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Second order phase transitions
- Subject: Second order phase transitions
- From: Arun Gupta <gupta@tlctest.att.com>
- Date: 13 Aug 1999 00:00:00 GMT
- Approved: baez@math.ucr.edu
- Newsgroups: sci.physics.research
- Organization: ATT
- References: <7opkoi49ドル@charity.ucr.edu> <7oqcp5$prg@charity.ucr.edu>
Nathan Urban wrote:
>John Baez wrote:
>> (In physics jargon, one wants a 2nd-order phase transition, [...])
>Why?
Let us do statistical mechanics on a lattice. We have Wick-rotated our
space-time, so that computing QFT path integrals has turned into computing
partition functions. We have discretized our space because of all the
mathematical problems we have found doing QFT in continuous space-times.
For example, we could have a cubical lattice, with a spin residing at
each vertex in the lattice. The spins take values +1 and -1, and
neighboring spins interact. An Hamiltonian describes the spin interactions,
summed up over all the lattice vertices. For instance, there may be a
next-neighbor interaction of the form : M spin(n) spin(n+1) where
M is a positive number, and spin(n) and spin(n+1) are the values of
the spin on sites n and n+1 on the lattice. In this instance, we get
a positive addition to the energy when the two spins are the same.
The magnitude of this is determined by M, which is a coupling constant.
Likewise, you can imagine more complicated interactions involving more
spins and sites on the lattice.
Given this, we can compute all kinds of statistical mechanical properties
of this model. For example, we can compute a correlation function :
given that a spin at site X has the value +1, what is the likely value
for some given values of coupling constants, of the spin at site Y ?
Conceptually, we compute this by examining each configuration and doing
a Boltmann-factor weighted average. The coupling mentioned above tends
to make neighboring spins take on opposite values. But we expect that
far away from X, the spin is about equally likely to be +1 as -1, pretty
much uncorrelated. Typically any correlation dies down exponentially
in the distance, measured in lattice units. Exactly how fast this
exponential decays introduces a length scale (in lattice units) -- let
us call this the correlation length. The correlation length is typically
a just a few lattice lengths.
Now, what we want to do is to see if there is a limiting procedure --
the limit we want is where the lattice size is vanishingly small
compared to the length scale on which we are doing physics, so that
somehow the lattice model provides an approximation to continuum physics.
Conceptually, what we hope for is that our lattice model shows features
or structure on a scale of many, many lattice lengths, that we can
identify with features of our physical system that we are dealing with.
For example, maybe our lattice model, for some value of coupling constants,
displays islands of all +1 spins or -1 spins that are hundreds of thousands
of lattice lengths in size. In terms of the island size, which we identify
with something we see at the length scale that we are doing physics, the
lattice size is very small, and we may be close to approximating a continuum
model with a lattice model.
But how can this happen when the correlation length is typically just
a few lattice lengths? Usually, it cannot. However, sometimes, for
certain values of coupling constants something miraculous happens.
As coupling constants approach some value, the correlation length tends
to grow and grow, approaching infinity. It turns out, that in statistical
mechanics, this is what happens in a second order phase transition.
Correlations no longer die out exponentially but rather by
some power law in lattice units, for example. Then, the lattice scale
can be made infinitesimally small compared to the correlation scale,
simply by tuning the parameters in our Hamiltonian appropriately.
If we identify the length scale we are interested in with the
correlation length, then the lattice size, in physical scale units,
has become infinitesimally small.
To put it another way, when a lattice model is close to a second order
phase transition, then you know exactly how to shrink the lattice and at the
same time modify coupling constants as a function of the lattice size,
so that a physical scale -- the correlation length -- remains constant.
Moreover, this series of lattice models all corresponds to the same
physical model, because this series of models all correspond to the
large scale behavior of the same second-order phase transition.
In the language of earlier articles : the cut-off scale is simply the
lattice length. Physics in most models is all at the cut-off scale --
of the order of a few lattice lengths. In a few models, with an
ultraviolet fixed point (the value of couplings that give a second
order phase transition) we get a theory that is non-trivial at the
the physical length scale (which is many many times larger than the
cutoff scale). Of course, there is no logical reason that the theory
derived from the lattice model in this manner has to be perturbatively
renormalizable.
-arun gupta