Two-dimensional phase transitions
Malte Henkel
aComputational physics for engineering materials, Institute for Building Materials (IfB), ETH Zurich, Switzerland
bGroupe de Physique Statistique, DP2M,
Institut Jean Lamour(CNRS UMR 7198), Universit´e de Lorraine Nancy, France
Course Spring semester 2017, ETH Z¨urich
Some further reading
Nishimori & Ortiz, Elements of phase transitions and critical phenomena, Oxford Univ. Press (2011)
Cardy, Scaling and renormalisation in statistical physics, Oxford Univ.
Press (1996)
di Francesco, Mathieu, S´en´echal, Conformal field-theory, Springer (1997) mh, Conformal invariance and critical phenomena, Springer (1999) Blumenhagen, Plauschinn, Introduction to conformal field-theory,
Springer (2009)
mh, Karevski (´eds), Conformal invariance . . . , Springer (2012)
Overview :
Lecture I :Historical introduction to critical phenomena ; 2D experiments
Lecture II :Widom scaling ; Correlators and scale-invariance ; conformal invariance
Lecture III :conformal invariance ; co-variant correlators
Lecture IV :Finite-size scaling & conformal invariance
Lecture III
Some historic dates
1953Stueckelberg & Peterman : concept of renormalisation group
Helv. Phys. Acta 26, 499 (1953)
1954 Gell-Mann & Low : renormalisation group in QED 1965 Widom : scaling laws
1966 Kadanoff : scaling, block spins, operator algebras 1970 Callan & Szymanzik : concept of the β-function
1970 Polyakov : conformal invariance for d > 2
1971 Wilson : reormalisation group & Kondo problem 1972 Wilson & Fisher : ε-expansions
1972 Fisher & Barber : finite-size scaling (fss)
1973 Gross, Politzer, Wilczek : asymptotic freedom in QCD 1981 Fr¨ohlich, Aizenman : triviality of φ4-theory for d > 4 1984Belavin, Polyakov, Zamolodchikov :
2D conformal field-theory
1984/86 Cardy : fss and conformal invariance, modular invariance 1987 Cappelli, Itzykson, Zuber :
ADE classification of modular invariants
1989 Zamolodchikov : integrability of 2D Ising model with T = Tc, h 6= 0 2001/02 Lawler, Schramm, Smirnov, Werner : sle & conformal invariance 2012 . . . Rychkov et al. : 3D conformal bootstrap
Rappel :
Ward identity and conformal invariance
entire phenomenology of phase transitions (‘Widom scaling’) can be reduced to co-variance of scaling operatorsφi(r) = b−xiφi(r/b) infinitesimal change of coordinates r 7→ r + α(r)
Definition : a (scalar) scaling operator φ isquasi-primaryif δφ = dx∇ · α + α · ∇ φ = 0.
Invariance of n-point correlators of quasi-primary scaling operators gives
n
X
p=1
φ1(r1) . . . δtotφp(rp) . . . φn(rn) !
= 0
=
n
X
p=1
D
φ1(r1) . . .
α(rp) · ∇p+xp
d ∇p· α(rp)
φp(rp) . . . φn(rn) E
+cd
Z
dr hφ1(r1) . . . φn(rn)Tµνi ∂µαν(r)
globalWard identity ⇒essential basis for space-symmetries
theenergy-momentum tensor Tµν must obey the properties : (1) conservation∂µTµν = 0
(2) symmetry, from rotation-invariance Tµν = Tνµ (3) tracelessness, from scale-invarianceTµµ= 0
for the ‘special’ conformal transformation r0/r02= r/r2+ b, find
⇒ Tµν∂µαν = 2 Tµν
|{z}
symm.
(rµbν − bµrν)
| {z }
anti−symm.
| {z }
=0
−2 Tµµ
|{z}
=0
b · r = 0
Theorem :
translation inv.
rotation inv.
dilatation inv.
Ward identity
=⇒special conformal invariance
Callan, Coleman, Jackiw, Ann. of Phys. 59, 42 (1970)
Conformal invariance I - definition
distancesds are described in terms of a metric tensor gµν
ds2 =gµνdrµdrν under a coordinate change r 7→ r0, this transforms as :
gµν(r) 7→gµν0 (r0) = ∂rα
∂r0µ
∂rβ
∂r0νgαβ(r) Definition : A coordinate change r 7→ r0 isconformal, if
gµν0 (r0) = Ω(r)gµν(r), with Ω(r) > 0.
(a) 7→ (b) is conformal, but(a) 7→ (c) is not conformal(there is ashear)
Interpretation : angle θbetween two vectors a,b : cos θ := √a·b
a2b2
under a conformal transformation cos θ0 = gµν0 a0µb0ν
q
gµν0 a0µa0νgκλ0 b0κb0λ
= Ω
√ Ω2
gµνaµbν
pgµνaµaνgκλbκbλ =cos θ
⇒ conformal transformations preserve angles e.g. Mercator-projection
Theorem :In d > 2 dimensions, a transformation r 7→ r0 is conformal, iff it is one of the following :
1 a translationr0= r + a
2 a rotationr0 = Rr
3 a dilatationr0= b−1r
4 a special conformal transformation rr020 = rr2 + b
The conformal Lie group in d dimensions has 12(d + 1)(d + 2) transformations.
⇒ ‘position-dependent, angle-preserving, dilatations’ r 7→ r/b(r)
Conformal invariance II - classical field-theory
classical field-theories are specified by an actionS =R dr L
with aLagrangian L = L(φ, ∂µφ) (unique up to total divergencies) field-theory textbooks give a ‘canonical’ energy-momentum tensor
Tµν[can] = ∂L
∂(∂µφ)∂νφ − δµνL
Example : the free fieldL = 12(∂µφ)2 ⇒ eqn. of motion ∂µ∂µφ = 0
⇒ Tµν = Tµν[can]= ∂µφ∂νφ − 12δµν(∂σφ)2
test of the three properties : conservation, symmetry, traceless-ness : (i) conservation : ∂µTµν = (∂µ∂µφ) ∂νφ = 0
(ii) symmetry : Tµν = Tνµ is obvious
(iii) trace: Tµµ= 1 −d2 [∂µ(φ∂µφ)− φ (∂µ∂µφ)] 6= 0
⇒ must add a divergenceto Land construct the
‘improved’ energy-momentum tensor Tµν[imp]:= ∂µφ∂νφ −21δµν(∂σφ)2+ 1
4 d − 2
d − 1(δµν∂σ∂σ− ∂µ∂ν) φ2
Conformal invariance III -
two-point correlator Polyakov 1970? conformal-invariance prediction for the two-point correlatorC12= hφ1φ2i ?
? two points r1, r2 can always be brought onto any prescribed line
? three points r1, r2, r3 can always be brought onto any prescribed plane
⇒ can restrict to two dimensions, denoter =
t r
,
t ‘time’
r ‘space’
write infinitesimal Lie algebra generatorsfor conformal transformations (1) translations in time and space :−∂t∂ and−∂r∂
(2) time-space rotations :−r∂t∂ + t∂r∂ assume scalars
(3) dilatations :−t∂t∂ − r∂r∂ − x (4) ‘special’ : − t2− r2 ∂
∂t − 2tr∂r∂ − 2tx and−2tr∂t∂ + t2− r2 ∂
∂r − 2rx
2-point correlator :4variables (t1, t2, r1, r2),6 conditions ⇒ over-determined 3-point correlator :6variables (t1, t2, t3, r1, r2, r3),6 conditions⇒ unique solution 4-point correlator :8variables (t1, . . . t4, r1, . . . r4),6 conditions ⇒ 2 variables left free
? how do these Lie algebra generators arise ?
Example 1 : translations |at|, |ar| 1
r0 =
t0 r0
=
t + at
r + ar
= r + a
⇒ δr = a=⇒ δt = at,δr = ar
Lie algebra generators
time translation Xt= − ∂t0(t, r ; at, ar)
∂at a=0
∂
∂t = − ∂
∂t space translation Xr = − ∂r0(t, r ; at, ar)
∂ar a=0
∂
∂r = −∂
∂r Lie algebra given by commutator (‘abelian Lie algebra’)
[Xt, Xr] := XtXr − XrXt = ∂2
∂t∂r − ∂2
∂r ∂t = 0
Example 2 : rotations |ϕ| 1 r0 =
t0 r0
=
cos ϕ sin ϕ
− sin ϕ cos ϕ
t r
=
cos ϕ t + sin ϕ r
− sin ϕ t + cos ϕ r
'
t + ϕr
−ϕt + r
=⇒ δt = ϕr,δr = −ϕt
Lie algebra generatorof rotations XR= − ∂t0(t, r ; ϕ)
∂ϕ ϕ=0
∂
∂t − ∂r0(t, r ; ϕ)
∂ϕ ϕ=0
∂
∂r = −r ∂
∂t + t ∂
∂r
‘Euclidean Lie algebra’ given by non-vanishing commutators [XR, Xr] = −∂
∂t = Xt , [XR, Xt] = ∂
∂r = −Xr
Remark : very useful simple identities for computing complicated commutators [A, BC ] = [A, B] C + B [A, C ]
[AB, CD] = A [B, C ] D + [A, C ] BD + CA [B, D] + C [A, D] B
Example 3 : dilatations |ε| 1
r0=
t0 r0
= (1 + ε)
t r
=
t + εt r + εr
=⇒ δt = εt,δr = εr
Lie algebra generatorof dilatations XD = − ∂t0(t, r ; ε)
∂ε ε=0
∂
∂t − ∂r0(t, r ; ε)
∂ε ε=0
∂
∂r = −t ∂
∂t − r ∂
∂r XD should act on scaling operators. For quasi-primary scaling operators, add contribution from Jacobi determinant (contains scaling dimensionx)
XD = −t ∂
∂t − r ∂
∂r −x
‘scaling Euclidean Lie algebra’ given by further non-vanishing commutators [XD, Xt] = Xt , [XD, Xr] = Xr , [XD, XR] =
−t ∂
∂t − r ∂
∂r, −r ∂
∂t + t ∂
∂r
= 0
Example 4 : ‘special’ conformal |bt|, |br| 1
r0 =
t0 r0
'
t (1 + 2(btt + brr )) − t2+ r2 bt r (1 + 2(btt + brr )) − t2+ r2 br
⇒ δt = 2t(btt + brr ) − t2+ r2 bt,δr = 2r (btt + brr ) − t2+ r2 br Lie algebra generatorsof special conformal transformations
XSt = − ∂t0(t, r ; bt, br)
∂bt b=0
∂
∂t − ∂r0(t, r ; bt, br)
∂bt b=0
∂
∂r
= − t2− r2 ∂
∂t − 2tr ∂
∂r XSr = − ∂t0(t, r ; bt, br)
∂br
b=0
∂
∂t − ∂r0(t, r ; bt, br)
∂br
b=0
∂
∂r
= −2tr ∂
∂t + t2− r2 ∂
∂r
XSt, XSr should act on scaling operators. For quasi-primary scaling operators, add contribution from Jacobi determinant (contains scaling dimensionx)
XSt = − t2− r2 ∂
∂t − 2tr ∂
∂r − 2tx, XSr = −2tr ∂
∂t + t2− r2 ∂
∂t − 2rx conformal Lie algebragiven by further non-vanishing commutators
[XSt, Xt] = [XSr, Xr] = 2XD , [XSt, Xr] = − [XSr, Xt] = 2XR , [XSt, XD] = − [XSr, XR] = XSt , [XSr, XD] = [XSt, XR] = XSr ,
[XSt, XSr] = 0
notation : ∂i := ∂t∂
i,Di := ∂r∂
i
conditions of conformal co-variance : ⇒ the 2-point correlator C = C (t1, t2; r1, r2) mustvanish under the action of all
two-body Lie algebra generators
(∂1+ ∂2) C = 0 (D1+ D2) C = 0 (r1∂1+ r2∂2− t1D1− t2D2) C = 0 (t1∂1+ t2∂2+ r1D1+ r2D2+ x1+ x2) C = 0 t12− r12 ∂1+ t22− r22 ∂2+ 2t1r1D1+ 2t2r2D2+ 2x1t1+ 2x2t2 C = 0 2t1r1∂1+ 2t2r2∂2− t12− r12 D1− t22− r22 D2+ 2x1r1+ 2x2r2 C = 0 set of 6 coupled linear partial differential equations (PDE) of first order
how to solve such systems efficiently : use partial results to reduce the PDEs
time- and space-translation-invariance :C = C (t, r ),t = t1− t2,r = r1− r2
⇒ the remaining PDEs must only contain t, r Rotations :
[(r1− r2) ∂t− (t1− t2) ∂r] C = 0 ⇒ [r ∂t− t∂r] C = 0 Dilatations :
[(t1− t2) ∂t+ (r1− r2) ∂r + x1+ x2] C = 0 ⇒ [t∂t+ r ∂r + x1+ x2] C = 0 first special transformation :
t12− r12 − t22− r22 ∂t+ 2 [t1r1− t2r2] ∂r + 2x1t1+ 2x2t2 C = 0 express all variables in the PDE in terms of only t, r
nh
(t1− t2)2+ 2t1t2− 2t22
−
(r1− r2)2+ 2r1r2− 2r22i
∂t +2 [(t1− t2) (r1− r2) + t1r2+ t2r1− 2t2r2] ∂r
+2x1(t1− t2) + 2(x1+ x2)t2 C = 0 these terms are collected as follows
(t2− r2)∂t+ 2tr ∂r + 2x1t
| {z }
requested terms +2t2[(t1− t2)∂t+ (r1− r2)∂r + x1+ x2]
| {z }
=0, dilatation-inv.
+ 2r2[(r1− r2)∂t− (t1− t2)∂r]
| {z }
=0, rotation-inv.
C = 0
in this way, the two special conformal transformations lead to
(t2− r2)∂t+ 2tr ∂r + 2x1t C = 0, 2tr ∂t− (t2− r2)∂r + 2x1r C = 0
? compare the first of these with dilation-invariance :
t2∂t+ tr ∂r + t(x1+ x2) C = 0
=⇒−r2∂t+ tr ∂r + t (x1− x2) C = 0
? compare with rotation-invariance : r2∂t− tr ∂r C = 0. This gives t (x1− x2) C = 0 =⇒ constraint : x1= x2
? same constraintalso from the second special transformation.
The conformally co-variant 2-point correlator C = C (t, r )obeys [r ∂t− t∂r] C = 0
| {z }
rotations
, [t∂t+ r ∂r + 2x1] C = 0
| {z }
dilatations
, constraint : x1 = x2
| {z }
special conformal
PDEs of this kind are very often solved through two simple results : Lemma 1 : If f = f (x , y ) is a solution of the PDE
[a(x , y )∂x + b(x , y )∂y] f (x , y ) = 0, then F (f (x , y ))is also a solution, where F is an arbitrary differentiable function.
Lemma 2 : The PDE[a(x )∂x+ b(y )∂y] f (x , y ) = 0 has the solution f (x , y ) =
Z x dx0 a(x0) −
Z y dy0
b(y0) , a(x ) 6= 0 and b(y ) 6= 0
e.g. Kamke, Differentialgleichungen : L¨osungsmethoden und L¨osungen, Bd. 2 (Teubner)
Apply this to conformal invariance : make the ansatzC = C (t, ρ), with ρ = t2+ r2 (which clearly solves rotation-invariance). Then
∂tC = 0 , [t∂t+ 2ρ∂ρ+ 2x1] C = 0 =⇒ C = C0ρ−x1 Final result :for quasi-primary scalar scaling operators, Φ0= cste.
hφ1(t1, r1)φ2(t2, r2)i = δx1,x2Φ0 (t1− t2)2+ (r1− r2)2−x1
this can be re-written in a vector form : let r12= |r12| = r1− r2. Then hφ1(r1)φ2(r2)i = δx1,x2r12−2x1
hφ1(r1)φ2(r2)φ3(r3)i = C123
r12x1+x2−x3r23x2+x3−x1r31x3+x1−x2 hφ1(r1)φ2(r2)φ3(r3)φ4(r4)i = Y
i <j
rijx /3−xi−xjf r12r34 r13r24
,r12r34 r23r14
whereC123 is anuniversalconstant (with chosen normalisation of hφ1φ2i), x = x1+ x2+ x3+ x4 andf an arbitrary function.
two-point function the same as from scale-invariance alone & extra constraint x1= x2 Ising model : hσσi, hεεi 6= 0, but hσεi = 0 expected from Z2-symmetry.
further information required to fix the shape of f ⇒ must restrict to d = 2 dimensions, where conformal Lie algebra is infinite-dimensional
2D conformal invariance
notation : complex coordinates z = t + ir,¯z = t − ir ⇒ ds2= dzd¯z
conformal transformations for d > 2 can be re-written as (projective) z 7→z0 = αz + β
γz + δ , ¯z 7→z¯0 = α¯¯z + ¯β
¯
γ¯z + ¯δ , αδ − βγ = 1 , ¯α¯δ − ¯β ¯γ = 1 these are maps C → C (including the point z = ∞)
in 2D all analytic (or anti-analytic) transformations are conformal z 7→ w = f (z) , ¯z 7→ ¯w = ¯f (¯z) =⇒ ds02= dw d ¯w =dwdz d ¯d¯wzdzd¯z= |w0(z)|2ds2 Lie algebra generators(the Lie algebra has dim vect(S1) ⊕ vect(S1) = ∞)
`n= −zn+1 ∂
∂z , `¯n= −¯zn+1 ∂
∂¯z , n ∈ Z
with the commutators E. Cartan (1909)´
[`n, `m] = (n − m)`n+m , `¯n, ¯`m = (n − m)¯`n+m , `n, ¯`m = 0
Definition : A scaling operator φ = φ(z, ¯z) isquasi-primary, if under a projective transformation z 7→ wp(z) = αz+βγz+δ it transforms
φ0(z, ¯z) = ∂wp(z)
∂z
∆
∂ ¯wp(¯z)
∂¯z
∆
φ(wp(z), ¯wp(¯z)) The constants ∆and∆are called conformal weights.
Example 1 : dilatation z 7→ λz, ¯z 7→ λ¯z ⇒φ0(z, ¯z) = λ∆+∆φ(λz, λ¯z)
=⇒ scaling dimensionx = xφ= ∆ + ∆
Example 2 : rotation z 7→ eiϕz, ¯z 7→ e−iϕ¯z ⇒ φ0(z, ¯z) = eiϕ(∆−∆)φ(eiϕz, e−iϕ¯z)
=⇒ spins = sφ= ∆ − ∆
Definition : A scaling operator φ = φ(z, ¯z) isprimary, if under any conformal transformation z 7→ w (z) it transforms
φ0(z, ¯z) = ∂w (z)
∂z
∆ ∂ ¯w (¯z)
∂¯z
∆
φ(w (z), ¯w (¯z))
Belavin, Polyakov, Zamolodchikov 1984
Lie algebra generators `n= −zn+1 ∂∂z − ∆(n + 1)zn
co-variance under projective transformations gives quasi-primary correlators hφ1(z1, ¯z1)φ2(z2, ¯z2)i = δ∆1,∆2δ∆
1,∆2(z1− z2)−2∆1(¯z1− ¯z2)−2∆1 hφ1(z1, ¯z1)φ2(z2, ¯z2)φ3(z3, ¯z3)i = C123z12−∆12,3z23−∆23,1z31−∆31,2
×C123¯z12−∆12,3z¯23−∆23,1z¯31−∆31,2 hφ1(z1, ¯z1) . . . φ4(z4, ¯z4)i =
Y
i <j
zij−γij¯zij−γij
F (η, ¯η) wherezij := zi − zj,∆12,3:= ∆1+ ∆2− ∆3
C123 is called an operator product expansion (ope) coefficient P
iγij = 2∆j,η := zz12z34
23z14
F cannot be found from projective transformations alone
N.B. the variable ¯z etc. is only written explicitly if really needed
Ortho- and meta-conformal invariance
? is there a single kind of conformal invariance ?
here : attach the word ‘conformal’ to the Lie algebra conf(2) Definition 1 : meta-conformal transformations
greek : ‘µετ α’ of secondary rank transformationsM (t, r) whose Lie algebra is isomorphic to conf(2) Definition 2 : ortho-conformal transformations
greek : ‘o%θo’ right, standard tranformationsO(t, r) which are metaconformal and angle-perserving N.B. : ortho-conformal invariance = habitual conformal invariance
(A) Standard, ortho-conformal invariance at equilibrium label coordinates as ‘time’t and ‘space’r
in (1 + 1)D use complex variables z = t + ir and ¯z = t − ir
Extend global dynamical scaling to local, projective transformations z 7→ αz + β
γz + δ , ¯z 7→ α¯¯z + ¯β
¯
γ¯z + ¯δ , αδ − βγ = 1 , ¯α¯δ − ¯β ¯γ = 1 Transformation of primary scaling operators with ˙f (z0) ≥ 0
z = f (z0) , φ(z, ¯z) = df (z0) dz0
−∆
d¯f (¯z0) d¯z0
−∆¯
φ0(z0, ¯z0) with x = ∆ + ¯∆scaling dimension,s = ∆ − ¯∆ spin(‘usually’ s = 0)
at equilibrium, scalar φ has asinglescaling dimension x.
infinitesimal generators `n= −zn+1∂z− ∆(n + 1)zn
generators Xn= `n+ ¯`n andYn= `n− ¯`n spanconformal Lie algebraconf(2) [Xn, Xm] = (n − m)Xn+m, [Xn, Ym] = (n − m)Yn+m, [Yn, Ym] = (n − m)Xn+m
(C) InvariantSchr¨odinger operator(Laplacian) S = ∂t2+ ∂r2 = 4∂z∂¯z
[S, X−1] = [S, Y−1] = [S, Y0] = 0
[S, X0] = −S , [S, X1] = −2(z + ¯z)S − 8(∆∂¯z+ ¯∆∂z) [S, Y1] = −2(z − ¯z)S − 8(∆∂¯z− ¯∆∂z) Lemma :If Sφ = 0 and ∆ = ¯∆ = 0, then S(X φ) = 0. Cartan 1909
conf(d ) maps solutions of Sφ = 0 onto solutions.
Co-varianttwo-point correlator Polyakov 70
hφ1(t, r )φ2(0, 0)i = δx1,x2 t2+ r2−x1
= δx1,x2t−2x1
1 +r
t
2−x1
(P)
(B)Meta-conformal invariance, in (1 + 1)D mh 02
Xn = −tn+1∂t−(t + r )n+1− tn+1 ∂r− (n + 1)xtn− (n + 1)ξ [(t + r )n− tn] Yn = −(t + r )n+1∂r− ξ(n + 1)(t + r )n
Lie algebra hX±1,0, Y±1,0i is isomorphic to Lie algebraconf(2) (C)
[Xn, Xm] = (n − m)Xn+m, [Xn, Ym] = (n − m)Yn+m, [Yn, Ym] = (n − m)Yn+m
InvariantSchr¨odinger operator(ballistic transport)S = −∂t+ ∂r
[S, X0] = −S , [S, X1] = −2tS + 2(x − ξ) Lemma :If Sφ = 0 and x = ξ, then S(X φ) = 0.
Lie algebrahXn, Ynin∈Z maps solutions of Sφ = 0 onto solutions. scaling operator φ has scaling dimension x and rapidityξ.
co-varianttwo-point correlator : form distinct from (P) mh, Stoimenov 16
hφ1(t, r )φ2(0, 0)i = δx1,x2δξ1,ξ2|t|−2x1
1 + 1
ξ1
ξ1· r t
−2ξ1
(P’)
Comparison : ortho- vs meta-conformal two-point function
hφ(t1, r1)φ2(t2, r2)i =δx1,x2δξ1,ξ2(t1− t2)−2x1f r1− r2 t1− t2
f (u) = (
1 + u2−x1
ortho-conformal
(1 + |u|)−2ξ1 meta-conformal, here x1= ξ
x1= ξ1= 12
o%θo : of first rank, standard µετ α : of secondary rank
Lecture IV
Rappel :
Conformal invariance in 2D
? usecomplex coordinates z = t + ir,z = t − ir¯
? all analytic (or anti-analytic) transformations are conformal(= angle-preserving)
z 7→ w = f (z) , ¯z 7→ ¯w = ¯f (¯z)
? primary scaling operatorstransform as ∆, ∆areconformal weights
φ0(z, ¯z) = ∂w (z)
∂z
∆
∂ ¯w (¯z)
∂¯z
∆
φ(w (z), ¯w (¯z))
? quasi-primary scaling operatorsφ(z, ¯z)only transform co-variantly under projective transformationsz 7→ wp(z) = αz+βγz+δ, αδ − βγ = 1
? scaling dimension x = ∆ + ∆,spins = ∆ − ∆
Lie algebra generators `n= −zn+1 ∂∂z − ∆(n + 1)zn , obeyloop algebra [`n, `m] = (n − m)`n+m , `¯n, ¯`m = (n − m)¯`n+m , `n, ¯`m = 0
co-variance under projective transformations gives quasi-primary correlators hφ1(z1, ¯z1)φ2(z2, ¯z2)i = δ∆1,∆2δ∆
1,∆2z12−2∆1¯z12−2∆1 hφ1(z1, ¯z1)φ2(z2, ¯z2)φ3(z3, ¯z3)i = C123z12−∆12,3z23−∆23,1z31−∆31,2
×C123¯z12−∆12,3z¯23−∆23,1¯z31−∆31,2 hφ1(z1, ¯z1) . . . φ4(z4, ¯z4)i =
Y
i <j
zij−γij¯zij−γij
F (η, ¯η) wherezij := zi − zj,∆12,3:= ∆1+ ∆2− ∆3
C123is called anoperator product expansion (ope) coefficient(universal) P
iγij= 2∆j,η :=zz12z34
23z14
F cannot be found from projective transformations alone
possibility of several ‘variants’ of conformal invariance (‘ortho’, ‘meta’)
Finite-size scaling (fss)
Fisher & Barber 1971/72experiments often in finite-size geometry if L ξ ⇒ bulk behaviour
if ξ L⇒ finite-size effects
correlation length ξ of 2D Ising model, strip geometryL × ∞
? if L → ∞: divergence ξ ∼ |t|−ν
? if L < ∞ :
(1) singularityroundedto a finite maximum (2) shiftof lieu of maximum
(3) maximum increases withL
from thermodynamics, Gibbs free energy V =volume,A=surface, . . .
G (T , V , N ) = V g (T , ρ)
| {z } volume part
+ A g1(T , ρ)
| {z } surface part
+ . . . , ρ = N V
the surface part can have different singularities from those of the bulk
⇒surface critical phenomena
If T 6= Tc, these potentials should be given byMinkowski functionals
Mecke et al. 04
AMinkowski functional M : Ω → R,Ω ⊂ Rd is invariant under translations and rotations and isadditive, i.e.M(Ω1∪ Ω2) + M(Ω1∩ Ω2) = M(Ω1) + M(Ω2).
Theorem :Any invariant and additive functional f (Ω) is a finite sum over Minkowski functionals f (Ω) =Pd
n=0fnMn(Ω) Hadwiger 1957
the Mn in 2D : area, circumference, Euler characteristic
the Mn in 3D : volume, surface, mean integral curvature, Euler characteristic
Mecke & Wagner 1991
near a bulk critical point, observe the following finite-size effects
(1) shift of Tc : specific heats, susceptibilities, correlation lengths, . . . have maximum at pseudocritical temperatureTc(L):
Tc(L) − Tc(∞) ∼ L−λ , λ =shift exponent
Ni/W(110)
λ = 1.4(1)
(2) rounding : e.g. finite-size susceptibility χL(T ) ' χ∞(T ) if
|T − Tc| ≥ |T∗(L) − Tc|and
T∗(L) − Tc(∞) ∼ L−θ , θ =rounding exponent
(3) height of maximum : will follow from finite-size scaling theory . . .
Finite-size scaling hypothesis
finite-size scaling limit: T → Tc,L → ∞,z = L/ξ(T ) fixed in this limit, e.g. the susceptibility should have the form
χL(T ) = Lω¯Q(z) = L¯ ω¯Q
L1/νt
, t = T − Tc
Tc direct consequence : λ = θ = 1ν
consistency : forx → ∞ :Q(x ) ' q∞±|x|−ρ ⇒ χL(T ) ' q∞±Lω−ρ/ν¯ t−ρ
⇒ ρ = γ andω¯ = ρν = γν =⇒ att = 0, have χL∼ Lγ/ν bulk finite-size
specific heat c ∼ t−α Lα/ν susceptibility χ ∼ t−γ Lγ/ν correlation length ξ ∼ t−ν L1 Gibbs free energy gsin ∼ t2−α L−d magnetisation M ∼ tβ L−β/ν latent heat `h ∼ t1−α L−(1−α)/ν
efficient technique for extracting exponents from simulations precautions required to measure M, `h on finite lattices
Example : thin films of 4He Gasparini et al., Rev. Mod. Phys. 80, 1009 (2008)
schematic specific heat
breakdown of standard thermodynamics for T ≈ Tλ
cp raw data,t = 1 − T /Tλ> 0 scaling for T > Tλ scaling for T < Tλ
finite-thickness shift in Tλ
1 −TmaxT
λ ∼ L−1/ν, Tmax from max. of cp
ν = 1λ = 0.669(4)
Finite-size scaling in systems withO(n)- orPotts-q symmetry :shift exponent (a)few-monolayers magnets/liquid :
n Substance λexp λth= 1/ν 1 FeF2/ZnF2 1.56(10) 1.588
CoO/SiO2 1.55(5)
2 4He 1.4910(4) 1.495
1.495(9)
3 Ni/Cu 1.44(20) 1.417
Ni/W 1.4(1) Ni/Cu 1.32(14) Fe/Cr 1.4(3) Cr2O3 1.34(7)
(b) percolating resistor network/sub-monoloyer magnetisation :
q Substance λexp λth= 1/ν
1 nanowire Ag 0.75(2) 0.75
2 Fe(110)/W(110) 1.03(14) 1.00 nanowire Ni 0.94
the transitions 3D → 2D and2D → 1D have clearly separate exponents
in certain magnetic films one finds much larger effective shift exponents :
n ∆1 1/ν λth λexp system
1 0.508(7) 1.588(3) 3.20(3) 3.4(3) CoO/SiO2
2 0.533(8) 1.495(4) 3.09(3)
3 0.556(10) 1.417(7) 2.99(3) 3.15(15) Fe/Ir(100) 2.8(3) Gd(0001)
can be explained by taking the leading corrections to scaling into account :
∆1 = ων is the leading correction exponent
for magnets with up/down symmetry, the first order vanishes
⇒ go to 2nd order =⇒ λth = 1ν(1 + 2∆1) Phys. Rev. Lett. 80, 4783 (1997)
Without the magnetic up-down symmetry, the first order remains :
⇒ λ = ν1(1 + ∆1) = 2.38(4) for Ising 3D liquide/gaz.
In SF6, one finds λeff ≈ 2.5, Thommes & Findenegg, Langmuir 10, 4270 (1998)
‘apply fss to develop an industrially relevant design rule in nanowire films’
Large et al., Nanoscale 8, 13701 (16) Source : mh, Conformal Invariance and Critical Phenomena, Springer (1999), and refs. therein.
return to full finite-size scaling forms. Have seen that
g (t, h; L−1) = b−dg (bytt, byhh; bL−1) , ξi(t, h; L−1) = bξi(bytt, byhh; bL−1) L−1 acts as relevant scaling field, withyL = 1
recast as follows, which traces the non-universalmetric factorsC1,2
g (t, h; L) = L−dY
C1tL1/ν, C2hL(β+γ)/ν ξ−1i (t, h; L) = L−1Si
C1tL1/ν, C2hL(β+γ)/ν
where Y (x1, x2) andSi(x1, x2) areuniversal scaling functions
and the same metric factors C1, C2 in all quantities ! Privman & Fisher 1984
in 2D, use conformal invariance to express the universal constants Si(0, 0) andY (0, 0) in terms of universal exponents, . . .
for d > 4 : Berche, Kenna et al., Nucl. Phys. B865, 115 (12) ; Europhys. Lett. 105, 26005 (14) ; Eur. Phys. J. B88, 28 (15) . . .
Finite-size scaling in strips and conformal invariance
Cardy 1984non-projective conformal transformations change the geometry
thelogarithmic transformation w = 2πL ln z maps the complex plane C onto an infinitely long cylinder of circumference L
z = ρeiϕ ⇒ w = u + iv = 2πL (ln ρ + iϕ) ρ ≥ 0, 0 ≤ ϕ < 2π
periodic boundary conditions on the cylinder (or strip) are implied
Kober, Dictionary of conformal representations, British Admirality (1948) & Dover (N.Y. 1957)
transformation of aprimarytwo-point correlator w1= w (z1),. . .
hφ(z1, ¯z1)φ(z2, ¯z2)iz = dw1 dz1
dw2
dz2
∆ d ¯w1
d¯z1
d ¯w2
d¯z2
∆
hφ(w1, ¯w1)φ(w2, ¯w2)iw
in the plane, have hφ(z1, ¯z1)φ(z2, ¯z2)iz = (z1− z2)−2∆
from the logarithmic transformation z = exp(2πL−1w ) = exp(2πL−1(u + iv ))
hφ(w1, ¯w1)φ(w2, ¯w2)iw = 2π L
2∆+2∆
z11/2z21/2 z1− z2
!2∆
¯ z11/2¯z21/2
¯ z1− ¯z2
!2∆
= 2π
L
exp[πL(w1+ w2)]
exp(2πL w1) − exp(2πL w2)
!2∆
· 2π L
exp[πL( ¯w1+ ¯w2)]
exp(2πLw¯1) − exp(2πLw¯2)
!2∆
= π L
1
sinh[πL(w1− w2)]
2∆
· π L
1
sinh[πL( ¯w1− ¯w2)]
2∆
wherew1− w2 = (u1− u2) + i(v1− v2).
evaluation of this result :
(1) if |w1− w2| L: ⇒ recover the two-point function in the plane C.
(2) if |w1− w2| L:
take v1= v2, have |u1− u2| L, read off asymptotic exponential decay hφ(u1, 0), φ(u2, 0)istrip' 2π
L
2x
exp
−2π
L (∆ + ∆ )(u1− u2)
identify correlation length ξ, via hφ(u, 0)φ(0, 0)i ∼ e−u/ξ
Recall x = ∆ + ∆, read off final result Cardy 1984
in the periodic L × ∞ strip ξi−1= L−1(2πxi)=⇒Si(0, 0) = 2πxi
!THE basic result for the application of conformal invariance to 2D phase transitions !
quantum chains : ξi−1= Ei − E0 find Ei from diagonalising H
⇒ very efficient method to findall scaling dimensions !
?andY (0, 0) ?Must work much more & much harder . . .
? analogous results for dimensions d > 2 ?
? consider a L × L × ∞ column geometry
? correlations lengthsξσ,ξε of the spin-spin and energy-energy correlators The ratio of the critical finite-size scaling amplitudesis
Ξ := ξε(0, 0) ξσ(0, 0)
−1
= Sε(0, 0)
Sσ(0, 0) at T = Tc
Observation : for anti-periodic boundary conditions, find ΞA = xε/xσ for the 3D Ising andsphericaluniversality classes mh, J. Phys. A20, L769 (1987)
tested with increased precision in 3D O(n)-symmetric models
Weigel & Janke, Ann. Physik 7, 575 (1998) ; Phys. Rev. Lett. 82, 2318 (1999) ; Phys. Rev. B62, 6343 (2000)
n 1/Tc ΞP ΞA xε/xσ
1 0.2216544(3) 3.67(3) 2.736(13) 2.7264(13) 2 0.454167(3) 3.97(3) 2.93(5) 2.914(4) 3 0.693004(7) 4.248(9) 3.08(8) 3.089(8) 10 2.42792(8) 4.97(8) 3.65(6) 3.62(7)
∞ 2 2
good evidencefor ΞA = xxε
σ
hints forΞP = 43xxε
σ
N.B.purely numerical observation, no explanation known
The 2D energy-momentum tensor I
We already know : the energy-momentum tensorTµν must obey : (1) conservation∂µTµν = 0
(2) symmetry, from rotation-invariance Tµν = Tνµ
(3) tracelessness, from scale-invarianceTµµ= 0
in 2D, Tµν is a 2 × 2 matrix ⇒ 2 independent components! these are chosen as
T := Tzz = 12(T11− iT12) , T := T¯ ¯z ¯z = 12(T11+ iT12) the conservation law becomes in complex coordinates
∂¯zT = 0 , ∂zT = 0¯ =⇒ T = T (z) , T = ¯¯ T (¯z)
these are the Cauchy-Riemann conditions from complex analysis
T = T (z)is a holomorphic function of z and T = ¯¯ T (¯z)is an anti-holomorphic function of ¯z
Theorem : (Liouville) A holomorphic function f : C → C which is also bounded, i.e. |f (z)| ≤ M < ∞, must be a constant.
⇒ there are no non-constant ‘infinitesimal holomorphic functions’ ε(z) re-write Ward identity for n-point correlator hφ1(z1) . . . φn(zn)i:
choose contourC which surrounds pointsz1, . . . ,zn
interior domainD1 := int C :|ε(z)| 1holomorphic exterior domain D2:= C − D1 :|ε(z, ¯z)| 1
consider ‘infinitesimal’ coordinate transformation r 7→ r + ε(r), or z 7→ z + ε(z, ¯z), and recall 2D global Ward identity
δεhφ1. . . φni =
n
X
p=1
D
φ1(r1) . . .
ε(rp) · ∇p+xp
d ∇p· ε(rp)
φp(rp) . . . φn(rn)E
= −c2 Z
dr hφ1(r1) . . . φn(rn)Tµνi ∂µεν(r) c2= (2π)−1
