Dimerization in quantum spin chains with O(n) symmetry
Jakob E. Björnberg, Peter Mühlbacher, Bruno Nachtergaele, Daniel Ueltschi
224 January 2021
DIMERIZATION IN QUANTUM SPIN CHAINS WITH O ( n ) SYMMETRY
JAKOB E. BJ ¨ORNBERG, PETER M ¨UHLBACHER, BRUNO NACHTERGAELE,AND DANIEL UELTSCHI
Abstract.
We consider quantum spins with S ≥
1, and two-body interactions with O (2 S +1) symmetry. We discuss the ground state phase diagram of the one-dimensional system.We give a rigorous proof of dimerization for an open region of the phase diagram, for S sufficiently large. Contents
1. Introduction 11.1. A family of quantum spin chains with O ( n )-invariant interactions 21.2. Ground state phase diagram for general n ≥ S = 1 model ( n = 3) 41.4. Our result about dimerization 62. Graphical representation for O ( n ) models 73. The contour model 93.1. Contours 93.2. Domains and admissibility of contours 143.3. Decomposition of H ( ω ) 174. Proof of dimerization 194.1. Cluster expansion for the partition function 194.2. Dimerization 23Appendix A. The interaction uT + vP when n is even 25 References
Introduction
Over the course of almost a century of studying quantum spin chains, physicists and math-ematicians have uncovered a wide variety of interesting physical phenomena and in the processinvented an impressive arsenal of new mathematical techniques and structures. Nevertheless,our understanding of these simplest of quantum many-body systems is still far from complete.For many models of interest we have only partial information about the ground state phasediagram, the nature of the phase transitions, and the spectrum of excitations. We considerhere a family of spin systems with two-body interactions, where interactions are translationinvariant and O (2 S + 1) invariant. We investigate the ground state phase diagram, looking forground states that possess less symmetry than the interactions. Our main result is a rigorousproof of dimerization (where translation invariance is broken) in a region of the phase diagramwith S large enough. Mathematics Subject Classification. © a r X i v : . [ m a t h - ph ] J a n J.E. BJ ¨ORNBERG, P. M¨UHLBACHER, B. NACHTERGAELE, AND D. UELTSCHI
The family of models is introduced in Section 1.1; the phase diagram for general S ≥ S = 1 has received a lot of attention and we discuss itexplicitly in Section 1.3; our result about dimerization is stated in Section 1.4.The O ( n ) models have a graphical representation which we describe in Section 2. We use itto define a “contour model” in Section 3 where contours are shown to have small weights. Thisallows to use the method of cluster expansion and prove dimerization in Section 4.1.1. A family of quantum spin chains with O ( n ) -invariant interactions. We considera family of quantum spin chains consisting of 2 (cid:96) spins of magnitude S defined by a nearest-neighbor Hamiltonian H (cid:96) acting on the Hilbert space H (cid:96) = ( C n ) ⊗ (cid:96) , with n = 2 S + 1 ≥
2, ofthe form H (cid:96) = (cid:96) − (cid:88) x = − (cid:96) +1 h x,x +1 , (1.1) where h x,x +1 denotes a copy of h = h ∗ ∈ M n ( C ) ⊗ M n ( C ) acting on the nearest neighbor pairat sites x and x + 1.We are interested in the family of interactions h = uT + vQ, u, v ∈ R , (1.2) where T is the transposition operator defined by T ( φ ⊗ ϕ ) = ϕ ⊗ φ , for φ, ϕ ∈ C n , and Q isthe orthogonal projection onto the one-dimensional subspace of C n ⊗ C n spanned by a vectorof the form ψ = 1 √ n n (cid:88) α =1 e α ⊗ e α , (1.3) for some orthornormal basis { e α | α = 1 , . . . , n } of C n .The spectrum of h is easy to find. T has the eigenvalues 1 and −
1, corresponding to thesymmetric and antisymmetric subspaces of C n ⊗ C n , whose dimensions are n ( n + 1) / n ( n − /
2, respectively. Since ψ is symmetric, the eigenvalues of h are u + v, u, − u .Let R be a linear transformation represented by an orthogonal matrix in the basis { e α } ,meaning (cid:104) e α , RR T e β (cid:105) = δ αβ . This amounts to defining a specific representation of O ( n ) on thesystem under consideration. It is then straightforward to check ( R ⊗ R ) ψ = ψ . It follows that R ⊗ R commutes with Q = | ψ (cid:105)(cid:104) ψ | . Since T also commutes with R ⊗ R , the Hamiltonians withinteraction h given in (1.2) have a local O ( n ) symmetry. This family of models is in fact, up toa trivial additive constant, the most general translation-invariant nearest neighbor Hamiltonianfor spins of dimension n and with a translation-invariant local O ( n ) symmetry.To make contact with previous results in the literature, it is useful to note a couple ofequivalent forms of the spin chains we consider. First, for integer values of S , that is odddimensions n , consider the orthonormal basis { e α } , relabeled by α = − S, . . . , S , and related tothe standard eigenbasis of the third spin matrix S (3) , satisfying S | α (cid:105) = α | α (cid:105) , as follows: for α = 0 take e = i S | (cid:105) , and for α > e α = i S − α √ (cid:0) | α (cid:105) + | − α (cid:105) (cid:1) , e − α = i S − α +1 √ (cid:0) | α (cid:105) − | − α (cid:105) (cid:1) . (1.4) Then, we have ψ = φ := 1 √ n S (cid:88) α = − S ( − S − α | α, − α (cid:105) , (1.5) which is the SU (2) singlet vector in the standard spin basis. The transposition operator T is ofcourse not affected by any translation-invariant local basis change. Therefore, for odd n , and IMERIZATION IN QUANTUM SPIN CHAINS WITH O ( n ) SYMMETRY 3 uv AA’ BB’ CReshetikhin v = − nn − u ferromagneticdimerization incommensuratephase correlationsMatrix-product state v = − u Figure 1.
Ground state phase diagram for the chain with nearest-neighborinteractions uT + vQ for n ≥
3. Our main result, Theorem 1.1, is a proof ofdimerization in an open region around the point B’.with a simple change of basis, the family of interactions (1.2) is seen to be equivalent to˜ h = uT + vP, u, v ∈ R , (1.6) where P is the orthogonal projection onto the singlet state φ .The case of even n is different. Interactions h and ˜ h are not unitarily equivalent. But themodel with interaction ˜ h is nonetheless interesting and we discuss it in Appendix A. We alsoprove dimerization in this case, see Theorem 1.2.For n ≥ u = 0, and v = −
1, this is the much studied − P (0) spin chain [6, 1, 10, 4, 15, 14, 3].1.2. Ground state phase diagram for general n ≥ . We start with the phase diagramfor arbitrary n ≥ n = 3 in Section 1.3. The ground state phasediagram of the spin chain with nearest-neighbor interactions h x,x +1 = uT x,x +1 + vQ x,x +1 isdepicted in Fig. 1. It can be broadly divided into four domains.The domain formed by the quadrant u ≤ , v ≥ h x,x +1 for all x ; that is, they are frustration-free. The ground state energy per bond is equal to u . Indeed, let ϕ = (cid:80) α c α e α with (cid:80) α | c α | =1. It is clear that | ϕ ⊗ ϕ (cid:105) is eigenstate of T with eigenvalue 1; further, we have (cid:104) ϕ ⊗ ϕ | Q | ϕ ⊗ ϕ (cid:105) = 1 n (cid:12)(cid:12)(cid:12)(cid:88) α c α (cid:12)(cid:12)(cid:12) . (1.7) The latter is zero when (cid:80) α c α = 0. Since Q is a projector, such a state is eigenstate with eigen-value 0. Notice that the state Rϕ also satisfies this condition, for all orthogonal transformation R . The product state ⊗ (cid:96)x = − (cid:96) +1 ϕ is then a ground state of h x,x +1 with eigenvalue u , for all x .In addition to these product states, we can obviously take linear combinations. J.E. BJ ¨ORNBERG, P. M¨UHLBACHER, B. NACHTERGAELE, AND D. UELTSCHI
The next domain is the arc-circle between ( u, v ) = ( − ,
0) and the “Reshetikhin point”with v = − nn − u (yellow region in Fig. 1), which features dimerization. In order to see thatdimerization is plausible as soon as v <
0, let ϕ x,x +1 = (cid:112) − ε | S, S (cid:105) + ε √ n − S − (cid:88) α = − S | α, α (cid:105) . (1.8) and consider consider the (partially) dimerized state ϕ (cid:96) +1 , − (cid:96) +2 ⊗ ϕ − (cid:96) +3 , − (cid:96) +4 ⊗ . . . . For ε = 0,this is a product state, but for ε (cid:54) = 0 it is not. Roughly half the edges, namely the edges x, x + 1with x = − (cid:96) + 1 , − (cid:96) + 3 , . . . , are dimerized and their energy is (cid:104) ϕ x,x +1 | uT x,x +1 + vQ x,x +1 | ϕ x,x +1 (cid:105) = u + vn (1 + 2 √ n − ε ) + O ( ε ) . (1.9) The non-dimerized edges contribute (cid:104) ϕ x − ,x ⊗ ϕ x +1 ,x +2 | uT x,x +1 + vQ x,x +1 | ϕ x − ,x ⊗ ϕ x +1 ,x +2 (cid:105) = u + vn + O ( ε ) . (1.10) The average energy per bond of the state ϕ x,x +1 is then u + vn + v √ n − n ε , up to O ( ε ) corrections.When v < u + vn (using (1.7) with (cid:80) α c α = 1), so thepartially dimerized state (1.8) has lower energy when ε is positive and small.Our main result is that dimerization does occur in an open domain around the point B’,provided n is sufficiently large, see Theorem 1.1. This extends the results of [14, 3], valid atthe point B’.Then comes the domain formed by the arc-circle between the Reshetikhin point v = − nn − u and ( u, v ) = (1 ,
0) (red region in Fig. 1). For n odd a unique translation-invariant ground stateis expected.This domain contains several interesting special cases. The direction ( u = 1 , v = 0) is thethe SU ( n ) generalization of the spin-1/2 Bethe-ansatz solvable Heisenberg model studied bySutherland and others [19]. The direction v = − nn − u was solved by Reshetikhin [17] (thisgeneralizes the Takhtajan–Babujian model for n = 3). These models are gapless. The direction v = − u is a frustration free point and the ground states are given matrix-product states. Forodd n , these are generalizations if the AKLT model. The ground state for the infinite chain isunique and is in the Haldane phase. For even n , there are two matrix-product ground statesthat break the translation invariance of the chain down to period 2 [23].The final domain is the quadrant u, v >
0. The ground states are expected to have slowdecaying correlations with incommensurate phase correlations. That is, spin-spin correlationsbetween sites 0 and x are expected to be of the form | x | − r cos( ω | x | ) for | x | large, and where r, ω depend on the parameters u, v [9].It is perhaps worth mentioning that the phase diagram for spatial dimensions other than 1is quite different. Dimerization is not expected. Instead, the system displays various forms ofmagnetic long-range orders (ferromagnetic, spin nematic, N´eel, . . . ). See [25] for results aboutmagnetic ordering for all n ≥ The S = 1 model ( n = 3 ). For n = 3, the family of models is equivalent to the familiarspin-1 chain with bilinear and biquadratic interactions. The latter is most often parametrizedby an angle φ as follows:cos φ (cid:126)S x · (cid:126)S x +1 + sin φ ( (cid:126)S x · (cid:126)S x +1 ) = 3(sin φ − cos φ ) P + cos φ T + sin φ I. (1.11) We can apply the change of basis that is the inverse of Eq. (1.4), namely | (cid:105) = − i e , | (cid:105) = √ ( e − i e − ) , | − (cid:105) = √ ( e + i e − ) . (1.12)IMERIZATION IN QUANTUM SPIN CHAINS WITH O ( n ) SYMMETRY 5 cos φ sin φ A SutherlandA’ BB’ C Takhtajan-BabujianAKLT (tan φ = )ferromagnetic dimerizationincommensuratephase correlations Figure 2.
Ground state phase diagram for the S = 1 chain with nearest-neighbor interactions cos φ(cid:126)S x · (cid:126)S x +1 + sin φ ( (cid:126)S x · (cid:126)S x +1 ) . The domains and thepoints are the same as those in Fig. 1.Then the interaction is given by (1.11) but with the operator Q instead of P .The ground state phase diagram with parameter φ is depicted in Fig. 2. The domains andthe points are the same as that of Fig. 1. The ferromagnetic domain corresponds to φ ∈ ( π , π ),and the model is frustration-free in this range. Among the ground states, there is a family ofproduct states that shows that the O (3) symmetry of the Hamiltonian is spontaneously broken.As a consequence, the Goldstone Theorem [12] implies that there are gapless excitations abovethe ground state in this region. The dimerization domain is φ ∈ ( π , π ). The next domain is φ ∈ ( − π , π ) with unique, translation-invariant ground states. Finally, the domain φ ∈ ( π , π )is expected to display states with slow decay of correlations, with incommensurate phase.There are several points where exact and/or rigorous information is available: (i) φ ∈ [0 , π/ φ = 1 /
3, it is the spin-1 AKLT chain [2] with interaction ˜ h given by the orthogonalprojection on the spin-2 states. In the thermodynamic limit, it has a unique ground stateof Matrix Product form with a non-vanishing spectral gap and exact exponential decay ofcorrelations; (ii) the two points with tan φ = 1, A and A’ in Fig. 2, have SU (3) symmetryand are often referred to as the Sutherland model [19]. An exact solution for the ground stateat φ = − π/ φ = π/ φ = − π/ φ = − π/
2, is the − P (0) spin-1 chain, already mentioned above. Aizenman, Duminil-Copin, and Warzel proved thatit has two dimerized (2-periodic) ground states with exponential decay of correlations [3]; allevidence indicates that these states are gapped.Let us briefly comment on higher spatial dimensions. Dimerization is not expected. Variousrigorous results about long-range order have been established: for φ = 0 [8]; for φ (cid:38) π [21, 25];and for φ (cid:46) n ) group. It is possiblethat these methods could be applied to the chain. J.E. BJ ¨ORNBERG, P. M¨UHLBACHER, B. NACHTERGAELE, AND D. UELTSCHI
Our result about dimerization.
Let R x,y = T x,y + nQ x,y , and let (cid:104) R x,y (cid:105) (cid:96),β,u = 1Tr e − βH (cid:96) Tr R x,y e − βH (cid:96) , (1.13) where the hamiltonian is given by (1.1) with h x,x +1 = uT x,x +1 − Q x,x +1 . (Thus we take v = − Theorem 1.1.
There exist constants n , c > such that for n > n and | u | < n − / , we have lim inf β →∞ (cid:104) (cid:104) R , (cid:105) (cid:96),β,u − (cid:104) R − , (cid:105) (cid:96),β,u (cid:105) > c for all (cid:96) odd; lim sup β →∞ (cid:104) (cid:104) R , (cid:105) (cid:96),β,u − (cid:104) R − , (cid:105) (cid:96),β,u (cid:105) < − c for all (cid:96) even. -3-4 -2 -1 0 1 2 3 4 5-3 -2 -1 0 2 3 4 (cid:96) = 5, odd (cid:96) = 4, even Figure 3.
Illustration for dimerization. Depending on whether (cid:96) is even orodd, the site x = 0 is more entangled with its left or its right neighbor.This establishes the existence of at least two distinct infinite-volume ground states, close tothe point B’ of the phase diagram (see Fig. 3). Notice that the same result holds if we replacethe operators R x,y with spin operators S (3) x S (3) y , diagonal in the basis { e α } .We expect that there are exactly two extremal ground states, precisely given by limits (cid:96) → ∞ along odd or even integers. We also expect that, if we take the chain to be {− (cid:96), − (cid:96) + 1 , . . . , (cid:96) } ,the corresponding infinite-volume ground state is equal to the average of the two extremalstates.Dimerization has been established in [14, 3] at the point B’ in the phase diagrams of Figs1 and 2. The earlier result [14] uses the loop representation of [4] combined with a Peierlsargument; it holds for S ≥ n ≥ S ≥ n ≥ S (or n ) wheredimerization is expected. It uses the loop representation and random cluster representation of[4] as well as recent results for the two-dimensional random cluster model [7, 16].Away from the point B’ these methods do not quite apply. Here we use the loop repre-sentation of [25], which combines those of [22, 4], in order to get a contour model. The looprepresentation involves a probability measure for u, v ≤ u .These steps are performed in the rest of this article. We introduce the loop model in the nextsection; the relation with quantum spins is stated in Theorem 2.1. We establish dimerizationin the loop model in Theorem 4.6 below, which is equivalent to Theorem 1.1, thus proving ourmain result.We now discuss the case of the spin chain with Hamiltonian˜ H (cid:96) = (cid:96) − (cid:88) x = − (cid:96) +1 (cid:0) uT x,x +1 + vP x,x +1 (cid:1) , (1.14)IMERIZATION IN QUANTUM SPIN CHAINS WITH O ( n ) SYMMETRY 7 where P is projection onto the singlet state (recall (1.6)). We have a similar result aboutdimerization. Let ˜ R x,y be “suitable” operators at sites x, y (these operators are defined in theappendix). Let (cid:104) ˜ R x,y (cid:105) ˜ (cid:96),β,u = 1Tr e − β ˜ H (cid:96) Tr ˜ R x,y e − β ˜ H (cid:96) , (1.15) Theorem 1.2.
Let v = − . There exist constants n , c > such that for n > n and | u |
0. To each vertex and edge ofthis graph we associate a periodic time interval T β = ( − β, β ) per to obtain a set of space-timevertices V (cid:96),β := V (cid:96) × T β as well as a set of space-time edges E (cid:96),β := E (cid:96) × T β .By a configuration ω we mean a finite subset of E (cid:96),β , each point of ω receiving a mark or. The points of ω will collectively be called links , those marked being referred to as crosses and those marked as double-bars . We write ω = ( ω , ω ) and denote the set of all such (link)configurations Ω (cid:96),β .To every configuration ω ∈ Ω (cid:96),β corresponds a set of loops ; see Fig. 4 for an illustration. Aloop l is a closed, injective trajectory[0 , L ] per → V (cid:96) t (cid:55)→ l ( t ) = ( v ( t ) , T ( t )) , such that x ( t ) is piecewise constant and T (cid:48) ( t ) ∈ {± } . We call L ≡ | l | the length of l , thatis the smallest L > t ∈ [0 , L ] provided that { x ( t − ) , x ( t +) } × T ( t ) contains a link. We have T (cid:48) ( t +) = − T (cid:48) ( t − ) in case that link is a doublebar and T (cid:48) ( t +) = T (cid:48) ( t − ) in case it is a cross. We identify loops with identical support and weoccasionally abuse notation and identify a loop with the set of links it traverses. The numberof loops in a configuration ω is denoted L ( ω ).For u ∈ R , we define the following signed measure on the set Ω (cid:96),β of link configurations ω :d¯ ρ u ( ω ) = u | ω | d ⊗ n x, (2.1) where n = | ω | is the number of links in ω , and d x is the Lebesgue measure on E (cid:96),β . We alsointroduce the following normalized measure ρ u , satisfying ρ u (Ω (cid:96),β ) = 1:d ρ u ( ω ) = e − (1+ u )2 β | E (cid:96) | d¯ ρ u ( ω ) (2.2) If u is positive, the measure ρ u is a positive measure and hence a probability measure; in fact,under this measure ω has the distribution of a Poisson point process with intensity u for crossesand intensity 1 for double-bars . But we also allow small, negative u . Let Z (cid:96),β,n,u := (cid:90) Ω (cid:96),β d ρ u ( ω ) n L ( ω ) −| ω | . (2.3) J.E. BJ ¨ORNBERG, P. M¨UHLBACHER, B. NACHTERGAELE, AND D. UELTSCHI This loop model is equivalent to the quantum spin system, and the next result is an instanceof this equivalence. The equivalence goes back to T´oth [22] and Aizenman–Nachtergaele [4] forspecial choices of the parameters; the general case of the interaction (1.2) is due to [25]. Notethat it holds for arbitrary finite graphs, not only for chains. Recall that R x,y = T x,y + nQ x,y . Theorem 2.1.
For the Hamiltonian (1.1) with h x,x +1 = − uT x,x +1 − Q x,x +1 , we have that(a) Tr e − βH (cid:96) = e β (1+ u ) | E (cid:96) | Z (cid:96),β,n,u .(b) Tr R x,y e − βH (cid:96) = e β (1+ u ) | E (cid:96) | (cid:90) Ω (cid:96),β d ρ u ( ω ) n L ( ω ) −| ω | (cid:0) n + ( n + 1 − n )1l[ x → y ] (cid:1) . Here x ↔ y characterizes the set of configurations ω where ( x, and ( y, belong to the sameloop. The sign of the parameter u in the definition of the interaction has indeed changed; but thetheorem holds for arbitrary real (or even complex) parameters. Proof.
The proof of (a) can be found in [25, Theorem 3.2] so we only prove (b). Proceeding inthe same way, we getTr T x,y e − βH (cid:96) = e β (1+ u ) | E (cid:96) | (cid:90) Ω (cid:96),β d ρ u ( ω ) n L (˜ ω ) −| ˜ ω | , (2.4) where ˜ ω is the configuration ω with an additional cross at sites x, y and time 0. We write x (cid:54)↔ y to characterize the set of configurations ω where ( x,
0) and ( y,
0) belong to distinctloops; x + ←→ y where the top of ( x,
0) is connected to the bottom of ( y, x − ←→ y wherethe top of ( x,
0) is connected to the top of ( y,
0) (see [25, Fig. 2] for an illustration). We thenhave L (˜ ω ) = L ( ω ) − x (cid:54)↔ y, L ( ω ) + 1 if x + ←→ y, L ( ω ) if x − ←→ y. (2.5) ThenTr T x,y e − βH (cid:96) = e β (1+ u ) | E (cid:96) | (cid:90) Ω (cid:96),β d ρ a,b ( ω ) n L ( ω ) −| ω | (cid:16) n x (cid:54)↔ y ]+ n x + ←→ y ]+1l[ x − ←→ y ] (cid:17) . (2.6) In a similar fashion, we have thatTr Q x,y e − βH (cid:96) = e β (1+ u ) | E (cid:96) | (cid:90) Ω (cid:96),β d ρ a,b ( ω ) n L ( ω ) −| ω | (cid:16) n x (cid:54)↔ y ]+ 1 n x + ←→ y ]+1l[ x − ←→ y ] (cid:17) . (2.7) We get the claim (b) by grouping these two expressions, using that 1l[ x + ←→ y ] + 1l[ x − ←→ y ] =1l[ x ↔ y ]. (cid:3) From now on and to the end of this article we work with the loop model.
Remark 2.1 (Intuition) . It is helpful to think of ρ u as an a-priori measure on a gas of loops,and rewrite the integrand n L ( ω ) −| ω | as e − (log n ) H ( ω ) , with ‘Hamiltonian’ − H ( ω ) := L ( ω ) − | ω | , (2.8) and inverse temperature log n . Thinking of n as large, the Laplace principle tells us that ‘typ-ical’ configurations should maximise n L ( ω ) −| ω | . Our goal is to write Z (cid:96),β,n,u as a dominantcontribution from such maximizers, and some excitations. IMERIZATION IN QUANTUM SPIN CHAINS WITH O ( n ) SYMMETRY 9 Using language suggested by a Peierls argument, excitations will be encoded in contours , aswill be defined below.In light of Theorem 2.1, we introduce the following measure: µ (cid:96),β,n,u ( f ) = Z (cid:96),β,n,u [ f ] Z (cid:96),β,n,u , where Z (cid:96),β,n,u [ f ] := (cid:90) Ω (cid:96),β d ρ u ( ω ) n L ( ω ) −| ω | f ( ω ) . (2.9) In terms of this measure, and using Theorem 2.1, the quantity (cid:104) R , (cid:105) (cid:96),β,u − (cid:104) R − , (cid:105) (cid:96),β,u appear-ing in Theorem 1.1 may be written as (dispensing with subindices) (cid:104) R , (cid:105) − (cid:104) R − , (cid:105) = ( n + 1 − n ) (cid:0) µ (0 ↔ − µ (0 ↔ − (cid:1) . (2.10) In proving Theorem 1.1, we must therefore show that the origin is either more likely to beconnected to its left neighbor or to its right neighbor, depending on the parity of (cid:96) .We end this section with the following remark about working with a signed measure. Sincethe (possibly signed) measure ρ u is closely related to the probability measure ρ , it is easy tosee that any event A satisfying ρ ( A ) = 0 also has zero measure under ρ u . In fact, we need thefollowing slightly stronger property: Lemma 2.2. If A is an event such that ρ ( A ) = 0 and f : Ω (cid:96),β → R is a ρ -integrable function,then for any u ∈ R we have that (cid:90) A d ρ u ( ω ) f ( ω ) = 0 . (2.11) Proof.
Using (2.1) and (2.2) it is easy to see that (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) A d ρ u ( ω ) f ( ω ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:90) A d ρ ( ω ) | f ( ω ) | = 0 , (2.12) for some finite constant C depending only on u, (cid:96), β . (cid:3) As a consequence, we may assume that crosses and double-bars occur at different times, alsowhen u < ρ u carries signs. We implicitly used this property when definingloops. 3. The contour model
Contours.
We classify loops as follows, see Fig. 4. A loop is long if it visits three or moredistinct vertices and short otherwise. A loop is contractible if it can be continuously deformedto a point and winding otherwise. Not all loops are contractible since our time interval T β isperiodic.We define a canonical orientation of the space-time vertices V (cid:96),β , using the directions up( ↑ ) and down ( ↓ ), by orienting the leftmost space-time vertex {− (cid:96) + 1 } × T β down ↓ andrequiring that neighbouring space-time vertices have opposite orientations; see Fig. 5. We write V ↑ (cid:96) := { x ∈ V (cid:96) : x + (cid:96) is even } for the set of vertices with up-orientation, and V ↓ (cid:96) := { x ∈ V (cid:96) : x + (cid:96) is odd } for the set of vertices with down-orientation, and introduce the following subsetsof the edge-set E (cid:96) : E + (cid:96) := (cid:8) ( x, x + 1) ∈ E (cid:96) : x ∈ V ↓ (cid:96) , x + 1 ∈ V ↑ (cid:96) (cid:9) ,E − (cid:96) := E (cid:96) \ E + (cid:96) = (cid:8) ( x, x + 1) ∈ E (cid:96) : x ∈ V ↑ (cid:96) , x + 1 ∈ V ↓ (cid:96) (cid:9) . (3.1) We define E + (cid:96),β and E − (cid:96),β , as well as V ↑ (cid:96),β and V ↓ (cid:96),β , analogously.These definitions are motivated as follows. We expect that ‘typical’ configurations ω containmany short, contractible loops. To maximize the number of short contractible loops one placesonly double-bars in E + (cid:96) , as in Fig. 5. The canonical orientation is chosen so that all the short (cid:96)β − β − (cid:96) +1 Figure 4.
A configuration ω consisting of three short loops (green, brown,purple), one long loop (red) and two winding loops (blue, orange). (a) (b) -2 -2-1 -10 01 12 23 3 Figure 5. (a) The canonical orientation of V (cid:96),β with the set E + (cid:96) highlightedred. (b) a configuration ω with many short, contractible loops; these are posi-tively oriented under the canonical orientation.loops in such a configuration are positively oriented (i.e. counter-clockwise). The canonicalorientation will be useful in classifying the excitations away from such ‘typical’ ω . Also notethat if the origin 0 belongs to a short, positively oriented loop, then we have 0 ↔ (cid:96) oddand 0 ↔ − (cid:96) even. To prove our main result Theorem 1.1, in the form (2.10), we willessentially argue that the origin is likely to belong to a short, positively oriented loop.Given a loop l in a configuration ω , we define a segment of l as a trajectory of l between twotimes 0 ≤ s < s ≤ L ( l ) when l passes through height β . That is to say, l ( s ) = ( v , β ) , l ( s ) =( v , β ) for some v i ∈ V (cid:96) , while l does not pass through height β in times t ∈ ( s , s ). We saythat a segment is spanning if for every t ∈ T β there exists a v = v ( t ) ∈ V (cid:96) such that the segmenttraverses ( v, t ). Note that a spanning segment is not necessarily part of a winding loop. SeeFig. 6. Definition 3.1 (Pre-contours) . We say that two loops are connected if they share a link. A pre-contour is then a maximally connected set of long or winding loops.
We call a pre-contour odd if it has an odd number of spanning segments, otherwise we callit even . IMERIZATION IN QUANTUM SPIN CHAINS WITH O ( n ) SYMMETRY 11 + β − β Figure 6.
The leftmost and rightmost loops are winding loops with one span-ning segment each. The loop in the middle is contractible. There are fourspanning segments in total.
Lemma 3.2.
Let ω ∈ Ω (cid:96),β ; then (i) the number of spanning segments is even and (ii) thenumber of odd pre-contours is even.Proof. Any segment contains either 1 or 2 points of the form ( v, β ), and it is 1 if and only ifthe segment is spanning. The first claim follows from the fact that the total number of pointsof the form ( v, β ) is even (it is equal to 2 (cid:96) ). The second claim follows from the first togetherwith the fact that even pre-contours account for an even number of spanning segments. (cid:3)
Remark 3.3.
For later reference, we note here that any cross which is traversed by someloop in a pre-contour is necessarily traversed both ways by the pre-contour; see Fig. 7. γ Figure 7.
A cross is traversed by a pre-contour γ . If the red loop visits athird vertex, it is a long loop; otherwise it must be a winding loop. In bothcases, it is actually part of γ .A pre-contour which contains at least one winding loop will be called a winding pre-contour .Note that the set of winding pre-contours admits a natural ordering from left to right. This isbecause any two winding loops which cross each other necessarily belong to the same contour.In particular, since an odd pre-contour is necessarily also a winding pre-contour, the set of odd pre-contours can be ordered from left to right. We use this ordering to pair odd pre-contours:first the leftmost odd pre-contour is paired with the rightmost one, then the leftmost unpairedodd pre-contour is paired with the rightmost unpaired odd pre-contour, etc. See Fig. 8. Definition 3.4 (Contours) . A contour γ is defined as an even pre-contour or two paired oddpre-contours. + β − β Figure 8.
Two contours: one consisting of two paired odd pre-contours, andone consisting of four loops.Define the (vertical) length of a contour as the sum of the (vertical) lengths of its loops, | γ | := (cid:80) l ∈ γ | l | .We need a notion of interior of a contour, and for this it is useful to regard our configuration ω as living in the bi-infinite cylinder C β = T β × R . More precisely, given ω we consider thesubset ω of C β obtained as the union of (i) V β embedded in C β in the natural way, and (ii) thelinks of ω embedded as straight line segments connecting adjacent points of V β . Note that, inthe embedding ω , crosses and double-bars are embedded in the same way. For a loop l of ω ,define its support S ( l ) as the subset of ω traced out by l , meaning the union of the vertical andhorizontal line segments of ω corresponding to the intervals of V β and the links of ω traversedby l . For a contour γ of ω we then make the following definitions. • The support S ( γ ) is the union of the supports S ( l ) of the loops l belonging to γ . Notethat S ( γ ) is a closed subset of C β . • The exterior E ( γ ) is the union of the unbounded connected components of C β \ S ( γ ).Note that E ( γ ) is open. • The interior I ( γ ) := C β \ E ( γ ). Note that I ( γ ) is an open set. • The boundary B ( γ ) := E ( γ ) \ E ( γ ) which is a closed set.These notions are illustrated in Figs 9–11.Having defined I ( γ ) as a subset of the cylinder C β , we may also regard I ( γ ) (or more precisely,its closure I ( γ )) as a subset of E (cid:96),β by identifying a point ( x, x + 1) × { t } ∈ E (cid:96),β with the closedline-segment from ( x, t ) to ( x +1 , t ) in C β . Similarly, S ( γ ) and B ( γ ) may be regarded as subsetsof V (cid:96),β ∪ ω . We freely switch between these points of view.Fixing a contour γ , note that the boundary B ( γ ) consists of a collection of closed curves andhorizontal line segments (of length 1). We use the canonical orientation of V (cid:96),β to orient eachvertical segment of B ( γ ). It is not hard to see that this gives a consistent orientation of all theclosed curves constituting B ( γ ). (This follows from Remark 3.3.) Recall the standard notionof a positively oriented curve as one whose interior is always on the left. IMERIZATION IN QUANTUM SPIN CHAINS WITH O ( n ) SYMMETRY 13 Figure 9.
A configuration ω with three contours highlighted green, blue andred. The green contour consists of a pair of odd pre-contours. Figure 10.
The corresponding embedding ω ⊆ C β = T β × R , with the sup-ports S ( γ ) of the contours highlighted with the corresponding colors. Figure 11.
The interiors of the corresponding contours with the boundaries B ( γ ) receiving the canonical orientation. The green and blue contours are ofpositive type (interiors I ( γ ) on the left) while the red contour is of negativetype (interior on the right). Definition 3.5 (Type of a contour) . We say that the contour γ is of positive type if thecanonical orientation of B ( γ ) is positive in the sense that I ( γ ) is on the left of each closed curve of B ( γ ) . Otherwise we say that γ is of negative type (being of negative type is equivalentto the interior being on the right). Remark 3.6.
Suppose that ω ∈ Ω (cid:96),β is such that a given point ¯ v ∈ V (cid:96),β is not on or insideany contour, that is to say ¯ v ∈ (cid:92) γ ∈ Γ( ω ) E ( γ ) , (3.2) where E ( γ ) is the exterior of γ defined above. Then we have that ¯ v is on a positively oriented short loop. Indeed, this is related to the fact that all external contours are of positive type, seeLemma 3.8. Domains and admissibility of contours.
We now introduce several notations anddefinitions pertaining to contours and how they relate to each other. First, given ω ∈ Ω (cid:96),β we define Γ( ω ) = { γ , . . . , γ k } as the set of contours in the configuration ω . Here, and inwhat follows, a contour may be identified with the set of links it traverses. The collection ofall possible contours will be denoted X (cid:96),β = (cid:83) ω ∈ Ω (cid:96),β Γ( ω ), and we write X + (cid:96),β ⊆ X (cid:96),β for thecollection of positive-type contours. We write X (cid:96),β = (cid:91) k ≥ (cid:18) X (cid:96),β k (cid:19) and X + (cid:96),β = (cid:91) k ≥ (cid:18) X + (cid:96),β k (cid:19) (3.3) for the set of finite collections of contours, respectively positive-type contours. Elements of X (cid:96),β and of X + (cid:96),β will usually be denoted by Γ. It is important to note that far from every such setΓ of contours can be obtained as Γ( ω ) for some ω ∈ Ω (cid:96),β ; in fact, we will devote some effort toidentifying criteria under which such an ω does indeed exist. We say that Γ ∈ X (cid:96),β is admissible if Γ = Γ( ω ) for some ω ∈ Ω (cid:96),β , and write A (cid:96),β = Γ(Ω (cid:96),β ) for the collection of admissible sets ofcontours.Recall that the interior I ( γ ) of a contour γ is by definition an open subset of the cylinder C β .Also recall that we regard E (cid:96),β as a closed subset of C β by identifying a point ( x, x + 1) × { t } ∈ E (cid:96),β with the closed line-segment from ( x, t ) to ( x + 1 , t ). We now define the (interior) domains of γ as follows. Definition 3.7.
A domain D of γ is a subset of E (cid:96),β ∩ I ( γ ) which, when regarded as a subsetof C β as above, is connected, satisfies D ∩ S ( γ ) = ∅ , and is maximal with these properties. We define the type of a domain in a similar way to the type of a contour. Namely, we orientthe (topological) boundary of D consistenly with the canonical orientation of V (cid:96),β and say that D is of positive type if this is a positive orientation (interior on the left), and of negative type otherwise. See Fig. 12.Given two contours γ and γ (cid:48) , we say that γ is a descendant of γ (cid:48) , writing γ ≺ γ (cid:48) , if S ( γ ) ⊆ D for some domain D of γ (cid:48) . Given Γ ∈ X (cid:96),β and γ, γ (cid:48) ∈ Γ, we say that γ is an immediatedescendant of γ (cid:48) in Γ if γ ≺ γ (cid:48) and there is no γ ∈ Γ satisfying both γ ≺ γ and γ ≺ γ (cid:48) . Itis important to note that the notion of being an immediate descendant depends not only onthe two contours γ and γ (cid:48) but on the set Γ; in other words, immediate descendancy cannot bechecked in a pairwise manner. If γ ∈ Γ is not the descendant of any other contour γ (cid:48) ∈ Γ thenwe say that γ is an external contour; this notion is also dependent on the set Γ. Lemma 3.8.
Fix Γ ∈ X (cid:96),β . Then Γ is admissible, i.e. Γ ∈ A (cid:96),β , if and only if the followinghold:(1) all external contours in Γ are of positive type;(2) for any pair of distinct contours γ, γ (cid:48) ∈ Γ we have that either I ( γ ) ∩ I ( γ (cid:48) ) = ∅ or γ ≺ γ (cid:48) or γ (cid:48) ≺ γ ; IMERIZATION IN QUANTUM SPIN CHAINS WITH O ( n ) SYMMETRY 15 D D D Figure 12.
A contour γ of positive type, containing three domains D , D , D .Domains D and D are of negative type, while D is of positive type. (3) if γ is an immediate descendant of γ (cid:48) , in a domain D of γ (cid:48) , then the types of γ and of D coincide.Proof. It is easy to see that the three conditions above hold for any admissible Γ = Γ( ω ). Toshow the converse, we construct an explicit ω ∈ Ω (cid:96),β with Γ( ω ) = Γ. Starting from the emptyconfiguration ω = ∅ ∈ Ω (cid:96),β , add all links of all external contours and then place a double barat height 0, say, on each e ∈ E + (cid:96) that does not have any link on it. This defines ω such thatΓ( ω ) is precisely the set of external contours of Γ. Next, add the links of all contours whichare immediate descendants of external contours. This does not create any new long loops apartfrom those in these contours because their types coincide with those of the domains they arein. Iterate this procedure until there are no more contours left to add. (cid:3) An important prerequisite for applying a cluster expansion is to be able to verify the admis-sibility of a set of contours in a pairwise manner. As indicated above, and in the light of Lemma3.8, this is not directly possible since the notion of being an immediate descendant depends onthe whole set Γ. We get around this issue by introducing a notion of compatibility which appliesto sets of positive-type contours Γ ∈ X + (cid:96),β , and which can be checked in a pairwise manner. Wethen show that there is a bijective correspondence between admissible and compatible sets ofcontours.The bijective correspondence referred to above involves shifting contours and rests on thesimple observation that if γ is a negative-type contour, then γ (cid:48) = γ + (1 ,
0) (i.e. γ translated tothe right one unit) is a positive-type contour.Given a positive-type contour γ ∈ X + (cid:96),β with domains D ( γ ) , . . . , D k ( γ ) ⊆ I ( γ ), we definethe appropriately shifted domains D + i ( γ ) of γ by D + i ( γ ) = (cid:26) D i ( γ ) , if D i ( γ ) is of positive type ,D i ( γ ) + (1 , , otherwise . (3.4) Note that while D + i ( γ ) ⊆ I ( γ ), a shifted domain may intersect the boundary B ( γ ). See Fig.13. Definition 3.9.
Given two positive-type contours γ, γ (cid:48) ∈ X + (cid:96),β , we say that γ and γ (cid:48) are com-patible if one of the following hold:(1) I ( γ ) ∩ I ( γ (cid:48) ) = ∅ , or(2) S ( γ ) ⊆ D + i ( γ (cid:48) ) for some i , or(3) S ( γ (cid:48) ) ⊆ D + i ( γ ) for some i . Figure 13.
A contour γ and its two appropriately shifted domains (shadedareas). The lower one was not moved since it already was positive type. Theupper one was shifted one column to the right. If a γ (cid:48) ∈ X (cid:96),β gets placed insideit, S − (Γ = ( γ, γ (cid:48) )) will return an admissible collection of contours. We define δ ( γ, γ (cid:48) ) = (cid:40) if γ, γ (cid:48) are compatible otherwise. (3.5) Finally, we let C + (cid:96),β ⊆ X + (cid:96),β to be the collection of all pairwise compatible sets of positive-typecontours; that is, Γ = { γ , . . . , γ k } ∈ X + (cid:96),β belongs to C + (cid:96),β if (cid:81) ≤ i Σ : C + (cid:96),β → X (cid:96),β as follows. First, given Γ ∈ C + (cid:96),β and γ ∈ Γ,write σ Γ ( γ ) for the number of contours γ (cid:48) ∈ Γ \ { γ } such that γ ⊆ D + i ( γ (cid:48) ) (cid:54) = D i ( γ (cid:48) ). Thisrepresents the number of times γ is shifted to the right in order to obtain the compatible set Γfrom an admissible set of contours. We defineΣ(Γ) = { γ − ( σ Γ ( γ ) , 0) : γ ∈ Γ } . (3.6) Lemma 3.10. The shift Σ is a bijection from C + (cid:96),β , the collection of compatible sets contours,to A (cid:96),β , the collection of admissible sets of contours.Proof. It is easy to construct an inverse Σ − of Σ on A (cid:96),β , as follows. Given Γ ∈ A (cid:96),β , startwith an external contour γ (which is of positive type by Lemma 3.8) and form its appropriatelyshifted domains D + i ( γ ). In doing so, shift also the descendants of γ along with their domains.Note that all the immediate descendants of γ are then mapped to positive type contours. Theniteratively continue this procedure for the (shifted) immediate descendants of γ . The resultingset Σ − (Γ) then satisfies Definition 3.9.It remains to show that Σ(Γ) ∈ A (cid:96),β for all Γ ∈ C + (cid:96),β , i.e. that Σ(Γ) satisfies Lemma 3.8. Itis clear that external contours are of positive type since they are not shifted. For γ, γ (cid:48) withdisjoint interiors, this property is preserved by Σ; if γ ⊆ D + i ( γ (cid:48) ) then the shifting ensures thatthe images of γ, γ (cid:48) under Σ satisfy γ ≺ γ (cid:48) , while the relative amounts by which the contoursare shifted ensures that the types of immediate descendants in Σ(Γ) coincide with the types ofthe relevant domains. (cid:3) We close this subsection with a simple lemma about counting the amount of ‘available space’for short loops in a configuration ω , in terms of the lengths of the contours. For Γ ∈ A (cid:96),β ,we define the free set F (Γ) ⊆ E (cid:96),β as the space-time edges where we can add links withoutmodifying the contours in Γ or creating new ones. Lemma 3.11. Let Γ ∈ A (cid:96),β be an admissible set of contours. Then | F (Γ) | = | E + (cid:96) | − (cid:80) γ ∈ Γ | γ | . (3.7) Proof. We need to show that 2 | E + (cid:96),β | = 2 | F (Γ) | + (cid:80) γ ∈ Γ | γ | . Note that 2 | E + (cid:96),β | = | V (cid:96),β | and that2 | F (Γ) | equals the total length of all the short loops. But any point in V (cid:96),β lies either on acontour or on a short loop, thus | V (cid:96),β | = 2 | F (Γ) | + (cid:80) γ ∈ Γ | γ | , as required. (cid:3) Decomposition of H ( ω ) . Recall from (2.8) the quantity − H ( ω ) = L ( ω ) − | ω | . Wenow show that H ( ω ) can be decomposed as a sum over contours and we prove bounds onthe summands. To this end, for a loop l let T ( l ) denote the number of turns that l makes;symbolically T ( l ) = γ , write T ( γ ) for the total number of turns ofall loops in γ . Next define the function h : X (cid:96),β → Z by h ( γ ) = L ( γ ) − T ( γ ) (3.8) where L ( γ ) denotes the number of loops in the contour γ . Lemma 3.12. For ω ∈ Ω (cid:96),β with contours Γ = Γ( ω ) we have − H ( ω ) = (cid:80) γ ∈ Γ h ( γ ) .Proof. Since every double-bar of ω accounts for exactly two turns (of either one or two loops),we have − H ( ω ) = (cid:88) l (cid:0) − T ( l ) (cid:1) (3.9) where the sum is over all loops l in the configuration ω . The result now follows from theobservation that short, non-winding loops make exactly two turns. (cid:3) Write γ for the number of double-bars visited by γ and γ for the number of crosses. Lemma 3.13. The function h : X (cid:96),β → Z satisfies h ( γ ) ≤ − γ + γ + 2 (cid:96) { γ has a spanning segment } . (3.10) Note that the constant − is tight for the smallest non-winding contours with six double-barsand no crosses, while for larger contours the constant may be taken closer to − . The constant1 for γ is not optimal, but only affects the range of u which we may consider later on. Asto the indicator function, we will see that contours containing spanning segments become veryrare asymptotically. Proof. For a loop l , write C ( l ) for the number of traversals of crosses by l , symbolically C ( l ) = C ( γ ) = (cid:80) l ∈ γ C ( l ). Write W ( l ) for the number of winding segments in l and W ( γ ) = (cid:80) l ∈ γ W ( l ). We claim that it suffices to show that h ( γ ) ≤ r ( γ ) where r ( γ ) = − T ( γ ) + C ( γ ) + W ( γ ) . (3.11) Indeed, r ( γ ) is bounded above by the right-hand-side of (3.10) for the following reasons: • double-bars visited twice by γ count twice in T ( γ ) but only once in γ , while thosevisited once by γ count once in both, meaning that T ( γ ) ≥ | γ | ; • C ( γ ) = 2 γ since γ visits each cross exactly twice, by Remark 3.3; • W ( γ ) ≤ (cid:96) { γ has a spanning segment } since each point of the form ( x, ∈ V (cid:96),β isvisited by at most one winding segment.Next, the claimed inequality h ( γ ) ≤ r ( γ ) is equivalent to: T ( γ ) + 3 C ( γ ) + 6 W ( γ ) ≥ L ( γ ) . (3.12) To establish (3.12), first note that both sides are additive over loops. Thus it suffices to showthat any long or winding loop l satisfies T ( l ) + 3 C ( l ) + 6 W ( l ) ≥ . (3.13) If W ( l ) ≥ C ( l ) isnecessarily even, by parity constraints, meaning that we may also assume that C ( l ) = 0, i.e. thatthe loop traverses only double-bars. A long, non-winding loop which traverses only double-barsnecessarily makes at least 6 turns, see Fig. 15. This proves (3.13) and hence the claim. (cid:3) Figure 15. A long, non-winding loop makes at least 6 turns. IMERIZATION IN QUANTUM SPIN CHAINS WITH O ( n ) SYMMETRY 19 Proof of dimerization Cluster expansion for the partition function. Recall the partition function from (2.3) Z (cid:96),β,θ,u = e − (1+ u ) | E (cid:96),β | (cid:90) Ω (cid:96),β d¯ ρ u ( ω ) n L ( ω ) −| ω | (4.1) where d¯ ρ u ( ω ) = u | ω | d ⊗ n x is given in (2.1). Since we identify contours γ ∈ X (cid:96) with the linksthey are made up of, d¯ ρ u ( γ ) is also well defined. Let w ( γ ) := e − (1+ u ) | γ | n h ( γ ) u | γ | , (4.2) where h ( γ ) is defined in (3.8). Proposition 4.1. We have, for any u ∈ R , Z (cid:96),β,θ,u = e − (1+ u ) | E − (cid:96) | (cid:88) k ≥ k ! (cid:90) X + (cid:96) d¯ ρ ( γ ) · · · (cid:90) X + (cid:96) d¯ ρ ( γ k ) (cid:16) k (cid:89) i =1 w ( γ i ) (cid:17)(cid:16) (cid:89) ≤ i Fix any (cid:96) ∈ N , β > c > and points ( v, s ) , ( v, t ) ∈ V (cid:96) with s < t . Write I = [( v, s ) , ( v, t )] . Then we have (cid:90) X nw (cid:96),β ( I,k ) d¯ ρ ( γ ) e − cL ( γ ) ≤ k − c − k (cid:0) c | I | (cid:1) , (4.8) (cid:90) X w (cid:96),β ( k ) d¯ ρ ( γ ) e − cL ( γ ) ≤ k(cid:96) k c − k . (4.9) Proof. Let us start with the case of non-winding contours and the case when I = { ¯ v } containsonly one point. Elements γ ∈ X nw (cid:96),β (¯ v, k ) may be encoded using tuples ( t , . . . , t k , l , . . . , l k − ) ∈ R k + × ( { , } × { L , R } × { , } ) k − , as follows. • Consider a walker started at ¯ v and travelling upwards until it first encounters the endpointof a link; store the vertical distance traversed as t . • This link can go to the left, L , or to the right R ; it can be a double bar or a cross ; andit can be traversed by loops in γ once, , or twice, . Store this information as l . • Having crossed the link, our walker follows γ and records vertical distances until previouslyunexplored links as t i and information about those links as l i , as before. • If a loop is closed and there are still links that are traversed twice by loops in γ , but have onlybeen visited once by our walker, the walker continues walking from such a link and recording t i and l i as before (we fix some arbitrary rule for selecting the link and the direction of travel). • This procedure is iterated until the entire contour has been traversed.See Fig. 16 for an illustration.Noting that t + · · · + t k ≤ | γ | and that the number of options for { l i } k − i =1 is bounded by8 k − , we get (cid:90) X nw (cid:96),β (¯ v,k ) d¯ ρ ( γ ) e − c | γ | ≤ k − (cid:90) ∞ dt · · · (cid:90) ∞ d t k e − c ( t + ··· + t k ) = 8 k − c − k . (4.10) Next, we may apply a similar argument to obtain that, for ε > (cid:90) X nw (cid:96),β (( v,t ) ,k ) \ X nw (cid:96),β (( v,t + ε ) ,k ) d¯ ρ ( γ ) e − cL ( γ ) ≤ k − c − k +1 1 − e − εc c ≤ k − c − k +1 ε. (4.11) Indeed, for a contour γ which visits ( v, t ) but not ( v, t + ε ), we must have t ≤ ε in the encodingabove, and replacing the integral over t ∈ [0 , ∞ ) with an integral over t ∈ [0 , ε ] gives the claim.Next, to deduce (4.8) from (4.10) and (4.11), we argue as follows. If γ visits I = [( v, s ) , ( v, t )],then either γ contains the endpoint ( v, t ), or there are r ∈ ( s, t ) and ε > v, r ) ∈ γ but ( v, r + ε ) (cid:54)∈ γ . Using (4.10), the first possibility accounts for the first term 8 k − c − k in(4.8). The other possibility accounts for the second term, which one may, for example, see by IMERIZATION IN QUANTUM SPIN CHAINS WITH O ( n ) SYMMETRY 21 ¯ vt t t t t t t t t t t l l l l l l l l l l Figure 16. Illustration of t , . . . , t k , l , . . . , l k − . Note, for example, that t isnot merely the distance between the fourth and fifth links, and that t doesnot readily admit an interpretation as distance between links at all.using a fine dyadic discretization of the interval I and passing to the limit using (4.11) and themonotone convergence theorem.For winding contours γ , recall that they consist of pairs of winding pre-contours. Let usdenote these by γ and γ , and the numbers of links they each visit by k and k , respectively,where k = k + k . There are two vertices v , v ∈ V (cid:96) such that γ visits ( v , 0) and γ visits( v , v , v as well as k , k , and applying the argumentfor (4.10) to γ and γ , respectively, we obtain (cid:90) X w (cid:96),β ( k ) d¯ ρ ( γ ) e − c | γ | ≤ (cid:18) (cid:96) + 12 (cid:19) ( k + 1)8 k − c − k ≤ k(cid:96) k c − k . (4.12) (cid:3) In order to ensure the convergence of the cluster expansion, we need to check that interactionsbetween contours are small in a suitable way. A sufficient condition is the “Koteck´y-Preisscriterion” [11] (see [24] for a version that allows continuous polymers with signed measure). Letus introduce a ( γ ) = a n ( γ ) = a | γ | + a γ (4.13) where γ denotes the number of links visited by γ , a = ε − u for some 0 < ε < u / 3, and a = δ log θ for some δ ∈ (0 , / Lemma 4.3 (Koteck´y–Preiss criterion) . Let w ( γ ) be as in (4.2) and let u := n − / . Thenfor | u | < u , and any γ ∈ X + (cid:96),β , there exists n > such that for n > n and for all β largeenough we have (cid:90) X + (cid:96),β d¯ ρ ( γ ) | w ( γ ) | e a ( γ ) (1 − δ ( γ, γ )) ≤ a ( γ ) . (4.14) Remark 4.4 (Our condition for the range of the parameter u ) . The range of u could have beenexpected to increase as n increases, since this tilts the measure towards shorter loops. Indeed,as in the Peierls’ argument, the weight of a loop decays exponentially in its “length”, i.e. in its vertical length plus the number of links it traverses. However, there are loops that saturate thebounds in Lemma 3.13, so this is not actually true for all loops and instead their weight canincrease in the number of crosses – at least for few crosses. But we also get a factor of u forevery cross a loop traverses, so the key idea is to make | u | sufficiently small to make up for this.Proof. We start by bounding ¯ w ( γ ) := | w ( γ ) | e a ( γ ) . Using Lemma 3.13 and the fact that1I { γ has a spanning segment } ≤ | γ | / (2 β ) , (4.15) we get¯ w ( γ ) ≤ exp (cid:0) − ( u − a − log n (cid:96)β ) | γ | (cid:1) exp (cid:0) − ( log n − a ) γ (cid:1) exp (cid:0) − (log( u ) − a − log n ) γ (cid:1) ≤ exp (cid:0) − ( − u − a − log n (cid:96)β ) | γ | (cid:1) exp (cid:0) − ( − δ ) log n γ (cid:1) ≤ e − c | γ | e − c γ . (4.16) Here we let c = − ε/ and c = (1 / − δ ) log n . We used that | u | < u and log n(cid:96)/β < ε/ β large enough). Clearly, (cid:90) X + (cid:96),β d¯ ρ ( γ ) ¯ w ( γ )(1 − δ ( γ, γ )) ≤ (cid:90) X w (cid:96),β d¯ ρ ( γ ) ¯ w ( γ ) + (cid:90) X nw (cid:96),β d¯ ρ ( γ ) ¯ w ( γ )(1 − δ ( γ, γ )) . (4.17) Using Lemma 4.2 and the fact that a winding contour γ satisfies | γ | ≥ | γ | + β , the first termsatisfies (cid:90) X w (cid:96),β d¯ ρ ( γ ) ¯ w ( γ ) ≤ e − c β (cid:88) k ≥ e − c k (cid:90) X w (cid:96),β ( k ) d¯ ρ ( γ ) e − c L ( γ ) ≤ (cid:96) e − c β (cid:88) k ≥ k (cid:0) c e − c (cid:1) k . (4.18) The right-hand-side is arbitrarily small if n and β are large enough.To proceed, note that if δ ( γ, γ ) = 0 then γ and γ intersect somewhere on V (cid:96),β . We maydecompose the subset of V (cid:96),β visited by γ as a union of closed intervals I , . . . , I m where m ≤ γ , the number of links of γ . Noting also that a non-winding contour γ has at least 5links, we obtain from Lemma 4.2 that (cid:90) X nw (cid:96),β d¯ ρ ( γ ) ¯ w ( γ )(1 − δ ( γ, γ )) ≤ m (cid:88) j =1 (cid:88) k ≥ e − c k (cid:90) X nw (cid:96),β ( I j ,k ) d¯ ρ ( γ ) e − c | γ | ≤ m (cid:88) j =1 (cid:88) k ≥ e − c k k − c − k (cid:0) c | I j | (cid:1) ≤ (cid:0) γ + c | γ | (cid:1) (cid:88) k ≥ (cid:0) c e − c (cid:1) k . (4.19) Recalling that c = − ε/ and e − c = n − (1 / − δ ) , we conclude that the claimed bound holdsprovided • − ε/ (cid:80) k ≥ ( − ε/ n − (1 / − δ ) ) k ≤ a = ε − n − / , and • (cid:80) k ≥ ( − ε/ n − (1 / − δ ) ) k ≤ a = δ log n .Clearly both inequalities are satisfied if n is large enough. (cid:3) IMERIZATION IN QUANTUM SPIN CHAINS WITH O ( n ) SYMMETRY 23 Let C k denote the set of connected (undirected) graphs with vertex set { , . . . , k } and define ϕ ( γ , . . . , γ k ) = (cid:26) , if k = 1 , k ! (cid:80) G ∈C k (cid:81) ij ∈ G ( δ ( γ i , γ j ) − , if k ≥ , (4.20) where the product in the second line is over the edges of G . The following is the main conse-quence of Lemma 4.3. Proposition 4.5 (Cluster expansion of the partition function) . For parameters as in Lemma4.3, the following sum converges absolutely: Ξ (cid:96),β := (cid:88) m ≥ (cid:90) X + (cid:96),β d¯ ρ ( γ ) · · · (cid:90) X + (cid:96),β d¯ ρ ( γ k ) (cid:16) k (cid:89) i =1 w ( γ i ) (cid:17) ϕ ( γ , . . . , γ m ) , (4.21) and we have that e (1+ u ) | E − (cid:96) | Z (cid:96),β (cid:26) ≤ exp (cid:0) Ξ (cid:96),β (cid:1) , ≥ (1 − e − β ) (cid:96) exp (cid:0) Ξ (cid:96),β (cid:1) . (4.22) Proof. Using Lemma 4.3 and the upper and lower bounds on the factor R ( γ , . . . , γ k ) that arestated there, the result follows from [24, Theorem 1]. (cid:3) Dimerization. Recall the (signed) measure µ (cid:96),β,n,u defined in (2.9). The next theorem isequivalent to our main result but it is formulated in terms of loops (recall (2.10)). Its proof isthen also a proof of Theorem 1.1. Theorem 4.6. There exists n > such that for all n > n , | u | < u = n − / lim inf β →∞ (cid:0) µ (cid:96),β,n,u (0 ↔ − µ (cid:96),β,n,u (0 ↔ − (cid:1) > if (cid:96) is odd, lim sup β →∞ (cid:0) µ (cid:96),β,n,u (0 ↔ − µ (cid:96),β,n,u (0 ↔ − (cid:1) < if (cid:96) is even.Proof. Let O (for outside ) denote the event that (0 , 0) is not on or inside any contour, that is O = (cid:110) ω ∈ Ω (cid:96),β : (0 , ∈ (cid:92) γ ∈ Γ( ω ) E ( γ ) (cid:111) . (4.23) By Remark 3.6 and the fact that µ (cid:96),β (1) = 1, µ (cid:96),β,n,u (0 ↔ − µ (cid:96),β,n,u (0 ↔ − 1) = 2 µ (cid:96),β,n,u ( O ) − . (4.24) It thus suffices to show that, for n large enough, there is some c > (cid:96) or β , such that µ (cid:96),β,n,u ( O ) ≥ c for large enough β . To do this we use the cluster expansionprovided by Proposition 4.5.First note that Z (cid:96),β,n,u [1I O ] may be written as in Proposition 4.1, but with the weights w ( γ i )replaced by w ( γ i )1I { (0 , ∈ E ( γ i ) } . Since | w ( γ i )1I { (0 , ∈ E ( γ i ) }| ≤ | w ( γ i ) | , Lemma 4.3 holdsfor the modified weights too, and therefore also the suitable modification of Proposition 4.5.More precisely, introducingΞ (cid:96),β ( O ) := (cid:88) m ≥ (cid:90) X + (cid:96),β d¯ ρ ( γ ) · · · (cid:90) X + (cid:96),β d¯ ρ ( γ k ) (cid:16) k (cid:89) i =1 w ( γ i )1I { (0 , ∈ E ( γ i ) } (cid:17) ϕ ( γ , . . . , γ m ) , (4.25) we have that the sum in Ξ (cid:96),β ( O ) converges absolutely, and that e (1+ u ) | E − (cid:96) | Z (cid:96),β [1I O ] (cid:26) ≤ exp (cid:0) Ξ (cid:96),β ( O ) (cid:1) , ≥ (1 − e − β ) (cid:96) exp (cid:0) Ξ (cid:96),β ( O ) (cid:1) . (4.26) Writing I ( γ , . . . , γ k ) = 1I {∃ ≤ i ≤ k : (0 , ∈ I ( γ i ) } , (4.27)4 J.E. BJ ¨ORNBERG, P. M¨UHLBACHER, B. NACHTERGAELE, AND D. UELTSCHI where I ( γ i ) is the interior of γ i , defined in Section 3, it follows that µ (cid:96),β,n,u ( O ) = Z (cid:96),β [1I O ] Z (cid:96),β = (1 + o (1)) exp (cid:110) − (cid:88) k ≥ (cid:90) X + (cid:96),β d¯ ρ ( γ ) · · · (cid:90) X + (cid:96),β d¯ ρ ( γ k ) (cid:16) k (cid:89) i =1 w ( γ i ) (cid:17) ϕ ( γ , . . . , γ k ) I ( γ , . . . , γ k ) (cid:111) (4.28) where the 1 + o (1) factor is bounded between (1 − e − β ) ± (cid:96) and thus goes to 1 as β → ∞ .To bound µ (cid:96),β,n,u ( O ) from below, we bound the sum in the exponential from above. Wehave that (cid:12)(cid:12)(cid:12)(cid:12) (cid:88) k ≥ (cid:90) X + (cid:96) d¯ ρ ( γ ) · · · (cid:90) X + (cid:96),β d¯ ρ ( γ k ) (cid:16) k (cid:89) i =1 w ( γ i ) (cid:17) ϕ ( γ , . . . , γ k ) I ( γ , . . . , γ k ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:88) k ≥ m (cid:90) X + (cid:96),β d¯ ρ ( γ ) ¯ w ( γ )1I { (0 , ∈ I ( γ ) } (cid:90) X + (cid:96),β d¯ ρ ( γ ) ¯ w ( γ ) · · · (cid:90) X + (cid:96),β d¯ ρ ( γ k ) ¯ w ( γ k ) | ϕ ( γ , . . . , γ k ) | = (cid:90) X + (cid:96),β d¯ ρ ( γ ) ¯ w ( γ )1I { (0 , ∈ I ( γ ) } (cid:16) (cid:88) k ≥ k (cid:90) X + (cid:96),β d¯ ρ ( γ ) ¯ w ( γ ) · · · (cid:90) X + (cid:96),β d¯ ρ ( γ k ) ¯ w ( γ k ) | ϕ ( γ , . . . , γ m ) | (cid:17) ≤ (cid:90) X + (cid:96),β d¯ ρ ( γ ) ¯ w ( γ ) e a ( γ ) { (0 , ∈ I ( γ ) } . (4.29) The last inequality follows from [24, Eq (4)]. To proceed, we use Lemma 4.2. Note that if(0 , ∈ I ( γ ) then there is some 0 ≤ v ≤ (cid:96) such that γ contains the point ( v, γ is non-winding then necessarily γ traverses at least v ∨ w ( γ ) ≤ e − c | γ |− c γ and that a ( γ ) = a | γ | + a γ , we obtain (cid:90) X + (cid:96),β d¯ ρ ( γ ) ¯ w ( γ )1I { (0 , ∈ I ( γ ) }≤ e − ( c − a ) β (cid:88) k ≥ (cid:90) X w (cid:96),β ( k ) d¯ ρ ( γ ) e − ( c − a ) k e − 12 ( c − a ) | γ | + (cid:96) (cid:88) v =0 (cid:88) k ≥ v ∨ (cid:90) X nw (cid:96),β (( v, ,k ) d¯ ρ ( γ ) e − ( c − a ) k e − ( c − a ) | γ | ≤ e − ( c − a ) β (cid:88) k ≥ e − ( c − a ) k k(cid:96) k (cid:0) c − a (cid:1) − k + (cid:96) (cid:88) v =0 (cid:88) k ≥ v ∨ e − ( c − a ) k k − ( c − a ) − k . (4.30) In the first inequality, we used that a winding contour satisfies | γ | ≥ | γ | + β .To proceed, we recall that • u = n − / , • a = ε − u for 0 < ε < u / • a = δ log n for δ ∈ (0 , / IMERIZATION IN QUANTUM SPIN CHAINS WITH O ( n ) SYMMETRY 25 • c = − ε/ , • c = (1 / − δ ) log n .Writing q := 16 n − / δ − n − / ≥ − ( c − a ) c − a , (4.31) we find that the first term of the last line of (4.30) satisfiese − ( c − a ) β (cid:88) k ≥ e − ( c − a ) k k(cid:96) k (cid:0) c − a (cid:1) − k ≤ (cid:96) e − ( c − a ) β (cid:88) k ≥ k (2 q ) k , which is finite if q < / 2. This holds for n large enough, since δ < / 6. It vanishes in the limit β → ∞ .The second term of the last line of (4.30) satisfies (cid:96) (cid:88) v =0 (cid:88) k ≥ v ∨ e − ( c − a ) k k − ( c − a ) − k = (cid:96) (cid:88) v =0 (cid:88) k ≥ v ∨ e − ( c − a ) k k − ( c − a ) − k ≤ q − q ) (cid:16) − q (cid:96) − − q (cid:17) . (4.32) This can be made arbitrarily small by taking n large enough.To wrap up, we have that µ (cid:96),β,n,u ( O ) ≥ (1 + o (1)) exp (cid:8) − ( o (1) + ε ) (cid:9) , (4.33) where both o (1) terms vanish as β → ∞ , and ε can be made arbitrarily small. The resultfollows. (cid:3) Appendix A. The interaction uT + vP when n is even For n odd, the interactions uT + vP and uT + vQ are related by the unitary transformationof Eq. (1.4). This holds for models defined on arbitrary graphs or lattices.We now discuss the case of n even. As we shall see, we need to restrict ourselves to bipartitegraphs (of which the chain is of course an example). We work with the S -eigenbasis e α := | α (cid:105) with α = − S, − S + 1 , . . . , S . To begin, we define a unitary V by setting V | α (cid:105) = ( − S − α | − α (cid:105) . (A.1) With ψ the vector of (1.3) and φ the vector of (1.5), we have φ = (1I ⊗ V ) ψ. (A.2) Therefore, since P is the projection onto φ , P = (1I ⊗ V ) Q (1I ⊗ V ∗ ) . (A.3) Since T ψ = ψ and T φ = − φ , we have T QT = Q and T P T = P . Using these properties we find( V ⊗ Q ( V ∗ ⊗ V ⊗ T QT ( V ∗ ⊗ T (1I ⊗ V ) Q (1I ⊗ V ∗ ) T = P. (A.4) Both models are translation-invariant although the unitary that relates them is not:( V ⊗ ⊗ V · · · ⊗ (cid:34) (cid:96) − (cid:88) x = − (cid:96) +1 Q x,x +1 (cid:35) ( V ∗ ⊗ ⊗ V ∗ · · · ⊗ (cid:96) − (cid:88) x = − (cid:96) +1 P x,x +1 . (A.5) Let ˜ T be the transformation of the operator T . We have˜ T = (1I ⊗ V ) T (1I ⊗ V ∗ ) = (1I ⊗ V )( V ∗ ⊗ T = − ( V ⊗ V ) T. (A.6)6 J.E. BJ ¨ORNBERG, P. M¨UHLBACHER, B. NACHTERGAELE, AND D. UELTSCHI Let us summarize the above considerations by the following proposition. We define the newHamiltonian H (cid:48) (cid:96) = (cid:80) (cid:96) − x = − (cid:96) +1 (cid:0) u ˜ T x,x +1 + vQ x,x +1 (cid:1) . Proposition A.1. For n even, the interaction uT + vP is unitarily equivalent with u ˜ T + vQ .The Hamiltonian ˜ H (cid:96) defined in (1.14) is unitarily equivalent to H (cid:48) (cid:96) . Notice that, when u = 0, the Q -model and P -model are unitarily equivalent for all n . Theproposition is stated for chains, but it clearly holds for arbitrary bipartite graphs.Next, we derive a loop representation for the model H (cid:48) (cid:96) . Let us introduce the diagonaloperator R (cid:48) | α (cid:105) = | α (cid:105) if | α | = S, −| α (cid:105) if | α | = S − , (A.7) Then R (cid:48) x is the operator R (cid:48) acting on the spin at site x . Proposition A.2. There exists a function s ( l ) from the set of loops to ± such that for alleven n ≥ ,(a) Tr e − βH (cid:48) (cid:96) = e β (1+ u ) | E (cid:96) | (cid:90) d ρ u ( ω ) n L ( ω ) −| ω | (cid:89) loop l in ω s ( l ) .(b) Tr R (cid:48) x R (cid:48) y e − βH (cid:48) (cid:96) = n e β (1+ u ) | E (cid:96) | (cid:90) d ρ u ( ω ) n L ( ω ) −| ω | x ↔ y ] (cid:89) loop l in ω s ( l )Notice that (a) holds for n = 2 as well, but the operators R (cid:48) x are defined for n ≥ Proof. The expansion of the operator e − βH (cid:48) (cid:96) can be made in terms of configurations ω , andof “space-time spin configurations” (see e.g. [25]). The space-time spin configurations that arecompatible with ω have the property that their value on a loop is ± α for some α = − S, . . . , S ,the changes of signs occurring when traversing crosses (and any such choice results in a possiblespace-time spin configuration).Indeed, proceeding as in Theorem 2.1, we find thatTr e − βH (cid:48) (cid:96) = e β (1+ u ) | E (cid:96) | (cid:90) d ρ u ( ω ) n L ( ω ) −| ω | s ( ω ) , (A.8) where s ( ω ) is an overall sign: s ( ω ) = ± 1. Notice that, since ˜ T involves a minus sign, there isno need to change the sign of u in the interaction as in Theorem 2.1. In order to see that s ( ω )factorizes according to loops, first observe that the number of crosses along the trajectory of aloop, is even. Indeed, the total number of crosses and double-bars along the trajectory is evenbecause the graph is bipartite; and the number of double-bars is even because the number ofchanges in vertical direction is even; so the number of crosses is also even.The signs are due to the action of operators V . We can collect the signs for each loopindividually. Consider two successive crosses. If the vertical direction is the same (which is thecase if there is an even number of double-bars between them), we get the factor( − S − α ( − S + α = ( − S = − . (A.9) If the vertical direction is opposite (which is the case if there is an odd number of double-barsbetween them), the factor is ( − S − α ( − S − α = 1 . (A.10) This is illustrated in Fig. 17. The value of s ( l ) is the product of these factors. Notice that thesign does not depend on the value of α in the loop. This proves item (a) of the proposition. IMERIZATION IN QUANTUM SPIN CHAINS WITH O ( n ) SYMMETRY 27 ααα α − α − α − α − α − α factor (cid:104) α | V | − α (cid:105) = ( − S + α factor (cid:104)− α | V | α (cid:105) = ( − S − α Figure 17. Signs arising when traversing crosses. Left: the crossesare separated by an even number of double bars which yields the factor( − S − α ( − S + α = − 1. Right: the crosses are separated by an odd num-ber of double bars which yields the factor 1.The expectation involving the operators R (cid:48) x , R (cid:48) y can also be understood using space-timespin configurations. Assume that x and y (at time 0) do not belong to the same loop andconsider the loop containing ( x, α on this loop, we get 0 since thecontributions of | α | = S (factor +1) cancel those of | α | = S − − x and y belong to the same loop; the factors are (+1) or ( − , so the contributions do notcancel. Apart from the sign s ( l ), we get a factor 4 because of the 4 possible values of α , insteadof the factor n as in the other loops. This gives the term n and completes the proof of item(b). (cid:3) We can now prove Theorem 1.2. Proof of Theorem 1.2. We take ˜ R x,y = R (cid:48) x R (cid:48) y (notice that the latter operators are invariantunder the transformation (A.1)). By Proposition A.1, the claims of Theorem 1.2 are equivalentto proving dimerization in the model with Hamiltonian H (cid:48) (cid:96) . We use the loop representationof Proposition A.2. We can then retrace the steps of the proof of Theorem 4.6. In doing so,note that any short loop l has s ( l ) = +1, while long or winding loops have s ( l ) = ± 1. Weincorporate the latter factors in the weights w ( γ ) (see (4.2)) of the contours. Therefore, theonly difference is that the weights of contours have possibly other signs. All bounds are thesame, though, and the cluster expansion gives the same result. (cid:3) Acknowledgements: We are grateful to Vojkan Jakˇsi´c and the Centre de Recherches Math´e-matiques of Montreal for hosting us during the thematic semester “Mathematical challengesin many-body physics and quantum information”, with support from the Simons Foundationthrough the Simons–CRM scholar-in-residence program.JEB gratefully acknowledges support from Vetenskapsr˚adet grants 2015-0519 and 2019-04185as well as Ruth och Nils-Erik Stenb¨acks stiftelse .Based upon work supported in part by the National Science Foundation under grant DMS-1813149 (BN). References [1] I. Affleck, Exact results on the dimerisation tran-sition in su ( n ) antiferromagnetic chains , J. Phys.:Condens. Matter 2, 405–415 (1990)[2] I. Affleck, T. Kennedy, E.H. Lieb, H. Tasaki, Valencebond ground states in isotropic quantum antiferro-magnets , Comm. Math. Phys. 115, 477-528 (1988) [3] M Aizenman, H. Duminil-Copin, S. 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Phys. 54, 083301, 1–40(2013) Department of Mathematics, Chalmers University of Technology and the University of Gothen-burg, Sweden Email address : [email protected] Department of Mathematics, University of Warwick, Coventry, CV4 7AL, United Kingdom Email address : [email protected] Department of Mathematics, University of California, One Shields Ave, Davis, CA 95616, UnitedStates Email address : [email protected] Department of Mathematics, University of Warwick, Coventry, CV4 7AL, United Kingdom Email address ::