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Inhomogeneous conformal field theory out of equilibrium

Per Moosavi

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 $β(x)$ and $μ(x)$. 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.

Inhomogeneous conformal field theory out of equilibrium

Abstract

We study the non-equilibrium dynamics of conformal field theory (CFT) in 1+1 dimensions with a smooth position-dependent velocity 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 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 and . 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 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.

Paper Structure

This paper contains 34 sections, 4 theorems, 129 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Projective unitary representations and a non-equilibrium recipe
  3. Summary of results and Wiedemann-Franz law for finite times
  4. Organization of the paper
  5. Prerequisites
  6. Conformal transformations
  7. Gauge transformations
  8. Examples
  9. Non-equilibrium recipe
  10. Applications
  11. Densities and currents
  12. Correlation functions
  13. Conductivities
  14. Dynamical approach
  15. Green-Kubo approach
  16. ...and 19 more sections

Key Result

Proposition 3.1

Define $H$, $Q$, and $G$ as in H_iCFT--G_iCFT as well as $f(x)$, $g(x)$, $h(x)$, $v_{0}$, $\beta_{0}$, and $\mu_{0}$ as in fgh. Let This is to avoid any confusion of notation, reserving $\mathcal{O}(\cdot, \cdot)$ to the dependence on the right- and left-moving coordinates, cf. the primary fields in denote the inhomogeneous time evolution for any local operator $\mathcal{O}(x)$ and let denote the

Figures (1)

  • Figure 1: Illustration of an inhomogeneous CFT with a fixed position-dependent velocity $v(x)$ effectively describing a quantum $XXZ$ spin chain with couplings $J_{j}^{x} = J_{j}^{y} = J_{j}$ and $J_{j}^{z} = J_{j} \Delta$ (for constant $\Delta$) between spins on adjacent sites at $x_{j}$ and $x_{j+1}$ uniformly varying on mesoscopic length scales much larger than the lattice spacing but much smaller than the system size. The spatial dependence of $v(x)$ is directly related to that of the couplings $J_{j}$ (see Section 5.2 in Moo), and the color and size of the dots indicate the magnitude of the latter.

Theorems & Definitions (17)

  • Example 2.1: Free massless fermions on the circle
  • Example 2.2: Free massless bosons on the circle
  • Remark 2.3
  • Example 2.4: Local Luttinger model
  • Proposition 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 4.1
  • Remark 4.2: Interpretations of \ref{['iCFT_cEcJ_cont_eq_2']} and \ref{['iCFT_rj_cont_eq_2']}
  • Remark 5.1: Sugawara construction
  • ...and 7 more