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CRITICAL CENTER MAGNETIZATION IN THE PLANAR XY MODEL
A reader's guide for a high-school student
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This document explains a theorem about a famous model of statistical
mechanics: the two-dimensional XY model, and its critical point. It is
written to be read on its own. You should be able to follow everything
with basic trigonometry, basic probability, and the ideas of "scaling"
and "logarithm" that you already know from compounding interest.
The theorem states the exact shape of the magnetization at the center
of a very large square whose boundary spins are all pinned to point
the same way. On the way we will explain what "critical point" means,
why two thirds of the answer is a classic scaling story, and why the
last third is a subtle logarithm that required a serious new argument.
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1. WHAT THE MODEL LOOKS LIKE
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Imagine a very large chessboard. On every square, plant a small arrow.
The arrow can point in any direction on the plane, so it is described
by a single angle theta. Two angles that differ by a full turn (2*pi)
describe the same arrow.
Two arrows on neighboring squares "want" to point in the same direction.
The energy of a whole board configuration is
E(theta) = - sum over neighbor pairs {x,y} of cos(theta_x - theta_y)
Each pair of neighbors is counted once. If all arrows are aligned the
energy is as negative as possible, so that configuration is favored.
If members of a pair point in opposite directions, cos = -1, and the
energy is large.
Now put the board in a heat bath. At "inverse temperature" b > 0, a
configuration gets probability proportional to
exp( -b * E(theta) )
joined with the uniform measure on angles. Big b corresponds to cold:
the arrows want to align. Small b corresponds to hot: the arrows
jiggle wildly.
Two boundary conditions are used in this paper:
* Free: no condition on any spin outside the square.
* Aligned: every spin on the boundary of the square is set to 0.
The free system is the one whose "critical point" is defined. The
aligned system is the one whose center magnetization a_n is studied.
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2. THE CRITICAL POINT AND THE BKT TRANSITION
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On the free system, define the two-point correlation function at
distance r along the horizontal axis:
C_b(r) = E[ cos(theta_0 - theta_{r*e_1}) ]
It measures how much two distant arrows still tend to agree.
In the "ordered" phase (very cold), C_b(r) tends to a positive
constant as r grows. In the "disordered" phase (very hot), C_b(r)
decays exponentially fast. Between those two regimes the decay is a
power law: C_b(r) ~ 1 / r^something.
Define the mass gap
m(b) = - lim_{r -> infinity} (1/r) * log C_b(r).
Then m(b) = 0 exactly when the decay is a power law, not exponential.
The critical inverse temperature b_c is the smallest b for which
m(b) = 0. This is the mass-defined critical point.
Physicists know this transition well. It is called the
Berezinskii-Kosterlitz-Thouless (BKT) transition, invented around 1970.
It is not like the water/ice transition, because the arrows do not
"freeze" in place. They always twist and swirl; in two dimensions you
cannot have true long-range order, by the Mermin-Wagner theorem. The
distinction between the phases is instead about "vortices". A vortex
is a small whirlpool where the arrows wind once as you go around a
closed loop. At low temperature vortices only exist in closely bound
pairs. Above b_c, they unbind and proliferate, and the long-range order
is destroyed. That is the BKT mechanism.
Special feature of BKT: at the critical point, one parameter of the
theory, the "stiffness", is marginal. Marginal means it has no
definite scaling dimension and it changes with scale very slowly,
in a way that produces logarithms. This is the reason we eventually
see a (log n) in the answer.
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3. WHAT a_n MEASURES
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Put the XY model on the square [ -n, n ]^2 of the integer lattice.
The two sides of the square have length 2n, and the origin sits at
the exact center. Pin every spin on the boundary of the square to
point to the right: theta_x = 0 for |x|_infinity = n. This is the
aligned boundary condition.
The center spin theta_0 is still random; the boundary alignment only
biases it toward 0. Define
a_n = E[ cos(theta_0) ] at b = b_c, on the aligned square.
If the boundary spins had no effect, a_n would be 0. If the system
were ordered, a_n would be a positive constant. At criticality the
two effects balance and a_n decays as a power of n.
This is the observable the theorem is about.
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4. THEOREM
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Let b_c be the mass-defined critical inverse temperature (just defined).
Then, as n -> infinity,
a_n = A_XY * n^(-1/8) * (log n)^(1/16) * (1 + o(1)).
Here A_XY is a finite, strictly positive constant. It depends on the
microscopic model but not on n. Equivalently, the limit
B_inf = lim_{n -> infinity} [ log a_n + (1/8) log n
- (1/16) log log n ]
exists and is finite, and A_XY = exp(B_inf).
The value of A_XY is not given in closed form. It is not believed to
be expressible in terms of familiar constants. The theorem simply
says it is a genuine, finite, positive number.
The theorem is proved under three companion-paper inputs:
* Paper X: the critical Gaussian coefficient of the height model
equals 8*pi at b_c.
* Paper B: the local renormalization maps, the exact recurrences
for the gradient and fundamental coordinates, and the constants
H > 0 and b_* = 2*log L.
* Paper S: a rough lower bound a_n >= c * n^(-1/2), and a
"regular-variation" ratio limit that extends the result from
geometric sizes L^J/2 to all integers n.
Everything else in the proof is new to this paper.
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5. WHY n^(-1/8), AND WHY (log n)^(1/16)
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The answer has two parts, and each part has a different origin.
5.1 The exponent 1/8
The boundary alignment is a magnetic source. In the critical theory,
the spin field has scaling dimension Delta = 1/8. The standard scaling
hypothesis says that a one-point function of a spin field of dimension
Delta, in a domain of size L, scales like L^(-Delta). Here L = 2n, and
the 2 factors are absorbed into A_XY, so
a_n ~ n^(-1/8) * (slowly varying factor).
This is exactly the "eta/2" piece from the 2D XY critical exponents,
where eta = 1/4 is the two-point correlation exponent. Making it
rigorous is the content of papers X and S.
5.2 The prefactor (log n)^(1/16)
The BKT point is special because of a marginal direction. The
gradient stiffness of the height model is not a fixed number at
criticality; it runs with scale. Let t_j be the running gradient
coefficient after j block-renormalization steps. Then t_j tends to
zero, but only slowly: t_j ~ 1 / (2*j*log L) for a fixed block factor
L, which is like 1/j up to constants.
That 1/j comes from the interaction between the two marginal
coordinates, u_j and Z_j. They obey
u_{j+1} - u_j = - H * Z_j^2 + smaller
Z_{j+1} - Z_j = - b * u_j * Z_j + smaller
with H > 0 and b = 2*log L. Solving them together gives u_j ~ 1/j
and Z_j ~ 1/j, exactly the marginal behavior.
At the end, this running t_j gets multiplied by the "connection
energy difference" between two squares and produces
log a_{Ln} - log a_n = - log L / 8 + (1/16) * (1/j) + ...
Summing 1/j over j = 1, ..., J gives (1/16) log J. Since log n ~ J*log L,
that is (1/16) log log n, which exponentiates to a (log n)^(1/16)
prefactor. The 1/16 is therefore a universal number written into the
answer by the marginal running.
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6. HOW THE PROOF WORKS: A GUIDED TOUR
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The full proof is about 5000 lines of LaTeX. Here is the skeleton.
6.1 Fourier duality (angles -> heights)
By an exact Fourier transform, the XY model is dual to a "height
model" in which the spins are integers rather than angles. The
Boltzmann weight of an edge of height difference 2*pi*k is a Bessel
function I_k(b). At b = b_c this becomes a critical lattice field
with a Gaussian coefficient equal to 8*pi.
The big win is that the center magnetization becomes a ratio of two
partition functions:
a_n = Z_n[ center flux ] / Z_n[ no flux ].
The numerator has a single 2*pi circulation of the height around the
origin (a "vortex" pinned at the center). The denominator has none.
This is the magnetic duality trick. From now on everything is said
in the height language.
6.2 Entering the renormalization at a small scale
The height interaction is not small at criticality, so the
renormalization-group maps do not apply directly. The paper solves
this by "entering" the model at a very large observation scale.
Fix a large index J, and partition the lattice into observation cells
of side s ~ J^3. On each cell boundary, multiply the interaction by a
smooth cutoff chi(d) = (1/sqrt(2*pi)) * int_{-T}^{T} exp(-(d-x)^2/2) dx
with T = sqrt(s). It is a smoothed step function that damps large
observation differences d. The effect is that the leftover local
activity in the height model has norm < epsilon (any prescribed
small epsilon).
The only exception is the single cell that carries the magnetic
insertion. That exception is handled separately, and its amplitude is
never estimated. The entry scale is L^i with i = O(log J), far below
the target scale J.
The cutoff itself changes finite-volume quantities only by
exponentially small amounts: the error is O( (n/s)^2 * e^{-c*s} ),
exponentially small in J^3.
6.3 A trajectory that runs forever
At every scale L^j the height activity has three principal coordinates:
t_j : the gradient coordinate (scalar)
Z_j : the fundamental coordinate (scalar, positive)
R_j : the remainder (a function of the field)
Balancing the gradient and fundamental coordinates with the Gaussian
curves also produces a scalar u_j, which agrees with t_j up to
quadratic error: t_j = u_j + O(u_j^2 + Z_j^2). The renormalization
acts on (u_j, Z_j, R_j) by the recurrences in section 5.2, plus a
contraction of R_j. Two lemmas in the paper
ensure the entire finite history remains small:
* "No exit" (Lemma 4.5): the entry can be chosen so small that
every finite history remains in the analytic tube through index 8J.
* "Positive odd coordinate" (Lemma 4.6): Z_j does not decay
exponentially; instead Z_k > 0 and -log Z_k = o(J). This is
proved by a lower bound on the physical mean-phase expectation
of the periodic model.
The result is a finite trajectory (u_j, Z_j, R_j) on each J-history,
with
u_j ~ 1 / (2 j log L),
| Z_j | ~ bounded / j,
|| R_j || = O(1/j^2).
6.4 Stitching different histories by physical observables
Different J-histories start at different entry scales, so their
activities are not identical. But there are two observables of the
*original* periodic height model that read out (u_j, Z_j) from
physical quantities at scale L^j:
U_j = Var( long-wave smear ) / g
Y_j = E[ exp( i * average of heights ) ]
U_j and Y_j do not depend on the history. They are properties of the
model at the absolute scale L^j. Comparing them for two histories
lets us match the two trajectories at overlapping scales. The
mismatch between them is of order Z^2 in u and |u|Z in Z, exactly
the size that the abstract lemma on "trajectories with geometrically
separated jumps" can absorb.
Consequence: the paper produces a single effective trajectory t_j,
defined for all j, satisfying
t_j = 1 / (2 j log L) + O( log j / j^2 ).
No rate of convergence for the height-model scaling limit is used.
6.5 Comparing two squares with a magnetic source
Everything up to now concerned the height model without a magnetic
source. To reach a_n we need the source.
Fix an index j, and consider two squares with sides M = L^j and
LM = L^{j+1}, both centered at the source. Choose the J-history
consistent with the target (namely J with J <= j < 2J). Run the
height renormalization in both squares:
* from the same entry scale L^i (same i, same history);
* to the same tiny stop scale v = L^h, where h = j - O(log j).
The stop is chosen so that the number of stop-blocks D = M/v in the
square is a small positive power of j, roughly j^(1/10). Concretely,
j^{1/10} <= D < L * j^{1/10}. The terminal errors D^4/j^2 = O(j^{-8/5})
and 1/(D*j) = O(j^{-11/10}) both go to zero, so the terminal Gaussian
calculation is in the correct window.
Now comes the key trick of the paper. Both squares see the source in
the same way. The exact scalar normalizations that appear near the
source in the renormalization are literally equal in the two squares.
Therefore in the ratio a_{LM}/a_M they cancel. The paper never has to
estimate the (possibly polynomially large) amplitude of the magnetic
insertion.
What does NOT cancel is a residual insertion activity. Its norm is
bounded by
|| I_{h,Q} || <= C_R * s^{C_R} * L^{-2(h-i)/3}.
Iterating this contraction from entry down to stop gives a residual
activity exponentially small in i - h. Since the rough lower bound
a_n >= c*n^{-1/2} from paper S is polynomial, the residual activity
is negligible relative to a_{LM} in the ratio. So a_{LM}/a_M is
determined by the cancelling scalars and the deterministic pieces.
6.6 The terminal Green-function computation
At the stop scale v, the remaining activity is a small Gaussian.
Working out its contribution gives
log a_{LM} - log a_M
= - (1 - t_h) / (2 * alpha^2) * (E_{LM} - E_M) + small
where alpha^2 = 8*pi, t_h is the running gradient coefficient at the
stop (which we already computed), and E_Q is the energy of the
minimal divergence-free connection eta_l,Q in the square of side Q.
The classical Green-function identity says
E_Q = (2*pi)^2 * G_{Q/2}(0,0),
where G_n is the Green function of the killed unit-edge Laplacian on
the spin square of half-side n. Comparing G_{LM/2} and G_{M/2} and
using the low-pass approximation to the discrete kernel gives
E_{LM} - E_M = 2*pi * log L + O(L^{-c})
for some fixed c > 0. This is the classical computation that pins
down the leading behavior.
Putting the pieces together:
log a_{LM} - log a_M
= - (1 - t_h) / (16*pi) * 2*pi*log L + small
= - log L / 8 + t_h * log L / 8 + small
= - log L / 8 + 1 / (16 h) + O( h^{-2} )
= - log L / 8 + 1 / (16 j) + O( j^{-1-epsilon} ).
The last equality uses h = j - O(log j).
6.7 Summing the ratio
The right-hand side is summable in j. Sum from j = j_0 to j = J-1:
log a_{L^J / 2} - log a_{L^{j_0} / 2}
= - (1/8) * (J - j_0) * log L
+ (1/16) * (log J - log j_0)
+ O(1).
Exponentiating:
a_{L^J / 2}
= A_XY * (L^J / 2)^{-1/8} * J^{1/16} * (1 + small).
Now L^J/2 = n, and J = (log n + log 2) / log L, so J^{1/16} is
(log n)^{1/16} up to a constant factor absorbed into A_XY. The
limit theorem now holds along geometric sizes n = L^J / 2.
6.8 Extending to all n
Paper S provides the regular-variation limit: if n'/n -> d in
(0, infinity), then a_{n'}/a_n -> d^{-1/8}. Any integer n lies
between consecutive L^J/2 sizes. The ratio limit and the slow
variation of the log correction force the same asymptotic for all
n -> infinity. This completes the theorem.
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7. WHAT IS NEW, AND WHAT IS BORROWED
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Borrowed:
* The value 8*pi of the critical Gaussian coefficient in the height
model (paper X).
* The block geometry, analytic norms, parity-resolved recurrences
with H > 0 and b_* = 2*log L, and the exact Gaussian curves
(paper B).
* A rough lower bound a_n >= c*n^{-1/2} and the geometric ratio
limit (paper S).
New in this paper:
* An analytic entry construction with one magnetic insertion, and
the insertion bound that survives all renormalization steps.
* A stitching of histories at different entry scales by physical
observables, with no rate assumed for the height scaling limit.
* Exact cancellation of the magnetic scalar normalization between
two squares of sides M and LM.
* The terminal Green-function computation and the 1/16 coefficient.
* Lemmas that keep the finite histories in a small tube ("no exit")
and that give a subexponential lower bound on the odd coordinate
Z_k (the "positive odd" lemma).
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8. WHAT TO REMEMBER
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At criticality of the two-dimensional XY model, the center magnetization
with aligned boundary in a square of half-side n is
a_n = A_XY * n^(-1/8) * (log n)^(1/16) * (1 + o(1)).
The n^(-1/8) comes from the magnetic scaling dimension of the spin
field.
The (log n)^(1/16) comes from the marginal running of the gradient
stiffness under renormalization: its reciprocal grows like 2*log L
per doubling of scale, and integrated over all scales produces a
logarithm of a logarithm, hence a power of log n.
The proof combines:
* Fourier duality from spins to heights;
* a smooth smearing to make the local activity small enough for
the renormalization maps;
* the finite trajectories of paper B;
* exact cancellation of magnetic scalars between two squares;
* a discrete Green-function computation for the terminal energy;
* summation of a summable telescoping series.
The constant A_XY is not given in closed form, but it is a genuine,
finite, positive number. That is the theorem.