Abstract
We study the non-equilibrium dynamics of conformal field theory (CFT) in 1+1 dimensions with a smooth position-dependent velocity v(x) explicitly breaking translation invariance. Such inhomogeneous CFT is argued to effectively describe 1+1-dimensional quantum many-body systems with certain inhomogeneities varying on mesoscopic scales. Both heat and charge transport are studied, where, for concreteness, we suppose that our CFT has a conserved U(1) current. Based on projective unitary representations of diffeomorphisms and smooth maps in Minkowskian CFT, we obtain a recipe for computing the exact non-equilibrium dynamics in inhomogeneous CFT when evolving from initial states defined by smooth inverse-temperature and chemical-potential profiles β(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta (x)$$\\end{document} and μ(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu (x)$$\\end{document}. Using this recipe, the following exact analytical results are obtained: (i) the full time evolution of densities and currents for heat and charge transport, (ii) correlation functions for components of the energy–momentum tensor and the U(1) current as well as for any primary field, and (iii) the thermal and electrical conductivities. The latter are computed by direct dynamical considerations and alternatively using a Green–Kubo formula. Both give the same explicit expressions for the conductivities, which reveal how inhomogeneous dynamics opens up the possibility for diffusion as well as implies a generalization of the Wiedemann–Franz law to finite times within CFT.
Highlights
Conformal field theory (CFT) is routinely used to effectively describe universal properties of quantum many-body systems in equilibrium [1]
This makes clear that the generalization to the inhomogeneous dynamics given by H in (1.1) is important as it opens up a mechanism for diffusion within CFT
The conductivities were computed in two ways: The first based on a dynamical approach and the second using a Green–Kubo formula that we derived
Summary
Conformal field theory (CFT) is routinely used to effectively describe universal properties of quantum many-body systems in equilibrium [1]. One such example is the partitioning protocol, where the time evolution is studied starting from an initial state produced by glueing together two semi-infinite systems independently in equilibrium with different thermodynamic variables, such as different temperatures and/or chemical potentials This was studied within CFT in, e.g., [3,4,5,6,7,8] among others. We mention that the diffusive effect due to the type of randomness in [22] was recently demonstrated numerically for random quantum spin chains in [30] using generalized hydrodynamics [31,32] This makes clear that the generalization to the inhomogeneous dynamics given by H in (1.1) is important as it opens up a mechanism for diffusion within CFT. One supplementary purpose of this paper is to demonstrate the simplicity and beauty of the Minkowskian theory, which are true when one studies non-equilibrium properties
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