We compute the continuum limit of the spectra for the XX model with arbitrary complex boundary fields. In the case of Hermitian boundary terms one obtains the partition functions of the free compactified boson field on a cylinder with Neumann-Neumann, Dirichlet-Neumann or Dirichlet-Dirichlet boundary conditions. This also applies for certain non-Hermitian boundaries. For special cases we also compute the free surface energy. For certain non-Hermitian boundary terms the results are more complex. Here one obtains logarithmic corrections to the free surface energy. The asymmetric version of the XX model with boundaries (this includes the Dzyaloshinsky-Moriya interaction) is also discussed.
We compute the continuum limit of the spectra for the XX model with arbitrary complex boundary fields. In the case of Hermitian boundary terms one obtains the partition functions of the free compactified boson field on a cylinder with Neumann-Neumann, Dirichlet-Neumann or Dirichlet-Dirichlet boundary conditions. This also applies for certain non-Hermitian boundaries. For special cases we also compute the free surface energy. For certain non-Hermitian boundary terms the results are more complex. Here one obtains logarithmic corrections to the free surface energy. The asymmetric version of the XX model with boundaries (this includes the Dzyaloshinsky-Moriya interaction) is also discussed.
A detailed analysis of the finite-size effects on the bulk critical behaviour of the d -dimensional mean spherical model confined to a film geometry with finite thickness L is reported. Along the finite direction different kinds of boundary conditions are applied: periodic ( p ), antiperiodic ( a ) and free surfaces with Dirichlet ( D ), Neumann ( N ) and a combination of Neumann and Dirichlet ( ND ) on both surfaces. A systematic method for the evaluation of the finite-size corrections to the free energy for the different types of boundary conditions is proposed. The free energy density and the equation for the spherical field are computed for arbitrary d . It is found, for 2 < d < 4, that the singular part of the free energy has the required finite-size scaling form at the bulk critical temperature only for ( p ) and ( a ). For the remaining boundary conditions the standard finite-size scaling hypothesis is not valid. At d = 3, the critical amplitude of the singular part of the free energy (related to the so-called Casimir amplitude) is estimated. We obtain Δ ( p ) = −2ζ(3)/(5π) = −0.153 051..., Δ ( a ) = 0.274 543... and Δ ( ND ) = 0.019 22..., implying a fluctuation-induced attraction between the surfaces for ( p ) and repulsion in the other two cases. For ( D ) and ( N ) we find a logarithmic dependence on L .
A detailed analysis of the finite–size effects on the bulk critical behaviour of the d–dimensional mean spherical model confined to a film geometry with finite thickness L is reported. Along the finite direction different boundary conditions are applied: periodic (p), antiperiodic (a) and free surfaces with Dirichlet (D), Neumann (N) and a combination of Neumann and Dirichlet (ND) on both surfaces. A systematic method for the evaluation of the finite–size corrections to the free energy for the different types of boundary conditions is proposed. The free energy density and the equation for the spherical field are computed for arbitrary d. It is found, for 2 < d < 4, that the singular part of the free energy has the required finite–size scaling form at the bulk critical temperature only for (p) and (a). For the remaining boundary conditions the standard finite–size scaling hypothesis is not valid. At d = 3, the critical amplitude (related to the so called Casimir amplitude) is estimated. We obtain
We extend the well-known method of calculating bulk correlation functions of the conformal Ising model via bosonisation to situations with boundaries. Oshikawa and Affleck have found the boundary states of two decoupled Ising models in terms of the orbifold of a single free boson compactified on a circle of radius r=1; we adapt their results to include disorder operators. Using these boundary states we calculate the expectation value of a single disorder field on a cylinder with free boundary conditions and show that in the appropriate limits we recover the standard and frustrated partition functions. We also show how to calculate Ising correlation functions on the upper half plane. 1
In quantum field theories defined on a space-time with boundaries, the entanglement entropy exhibits subleading, boundary-induced corrections to the ubiquitous area law. At critical points described by conformal field theories (CFTs), and when the entangling surface intersects the physical boundary of the space, new universal terms appear in the entropy and encode valuable information about the boundary CFT. In 2+1 dimensions, the universal subleading boundary term is logarithmic with coefficient $b(\ensuremath{\theta})$ depending on the angle $\ensuremath{\theta}$ at which the entangling surface intersects the boundary, as well as on the boundary conditions (BCs). In this paper, we conduct a numerical study of $b(\ensuremath{\theta})$ for free bosons on finite-size square lattices. We find a surprisingly accurate fit between our lattice results and the corresponding holographic function available in the literature. We also comment on the ratio ${b}^{\ensuremath{'}\ensuremath{'}}(\ensuremath{\pi}/2)/{A}_{T}$, where AT is the central charge in the near boundary expansion of the stress tensor, for which a holographic analysis suggests that it may be a universal quantity. Though we show evidence that this ratio is violated for the free boson with Dirichlet BCs, we conjecture its validity for free bosons evenly split between Dirichlet and Neumann BCs.
Finite-size effects are investigated in the Gaussian model with isotropic and anisotropic short-range interactions in film geometry with nonperiodic boundary conditions (bc) above, at, and below the bulk critical temperature Tc. We have obtained exact results for the free energy and the Casimir force for antiperiodic, Neumann, Dirichlet, and Neumann-Dirichlet mixed bc in 1<d<4 dimensions. For the Casimir force, finite-size scaling is found to be valid for all bc. For the free energy, finite-size scaling is valid in 1<d<3 and 3<d<4 dimensions for antiperiodic, Neumann, and Dirichlet bc, but logarithmic deviations from finite-size scaling exist in d=3 dimensions for Neumann and Dirichlet bc. This is explained in terms of the borderline dimension d*=3 , where the critical exponent 1-α-ν=(d-3)∕2 of the Gaussian surface energy density vanishes. For Neumann-Dirichlet bc, finite-size scaling is strongly violated above Tc for 1<d<4 because of a cancelation of the leading scaling terms. For antiperiodic, Dirichlet, and Neumann-Dirichlet bc, a finite film critical temperature Tc,film(L)<Tc exists at finite film thickness L . Our results include an exact description of the dimensional crossover between the d -dimensional finite-size critical behavior near bulk Tc and the (d-1) -dimensional critical behavior near Tc,film(L). This dimensional crossover is illustrated for the critical behavior of the specific heat. Particular attention is paid to an appropriate representation of the free energy in the region Tc,film(L)≤T≤Tc. For 2<d<4 , the Gaussian results are renormalized and reformulated as one-loop contributions of the φ4 field theory at fixed dimension d and are then compared with the ε=4-d expansion results at ε=1 as well as with d=3 Monte Carlo data. For d=2 , the Gaussian results for the Casimir force scaling function are compared with those for the Ising model with periodic, antiperiodic, and free bc; unexpected exact relations are found between the Gaussian and Ising scaling functions. For both the d -dimensional Gaussian model and the two-dimensional Ising model it is shown that anisotropic couplings imply nonuniversal scaling functions of the Casimir force that depend explicitly on microscopic couplings. Our Gaussian results provide the basis for the investigation of finite-size effects of the mean spherical model in film geometry with nonperiodic bc above, at, and below the bulk critical temperature.
Finite-size corrections to the energy levels of the asymmetric six-vertex model transfer matrix are considered using the Bethe ansatz solution for the critical region. The nonuniversal complex anisotropy factor is related to the bulk susceptibilities. The universal Gaussian coupling constant g is also related to the bulk susceptibilities as g=2${\mathit{H}}^{\mathrm{\ensuremath{-}}1/2}$/\ensuremath{\pi}, H being the Hessian of the bulk free energy surface viewed as a function of the two fields. The modular covariant toroidal partition function is derived in the form of the modified Coulombic partition function which embodies the effect of incommensurability through two mismatch parameters. The effect of twisted boundary conditions is also considered. \textcopyright{} 1996 The American Physical Society.