Part 8 Beyond Mean Field

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Perimeter Institute Lecture Notes on Statistical Physics part 8: Beyond Mean Field Theory Version 1.7 9/12/09 LeoKadanof 1

Part 8: Phase Transitions: Beyond Mean Field Theory

PeriodizationWorries about Mean Field Theory

fluid experiment 1fluid experiment 2earlier hintstheoretical workuniversalityturbulencemore rumbles

Next Stepstoward the revolutionadditional phenomenology

Block Spinless is the samerenormalization for Ising modelother renormalizationsfields relevant, irrelevant, ...

a worthwhile phenomenologyThe Wilson Revolution

particle RG before Wilsonthe renormalization revolutioncrucial ideas for the revolutionoutcome of revolutionconceptual advances


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After the Revolution2d XY modelRG point of view is absorbed into particle physics

Coulomb gasFeigenbaum and routes to chaosoperator product expansionfield theorysummary

Conformal Field TheoryPolyakov, Virasoro algebra

Friedan, Qiu, Shenkercorrelation functionsmany exact calculationsquantum gravity

SLESchrammcritical shapes

ensemble of critical shapesensemble of critical shapes

SummaryReferences


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Perimeter Institute Lecture Notes on Statistical Physics part 8: Beyond Mean Field Theory Version 1.7 9/12/09 LeoKadanof

Periodization

Early period: look at the whole phase diagram,glance at critical region. I talked about this in aprevious part. During this period mean fieldtheory was developed.

Period of unrest. focus on critical region. meanfield theory is not working, what to do? Developphenomenological theory. Ill talk about this now.

The Revolution: Wilson put forwardrenormalization group theory of critical

phenomena

Afterward: various mathematical/theoretical

expressions and extensions of theory.

3

1860 to 1937

1937 or 1963 to 1971

1971

1971-now


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A worry?

Mean field theorygivesand = 1/2This power is,however, wrong.Experiments arecloser to

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M~ (TcT)

M (Tc T)1/3 in 3-D

order parameter: density versusTemperature in liquid gas phasetransition. AfterE. A. GuggenheimJ.Chem. Phys. 13 253 (1945)

1880-1960: No oneworries much aboutdiscrepancies


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Perimeter Institute Lecture Notes on Statistical Physics part 8: Beyond Mean Field Theory Version 1.7 9/12/09 LeoKadanof5

vapor liquid


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Earlier Hints of Trouble

van der Waals had gotten something very much like = 1/3 fromanalysis ofAndrews data in the critical region. This worried him.He did not know what to do. He had no theory or model whichgave anything like this. He believed that this was important, but hehad no place to go with it.

Gugenheim was ignored.

6


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Theoretical workOnsager solution (1943) for 2D Ising model gives infinity

in specific heat and order parameter index =1/8(Yang)--contradicts Landau theory which has =1/2.Landau theory has jump in specific heat but no infinity.

Kings College school (CyrilDomb, Martin Sykes,Michael Fisher, .... (1949- )) calculates indices using seriesexpansion method. Gets values close to =1/8 in two

dimensions and =1/3 in three and not the Landau\vander Waals value, =1/2. They emphasize that mean fieldtheory is incorrect.

Kramers recognizes that phase transitions require aninfinite system. Mean field theory does not require an

infinite system for its phase transitions.Still Landaus no-fluctuation theory of phase transitionsstands.

L. Onsager, Phys. Rev.65 117 (1944)

C.N, Yang, Phys. Rev.85 808 (1952)

C. Domb, The CriticalPoint, Taylor andFrancis, 1993


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Universality

Van der Waals and Uhlenbecklook for universal theory applying to

entire phase diagram of many different fluids. This desire ismisdirected. Experimental work on fluids shows that it is wrong. Thisconclusion also arises from theories of the Kirkwood school ofphysical chemists, which produces a non-universal theory of linkedcluster expansion for fluids.

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In the meantime, hints of another universality-- one near the criticalpoint-- arises through work of Landau on mean field theory, whichproves incorrect, but also through the numerical work of the KingsCollege School which sees different critical points on different lattices

as quite alike. Brian Pippard brings this up in a more theoretical vein.These studies give hope of a universal behavior in critical region.Universality is a most attractive idea, drawing people to the subject.


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Turbulence Work advances scaling ideas

Kolmogorov theory (1941) uses a meanfield argument to predict velocity in cascadeof energy toward small scales.

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Result: velocity difference at scale r behaves as

(N.B. One of first Scaling theories) Landaucriticizes Ks work for leaving out fluctuations.Kolmogorov modifies theory (1953) byassuming rather strong fluctuations in velocity.

Landau & Lifshitz worry in print about Landautheory of phase transition.

v(r) (r)1

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Still Landaus no-fluctuation theory of phasetransitions stands

Andrey Nikolaevich Kolmogorov


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More rumbles before the revolution

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US NBS conference 1965 helps people recognize that criticalphenomena is a subject .

Focus changes: Dont look at the entire phase diagram, examine onlythe region near the critical point.

Get a whole host of new experiments,

embrace a new phenomenology,

Moldover and Little (1966)

Significant earlier work: Voronel et. al (1960)specific heat of near-critical Argon.

superfluid transition helium, Kellers PhDthesis (1960) Stanford.

In each case, mean field theory saysspecific heat should remain finite but havea discontinuous jump at critical point. Datasuggests an infinity at critical point.


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Kadanoff(1966) .....

Toward the revolution

Ben Widom noticed the most significantscaling properties of critical phenomena,but did not detail where they might havecome from. B. Widom, J. Chem. Phys.433892 and 3896 (1965).

.

The phenomenology

Widom 1965: scaling result F(t,h)=Fns + td f*(h/t):He focuses attention on scaling near critical point. In this region, averages andfluctuations have a characteristic size, for examplefree energy ~ (coherence length)-d , i.e. there is a free energy of kT per coherencevolume

magnetization ~ (-t)

when h=0magnetization ~ (h)1/ when t=0 implies h ~ (-t) implies=therefore scaling for free energy

and magic relations e.g. 2- = d


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additional phenomenologyPokrovsky & Patashinskii study correlation functions, build uponWidoms work

They have scaling ideas (r)~1/rx

orders of magnitude from field theory gives

< (r1) (r2).....(rm)> ~ 1/rmx

1964, 1966: This is a good idea but produces a partially wronganswer x 3/2. (It is actually close to 1/2.)

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Before Widom, Michael E. Fisher introduces scaling ideas, and the two basicindices in his 1965 paper in the University of Kentucky conference on phasetransitions. He bases his approach upon an insightful view of droplets of the

different phases driving the thermodynamics. However, he misses therelation between correlation length and thermodynamics.

Kadanoff(1966) .....


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Kadanoff 1966

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Kadanoffconsiders invariance properties ofcritical point and asks how descriptionmight change if one replaced a block of

spins by a single spin, changing the lengthscale, and having fewer degrees of freedom.

Answer: There are new effective values of(T-Tc)=t, magnetic field=h, and free energyper spin K0. These describe the system justas well as the old values. Fewer degrees offreedom imply, new couplings, but no changeat all in the physics. This result incorporatesboth scale-invariance and universality. Thisapproach justifies the phenomenology of

Widom and provides the outlines of atheory which might give the correlationfunctions ofPataskinskii & Pokrovsky.

Less is the same, too.

fewer degrees of freedom

produces block

renormalization


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Renormalization for Ising model in any dimension; reprise

fewer degrees of freedomproduces blockrenormalization

Z=Trace{} exp(WK{})

Each box in the picture has in it a variable called R,

where the Rs are a set of new lattice sites withnearest neighbor separation 3a. Each new variable istied to an old ones via a renormalization matrixG{,}= g(R,{}) where g couples the R to

R

the s in the corresponding box. We take each R to

be 1 and define g so that, g(,{s}) =1.

Notice that Z=Trace{} exp(W{})

If we could ask our fairy god-mother what we wished for now, it would bethat we came back to the same problem as we had at the beginning: W{}=WK{} where the subscript represents the three relevant couplings inthe system: t , h, K0.

Now we are ready. Define exp(W{})= Trace{} G{,} exp(WK{})


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Transformations: a--> 3a =a WK{}--> WK{} Z=Z K=R(K)

Scale Invariance at the critical point:--> Kc=R(Kc)

Temperature Deviation:K=Kc+t K=Kc+tcritical point: ift=0 then t=0

coexistence (ordered) region: if (t0, h=0)

ift is small, t=bt. b=(a/a)x definesx=xt = critical index fortemperature. b>1 implies motion away from critical point

near critical point, h=bh h ln bh =xh ln (a/a) definesxh. which

then describes renormalization of magnetic field.

bs can be found through a numerical calculation.

Renormalization


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Free energy: F = ....+Nfc(t)= F= ...+Nfc(t)fc(t) = fc0 tdx

Specific heat: C=d2F/dt2~ tdx-2 = t- form of singularity determined by x

coherence length: =0 a t- 2d Ising has =1; 3d has 0.64....

= 0 a t- = 0 a(t)-so =1/x

number of lattice sites: N =/ad N=/ad

N/N= ad /ad = (a/a)-d

Other Renormalizations


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Fields: Relevant, Irrelevant, marginal, ....

there are a few relevant fields: like t,h, K0, whichgrow at larger length scales,

completely dominate large-scale behavior

there are many irrelevant fields: they have |b|


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A Worthwhile Phenomenology

Weaknesses of phenomenology:i. One cannot calculate everything: value of xs unknown

ii. One cannot be sure about what parts of theory are right, whatparts wrong.

iii. cannot determine universality classes from theory

iv. Does not provide lots of indication of what one should do tomake the next step.

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Particle Physics RG before Wilson

idea: masses, coupling constants, etc. in Hamiltonian description ofproblem different from observed masses, coupling constants, etc. .They change with distance scale as particles are dressed byeffects of the interaction.

A good phenomenological idea, used in quantum electrodynamicsbut not really crucial to particle physics.

Not used in statistical physics.

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The Renormalization Revolution:

Kenneth G. Wilsonsynthesizes new theory

Wilson converts a phenomenology into a calculationalmethod.

Instead of using a few fields (T-Tc, h), he conceptualizesthe use of a whole host of fields K =R(K)

Adds concept of fixed point Kc =R(Kc)

He considers repeated transformations

This then provides a method which can be used by manypeople for many real calculations. The most notable is theepsilon (4-d) expansion of Wilson and Fisher.

This theory permits one to calculate everything peoplewanted to know at the time and fully matchesexperiments.

Everything in critical phenomena seems to be explained.

* In both condensed matter and particle physics.


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The Crucial Ideas-for the revolution

Ideas: Criticality: recognized as a subject in itself Scaling: Behavior has invariance as length scale is changed

Universality: Expect that critical phenomena problems can bedivided into different universality classes

Running Couplings: Depend on scale. Cf. standard modelbased on effective couplings ofLandau & others.

effective fields of all sorts: Running couplings are but oneexample of this.

Fixed Point: Singularities when couplings stop running. K. Wilson

Renormalization Group:K. Wilson (1971), calculational methodbased on ideas above.

Each Item, except RG, is a consensus product of many minds


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The Outcome of RevolutionExcellent quantitative and qualitative understanding of

phase transitions in all dimensions. Information about

Universality Classes

All problems divided into Universality Classes based upon

dimension, symmetry of order parameter, ....

Different Universality Classes have different critical

behavior

e.g. Ising model, ferromagnet, liquid-gas are in same class

XYZ model, with a 3-component spin, is in different class

To get properties of a particular universality class you need

only solve one, perhaps very simplified, problem in that

class.

moral: theorists should study simplified models. Theyare close to the problems we wish to understand


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Conceptual Advances

First order phase transition represent a choice amongseveral available states or phases. This choice is made by theentire thermodynamic system.

Critical phenomena are the vacillations in decision making asthe system chooses its phase.

Information is transferred from place to place via localvalues of the order parameter.

There are natural thermodynamic variables to describe theprocess. The system is best described using these variable.

Each variable obeys a simple scaling.

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Next: After the Revolution:


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2D XY model-

Kosterlitz & Thouless (1973)

The XY model is a set of two-dimensionalspins, described by an angle and a nearest

neighbor couplingK cos ( - ).

There is an elegant description in terms offree charges and monopoles with

electromagnetic interactions between them.

Thisworkisalsoimportantbecauseitisa

successfulcalculationinvolvingtopological

excitations. Milestone.

s= (cos , sin ),


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The RG point of view is Fully Absorbedinto Particle Physics

Running coupling constants help define the

standard model.

The model is extrapolated back to when weak,

electromagnetic and strong interactions are all

equal.

Asymptotic freedom--weakening of strong

interactions at small distances--permits high

energy calculations. Gross, Wilczek, Politzer

(1973).

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Feigenbaum used RG, Universality,and Scaling concepts to investigatethe period doubling route to chaosvia the discrete equation

x(t+1) =r x(t) (1-x(t))

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Feigenbaum: Analyzes Route to Chaos (1978)

long-term x-values versus r


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Operator Product Expansion: Wilson, Kadanoff

Local operators proved particularly interestingProducts of nearby operators O(R+r/2)O(R-r/2)

can be expanded in terms of the local operators at R.

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O(R+ r/2)O(R+ r/2)

=

A(r)O(R)

This operator product expansion particularly suggests that theoperators obey a kind of algebra. Working from the algebra,something deeper might be found.


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Field TheoryPre-Revolutionary Period:

Onsager solution of 2-D Ising model is free fermion theory

Revolutionary Period:

The connection: Wilson & Fisher(1972) use Ginzburg Landau freeenergy formulation plus path integral formulation of quantum

mechanics to describe critical phenomena theory as a closerelative of quantum field theory.

After the Revolution: Polyakov and others reformulate criticalpoint theory as Conformal Field Theory, a field theory forsituations invariant under scale transformations but not shear-typedistortions


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After the Revolution:New Ideas-mostly for d=2

XY model- Kosterlitz & Thouless

Coulomb gas: B. Nienhuis

Conformal Field Theory: A. Polykov Quantum Gravity: B. Duplantier

SLE: Oded Schramm


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Conformal Field Theory: IA. Polyakov emphasized that there is a special form of fieldtheory which holds at critical points, i.e. places in which there

is full scale invariance. In two dimensions this is super-specialbecause the invariance includes all kinds ofconformal (anglepreserving) transformations which can then be studiedthrough the use of complex variable methods.

Space distortions are based upon stress tensor operators,with theVirasoro algebra being the algebra of local stresstensor densities. Just as spinors, vectors and tensors arederived as representations of the rotation group algebra,equally the local operators of critical phenomena haveproperties, including critical indices, derivable from the fact

that they are representations of the Virasoro algebra.

Continuously varying families of solutions are generated inthis fashion


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Conformal Field Theory: II

Friedan Qiu & Shenker(1984) showed that

unitarity, a quality of all field theoriesrepresenting possible quantum processes,

limits the domain of field theories to include

all the known critical 2-D models, e.g. q=2,3,

4 Potts models, but not others (e.g q=1.5).Further we get a quite different algebra for

each model.

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Friedan Qiu & ShenkerPRL 52 1575 (1984)

For monographmaterial seeDiFrancesco,Mathieu, SenechalConformal Field

Theory, Springer,1997


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Conformal Field Theory: III

Compared to general situations, all thefamiliar statistical mechanical models are alldegenerate and truncated, and permitconsiderable calculation of correlationfunctions.

Further conformal transformation permitthe calculation of correlations in all kinds ofshapes from just a few shapes: plane, half-plane, interior of circle.

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Many Exact Calculations

For Ising model, x=1/8

and .= 134

r1

r2

< (r1)(r2)(r3) > = Cr12x2x(r23r13)x

r3

x

find correlation of two spins(local magnetization densities)and a local energy density


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Quantum Gravity:

The next step beyond scale invariance is no scale at all. One canformulate quantum gravity theory as a classical theory in whichone sums over all possible metrics on a given space.

In two dimensions, there are no physical degrees of freedom ingravity theory, and the summation can be carried out exactly.(Gross& Migdal, Douglas & Shenker, Brezin & Kazakov.) In addition to being a solvablegravity model, this approach offers a good start for criticalproblems. For example B. Duplantier carried out a calculation inwhich he calculated the spectrum of electric fields in theneighborhood

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SLE=Schramm-Loewner-Evolution

Conformal field theory took us to a

point at which we could formulatecritical phenomena problems on

surfaces of various shapes. The most

recent area of progress arises from

work of Oded Schramm, who combinedthe complex analytic techniques of

Loewner with methods of modern

mathematical probability theory to

gain new insight into the shapes which

arise in critical phenomena.

O. Shramm, Israel J. Math. 118 (2000), 221--288.

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Oded Schramm


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From Schramm to Critical Shapes

The work of the 20th. Century oncritical phenomena was centeredon thermodynamics, andcorrelation functions.

We depicted but did not calculate

the shapes of the correlatedfractal clusters arising in criticalsituations. Schramm provided aconstructive technique, involving adifferential equation, for making

the ensemble of such clusters atcritical points.

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after Duplantier


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SLE-IIDefine a critical cluster as the shape

of cluster of spins pointed in thesame direction in an Ising model, orof a connected set of occupied sitesin a percolation problem.

Problem: Define the ensemble of

cluster shapes for any criticalsituation. Before Schramms workwe ignored this aspect of criticalproblems.

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percolation cluster after Duplantier


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Problem: Define the ensemble of cluster shapes for any criticalsituation.

Answer: Look for the ensemble of

shapes formed from thesingularities in the solution to thedifferential equation

where is a random walk on aline with correlations

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after Duplantier


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SummaryCritical behavior occurs at but one point of thephase diagram of a typical system. It is anomalousin that it is usually dominated by fluctuationsrather than average values. These two factsprovide a partial explanation of why it took untilthe 1960s before it became a major scientificconcern. Nonetheless most of the ideas used inthe eventual theoretical synthesis were generated

in this early period.

Around 1970, these concepts were combinedwith experimental and numerical results toproduce a complete and beautiful theory of

critical point behavior.In the subsequent period the revolutionarysynthesis radiated outward to (further) informparticle physics, mathematical statistics, variousdynamical theories....

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JW Gibbs


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References

Cyril Domb, The Critical Point, Taylor and Francis, 1993

L.P. Kadanoff, W. Gotze, D. Hamblen, R. Hecht, E.A.S. Lewis, V.V.Palciauskas, M. Rayl, J. Swift, D. Aspnes, and J.W. Kane, Static

Phenomena Near Critical Points: Theory and Experiment, Rev. Mod Phys.39 395 (1967).

G Falkovich & K. Sreenivasan, Lessons from Hydrodynamic Turbulence,preprint (2005).

Debra Daugherty, PhD Thesis, University of Chicago, in preparation.Contains ideas about Ehrenfest and Landau.

Critical Phenomena, Proceedings of a Conference, National Bureau ofStandards, 1965.

A. Einstein, Annals of Physics Leipzig 33, 1276.

More references


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More references

M. Smoluchowski, Ann. Phys. Leipzig 25, 205 (1912).

L.S. Ornstein and F. Zernike, Proc, Acad Amsterdam 17 793 (1914).

E.C.G. Stueckelberg, A. Peterman (1953): Helv. Phys. Acta, 26, 499. Murray Gell-Mann, F.E. Low (1954): Phys. Rev. 95, 5, 1300. The origin of

renormalization group N.N. Bogoliubov, D.V. Shirkov (1959): The Theory of Quantized Fields. New

York, Interscience. The first text-book on the renormalization group method. L.P. Kadanoff (1966): "Scaling laws for Ising models near Tc", Physics (Long IslandCity, N.Y.) 2, 263. The new blocking picture.

C.G. Callan (1970): Phys. Rev. D 2, 1541.[2] K. Symanzik (1970): Comm. Math.Phys. 18, 227.[3] The new view on momentum-space RG.

K.G. Wilson (1975): The renormalization group: critical phenomena and theKondo problem, Rev. Mod. Phys. 47, 4, 773.[4] The main success of the newpicture.

S.R. White (1992): Density matrix formulation for quantum renormalizationgroups, Phys. Rev. Lett. 69, 2863. The most successful variational RG method.
http://prola.aps.org/abstract/RMP/v47/i4/p773_1http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1103842537http://prola.aps.org/abstract/PRD/v2/i8/p1541_1http://en.wikipedia.org/wiki/Nikolay_Bogolyubovhttp://en.wikipedia.org/wiki/Nikolay_Bogolyubovhttp://en.wikipedia.org/wiki/Nikolay_Bogolyubovhttp://en.wikipedia.org/wiki/Dmitry_Shirkovhttp://prola.aps.org/abstract/RMP/v47/i4/p773_1http://prola.aps.org/abstract/RMP/v47/i4/p773_1http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1103842537http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1103842537http://prola.aps.org/abstract/PRD/v2/i8/p1541_1http://prola.aps.org/abstract/PRD/v2/i8/p1541_1http://en.wikipedia.org/wiki/Dmitry_Shirkovhttp://en.wikipedia.org/wiki/Dmitry_Shirkovhttp://en.wikipedia.org/wiki/Nikolay_Bogolyubovhttp://en.wikipedia.org/wiki/Nikolay_Bogolyubovhttp://en.wikipedia.org/wiki/Murray_Gell-Mannhttp://en.wikipedia.org/wiki/Murray_Gell-Mann

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