A Brief Introduction to the Renormalization Group

Dimitri D. VvedenskyThe Blackett Laboratory, Imperial College London

Christoph A. HaselwandterDepartment of Physics, CalTech

1. Origin of RG: Discovery that continuous phase transitions and quantum fieldtheory are connected on a very fundamental level (1970s, K. G. Wilson): Essential property is the large number of degrees of freedom, details of underlying physics irrelevant.

2. RG is not a descriptive but a constructive physical theory. The RG is a general method for constructing theories at different scales.

3. Starting point is a physical description formulated at a given scale. RG thenprovides qualitative information about the system at arbitrary scales, and allow the calculation of quantities involving many degrees of freedom. Thus, using the RG one can

• construct descriptions appropriate for different scales.

• relate behavior at different scales in a quantitative fashion.

• understand how collective properties of systems arise out of theirconstituent elements.

What is the Renormalization Group?

1. K. G. Wilson, Problems in Physics with Many Scales of Length, in Scientific American (August 1979), p. 158.

2. K. G. Wilson and J. Kogut, Phys. Rep. 12, 75 (1974). K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975). K. G. Wilson, Rev. Mod. Phys. 55, 583 (1983).

3. D. J. Wallace and R. K. P. Zia, Rep. Prog. Phys. 41, 1 (1978).

4. D. J. Amit and V. Martin-Mayor, Field Theory, the Renormalization Group, and Critical Phenomena (World Scientific, Singapore, 3rd ed., 2005).

5. S.-K. Ma, Modern Theory of Critical Phenomena (Benjamin/Cummings PublishingCompany, Reading, 1976).

6. My course notes (available upon request – d.vvedensky@imperial.ac.uk)

Background References

Everyday life is built upon a separation of length and time scales – it revolves around objects that have the essential property of stability towards internal & external perturbations. Similarly, science relies on a separation of scales: e.g.

1. Thermodynamics is independent of the existence of atoms.

2. Cosmological descriptions based on general relativity incorporate only gravity and none of the other forces.

3. Atoms can be described to unprecedented accuracy on the basis of quantum mechanics, ignoring gravity altogether.

4. Quantum mechanics is the large-scale limit of…?

A successful strategy in science is to partition reality into essentially independent “worlds” associated with different scales. Thus, we can formulate theories for a given scale, without recourse to smaller or larger scales..

Separation of Scales

Separation of Scales – Powers of 10

1. What is the basis for the divide and conquer approach of science? Distinguish between two different kinds of phenomena:

• Correlation length x small: Split system into independent subsystems, traditional divide and conquer approach applies, physics up to 20th century.

• Correlation length x large: System cannot be reduced, collective behavior,many degrees of freedom, many body problems.

2. x large ↔ many different scales are interrelated. RG developed to deal with situations in which x is large. RG can be seen as a “meta-theory” which connects theories at different scales.

3. Origin of name “renormalization-group” lies in quantum field theory, and the hope that fundamental physics reduces to group theory. Not very helpful to think of RG as a group in mathematical sense.

RG = set of physical ideas + vast array of methods for calculations. Qualitative picture of RG is generic; quantitative success of RG depends on methods used and problem at hand.

Origin of Renormalization Group

Theories of nature are only defined at specific scales. As a result, all parameters in the theory are inherently scale-dependent.

RG analysis: Resolution of theory is successively decreased such that large-wavelength properties unchanged, short-wavelength properties are absorbed into effective large-wavelength interactions.

Outcome of RG analysis: Set of equations which describe how the parameters change with scale. Typically new parameters are generated.

Fixed points: For large-scale features often sufficient and much simpler to calculate with fixed-point version of theories. Especially true for phenomena with large x.

RG flow/RG trajectory: Flow of theory under scale changes in the space of parameters associated with all interactions.

RG flow provides complete description of multiscale properties. But much can already be deduced from fixed point structure of parameter space and basic form of RG flow.

Essential Elements of the Renormalization Group

C. H. K. Williamson, Cornell (Source: http://www.efluids.com)

• Free gliding of delta-wing in water

• Fluorescent dye illuminated by laser

• Vortices in near field

• Turbulence in far field

• Both panels have same scale

• Energy cascades from large to small scales

Multiscale Physics: Turbulence

F. F. Abraham, D. Brodbeck, R. A. Rafey, andW. E. Rudge, Phys. Rev. Lett. 73, 272 (1994).

• 2D simulation of 106- atom system

• Bond-breaking at crack tip

• Dislocation emission blunts crack tip

• Feed-back between atomic and continuum modes

Multicsale Physics: Fracture

J. D. Noh, Chungham National University, Korea

Multiscale Physics: Critical Phenomena in the 2D Ising Model

(Courtesy, M. Scheffler)

The Multiscale Paradigm

1. The RG is a theory of coarse-graining which can be used to construct descriptionsappropriate for different scales.

2. RG theory is a set of generic qualitative ideas coupled with a vast array of analyticand numerical methods for doing specific calculations.

3. In a RG analysis, one absorbs small-scale degrees of freedom into effective largerscale interactions, and rescales the system to allow the iteration of this procedure.

4. RG was developed to bring about an understanding of phenomena with largecorrelation lengths (many degrees of freedom). In particular,

• Understand why divergences occur in quantum field theory (i.e. why parameters like the electron mass are scale dependent).

• Understand universality and scaling in continuous phase transitions (systems with infinite correlation lengths are at “fixed points” of the RG).

Some General Features of the Renormalization Group

Step 1: “Integrate out ” or “thin out” degrees of freedom associated with small scales, and absorb contributions into effective interactions between the remaining degrees of freedom. Real-space RG: Elimination of small-wavelength degrees of freedom is carried out in real space, thinning out amounts to summations in the case of lattice systems. Momentum-space RG: Integrate over large-k Fourier components.

Step 2: Restore the original range of the physical quantities in terms of which the description of the system is cast: “rescaling”. For instance, thinning out of degrees of freedom can increase the spacing between lattice sites by a factor b = 2 to 2a, where a is the original lattice spacing. Original range of spatial variable x, a≤x≤∞, is restored by replacing x by bx.

Result of RG procedure: A renormalized system which has the same long-wavelength properties as the original system, but fewer degrees of freedom.

The Two Steps of the Renormalization Group

Renormalization Group Transformations

Original system Coarse-grained Rescaled

Realspace

Momentumspace

Real-Space Renormalization of the 2D Ising Model

The Hamiltonian:

The partition function:

Elimination of degrees of freedom by “decimation”:

2D Ising Model: Decimation and Rescaling

Original system After decimationAfter renormalization

(decimation and rescaling).Additional lines between

spins represent newinteractions.

Summation over alternate spins(decimation)

Rescale by b=√2 and rotate by π/4

Decimation: The Nuts and Bolts1. Sum over alternate spins in the partition function.

2. What is the form of the renormalized Hamiltonian? Pick an arbitrary spin (s5, say) and evaluate its effect on the partition function:

Renormalized H should ideally have the same form as the original H, but

Produce different values of

3. What interactions are generated by decimation?• Sampling over s5 mediates interactions between s1, s2, s3, and s4.• Assume the presence of nearest and second-nearest nearest-neighbor

interactions in the renormalized system• State of s1 influenced by state of s2, s3, and s4, which implies the

presence of 4-spin interactions

Decimation: The Nuts and Bolts (cont’d)

• The RG procedure is completed by relabelling the lattice sites and rescaling the

lattice spacing by a factor b=√2.

• The generation of new interactions preempts the generalization of theresulting recursion relations to RG equations – each successive RGtransformation generates longer-range interactions.

• New interactions are generated due to the higher connectivity of the 2Dlattice, and this is the usual state of affairs.

• The real-space RG scheme proceeds by making approximations based onintuition. Only consider nearest-neighbor interactions? Ignoring all

interactions apart from K1 predicts the same RG flow as for the 1D Isingmodel (i.e., no phase transition). But next-nearest neighbors H’ in arenearest-neighbors in H.

Inclusion of Higher Connectivity

• To take into account correspondence between nearest-neighbor and next-nearest

neighbor interactions, increase K1 to a function K of K1 and K2.

• In a cubic 2D system with N/2 sites, there are N nearest-neighbor bonds and N

next-nearest-neighbor bonds (these are pair-bonds). Total energy of perfectly aligned configuration is –NK1–NK2

• To allow for the increase in cooperative behavior due to increased connectivity,

choose K(K1,K2) such that same energy is obtained for aligned state: K=K1+K2

(for non-aligned states this amounts to only a very crude approximation).

Renormalization Group Flow of the 2D Ising Model

Kc

K = 0 is a stable fixed point

K ∞ is a stable fixed point

K = Kc is an unstable fixed point.

Kc corresponds to the criticalpoint. The critical value of

Kc = 0.50698

is close to the exact result of 0.44069 first obtained by Onsager.

Physical Interpretation of RG Transformations

Real-space RG conceptually transparent, but has several shortcomings:

1. Precise form of real-space RG transformation must be found from physical intuition based on specific properties of system – physical predictions independent of specific implementation

2. Real-space RG involves finite scale changes by b > 1 – physical predictions independent of value of b.

3. Which interactions generated under the real-space RG are relevant, and which interactions are irrelevant

Momentum-space RG is a more systematic approach, which provides aformulation of the RG for general systems:

4. Momentum-space RG can be applied to continuous systems formulated in terms of general functions of x and t (lattice systems can often be recast as continuous systems and vice versa).

5. In momentum-space RG, scale changes can be infinitesimal.

6. Letting the dimension of the system take non-integer values, one can make systematic statements about relevance or irrelevance.

Real- versus Momentum-Space Renormalization

Momentum-space RG is complementary to real-space RG – real-space RG suitable for numerical calculations, momentum-space RG allows analytic studies of RG flows and fixed points.

Several mathematical formulations of the momentum-space RG have been developed: Integration over finite or infinitesimal momentum shells. Formulation based on functional calculus, dimensional regularization, counterterms,...

Quantum field theory: Extensive mathematical framework for doing RG calculations, connections to deep mathematical problems. Relation to other fields often not obvious!

Statistical field theory: Analogous methods to quantum field theory, but very different physics. Continuous phase transitions.

Nonequilibrium physics: Time-dependent and stochastic systems (“dynamic RG”), mapping to field theory possible, intuitive formulations based on Feynman graphs.

Momentum-Space Renormalization

Most common formulation of quantum field theory is in terms of functional integrals: Infinite number of degrees of freedom of quantum field are the variables of integration.

Central quantity in functional formulation is the generating functional, which is analogous to the partition function in statistical mechanics:

This is the continuous formulation of the Ising model. The correspondence betweenMagnetic and quantum field systems is

Continuous Formulation of the Ising Model