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using Plots

, LinearAlgebra

, ForwardDiff

, Roots

, PlutoUI

1.9 s

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TableOfContents(title="Table of Contents", indent=true)

29.5 μs

Plots.PlotlyBackendPlotlyBackend

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plotly()

For saving to png with the Plotly backend PlotlyBase has to be installed.

112 ms

Ising Model and Z2 Gauge Theory

Topics

Duality and Wilson Loops
Gauge Field Theory
Legendre Transform

Reading

M. Kardar Stat. Theory of Fields ch. 7
J. Greensite: Center vortices, and other scenarios of quark confinement
Kohsuke Yagi, Tetsuo Hatsuda, and Yasuo Miake: QGP: From Big Bang to Little Bang
Cheng & Li: Gauge theory of elementary particle physics

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md"

# Ising Model and Z2 Gauge Theory

# Topics

* Duality and Wilson Loops

* Gauge Field Theory

* Legendre Transform

# Reading

* M. Kardar Stat. Theory of Fields ch. 7

* [J. Greensite: Center vortices, and other scenarios of quark confinement](https://link.springer.com/content/pdf/10.1140/epjst/e2007-00002-6.pdf)

* Kohsuke Yagi, Tetsuo Hatsuda, and Yasuo Miake: QGP: From Big Bang to Little Bang

* Cheng & Li: Gauge theory of elementary particle physics

1.0 ms

Story Telling Time:

From spins to links and plaquettes

Self-Duality does NOT work in 3D Ising: Low and High T expansions do not match
Low T: theory of faces
total no. of faces = Area

$(e^{- 2 K})^{A r e a}$

High T: t-theory of links, forming loops / plaquettes
counting no. of links: i.e. perimeter of loop

$t^{P e r i m e t e r}$

this will NOT match!

Find a new theory with High T expansion compatible with Low T 3D Ising Model

new variables: links $u_{P}$ = dual spins
unit = face at low T 3d Ising, surrounding islands of - in sea of +'s
face formed from links: u, e.g. 4 u's make a plaquette.
glue (via u) faces together to form a cube = low T island

$Z \sim \prod_{p l a q .} (1 + \tilde{t} u_{P} u_{P} u_{P} u_{P}) \propto e^{\tilde{K} \sum_{p l a q .} u_{P} u_{P} u_{P} u_{P}}$

this describes a Z(2) gauge theory

and to Wilson loops

The new theory is NOT supposed to have no Spontaneous Symmetry Breaking.
global Z(2): all $u \to - u$
local gauge symmetry: pick a site and flip all the relevant $u$ 's
local order parameter like $⟨ σ_{i} ⟩$ will not work!
but ''extended'' object like a Wilson loop could...

Wilson loop:

High Temperature

rough notes:

need to cancel out the links of C,
draw plaquettes inside,
each brings in a factor of $\tilde{t}$ , Area times!

$\begin{aligned} ⟨ \prod_{C} u_{C} ⟩ & \to \frac{\sum_{{u_{P}}} \prod_{C} u_{C} \prod_{P} (1 + \tilde{t} u_{P} u_{P} u_{P} u_{P})}{\sum_{{u_{P}}} \prod_{P} (1 + \tilde{t} u_{P} u_{P} u_{P} u_{P})} \sim {\tilde{t}}^{A_{C}} \\ \sim e^{- A_{C} #} \end{aligned}$

Low Temperature

rough notes:

start with all links +'s, including those in C, this gives 1.
Then flip one link to -1, 4 surfaces are affected, hence factor of $e^{- 2 \tilde{K} 4}$ .
For C, if the flipped link is not one of the $P_{C}$ links in C, value of loop unchanged,
otherwise it flips the value also.

$\begin{aligned} ⟨ \prod_{C} u_{C} ⟩ & \to \frac{N_{G} e^{3 \tilde{K} N_{s}^{d}} (1 + (3 N_{s}^{d} - P_{C}) e^{- 2 \tilde{K} 4} + (- 1) \times P_{C} e^{- 2 \tilde{K} 4})}{N_{G} e^{3 \tilde{K} N_{s}^{d}} (1 + 3 N_{s}^{d} e^{- 2 \tilde{K} 4})} \sim 1 - 2 P_{C} e^{- 2 \tilde{K} 4} \\ \sim e^{- P_{C} \times 2 e^{- 2 \tilde{K} 4}} \end{aligned}$

Spatial Wilson loop: Perimeter law -> Area law from low to high T: sensitive to the phase transition in between!

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22.2 ms

A glimpse of Gauge Field Theory

from continuum to lattice
pure gauge theory
confinement

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318 μs

Z2 gauge theory in Julia

one of the simplest gauge theory, closely related to 3D (and above) Ising Model

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202 μs

Z2gauge (generic function with 3 methods)

function Z2gauge(beta, Ns = 20, dim = 4)

latt = ones(Int64, ntuple(i1 -> i1 <= dim ? Ns : dim, dim + 1))

links = CartesianIndices(latt)

function mvup(x::CartesianIndex, d::Int64)

local dim::Int64 = length(x)

i_tmp::Int64 = x[d] == Ns ? 1 : x[d] + 1

return CartesianIndex(ntuple(i1-> i1!=d ? x[i1] : i_tmp, dim))

end

function mvdown(x::CartesianIndex, d::Int64)

local dim::Int64 = length(x)

i_tmp::Int64 = x[d] == 1 ? Ns : x[d] - 1

return CartesianIndex(ntuple(i1-> i1!=d ? x[i1] : i_tmp, dim))

end

function coldstart()

latt .= 1

end

function randomstart()

latt .= rand([-1, 1])

end

function staplecal(link::CartesianIndex)::Int64

x = CartesianIndex(link.I[1:end-1])

d = link.I[end]

# following M. Creutz

# staples around link(1->4)

# dperp 6--5

# ^ | |

# | 1--4

# | | |

# -----> d 2--3

staplesum = 0

for dperp = 1:length(x)

if dperp != d

# plaquette 1234

x = mvdown(x, dperp)

linkA = latt[x, dperp]

linkA *= latt[x, d]

x = mvup(x, d)

linkA *= latt[x, dperp]

x = mvup(x, dperp)

# plaquette 4561

linkB = latt[x, dperp]

x = mvup(x, dperp)

x = mvdown(x, d)

linkB *= latt[x, d]

x = mvdown(x, dperp)

linkB *= latt[x, dperp]

staplesum += linkA + linkB

end

return staplesum

end

function sweep1!()

# heatbath

action = 0.0

for link in links

staplesum = staplecal(link)

# calculate the Boltzmann weight

bplus = exp(beta * staplesum * 1)

bplus /= (bplus + 1 / bplus)

# the heatbath algorithm

if rand() < bplus

latt[link] = 1

action += staplesum

else

latt[link] = -1

action -= staplesum

end

# return 1.0 - action / Ns^dim / dim / 6.0

end

function sweep!()

# metropolis

action = 0

for link in links

# Esite = staplecal(link)

# dE = -2 * (Const. - latt[link] * staplecal(link))

if rand() < exp(-beta * 2*latt[link] * staplecal(link))

latt[link] *= -1

end

# return 1.0 - action / Ns^dim / dim / 6.0

end

function update!(N = 40)

for i1 = 1:N

sweep!()

end

function plaqcal(x::CartesianIndex, d, dperp)::Int64

# compute plaq

2-- link2 --x

link3 link1

0-- LINK --1

if dperp != d

ret = latt[x, d]

x = mvup(x, d)

ret *= latt[x, dperp]

x = mvup(x, dperp)

x = mvdown(x, d)

ret *= latt[x, d]

x = mvdown(x, dperp)

ret *= latt[x, dperp]

else

ret = 0

end

return ret

end

function measure()

plaqval = 0

Eval = 0

for link in links

x = CartesianIndex(link.I[1:end-1])

d = link.I[end]

Eval += latt[link] * staplecal(link)

for dperp in 1:length(x)

if dperp > d

plaqval += plaqcal(x, d, dperp)

end

# per X

Npair = dim*(dim-1)/2

plaqval /= Ns^dim * Npair

Eval /= Ns^dim * dim * Npair

return plaqval, Eval

end

# stir well

update!(180)

plaq = 0.0

Eav = 0.0

Nm = 25

for i1 = 1:Nm

_p, _E = measure()

plaq += _p / Nm

Eav += _E / Nm

update!(10)

end

return plaq, Eav

end

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12.8 ms

plaq_exact (generic function with 2 methods)

function plaq_exact(beta, dim=4)

K = beta * 1.0

highB = exp(-2*4*exp(-4*(dim-1)*K))

lowB = tanh(K)

return lowB, highB

end

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546 μs

let

dim = 4

betaC = 0.43

beta_arr = range(0.45*betaC, 1.5*betaC, length = 14)

obs = []

for beta in beta_arr

push!(obs, [beta, Z2gauge(beta, 10, dim)..., plaq_exact(beta, dim)...]')

end

obs = vcat(obs...)

for i1 in eachindex(obs[:, 1])

obs[i1, 4] = obs[i1, 1] < 1.3 * betaC ? obs[i1, 4] : Inf

obs[i1, 5] = obs[i1, 1] > 0.7 * betaC ? obs[i1, 5] : Inf

end

scatter(

obs[:,1], obs[:,2],

label="plaq",

xlabel="beta",

title="Z2 gauge theory (4D)"

)

plot!(obs[:, 1], obs[:, 4], label = "low beta", line = (dash = :dash), color = "blue")

plot!(obs[:, 1], obs[:, 5], label = "high beta", line = (dash = :dash), color = "red")

end

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18.8 s

let

dim = 3

betaC = 0.755

beta_arr = range(0.3*betaC, 1.5*betaC, length = 20)

obs = []

for beta in beta_arr

push!(obs, [beta, Z2gauge(beta, 14, dim)..., plaq_exact(beta, dim)...]')

end

obs = vcat(obs...)

for i1 in eachindex(obs[:, 1])

obs[i1, 4] = obs[i1, 1] < 1.3 * betaC ? obs[i1, 4] : Inf

obs[i1, 5] = obs[i1, 1] > 0.7 * betaC ? obs[i1, 5] : Inf

end

scatter(

obs[:,1], obs[:,2],

label="plaq",

xlabel="beta",

title="Z2 gauge theory (3D)"

)

plot!(obs[:, 1], obs[:, 4], label = "low beta", line = (dash = :dash), color = "blue")

plot!(obs[:, 1], obs[:, 5], label = "high beta", line = (dash = :dash), color = "red")

end

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3.3 s

The Polyakov Loops and the Susceptibilities

An order parameter for Z(Nc) SSB
effective order parameter for deconfinement
order parameter is NOT Unique: SUS can also do the job! (and better)

$L_{\vec{x}} = \prod_{x_{4}} U_{4} (x_{4}; \vec{x})$

In the end, we will measure

$⟨ L ⟩ = \frac{1}{N_{s}^{3}} \sum L_{\vec{x}}$

The Polyakov loop susceptibility is constructed via

$χ_{L} = N_{s}^{3} (⟨ L^{2} ⟩ - ⟨ L ⟩^{2}) .$

Why do we need the factor of $N_{s}^{3}$ ?

(we may need to multiply beta's for better rendering)

How to (re)normalize the Polyakov loop susceptiblity is a research topic!!

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488 μs

Z2gaugePL (generic function with 3 methods)

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17.3 ms

let

dim = 4

betaC = 0.43

beta_arr = range(0.7*betaC, 1.3*betaC, length = 25)

obs = []

for beta in beta_arr

push!(obs, [beta, Z2gaugePL(beta, 10)...]')

end

obs = vcat(obs...)

plot(obs[:,1], obs[:,2], label="<L>", xlabel="beta",

title="Z2 gauge theory (4D)")

plot!(obs[:,1], obs[:,3], label="<|L|>")

scatter!(obs[:,1], obs[:,4], label="susL")

scatter!(obs[:,1], obs[:,5], label="susA")

_f(x) = x < betaC ? 1-2/pi : Inf

plot!(obs[:,1], _f, label="PML ansatz")

end

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93.9 s

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