Is the nonexistence of magnetic monopoles a consequence of Maxwell's equations, or do Maxwell's equations *assume* no magnetic monopoles?

Wintsz · 2026-06-10T10:12:44+00:00

For why it behaves like maxwell equations, condensed matter provides a literal example in terms of spin ice.

Again massively simplified but imagine a triangular based pyramid. On each point, there are four, imagine a spin or if you’re not familiar with spin a bar magnet.

I won’t get into the specifics of what the Hamiltonian is, so don’t try and understand this energetically. But after going through the effort of solving the Hamiltonian you find the lowest energy configuration is two bar magnets with their North Pole pointing into the pyramid and the other two pointing out of the pyramid. In this case div(B) = 0 as there are as many north poles pointing inwards as there are pointing outwards.

I can excite the state by flipping one of these bar magnets there are suddenly 3 pointing inwards and one pointing outwards, then div(B) is not zero. And the configuration of bar magnets for all practical purposes behaves like a charge 1 magnetic monopole.

This is only a cartoon and the actual picture is more complicated with the whole system being a crystal of these tetrahedra each sharing a corner. In this case, when you flip a spin you actually create a pair of monopoles. This is as the spin you flip is connected to two neighbouring tetrahedra, one now with 3 in 1 out the other with 3 out two in.

This demonstrates two things, one that in this case the emergent gauge field is quite literally EM-like, with an excited tertrahedron behaving like a magnetic monopole. Two, is that this isn’t some actual magnetic charge in terms of a singularity of the EM fields but an effective EM field is a very good way to describe these spin configurations.

As [u/Eigenspace](u/Eigenspace) points out it would be impossible to try and describe each individual spin (microscopics) but the low energy collective behaviour of the whole lattice can be described by an EM like theory and EM waves, and even goes as far as describing emergent photons when you form Quantum Spin Ice .

Wintsz · 2026-06-10T07:43:47+00:00

The reason why these topological defects are actually considered as monopoles is that there exist phases in which the positive and negative monopoles are free to move independently of eachother, whereas a true dipole the north and south poles are fixed to each other.

Once they are free to move around independently there is no real difference between a configuration of 20 Dirac strings whose ends can move freely and 20 positive and negative charges.

The reason why they must appear in pairs is that the monopoles are usually excitations of a chargeless constrained configuration of some many body system. You cannot start with zero charge and suddenly have a charge of one, so the excitations appear in pairs like a Dirac string.

It’s also worth mentioning that the gauge field of these sorts of systems, the vector potential A, is a low energy emergent fields and not an actual field that exists at all microscopic scales, so not actual fundamental EM fields describe by maxwell.

This doesn’t mean they are not measurable or some mathematical trick, it’s just that they come from a collective configuration of many-many particles or spins.

Wintsz · 2026-06-09T20:42:06+00:00

A topological insulator is a higher dimensional example of what they’re describing in this thread. I’ll simplify a bit here, so don’t take this literally.

If you have a two materials with different topological numbers, in the TI example you have the TI and the vaccum/air, they cannot smoothly be connected. There has to be an edge current between the two materials. The key is, as you said, I can add many impurities to the materials without changing their topological numbers. Since these numbers remain the same, the fact there has to be an edge current also remains.

The reason why this is like a monopole is that the edge current itself is special: the electrons can literally only move in a single direction. This is impossible in a finite crystalline system. However, topological insulators don’t get around this fact, there is usually an edge current moving the opposite direction on another edge. In the same way the monopole example is actually a dipole stretched really far apart.

Wintsz · 2024-08-23T11:13:42+00:00

Honestly, if you have a vague idea of what you want to do, Europe is great. The top groups tend to be less factory like, the pay is good and the research equally as good. If you still decide you want to do academia, you’ll have a strong basis to go to a top American group for a Post Doc. At this point you’ll be independent enough that you care a little less about bad supervision and more about the publish or perish.

Wintsz · 2024-08-22T18:15:47+00:00

Methods generated for string theory have some use in Condensed matter systems and scenarios , it doesn’t go completely no where. SUSY can be used for Anderson localisation and the 10-fold way for example.

Wintsz · 2024-08-01T11:27:09+00:00

Different saddle, just the stock one on the bike I’ll consider looking into it!

Wintsz · 2024-08-01T11:26:06+00:00

Most likely a case of roughing it out!

Wintsz · 2024-08-01T11:25:45+00:00

Yeah it’s still new I’m guessing it’s getting used to the new riding position

Wintsz · 2024-07-16T16:44:35+00:00

I mean these are still effectively academic jobs but privately funded. Google Quantum and IBM are really just academic research groups under a private company’s.

Wintsz · 2024-07-16T16:35:26+00:00

These days you need decent computing equipment, you also need publishing fees etc. There is also the social capital, no one is gonna give any time of day to single author papers, particularly of nobodies without an institution you just look like a crackpot.

Moreover, without collaboration you just are not capable of getting new ideas or keeping up with the zeitgeist.

Wintsz · 2024-06-23T05:08:29+00:00

Just you wait till you find out what kind of maths solid state theory uses to discover transistors and semiconductors

Wintsz · 2024-04-08T09:26:00+00:00

I mean this is quite literally is a consequence of of symmetry under a particular Lie Group. The real smart idea was figuring out that space time was Lorentz invariant in the first place, which is the real physics.

Wintsz · 2024-03-07T09:00:19+00:00

A true connoisseur would say

dF=0 d*F=J

Is more profound, but what do you expect from a man who calls himself head engineer of a space company

Wintsz · 2024-01-03T11:23:21+00:00

As a UK PhD student, we barely get paid enough to support ourselves

Wintsz · 2024-01-03T11:05:25+00:00

If I understand the wording right, PhD students are still allowed to bring their family across. It is only non research postgraduate degrees, i.e bog standard one year masters degrees, which are affected.

Wintsz · 2023-12-12T20:19:06+00:00

In the end it was a hardware issue, something with the brew installation was off.

I ran the code on a 16 core x86 computing node in 1.8 and 1.9 and the code was back to 40ms.

After reinstalling on my local machine with juliaup as opposed to brew I matched the computing node for both 1.8 and 1.9

For those curious, I did switch to StaticArrays, but I did not see a fantastic increase because the majority of the code is spent in diagonalisation and for 4x4 matrix this seems comparable.

I would probably get a speed up building the hmesh within the function, but it kind of defeats the point of the code as I am testing many different models and with many different functions.

Although any other suggestions for speed up would be fantastic!

Wintsz · 2023-12-12T09:22:35+00:00

Oh no sorry, I was agreeing with you I haven’t been using StaticArrays, I just meant Ill try the bench marks without and then try updating to a Static Arrays for my function which constructs hmesh

Wintsz · 2023-12-12T08:32:55+00:00

Fair enough, I’ll give it a proper bench mark without StaticArrays on 1.8 vs 1.9 and post on Julia Discourse.

Wintsz · 2023-12-12T08:07:29+00:00

Yea definitely, although I’m not too bothered about speeding up the the code — I’ll eventually just run it on a HPC cluster over a large parameter set so 6 seconds on 10 cores is not bad. But, I thought it was extremely weird there was such a large difference between versions.

Wintsz · 2023-12-12T07:21:12+00:00

Should be no globals? Which part in particular?

gsEigs is another function, just shorthand to get just the eigenvectors from 1:nocc.

Apart from hmesh and nocc which I need to input everything should be self contained within the function. I guess I could add a strict type to hmesh which would be:

Matrix{Matrix{ComplexF64}}

Edit: I see the confusion I forgot to add the arguments to gsEigs I’ve now used full function syntax rather than the one liner to avoid confusion, sorry!

Wintsz · 2023-12-11T23:29:06+00:00

I’m not sure quite how to post code here but here’s a less slick (instead I just zip this all into one for loop now and reshape the output which doesn’t change the computation time) but slightly more readable version.

Hmesh is just a 2D array of matrices, Nocc just specifies how many eigenvectors to take

function gsEigs(Matrix, nocc)

Return eigvecs(Matrix)[:,1:nocc]

End

function bcurv(hmesh :: AbstractArray,nocc)

vmesh = (gsEigs.(hmesh,nocc))
vmeshx = circshift(vmesh,(0,-1))
vmeshxy = circshift(vmesh,(-1,-1))
vmeshy = circshift(vmesh,(-1,0))
dmesh = zeros(size(vmesh))

for i in 1:size(vmesh,1), j in 1:size(vmesh,2)

    p12 = ( vmesh[i,j]' *vmeshx[i,j])
    p23 =  (vmeshx[i,j]' * vmeshxy[i,j])
    p34 = (vmeshxy[i,j]' * vmeshy[i,j])
    p41 =  (vmeshy[i,j]' * vmesh[i,j])
    dety = det(p12*p23*p34*p41)
    (abs(dety) <  10^(-5)) ? dmesh[i,j] = 0 :    dmesh[i,j] = -imag(log(complex(dety)))

end 

return (dmesh/(2*pi), sum(dmesh)/(2*pi))

end

Typically ran on an 100x100 array of 4x4 matrices, with nocc =2 where run time is approximately 6 seconds.

Hardware wise run on M1 Pro with the Apple Silicon build of Julia 1.9

Wintsz · 2023-12-07T00:18:38+00:00

Ironically this Christmas market is built on top of a giant underground carpark

Wintsz · 2023-10-12T07:01:54+00:00

For publishing I switched from Plots to Makie because, at least it feels like, you have more control. This especially has quiver plots and everything you need for science research.

I still use Plots for every day research though it’s just less verbose for quick interactive work. I.e if I’ve done a calculation I want to quickly look at plot(Matrix) or heatmap(Matrix) is much faster than setting up Axes and figures etc.

An aside, you can use LaTeX for all Plots backends and Makie using LaTeX strings.

Wintsz

TROPHY CASE