Green function in solid-state physics
GPU meeting
Shuxiang Yang
October 25, 2012
Outline
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Introduction
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Green function
●
●
–
Definition (mathematical and physical)
–
Properties
Diagrammatic approach to calculate G
–
Single-particle level
–
Two-particle level
Conclusion
2
Introduction
3
Introduction
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Physics research
Materials
Physics experiments
Modeling
Theoretical & numerical analysis
Sovle model
Interpret results
4
Modeling
U
t
(Zhang and Rice, PRB 1988,
P.W. Anderson)
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Models: Ising model, Heisenberg model, Hubbard model, etc.
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2-D Hubbard model (quantum system model)
Simplest model able to capture the low energy physics of cuprates
5
Solve the model
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We need to solve the Schrodinger equation
for quantum system
Equation of motion
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Note that the hamiltonian is time-independent, we can
separate the two variables as
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Then we just need to solve a simpler equation
6
Solve the model
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How to solve this equation
operator??
scalar
vector
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It can be represented as a matrix if a basis is chosen
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Four possible configurations per-site
(occupation number representation)
U
●
t
Number of configurations: 4^Nc
7
Solve the model
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Eigen-value and-vector problem
matrix
scalar
vector
–
Wave-function based
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H: linear dimension: 4^Nc
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Nc: number of sites (atoms), 10^23 for real materials
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This puts a severe constraint on the solving of this equation
both in computational time and in memory
e.g. Nc=16, memory requirement for
wave function vector:16GB; Hamiltonian matrix: 7x1010GB !
We need a smarter way of solving this model
8
Outline
●
Introduction
●
Green function
●
●
–
Definition (mathematical and physical)
–
Properties
Diagrammatic approach to calculate G
–
Single-particle level
–
Two-particle level
Conclusion
9
Green function in mathematics
Named after the British mathematical physicist George Green.
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Inhomogeneous differential equation
Linear differential operator
examples:
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Green function is defined as
George Green
(07/14/1793~03/31/1841)
10
Green function in mathematics
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Once the Green function is calculated,
the solution of equation
is
This can be checked by applying L on both sides and
using the definition of G
11
Green function in physics
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We only need the following replacement
Green function in physics is defined as (frequency space)
solution:
●
for the time space, it is
12
Green function in physics
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Formal definition in physics
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Operator in the Heisenberg representation
–
Measurement
Other names
propagator, correlation function
13
Physical meaning of Green function
Represents the phase accumulated when particles move
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Green function
–
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Self-energy
–
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long-ranged
Short-ranged
Related by Dyson eq
14
Different Green functions
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Real time/frequency vs imaginary time/frequency
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Real space vs momentum space
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Single-particle vs two-particle
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Homogeneous vs inhomogeneous system
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...
15
Examples of Green function
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Bare (non-interacting) Green function
Dressed (interacting) Green function
self-energy
16
Outline
●
Introduction
●
Green function
●
●
–
Definition (mathematical and physical)
–
Properties
Diagrammatic approach to calculate G
–
Single-particle level
–
Two-particle level
Conclusion
17
Dyson equation
other forms:
Dyson equation
Taylor expansion
18
Diagrammatic representation
●
Dyson equation
(1)
(2)
(3)
(1)
(2)
(3)
19
Diagrammatic representation
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Two elements:
–
Bare Green function
Contains frequency and dispersion information
–
Interaction:
20
contains Coulomb interaction effect
Green function diagram examples
Question: how to generate and sum these diagrams efficiently?
21
Self-consistent calculation
Approximation on
Green function
Approximation on
self-energy
Self-consistent
calculation
22
Two-Particle Quantities
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Scattering Process and Vertex
23
Parquet Formalism
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1-particle formalism + 2-particle formalism
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Vector: G, T, Σ
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Rank-3 Tensor: χ, F, Γ, Λ
24
Hierarchy of Approximate Methods
HPC
needed
25
Conclusion
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Green function
–
Phase accumulated when particles move
–
A language to describe and a tool to solve physical system
Diagrammatic approach to sum up diagrams
–
Single-particle level
–
Two-particle level
26
Thank you!
Questions??
27