On magnetization in holographic models International Workshop for String Theory and Cosmology (STCOS 2015) June 29 (Mon), 2015 ~ July 01 (Wed), 2015 Kyung Kiu Kim(GIST) with San-Jin Sin(Hanyang Univ.), Yunseok Seo(Hanyang Univ.) and Keun-Young Kim (GIST)

Motivation AdS/CMT Ferromagnetism High Tc superconductors Anti-ferromagnetism Meissner Effect Nernst Effect, Hall angle, ….

Contents Why is magnetization important in AdS/CMT ? -Ferromagnetism -High Tc superconductor phase diagram ▷Superconducting phase 》 Holographic superconductor with Meissner effect ▷Normal phase of superconductor 》 Magnetic property (Nernst Effect, Hall angle) ▷Insulator phase of superconductor 》 Anti-ferromagnetism (Mott Insulator) -Magnetic Impurity ▷ Anomalous Hall effect, Magneto resistance, … Holographic models to describe these physics - Magnetic impurity model in more detail - Real scalar spontaneous magnetization model in more detail Summary

Ferromagnetism An applied magnetic field classifies materials. Also, turning off the magnetic field

Ferromagnetism Magnetization without External magnetic field < M > ≠ 0 with B = 0

Ferromagnetism Hysteresis Curve

Ferromagnetism Phase transition can be descried by the Landau-Ginzburg model with a real scalar. The model is given by

Phase transition with an effective model

A comment on 2+1 dimension In the real condensed matter: 2+1 dimension is not 2+1 dimension ▶ 2+1 dimensional material with 3+1 dimensional theory Introduced in Cai and Sin’s talk In 2+1 dimension L_{12} is like a helicity in 3+1 dim…

High Tc superconductor Demilitarized zone(DMZ) of phase diagram Magnetic mines are spread all over the phase diagram

A digression Typical properties of superconductor -Fundamental property : Meissner effect -Subsidiariness : Infinite conductivity Orignal holographic superconductor has a problem. Normal state : RN black hole Superconducting state : Hairy black hole Holographic DC conductivity - RN black hole(Infinite conductivity) from translational invariance - Hairy BH(Infinite conductivity) from translational invariance + superfluid density

Recent development : breaking translational invariance by Explicit space coordinate dependence -Explicit Lattice : PDE problem -> Horowitz, Santos and Tong(2012) -Massive gravity : Breaking diffeomorphism by mass term of graviton (ODE) -> Vegh(2013) -Axion model : Linear massless scalar to avoid PDE (ODE) -> Andrade and Withers(2013) -Q-Lattice model : more general model than the Axion model (ODE) -> Donos and Gauntlett(2014)

Holographic superconductors in every version -Explicit Lattice : Horowitz and Santos(2013) -Massive gravity : Cheng and Wu(2014) -Q-Lattice : Liu, Niu, Wu and Xian(2014) -Axion Model : Andrade and Gentle(2014) Kim, Kim and Park(2015) Typical AC conductivity

Meissner Effect of Holographic superconductor(Hartnoll, Herzog and Horowitz 2018) : We need Magnetization for strict Meissner effect

Normal phase of the superconductor Strange Metal Nernst Signal

Nernst signal becomes stronger as the paramaters(Tem perature, Dopping) goes to Quantum Critical Point Special shape of Signal to magnetic field by vortex fluid

Anti-ferromagnetism region in high Tc SC -> Cai’s talk

Magnetic Impurity -Anomalous Hall effect -Magneto resistance -Magneto electric effect -…

Holographic model of magnetic Impurities When = 0 The Axion field plays a role of impurity (Kim, Kim, Seo and Sin (2014))

Why ? Back to a model without impurity Background solution is Dyonic BH(Hartnoll Kovtun Muller and Sachdev 2007) Thermodynamic potential is the on-shell action by AdS/CFT dic. . The magnetization is defined as

Considering a fluctuation solution One can compute This is not ferromagnetic material because B=0 -> M =0 If there is a term There is a possibility for the spontaneous magnetization M without B

Possible terms Cai’s approach and Sin’s approach in their talks - M tensor is proportional to a polarization tensor. -Magnetization is proportional to the charge density because F_{rt} is the charge density by AdS/CFT dic.. Dual theory is like a Chen-Simon matter theory.

Without -> A model with impurities With -> A model with magnetic impurities Exact solution

Charge density Magnetization Energy Thermodynamic stability is guaranteed for small

Magnetization

Also, this solution is an exact solution of a massive gravity.

The DC conductivity

An interesting magneto-resistance

Spontaneous magnetization model In our computation we took n=2. We are calculating for other cases. Without This is same with a holographic superconductor model with a real scalar. -> Spontaneous Z_2 symmetry breaking

Without There is a transition from RN BH to a hairy BH which is dual to a phase transition related to real order parameter. This operator expectation value has nothing to do with magnetization.

With Solve background and fluctuation numerically. We obtained a spontaneous magnetization using real scalar condensation.

Summary Magnetization is important in condensed matter physics. AdS/CMT has to reproduce phenomena related to magnetization. We constructed a magnetic impurity model and a spontaneous magnetization model.

Thank you very much !