, the NCCR SwissMAP, the Swiss NSF and an IDEX Chair from ParisSaclay. SW is supported by the DFG under EXC2111 – 390814868.
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Aizenman, M., DuminilCopin, H. & Warzel, S. Dimerization and Néel Order in Different Quantum Spin Chains Through a Shared Loop Representation. Ann. Henri Poincaré 21, 2737–2774 (2020). https://doi.org/10.1007/s00023020009242
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DOI: https://doi.org/10.1007/s00023020009242