A Holographic Realization of Ferromagnets

Transcription

1 A Holographic Realization of Ferromagnets Masafumi Ishihara ( AIMR, Tohoku University) Collaborators: Koji Sato ( AIMR, Tohoku University) Naoto Yokoi Eiji Saitoh ( IMR, Tohoku University) ( AIMR, IMR, Tohoku University ERATO, JST ASRC, JAEA) arxiv: [hep-th]

2 Holographic duality New Duality from string theory: Holographic Duality (Holography) J.M. Maldacena 1998,

3 Holographic Ferromagnet We construct the dual gravity model of ferromagnet by holography Rotational SU(2) sym. U(1)

4 Introduction Contents Ferromagnet (condensed matter theory) Ferromagnet (Holographic duality) Numerical Result Summary

5 Ginzburg-Landau Theory Ferromagnet : SU(2) is broken to U(1) GL theory: useful near critical temperature T T c F: Free energy M: Magnetization H: Magnetic field F = F a T T c M bm4 MH For H=0, F M = a T T c M + bm 3 = 0 M T T c 1 2 M = 0 for T < T c for T > T c For H 0, F M = a T T c M + bm 3 H = 0 M H 1 3 at T T c,

6 Curie-Weiss Law Susceptibility χ M H F M = a T T c M + bm 3 H = 0 a T T c χ + 3bM 2 χ 1 = 0 Curie-Weiss Law χ H=0 = 2C T T c C T c T (T > T c ) (T < T c ) C: constant

7 Low Temperature and magnons At low temperatures, magnetization is mostly aligned. elementary excitations: magnons (quantized spin wave) Reduction of magnetization is proportional to magnon density n : M = M 0 ΔM ΔM n Dispersion of magnons : ε k = Dk 2 + αh Magnon density : n = T 3 2e αh/k BT Bloch T 3/2 law : ΔM H 0 T 3 2

8 Prescription of Holography Find the 1-dimensional higher gravity action with the same symmetry (breaking) as the Ferromagnetic system. Solve the equation of motion from the gravitational action. Extract the physical quantities from the solution by using holographic dictionary.

9 Gravity action dual to Ferromagnet (3+1)D Ferromagnetic system : SU(2) symmetry which is spontaneously broken to U(1) (4+1)D Gravitational system with SU(2) fields which is spontaneously broken to U(1) S g = d 5 x g 1 2κ 2 1 R 2Λ 4e 2 G MNG MN 1 4g 2 F MN a F amn 1 2 D Mφ a 2 + V φ a F MN = M A a N N A a M + ε abc A b c M A N SU(2) gauge field G MN = M B N N B M U(1) gauge field a = 1, 2, 3 x M = (t, x, y, z, r) φ a = 0, 0, φ(r) : triplet scalar V = λ 4 φ 2 m2 λ 2 : potential for scalar

10 Dictionary GKP-Witten relation Z QFT J = e S gravity[j] J: source S.S. Gubser, I.R. Klebanov and A.M. Polyakov 1998, E.Witten 1998 (3+1)D Ferromagnet Magnetization M External Magnetic Field H Temperature T Charge current J μ Spin Current J μ a Holography (4+1)D gravity Scalar field φ Black Hole temperature T U(1) gauge field B M SU(2) gauge field A M a

11 Black Hole solution S g = d 5 x g 1 2κ 2 1 R 2Λ 4e 2 G MNG MN 1 4g 2 F MN a F amn 1 2 D Mφ a 2 + V φ Solution of EOM for φ = 0 (4+1)-Dim AdS charged Black Hole metric 2 ds AdS cbh = r2 f r l 2 dt2 + dx 2 + dy 2 + dz 2 + l2 f r dr 2 r 2 f r = 1 + Q 2 r H r Q 2 r H r 4 A 0 3 = μ s r H l 1 r H 2 r 2 Q 2 = 2κ2 3 μ 2 2 e 2 + μ s g 2 B 0 = μ r H l 1 r H 2 r 2 Black Hole Temperature : T = 2 Q2 2π Λ = 6 l 2 r H = l = 1 C. P. Herzog and S. S. Pufu (2009) N. Iqbal, H. Liu, M. Mezei, and Q. Si 2010

12 Action for φ Equation of motion for φ S φ = dr g AdS cbh 1 2 φ r 2 V φ r Equation of motion λφ 3 (r) m 2 φ(r) 5f(r)r + f r r 2 φ r f r r 2 φ"(r) = 0 f r = 1 + Q2 1+Q2 r6 r 4 T = 2 Q2 2π Asymptotic solution: φ r = H M r2 Δ + r 2+Δ + Δ 4 m m 2 4 H: External magnetic field M: Magnetization from GKP-Witten relations ( Z QFT [J] = e S gravity[j] )

13 Numerical method We solve the EOM numerically. (m 2 = 3.89, λ = 1) λφ r 3 m 2 φ(r) 5f(r)r + f r r 2 φ r f r r 2 φ"(r) = 0 f r = 1 + Q2 1+Q2 r6 r 4 T = 2 Q2 2π We will focus on Spontaneous magnetization (M when H = 0 ) φ r = H M r2 Δ + r 2+Δ + Δ 4 m2 H = r 2 Δ φ r r M = r2δ+1 2Δ d r 2 Δ φ r dr r

14 Spontanious Magnetization M Result: T~T c M 1 T T c 1 2 : result by holographic duality Magnetic susceptibility χ dm dh H=0 we can get Curie Weiss law χ = C + T/T c 1 T > T c c 1 T/T c (T < T c ) c + /c 2. 22

15 H: External magnetic field M: Magnetization Result : T T c M H 1 3 F: Free energy The scalar part of the on-shell action F T T c 2 Results near T c are consistent with Ginzburg-Landau Theory

16 Magnetization M Result: low temperature (T 0) we can reproduce the Bloch T 3 2 law M 1 C T T c 3 2 At low T, Results are consistent with magnons.

17 Result: low temperature (T 0) Magnetic susceptibility: χ χ χ 0 + D T T c 1 2 First term χ 0 : Pauli paramagnetic susceptibility from conduction electrons Second term: susceptibility from magnons F: Free energy F F 0 + E T T c γ T T c 2 γ: linear in T of the specific heat from conduction eletctrons

18 Summary We have constructed a holographic dual model of ferromagnet and found the holographic dictionary between ferromagnet and gravity. Using the dictionary, we analyzed the temperature dependence of Magnetization M, Susceptibility χ, Free energy F Our results are consistent with T 0 : Magnon + Conduction electron T T c : GL theory Black Hole captures the ferromagnetic system both near T c and low temperatures Outlook 1 Magnon dynamics 2 Correlation functions