Tutorial on Renormalization Group Applied to Classical and Quantum Critical Phenomena Branislav K
Tutorial on Renormalization Group Applied to Classical and Quantum Critical Phenomena Branislav K. Nikolić Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, U.S.A. http://wiki.physics.udel.edu/phys813
PHYS813: Quantum Statistical Mechanics Renormalization group The Task for Renormalization Group (RG) Theory
K. G. Wilson “Problems in physics with many scales of length,” Scientific American 241(2), 140 (1979)
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
Other problems in physics with many scales of length can be handled by RG
PHYS813: Quantum Statistical Mechanics Renormalization group Quick and Dirty Introduction Using Numerical Renormalization Group (RG) for Ising Model
Result of RG procedure: A renormalized system which has the same long- wavelength properties as the original system, but fewer degrees of freedom.
RG developed to treat situations with:
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
PHYS813: Quantum Statistical Mechanics Renormalization group RG Steps in Pictures
Original Coarse-grained Rescaled
Real-Space
Momentum-Space
VOCABULARY coarse-graining via partial −−H(,{ SSii′′} HS'({ i} CONVENTIONS ZHNN( ) = ∑∑ e= ∑ e = Z' ( H') evaluation of partition {SS} {}′′{S} function J ii i = RG recursion relations H=−=−∑∑ SSij K SSij H' RHb () kT ij ij B H''= RH ( ') =⋅= R RH ( ) R ( H ) RG is semi-group b2 b2 b 1 bb12 −HS(){ } i * ** n Ze= fixed point definition ∑ H= RHb( ), H= lim RHbc ( ) n→∞ {Si } −−1 dd r'= brN , ' = b NfH , ( ')= bfH ( ) rescaling
PHYS813: Quantum Statistical Mechanics Renormalization group Real-Space RG for 1D Ising Model
original S S S S S S S S 1 2 3 4 5 6 7 17 first renormalized S S S S S 1 3 5 7 17 second renormalized S S S S 1 5 9 17 third renormalized S S S17 1 9 20th renormalized ∞ S1 −6 , ferromagnet S1048576 KK20 <→10 ; 0, paramagnet
PHYS813: Quantum Statistical Mechanics Renormalization group RG Flow for 1D Ising Model
K( SS++ S S ) = 12 23 = KS12 S KS23 S KS 34 S KS 45 S ZKN ( ) ∑∑ e e e e e {SSii} { }
KS1+ KS 3 −− KS1 KS 3 KS 34 S KS 45 S = ∑ e+ e ee {SSi ≠ 2}
+ −− eKS1 KS 3+= e KS1 KS 3 AKe()BKSS()13 BK() SS13= =±⇒1 2cosh( 2K) = AKe ( ) 1 KKii+1 = ln cosh( 2 ) S=− S =±⇒=1 2 AKe ()−BK() 2 13 ⇓ ⇓ 1 AK( )= 2cosh2( KBK) ; ( )= lncosh2( KK) = **1 2 1 KK= ln cosh( 2 ) N 2 2 ZN () K= [ AK ()] ZN 2 ( K1 )
stable RG flow does not contain non-trivial fixed points, which means unstable fixed point 0 no phase transition at any finite temperature ∞ fixed point PHYS813: Quantum Statistical Mechanics Renormalization group Real-Space RG for 2D Ising Model
KSS( 01+++ SS 02 SS 03 SS 04) S2 ZKN ( ) = ∑ e {Si } +++ − +++ ⇒ KS()()1234 S S S KS1234 S S S S3 S0 S1 = ∑ ee+ {SSi ≠ 0}
S4
KS()()+++ S S S − KS +++ S S S BK()[ SS+++++ SS SS SS SS SS] + CK() SSSS e 1234+=e 1234 AKe() 12 23 34 41 13 24 1234 6()BK+ CK () SSSS1234= = = =±⇒1 2cosh 4K = AKe ( ) SSSS= = =− =±⇒1 2cosh 2K = AKe ( ) −CK() 123 4 If we set K2=K3=0, we −+2()BK CK () get similar trivial RG SSSS1234= =− =− =±⇒=1 2AKe () flow as in 1D Ising ⇓ model: 1 11 A(KKKB )= 2 cosh28 cosh4 ; (KK )= lncosh4 ;C (KKK )= lncosh4− lncosh2 8 82 ii+1 1 KK11= ln cosh 4 N 2 4 ZN () K= [ AK ()] ZN 2 ( K123 , K , K )
H=−2() B K∑∑ SSij −− B () K SSij C () K ∑ SSijkl S S ij [ij] square
PHYS813: Quantum Statistical Mechanics Renormalization group RG Flow for 2D Ising Model
stable 0 [a posteriori we indeed find K′3 (Kc )= − 0.05323] fixed point
unstable fixed point Both K1 and K2 are positive and thus favor the alignment of spins. Hence, we will omit the RG Kc = 0.50698 explicit presence of K2 in the renormalized exact energy, but increase K1 accordingly to a new K = 0.44069 value K’ = K’1 + K’2 so that the tendency toward c alignment is approximately the same
stable fixed point
PHYS813: Quantum Statistical Mechanics Renormalization group 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) close to critical point free energy can f(, th )= fsing (, th ) + freg (, th ) be decomposed in singular and regular part thermodynamic scaling relations which can be tested experimentally scaling hypothesis assumes that singular part −d yy fthbfbtbh(, )= (th , ) is generalized homogeneous function sing sing 22−=α βγ + b is an arbitrary (dimensionless) dy yt t h scale factor b=1 t ⇒ f (, th ) = t Φ± sing yyht −=α βδ + t and exponents yt 2 ( 1) and yh are characteristic of Φ=± (xf )sing ( ± 1, x ) universality class
2 D 1 ∂ fsing −−2 α d ≈ =yt = ⇒=−α C2 tt 2 Ttct∂ y
Dy− h ∂fsing β dy− m≈− ∝() − ttyt = () − ⇒β = h ∂ hyh=0 t t→0 →∞ x D d −1 Dy−−hhm<∞ dy −1 ∂fsing h y yh 1 δ y yy h h mth(, )≈− = t tt Φ′± ∝t x ⇒ mh(), ∝ h = h δ = yh ∂−h dyh t yt
2 Dy−2 h ∂ fsing −γ 2yd− χγ≈ ∝−yt =− ⇒ = h 2 ()tt () ∂h yt
PHYS813: Quantum Statistical Mechanics Renormalization group Scaling Hypothesis for Correlation Function
close to the critical point correlation function −−2(dyh ) r yth y is dominated by its singular part → actually valid only if Gsing (r ;, th )= b Gsing ;, b tb h the dimension of space of smaller than certain upper b critical dimension otherwise fails due to the so-called dangerously irrelevant couplings in RG 2(dy− ) −1 y h r h y bt= ⇒ Gsing (r ; t ,0) = t t Ψ± − 1 yt t t ≠0 r →∞ −−1 y ν − r ξ t four thermodynamic exponents expressed in terms Ψ± (xGx ) = ( ; ± 1, 0) ⇒ G (r ) ∝ e ⇒∝ξ t = t sing sing of the two correlation function exponents 1 hyperscaling relation because it connects ν= ⇒−2 αν =d singularities in thermodynamic observables with singularities related to the correlation function αν=2 − d yt x→∞ −−2(dyh ) t→0, Gsing (r ; t ,0)<∞ iff Ψ± ( xx ) ∝ ν − − − −+η 2(dyh ) ( d 2 ) βη=(2)d −+ G (rr ;0,0) ∝= r sing 2 2(dy−hh ) = d −+ 2 ηη ⇔ = d +−2 2 y γν=(2 − η ) another hyperscaling relation γν=(2 − η ) d +−2 η δ = d −+2 η
PHYS813: Quantum Statistical Mechanics Renormalization group 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 JJ δ KKK=−=c − ; focus on the singular kTB kT Bc part of free energy J TT− =c = Kt kT T linearize recursion Bc in RG flow
can be written like this ; ⇒ because of composition law of RG ;
but Kc is constant c ∂f(, th ) − mth( ,= 0) = ∝ t(dyh )/ y K remains finite ∂h h=0 at the critical point 2 ∂f(, th ) − χ(th ,= 0) = ∝ t(dy 2h )/2 y valid for any b, so choose this ∂h2 parameterization h=0 PHYS813: Quantum Statistical Mechanics Renormalization group Flow in the Case of More Than One Coupling Constant
PHYS813: Quantum Statistical Mechanics Renormalization group 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 yy 1 ,, n can be related to critical exponents
The exponents yy 1 ,, n determine the behavior of UU 1 ,, n under the repeated action of the linear RG recursion relations. If y i > 0 , U i is called relevant. If y i < 0 , U i is called irrelevant and if y i = 0 , U i 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. 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. PHYS813: Quantum Statistical Mechanics Renormalization group Do We Need Quantum Mechanics to Understand Phase Transitions at Finite Temperature?
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:
The decay time of temporal correlations for order-parameter fluctuations in dynamic (time-dependent) phenomena in the vicinity of critical point → critical slowing down
In quantum systems static and dynamic fluctuations are not independent because the Hamiltonian determines not only the partition function, but also the time evolution of any observable via the Heisenberg equation of motion
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
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
PHYS813: Quantum Statistical Mechanics Renormalization group Formal Definition of Quantum Phase Transitions Quantum systems have fluctuations driven by the Heisenberg uncertainty in the ground state, and these can drive non-trivial phase transitions at T = 0
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. 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
PHYS813: Quantum Statistical Mechanics Renormalization group Simple Theoretical Model I: Quantum
Criticality in Quantum Ising Chain (CoNb3O3) Phys. Today 64(2), 29 (2011) Quantum Ising chain in the transverse external magnetic field
Each ion has two possible states:
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
or
PHYS813: Quantum Statistical Mechanics Renormalization group Simple Theoretical Model II: Quantum
Criticality in Dimer Antiferromagnet (TiCuCl3) Experiment: Phys. Rev. Lett. 100, 205701 (2008) , 29 (2011) 64(2) Phys. Today
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
PHYS813: Quantum Statistical Mechanics Renormalization group 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.
, 29 (2011) Wavefunction at g=gc is a complex superposition of an exponentially large set of configurations
64(2) 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
Phys. Today separated at all length scales.
PHYS813: Quantum Statistical Mechanics Renormalization group Scaling in the Vicinity of Quantum Critical Points physics is dominated by thermal excitations of the quantum critical ground state
QPT in d dimensions is equivalent to classical transition in higher d+z dimensions Z= dpdqe−βHpq(,)= dpe −− ββKp() dqe Uq() dqe − βUq() ∫ ∫∫ ∫ statics and dynamics ⇒ ˆ ˆˆN Z=Tr e−βH = lim ee −− ββTN UN coupled in quantum N →∞ ( ) partition function CAVEATS: represents fluctuations of the order parameter and it • mapping holds for thermodynamics only depends on the imaginary time τ=-iβ/ħ which takes values • resulting classical system can be unusual and anisotropic in the interval [0,1/T]; the imaginary time direction acts • extra complications without classical counterpart may like an extra dimension, which becomes infinite for T→0 arise, such as Berry phases
PHYS813: Quantum Statistical Mechanics Renormalization group Example of Scaling Analysis: Superconductor-Insulator QPT in Thin Films Phys. Today 51(11), 39 (1998)
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
PHYS813: Quantum Statistical Mechanics Renormalization group RG Coarse Graining for 1D Quantum Ising Model in Transverse Magnetic Field 1 2 j j +1 N block I blockI + 1 ˆˆ ˆ ˆ zz x HH=∑∑block, j + Hcoupling, j Hblock =−− Jσσˆˆjj h σ ˆ j coupling , j jj∈∈odd even
↑↑ ↑↓ ↓↑ ↓↓ −+Jh221 −−Jh00 1 keep 1 and 2 only 22 − 22−+Jh Pˆ =11 + 2 2 ˆ 00Jh I Hblock, j = = −hJ003 22+ σ =11 − 2 2 Jh3 z 00−−hJ 4 22 σ x =12 + 21 Jh+ 4
ˆˆ ˆ ˆ projector onto h PP=⊗⊗12 P PN 2coarse-grained system z J z x σ σ + σ I I 22I 1 22 ˆˆ ˆ 22ˆ Jh+ Jh+ PHIblock, jI P=−+ J h PI ˆˆ ˆ ˆˆ ˆˆjj ˆ+1 ˆ ˆˆ j ˆ (PPHII⊗+1) coupling,j( PP I ⊗=− I+1 ) JPP( IzIσσˆˆ) ⊗( P I++1 z P I 1 ) − hPPP( IxI σ ˆ) ⊗ I+1
PHYS813: Quantum Statistical Mechanics Renormalization group RG Flow for 1D Quantum Ising Model in Transverse Magnetic Field
NN ˆ zz x original Hamiltonian for N spins HJ=−−∑∑σσˆˆjj+1 h σ ˆj jj=11=
NN2 2221− N2 coarse grained Hamiltonian ˆˆˆ 22 Jhzz x H= PHP =−+− J h P σσ+ − σ for N/2 spins after ∑∑I 22 II1 22 ∑I II=11Jh++= Jh I= 1one RG iteration
2 RG flow equations J * K = 0 stable → ferromagnet 22 2 ′ 2 fixed points J Jh+ hh′ 2 ** * = ⇔==KK′ = KK=( ) ⇒=∞ K stable → paramagnet ′ 2 ′ h J JJ * = = unstable → criticality KKc 1 Jh22+ extract critical exponents from RG flow
linearize RG equations close to Kc 1 const −v 1 ξ(KK ')= ξξ ( ) ⇒= ∝()KK −c ⇒=ν =1 dK′ b KK− cKy K′ −≈ Kcc( KK −) dK 1J 111 Kc σˆ = σσ= z22 zz2 ′ Jh++1 K dK yK λKK= ==⇒=21by −−11 1 dK mK()= bxx mb (yK K ) ⇒ b = ⇒=x,β =−= 1 x Kc 2 22 1+ Kc
PHYS813: Quantum Statistical Mechanics Renormalization group