Holographic fermions with lattices
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Transcript Holographic fermions with lattices
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:
S
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
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 m 2
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 k F ) ( , k )
R
Ce T ...
m*
e 0 AT 2 ...
The retarded Green function:
G R , 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,
z1 z 22 ...
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 ab 0
V '( ) 0
How to find a lattice background?
•
4D Framework:
Scalar field with periodic behavior:
2
V() 2
L
z 0,
z1 z 22 ...
1 ( x) A0 cos(k0 x)
Lattice constant
a 2 / k0
How to find a lattice background?
•
4D Setup :
z 0,
dS2 dt 2 dx 2 dy 2
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
?
2
Temperature:
RN black holes:
Qtt ( x, z ) Qzz ( x, z ) Qxx ( x, z ) Qyy ( x, z ) 1
Qxz ( x, z ) ( x, z ) 0,
12 z 3
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 ( ab) 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:
ds 2 gtt ( x, z)dt 2 g zz ( x, z)dz 2 g xx ( x, z)dx2 g yy ( x, z)dy 2 2 g xz ( 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
it iki xi
A
F
B
30 z 31
A1
B1
B1
B2
m i 0 i1 i 2 0
B1
A1
A1
A2
30 z 31
A2
B2
B2
B1
m i0 i1 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
)
n0,1,2,... ,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
F ,n
1
m 0
a ,n z b ,n z
0
1
m
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, k xy k xF 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, k xy k xF 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!