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Steklov Mathematical Institute Seminar
January 17, 2013 16:30, Moscow, Steklov Mathematical Institute of RAS, Conference Hall (8 Gubkina)
 


Form factor approach to the calculation of dynamical correlation functions in critical models

N. A. Slavnov
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N. A. Slavnov
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Abstract: The most known phenomenon of phase transition is a transition of matter from a liquid to a gas. Gradually varying the temperature leads to an abrupt change of the density at a certain value of $\mathbf T_0$. Other phenomenon of phase transition is an abrupt change in the magnetization of ferromagnets at small changes in the direction of a weak external magnetic field. Magnitude of the jump depends on the temperature of the ferromagnet. As the temperature increases, the value of the jump decreases, vanishing at some $\mathbf T=\mathbf T_{\mathrm{cr}}$ (Curie point). The value of $\mathbf T_{\mathrm{cr}}$ called a critical point. Physical models whose parameters equal to the critical values are called critical models.
The characteristic size of the fluctuations in critical models is greatly increased. It leads to the fact that all phenomena in the critical model are cooperative, that is, they are due to the properties of the entire set of particles, and not the individual properties of each particle. For example, in the models, describing crystals with nearest-neighbor interaction, one actually observes a long-range interaction (the correlation functions do not decay exponentially, but decay as a power law).
In our work, a method of calculating correlation functions in critical quantum integrable systems was developed. We compute correlation function of two spin operators, separated by spatial and temporal intervals. The method is based on the decomposition of the two-point correlation functions with respect to the matrix elements of an individual operator (form factors). In critical models, form factors in the infinite volume limit (thermodynamic limit) tend to zero as a power of this volume. We have introduced a notion of dressed form factor, which is the sum of the original form factors over all excitations with the same values of the excitation energy and momentum. Dressed form factor has a finite value in the thermodynamic limit. In the framework of this approach we have calculated the asymptotic behavior of the correlation functions in the XXZ Heisenberg spin chain at large distances and times.
It was shown that some asymptotics of correlation functions are not described conformal field theory, if the distance and time simultaneously go to infinity. We also obtained explicit formulas for the structure factors near the thresholds of the density of states. The calculated values of structure factors coincide with very good accuracy with the experimental data obtained in the measurement of the inelastic neutron scattering in one-dimensional crystals.

References
  1. N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras, “Form factor approach to dynamical correlation functions in critical models”, J. Stat. Mech. Theory Exp., 2012, P09001, 33 pp., arXiv: 1206.2630  crossref  mathscinet  isi  scopus
  2. N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras, “A form factor approach to the asymptotic behavior of correlation functions”, J. Stat. Mech. Theory Exp., 2011, P12010, 28 pp., arXiv: hep-th/1110.0803  crossref  isi  scopus
  3. N. Kitanine, K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras, “The thermodynamic limit of particle-hole form factors in the massless Heisenberg chain”, J. Stat. Mech. Theory Exp., 2011, P05028, 34 pp.  crossref  isi  scopus
  4. N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras, “On the thermodynamic limit of form factors in the massless $XXZ$ Heisenberg chain”, J. Math. Phys., 50:9 (2009), 095209, 24 pp.  crossref  mathscinet  zmath  adsnasa  isi  scopus


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