Tutorial on Renormalization Group Applied to Classical and ... · PHYS813: Quantum Statistical...

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


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)


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.


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


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− ∞

< →


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


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+ =


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


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


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


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

−

=

−

=

∂= = ∝

∂

∂= = ∝

∂


Flow in the Case of More Than One Coupling Constant


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.


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


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


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


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


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)


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


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)


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 ++


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

Tutorial on Renormalization Group Applied to Classical and ... · PHYS813: Quantum Statistical...

Documents

Transcript of Tutorial on Renormalization Group Applied to Classical and ... · PHYS813: Quantum Statistical...