Solvable critical dense polymers on the cylinder

Abstract

A lattice model of critical dense polymers is solved exactly on a cylinder with finitecircumference. The model is the first member of the Yang–Baxter integrable series of logarithmic minimal models.The cylinder topology allows for non-contractible loops with fugacityα that wind around the cylinder or for an arbitrary number of defects that propagate along the full length of the cylinder. Using an enlarged periodicTemperley–Lieb algebra, we set up commuting transfer matrices acting on states whose links areconsidered distinct with respect to connectivity around the front or back of the cylinder. Thesetransfer matrices satisfy a functional equation in the form of an inversion identity. For evenN, this involves a non-diagonalizable braid operatorJ and aninvolution R = − (J3 − 12J)/16 = (−1)F with eigenvalues . This is reminiscent of supersymmetry with a pair of defects interpreted as a fermion. Thenumber of defects thus separates the theory into Ramond ( even), Neveu–Schwarz ( odd) and ( odd) sectors. For the case of loop fugacityα = 2, the inversion identity is solved exactly sector by sector for the eigenvalues in finitegeometry. The eigenvalues are classified according to the physical combinatorics of thepatterns of zeros in the complex spectral-parameter plane. This yields selection rules forthe physically relevant solutions to the inversion identity. The finite-size correctionsare obtained from Euler–Maclaurin formula. In the scaling limit, we obtain theconformal partition functions as sesquilinear forms and confirm the central chargec = − 2 and conformal weights . Here and in the even sectors with Kac labels r = 1, 2, 3,...;s = 1, 2 while in the odd sectors. Strikingly, the odd sectors exhibit a -extended symmetry but the even sectors do not. Moreover, the naive trace summing over all even sectors does not yield a modular invariant.