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
主要参考文献:
G. Horowitz, J. Santos and D. Tong
Optical Conductivity with Holographic Lattices.
JHEP 1207 (2012) 168 ，ArXiv:1204.0519.
Further Evidence for Lattice-Induced Scaling.
JHEP 1211 (2012) 102, ArXiv:1209.1098.
G. Horowitz and J. Santos
General Relativity and the Cuprates
arXiv:1302.6586
凌意、牛超、吴健聘、冼卓宇、张宏宝
Holographic Fermionic Liquid with Lattices
arXiv:1304.2128
Outlines
I.
Preliminary: Applications of AdS/CFT to CMT
II.
Introduction: Why lattices?
III. How to find a lattice background?
IV. Holographic Fermions with lattices
V.
Prospects
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
R2
dS  2 (  dx  dx  dz 2 )
z
2
J. Mcgreevy arXiv:0909.0518
Applications of AdS/CFT to CMT
• Bulk/boundary correspondence
 LJ
Zb [ J ]  e 
 Z QG b.c. depends on J 
CFT
N 1

e
 S grav
EOM
• More specifically

iS [cl ]i d d xJ 
Zboundary [ J ]   D e
G ( x  y )  i T  ( x) ( y )
QFT
lim
z 0
cl ( z , x)
z

 J ( x)
 2 S[cl ]

 J ( x) J ( y )  ( z 0)  J

 ( z, x )  e
ik x 
f k ( z ), k x   t  k  x
f k ( z )  Ak z   Bk z 
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 
例：
Bulk里的动力学场
能动张量 T ：
引力子 g ab：
整体流 J  ：
Maxwell场 Aa ：

标量算子 B ：
标量场
费米算子  F ：
费米场 ：

：
1
6 1
4
ab
a
S
d
x

g
[
R


F
F

2


  4V ( )]
ab
a
2

16 G
L 2
全息引力在凝聚态理论的应用简介
•
Eg.1：Holographic superconductors

1 R
G  
i
G R     lim g (r )rAx Ax'
r 
The action of matter in the bulk ：
2
2
 1
Sm   d 5 x  g  F  F    iqA  m2  
 4

全息引力在凝聚态理论的应用简介
•
Holographic superconducting phase
全息引力在凝聚态理论的应用简介
•
Eg.2：Holographic (Non-)Fermi-like Liquid
Z
G  , k  
  F (k  kF )  (, k )
R
Ce   T  ...

m*
e  0  AT 2  ...
The retarded Green function：
GR , k   iS 0
  Ar m  Br  m1 ,   Cr m1  Dr  m
D  SA
i

2
i 2
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
 (r )   uk (r )eik r
k
uk (r )  uk (r  Rl )  uk (r  la),
l  0, 1, 2...
Introduction: Why lattices?
•
格点（周期势场）引入后导致的两个主要物理结果：
1.
能隙的出现与能带论
周期区图示
简约区图示
Introduction: Why lattices?
金属、绝缘体、半导体的能带特征
Introduction: Why lattices?
•
格点（周期势场）引入后导致的两个主要物理结果：
2.
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。
How to find a lattice background?
•
Two methods：
1、Scalar lattice: Simulating lattices with periodic scalar field
with potential
z  0,
  z1  z 22  ...
1 ( x)  A0 cos(k0 x)
2、Ionic lattice: directly introducing a periodic chemical potential
 ( x)  [1  A0 cos(k0 x)]
How to find a lattice background?
•
4D Framework：
1
6 1
4
ab
a
S
d
x

g
[
R


F
F

2


  4V ( )]
ab
a
2

16 G
L 2
Equations of motion:
Gab  Rab 
3
g ab  ....  0
2
L
a F a b  0
  V '()  0
How to find a lattice background?
•
4D Framework：
Scalar field with periodic behavior：
2
V( )   2
L
z  0,
  z1  z 22  ...
1 ( x)  A0 cos(k0 x)
Lattice constant
a  2 / k0
How to find a lattice background?
•
4D Setup ：
z  0,
dS2  dt 2  dx2  dy2
Ansatz of variables
Q ( x, z )
L2
ds  2 [(1  z ) P( z )Qtt ( x, z )dt 2  zz
dz 2
z
P( z )(1  z )
2
 Qxx ( x, z )[dx  z 2Qxz ( x, z )dz ]2  Qyy ( x, z ) dy 2 ]
A  (1  z ) ( x, z )dt
  z ( x, z)
No change!
P( z )  1  z  z 
2
12 z 3
Temperature:
RN black holes：
Qtt ( x, z )  Qzz ( x, z )  Qxx ( x, z )  Qyy ( x, z )  1
Qxz ( x, z )   ( x, z )  0,
?
2
P(1) 6  12
T

4 L
8
 ( x, z )    1
?:
Qtt ( x,1)  Qzz ( x,1)
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.
How to find a lattice background?
•
DeTurck method：
1、Einstein-DeTurck equation
GabH  Gab (ab)  0
a
a
 a : g cd [cd
( g )  cd
( g )]
g : a reference metric with the same asmptotics and horizon structures
Here a reference metric is the RN black hole:
Qtt ( x, z)  Qzz ( x, z)  Qxx ( x, z)  Qyy ( x, z)  1, Qxz ( x, z)  0
How to find a lattice background?
•
DeTurck method：
2、To guarantee the numerical result is a solution to Einstein equation:
a. The convergence of the solutions
a
10
b.  a  10
How to find a lattice background?
•
Boundary conditions：
1、Conformal symmetry at infinity (z=0):
Qtt ( x, 0)  Qzz ( x, 0)  Qxx ( x, 0)  Qyy ( x, 0)  1
Qxz ( x, z )  0,  ( x, 0)  1 ( x),
 ( x, z )   ( 1 )
Remark: Such an assignment must be consistent with the asymptotic
behavior of the EOM!
2、Regular conditions on horizon (z=1):
Qij ( x, z )  Qij0 ( x)  (1  z )Qij1 ( x)  (1  z ) 2 Qij2 ( x)  ...
 ( x, z )   0 ( x)  (1  z ) 1 ( x)  (1  z ) 2  2 ( x)  ...
Qtt ( x,1)  Qzz ( x,1)
 ( x, z )   0 ( x)  (1  z ) 1 ( x)  (1  z ) 2 2 ( x)  ...
Remark: To me it is not clear yet if such a regular condition will definitely
lead to a unique solution!
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
How to find a lattice background?
•
The numerical results: examples
1、Scalar lattice
k0  2, A0  1,   1.4, T /   0.1
How to find a lattice background?
•
The numerical results
2、charge density
z 0
 ( x, z)    [   ( x)]z  ....
k0  2,   2.35, T /   0.008
How to find a lattice background?
•
The numerical results: examples
2、Ionic lattice
k0  2, A0  0.1,   2.3, T /   0.01
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.
Holographic fermions with lattices
•
The setup
S D  i  d 4 x  g  ( a Da  m)
a Da  m  0
1
Da   a  ( )a    iqAa
4
( )a  (e )b a (e )b
Background:
ds2  gtt ( x, z)dt 2  gzz ( x, z)dz 2  gxx ( x, z)dx2  g yy ( x, z)dy2  2gxz ( x, z)dxdz
Aa  At ( x, z)dta
Remark: a) it is a linear, no need of Newton method.
b) it is first-order, only fixing the boundary condition on one side.
Holographic fermions with lattices
•
Writing down the Dirac equations explicitly
1
4
   gtt g xx g yy  F ( x, z)e

F  (F1, F2 )
T
it iki xi
 A 
F   
 B 
 30 z  31
 A1 
 B1 
 B1 
 B2 
m    i0    i1    i 2    0
 B1 
 A1 
 A1 
 A2 
 30 z  31
 A2 
 B2 
 B2 
 B1 
m    i 0    i1    i 2    0
 B2 
 A2 
 A2 
 A1 
Holographic fermions with lattices
•
The spectral method
 A ,n ( z )  inKx
 A ( x, z ) 

    B ( z ) e
B
(
x
,
z
)
 
 n0,1,2,...   ,n 
K
2
a
Boundary condition at the horizon (z=1)
 A ,n   1  B ,n 
z 


0
B
A
  ,n  4 T 1  z   ,n 
i

 B ,n   1
4 T

    (1  z )
 A ,n   i 
Qtt ( x,1)  Qzz ( x,1)
Holographic fermions with lattices
•
Read off the retarded Green function
The asymptotic behavior of EOM at infinity
 F1,n 
( z z  m )  
0
 F2,n 
3
1
 0
F ,n  a ,n z m    b ,n z  m  
 0
1
a ,n ( , l )  G ,n; ',n 'b ',n ' ( , l )
Holographic fermions with lattices
•
The numerical results
1、Parameters for the background
k0  2, A0  1.5,   2.35, T  0.0081
2、A parameter for perturbations
q  1.7
Holographic fermions with lattices
•
The numerical results
2
ky2
kx

1
2
2
1.8991 1.8511
Holographic fermions with lattices
•
The shape of the Fermi surface is ellipse!
Holographic fermions with lattices
•
Some other properties:
1. 耦合参数q增加，费米动量增加，格林函数幅值变尖锐；
2. 格点幅值增加， kxy  kxF  k yF 增加；
3. 温度降低，费米动量减小，格林函数幅值变尖锐；
4. 温度降低， kxy  kxF  k yF 增加；
Holographic fermions with lattices
•
The numerical results on band gap
Holographic fermions with lattices
•
The numerical results on band gap
Summary
•
New results on holographic fermions when lattice is introduced:
1. 费米面为一椭圆;
2. 在布里渊区与费米面交界处观测到了带隙。
Prospects
•
On holographic fermions with lattices:
1. 绝对零温和零温极限是一个主要问题；
2. 椭圆产生的机理；
•
On applications of lattices to other topics:
1. Weyl项在全息格点模型里对电导率的影响；
2. 全息格点与AdS3/CFT2(规范条件与渐进行为不匹配？）；
3. 全息格点与超导；
4. 全息格点与超导/绝缘体相变；
谢谢！