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

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

http://ja.wikipedia.org/wiki/%E7%A3%81%E7%9F%B3 http://en.wikipedia.org/wiki/Black_hole

Contents

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

✓

Ginzburg-Landau Theory

Ferromagnet : SU(2) is broken to U(1) 𝑮 = 𝑮𝟏 + 𝟐 𝟑 𝒃 𝑼 − 𝑼𝒅 𝑵𝟑 + 𝟐 𝟓 𝒄𝑵𝟓 − 𝑵𝑰

GL theory: useful near critical temperature 𝑼 ∼ 𝑼𝒅

For H=0, 𝝐𝑮

𝝐𝑵 = 𝒃 𝑼 − 𝑼𝒅 𝑵 + 𝒄𝑵𝟒 = 𝟏

𝑵 ∝ 𝑼 − 𝑼𝒅

𝟐 𝟑

for 𝑼 < 𝑼𝒅

↔

𝑵 = 𝟏 for 𝑼 > 𝑼𝒅 For 𝑰 ≠ 𝟏,

𝝐𝑮 𝝐𝑵 = 𝒃 𝑼 − 𝑼𝒅 𝑵 + 𝒄𝑵𝟒 − 𝑰 = 𝟏

𝑵 ∝ 𝑰

𝟐 𝟒

at 𝑼 ∼ 𝑼𝒅,

↔

F: Free energy M: Magnetization H: Magnetic field

Curie-Weiss Law

𝒃 𝑼 − 𝑼𝒅 𝝍 + 𝟒𝒄𝑵𝟑𝝍 − 𝟐 = 𝟏

↔

Susceptibility 𝝍 ≡ 𝝐𝑵

𝝐𝑰

𝝐𝑮 𝝐𝑵 = 𝒃 𝑼 − 𝑼𝒅 𝑵 + 𝒄𝑵𝟒 − 𝑰 = 𝟏 𝝍

𝑰=𝟏 =

𝟑𝑫 𝑼 − 𝑼𝒅 𝑫 𝑼𝒅 − 𝑼 (𝑼 > 𝑼𝒅) (𝑼 < 𝑼𝒅)

C: constant

Curie-Weiss Law

Low Temperature and magnons

Reduction of magnetization is proportional to magnon density n : 𝑵 = 𝑵𝟏 − 𝚬𝑵 𝚬𝑵 ∝ 𝒐 Magnon density : 𝒐 = 𝑼

𝟒 𝟑𝒇−𝜷𝑰/𝒍𝑪𝑼

Bloch 𝑼𝟒/𝟑 law :

𝚬𝑵 𝑰→𝟏 ∝ 𝑼

𝟒 𝟑

At low temperatures, magnetization is mostly aligned. elementary excitations: magnons (quantized spin wave) Dispersion of magnons : 𝝑𝒍 = 𝑬𝒍𝟑 + 𝜷𝑰

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)

𝑻𝒉 = ∫ 𝒆𝟔𝒚 −𝒉 𝟐 𝟑𝝀𝟑 𝑺 − 𝟑𝚳 − 𝟐 𝟓𝒇𝟑 𝑯𝑵𝑶𝑯𝑵𝑶 − 𝟐 𝟓𝒉𝟑 𝑮𝑵𝑶

𝒃

𝑮𝒃𝑵𝑶 − 𝟐 𝟑 𝑬𝑵𝝔𝒃 𝟑 + 𝑾 𝝔

𝑮𝑵𝑶

𝒃

= 𝝐𝑵𝑩𝑶

𝒃 − 𝝐𝑶𝑩𝑵 𝒃 + 𝝑𝒃𝒄𝒅𝑩𝑵 𝒄 𝑩𝑶 𝒅

SU(2) gauge field 𝝔𝒃 = 𝟏, 𝟏, 𝝔(𝒔) : triplet scalar 𝑯𝑵𝑶 = 𝝐𝑵𝑪𝑶 − 𝝐𝑶𝑪𝑵 U(1) gauge field

𝑾 = 𝝁

𝟓 𝝔 𝟑 − 𝒏𝟑 𝝁 𝟑

: potential for scalar

𝐲𝐍 = (𝒖, 𝒚, 𝒛, 𝒜, 𝒔) 𝒃 = 𝟐, 𝟑, 𝟒

Dictionary

(3+1)D Ferromagnet Magnetization M External Magnetic Field H Temperature 𝑼 Charge current 𝑲𝝂 Spin Current 𝑲𝝂

𝒃

Holography

http://ja.wikipedia.org/wiki/%E7%A3%81%E7%9F%B3 http://en.wikipedia.org/wiki/Black_hole

𝒂𝑹𝑮𝑼 𝑲 = 𝒇−𝑻𝒉𝒔𝒃𝒘𝒋𝒖𝒛[𝑲] J: source

GKP-Witten relation (4+1)D gravity Scalar field 𝝔 Black Hole temperature 𝑼 U(1) gauge field 𝑪𝑵 SU(2) gauge field 𝑩𝑵

𝒃

S.S. Gubser, I.R. Klebanov and A.M. Polyakov 1998 , E.Witten 1998

Black Hole solution

• C. P. Herzog and S. S. Pufu (2009) N. Iqbal, H. Liu, M. Mezei, and Q. Si 2010

Solution of EOM for 𝝔 = 𝟏 (4+1)-Dim AdS charged Black Hole metric

𝒆𝒕𝑩𝒆𝑻−𝒅𝑪𝑰

𝟑

=

𝒔𝟑 𝒎𝟑 −𝒈 𝒔 𝒆𝒖𝟑 + 𝒆𝒚𝟑 + 𝒆𝒛𝟑 + 𝒆𝒜𝟑 + 𝒎𝟑 𝒈 𝒔 𝒆𝒔𝟑 𝒔𝟑

𝒈 𝒔 = 𝟐 + 𝑹𝟑 𝒔𝑰 𝒔

𝟕

− 𝟐 + 𝑹𝟑 𝒔𝑰 𝒔

𝟓

Black Hole Temperature : 𝑼 =

𝟑−𝑹𝟑 𝟑𝝆

𝑹𝟑 = 𝟑𝝀𝟑 𝟒 𝝂𝟑 𝒇𝟑 + 𝝂𝒕

𝟑

𝒉𝟑 𝑪𝟏 = 𝝂 𝒔𝑰 𝒎 𝟐 − 𝒔𝑰

𝟑

𝒔𝟑 𝑩𝟏

𝟒 = 𝝂𝒕

𝒔𝑰 𝒎 𝟐 − 𝒔𝑰

𝟑

𝒔𝟑 𝑻𝒉 = ∫ 𝒆𝟔𝒚 −𝒉 𝟐 𝟑𝝀𝟑 𝑺 − 𝟑𝚳 − 𝟐 𝟓𝒇𝟑 𝑯𝑵𝑶𝑯𝑵𝑶 − 𝟐 𝟓𝒉𝟑 𝑮𝑵𝑶

𝒃

𝑮𝒃𝑵𝑶 − 𝟐 𝟑 𝑬𝑵𝝔𝒃 𝟑 + 𝑾 𝝔 𝚳 = − 𝟕 𝒎𝟑 𝒔𝑰 = 𝒎 = 𝟐

Equation of motion for 𝝔

H: External magnetic field M: Magnetization

Equation of motion 𝝁𝝔𝟒(𝒔) − 𝒏𝟑𝝔(𝒔) − 𝟔𝒈(𝒔)𝒔 + 𝒈′ 𝒔 𝒔𝟑 𝝔′ 𝒔 − 𝒈 𝒔 𝒔𝟑𝝔"(𝒔) = 𝟏 𝒈 𝒔 = 𝟐 + 𝑹𝟑

𝒔𝟕 − 𝟐+𝑹𝟑 𝒔𝟓

𝑼 = 𝟑−𝑹𝟑

𝟑𝝆

Asymptotic solution: 𝝔 𝒔 =

𝑰 𝒔𝟑−𝚬 + 𝑵 𝒔𝟑+𝚬 + ⋯

𝜠 ≡ 𝟓 − 𝒏𝟑

from GKP-Witten relations ( 𝒂𝑹𝑮𝑼[𝑲] = 𝒇−𝑻𝒉𝒔𝒃𝒘𝒋𝒖𝒛[𝑲] )

Acti tion for

• r 𝝔

𝐓𝝔 = ∫ 𝒆𝒔 −𝒉𝑩𝒆𝑻−𝒅𝑪𝑰 − 𝟐

𝟑 𝝐𝝔 𝒔 𝟑 − 𝑾 𝝔 𝒔

𝟒. 𝟔 ≤ 𝒏𝟑 ≤ 𝟓

Numerical method

We solve the EOM numerically. (𝑛2 = 3.89, 𝜇 = 1)

𝝁𝝔 𝒔 𝟒 − 𝒏𝟑𝝔(𝒔) − 𝟔𝒈(𝒔)𝒔 + 𝒈′ 𝒔 𝒔𝟑 𝝔′ 𝒔 − 𝒈 𝒔 𝒔𝟑𝝔"(𝒔) = 𝟏 𝒈 𝒔 = 𝟐 + 𝑹𝟑

𝒔𝟕 − 𝟐+𝑹𝟑 𝒔𝟓

𝑼 =

𝟑−𝑹𝟑 𝟑𝝆

We will focus on Spontaneous magnetization (M when 𝑰 = 𝟏 )

𝝔 𝒔 =

𝑰 𝒔𝟑−𝚬 + 𝑵 𝒔𝟑+𝚬 + ⋯ 𝜠 ≡ 𝟓 − 𝒏𝟑

𝑰 = 𝒔𝟑−𝚬𝝔 𝒔 𝒔→∞ 𝑵 =

𝒔𝟑𝜠+𝟐 −𝟑𝚬 𝒆 𝒔𝟑−𝚬𝝔 𝒔 𝒆𝒔

𝒔→∞

Result: 𝑼~𝑼𝒅

𝑫+ 𝑼/𝑼𝒅−𝟐

𝑼 > 𝑼𝒅

𝒅− 𝟐−𝑼/𝑼𝒅

(𝑼 < 𝑼𝒅) 𝝍 =

• : result by holographic duality

𝑵 ∝ 𝟐 − 𝑼 𝑼𝒅

𝟐 𝟑

we can get Curie–Weiss law

Magnetic susceptibility 𝝍 ≡ 𝒆𝑵

𝒆𝑰 𝑰=𝟏

Spontanious Magnetization 𝑵 𝒅+/𝒅− ∼ 𝟑. 𝟑𝟑

Result : 𝐔 ∼ 𝑼𝒅

Results near 𝑼𝒅 are consistent with Ginzburg-Landau Theory 𝑵 ∝ 𝑰

𝟐 𝟒

H: External magnetic field M: Magnetization

F: Free energy 𝑮 ∝ 𝑼 − 𝑼𝒅 𝟑 The scalar part of the on-shell action

Result: low temperature (𝑼 ∼ 𝟏)

At low T, Results are consistent with magnons. Magnetization 𝑵 we can reproduce the Bloch 𝑼

𝟒 𝟑 law

𝑵 ∝ 𝟐 − 𝑫

𝑼 𝑼𝒅

𝟒 𝟑

Result: low temperature (𝑼 ∼ 𝟏)

Magnetic susceptibility: 𝝍

𝝍 ∼ 𝝍𝟏 + 𝑬 𝑼 𝑼𝒅

𝟐 𝟑

F: Free energy

𝑮 ∼ −𝑮𝟏 + 𝑭 𝑼 𝑼𝒅 − 𝜹 𝑼 𝑼𝒅

𝟑

First term 𝝍𝟏: Pauli paramagnetic susceptibility from conduction electrons Second term: susceptibility from magnons 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.

T ∼ 𝑈

𝑑 : GL theory

T ∼ 0 : Magnon + Conduction electron Using the dictionary, we analyzed the temperature dependence of Magnetization M, Susceptibility 𝝍 , Free energy 𝑮 Black Hole captures the ferromagnetic system both near 𝑼𝒅 and low temperatures Our results are consistent with Outlook 1 Magnon dynamics 2 Correlation functions

http://en.wikipedia.org/wiki/Black_hole