A Holographic Realization of Ferromagnets Masafumi Ishihara ( - PowerPoint PPT Presentation

A Holographic Realization of Ferromagnets Masafumi Ishihara ( AIMR, Tohoku University) Collaborators: Koji Sato ( AIMR, Tohoku University) Naoto Yokoi ( IMR, Tohoku University) Eiji Saitoh ( AIMR, IMR, Tohoku University ERATO, JST ASRC, JAEA) arXiv:1508.01626 [hep-th]

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

Holographic Ferromagnet http://ja.wikipedia.org/wiki/%E7%A3%81%E7%9F%B3 http://en.wikipedia.org/wiki/Black_hole We construct the dual gravity model of ferromagnet by holography U(1) Rotational SU(2) sym.

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

Ginzburg-Landau Theory Ferromagnet : SU(2) is broken to U(1) GL theory : useful near critical temperature 𝑼 ∼ 𝑼 𝒅 F: Free energy M : Magnetization H : Magnetic field 𝑮 = 𝑮 𝟏 + 𝟐 𝟑 𝒃 𝑼 − 𝑼 𝒅 𝑵 𝟑 + 𝟐 𝟓 𝒄𝑵 𝟓 − 𝑵𝑰 For H=0 , 𝝐𝑮 𝝐𝑵 = 𝒃 𝑼 − 𝑼 𝒅 𝑵 + 𝒄𝑵 𝟒 = 𝟏 𝟐 𝑵 ∝ 𝑼 − 𝑼 𝒅 for 𝑼 < 𝑼 𝒅 𝟑 ↔ 𝑵 = 𝟏 for 𝑼 > 𝑼 𝒅 𝝐𝑵 = 𝒃 𝑼 − 𝑼 𝒅 𝑵 + 𝒄𝑵 𝟒 − 𝑰 = 𝟏 𝝐𝑮 Fo r 𝑰 ≠ 𝟏, 𝟐 ↔ 𝑵 ∝ 𝑰 at 𝑼 ∼ 𝑼 𝒅 , 𝟒

Curie-Weiss Law 𝝍 ≡ 𝝐𝑵 Susceptibility 𝝐𝑰 𝝐𝑮 𝝐𝑵 = 𝒃 𝑼 − 𝑼 𝒅 𝑵 + 𝒄𝑵 𝟒 − 𝑰 = 𝟏 𝒃 𝑼 − 𝑼 𝒅 𝝍 + 𝟒𝒄𝑵 𝟑 𝝍 − 𝟐 = 𝟏 ↔ Curie-Weiss Law 𝟑𝑫 (𝑼 > 𝑼 𝒅 ) 𝑼 − 𝑼 𝒅 𝝍 𝑰=𝟏 = 𝑫 (𝑼 < 𝑼 𝒅 ) 𝑼 𝒅 − 𝑼 C : constant

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 : 𝑵 = 𝑵 𝟏 − 𝚬𝑵 𝚬𝑵 ∝ 𝒐 Dispersion of magnons : 𝝑 𝒍 = 𝑬𝒍 𝟑 + 𝜷𝑰 𝟒 𝟑 𝒇 −𝜷𝑰/𝒍 𝑪 𝑼 Magnon density : 𝒐 = 𝑼 𝟒 Bloch 𝑼 𝟒/𝟑 law : 𝚬𝑵 𝑰→𝟏 ∝ 𝑼 𝟑

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”.

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) 𝟑𝝀 𝟑 𝑺 − 𝟑𝚳 − 𝟐 𝟐 𝟐 𝑮 𝒃𝑵𝑶 − 𝟐 𝟓𝒇 𝟑 𝑯 𝑵𝑶 𝑯 𝑵𝑶 − 𝟑 𝑬 𝑵 𝝔 𝒃 𝟑 + 𝑾 𝝔 𝒃 𝑻 𝒉 = ∫ 𝒆 𝟔 𝒚 −𝒉 𝟓𝒉 𝟑 𝑮 𝑵𝑶 𝒃 − 𝝐 𝑶 𝑩 𝑵 𝒃 + 𝝑 𝒃𝒄𝒅 𝑩 𝑵 𝒄 𝑩 𝑶 𝒃 𝒅 𝒃 = 𝟐, 𝟑, 𝟒 𝑮 𝑵𝑶 = 𝝐 𝑵 𝑩 𝑶 SU(2) gauge field 𝐲 𝐍 = (𝒖, 𝒚, 𝒛, 𝒜, 𝒔) 𝑯 𝑵𝑶 = 𝝐 𝑵 𝑪 𝑶 − 𝝐 𝑶 𝑪 𝑵 U(1) gauge field 𝝔 𝒃 = 𝟏, 𝟏, 𝝔(𝒔) : triplet scalar 𝟑 𝟓 𝝔 𝟑 − 𝒏 𝟑 𝑾 = 𝝁 : potential for scalar 𝝁

Dictionary GKP-Witten relation = 𝒇 −𝑻 𝒉𝒔𝒃𝒘𝒋𝒖𝒛 [𝑲] 𝒂 𝑹𝑮𝑼 𝑲 J : source S.S. Gubser, I.R. Klebanov and A.M. Polyakov 1998 , E.Witten 1998 (3+1)D Ferromagnet (4+1)D gravity Magnetization M Scalar field 𝝔 External Magnetic Field H Holography Temperature 𝑼 Black Hole temperature 𝑼 Charge current 𝑲 𝝂 U(1) gauge field 𝑪 𝑵 𝒃 𝒃 𝑲 𝝂 Spin Current SU(2) gauge field 𝑩 𝑵 http://ja.wikipedia.org/wiki/%E7%A3%81%E7%9F%B3 http://en.wikipedia.org/wiki/Black_hole

Black Hole solution 𝟑𝝀 𝟑 𝑺 − 𝟑𝚳 − 𝟐 𝟐 𝟓𝒇 𝟑 𝑯 𝑵𝑶 𝑯 𝑵𝑶 − 𝟐 𝑮 𝒃𝑵𝑶 − 𝟐 𝟑 𝑬 𝑵 𝝔 𝒃 𝟑 + 𝑾 𝝔 𝒃 𝑻 𝒉 = ∫ 𝒆 𝟔 𝒚 −𝒉 𝟓𝒉 𝟑 𝑮 𝑵𝑶 Solution of EOM for 𝝔 = 𝟏 (4+1)-Dim AdS charged Black Hole metric 𝒔 𝟑 𝒎 𝟑 𝒆𝒔 𝟑 𝒎 𝟑 −𝒈 𝒔 𝒆𝒖 𝟑 + 𝒆𝒚 𝟑 + 𝒆𝒛 𝟑 + 𝒆𝒜 𝟑 + 𝟑 𝒆𝒕 𝑩𝒆𝑻−𝒅𝑪𝑰 = 𝒔 𝟑 𝒈 𝒔 𝟑 𝟕 𝟓 𝒈 𝒔 = 𝟐 + 𝑹 𝟑 𝒔 𝑰 𝒔 𝑰 𝒔 𝑰 𝟐 − 𝒔 𝑰 𝟒 = 𝝂 𝒕 − 𝟐 + 𝑹 𝟑 𝑩 𝟏 𝒔 𝟑 𝒎 𝒔 𝒔 𝟑 𝑹 𝟑 = 𝟑𝝀 𝟑 𝝂 𝟑 𝟑 𝑪 𝟏 = 𝝂 𝒔 𝑰 𝟐 − 𝒔 𝑰 𝒇 𝟑 + 𝝂 𝒕 𝒔 𝟑 𝒉 𝟑 𝒎 𝟒 𝟑−𝑹 𝟑 𝚳 = − 𝟕 Black Hole Temperature : 𝑼 = 𝒔 𝑰 = 𝒎 = 𝟐 𝒎 𝟑 𝟑𝝆 C. P. Herzog and S. S. Pufu (2009) N. Iqbal, H. Liu, M. Mezei, and Q. Si 2010

Equation of motion for 𝝔 Acti tion for or 𝝔 𝟑 − 𝑾 𝝔 𝒔 𝐓 𝝔 = ∫ 𝒆𝒔 −𝒉 𝑩𝒆𝑻−𝒅𝑪𝑰 − 𝟐 𝟑 𝝐𝝔 𝒔 Equation of motion 𝝁𝝔 𝟒 (𝒔) − 𝒏 𝟑 𝝔(𝒔) − 𝟔𝒈(𝒔)𝒔 + 𝒈 ′ 𝒔 𝒔 𝟑 𝝔 ′ 𝒔 − 𝒈 𝒔 𝒔 𝟑 𝝔"(𝒔) = 𝟏 𝒈 𝒔 = 𝟐 + 𝑹 𝟑 𝒔 𝟕 − 𝟐+𝑹 𝟑 𝑼 = 𝟑−𝑹 𝟑 𝒔 𝟓 𝟑𝝆 Asymptotic solution: 𝑰 𝑵 𝟒. 𝟔 ≤ 𝒏 𝟑 ≤ 𝟓 𝟓 − 𝒏 𝟑 𝝔 𝒔 = 𝒔 𝟑−𝚬 + 𝒔 𝟑+𝚬 + ⋯ 𝜠 ≡ H : External magnetic field M : Magnetization from GKP-Witten relations ( 𝒂 𝑹𝑮𝑼 [𝑲] = 𝒇 −𝑻 𝒉𝒔𝒃𝒘𝒋𝒖𝒛 [𝑲] )

Numerical method We solve the EOM numerically. ( 𝑛 2 = 3.89, 𝜇 = 1 ) 𝝁𝝔 𝒔 𝟒 − 𝒏 𝟑 𝝔(𝒔) − 𝟔𝒈(𝒔)𝒔 + 𝒈 ′ 𝒔 𝒔 𝟑 𝝔 ′ 𝒔 − 𝒈 𝒔 𝒔 𝟑 𝝔"(𝒔) = 𝟏 𝒈 𝒔 = 𝟐 + 𝑹 𝟑 𝟐+𝑹 𝟑 𝟑−𝑹 𝟑 𝒔 𝟕 − 𝑼 = 𝒔 𝟓 𝟑𝝆 We will focus on Spontaneous magnetization ( M when 𝑰 = 𝟏 ) 𝑰 𝑵 𝝔 𝒔 = 𝒔 𝟑−𝚬 + 𝒔 𝟑+𝚬 + ⋯ 𝟓 − 𝒏 𝟑 𝜠 ≡ 𝒆 𝒔 𝟑−𝚬 𝝔 𝒔 𝒔 𝟑𝜠+𝟐 𝑰 = 𝒔 𝟑−𝚬 𝝔 𝒔 𝒔→∞ 𝑵 = 𝒔→∞ −𝟑𝚬 𝒆𝒔

Result: 𝑼~𝑼 𝒅 Spontanious Magnetization 𝑵 𝟐 𝟐 − 𝑼 𝟑 𝑵 ∝ 𝑼 𝒅 ● : result by holographic duality Magnetic susceptibility 𝝍 ≡ 𝒆𝑵 𝒆𝑰 𝑰=𝟏 we can get Curie – Weiss law 𝑫 + 𝑼 > 𝑼 𝒅 𝑼/𝑼 𝒅 −𝟐 𝝍 = 𝒅 − (𝑼 < 𝑼 𝒅 ) 𝟐−𝑼/𝑼 𝒅 𝒅 + /𝒅 − ∼ 𝟑. 𝟑𝟑

Result : 𝐔 ∼ 𝑼 𝒅 H : External magnetic field M : Magnetization 𝟐 𝑵 ∝ 𝑰 𝟒 F : Free energy The scalar part of the on-shell action 𝑮 ∝ 𝑼 − 𝑼 𝒅 𝟑 Results near 𝑼 𝒅 are consistent with Ginzburg-Landau Theory

Result: low temperature ( 𝑼 ∼ 𝟏 ) Magnetization 𝑵 𝟒 𝟑 law we can reproduce the Bloch 𝑼 𝟒 𝑼 𝟑 𝑵 ∝ 𝟐 − 𝑫 𝑼 𝒅 At low T , Results are consistent with magnons.

Result: low temperature ( 𝑼 ∼ 𝟏 ) Magnetic susceptibility: 𝝍 𝟐 𝑼 𝟑 𝝍 ∼ 𝝍 𝟏 + 𝑬 𝑼 𝒅 First term 𝝍 𝟏 : Pauli paramagnetic susceptibility from conduction electrons Second term: susceptibility from magnons F : Free energy 𝟑 𝑼 𝑼 𝑮 ∼ −𝑮 𝟏 + 𝑭 − 𝜹 𝑼 𝒅 𝑼 𝒅 𝜹 : linear in T of the specific heat from conduction eletctrons

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 𝝍 , Free energy 𝑮 Our results are consistent with T ∼ 0 : Magnon + Conduction electron T ∼ 𝑈 𝑑 : GL theory Black Hole captures the ferromagnetic system both near 𝑼 𝒅 and low temperatures Outlook 1 Magnon dynamics 2 Correlation functions http://en.wikipedia.org/wiki/Black_hole