RG flows from non-perturbative coarse graining

(Sethna, “Entropy, Order Parameters, and Complexity”, ex. XXX)
© 2024, James P. Sethna, all rights reserved.

This exercise is primarily analytical: only those parts with computational components are included in this file. The exercise will be available at https://sethna.lassp.cornell.edu/StatMech/SethnaExercises.pdf.

Long intro to non-perturbative RG and momentum-space RG...

Sparing ourselves this calculation, we shall quote the LPA approximation for the coarse-graining equations up to quartic terms:
𝑑𝐹/𝑑ℓ = −𝑘^3/(1+(𝐴/𝑘^2))
𝑑𝐴/𝑑ℓ = 3𝐺𝑘/(1+(𝐴/𝑘^2))^2
𝑑𝐺/𝑑ℓ = −18(𝐺^2/𝑘)/(1+(𝐴/𝑘^2)^3
𝑑𝑘/𝑑ℓ = -𝑘
 
These are not yet the renormalization-group flow equations: we have yet to rescale properly to see the fixed point. But we can use these as if we were the experimentalist, measuring universal power laws in the  “laboratory” provided by  𝑉∞(𝑀∞,𝐴∞,𝐺∞,𝐹∞).

We shall focus on the transition for  𝐺0=0.2, where the critical point lies at 
𝐴𝑐0≈−0.19053. At the critical point, the macroscopic potential develops a double well, leading to a finite magnetization. Hence we expect  𝐴∞=0 when  𝐴0 equals  𝐴𝑐0.

V[M_,F_,A_,G_]:= F + ... + (G/2)M^4

CoarseGrainingFlowEqns = 
{k'[ell] == -k[ell], 
A'[ell]==3 G[ell]k[ell]/(1+(A[ell]/k[ell]^2))^2,
G'[ell]==-18 (G[ell]^2/k[ell])/(1+A[ell]/k[ell]^2)^3,
F'[ell]==...}

(a) Numerically solve the flow equations starting at  ℓ=0,  𝑘=1,  𝐴0=𝐴𝑐0,  𝐺0=0.2, and  𝐹0=0, evaluating them at  ℓ ∈ [0,0.25,0.5,1,2,4]. Plot the potentials  𝑉_ℓ (𝑀). Does the potential appear to be at the critical point, where the bulk quadratic term  𝐴∞=0? Note also that the microscopic energy  𝑉0 has a double well, but the well disappears under coarse-graining. Why should this be expected, for a low enough barrier? (Hint: The barrier height in  𝑉0 is measured in units of the temperature. If it is much less than one, do you expect the system to stay in one well?)

Ac = -0.19053; 
CoarseGrainingInitialConditions = {A[0] == Ac, G[0] == ..., F[0] == 0, k[0] == 1};
eqns = Join[CoarseGrainingFlowEqns, CoarseGrainingInitialConditions];
ellFinal = 4;
ells2Plot = {0,0.25,0.5,...};
solCritical = NDSolve[eqns, {A[ell],..., F[ell], k[ell]}, {ell, 0,ellFinal}][[1]];
CurvesCritical = Table[V[M,F[ell],A[ell],...]/.solCritical,{ell, ells2Plot}];
Plot[CurvesCritical, {M,-4,4}, PlotRange->{-0.5,0.5},PlotLegends->LineLegend[Table[ell,{ell, ells2Plot}]]]

Your answer here (or in a separate writeup).

Since our free energy  𝑉(𝑀) is measured in units of the temperature, raising the temperature lowers  𝐴,  𝐺, and  𝐹. As  𝐴 directly controls the development of magnetization, let us mimic the experimental temperature by varying  𝐴0 through the critical value  𝐴𝑐0. We shall confine ourselves to the approach to  𝐴𝑐0 from above; more sophisticated NPFRG methods are needed to behave properly below the transition temperature.

Let us measure the macroscopic susceptibility  𝜒=∂𝑀bar /∂𝐻 | 𝐻=0, where  𝑀bar minimizes  𝑉∞(𝑀)−𝑀𝐻.

(b) Show that  𝜒=1/(2𝐴∞)  in the paramagnetic single-well phase when  𝐴0>𝐴𝑐0. (Hint: Find the equation satisfied by  𝑀bar, and take its derivative with respect to  𝐻.)

Your answer here (or in a separate writeup).

(c) As in part (a), plot the potential  𝑉_ℓ as it evolves starting at  𝐴𝑐0+0.05. Does it converge to a fixed potential, representing the bulk free energy? Measure the bulk quadratic term  𝐴∞ in the free energy for a range of microscopic values  𝐴0 close to  𝐴𝑐0, and estimate  𝛾. Go to larger l as your initial temperature gets close to Ac0. You can do a power-law fit, or you can just do a log-log plot of χ(A0) × (A0 − Ac0)^γtry vs. A0 − Ac0 and vary γtry until it has a flat region. Plot your result in either fit.  Compare your value  𝛾try to the value derived from conformal bootstrap,  𝛾boot∼1.237075.

CoarseGrainingInitialConditions = {A[0] == Ac+0.05, G[0] == ...};
eqns = Join[CoarseGrainingFlowEqns, CoarseGrainingInitialConditions];
solCritical = NDSolve[...][[1]];
CurvesCritical = Table[...,{ell, ells2Plot}];
Plot[...]

Clear[CoarseGrainingInitialConditions,eqnsVaryA,sols]

deltaAs =PowerRange[10^-4,10,10^(0.1)]; 
AInits = Ac + deltaAs;

eqnsVaryA[n_] := Join[CoarseGrainingFlowEqns, {A[0] == AInits[[n]], G[0] == 0.2, F[0] == ..., k[0] == ...}]
ellFinal = 4;
sols = Table[NDSolve[eqnsVaryA[n], {A[ell],G[ell], F[ell], k[ell]}, {ell, 0,ellFinal}][[1]], {n,1,Length[AInits]}];
chis = .../(...A[ell])/.sols /. ell->ellFinal;
γBootstrap =1.237075;
(* Vary γTry until plot is mostly horizontal *)
γTry = 0
points = Transpose[{deltaAs, chis deltaAs^γTry}];
ListLogLogPlot[points ]

Your answer here (or in a separate writeup).

Discussion ...

These rescaled equations for  𝑑=3 are (finally!) our RG flow equations
𝑑𝑎/𝑑ℓ =2𝑎+3𝑔/(1+𝑎)^2
𝑑𝑔/𝑑ℓ =𝑔−18𝑔^2/(1+𝑎)^3
𝑑𝑓/𝑑ℓ =3𝑓−1/(1+𝑎).
 
Part (d) deriving these, part (e) finding the fixed point...

What are  𝑎𝑐0 and  𝑔0, the rescaled parameters corresponding to the experiments we performed in parts(a) and(c)? Since  ℓ starts at zero with the microscopic Hamiltonian,  𝑘_ℓ=1, and thus the coarse-graining transition you studied happen at RG-flow initial conditions  𝑔0=𝐺0=0.2 and  𝑎𝑐0=𝐴𝑐0≈−0.1905316.

(f) Launch trajectories near this point, varying  𝑎0 slightly above and below  𝑎𝑐0, and plot the trajectories in the  𝑎0,𝑔0 plane. Show that they pass near to the fixed point before veering off to high or low rescaled temperatures.

RGFlowEqns = 
{a'[ell]==2a[ell] + 3 g[ell]/(1+a[ell])^2,
g'[ell]==g[ell]-18 g[ell]^2 /(1+a[ell])^3,
f'[ell]==...};

aStar = -1/13.; gStar = 96/2197.; fStar = ...;

deltaas =Range[-0.025,0.025,0.005]; 
aInits = Ac + deltaas;
ellFinal = 2;
eqnsVarya[n_] := Join[RGFlowEqns, {a[0] == aInits[[n]], g[0] == ..., f[0] == ...}]
sols = Table[NDSolve[eqnsVarya[n], {a[ell],g[ell], f[ell]}, {ell, 0,ellFinal},AccuracyGoal->13, PrecisionGoal->13][[1]], {n,1,Length[aInits]}];
flows ={g[ell],a[ell]}/.sols;
flowPlot = ParametricPlot[flows, {ell,0,ellFinal}, PlotRange->{{0,0.25},{-0.2,0.05}}];
pointsPlot = ListPlot[{{gStar,aStar}, {0.2, Ac}}, PlotStyle->{Red, PointSize[0.02]}];
Show[pointsPlot, flowPlot]

The next step is to linearize the flows for  𝑎 and  𝑔 at the fixed point  (𝑎∗,𝑔∗), to find the Jacobian
𝐽≡((∂𝑎˙/∂𝑎  ,  ∂𝑔˙/∂𝑎), (∂𝑎˙/∂𝑔 ,  ∂𝑔˙/∂𝑔)) = ((5/3, 24/169), (169/48, −1))
where, e.g.,  𝑎˙=∂𝑎/∂ℓ.

(g) Calculating all the components of  𝐽 is straightforward, but a bit tedious. Verify that the lower right element  ∂𝑔˙/∂𝑔 = −1. Then use  𝐽 to numerically find the eigenvalues and right eigenvectors. Add these to your plot of part (e), and verify that the flows approach the fixed point along the irrelevant eigendirection, and then veer away along the relevant eigendirection.

Your answer here (or in a separate writeup).

Warning: Eigenvectors are in (a,g,f) order. We want to plot (g, a).

J = {{5./3., 169./48.},{24./169.,...}};
{vals, vecs} = Eigensystem[J]
{λT,λU} = vals
{vecT, vecU} = vecs

eigPlot = ListPlot[{{{gStar,aStar},{gStar + vecT[[2]],aStar+vecT[[1]]}}, {{gStar,aStar},{gStar+vecU[[2]],aStar+vecU[[1]]}}},Joined->True,PlotRange->{{0,0.25},{-0.2,0.1}}, PlotStyle->Black];
Show[pointsPlot, flowPlot, eigPlot]

Calculating predicted critical exponents: parts (h), (i), (j), (k)