Quantum Integrability and Algebraic Geometry

Transcription

1 Quantum Integrability and Algebraic Geometry Marcio J. Martins Universidade Federal de São Carlos Departamento de Física July 218

2 References R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, 1982 J.H.H Perk and H. Au-Yang, Yang-Baxter Equations, Encyclopedia Vol.5, Elsevier, Amsterdam, 26 I.R. Shafarevich, Basic Algebraic Geometry, Springer-Verlag, New York, 1994 M.J. Martins, B. Nienhuis and R. Rietman, An intersecting loop model as a superspin chain, Phys.Rev.Lett. 81 (1998) 54 M.J. Martins, Algebraic Geometry methods associated to the one-dimensional Hubbard model, Nucl.Phys.B 97 (216) 479

3 Classical Lattice Models of Statistical Thermodynamics Is the framework of Statistical Thermodynamics able to handle Phase Transitions?

4 Brief History of Magnetic Phase Transition Pierre Curie, Paul Langevin and Pierre Weiss ( ): Boltzmann s framework for non-interacting micro-magnets. Wilhelm Lenz and Ernest Ising (192): Basic elements are dipoles which turn over among two positions, Z = σ 1 =± σ 2 =± σ N =± exp[ βe(σ 1,..., σ N )], β = 1 k B T Werner Heisenberg and Paul Dirac ( ): E = i,j J x ij σ x i σ x j + J y ij σy i σy j + J z ij σ z i σ z j Rudolf Pierls (1936): 2D Ising model display spontaneous magnetization for low temperatures prompting new interest.

5 Leo Kadanoff and Kenneth Wilson ( ): At criticality we have scale invariance and the Curie critical point is universal. Brief History of Magnetic Phase Transition Hendrick Kramers and Gregory Wannier (1941): In 2D derived the Curie temperature and a combinatorial sums translated into a linear algebra problem, Z = Tr[T N ] Lars Onsager (1944): 2D Lenz-Ising model is exactly solvable since Λ > Λ 1 >... was found, [ Z = Λ N 1 + ( Λ ] ) N +... Λ 1 The free-energy has no power law singularity as hypothesized, f s T T c 2 log( T T c )

6 Hamiltonian Limit Thermal fluctuations (D+1) classical system quantum effects D-spatial field theory, Path Integral with space-time lattice, = K(x a, t a, x b, t b ) = x a exp [ ih(t b t a )] x b dx N 1... dx 1 x b T x N 1 x N 1 T x N 2... x 1 T x a with t b t a = Nτ and T = exp( iτh) Amplitude with periodic boundary, Z = K(x, Nτ, x, )dx = Tr[T N ] Free energy density Correlation function Correlation lenght Vaccum energy density Propagator Inverse mass gap

7 Modeling Adsorption in Surfaces Greg Dash and Michael Bretz (1971): Thin films of gases adsorbed on regular crystal surfaces: graphite has a hexagonal lattice. The gas atoms slightly larger than a basic hexagon and two adjacent hexagons cannot both be occupied. C T T c α, α =.36

8 Tiling the Triangular Lattice Rodney Baxter (198): Consider a triangular lattice and place hexagonal tiles without overlapping Let g(m,n) be the number of ways of placing m hexagons on N sites, N/3 f (z) = lim log [Z N] /N, Z N = z m g(m, N) N m=

9 Interaction Around Face Model Spin σ =, 1 and instead sites of adsorption one use face sites σ i σ j = for all next-neighbors face variables z 1/4 z 1/4 z 1/4 z 1/4 z 1/2

10 The Ice Model Introduced by Pauling in 1935 to explain the experimental fact that certain phase of Ice has a residual entropy. The lattice sites are occupied by Oxygens O having four nearest neighbors Hydrogens H atoms:o-o >>> O-H.

11 The Vertex Representation To make water molecule H 2 O two Hydrogens are close to the central Oxygen and the other two are farther away. > > < < > > < < > < < > ω 1 ω 1 ω 2 ω 2 ω 3 ω 3 The statistical configuration sits on the edges and can be represented by an arrow whose tip points forwards the side where the Oxygen O is sited. The residual entropy can be computed, S = k B log [Λ ]

12 Brief History of Integrability Hans Bethe plane wave function (1931): H = i,j σ x i σ x j + σ y i σy j + σ z i σ z j Elliott Lieb for = 1/2 (1967): b 1 b 2 b 3 b 4 b N T b 1...b N c a 1...a N = 1 c 2 c 3 c 4 c c N c 1 a 1 a 2 a 3 a 4 a N Barry MacCoy and F. Wu (1968): [T, H] =, provided the weights sit in the quadric, ω1 2 + ω2 2 ω3 2 2 ω 1 ω 2 =.

13 Face Models Yang-Baxter W (a 1, a 2, d, c 1 )W (c 1, d, b 2, b 1 )W (d, a 2, c 2, b 2 ) d = W (c 1, a 1, d, b 1 )W (a 1, a 2, c 2, d)w (d, c 2, b 2, b 1 ) d

14 Onsager Star-Triangle Relation W h (c, a)w v (c, b)w h (b, a) = d W v (c, d)w h (d, a)w v (d, b) W h (a, b)w v (b, c)w h (a, c) = W v (b, d)w h (a, d)w v (d, c)

15 Vertex Models Yang-Baxter W c 2,c 3 a 2,a 3 W c 1,b 3 a 1,c 3 W b 1,b 2 c 1,c 2 = c 1,c 2,c 3 McGuire (1964), C.N. Yang (1968) W c 1,c 2 a 1,a 2 W b 1,c 3 c 1,a 3 W b 2,b 3 c 2,c 3 c 1,c 2,c 3