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Register a new userLogarithmic correlators or responses in non-relativistic analogues of conformal invariance
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Recent developments on emergence of logarithmic terms in correlators or response functions of models which exhibit dynamical symmetries analogous to conformal invariance in not necessarily relativistic systems are reviewed. The main examples of these are logarithmic Schrodinger-invariance and logarithmic conformal Galilean invariance. Some applications of these ideas to statistical physics are described.
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We show how logarithmic terms may arise in the correlators of fields which belong to the representation of the Schrodinger-Virasoro algebra (SV) or the affine Galilean Conformal Algebra (GCA). We show that in GCA, only scaling operator can have a Jordanian form and rapidity can not. We observe that in both algebras logarithmic dependence appears along the time direction alone.
Logarithmic representations of the conformal Galilean algebra (CGA) and the Exotic Conformal Galilean algebra ({sc ecga}) are constructed. This can be achieved by non-decomposable representations of the scaling dimensions or the rapidity indices, specific to conformal galilean algebras. Logarithmic representations of the non-exotic CGA lead to the expected constraints on scaling dimensions and rapidities and also on the logarithmic contributions in the co-variant two-point functions. On the other hand, the {sc ecga} admits several distinct situations which are distinguished by different sets of constraints and distinct scaling forms of the two-point functions. Two distinct realisations for the spatial rotations are identified as well. The first example of a reducible, but non-decomposable representation, without logarithmic terms in the two-point function is given.
The periodic sl(2|1) alternating spin chain encodes (some of) the properties of hulls of percolation clusters, and is described in the continuum limit by a logarithmic conformal field theory (LCFT) at central charge c=0. This theory corresponds to the strong coupling regime of a sigma model on the complex projective superspace $mathbb{CP}^{1|1} = mathrm{U}(2|1) / (mathrm{U}(1) times mathrm{U}(1|1))$, and the spectrum of critical exponents can be obtained exactly. In this paper we push the analysis further, and determine the main representation theoretic (logarithmic) features of this continuum limit by extending to the periodic case the approach of [N. Read and H. Saleur, Nucl. Phys. B 777 316 (2007)]. We first focus on determining the representation theory of the finite size spin chain with respect to the algebra of local energy densities provided by a representation of the affine Temperley-Lieb algebra at fugacity one. We then analyze how these algebraic properties carry over to the continuum limit to deduce the structure of the space of states as a representation over the product of left and right Virasoro algebras. Our main result is the full structure of the vacuum module of the theory, which exhibits Jordan cells of arbitrary rank for the Hamiltonian.
There exists a certain argument that in even dimensions, scale invariant quantum field theories are conformal invariant. We may try to extend the argument in $2n + epsilon$ dimensions, but the naive extension has a small loophole, which indeed shows an obstruction in non-linear sigma models in $2+epsilon$ dimensions. Even though it could have failed due to the loophole, we show that scale invariance does imply conformal invariance of non-linear sigma models in $2+epsilon$ dimension from the seminal work by Perelman on the Ricci flow.
We present an expression for the generating function of correlation functions of the sine-Gordon integrable field theory on a cylinder, with compact space. This is derived from the Destri-De Vega integrable lattice regularization of the theory, formulated as an inhomogeneous Heisenberg XXZ spin chain, and from more recent advances in the computations of spin form factors in the thermodynamic limit.
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