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Tutorial on Renormalization Group Applied to Classical and Quantum Critical Phenomena Branislav K

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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( K) ; BK ( )= 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
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