Transcript - Lorentz Center

A Brief Introduction to the Renormalization Group
Dimitri D. Vvedensky
The Blackett Laboratory, Imperial College London
Christoph A. Haselwandter
Department of Physics, CalTech
What is the Renormalization Group?
1.
Origin of RG: Discovery that continuous phase transitions and quantum field
theory 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 then
provides 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 their
constituent elements.
Background References
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 – [email protected])
Separation of Scales
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.
1. 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 – Powers of 10
Origin of Renormalization Group
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.
Essential Elements of the 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 largewavelength 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.
Multiscale Physics: Turbulence
• 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
C. H. K. Williamson, Cornell (Source: http://www.efluids.com)
Multicsale Physics: Fracture
• 2D simulation of 106- atom
system
• Bond-breaking at crack tip
• Dislocation emission blunts
crack tip
• Feed-back between atomic
and continuum modes
F. F. Abraham, D. Brodbeck, R. A. Rafey, and
W. E. Rudge, Phys. Rev. Lett. 73, 272 (1994).
Multiscale Physics: Critical Phenomena in the 2D Ising Model
J. D. Noh, Chungham National University, Korea
The Multiscale Paradigm
(Courtesy, M. Scheffler)
Some General Features of the Renormalization Group
1. The RG is a theory of coarse-graining which can be used to construct descriptions
appropriate for different scales.
2. RG theory is a set of generic qualitative ideas coupled with a vast array of
analytic
and numerical methods for doing specific calculations.
3. In a RG analysis, one absorbs small-scale degrees of freedom into effective larger
scale interactions, and rescales the system to allow the iteration of this
procedure.
4. RG was developed to bring about an understanding of phenomena with large
correlation 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).
The Two Steps 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.
Renormalization Group Transformations
Original system
Real
space
Momentum
space
Coarse-grained
Rescaled
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 decimation
Summation over alternate spins
(decimation)
After renormalization
(decimation and rescaling).
Additional lines between
spins represent new
interactions.
Rescale by b=√2
and rotate by π/4
Decimation: The Nuts and Bolts
1.
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 the
resulting recursion relations to RG equations – each successive RG
transformation generates longer-range interactions.
•
New interactions are generated due to the higher connectivity of the 2D
lattice, and this is the usual state of affairs.
•
The real-space RG scheme proceeds by making approximations based on
intuition. Only consider nearest-neighbor interactions? Ignoring all
interactions apart from K1 predicts the same RG flow as for the 1D Ising
model (i.e., no phase transition). But next-nearest neighbors H’ in are
nearest-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
K = 0 is a stable fixed point
K
∞ is a stable fixed point
K = Kc is an unstable fixed point.
Kc corresponds to the critical
point. The critical value of
Kc
Kc = 0.50698
is close to the exact result of
0.44069 first obtained by Onsager.
Physical Interpretation of RG Transformations
Real- versus Momentum-Space Renormalization
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 a
formulation 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.
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.
Continuous Formulation of the Ising Model
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 between
Magnetic and quantum field systems is