Address: Svobody Sq. 4, 61022, Kharkiv, Ukraine, Rooms. 5-46, 7-36, Phone: +38(057) , Outline 1/30 1. Spin models in Quantum theory of magnetism 2. Spin-1/2 XY chain and XX chain Hamiltonians as exactly solvable spin models 3. The exact energy spectrum of XY chain and XX chain 4. Quantum Phase Transition in XX chain at zero temperature 5.Low temperature thermodynamics of XX chain Spin-1/2 chain with anisotropic exchange and periodic boundary conditions Ising model in longitudinal field (1925) XXX-model: Bethe Ansatz (1931), Hulthn (1938) XY-model: Lieb, Shultz, Mattis (1961) Ising model in transverse field (similar to XY-model) XXZ-model: Orbach (1958), Yang, Yang (1966) XYZ-model: Baxter (1972), Takhtajan, Faddeev (1981) (Quantum inverse scattering method = algebraic Bethe ansatz ) EXACT RESULTS FOR HEISENBERG EXCHANGE MODEL 2/30 EXACT RESULTS FOR HEISENBERG EXCHANGE MODEL 3/30 Important commutation relations for Hamiltonian (1) For isotropic (XXX) model and for model with uniaxial anisotropy (XXZ) z-projection of total spin is the good quantum number. One can classify the stationary states by the number of inverted spins. For XXX model the total also is conserved. For fully anisotropic (XYZ) chain does not conserved! Spin-1/2 chain with anisotropic exchange and periodic boundary conditions Spin-1/2 XY chain. Ideal gas of spinless fermions. 4/30 Examples of XY chain real compounds It is important, that we consider open chain Spin-1/2 XY chain. Ideal gas of spinless fermions. 5/30 The Hamiltonian (1) can be rewritten in terms of ladder operators Spin-1/2 XY chain. Ideal gas of spinless fermions. Let us use Jordan-Wigner transformation (1928) to rewrite the Hamiltonian (2) in terms of creation and annihilation operators with Fermi commutation rules 6/30 Fourier transformation for the Hamiltonian (3): 7/30 Spin-1/2 XY chain. Ideal gas of spinless fermions. in case of closed chain in case of open chain 8/30 Spin-1/2 XY chain. Ideal gas of spinless fermions. Using (6), rewrite (4) in the form: Spin-1/2 XY chain. Ideal gas of spinless fermions. 9/30 Coefficients U, V are determined by Spin-1/2 XY chain. Ideal gas of spinless fermions. 10/30 (7) Anisotropic XY chain always has gapped spectrum Spin-1/2 XY chain. Ideal gas of spinless fermions. 11/30 Spin-1/2 XY chain. Ideal gas of spinless fermions. 12/30 Energy gap may vanish only, if in low magnetic field Energy spectrum of XY chain in the external magnetic field for different ratio Important remark about the energy spectrum of XX chain. Spin-1/2 XY chain. Ideal gas of spinless fermions. 13/30 Fourier transformation diagonalize the Hamiltonian (3) in case J x = J y = J. So, for XX chain we obtain reduces the Hamiltonian (8) to (7). It means that there are two types of excitations in case of H < H c Energy spectrum of XX chain in zero magnetic field corresponds to Tomonaga-Luttinger liquid with linear k dependence of energy spectrum for lowest excitations Spin-1/2 XY chain. Ideal gas of spinless fermions. 14/30 Lowest energies approximation : Spin-1/2 XY chain. Ideal gas of spinless fermions. 15/30 Isotropic XX chain may be gapless for low magnetic fields Energy spectrum of XX chain in the external magnetic field Spin-1/2 XY chain. Ideal gas of spinless fermions. 16/30 17/30 Spin-1/2 XY chain. Thermodynamics at low temperatures. Partition function: Free energy: Internal energy: Magnetization and susceptibility:Heat capacity; For XX chain all thermodynamic characteristics can be calculated by Hamiltonian either in the form (7), or in the form (8). 18/30 Spin-1/2 XY chain. Thermodynamics at low temperatures. Partition function: Free energy: Internal energy: Magnetization and susceptibility: Heat capacity; Spin-1/2 infinite XX chain. Second order quantum phase transition at T = 0. 19/30 Critical field Spin-1/2 XX chain. Field dependence of magnetization at low temperatures for finite and infinite chain 20/30 For finite chain at T=0 there are a number of first order phase transitions. Each jump corresponds to inverting of one spin. At very low temperatures M(H) demonstrates the remains of these transitions. Spin-1/2 XX chain. Field dependence of magnetization for finite chain N= 12 for different temperatures 21/30 Temperature increasing smooth M(H) rapidly. 22/30 Spin-1/2 XX chain. Field dependence of specific heat 23/30 Field dependence of specific heat for finite XX chain has multiple maxima and minima at very low temperatures Spin-1/2 XX chain. Field dependence of specific heat for finite chain N= 12 for different temperatures 24/30 Spin-1/2 XX chain. Temperature dependence of specific heat for finite chain N= 12 and for infinite chain for different fields 25/30 Spin-1/2 XX chain. Temperature dependence of specific heat for finite chain N= 12 for different fields 26/30 At H=0.92H c Sources D.Mattis, The Theory of Magnetism Made Simple: An Introduction to Physical Concepts and to Some Useful mathematical methods. World Scientific (2006) 565 p. A.A. Zvyagin. Quantum Theory of One-Dimensional Spin Systems, Cambridge Scientific Publishers, Cambridge (2010) 330p. Lieb E., Schultz T.D., Mattis D.C.. Two soluble models of an antiferromagnetic chain // Ann.Phys.(NY). V.16. P .., .. // .1966. .50, .5. .., .. // .1967. T. 52. c /30 29