Thermal conductivity of anisotropic spin ladders By :Hamed Rezania Razi University, Kermanshah, Iran
Magnetic Insulator In one dimensional is a good candidate for thermal conductivity due to magnetic excitation Partially filled electron shell : 3d,4d,4f,5f Heisenberg model Hamiltonian describes these material
Low dimensional quantum magnets 1)Heisenberg chains 2) Spin ladders 3) Spin Peierls: H=J H Heisenberg J i, j S. S i j 1)Finite energy gap in magnetic excitation spectrum Spin liquid phase S r. S 0 e r / 2)Spatial Exponential behaviour for spin correlation function Sr2CuO3, SrCuO2 Electrically Insulating compounds Heisenberg chain
Thermal conductivity Transport of heat in insulator 1)Phonons 2) In low dimensions: magnetic excitation Playground for magnetic studies of this material and quantum information processing and in electronic device Integrable one dimensional Ballistic Transport One dimensional quantum magnets show a unusually large thermal conductivity 1D spin ladder Nonvanishing of Drude weight Castella, etal: PRL 74, 972(1995) Nonitegrability with Luttinger fixed point S. Fujimoto, PRL 90,19 (2003)
Ballistic and Diffusive Transport Nonintegrable System Nonzero Drude weight Zero Drude Weight Integrable System Algebraic form for time correlation function between current functions Exponential Behavior for time correlation function between
Three main cases at finite temperature: Case (a) Infinite Conductivity and exactly conserved current Case (b) Case (c) Exponential decay for currentcurrent correlation function with full dissipation for current
Anisotropic spin ladder model Hamiltonian Because of crystal field effects and easy axis effect, we introduce two global and local anisotropies:, S are the spin operator of localized electrons on each chain and the coupling constants are antiferromagnetic type.
Bond operator Representation [ S S s t t t x y z, S ] i S 0 0 0 0 1 ( 2 ) 1 ( 2 ) i ( 2 ) 1 ( 2 ) s s t t t, t S. Sachdev and R. Bhatt, PRB 41, 9332(1990) 1 ( ) S s t t s i t t 2 1 ( s t t s i t t ) 2 1 Constraint
Bosonic Representation The parts of hamiltonian after applying Bond operator transformation With coefficients:
Green s function formalism Interacting one particle Green s function : Z ( A ( k,0)) ( B ( k,0)) G 2 2 k, k, k, n, k, a, 2 2 Z k, U k, Z k, V k, ( k, ) i i k, k, n, ( k,0), a, ( k,0) is normal and anomalous self energies and is renormalization constant. 1 k, Interacting Bogoliubov coefficients: Z 1 n, ( k, 0 ) Z k, U, V, 2 2 k, k, 1 A 2 2 k, n, k, ( k,0)
Finite temperature Calculations Dilution of triplet gas Matsubara s Green s functions Interacting one particle Green s function : renormalization constant: Interacting Bogoliubov coefficients:
Calculation of hard core self-energy Brueckner approach [Fetter & Walecka] for finding self energy of dilute Boson gas in the hard core condition :, ( K k 1 k 2 ) Vertex function (Scattering amplitude) Neglecting anomalous GF n i 1 N 2 t, it, i vq, q, 0.1 Hard core part of self energy:
Self-Consistent loop z Z u, v U, V n, ( k,0), a, ( k,0), Z ( k,0) k, U k,, V k,, k, After Convergency, ( k, ) Z n, a, ( k,0) ( k,0) ( k,0) U 4 a, ( k,0) ( k,0) 4 a, ( k,0)
Energy current and thermal conductance Energy current is obtained based on the following definition Local hamiltonian Energy conservation equation
Kubo Formula for thermal conductivity 1 21 22 1 J Q, L ( ev ) L T T Thermal conductivity
Spin susceptibility Spin susceptibility has two parts: 1)One particle bosonic Green s function 2)Two particle bosonic Green s function
Vertex correction Neglecting the below diagrams due to anomalous Green s function in the diagrams structures
Numerical Results After solving the equations self consistently Obtaining the Interacting green s function and thermal conductance Thermal conductivity of isotropic case versus temperature for various coupling constants. Monotonic decreasing with temperature Decreasing J conductance with
The effect of local anisotropy Noticeable change of conductivity with local anisotropy
The effect of global anisotropy There is no considerable change due to inter chain anisotropy
Acknowledgement Prof. Abdollah Langari Sharif university of technology In Collaboration with: Prof. Paul van Loosdrecht Cologne University Prof. Xenophon Zotos University of Crete, Greece
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