Holographic Fermions with Lattices

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1 Holographic Fermions with Lattices
凌意 中国科学院高能物理研究所 04/25/2013, 中科大交叉学科理论研究中心

2 主要参考文献: G. Horowitz, J. Santos and D. Tong
Optical Conductivity with Holographic Lattices. JHEP 1207 (2012) 168 ，ArXiv: Further Evidence for Lattice-Induced Scaling. JHEP 1211 (2012) 102, ArXiv: G. Horowitz and J. Santos General Relativity and the Cuprates arXiv: 凌意、牛超、吴健聘、冼卓宇、张宏宝 Holographic Fermionic Liquid with Lattices arXiv:  

3 Outlines Preliminary: Applications of AdS/CFT to CMT
Introduction: Why lattices? How to find a lattice background? Holographic Fermions with lattices Prospects

4 Applications of AdS/CFT to CMT
Theoretical foundation A p+2 dimensional theory of quantum gravity may be described by a p+1 dimensional quantum field theory without gravity. Large N gauge theories in D-dim (Semi-)Classical gravity in D+1-dim J. Mcgreevy arXiv:

5 Applications of AdS/CFT to CMT
Bulk/boundary correspondence More specifically

6 全息引力在凝聚态理论的应用简介 全息字典 量子场中规范不变算子 Bulk里的动力学场 例： 能动张量 ： 引力子 ： 整体流 ：
能动张量 ： 引力子 ： 整体流 ： Maxwell场 ： 标量算子 ： 标量场 ： 费米算子 ： 费米场 ：

7 全息引力在凝聚态理论的应用简介 Eg.1：Holographic superconductors
The action of matter in the bulk ：

8 全息引力在凝聚态理论的应用简介 Holographic superconducting phase

9 全息引力在凝聚态理论的应用简介 Eg.2：Holographic (Non-)Fermi-like Liquid
The retarded Green function：

10 Introduction: Why lattices?
动机与研究方案： 能带论是固体理论电子运动的一个理论基础，而采用具有晶格周期性的势场是得到能带的前提条件。在引力/凝聚态对偶中，引入周期性势场将为理论与实验的衔接起到至关重要的作用。 布洛赫定理与单电子周期势场示意图

11 Introduction: Why lattices?
格点（周期势场）引入后导致的两个主要物理结果： 能隙的出现与能带论 周期区图示 简约区图示

12 Introduction: Why lattices?
金属、绝缘体、半导体的能带特征

13 Introduction: Why lattices?
格点（周期势场）引入后导致的两个主要物理结果： 格点破坏平移不变性，将影响系统的低频行为 全息电导率中的一个普遍问题（现象）： 长波极限下，电导率虚部趋于无穷，（由Kramers-Kronig关系）意味着实部 在直流处始终存在一个delta函数。这与金属常温下的实际电导率不符。

14 How to find a lattice background?
Two methods： 1、Scalar lattice: Simulating lattices with periodic scalar field with potential 2、Ionic lattice: directly introducing a periodic chemical potential

15 How to find a lattice background?
4D Framework： Equations of motion:

16 How to find a lattice background?
4D Framework： Scalar field with periodic behavior： Lattice constant

17 How to find a lattice background?
4D Setup ： Ansatz of variables No change! ? RN black holes： Temperature: ?:

18 How to find a lattice background?
Crucial technical issues in AdS/CMT with lattices： 1、Numerically solve the background equations with appropriate boundary and gauge conditions; 2、Numerically solve the perturbation equations over the background.

19 How to find a lattice background?
DeTurck method： 1、Einstein-DeTurck equation Here a reference metric is the RN black hole:

20 How to find a lattice background?
DeTurck method： 2、To guarantee the numerical result is a solution to Einstein equation: The convergence of the solutions

21 How to find a lattice background?
Boundary conditions： 1、Conformal symmetry at infinity (z=0): Remark: Such an assignment must be consistent with the asymptotic behavior of the EOM! 2、Regular conditions on horizon (z=1): Remark: To me it is not clear yet if such a regular condition will definitely lead to a unique solution!

22 How to find a lattice background?
Numerical methods in solving equations： 1、(pseudo)spectral method Change the partial differential equations into nonlinear algebraic equations by pseudospectral collocation approximation X direction: Fourier series Z direction: Chebyshev polynomials 2、Newton-Raphson method Change nonlinear algebraic equations into linear algebraic equations and then solve then with simple command “Linearsolve” in Mathematica

23 How to find a lattice background?
The numerical results: examples 1、Scalar lattice

24 How to find a lattice background?
The numerical results 2、charge density

25 How to find a lattice background?
The numerical results: examples 2、Ionic lattice

26 Holographic fermions with lattices
Contents 1、Consider a Fermionic field over a lattice, solving the Dirac equations numerically. 2、Locating the position of the Fermi surface via the standard holographic dictionary.

27 Holographic fermions with lattices
The setup Background: Remark: a) it is a linear, no need of Newton method. b) it is first-order, only fixing the boundary condition on one side.

28 Holographic fermions with lattices
Writing down the Dirac equations explicitly

29 Holographic fermions with lattices
The spectral method Boundary condition at the horizon (z=1)

30 Holographic fermions with lattices
Read off the retarded Green function The asymptotic behavior of EOM at infinity

31 Holographic fermions with lattices
The numerical results 1、Parameters for the background 2、A parameter for perturbations

32 Holographic fermions with lattices
The numerical results

33 Holographic fermions with lattices
The shape of the Fermi surface is ellipse!

34 Holographic fermions with lattices
Some other properties: 耦合参数q增加，费米动量增加，格林函数幅值变尖锐； 格点幅值增加， 增加； 温度降低，费米动量减小，格林函数幅值变尖锐； 温度降低， 增加；

35 Holographic fermions with lattices
The numerical results on band gap

36 Holographic fermions with lattices
The numerical results on band gap

37 Summary New results on holographic fermions when lattice is introduced: 费米面为一椭圆; 在布里渊区与费米面交界处观测到了带隙。

38 Prospects On holographic fermions with lattices:
On applications of lattices to other topics: 绝对零温和零温极限是一个主要问题； 椭圆产生的机理； Weyl项在全息格点模型里对电导率的影响； 全息格点与AdS3/CFT2(规范条件与渐进行为不匹配？）； 全息格点与超导； 全息格点与超导/绝缘体相变；

39 谢谢！