Domains and Domain Walls in Quantum Spin Chains

Domains and Domain Walls in Quantum Spin Chains Statistical Interaction and Thermodynamics Ping Lu, Jared Vanasse, Christopher Piecuch Michael Karbach and Gerhard Müller 6/3/04 [qpis /5]

Domains and Domain Walls (Mesoscopic) Exchange interaction (strong, short-range) Magnetic dipolar interaction (weak, long range) Anisotropy, crystal field 6/3/04 [qpis /5]

Quantum Spin Chain CsCoCl 3 6/3/04 [qpis 3/5]

Quantum Spin Chain l l l + CsCoCl 3 6/3/04 [qpis 3/5]

Quantum Spin Chain l l l + CsCoCl 3 [Yoshizawa, Hirakawa, Satija, and Shirane 98] 6/3/04 [qpis 3/5]

Domains and Domain Walls (Microscopic) Spin-/ Ising chain: H / = J NX Sl z Sz l+ ; Sz l = +,, FM (J < 0), AFM (J > 0). l= vacuum domain...... vacuum vacuum vacuum...... 6/3/04 [qpis 4/5]

Nested Domains and Composite Domain Walls Spin- Ising chain: H = J NX Sl z Sz l+ ; Sz l = +,0, l= vac. outer domain vac....... inner dom. inner dom. vac. vac. vac. vac....... 6/3/04 [qpis3 5/5]

Statistical Interaction Generalized Pauli principle [Haldane 99] How is the number of states accessible to one particle of species i affected if other particles (of any species j) are added?. X d i = g ij N j g ij : coefficients of statistical interaction, A i : statistical capacity constants. j d i = A i X j g ij (N j δ ij ), Number of distinct many-particle states with composition {N i }: W({N i }) = Y i d i + N i N i! = Y i Γ(d i + N i ) Γ(N i + )Γ(d i ). Packing capacity limited by requirement d i > 0. 6/3/04 [qpis4 6/5]

Solitons in Spin-/ Ising Chain Soliton content and energy of nearest-neighbor bonds: bond N + 0 0 0 N 0 0 0 E/J 0 0 Solitons... have spin ±/, are fundamental, are independent, are statistically interacting. 6/3/04 [qpis5 7/5]

Solitons in Spin-/ Ising Chain Soliton content and energy of nearest-neighbor bonds: bond N + 0 0 0 N 0 0 0 E/J 0 0 M z \N A 0 4 6 3 6 6 9 6 5 0 6 0 9 6 5 6 6 3 30 30 64 Solitons... have spin ±/, are fundamental, are independent, are statistically interacting. 6/3/04 [qpis5 7/5]

Solitons in Spin-/ Ising Chain Soliton content and energy of nearest-neighbor bonds: bond N + 0 0 0 N 0 0 0 E/J Solitons... have spin ±/, are fundamental, are independent, 0 0 are statistically interacting. M z \N A 0 4 6 3 6 6 9 6 5 0 6 0 9 6 5 6 6 3 30 30 64 M z \N A 3 5 7 7/ 5/ 7 7 3/ 4 7 / 7 7 35 / 7 7 35 3/ 4 7 5/ 7 7 7/ 4 70 4 8 6/3/04 [qpis5 7/5]

Statistical Interaction of Solitons Number of Ising eigenstates with N A = N + + N solitons: W A (N +, N ) = N Y N N A σ=± d σ = A σ X σ g σσ (N σ δ σσ ) d σ + N σ N σ! A σ = (N ) g σσ = M z = (N + N ) ǫ ± = (J ± h) (AFM) X W A (N +, N ) = N N +,N 6/3/04 [qpis6 8/5]

Statistical Interaction of Solitons Number of Ising eigenstates with N A = N + + N solitons: W A (N +, N ) = N Y N N A σ=± d σ + N σ N σ! d σ = A σ X g σσ (N σ δ σσ ) σ A σ = (N ) g σσ = M z = (N + N ) ǫ ± = (J ± h) (AFM) X W A (N +, N ) = N N +,N particle n + n motif N + + N + 0 0 + A σ N N ǫ σ (J + h) (J h) g σσ + + 6/3/04 [qpis6 8/5]

Thermodynamics with Statistical Interaction Specifications of quantum many-body system: particle energies ǫ i, statistical interaction coefficients g ij, statistical capacity constants A i. Two tasks: combinatorial problem: W({N i }) = Y i extremum problem: δ(u TS µn) = 0. d i + N i N i!, 6/3/04 [qpis8 9/5]

Thermodynamics of Solitons Specifications: ǫ ± = (J ± h), g σσ =, A ± = (N ). Grand partition function: Z = «+ N/ «w+ + N/ w, w + w J ± h k B T = ln( + w ±) ln + w ± w ± ln + w w. 6/3/04 [qpis0 0/5]

Thermodynamics of Solitons Specifications: ǫ ± = (J ± h), g σσ =, A ± = (N ). Grand partition function: Z = «+ N/ «w+ + N/ w, w + w J ± h k B T = ln( + w ±) ln + w ± w ± ln + w w. Solution: w ± =» q e ±h/kbt + (e ±h/kbt ) + 4e (J±h)/k BT, Z = h i N e K cosh H + psinh H + e 4K. J, K = 4k B T, H =. h k B T. 6/3/04 [qpis0 0/5]

Transfer Matrix Solution Ising chain with spin s = / in a magnetic field: H = NX n=» JSnS z n+ z + h `S n z + Sn+ z. Canonical partition function (use σ n = S z n = ±): Z N = X σ,...,σ N exp NX n=» Kσ n σ n+ + H(σ n + σ n+ )! h i = Tr V N = λ N + " + λ λ + «N #. Transfer matrix [Kramers and Wannier 94]: V = e K+H e K e K e K H!, V (σ n, σ n+ ) = exp h i Eigenvalues: λ ± = e K cosh H ± psinh H + e 4K Kσ n σ n+ + «H(σ n + σ n+ ), K. = J 4k B T, H. = h k B T. 6/3/04 [qpis9 /5]

Soliton Occupancies Linear relations [Wu 994]: w i N i + X j g ij N j = A i, i =,,.... Solitons in s = / Ising chain at h 0: w ± + «N ± + N = N 0.5 0.4 -.5 N ± = Nw w + w + w + + w <N + > / N 0.3 0. -0.5-0.75 0. 0.75 0.5 h/j=0 0.5 0 4 6 8 0 J/k B T N ± N = hp i e ±H sinh H + e 4K ± sinh H sinh H + e 4K + cosh H p sinh H + e 4K h 0 / e K + 6/3/04 [qpis5 /5]

Solitons in Spin- Ising Chain Soliton content and energy of nearest-neighbor bonds: bond N + 0 0 0 0 0 N 0 0 0 0 0 E/J 0 0 6/3/04 [qpis 3/5]

Solitons in Spin- Ising Chain Soliton content and energy of nearest-neighbor bonds: bond N + 0 0 0 0 0 N 0 0 0 0 0 E/J 0 0 M z \N A 0 4 6 3 3 3 3 3 6 0 6 7 3 3 6 3 3 3 0 3 7 M z \N A 0 4 6 8 4 3 4 4 6 4 0 4 8 4 6 0 6 9 4 8 4 6 6 4 0 3 4 4 4 8 48 0 3 8 6/3/04 [qpis 3/5]

Soliton Pairs in Spin- Ising Chain particle r + r r 0 q + q q 0 motif, N + + N + 0 0 + + + 0 0 + + ǫ m J J J J J J Independent particles... are composite (soliton pairs), have spin +,0,, come in six species: - confined soliton pairs (r +, r ), - deconfined soliton pairs (q +, q 0, q ), - spacer particle (r 0 ). 6/3/04 [qpis 4/5]

Statistical Interaction of Soliton Pairs Multiplicity expression for soliton pairs: W 6 ({X m }) = N N N Σ 6Y m= d m = A m X m g mm (X m δ mm ) d m + X m X m! g mm 3 4 5 6 0 0 3 0 0 0 4 N Σ = X + X + X 3 + (X 4 + X 5 + X 6 ) 5 6 particle r + r r 0 q + q q 0 motif, N + + N + 0 0 + + + 0 0 + + m 3 4 5 6 A m N N 0 N N N ǫ m J J J J J J 6/3/04 [qpis3 5/5]

Thermodynamics of Soliton Pairs Six coupled equations for given ǫ i, g ij, and µ = 0: ǫ i µ k B T = ln( + w i) X j + wj g ji ln w j «, i =,,..., 6. Set K. = J/k B T. w = w, w 4 = w 5, + w 3 + w = e K, w 3 w 6 = e K, w 4 = + w 6, + w 6 w w 6 = e K. Solution: w 3 = cosh K + s cosh K «+. Grand partition function: Z = 6Y» + wi i= w i Ai = h ( + w 3 )e Ki N = ZN e NK. 6/3/04 [qpis4 6/5]

Soliton-Pair Occupancies at h = 0 Linear relations: w i N i + X j g ij N j = A i, i =,..., 6. Symmetry implies N = N, N 4 = N 5 = N 6. N = N = N N 3 = w 3 (w 3 + ek ) (w 3 + )(w 3 e K + 4w 3 + e K ), N(w 3 + e K ) (w 3 + )(w 3 e K + 4w 3 + e K ), N 4 = N 5 = 4 w 3 N 3, 0. N 6 = w 3 N 3. <N m > /N 0.05 m=, m=3 m=6 m=4,5 0 0 3 J/k B T 6/3/04 [qpis6 7/5]

Combinatorics of Domains Number of Ising states with {N, N,...} strings: W µ ({N µ }) = N N r Y d µ = A µ X µ g µµ (N µ δ µµ ) A µ = N µ ( µ, µ < µ g µµ = µ +, µ µ µ d µ + N µ N µ! { } +3 { } + 6 N = 6 { } + 6 N = { } + 6 N = { } + 3 N = 5 { } 0 6 N 3 = { } 0 6 N = N = { } 0 N = 3 M z = N r, r = X µ ǫ µ = J + µh (FM) X W µ ({N µ }) = N. {N µ } µn µ { } 0 6 N = N = 0 { } 3 N = { } 6 N = N 3 = { } 6 N 4 = 5 { } 6 N 5 = 6 { } 3 N 6 = 6/3/04 [qpis7 8/5]

Nested Domains in Spin- Ising Chain Nested two-level systems: µ-string vacuum: (FM ground state), ν-string vacuum: (µ-string particle). { } +4 { } +4, 4 { } +3 4 { } +3 4, { }+ 4, 8 4 { } + 4 { } + 4, { }+ 4, { }0 4, 6 4 { } + 4 { } + 4, { }0 4 3, { } 4 3, { } 4 3 { } 0 { } 0, { } 4 { } 4, { }, { } 3 4, { } 4, 6 { } + { } +, { }+ 4, { }0, 8 6 8 Link to nested coordinate Bethe ansatz. 6/3/04 [qpis7 9/5]

Combinatorics of Nested Domains Multiplicity expression: W {N µ }, {N ν (µ) } = N N r Y µ µ Y µ r µ ν d µ + N µ N µ! d (µ) ν + N (µ) ν N (µ) ν!, d µ = N µ X µ g µµ (N µ δ µµ ), d (µ) ν = µ ν X ν g νν (N (µ) ν δ νν ), r = X µ µn µ, r µ = X ν νn (µ) ν, g µµ = ( µ, µ < µ, µ + µ µ. Link to nested thermodynamic Bethe ansatz. 6/3/04 [qpis8 0/5]

Statistical Mechanics of Domains I Strings of restricted length: µ =,,..., M N. Grand potential: Ω M (T) = k B T MX µ= 4K µ = ln(w µ + ) A µ = N µ, ( µ, µ < µ g µµ = µ +, µ µ, K µ = ǫ µ k B T. wµ + A µ ln MX µ = w µ «. g µ µ ln w µ + w µ, µ =,..., M. Spin-/ ising chain: ǫ µ = J + µh. Thermodynamic limit: first N then M. 6/3/04 [qpis4 /5]

Statistical Mechanics of Domains II Solution for ǫ µ = J, finite M, and N. Introduce ξ µ. = ln(wµ + ), q. = e 4K, K = J /4k B T. Introduce Φ M. = 4K MX µ= ln e ξ µ. Grand potential per site: ω M (T). = lim N N Ω M(T) = J Φ M. Polynomial equation: qx (M+) + ( q)x x + = 0, x. = q Φ M <. 6/3/04 [qpis5 /5]

Entropy of Domains Entropy per site on infinite lattice: s M (T). = lim N N S M (T) = d dt ω M(T). s (K) k B s (K) k B p! + 4e = ln 4K + 8Ke 4K + + 4e 4K + p (-strings only), + 4e 4K = ln + e K + K e K (all strings allowed). + 0.6 M= M=3 M= 0.4 M= s M /k B 0. 0 0 0.5.5 k B T/ J 6/3/04 [qpis6 3/5]

Distribution of Domains 0 Normalized distribution: ˆn µ = n µ @ Solve the linear equations w µ n µ + MX n µ A µ = MX µ = µ n µ +, n µ. = lim N N µ N. µx n µ =, µ =,..., M. µ = 6/3/04 [qis7 4/5]

Distribution of Domains 0 Normalized distribution: ˆn µ = n µ @ Solve the linear equations w µ n µ + Consider the case M : w µ n µ + MX n µ A µ = MX µ = µ n µ +, n µ. = lim N N µ N. µx n µ =, µ =,..., M. µ = µx n ν = ζ, µ =,,... ν= where w µ = + ( + q) µ, ζ =. q q X µ = µ n µ. Result: ˆn µ = q ( + q) µ = e K ( + e K, µ =,,... [Denisov and Hänggi 005] ) µ 6/3/04 [qis7 4/5]

Open Questions Generalization to spin-3/ Ising chain: independent particles contain,3,...,6 solitons, confined soliton compounds (e.g., ), deconfined soliton compounds (e.g. ), spacer particles (e.g. ). Generalization to more complex spin-/ systems: diamond chain, spin ladder. Citeria for independent particle species. 6/3/04 [qpis8 5/5]