Robust Bounds for Emptiness Formation Probability for Dimers Shannon Starr University of Alabama at Birmingham October 9, 2016 Based on joint work with Scott Williams, UAB (undergraduate). Preprint, in progress. Shannon Starr – UAB Dimer Emptiness Formation Probability
Outline for talk ◮ Review Emptiness Formation Probability for the XXZ Chain. ◮ Lieb, Sutherland, Baxter relation to the 6-vertex model. ◮ Reflection positivity. ◮ Dimer model relation to the 6-vertex model. ◮ One-sided bound for the dimer model, and open problems. Shannon Starr – UAB Dimer Emptiness Formation Probability
Emptiness Formation Probability in the XXZ model Let Λ( N ) = Z / N Z be { 1 , . . . , N } with pbc ( N + 1 ≡ 1) N � k +1 + S y k S y ( S x k S x k +1 + ∆ S z k S z H Λ( N ) (∆) = k +1 ) . k =1 We restrict attention to even N . Using the unitary U N = exp( i π S z 2 k ) exp( i π S z 4 ) · · · exp( i π S z N ) , N � k +1 + S y k S y ( S x k S x k +1 − ∆ S z k S z U N H Λ( N ) (∆) U ∗ N = − k +1 ) k =1 N � 1 k +1 + 1 � � 2 S + k S + k +1 − ∆ S z k S z = − k S − 2 S − . k +1 k =1 Restricted to 0 magnetization, there is a unique energy minimizer. Shannon Starr – UAB Dimer Emptiness Formation Probability
For L ≤ N , the EFP projector is L � 1 � � 2 + S z Q L = . k k =1 Korepin, Lukyanov, Nishiyama and Shiroishi (Phys.Lett.A, 2003) argued, algebraically, tr[ Q L e − β H ∆ N ] ∼ AL − γ C − L 2 , N →∞ lim lim tr[ e − β H ∆ N ] β →∞ for C and γ which are explicit in ∆ ∈ ( − 1 , 1]. Shannon Starr – UAB Dimer Emptiness Formation Probability
With Nick Crawford and Stephen Ng (CMP, 2016) we proved Suppose ∆ > − 1 (where ∆ = − 1 is the Theorem. ferromagnet). Let �· · · � GS N , ∆ = the ground state expectation for H ∆ . N with magnetization 0 . Then there exist constants c 1 , C 1 , c 2 , C 2 ∈ (0 , ∞ ) with C 1 e − c 1 L 2 ≤ � Q L � GS N , ∆ ≤ C 2 e − c 2 L 2 , for all even N with L ≤ N / 2 . Shannon Starr – UAB Dimer Emptiness Formation Probability
6-vertex configurations = alternating sign matrices − 1 +1 0 0 0 0 0 0 0 0 +1 − 1 0 − 1 +1 − 1 +1 0 = 0 0 0 0 +1 − 1 − 1 0 0 0 0 +1 +1 − 1 0 +1 0 − 1 Shannon Starr – UAB Dimer Emptiness Formation Probability
In 1970 Sutherland showed that the row-to-row transfer matrix of the 6-vertex model commutes with the XXZ model . Weight of a vertex is e κ if it is a sink/source, 1 otherwise. Let A N ,κ be the row-to-row transfer matrix. Then A N ,κ and H ∆ N commute if ∆ = 1 2 e 2 κ − 1 . Note { κ ∈ ( −∞ , ∞ ) } ⇔ { ∆ ∈ ( − 1 , ∞ ) } . . In 1967 Lieb showed that the ground state for the XXZ model is the same as the invariant measure for the 6-vertex model. Need #rows → ∞ . Shannon Starr – UAB Dimer Emptiness Formation Probability
Reflection Positivity of the Six Vertex Model In 1980, Fr¨ ohlich, Israel, Lieb and Simon proved that the 6-vertex model is reflection positive. They actually showed that the 6-vertex model is reflection positive for 2 types of reflections: ◮ coordinate directions, and ◮ diagonal directions. Shannon Starr – UAB Dimer Emptiness Formation Probability
Shannon Starr – UAB Dimer Emptiness Formation Probability
R.J.Baxter (Ann.Phys. 1973) showed how to embed the dimer problem into the family of 8 vertex models on a diagonal lattice: Shannon Starr – UAB Dimer Emptiness Formation Probability
The EFP event defined on the set of dimers is that on a particular row, in a particular sub-interval, we see this Under reflections, this looks like this: Shannon Starr – UAB Dimer Emptiness Formation Probability
This would force at least a frozen region for a while. For example, the next row is this: And we can continue in this fashion. Using this and reflection positivity it is easy to derive upper bounds similar to those before. Shannon Starr – UAB Dimer Emptiness Formation Probability
Open Problems 1. Is there a quantum spin system that commutes with the dimer transfer matrix, in some sense? 2. How to get lower bounds? Shannon Starr – UAB Dimer Emptiness Formation Probability
A partial answer to question 1 is given in The problem is that the “integrals of motion” are not short-ranged quantum spin chains. Also, there is a question about b.c.’s. Shannon Starr – UAB Dimer Emptiness Formation Probability
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