Finite size effect

introduction

• Casimir effect in QED is one example of finite size effect
• the stress on the bounding surfaces when quantum field is confined to finite volume of space
• type of boundaries
  
  • real material media
  • interface between two different phases of the vacuum of a field theory such as QCD, in which case colored field may only exist in the interior region
  • topology of space
• the boundaries restrict the modes of the quantum fields
• give rise to measurable and important forces

how to compute the Casimir effect

• zero-point energy in the presence of the boundaries
  • sum over all modes
  • any kind of constraint or boudary conditions on the the zero-point modes of the quantum fields in question, including backgrounds such as gravity
  • In a model without boundary conditions, the Hamiltonian value associated wih the vacuum or ground state, called zero-point energy, is usually discarded because, despite being infinite, may be reabsorbed in a suitable redefinition of the energy origin
  • there are several ways to put such an adjustment into practice, normal ordering being oneof the most popular
• Green's functions method
  • represents the vacuum expectation value of the product of fields

finite size effect and central charge

• mass gap of order $1/N$ is the characteristic of conformal invariance
• finite-size correction term to the ground state energy

$$ E_0=N\epsilon_0-\frac{\pi c v_F}{6N} +O(\frac{1}{N^2} $$ where $N$ denotes the number of sites in the spin chain

• finite-size corrections to largest eigenvalue of the transfer matrix
• low temperature asymptotics of free energy of quantum system

$$ F(\beta)=F_0-\frac{\pi c}{6v_F}\beta^{-2}+O(\beta^{-2}) $$ where $\beta=T^{-1}$ is the inverse temperature

related items

• cosmological constant
  

books

• Kimball A. Milton The Casimir Effect: Physical Manifestations of Zero-Point Energy
  
• Claude Itzykson Conformal invariance and finite size effects in critical two dimensional statistical models
• Michael Krech Casimir effect in critical systems

encyclopedia

• http://ko.wikipedia.org/wiki/카시미르효과
• http://en.wikipedia.org/wiki/finite_size_effect
• http://en.wikipedia.org/wiki/Casimir_effect
• http://en.wikipedia.org/wiki/Vacuum_energy

articles

• Ian Affleck Universal term in the free energy at a critical point and the conformal anomaly, Phys. Rev. Lett. 56, 746–748 (1986)
• H. W. J. Blöte, J. Cardy and M. P. Nightingale Conformal invariance, the central charge, and universal finite-size amplitudes at criticality, Phys. Rev. Lett. 56, 742–745 (1986)
• Cardy, John L. 1986. “Operator Content of Two-dimensional Conformally Invariant Theories.” Nuclear Physics. B 270 (2): 186–204. doi:http://dx.doi.org/10.1016/0550-3213(86)90552-3.