Download Cobanera - DESY Theory Workshop

A Novel Algebraic Approach to
Quantum and Classical Dualities
Emilio Cobanera
Department of Physics - Indiana University
Gerardo Ortiz: Indiana University, Bloomington
Zohar Nussinov: Washington University - St. Louis
DESY Theory Workshop 2010
Why Dualities?
Quantum and Classical Stat Mech: Phase
Diagrams, Renormalization Group
Quantum Computation: Thermal Fragility
Lattice Field Theory
Dimensional Reduction
Why Dualities?
Quantum Field Theory:
Perturbation theory for strongly
coupled/correlated models
Montonen-Olive Conjecture
Seiberg-Witten
AdS-CFT
and Beyond
What are
Quantum Dualities?
Wisdom: Strong-coupling-to-Weak-coupling relations
Our bond-algebraic approach
Dualities Are Mappings
Between HamiltonianDependent Algebras
is dual to
if there is an
isomorphism between their bond algebras
Bond Algebras and Their
Symmetries
Quantum Hamiltonians are built as a sum of
quasi-local operators, its BONDS:
A bond algebra for H: the set of all linear
combinations of products of bonds
THE KEY TO (SELF-)DUALITIES
Self-Dualities are automorphims of bond algebras
that preserve the form of the Hamiltonian
In other words:
A Self-Dualities is a symmetry of the bond algebra
that preserves the form of the Hamiltonian
Quantum Mechanics requires these mappings to be
UNITARILY IMPLEMENTABLE
Example of Self-Duality: Ising
chain in a transverse field
BOND
ALGEBRA
Every bond
Every bond
anti-commutes with two bonds
anti-commutes with two bonds
SELF-DUALITY AUTOMORPHISM
Dual
Mapping is (almost)
Unitarily implementable
Ising chain in a transverse field is
self-dual, meaning:
Advantages:
Better suited for systematic
(ALGORITHMIC) search of
(self)dualities
Allows us to derive the (in general) nonlocal dual operator variables - the ones that
had to be guessed in the past
Parameter-Dep Bond Algebras
Bond algebra:
Automorphism:
Self-dual Hamiltonian:
Dual variables also depend on parameters
boundary terms
Classical Dualities
dom: Low-temperature-to-High-temperature relatio
Kramers-Wannier Self-Duality of
the D=2 Ising Model
in empty space (vacuum)
has a remarkable
property:
it is self-dual,
meaning…
KW SELF-DUALITY
RELATION
whene
ver
CONCEPT:
High-T
Low-T relation
The critical point is located at
the
self-dual point:
YES!!!!!!
The QUANTUM Self-Duality guarantees that
OR BETTER, IN TERMS
OF
CLASSICAL PARTITION
FUNCTIONS
Contrast: Quantum vs Classical
Quantum Self-duality relation
Classical Self-duality relation
Classical and Quantum (Self-)Dualities
are equivalent and in correspondence:
We have managed to UNIFY them.
ALL (EXACT) DUALITIES
WE KNOW OF
COVERED BY THE BOND
ALGEBRAIC APPROACH,
Plus Some NEW Self-Dual Field
Theories
E. Cobanera, G. Ortiz, Z. Nussinov,
Phys. Rev. Lett. 104, 020402 (2010),
[email protected]
Dualities and New
Symmetries
A self-duality is not a symmetry in
general, but
A self-duality is an emergent symmetry at
the self-dual point
Big Questions:
Non-Abelian Dualities
Is AdS-CFT an exact duality?
(Dimensional Reduction)
How to go beyond the Fourier transform
Bond-Algebraic Topological Excitations