A direct proof of dimerization in a family of SU(n)invariant quantum spin chains
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Abstract
We study the family of spinS quantum spin chains with a nearest neighbor interaction given by the negative of the singlet projection operator. Using a random loop representation of the partition function in the limit of zero temperature and standard techniques of classical statistical mechanics, we prove dimerization for all sufficiently large values of S.
Keywords
Quantum spin chain Dimerization SU(n)invariant chainsMathematics Subject Classification
82B10 82B20 82B261 Introduction
Dimerization is the most common type and the most common mechanism of spontaneous lattice translation symmetry breaking in the ground state of quantum spin systems. It is ubiquitous in one dimension due to the socalled (spin) Peierls instability [9]. In two and more dimensions it gives rise to columnar phases and other patterns of lattice symmetry breaking [12, 13, 23, 24]. In the ground states of quantum spin chains with isotropic antiferromagnetic interactions, longrange antiferromagnetic order, and the accompanying spontaneous breaking of the continuous rotation symmetry, is prevented by quantum fluctuations. Shortrange antiferromagnetic correlations, however, occur in several manifestations, some of which involve discrete symmetry breaking. Due to a result by Affleck and Lieb [3], halfinteger spin chains with a rotation symmetry and a unique ground state have a gapless excitation spectrum above the ground state. The opening up of a nonvanishing (in the thermodynamic limit) gap above the ground state requires that the ground state is degenerate, and for a class of reflection positive antiferromagnetic chains it is known that the nature of this degeneracy is dimerization, i.e., breaking of the translation invariance from \({\mathbb Z}\) to \(2{\mathbb Z}\) [4, 21].
The relationship between the spinS chain with Hamiltonian (1.1) with \(S\ge 1\) and the twodimensional qstate Potts model with \(q=(2S+1)^2\) at the selfdual point extends in a nontrivial way to the states: The two dimerized ground states correspond to the coexisting ordered and disordered phases of the qstate Potts model on the square lattice at its critical point. This coexistence (a firstorder phase transition) has been found by Baxter [6] for \(q\ge 5\) and has been rigorously established for all sufficiently large values of q [18]. Even if one accepts the identity of the spectra of the related Hamiltonians, relating order parameters or, in general, the states of these models is a subtle issue. In [4] it was shown that the dimerization order parameter for the spin chain and the order parameter of a suitable twodimensional ferromagnetic Potts model are bounded by a multiple of each other. This implies that one of them vanishes if and only if the other does. The Potts model for which this relationship is proved can be regarded as a strongly anisotropic limit (with respect to the two lattice directions), in which one of the lattice directions tends to the continuum. A similar result likely also holds for the standard Potts model on the square lattice, but this has not been worked out in the literature.
Due to the subtleties of the relationships between the different models discussed above, a complete proof of dimerization has been lacking. In view of the nontrivial nature of rigorous studies of the critical Potts model itself, it seemed worthwhile to look for a direct proof of the dimerization, bypassing establishing more details of the relationship between the various models. In this article we provide the first complete proof of the existence of dimerized ground states for this class of models at sufficiently large values of S. Moreover, by not relying on the Temperley–Lieb algebra and a relation to the Potts models, we open the possibility to study perturbations of the model away from the selfdual point in the phase diagram. This is important in the context of the current interest in stable gapped ground state phases of quantum lattice systems, which we briefly discuss in Sect. 5.
Our approach is based on a random loop representation for the partition function of the spin models (1.1) given in [4], which has been applied in recent years for a number of other rigorous results for quantum spin models [7, 8, 19]. For a review and extensions of the random loop representation for quantum spin models, see [25]. We give a precise statement of our results in the next section. A detailed description of the random loop representation and its properties is given in Sect. 3. In Sect. 4 we introduce a suitable notion of contours and the Peierls argument proof of our main result. We conclude with a short discussion.
2 Setting and results
Theorem 2.1
The theorem states that, for any \(x \in \{\ell +3, \ell +5, \dots ,\ell 1\}\), the probability that the bond \(\{x,x+1\}\) is in the singlet state (dimerized) exceeds the probability that \(\{x1,x\}\) is in the singlet state by a positive amount, uniformly in \(\ell \). See Fig. 1 for an illustration. Thus, in the limit \(\ell \rightarrow \infty \) along even values, one gets a ground state where \(\{1,0\}\) is more likely to be dimerized than \(\{0,1\}\). In the limit \(\ell \rightarrow \infty \) along odd values, the converse is true. This establishes the existence of two distinct, nontranslationinvariant ground states. By an averaging procedure, one sees that there are two periodic ground states of period 2. We conjecture that these are the only ground states of the infinite chain. The proof of Theorem 2.1 is found at the end of Sect. 4.
Our proof of dimerization is based on the random loop representation of [4]. We introduce excitation contours in a background of dimerized short loops, a setting that allows to use a Peierls argument. It is presented in Sect. 4. More precisely, Theorem 2.1 is an immediate consequence of Proposition 4.4 combined with Proposition 3.1, Lemma 3.2, and Proposition 3.3.
The ground state is no longer translation invariant, but it still has spin rotation invariance, as shown in the following theorem. In particular, there is no Néel order. In fact, this theorem implies that in the two periodic ground states all correlations (not just spinspin correlations) decay exponentially and therefore are extremal periodic ground states. This supports our conjecture that they are the only ground states of the infinite chain.
Theorem 2.2
The proof of this theorem is found at the end of Sect. 4; it is based on the same properties of the random loop representation that we prove for Theorem 2.1. Correlations between x and y can be expressed in terms of events in which loops (contours) connect x to y, and which require that these points are surrounded by contours of size larger than \(xy\). The probability of these contours decays exponentially fast with respect to their size. Because all correlation functions can be expressed in terms of loop connectivity (see [4]), it follows that all correlations decay exponentially. Notice that Theorem 2.2 holds for smaller S than Theorem 2.1; the reason is that it only involves large loops.
Theorem 2.2 also implies that the translation symmetry is spontaneously broken in the ground state for all \(S\ge 8\). This follows from [4, Theorem 6.1]. For \(S\ge 40\), Theorem 2.1 gives a quantitative estimate of this nontranslation invariance in terms of the probability that two nearest neighbor spins form a dimer.
3 Loop representation and contours
We now give a description of the random loop representation of the Gibbs states defined in (2.7). It is convenient to restrict to \(\beta \in {\mathbb N}\).
3.1 Loops
In contrast to previous applications of the random loop representation [4, 25], it will be convenient for us here to use a discrete version of the loop representation, defined as follows. Let \(n \in {\mathbb N}\), and consider the set \(T_{\beta ,n}\) of discrete times, \(T_{\beta ,n} = \frac{1}{n} {\mathbb Z}\cap [\beta , \beta ]\). A configuration \(\omega \) is a subset of \(E_\ell \times T_{\beta ,n}\); we say that a “double bar” is present at \(\{x,x+1\} \times t \in E_\ell \times T_{\beta ,n}\) whenever \(\{x,x+1\} \times t \in \omega \); there are no double bars at \(\{x,x+1\} \times t\) otherwise. We let \(\Omega _{\ell ,n}\) denote the set of configurations where no more than one double bar occurs at any given time; it is also useful to exclude double bars at time 0.
Proposition 3.1
 (a)
\(\displaystyle \mathrm{Tr}\,\,\mathrm{e}^{2\beta H_\ell }\, = \lim _{n\rightarrow \infty } Z_n(\beta ,\ell )\).
 (b)
\(\displaystyle \langle P_{x,x+1}^{(0)} \rangle _{2\beta ,\ell } = \tfrac{1}{(2S+1)^2} + \bigl ( 1  \tfrac{1}{(2S+1)^2} \bigr ) \lim _{n\rightarrow \infty } {\mathbb P}_{\beta , \ell ,n}(x \leftrightarrow x+1)\).
Proof
3.2 Simplified set of configurations without winding loops
As is usual, we will derive our estimates for finite systems, which in our case means finite chains at finite inverse temperature \(\beta \). The estimates will then carry over to the limit of infinite \(\beta \) and the infinite chain. In the limit \(\beta \rightarrow \infty \), the socalled winding loops will have vanishing probability. These winding loops are a complication for the Peierlstype argument we want to develop. Therefore, it will be helpful to work with a restricted set of configurations in which almost all winding loops have been eliminated. To do this we need to show that the probability of the configurations we ignore indeed vanishes in the limit \(\beta \rightarrow \infty \). This is the purpose of the next lemma.
Given an integer \(\alpha <\beta \), let \(\Omega _{\ell ,n}^\alpha \) denote the set of configurations where intervals \(\{x,x+1\} \times [\alpha ,\alpha +1]\) contain at least one double bar if \(\{x,x+1\} \in E_\ell ^1\), and none if \(\{x,x+1\} \in E_\ell ^2\). These configurations possess a convenient, spontaneous boundary condition in the time direction (this is depicted in Fig. 5). Almost all configurations have this property for some \(\alpha \in {\mathbb N}\):
Lemma 3.2
Proof
Loops visiting only one or two sites are called short loops. The loops shown in Fig. 3 are all short loops. We say that a loop is long if it is not short. Since the loops of configurations of \(\Omega _{\ell ,n}^\alpha \) do not wind, they have an interior in the sense of Jordan curves; we actually call “interior” its intersection with \(\{\ell +1,\dots ,\ell \} \times T_{\beta ,n}\). For \(\omega \in \Omega _{\ell ,n}^\alpha \), we introduce the event \(E_x^\circlearrowleft \) where (x, 0) belongs to a long loop, or to the interior of a long loop. See Eq. (4.1) for an equivalent definition that involves contours, to be defined below.
Proposition 3.3
Proof
There remains to show that \({\mathbb P}_{\beta ,\ell ,n}(E_x^\circlearrowleft  \Omega _{\ell ,n}^\alpha )\) is less than \(\frac{1}{2}\), uniformly in \(\beta ,\ell ,n\).
3.3 Blocks and domains
We define a “block” to be a set \(\{x,x+1\} \times I\), where \(\{x,x+1\} \in E_\ell ^1\) and I is a proper interval in \(T_{\beta ,n}\). A “domain” D is a finite collection of disjoint blocks.
Lemma 3.4
Proof
4 Contour representation
An edge in space time is of the form \((\{x,x+1\}, t)\), \(x\in [\ell , \ell 1]\subset \mathbb {Z}, t\in [\beta ,\beta ]\subset \mathbb {R}\). In figures it is more convenient to replace \(\{x,x+1\}\) by \([x,x+1]\subset {\mathbb R}\). In the same way, in figures we will depict a block as a rectangular region in \(\mathbb {R}^2\) of the form \([x,x+1]\times [a,b]\).

its number of double bars, \(\gamma \);

its 1interior \(\mathrm{int}_1\gamma \), that is, the set of edges \(\{x,x+1\} \times t\) inside the enclosed area, with \(\{x,x+1\} \in E_\ell ^1\);

its 2interior \(\mathrm{int}_2\gamma \), that is, the set of edges \(\{x,x+1\} \times t\) inside the enclosed area, with \(\{x,x+1\} \in E_\ell ^2\);

the length \(L(\gamma )\) such that \(\mathrm{int}_1\gamma   \mathrm{int}_2\gamma  = \tfrac{1}{2} n L(\gamma )\), where \(\cdot \) means cardinality. \(L(\gamma )\) is almost equal to the lengths of vertical legs of the loop (it is equal up to \(\frac{\gamma }{n}\)), because each stretch of the boundary of \(\mathrm{int}_1\gamma \) belongs either to the perimeter or to the vertical boundary of \(\mathrm{int}_2\gamma \);

its support \(\mathrm{supp}\,\gamma \), the set of edges of \(E_\ell \) with at least one endpoint on the loop;

its exterior \(\mathrm{ext}\,\gamma \), which is equal to the union of blocks outside the enclosed area;

it involves at least a jump \(\{x,x+1\} \in E_\ell ^2\);

it is not surrounded by another loop.
A difficulty with our Peierls argument is to control the change in the number of loops when erasing a contour. We find it convenient to condition on the configuration of external contours away from (x, 0); a similar idea was used in [14]. This gives a domain with complicated shape, but this is not a problem. The background configuration of loops is now simpler and this is useful. Given \(\omega \in \Omega _{D_\alpha }\), consider the set \(\Gamma \) of external contours that do not surround (x, 0). Let \(\mathrm{ext}\,\Gamma = D_\alpha {\setminus } \mathrm{int}_1 \Gamma \), and let \(X(\omega ) \subset \mathrm{ext}\,\Gamma \) denote the connected component that contains (x, 0). That is, the domain \(X(\omega )\) is a subset of \(D_\alpha \), and the graph whose vertices are the blocks, and with edges between blocks that can be connected by a double bar, is connected. This is illustrated in Fig. 6. Finally, let \(\widetilde{\Omega }_X \subset \Omega _X\) denote the configurations without contours, or with just one external contour that surrounds (x, 0); and let \(\widetilde{\mathbb P}_X(\cdot ) = {\mathbb P}_X(\cdot  \widetilde{\Omega }_X)\) denote the conditional probability.
Lemma 4.1
Proof
In view of Proposition 3.3, to prove Theorem 2.1 we need to show that, if \(S\ge 40\), we have \(\widetilde{\mathbb P}_X(E_x^\circlearrowleft ) < \frac{1}{2}\) for all X. This will be achieved in Proposition 4.4. In preparation for that estimate, we need two lemmas.
Lemma 4.2
Proof
The following lemma is motivated by a Peierls argument. The probability that (x, 0) is surrounded by a contour can be bounded by a sum over these contours with exponentially small weights.
Lemma 4.3
Proof
Proposition 4.4
Proof
Proof of Theorem 2.1
Proof of Theorem 2.2
5 Discussion
For the spin chain with Hamiltonian (1.1), we established the existence of dimerization when \(S\ge 8\): in the thermodynamic limit the model has at least two 2periodic ground states in which the translation invariance is broken. This follows directly from Theorem 2.1. We do not expect that these models have other ground states. In particular, based on what has been shown for the antiferromagnetic XXZ chain [10], it seems unlikely that domainwall ground states exist for this model. We also proved that in the two ground states we constructed, the SU(2)invariance remains unbroken.
As stated in Theorem 2.2, our proof of dimerization also implies exponential decay of correlations in the ground states. Following the arguments of Kennedy and Tasaki [15], exponential decay implies a spectral gap in this setting.
One may also ask about the stability of the dimerization under small translationinvariant perturbations of the interaction. Since the model is not frustration free and involves translation symmetry breaking, the result of Michalakis and Zwolak [20] does not apply but we expect that the random loop representation can be used as a starting point for studying perturbations of the model using traditional cluster expansion techniques as is done, e.g., in [15]. We have not pursued this possibility as it is beyond the scope of this work, but establishing stability under arbitrary, uniformly bounded, and sufficiently small perturbation of the interactions would certainly be un important contribution to understanding the phase diagram of quantum spin chains [2].
Notes
Acknowledgements
The research reported in this article was completed during the workshop ManyBody Quantum Systems and Effective Theories at the Mathematisches Forschungsinstitut Oberwolfach, September 11–17, 2016. We thank the organizers and the institute for the great hospitality and the stimulating program. One referee made very useful comments. The work of BN was supported in part by the National Science Foundation under Grant DMS1515850.
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