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Renormalization groupPHYS813: Quantum Statistical Mechanics
Tutorial on Renormalization Group Applied to Classical and Quantum
Critical Phenomena
Branislav K. NikolićDepartment of Physics and Astronomy, University of Delaware, Newark, DE 19716, U.S.A.
http://wiki.physics.udel.edu/phys813
Renormalization groupPHYS813: Quantum Statistical Mechanics
The Task for Renormalization Group (RG) Theory
Other problems in physics with many
scales of length can be handled by RG
RG was developed to understand:why divergences occur in quantum field theory (i.e., why parameters like the electron mass are scale dependent).universality and scaling in second-order or continuous phase transitions
Turbulence
K. G. Wilson “Problems in physics with many scales of length,”
Scientific American 241(2), 140 (1979)
Renormalization groupPHYS813: Quantum Statistical Mechanics
Quick and Dirty Introduction Using Numerical Renormalization Group (RG) for Ising Model
Block spin transformation leads to RG flow
The RG flow changes the effective temperature, where the critical point
corresponds to the unstable point of the flow
RG developed to treat situations with:
Result of RG procedure:A renormalized system
which has the same long-wavelength properties as the original system, but
fewer degrees of freedom.
Renormalization groupPHYS813: Quantum Statistical Mechanics
RG Steps in Pictures Original Coarse-grained Rescaled
Real-Space
Momentum-Space
{ }
{ }
( )i
i
i j i jij ijB
H S
S
JH S S K S Sk T
Z e−
= − = −
=
∑ ∑
∑
( ) { }
{ }
{ }
{ }( )
2 2 1 1 2
( , '('
{ }
* * *
1
'
' ( )'' ( ') ( ) ( )
( ), lim ( )
' , ' , ( ') ( )
i i i
i i i
H S S H SN N
S S S
b
b b b b b
nb b cn
d d
Z H e e Z H
H R HH R H R R H R H
H R H H R H
r b r N b N f H b f H
′ ′− −
′ ′
→∞
− −
= = =
== = ⋅ =
= =
= = =
∑∑ ∑RG recursion relations
RG is semi-group
CONVENTIONS
fixed point definition
rescaling
coarse-graining via partial evaluation of partition
function
VOCABULARY
Renormalization groupPHYS813: Quantum Statistical Mechanics
Real-Space RG for 1D Ising Model
S1 S2 S6S4S3 S17S5
S1
S7
S3 S17S5 S7
S1 S17S5 S9
S1 S17S9
original
S1
first renormalized
second renormalized
thirdrenormalized
20th
renormalized S1048576
620
, ferromagnet10 ;
0, paramagnetK K− ∞
< →
Renormalization groupPHYS813: Quantum Statistical Mechanics
RG Flow for 1D Ising Model
( ) ( )
{ } { }
{ }
1 2 2 3 2 3 3 4 4 51 2
1 3 1 3 3 4 4 5
2
i i
i
K S S S S KS S KS S KS SKS SN
S S
KS KS KS KS KS S KS S
S S
Z K e e e e e
e e e e
+ +
+ − −
≠
= =
= +
∑ ∑
∑
( )
( ) ( )
[ ]
1 3 1 3 1 3
2
( )
( )1 3
( )1 3
1
21
( )1 2cosh 2 ( )
1 2 ( )
1( ) 2 cosh 2 ; ( ) ln cosh 22
( ) ( ) ( )N
KS KS KS KS B K S S
B K
B K
NN
e e A K eS S K A K e
S S A K e
A K K B K K K
Z K A K Z K
+ − −
−
⇓
+ =
= = ± ⇒ =
= − = ± ⇒ =
= = =
=
( )
( )
1
* *
1 ln cosh 22
1 ln cosh 22
i iK K
K K
+
⇓
=
=
RG flow does not contain non-trivial fixed points, which means no phase transition at any finite temperature0 ∞stable
fixed pointunstable
fixed point
Renormalization groupPHYS813: Quantum Statistical Mechanics
Real-Space RG for 2D Ising Model
⇒( ) ( )
{ }
{ }
0 1 0 2 0 3 0 4
1 2 3 4 1 2 3 4
0
( ) ( )
i
i
K S S S S S S S SN
S
K S S S S K S S S S
S S
Z K e
e e
+ + +
+ + + − + + +
≠
=
= +
∑
∑
S0
S2
S4
S1S3
[ ]1 2 2 3 3 4 4 1 1 3 2 4 1 2 3 41 2 3 4 1 2 3 4 ( ) ( )( ) ( )
6 ( ) ( )1 2 3 4
( )1 2 3 4
2 ( ) ( )1 2 3 4
( )1 2cosh 4 ( )
1 2cosh 2 ( )1 2 ( )
B K S S S S S S S S S S S S C K S S S SK S S S S K S S S S
B K C K
C K
B K C K
e e A K eS S S S K A K eS S S S K A K eS S S S A K e
+ + + + + ++ + + − + + +
+
−
− +
⇓
+ =
= = = = ± ⇒ =
= = = − = ± ⇒ =
= = − = − = ± ⇒ =
[ ]
[ ]
2
8
21 2 3
1 1 1( ) 2 cosh 2 cosh 4 ; ( ) ln cosh 4 ; ( ) ln cosh 4 ln cosh 28 8 2
( ) ( ) ( , , )
2 ( ) ( ) ( )N
NN
i j i j i j k lij ij square
A K K K B K K C K K K
Z K A K Z K K K
H B K S S B K S S C K S S S S
= = = −
=
= − − −∑ ∑ ∑
If we set K2=K3=0, we get similar trivial RG flow as in 1D Ising
model:
11 1
1 ln cosh 44
i iK K+ =
Renormalization groupPHYS813: Quantum Statistical Mechanics
RG Flow for 2D Ising Model
30 [a posteriori we indeed find K ( ) 0.05323]cK′ = −
RG
exact
0.506980.44069
c
c
KK
=
=
Both K1 and K2 are positive and thus favor the alignment of spins. Hence, we will omit the explicit presence of K2 in the renormalized energy, but increase K1 accordingly to a new
value K’ = K’1 + K’2 so that the tendency toward alignment is approximately the same
stable fixed point
stable fixed point
unstable fixed point
Renormalization groupPHYS813: Quantum Statistical Mechanics
Scaling Hypothesis for Free Energy Thermodynamic exponents α, β, γ, δ are not independent and can be obtained from exponents ν and η
governing the scaling of the correlation function (Widom 1965 and Kadanoff 1966 prior to RG)
sing reg
sing sing
sing
sing
( , ) ( , ) ( , )
( , ) ( , )
1 ( , )
( ) ( 1, )
t h
tt
h t
y yd
d yyy y
f t h f t h f t h
f t h b f b t b h
hb t f t h tt
x f x
−
±
±
= +
=
= ⇒ = Φ
Φ = ±
close to critical point free energy can be decomposed in singular and regular part
scaling hypothesis assumes that singular part is generalized homogeneous function
b is an arbitrary (dimensionless)
scale factor and exponents yt
and yh are characteristic of universality class
1
22sing
2
sing
00
1sing 1
22sing2
1 2
( ) ( )
( , ) ( ) ,
2( ) ( )
t
h
t
Dh h yh
ht t
h
t
h
t
Dy
c tD y
y h
thtx dD y d ym
y hy yy
hy
D yy
f dC t tT t y
f d ym t th y
f yhm t h t t x m h h hh d yt
ft t
h
α
β
δ
γ
α
β
δ
χ γ
−
− −
−
=
→→∞
− −<∞ −
±
−−
∂≈ = = ⇒ = −
∂
∂ −≈ − ∝ − = − ⇒ =
∂
∂ ′≈ − = Φ ∝ ⇒ ∝ = = ∂ −
∂≈ ∝ − = − ⇒ =
∂h
t
y dy−
2 22 ( 1)
α β γα β δ
− = +− = +
thermodynamic scaling relations which can be tested experimentally
Renormalization groupPHYS813: Quantum Statistical Mechanics
Scaling Hypothesis for Correlation Function
2( )sing sing
2( )1sing 1
0
1sing sing
2( )sing
sing
( ; , ) ; ,
( ; ,0)
( ) ( ; 1,0) ( )1 2
0, ( ; ,0) iff ( )
h t h
hh
tt
t
h
d y y y
d yyy
y
t
y
tx d y
G t h b G b t b hb
b t G t tt
x G x G e t t
dy
t G t x x
G
νξ ξ
ν α ν
− −
−−
± −
≠→∞
− −−±
→∞ − −±
=
= ⇒ = Ψ
Ψ = ± ⇒ ∝ ⇒ ∝ =
= ⇒ − =
→ < ∞ Ψ ∝
rr
rr
rr
r
r2( ) ( 2 )( ;0,0)
2( ) 2 2 2(2 )
hd y d
h hd y d d y
η
η ηγ ν η
− − − − +∝ =
− = − + ⇔ = + −= −
r r r
hyperscaling relation because it connects singularities in thermodynamic observables
with singularities related to the correlation function
another hyperscaling relation
close to the critical point correlation function is dominated by its singular part → actually valid only if
the dimension of space of smaller than certain upper critical dimension otherwise fails due to the so-called
dangerously irrelevant couplings in RG
2
( 2 )2(2 )
22
d
d
dd
α ννβ η
γ ν ηηδη
= −
= − +
= −+ −
=− +
four thermodynamic exponents expressed in termsof the two correlation function exponents
Renormalization groupPHYS813: Quantum Statistical Mechanics
Scaling Hypothesis Applied to Extract Critical Exponents via RG for 2D Ising Model
⇒removal of short-range degrees of freedom results in an expression that is analytic in K
focus on the singular part of free energy
linearize recursion in RG flow
;
;
;
⇒ can be written like this because of composition
law of RG
but Kc is constant
valid for any b, so choose this parameterization
cB B c
c
B c
J JK K Kk T k T
T TJ Ktk T T
δ = − = −
−= =
c
K remains finite at the critical point
( )/
0
2( 2 )/2
20
( , )( , 0)
( , )( , 0)
h
h
d y y
h
d y y
h
f t hm t h th
f t ht h th
χ
−
=
−
=
∂= = ∝
∂
∂= = ∝
∂
Renormalization groupPHYS813: Quantum Statistical Mechanics
Flow in the Case of More Than One Coupling Constant
Renormalization groupPHYS813: Quantum Statistical Mechanics
Relevant, Irrelevant and Marginal Variables in RG Flow
Critical exponents associated with a fixed point are calculated by linearizing RG recursion relations about that fixed point:
The form of the singular part of the free energy is a generalized homogeneous function, where exponents can be related to critical exponents1, , ny y
The exponents determine the behavior of under the repeated action of the linear RG recursion relations. If , is called relevant. If , is called irrelevant and if , is called marginal. We see that if a relevant scaling field is non-zero initially, then the linear recursion relations will transform this quantity away from the critical point. Alternatively, an irrelevant scaling field will transform this quantity toward the critical point, while marginal variable will be left invariant. Marginal variable is associated with logarithmic corrections to scaling.
1, , ny y 1, , nU U
0iy > iU 0iy < iU0iy = iU
The existence of the relevant, irrelevant, and marginal variables explains the observation of universality, namely, that ostensibly disparate systems (e.g., fluids and magnets) show the same critical behavior near a second-order phase transition, including the same exponents for analogous physical quantities. In the RG picture, critical behavior is described entirely in terms of the relevant variables, while the microscopic differences between systems is due to irrelevant variables.
Renormalization groupPHYS813: Quantum Statistical Mechanics
Do We Need Quantum Mechanics to Understand Phase Transitions at Finite Temperature?
The decay time of temporal correlations for order-parameter fluctuations in dynamic (time-dependent) phenomena in the vicinity of critical point
Although quantum mechanics is essential to understand the existence of ordered phases of matter (e.g., superconductivity and magnetism are genuine quantum effects), it turns out that quantum mechanics does not influence asymptotic critical behavior of finite temperature phase transitions:
→ critical slowing down
Thus, in quantum systems energy associated with the correlation time is also the typical fluctuation energy for static fluctuations, and it vanishes in the vicinity of a continuous phase transition as a power law
In quantum systems static and dynamic fluctuations are not independent because the Hamiltonian determines not only the partition function, but also the time evolution ofany observable via the Heisenberg equation of motion
This condition is always satisfied sufficiently close to Tc, so that quantum effects are washed out by thermal excitations and a purely classical description of order parameter fluctuations is sufficient to calculate critical exponents
Phase transitions in classical models are driven only by thermal fluctuations,as classical systems usually freeze into a fluctuationless ground state at T = 0
Renormalization groupPHYS813: Quantum Statistical Mechanics
Formal Definition of Quantum Phase Transitions
DEFINITION: Any point of nonanalyticity in the ground state energy of the infinite lattice system signifies quantum phase transition. the nonanalyticity could be either the limiting case of an avoided level-crossing or an actual level-crossing.QPT is usually accompanied by a qualitative change in the nature of the correlations in the ground state as one changes parameter in the Hamiltonian
An avoided level-crossing between the ground and an excited state in a finite lattice could become progressively sharper as the lattice size increases, leading to a nonanalyticity at in the infinite lattice limit.
Quantum systems have fluctuations driven by the Heisenberg uncertainty in the ground state, and these can drive non-trivial phase transitions at T = 0
Renormalization groupPHYS813: Quantum Statistical Mechanics
Simple Theoretical Model I: Quantum Criticality in Quantum Ising Chain (CoNb3O3)
Quantum Ising chain in the transverse external magnetic field
The first term in the Hamiltonian prefers that the spins on neighboring ions are parallel to each other, whereas the second allows quantum tunneling between the and states with amplitude proportional to g
Phys. Today 64(2), 29 (2011)
Each ion has two possible states:
or
Renormalization groupPHYS813: Quantum Statistical Mechanics
Simple Theoretical Model II: Quantum Criticality in Dimer Antiferromagnet (TiCuCl3)
Phys
. Tod
ay 6
4(2)
, 29
(201
1) Experiment: Phys. Rev. Lett. 100, 205701 (2008)
The two noncritical ground states of the dimer antiferromagnet have very different excitation spectra:
spin waves with nearly zero energy and oscillations of the magnitude of
local magnetization
Renormalization groupPHYS813: Quantum Statistical Mechanics
How Quantum Criticality Extends to Non-Zero Temperatures
For small g, thermal effects induce spin waves that distort the Néel antiferromagnetic ordering. For large g, thermal fluctuations break dimers in the blue region and form quasiparticles called triplons. The dynamics of both types of excitations can be described quasi-classically.
Quantum criticality appears in the intermediate orange region, where there is no description of the dynamics in terms of either classical particles or waves. Instead, the system exhibits the strongly coupled dynamics of nontrivial entangled quantum excitations of the quantum critical point gc.
Wavefunction at g=gc is a complex superposition of an exponentially large set of configurations fluctuating at all length scales → thus, it cannot be written down explicitly due to long-range quantum entanglement which emerges for a very large number of electrons and between electrons separated at all length scales.Ph
ys. T
oday
64(
2), 2
9 (2
011)
Renormalization groupPHYS813: Quantum Statistical Mechanics
Scaling in the Vicinity of Quantum Critical Points
represents fluctuations of the order parameter and it depends on the imaginary time τ=-iβ/ħ which takes values in the interval [0,1/T]; the imaginary time direction acts like an extra dimension, which becomes infinite for T→0
physics is dominated by thermal excitations of the quantum critical ground state
( )
( , ) ( ) ( ) ( )
ˆ ˆ ˆTr lim
H p q K p U q U q
NH T N U N
N
Z dpdqe dpe dqe dqe
Z e e e
β β β β
β β β
− − − −
− − −
→∞
= = ⇒= =
∫ ∫ ∫ ∫statics and dynamics coupled in quantum partition function
CAVEATS:• mapping holds for thermodynamics only• resulting classical system can be unusual and anisotropic• extra complications without classical counterpart may
arise, such as Berry phases
QPT in d dimensions is equivalent to classical transition in higher d+z dimensions
Renormalization groupPHYS813: Quantum Statistical Mechanics
Example of Scaling Analysis: Superconductor-Insulator QPT in Thin Films
The success of finite-size scaling analyses of the
superconductor-insulator transitions as a
function of film thickness or applied magnetic field
provides strong evidence that T = 0 quantum phase transitions are occurring
Phys. Today 51(11), 39 (1998)
Renormalization groupPHYS813: Quantum Statistical Mechanics
RG Coarse Graining for 1D Quantum Ising Model in Transverse Magnetic Field
blockoupling
block, coupling, ,odd even
ˆ ˆ ˆ ˆ ˆ ˆ ˆc
z z xj j j j jj
j jH H H H J hσ σ σ
∈ ∈
= + = − −∑ ∑
block I block 1I +
j 1j +1 2 N
2 2
2 2
block,2 2
2 2
110 02 20 0ˆ
0 0 3 30 0 4 4
j
J hJ hJ hJ h
Hh J J h
h JJ h
− +− − − +− = = − + − − +
↑↑ ↑↓ ↓↑ ↓↓
keep 1 and 2 onlyˆ 1 1 2 2
1 1 2 2
1 2 2 1
I
z
x
P
σ
σ
= +
= −
= +
( ) ( ) ( ) ( ) ( )
1 2 2
2 2block,
11 coupling, 1 1 1 1
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ
N
I j I I
j j jI I j I I I z I I z I I x I I
P P P P
P H P J h P
P P H P P J P P P P h P P Pσ σ σ++ + + + +
= ⊗ ⊗
= − +
⊗ ⊗ = − ⊗ − ⊗
projector onto coarse-grained system z
Iσ2 2
xI
hJ h
σ+
12 2
zI
JJ h
σ ++
Renormalization groupPHYS813: Quantum Statistical Mechanics
RG Flow for 1D Quantum Ising Model in Transverse Magnetic Field
11 1
2 22 2 1 22 2
12 2 2 21 1 1
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ
N Nz z xj j j
j j
N N Nz z x
I I I II I I
H J h
J hH PHP J h PJ h J h
σ σ σ
σ σ σ
+= =
−
+= = =
= − −
= = − + − −+ +
∑ ∑
∑ ∑ ∑
coarse grained Hamiltonian for N/2 spins after
one RG iteration
2
22 22
2
2 2
JJ h hJ h K Kh J JJ
J h
′ ′ + ′= ⇔ = = = ′ ′ +
RG flow equations
( )*
fixed points2* * *
*
0
1c
KK K K
K K
=
= ⇒ = ∞ = =
stable → ferromagnet
stable → paramagnet
unstable → criticality
( )
2 1
c
K
c
c cK
yK K
K
dKK K K KdK
dK b ydK
λ
′′ − ≈ −
′= = = ⇒ =
original Hamiltonian for N spins
linearize RG equations close to Kc
1 1 1
2 2 2
2
1 const 1( ') ( ) ( ) 1
1ˆ1
1 1 1( ) ( ) , 12 21
K
vc
c K
z z z
yx x
c
K K K Kb K K y
JJ h K
m K b m b K b x xK
ξ ξ ξ ν
σ σ σ
β
−
− −
= ⇒ = ∝ − ⇒ = =−
= =+ +
= ⇒ = ⇒ = = − =+
extract critical exponents from RG flow