Quantum Integrability and Algebraic Geometry Marcio J. Martins Universidade Federal de São Carlos Departamento de Física July 218
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
Classical Lattice Models of Statistical Thermodynamics Is the framework of Statistical Thermodynamics able to handle Phase Transitions?
Brief History of Magnetic Phase Transition Pierre Curie, Paul Langevin and Pierre Weiss (1895-191): 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 (1928-1929): 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.
Leo Kadanoff and Kenneth Wilson (1963-1975): 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 )
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
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
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=
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 1 1 1 1 1 1 1 z 1/4 z 1/4 z 1/4 z 1/4 z 1/2
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.
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 ]
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 5...... 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 =.
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
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)
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