Quantum integrable combinatorics, Slides of Quantum Mechanics

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Quantum integrable combinatorics

Gus Schrader

Michael Zhao Memorial Colloquium October 30, 2019

Alternating sign matrices

An n × n alternating sign matrix is a matrix such that each entry of the matrix is either 0, 1 or −1. the sum of entries in each row and column is 1, and the non-zero entries in each row and column alternate in sign.

ASM 2 :

ASM 3 :

The ASM conjecture

Conjecture (Mills, Robbins, Rumsey, early 80s) The number of n × n alternating sign matrices is given by

|ASMn| =

n∏− 1

k=

(3k + 1)! (n + k)!

|ASM 2 | =

|ASM 3 | =

|ASM 4 | =

10 × 9 × 8 × 7

5 × 3 × 4 × 2

History

A proof of the ASM conjecture was announced by Doron Zeilberger in December 1992 – by mid 1994 all gaps in the 84 page paper had been filled and cases checked:

The six vertex model

The six vertex model is a statistical mechanics model of ice crystals.

Oxygen atoms sit at vertices of a square lattice; Hydrogen atoms live on the edges, and can be in one of two energy minima (closer to one of the two oxygen atoms)

The six vertex model

But each Oxygen atom must have exactly two Hydrogens close to it to form a H 2 O molecule!

Vertex weights

If we assign a weight wi to each of the 6 possible vertex types, define the weight of an arrow configuration σ on a finite domain D to be the product of all weights of vertices in σ.

The partition function of the model on the domain D is then the weighted enumeration

Z (D) =

all states σ

weight(σ).

Path interpretation of six vertex model states

Six vertex model can also be thought of as a model of interacting paths:

Non-interacting paths and free fermions

It’s the interaction between the paths in the 6-vertex model that makes it tricky to analyze.

After all, the number of non-intersecting ensembles of n paths in any acyclic directed graph Γ starting at vertices A = {a 1 ,... , an} and ending at vertices B = {b 1 ,... , bn} is computed by the Lindst¨om-Gessel-Viennot Lemma:

#paths (A → B) = det [E (ai , bj )]ni,j=1 ,

where E (ai , bj ) = #paths from ai → bj. (cf Wick’s theorem for free fermions, Kasteleyn determinant for dimers, etc...)

Interacting path models?

The 6-vertex is more subtle than a model of free fermions, and for general boundary conditions its partition function does not have a determinant representation.

Nonetheless, for a special type of boundary condition on the square grid called the domain wall, one can use the integrability of the model to exactly calculate the partition function.

Observation: (Elkies-Kuperberg-Larsen-Propp ’92) The assignment

defines a bijection

{n × n ASM’s} ↔ {6-vertex states on n × n grid with DWBC. }

Bijection with ASM

Integrable weights for the 6-vertex model

Baxter’s discovery: suppose q, x are nonzero complex numbers, and we choose vertex weights of the form

Algebraic perspective on six vertex model

Consider V = C^2 ' V ∗^ with orthonormal basis {e 1 , e 2 }.

We can think of a vertex as representing a 4-legged tensor in V ⊗ V ⊗ V ∗^ ⊗ V ∗, whose matrix elements with respect to the basis {e 1 ⊗ e 1 , e 1 ⊗ e 2 , e 2 ⊗ e 1 , e 2 ⊗ e 2 } are given by the vertex weights: