Domains and Domain Walls in Quantum Spin Chains

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1 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]

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

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

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

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

6 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 /3/04 [qpis 4/5]

7 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 /3/04 [qpis3 5/5]

8 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 ), 6/3/04 [qpis4 6/5]

9 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]

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

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

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

13 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]

14 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 A σ N N ǫ σ (J + h) (J h) g σσ + + 6/3/04 [qpis6 8/5]

15 Thermodynamics with Statistical Interaction Specifications of quantum many-body system: particle energies ǫ i, statistical interaction coefficients g ij, statistical capacity constants A i. 6/3/04 [qpis8 9/5]

16 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]

17 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!, Grand potential [Wu 994]: Ω = k B T ln Z, ǫ i µ k B T = ln( + w i) X j «+ Ai wi, Z = Y w i i «+ wi g ji ln. w i 6/3/04 [qpis8 9/5]

18 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]

19 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]

20 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. Comparison with transfer matrix solution: Z = Z N e NK. 6/3/04 [qpis0 0/5]

21 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]

22 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 N ± = Nw w + w + w + + w <N + > / N h/j= 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]

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

24 Solitons in Spin- Ising Chain Soliton content and energy of nearest-neighbor bonds: bond N N E/J 0 0 M z \N A M z \N A /3/04 [qpis 3/5]

25 Soliton Pairs in Spin- Ising Chain particle r + r r 0 q + q q 0 motif, N + + N 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]

26 Soliton Pairs in Spin- Ising Chain particle r + r r 0 q + q q 0 motif, N + + N 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 ). Sample configurations: : r + r +, : q + + r 0 : q 0 + r 0 + r 6/3/04 [qpis 4/5]

27 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 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 m A m N N 0 N N N ǫ m J J J J J J 6/3/04 [qpis3 5/5]

28 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. 6/3/04 [qpis4 6/5]

29 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]

30 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, J/k B T 6/3/04 [qpis6 7/5]

31 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]

32 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, Link to nested coordinate Bethe ansatz. 6/3/04 [qpis7 9/5]

33 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]

34 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]

35 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]

36 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 <. Solution for spin-/ Ising chain (M ): x = + q. ω (T) = k B T ln + e J /k BT. 6/3/04 [qpis5 /5]

37 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) M= M=3 M= 0.4 M= s M /k B k B T/ J 6/3/04 [qpis6 3/5]

38 Distribution of Domains 0 Normalized distribution: ˆn µ = n 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]

39 Distribution of Domains 0 Normalized distribution: ˆn µ = n 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. 6/3/04 [qis7 4/5]

40 Distribution of Domains 0 Normalized distribution: ˆn µ = n 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]

41 Distribution of Domains 0 Normalized distribution: ˆn µ = n 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 Pascal s distribution: P(µ) = γ( γ) µ, γ = e K e K + e K. 6/3/04 [qis7 4/5]

42 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. ). 6/3/04 [qpis8 5/5]

43 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. 6/3/04 [qpis8 5/5]

44 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]

45 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. Links to coordinate Bethe ansatz and thermodynamic Bethe ansatz (including nesting). 6/3/04 [qpis8 5/5]

The Ising model Summary of L12

The Ising model Summary of L12 The Ising model Summary of L2 Aim: Study connections between macroscopic phenomena and the underlying microscopic world for a ferromagnet. How: Study the simplest possible model of a ferromagnet containing