Physical Review Letters

Conformal invariance, the central charge, and universal finite-size amplitudes at criticality

Henk W. J. Blöte, John Cardy, M. P. Nightingale

Published Feb 17, 1986• 1,545 citations

Abstract

We show that for conformally invariant two-dimensional systems, the amplitude of the finite-size corrections to the free energy of an infinitely long strip of width L at criticality is linearly related to the conformal anomaly number c, for various boundary conditions. The result is confirmed by renormalization-group arguments and numerical calculations. It is also related to the magnitude of the Casimir effect in an interacting one-dimensional field theory, and to the low-temperature specific heat in quantum chains.

DOI: 10.1103/physrevlett.56.742

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Physical Review Letters

Conformal invariance, the central charge, and universal finite-size amplitudes at criticality

Henk W. J. Blöte, John Cardy, M. P. Nightingale

Published Feb 17, 1986• 1,545 citations

Publisher

Abstract

DOI: 10.1103/physrevlett.56.742

Related papers

Universal term in the free energy at a critical point and the conformal anomaly

Ian K. Affleck

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{)}^{2}

\ensuremath{\Elzxh}v, where v is the effective ``velocity of light.'' The latter formula is used to check recently derived critical theories of spin-s quantum chains against Bethe-Ansatz solutions.

The conformal-invariance approach is extended to two-dimensional critical systems with mixed ab boundary conditions. The profiles 〈\ensuremath{\varphi}(r)

{\mathrm{〉}}_{\mathit{a}\mathit{b}}

of the magnetization and the energy density in Ising and Q-state Potts strips with +- and fixed-free boundary conditions are derived. The profile of an arbritrary operator degenerate at level 2 is expressed in terms of hypergeometric functions. The universal amplitudes

{\mathit{A}}_{\mathit{a}\mathit{b}}

{\mathit{B}}_{\mathit{a}\mathit{b}}

of the Casimir force and the Fisher--de Gennes distant-wall correction to the profile satisfy

{\mathit{B}}_{\mathit{a}\mathit{b}}

=-4\ensuremath{\pi}${\mathit{x}}_{\mathrm{\ensuremath{\varphi}}}$${\mathit{c}}^{\mathrm{\ensuremath{-}}1}$

{\mathit{A}}_{\mathit{a}\mathit{b}}

. Here ${\mathit{x}}_{\mathrm{\ensuremath{\varphi}}}$ is the scaling dimension of \ensuremath{\varphi}(r) and c is the central charge.

Quantum one-dimensional criticality with conformal invariance is known to be characterized by conformal field theory with the inherent central charge (

c

), and universal finite-size amplitudes. Here, the authors demonstrate that around a quantum criticality with

c

=\textonehalf{} and two Majorana boundary modes, the symmetry-breaking effect modifies the finite-size amplitudes from a universal constant to be a universal function of an emergent scale. This finding provides a new perspective to study the interplay of topology and quantum criticality.