Many-body physics 1: Basics
Hongki Min
Department of Physics and Astronomy, Seoul National University, Korea
Orientation, March 9, 2015
− 다체계 물리: 기본
References
[1] Condensed matter field theory
Alexander Altland and Ben Simons
[2] Many-particle physics (3rd Ed.)
Gerald D. Mahan
[3] Quantum theory of the electron liquid
Gabriele Giuliani and Giovanni Vignale
[4] Introduction to Many Body Physics
Piers Coleman
References
[5] Quantum theory of many-particle systems
Alexander L. Fetter and John Dirk Walecka
[6] A guide to Feynman diagrams in
the many-body problem, Richard D. Mattuck
[7] Quantum many-particle systems
John W. Negele and Henri Orland
[8] Quantum field theory in condensed
matter physics, Naoto Nagaosa
Special topics [1] Renormalization-group approach to interacting
fermions, R. Shankar, Rev. Mod. Phys. 66, 129 (1994)
[2] Lectures on phase transitions and
the renormalization group, Nigel Goldenfeld
[3] Introduction to superconductivity
Michael Tinkham
[4] Superconductivity, superfluids, and condensates
James F. Annett
[5] The Quantum Hall effect
Daijiro Yoshioka
Special topics [6] Interacting electrons and quantum magnetism
Assa Auerbach
[7] Quantum phase transitions
Subir Sachdev
[8] Quantum physics in one dimension
Thierry Giamarchi
[9] Geometry, topology, and physics
Mikio Nakahara
[10] The geometric phase in quantum systems
Bohm, Mostafazadeh, Koizumi, Niu, and Zwanziger
Many-body problem
System of interacting electrons and ions
• Electron Hamiltonian
ji jii
iel
e
mH
rr
22
2
2
1
2
ionionelel HHHH
• Electron-ion interaction
Ii Ii
Iionel
eZH
,
2
Rr
• Ion Hamiltonian
JI JI
JI
I
I
I
ion
eZZ
MH
RR
22
2
2
1
2
Statistics, symmetries, effective low-energy theories
See Altland
and Simons, Ch.1
[1] Quasiparticle self-energy
휀 𝑘 = 휀0 𝑘
Corrections to the particle’s energy due to interactions
휀 휀0(𝒌)
non-interacting
𝐴 𝒌, 휀 ~Im1
휀 − 휀(𝒌)
See Mahan Ch.5.8
𝐴 𝒌, 휀 ~Im1
휀 − 휀(𝒌)
휀 휀0(𝒌)
non-interacting
ReΣ~𝛿휀
ImΣ~ℏ/2𝜏
[1] Quasiparticle self-energy
Interactions renormalize the energy dispersion
and gives rise to lifetime.
ReΣ(𝑘)~𝛿휀(𝑘)
ImΣ(𝑘)~ℏ
2𝜏𝑘
휀 𝑘 = 휀0 𝑘
Σ 𝑘 = Σ𝑒−𝑒 𝑘 + Σ𝑒−𝑝ℎ 𝑘 +⋯
Corrections to the particle’s energy due to interactions
+Σ(𝑘)
interacting
See Mahan Ch.5.8
[2] Response function
ext~ Vn n
Linking measurements to correlations
nnn ~
ext~ EJ JJ ~
ext~ HM M MMM ~
Response to the experimental probes can be expressed
in terms of correlation functions, which contain
information of the unperturbed system.
See Giuliani and Vignale, Ch.3~5
Occupation number representation
Identical particles and indistinguishability
𝑛𝜆1 = 1, 𝑛𝜆2 = 1
[𝑃12, 𝐻] = 0, 𝑃122 = 1
⇒ 𝑃12 = ±1 Boson/Fermion
𝜆1𝜆2 ± =1
2 𝜆1 1 𝜆2 2 ± 𝜆2 1 𝜆1 2
Example: Two identical particles
𝑛𝜆1 , 𝑛𝜆2 , 𝑛𝜆3 , … In general, Occupation number
representation
𝑃12 𝜆 1 𝜆′ 2 = 𝜆′ 1 𝜆 2
Many-body operators
𝑉 =1
2 𝑣 𝑖𝑗𝑖,𝑗
→1
2 𝑎
𝜆1′† 𝑎
𝜆2′† 𝑣𝜆1′ 𝜆2′ 𝜆1𝜆2𝑎 𝜆2𝑎 𝜆1
𝜆,𝜆′
𝐹 = 𝑓 𝑖𝑖
𝑓𝜆′𝜆= 𝜆′ 𝑓 𝜆
𝑣𝜆1′ 𝜆2′ 𝜆1𝜆2= 𝜆1′ 𝜆2
′ 𝑣 𝜆1𝜆2
Operators in the occupation number representation
One-body operators: Density, current, …
Two-body operators: Coulomb interaction, …
→ 𝑎 𝜆′† 𝑓𝜆′𝜆𝑎 𝜆
𝜆,𝜆′
See Giuliani and Vignale, App.2
Linear response theory
ext~ Vn n
Linking measurements to correlations
Experimental probes can be regarded as small perturbations
nnn ~
ext~ EJ JJ ~
ext~ HM M MMM ~
Response to the experimental probes can be expressed
in terms of correlation functions, which contain
information of the unperturbed system.
Linear response theory
)(ˆˆ)(ˆext0 tHHtH
Linear response to an external perturbation
Response function : Retarded correlation function
)()(ˆ)(ˆext tFtBtH
)(ˆ),(ˆ)()( tBtAtttti AB
)(ˆ)(ˆ)(ˆ)(ˆext
tUtAtUtA
)(ˆ),(ˆ)(ˆext tHtAtd
itA
t
)()()(ˆ
tFtttdtA AB
𝑒𝑥 ≈ 1 + 𝑥 for 𝑥 ≪ 1
See Giuliani and
Vignale, Ch.3.2
)(ˆ1)(ˆext tHtd
itU
t
Linear response theory
Response function in frequency space
ti
mn
titi mnnm eAenAmentAm
ˆ)(ˆ
)()( tedt AB
ti
AB
)(ˆ),(ˆ)()( tBtAtttti AB
nm
nmmn
nm
nmAB BA
i
PP
, 0)(
Pn: Occupation probability for state n
Non-interacting response function
Density response of non-interacting electrons
k
k+q
𝑛𝒌 Fermi distribution function for k
𝑔 spin/valley degeneracy factor
𝑠 band index
00 )0,0( Nq
N0: DOS at EF
𝜒0 𝑞, 𝜔 = 𝑔 ∫𝑑2𝑘
2𝜋 2𝑠,𝑠′
𝑛𝒌,𝑠 − 𝑛𝒌+𝒒,𝑠′
ℏ𝜔 + 휀𝒌,𝑠 − 휀𝒌+𝒒,𝑠′ + 𝑖0+
Linear response theory : Interacting system
extV
n
eff
0V
n
intexteff VVV
ext
eff
eff V
V
V
n
Density response of interacting electrons
Response of interacting system to an external field
can be obtained from the response of non-interacting
system to an effective self-consistent field
Veff: Effective self-consistent field acting on each
particle independently
Vint: Internal field generated by the density deviations
0int
0int0
11
int
ext
int0 1
V
n
n
V
xcCint
int
n
VRandom phase approximation
Polarization and dielectric function
Linear response theory : Interacting system
• Transport properties
― Screening
• Collective modes
― Ex: Plasmon
Collective density oscillations
• Correlation
― Electron motion
↔ Surrounding electrons
+ –
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
Single particle Green’s function (I)
𝑖𝐺 𝑘, 𝑡 = Ω 𝑇𝑐 𝐻𝑘 𝑡 𝑐 𝐻𝑘† (0) Ω
𝑇𝐴 𝑡 𝐵(𝑡′) 𝐴 𝑡 𝐵(𝑡′) 𝑡 > 𝑡′
±𝐵(𝑡′)𝐴 𝑡 𝑡′ > 𝑡
𝑇: Time-ordering operator
𝐻: Heisenberg picture
Interaction picture
𝑖ℏ𝜕
𝜕𝑡 𝜓𝑆(𝑡) =𝐻 (𝑡) 𝜓𝑆(𝑡)
𝐻 𝑡 = 𝐻 0 + 𝑉 𝑡
𝜓𝐼(𝑡) = 𝑒𝑖ℏ𝐻 0𝑡 𝜓𝑆(𝑡)
See Sakurai Ch.5.5
𝑂 𝐼(𝑡) = 𝑒𝑖
ℏ𝐻 0𝑡𝑂 𝑆 𝑒
−𝑖
ℏ𝐻 0𝑡
𝑖ℏ𝜕
𝜕𝑡 𝜓𝐼(𝑡) =𝑉 𝐼(𝑡) 𝜓𝐼(𝑡)
𝑖ℏ𝜕
𝜕𝑡𝑂 𝐼(𝑡)=[𝑂 𝐼(𝑡), 𝐻 0]
Time dependence due to interactions → state vectors
Time dependence of non-interacting systems → operators
Single particle Green’s function (II)
𝑖𝐺 𝑘, 𝑡 = Ω 𝑇𝑐 𝐻𝑘 𝑡 𝑐 𝐻𝑘† (0) Ω
=0 𝑇𝑐 𝑘 𝑡 𝑐 𝑘
†0 𝑈 𝐼(∞,−∞) 0
0 𝑈 𝐼(∞,−∞) 0
= 0 𝑇𝑐 𝑘 𝑡 𝑐 𝑘† 0 𝑈 𝐼 ∞,−∞ 0
𝑐𝑜𝑛𝑛
Interaction
picture
Connected
diagrams
Linked cluster theorem; Wick’s theorem;
Perturbation theory; Feynman diagrams; Self-energy
See Mahan Ch.2 and
Fetter and Walecka, Ch.3
𝑈 𝐼 𝑡, 𝑡0 = 𝑇exp −𝑖
ℏ 𝑑𝑡′𝑡
𝑡0
𝑉𝐼(𝑡′)
Single particle Green’s function (II)
Corrections to the particle’s energy due to interactions
휀 휀0(𝒌)
non-interacting
𝐴 𝒌, 휀 ~Im𝐺(𝒌, 휀)
See Mahan Ch.5.8
𝐺(0) 𝒌, 휀 =1
휀 − 휀𝒌 + 𝑖𝜂𝒌
𝐴 𝒌, 휀 ~Im𝐺(𝒌, 휀)
휀 휀0(𝒌)
non-interacting
ReΣ~𝛿휀
ImΣ~ℏ/2𝜏
Single particle Green’s function (II)
Interactions renormalize the energy dispersion
and gives rise to lifetime.
ReΣ(𝒌)~𝛿휀(𝒌)
ImΣ(𝒌)~ℏ
2𝜏𝒌
Corrections to the particle’s energy due to interactions
interacting
See Mahan Ch.5.8
𝐺 𝒌, 휀 =1
휀 − 휀𝒌 − Σ(𝒌, 휀)
Finite temperature formalism
𝑍𝐺 = Tr[𝑒−𝛽𝐾 ]
𝐾 = 𝐻 − 𝜇𝑁
−𝑔 𝑘, 𝜏 = Tr[𝜌 𝐺𝑇𝜏𝑐 𝐻𝑘 𝜏 𝑐 𝐻𝑘† (0)]
Grand partition function
Grand canonical Hamiltonian
≡ 𝑇𝜏𝑐 𝐻𝑘 𝜏 𝑐 𝐻𝑘† (0)
𝜏 = 𝑖𝑡 Imaginary time
𝜌 𝐺 = 𝑒−𝛽𝐾 /𝑍𝐺
Analytic continuation
The retarded Green’s function can be obtained from the
finite-temperature Green's function through the analytic
continuation 𝑖𝜔𝑛 → 𝜔 + 𝑖0+.
See Fetter and Walecka, Ch.9
𝐺 𝑅 𝑘, 𝜔 = 𝑑𝜔′
2𝜋
∞
−∞
𝜌(𝑘, 𝜔′)
𝜔 − 𝜔′ + 𝑖0+
𝑔 𝑘,𝜔𝑛 = 𝑑𝜔′
2𝜋
∞
−∞
𝜌(𝑘, 𝜔′)
𝑖𝜔𝑛 −𝜔′
Path integral methods
See Sakurai Ch.2.5
𝑈 𝑥𝑓, 𝑡𝑓; 𝑥𝑖 , 𝑡𝑖 = 𝑥𝑓 𝑒−𝑖ℏ(𝑡𝑓−𝑡𝑖)𝐻 𝑥𝑖
= 𝐷𝑥𝑥𝑓,𝑡𝑓
𝑥𝑖,𝑡𝑖
𝑒𝑖ℏ ∫
𝐿(𝑥,𝑥) 𝑡𝑓𝑡𝑖 𝐿 𝑥, 𝑥 =
1
2𝑚𝑥 2 − 𝑉(𝑥)
Functional integral methods
𝑍𝐺 = Tr[𝑒−𝛽𝐾 ]
= 𝐷𝜙∗𝐷𝜙 𝑒−𝑆[𝜙∗,𝜙]
𝑆[𝜙∗, 𝜙] = 𝑑𝜏𝛽
0
𝜙∗𝜕
𝜕𝜏𝜙 + 𝐾(𝜙∗, 𝜙)
𝑎 𝜙 = 𝜙 𝜙 coherent states
See Negele and Orland, Ch.2.2
Gaussian integral
𝑑𝑥∞
−∞
𝑒−12𝑎𝑥2+𝑏𝑥 =
2𝜋
𝑎𝑒𝑏2
2𝑎
The quadratic 𝑏2 term can be expressed as an integral
over the auxiliary term 𝑥 which couples linearly with 𝑏.
Question: How can we treat two-body interaction terms?
Hubbard-Stratonovich transformation
Two-body fermionic operators can be expressed as an
integral over the auxiliary field which couples linearly
with the ferminonic operators.
This transformation represents the interactions between
fermions in terms of an exchange boson.
By integrating out the microscopic fermions, we can
rewrite the problem as an effective field theory of the
bosonic order parameter 𝜙.
See Coleman Ch.13.5
𝜌 = 𝜓 𝜓
Landau-Ginzburg theory
𝑓 𝜙 =1
2𝛻𝜙 2 +
𝑟
2𝜙2 +
𝑢
4𝜙4 − ℎ𝜙
Express the free energy as a function of the order
parameter near the critical point taking into account the
inhomogeneity in space and the symmetry of the system.
𝑟 and 𝑢 are unknown constants depending on physics at
the atomic scale while the generic power-law dependence
follows from the structure of the free energy independent
of the microscopic details showing a universal behavior.
𝛿𝑓
𝛿𝜙= 0 → ℎ 𝜙 𝜒 =
𝛿𝜙
𝛿ℎ
Scaling theory
Close to the critical point, a correlation length diverges
and there is no characteristic length scale other than the
correlation length.
There is a relationship between the coupling constants
of an effective Hamiltonian and the length scale over
which the order parameter is defined.
𝑔𝑠𝑙 = 𝑅𝑠(𝑔𝑙) 𝑠: scaling factor
𝛽(𝑔) =𝑑𝑔
𝑑ln𝑙 𝛽: beta function
𝑔∗ = 𝑅𝑠(𝑔∗) 𝑔∗: fixed point
See Goldenfeld, Ch.9
𝑍 = 𝐷𝜙< 𝑒−𝑆0[𝜙<] 𝐷𝜙> 𝑒
−𝑆0 𝜙> 𝑒−𝑆𝐼[𝜙<,𝜙>]
Renormalization group
See R. Shankar,
Rev. Mod. Phys. 66, 129 (1994) 𝑍 = 𝐷𝜙(𝑘) 𝑒−𝑆[𝜙 𝑘 ]
𝜙<(𝑘) = 𝜙(0 < 𝑘 < Λ/𝑠) : slow modes
𝜙>(𝑘) = 𝜙(Λ/𝑠 < 𝑘 < Λ) : fast modes
Integrate out the fast modes to obtain the partition
function with an effective action over the slow modes.
For 𝑆 𝜙 = 𝑆0 𝜙< + 𝑆0 𝜙> + 𝑆𝐼 𝜙<, 𝜙>
𝑒−𝑆𝑒𝑓𝑓[𝜙<]
φ4 theory: Effective theory of Ising spins
See Altland and Simons, Ch.8.4
𝐻 = −1
2 𝐽𝑖𝑗𝑆𝑖𝑆𝑗
𝑖,𝑗∈𝑁.𝑁.
𝑆𝑖 = ±1
𝑆 𝜙 = 1
2𝛻𝜙 2 +
𝑟
2𝜙2 +
𝑢
4𝜙4
Semi-classical approach
An electron gas can support longitudinal oscillations due
to the restoring force by a self-consistent electric field
generated by local excessive charges, called plasmons.
𝑁𝑚𝑥 = 𝑁(−𝑒)𝐸
𝐸 = 4𝜋𝜎 = 4𝜋𝑛𝑒𝑥 𝜔𝑝𝑙 =4𝜋𝑛𝑒2
𝑚
See Ashcroft &
Mermin, Ch.1
𝑥 + 𝜔𝑝𝑙2 𝑥 = 0
+
+
+
+
−
−
−
−
Displaced electron gas with the positive background of ions
𝑥
𝐸
Many-body approach
The frequencies of collective modes in a many-body system
are determined by the poles of the response function.
𝜒𝑛 =𝛿𝑛
𝛿𝑉𝑒𝑥𝑡~ 𝑛 𝑛
Response ~ Correlation
See Giuliani and Vignale, Ch.5
~√𝑞(𝑐1 + 𝑐2𝑞)
~𝑐1 + 𝑐2𝑞2
𝜔𝑝𝑙 =4𝜋𝑛𝑒2
𝑚1 + 𝒪(𝑞2)
𝜒𝑛 → ∞ Collective modes