1. A Novel Algebraic Approach to Quantum and Classical Dualities Emilio Cobanera Department of Physics - Indiana University DESY Theory Workshop 2010 Gerardo Ortiz: Indiana University, Bloomington Zohar Nussinov: Washington University - St. Louis

2. Why Dualities? Quantum and Classical Stat Mech: Phase Diagrams, Renormalization Group Quantum Computation: Thermal Fragility Lattice Field Theory Dimensional Reduction

3. Why Dualities? Quantum Field Theory: Perturbation theory for strongly coupled/correlated models Montonen-Olive Conjecture Seiberg-Witten AdS-CFT and Beyond

4. What are Quantum Dualities? What are Quantum Dualities? Wisdom: Strong-coupling-to-Weak-coupling relations

5. Our bond-algebraic approach

6. is dual to if there is an is dual to if there is an isomorphism between their bond algebras Dualities Are Mappings Between Hamiltonian- Dependent Algebras

7. BONDS 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 Bond Algebras and Their Symmetries

8. Quantum Mechanics requires these mappings to be UNITARILY IMPLEMENTABLE Self-Dualities are automorphims of bond algebras Self-Dualities are automorphims of bond algebras that preserve the form of the Hamiltonian A Self-Dualities is a symmetry of the bond algebra that preserves the form of the Hamiltonian In other words:

9. Duality mappings: Non-local Kitaev’s toric code model: Wen’s plaquette model: 2 Ising chains: (Nussinov-Ortiz 2006) (Identical spectra)

10. Example of Self-Duality: Ising chain in a transverse field

11. BOND ALGEBRA Every bond anti-commutes with two bonds Every bond anti-commutes with two bonds

12. SELF-DUALITY AUTOMORPHISM Dual

13. Mapping is (almost) Unitarily implementable Mapping is (almost) Unitarily implementable Ising chain in a transverse field is self-dual, meaning:

14. Advantages: Better suited for systematic (ALGORITHMIC) search of (self)dualities Allows us to derive the (in general) non- local dual operator variables - the ones that had to be guessed in the past

15. Dualities in finite systems (Role of boundary terms) xx xx x xx xx x x It is not self-dual: It is self-dual: xx x x x x String

16. Parameter-Dep Bond Algebras Bond algebra: Automorphism: Self-dual Hamiltonian: boundary terms Dual variables also depend on parameters

17. Classical Dualities Wisdom: Low-temperature-to-High-temperature relations

18. has a remarkable property: it is self-dual, meaning… Kramers-Wannier Self-Duality of the D =2 Ising Model in empty space (vacuum)

19. whene ver KW SELF-DUALITY RELATION The critical point is located at the self-dual point:CONCEPT: High-T Low-T relation High-T Low-T relationCONCEPT:

20. YES!!!!!! The QUANTUM Self-Duality guarantees that OR BETTER, IN TERMS OF CLASSICAL PARTITION FUNCTIONS

21. Contrast: Quantum vs Classical Quantum Self-duality relation Classical Self-duality relation

22. Classical and Quantum (Self-)Dualities are equivalent and in correspondence: We have managed to UNIFY them.

23. 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), ecobaner@indiana.edu

24. A self-duality is not a symmetry in general, but Dualities and New Symmetries A self-duality is an emergent symmetry at the self-dual point

25. Non-Abelian Dualities Is AdS-CFT an exact duality? (Dimensional Reduction) How to go beyond the Fourier transform Bond-Algebraic Topological Excitations Big Questions: