(the lone magnetic pole nobody has ever found 🧲) turns up in a cold crystal by the billion billion.
In one family of crystals, chilled to about 1 degree above absolute zero, the atomic magnets obey the same rule as the hydrogen in an ice cube. Break that rule once and the break splits into a north pole and a south pole that wander off on their own. They aren't the monopole physicists are still hunting. They're stranger: magnetic charges that exist only inside the crystal, joined by a real trail of flipped magnets. Neutrons have seen their fingerprints, and you can even listen to them.
Each half has its own north end and its own south end. Snap it again and again, down to a single atom, and both ends always come along. A lone north pole, a monopole, has never been found.
But in one kind of crystal, things that behave like lone north poles and lone south poles wander around in huge numbers. Each is about 1/8000 as strong as a true monopole would be. That odd fraction turns out to be the best part of the story. 🧲
🧲Chapter 1
Two in, two out
In two crystals, dysprosium titanate and holmium titanate, the atoms sit on tiny four-cornered pyramids joined corner to corner, and every corner holds a strong little atomic magnet.
Each atomic magnet is shared by two pyramids and can point only into one and out of the other. Cool the crystal to about 1 degree above absolute zero (1 kelvin, written 1 K: colder than deep space) and the magnets settle into a rule. In every pyramid:
Two in, two out.
Now break it:
Flip one magnet. One of its two pyramids now has three magnets pointing in: it acts like a north pole. The other has three pointing out: a south pole.
Flip another magnet of the north pole's pyramid, one that points in. The pole moves a step.
Keep going. The poles walk apart, leaving a trail of flipped magnets.
Here's a flat cartoon of the crystal (the real one is 3D). Drag a glowing pole away from its partner, or tap any arrow to make a new pair, and watch the trail 👇
Did you notice? The trail grows for free. Only the distance between the poles matters: their pull fades as they part. Bring them back together and they vanish. (The figure gives their pull as a temperature, physicists' shorthand for an amount of energy: about 3 K when the poles touch, a third of that when they are three spacings apart.)
Physicists call them emergent monopoles: not new particles, but patterns in how billions of atomic magnets point, existing only inside the crystal. A pole's "charge" is how strong it is, and each carries about 1/8000 of the charge of the true monopole Paul Dirac imagined. With the 2008 paper's own numbers it comes to about 1/7,700; with newer values for the crystal, about 1/7,900. The simple picture of them works best in a narrow window of cold, roughly between half a degree and 2 degrees above absolute zero. Colder than that they are still there, but they get stuck.
📜Chapter 2
Dirac's bet
If you read Nº 0, you met the monopole as a villain: as heavy as a grain of pollen, able to make protons fall apart. Here's why physicists wanted one in the first place.
Every electric charge ever measured is a whole number of one basic unit (quarks carry thirds, still whole numbers of a smaller unit). Nobody knew why. In 1931 Paul Dirac showed that a single magnetic monopole, anywhere in the universe, would force it to be so.
There was a price. In Dirac's quantum maths a monopole comes with a "string", a line that carries its magnetism away to infinity or to an opposite pole. That string must be completely invisible: no experiment may ever detect it. And that demand fixes the monopole's charge to exact steps, and even the smallest step is huge. Two such monopoles would push or pull on each other about 4,700 times harder than two electrons at the same distance do (this page's arithmetic).
On 14 February 1982 a superconducting loop at Stanford recorded one signal of just the right size, the "Valentine's Day monopole". It was never seen again. Searches since, in cosmic rays and at colliders, have come up empty. Even IceCube, a cubic kilometre of Antarctic ice wired up to catch particles from space, has looked for fast monopoles and found none; Francis Halzen, the physicist behind it, won the 2026 Nobel Prize in Physics for the neutrinos from space it caught. As of 24 October 2026, Dirac's monopole has not been found.
Remember the shape of the bet: invisible string, fixed charge. Spin ice runs it backwards.
❄️Chapter 3
The secret rule of ice
In an ice cube, each oxygen atom has four oxygen neighbours, with one hydrogen atom on each link between them. In 1933 John Bernal and Ralph Fowler worked out the rule: every oxygen keeps two hydrogens close and two farther away. Draw each link as an arrow pointing towards the oxygen its hydrogen sits near, and the rule becomes: two in, two out. Sound familiar?
The rule allows a staggering number of arrangements. In 1935 Linus Pauling estimated how many: only 6 of the 16 possible arrangements at each oxygen obey the rule, which leaves about 1.5 choices per water molecule, all multiplied together. For a 1-gram ice cube, just writing that number down would take about 5,900 billion billion digits (this page's arithmetic). A count of arrangements like this, measured by how many digits it takes to write down, is what physicists call entropy (Nº 3 puts a price on it).
When ice is cooled, its hydrogens get stuck, so some of that disorder never leaves, even at the lowest temperatures anyone has reached. Measurements in 1936 found just about the leftover amount Pauling predicted. No law of physics is broken: the disorder simply froze before it could tidy itself up.
💎Chapter 4
The same rule in a magnet
Back to the crystal of chapter 1: the pyramids' centres play the oxygens, and the atomic magnets play the hydrogens, each pointing into one pyramid and out of the other. Same rule.
A pyramid's four magnets can point in 16 ways, and only 6 obey. If they pointed at random, 10 pyramids in 16 would break the rule. In the real crystal at 1 degree, only about 1 in 18 does (this page's model), and far fewer when it's colder.
The puzzle was first worked out on paper in 1956, but no material was known to behave this way until 1997, when Mark Harris, Steven Bramwell and colleagues found that holmium titanate refused to line up as a magnet should. They named it spin ice.
In 1999 Arthur Ramirez and colleagues found close to Pauling's leftover disorder in dysprosium titanate: ice's frozen-in mess, inside a magnet. It's been argued about since. A 2013 experiment that waited longer reported it gone, a 2018 re-check found it still there, and in March 2026 two physicists proposed a new test (not yet peer-reviewed). The safe summary: leftover disorder at the lowest temperatures anyone has reached; whether it would drain away if you could wait long enough is still open.
🏋️Chapter 5
Every magnet is two blobs
In 2008 Claudio Castelnovo, Roderich Moessner and Shivaji Sondhi found a trick that turned all this into a story about charges.
Picture each atomic magnet as a dumbbell: a + blob at the end it points to and a − blob at the other, stretched so that the blobs sit at the centres of the two pyramids it joins. The forces between magnets come out almost exactly the same; it's just a cleverer drawing.
Now count the blobs in one pyramid: two + from the magnets pointing in, two − from those pointing out. They cancel. The ice rule is simply: every pyramid adds up to zero.
Flip one magnet and one pyramid has three + blobs against one −: two left over, a north pole. Its neighbour has two − left over, a south pole. That leftover is the pole's charge. It depends only on how strong each atomic magnet is and how far apart the pyramids are, and for these crystals it comes to about 1/7,700 of Dirac's charge, which the 2008 paper rounded to 1/8000.
Turn the two pyramids, flip the shared magnet, and squeeze the crystal 👇
The charge appears only where the rule breaks, and it belongs to the crystal: squeeze it by 1% and the charge would grow by about 1% (this page's arithmetic; nobody has measured it yet). Dirac's charge could never budge.
🧵Chapter 6
A string that costs nothing
Every pyramid along the trail still keeps two in, two out, and every arrangement that keeps the rule has almost exactly the same energy, so the trail is free. Pulling the poles all the way apart costs only a small, fixed amount. In an ordinary magnet, a line of flipped atoms costs energy at every step, like a stretched rubber band, so its ends stay tied. In spin ice the poles are free: deconfined.
Is a law of nature broken? Magnetic field lines never end; they always loop back. A true monopole would break that rule. These poles don't: the trail of flipped magnets carries the field back from one pole to the other, like a pipe made of magnets. (In physicists' words, the poles are sources of the field called H, not of the field called B.)
That trail is a Dirac string, the very thing Dirac needed to be invisible. Except this one is made of real flipped magnets, and experiments can detect such trails.
The poles don't roll or coast like balls. A pole moves only when a magnet next to it flips, and at these temperatures each flip is a quantum tunnel: the magnet slips through an energy barrier instead of climbing over it. In dysprosium titanate a pole hops about 2,400 times a second (measured from about 2 degrees up; this page assumes the same rate when colder), each time in a random direction among those open to it, so it wanders like a lost tourist: about 21 nanometres in a second, 1.3 micrometres in an hour, if nothing got in its way (this page's arithmetic). Something always does: long before that, it meets another pole.
🔄Chapter 7
Dirac, backwards
Dirac's charge is fixed because his string must be invisible: hide the string, and the charge can only come in exact steps.
In spin ice the string is real, and neutrons can detect the pattern such strings make (next chapter). With a visible string, nothing forces the charge to any particular value. It comes out at whatever the crystal makes it: about 1/7,700 of Dirac's here, something else in another crystal, and in principle something else again under pressure.
Dirac's logic runs in reverse, and the odd fraction is the clearest sign that these aren't his monopoles.
📜Dirac's monopole: invisible string, fixed charge. Could be anywhere in the universe. Not found.
💎A spin-ice pole: real string, a charge set by the crystal. Lives only inside it. Seen, as a pattern.
And they can't leave. A spin-ice pole is a pattern in the magnets, the way a traffic jam is a pattern in cars. Take the crystal away and nothing is left to carry the charge.
❄️ Your freezer does it too
Push one extra hydrogen nucleus into perfect water ice and it splits into two defects that wander off separately. One carries about 0.62 of an electron's charge, the other about 0.38: together, exactly 1. Nobody can catch a 0.62 particle on its own; the fractions only make sense inside the crystal. The spin-ice poles are the magnetic twins of the 0.62 kind.
🔦Chapter 8
The fingerprint
How do you see something that exists only as a pattern? Shine neutrons on it. Neutrons are tiny magnets themselves, so they bounce off a crystal's atomic magnets, and where they land shows how those magnets line up with each other.
If every pyramid keeps two in, two out, the magnets together act like a magnetic field whose lines never start or stop, and that leaves a particular fingerprint: bow ties, bright one way, dark the other, pinched to a point.
On 16 October 2009 the same issue of the journal Science carried two experiments (both had appeared online that September). Fennell and colleagues saw the bow ties in holmium titanate, but only by sorting the neutrons according to whether their own tiny magnets had flipped on the way through. Morris and colleagues put dysprosium titanate in a strong magnetic field and caught the strings, as cone-shaped patterns from trails of about 50 flipped magnets.
Add monopoles and watch the bow ties blur. Then switch to "All together", counting every neutron whatever happened to its spin, and watch them vanish 👇
Each broken pyramid spoils the rule around it, so the bow ties blur. And the neutrons never show a single pole: they see the fingerprint of the rule and the strings. (In arrays of tiny man-made magnets, scientists have imaged single poles and their strings.)
🌡️Chapter 9
A gas of poles
Above absolute zero a few magnets are always flipped the wrong way, so the crystal holds a thin gas of poles, and the colder it is, the fewer. At 1 degree, about 1 pyramid in 18 holds a pole. At half a degree, about 1 in 2,000. At 0.3 degrees, about 1 in a million (this page's model). Their pull on each other makes them more common: around 1 degree it roughly doubles their number, though in the deep cold, where poles are rare, the effect fades.
Cool the crystal and count the poles. Then switch their pull off and see how many fewer there would be 👇
Fewer poles means a slower magnet, because walking poles are how it adjusts. It settles in about 2 thousandths of a second at 1 degree and about a tenth of a second at half a degree in this page's simple model; real crystals slow down even faster as they cool, partly because the poles' paths are cramped (chapter 10). Below about half a degree real crystals freeze before they can settle; above about 2 degrees there are so many poles that the idea of a thin gas stops helping.
One claim is still disputed. In 2009 a team used particles called muons to measure the poles' charge and flow ("magnetricity"); in 2011 another team challenged the method, and as far as we can find, it was never settled in print. The case for the poles doesn't rest on it.
🎧Chapter 10
The hiss
Remember the Valentine's Day loop? In 2008 the spin-ice theorists suggested the same kind of instrument for their poles, and in 2019 Ritika Dusad and colleagues put a crystal inside a coil wired to a SQUID, an extremely sensitive magnetic detector.
One pole is far too weak to make a click: it would take about 7,700 of them to make the jump of one Dirac monopole. But at the temperatures of that experiment, a cubic centimetre holds something like a billion trillion of them (this page's model). So instead of clicks there's a hiss, as poles are born, wander and meet, flickering over thousandths of a second. Amplified, you can hear it.
And it sounded wrong. Free, independent poles would make a noise that dies away steeply at high pitch. The real hiss kept more of its high notes.
Press Listen, then switch between "Free poles" and "Measured": the real poles hiss brighter 👇
Why? In 2022 Jonathan Hallén and colleagues found that the poles can't walk freely. Their neighbours block some of their steps, so their paths are tangled and branching, like a fractal: within 14 hops a pole can reach only about 130 of the 2,000 or so places a free walker could. That cramped walk shapes the hiss.
🌌Chapter 11
A universe inside
So far the magnets flip one at a time. Now imagine a crystal where rings of six magnets flip together by quantum tunnelling. Theory from the early 2000s says it then becomes a tiny universe: our poles play its electric charges, waves in the magnets play its light, and its own monopoles obey Dirac's rule. The rule the ordinary crystal escaped comes back, one level down.
A 2021 calculation puts its speed of light at about 1 metre per second, slower than you walk, and the number that sets how strongly its charges feel light (the fine-structure constant) near 1/10. Ours is 1/137.
Race the crystal's light against a person walking. Turn up the tunnelling and light wins 👇
Is it real? Crystals containing cerium gave what the teams studying them call strong evidence in 2025 and 2026: neutron signals that look like the predicted light and charges. But a paper from June 2026 still calls decisive confirmation an outstanding challenge, and as of 24 October 2026 it isn't settled. The walking-pace light and the 1/10 are theory, not measurements.
Dirac's monopole is still missing. Meanwhile, a crystal a few millimetres across lets us play with the world he imagined, and turn its knobs. Let's add it up. 🧾
🧾 One crystal, itemised
Ways a pyramid's magnets can pointout of 166 obey
Charge of one polevs Dirac's monopole1/7,700
Pyramids holding a poleat 1 degree, this page's model1 in 18
Hops per secondeach one a quantum tunnel2,400
Distance wandered in an hourfree walk, this page's arithmetic1.3 µm
Light in quantum spin icetheory, not measured~1 m/s
Dirac monopoles foundas of 24 October 20260
two in, two out ✦ the string is free ✦ the charge is not fixed
Somewhere in a lab tonight, a small crystal sits 1 degree above absolute zero.
Inside it, billions of billions of tiny magnets keep a rule they share with ice: two in, two out.
Now and then one of them tunnels the wrong way, and the rule breaks in two places at once.
A north and a south slip apart, hopping a few thousand times a second, trailing a thread of flipped magnets that costs them nothing.
They will never leave the crystal. And the monopole Dirac imagined, the one that would explain why every charge comes in whole units, is still out there, or nowhere.
Dirac's string had to be invisible, so his charge is fixed. In spin ice the string is real, so the charge is whatever the crystal makes it.
Nº 14 · ∫ Full version
One eight-thousandth
In two crystals cooled to about one kelvin, dysprosium titanate and holmium titanate, the atomic magnets obey the same rule as the protons in water ice: at every small tetrahedron, two point in and two point out. Break the rule once and the break splits in two. The halves walk apart and act as point sources of the magnetic field H: a north and a south magnetic charge of ±2μ/ad, about one eight-thousandth of the charge h/(μ0e) that Dirac's monopole would carry. The field B still has no sources: its flux returns along the chain of flipped magnets the pair leaves behind. That chain is a Dirac string made of real flipped magnets, and because it can be observed, nothing forces the charge to be quantised. These are emergent monopoles. They exist only inside the crystal, and neutrons saw their fingerprints in 2009.
This page explains Dirac's argument for why one monopole would force every electric charge to come in whole units, the ice rule and the entropy it leaves behind, spin ice, the dumbbell model that turns every atomic magnet into two charges, the string that costs nothing, water ice's own fractional charges, what neutrons saw, the monopole gas and its contested "magnetricity", the hiss of monopoles in a SQUID and the fractal they walk on, and quantum spin ice, a crystal that may have its own light.
This is the ∫ Full version, with the equations, caveats and sources. New to the idea? Start with the ✨ Simple version.
Numbers marked "this page's arithmetic" or "this page's model" were computed for this page from the cited inputs; they are not published results. The charge uses the inputs of Castelnovo, Moessner and Sondhi (2008); the monopole gas uses their Debye–Hückel theory (2011) with the hop rate measured by Bovo et al. (2013). Results and preprints are as of 24 October 2026.
Part IThe pole nobody has found
Chapter 1One eight-thousandth
This page is about one eight-thousandth of the magnetic charge that Paul Dirac's monopole would carry, if it exists. Nobody has found Dirac's monopole.1 But in two crystals cooled to about one kelvin, dysprosium titanate (Dy₂Ti₂O₇) and holmium titanate (Ho₂Ti₂O₇), something behaves like a gas of separate north and south magnetic charges, each about one eight-thousandth as strong as Dirac's.2
Both crystals are built of tetrahedra that share corners, with a strongly magnetic rare-earth ion at every corner. Each ion belongs to two tetrahedra, and its magnetic moment, about 10 Bohr magnetons (μB), can point only along the line between their centres: into one and out of the other. At low temperature the interactions favour the arrangements with two moments pointing in and two pointing out of every tetrahedron. It is the rule the hydrogen atoms obey around each oxygen in water ice, which is why these crystals are called spin ice.3
Flip one moment and the two tetrahedra it joins break the rule: one becomes three in and one out, the other one in and three out. Flip a suitable neighbour and one of the two rule-breakers moves one step, while every tetrahedron in between keeps the rule. Keep flipping and the pair walks apart, leaving a trail of flipped moments behind. Figure 1 lets you do this on a flat cartoon of the crystal: drag a pole, or tap an arrow to make a new pair, and watch the readouts for the string and the pull.
Figure 1🧩 Try it
Two in, two out, and one rule broken
Drag a glowing pole
Show as
Rule-breakers
String
Apart
Pull between themEach defect carriesCost of the pair0.62 e
A flat cartoon of spin ice: arrows on a square lattice (the six-vertex or square-ice model; Lieb 1967) in a disordered state that obeys the ice rule, two arrows in and two out at every crossing. In real spin ice the arrows are atomic magnets on corner-sharing tetrahedra (Chapter 4), and a plain two-dimensional square array of nanomagnets orders instead, because its six ice crossings are not equivalent (Wang et al. 2006), unless it is specially built to make them equal (Perrin et al. 2016). A crossing with three arrows in is a north pole (+), one in is a south pole (−). Moves follow the rule that a flip changes the charge at its two ends; double charges are refused, and the arrows at the edge are frozen. 'String' counts the arrows that differ from the ice-rule state the first pair was made from (so it starts at 4). 'Pull' is the three-dimensional dumbbell Coulomb law, −3.06 K × (one spacing ÷ distance), used here as an illustration (Castelnovo, Moessner and Sondhi 2011). The water-ice view draws the same state as oxygens and protons: three near protons is an H₃O⁺ defect, one is OH⁻, each carrying an effective charge of about 0.62 e (Petrenko and Whitworth, via De Poorter 2019). 'Ordinary magnet' charges an illustrative 1 K for every flipped arrow, as a ferromagnet charges for a line of flipped spins (Chapter 6): 'Cost of the pair' is then n × 1 K − 3.06 K ÷ r, for n flipped arrows and poles r spacings apart.A flat cartoon of the crystal: every crossing has two arrows in and two out. Drag a glowing pole and watch the trail of flipped arrows grow behind it. Making the pair costs energy, but the trail is free: after that, only the pull between the two poles changes the energy, and the pull fades as they part. Bring them back together and they vanish. Switch to 'Water ice' to see the same puzzle as the hydrogen in an ice cube. Tap 'Ordinary magnet' and every flipped arrow costs energy, as it would in a lump of iron: the trail becomes a stretched rubber band and the poles stay tied.
The trail grows and shrinks for free; only the distance between its two ends changes the energy. The ends behave as a north pole and a south pole. In 2008 Claudio Castelnovo, Roderich Moessner and Shivaji Sondhi showed that in the real three-dimensional crystal they are point charges of the magnetic field H, attract each other by a Coulomb law and are free to separate: emergent magnetic monopoles.2 Their charge is
q = 2μ/ad ≈ 4.6 μB/Å ≈ 4.3 × 10−13 A·m, qD/q ≈ 7,700μ ≈ 10 μB is the moment of each ion; ad ≈ 4.34 Å is the distance between the centres of neighbouring tetrahedra; qD = h/(μ0e) is Dirac's charge. Chapter 2 explains qD, Chapter 5 derives q.
Castelnovo, Moessner and Sondhi (CMS from here on) rounded the ratio to 8,000 in their paper. With their inputs it comes out at 7,693 (this page's arithmetic); with the refined values for Dy₂Ti₂O₇ that CMS used in 2011 (9.87 μB, ad = 4.38 Å, q ≈ 4.5 μB/Å) it is about 7,900.4 "One eight-thousandth" is a fair rounding; this page uses 1/7,700, the value for the 2008 inputs. The ratio is not a constant of nature. It is set by the size of the atomic moments and the spacing of the crystal, and Chapter 7 explains why nothing forces it to take any particular value.
Four qualifiers go with every sentence on this page. These are emergent monopoles: patterns in the arrangement of atomic magnets, not particles that could be pulled out of the crystal. They are sources of H, not of B, and Maxwell's ∇·B = 0 holds inside spin ice. Their charge is not quantised. And the simple picture of them works in a narrow window: below about 0.5–0.6 K the crystals fall out of equilibrium, and above about 2 K there are so many rule-breakers that the picture of a dilute gas of charges stops helping.5,4
The evidence for the picture is strong. Neutrons saw the pattern the ice rule makes and the strings the monopoles leave, in two papers published on the same day in 2009,5,6 and a SQUID heard the monopoles' magnetic noise in 2019.7 The page first tells Dirac's argument (Chapter 2), then the ice rule (Chapters 3 and 4), then how breaking it makes charges and why their charge is a fraction (Chapters 5 to 8), and finally how the charges were seen and heard, and what a quantum version of the crystal might hold (Chapters 9 to 12).
Chapter 2Dirac's bargain
Nº 0 met the grand-unified monopole, a particle as heavy as a pollen grain (20 to 200 nanograms) that could make protons decay. Here is the older argument for why anyone expected monopoles at all. It is Paul Dirac's, from 1931, and it is short.8
Classical electromagnetism does not mind a magnetic charge. Add a magnetic charge density to ∇·B = 0 and a magnetic current to Faraday's law, and Maxwell's equations become more symmetric, not less. Quantum mechanics minds. A charged particle feels electromagnetism through the vector potential A, with B = ∇ × A. If B has a point source, A cannot be smooth all the way around it: it has to misbehave along a line that runs from the monopole to infinity, or to an antimonopole. Dirac called that line a string. Picture an infinitely thin solenoid that brings the monopole's flux in from far away and releases it at its end.
An electron that goes once around a thin solenoid picks up a quantum phase eΦ/ħ, where Φ is the flux inside, even though it never touches the field. For the string to be undetectable by any experiment, that phase must be a whole multiple of 2π. The string carries the monopole's entire flux, Φ = μ0q, so
μ0qe = nh, so qD = h/(μ0e) ≈ 3.29 × 10−9 A·mSI units, in which a magnetic charge q makes a field B = μ0q/(4πr2). In Gaussian units the condition reads g = ne/(2α), so the smallest Dirac charge is about 68.5 e. Its flux is h/e, two superconducting flux quanta.
Read it the other way round and you have the bargain. If even one monopole exists anywhere in the universe, every electric charge must be a whole multiple of h/(μ0q). Every charge ever measured is a multiple of one unit (quarks carry thirds of e, still multiples of a single unit), a regularity that otherwise needs a separate explanation. One monopole would buy the quantisation of charge.
The price is a strong charge. The force between two Dirac monopoles would be 1/(4α2) ≈ 4,695 times the force between two electrons at the same distance (this page's arithmetic). And the argument says nothing about whether monopoles exist, how heavy they would be or how many there are.
Grand unified theories do predict monopoles, with the enormous masses Nº 0 quotes, and the searches have been long. Blas Cabrera's superconducting loop at Stanford recorded a single event on 14 February 1982 with the flux step a Dirac monopole would make; it was never seen again.9 Searches in cosmic rays and at colliders have found nothing since.1 In 2022 the MoEDAL experiment at CERN looked for monopoles produced by the strongest magnetic fields known, those of colliding lead nuclei, and excluded monopoles carrying one to three Dirac charges with masses up to 75 GeV for that production mechanism.10 In 2024 MoEDAL scanned a beam pipe that had surrounded lead–lead collisions in the CMS experiment in 2011, and excluded monopoles carrying 2 to 45 Dirac charges with masses up to 80 GeV, again for that mechanism.11 For fast monopoles arriving from space, IceCube, a cubic kilometre of instrumented ice at the South Pole, set in 2022 what its authors called the strictest limit to date: fewer than about 2 × 10−19 per square centimetre per second per steradian over most of the speed range from 0.75 to 0.995 of the speed of light.12 IceCube's principal investigator, Francis Halzen, was awarded the 2026 Nobel Prize in Physics, announced on 6 October 2026, for the observatory and its discovery of high-energy neutrinos from space.13 As of 24 October 2026, Dirac's monopole has not been found.
Keep the shape of the argument in mind, because spin ice runs it backwards. In Dirac's version the string is invisible, so the charge is fixed. Chapter 7 asks what happens when the string can be seen.
Part IITwo in, two out
Chapter 3A rule from the freezer
In ordinary ice each oxygen atom has four neighbouring oxygens at the corners of a tetrahedron, and a hydrogen atom sits on each of the four bonds to them. In 1933 John Desmond Bernal and Ralph Fowler proposed the rules: each bond carries exactly one hydrogen, and each oxygen has two hydrogens close to it and two farther away, so that the crystal is made of whole water molecules.14 These are the ice rules. Draw each bond as an arrow pointing towards the oxygen its hydrogen sits near, and the rule becomes: two in, two out.
The rules do not pick one arrangement. In 1935 Linus Pauling estimated how many they allow. Each of the 2N hydrogens in N molecules can sit near one end of its bond or the other, 22N = 4N choices in all. At each oxygen only 6 of the 16 local arrangements obey the rule. Treating the oxygens as independent gives15
W ≈ 4N × (6/16)N = (3/2)N, S = kB ln W = R ln(3/2) ≈ 3.37 J/(K mol)Per mole of water, or 0.806 cal/(K mol). Pauling's count ignores the correlations around closed rings of bonds; a series expansion by John Nagle (1966) gives 0.8145 cal/(K mol).
That entropy stays when the ice is cooled, because the protons stop moving between arrangements long before the lowest temperatures are reached. Calorimetry by William Giauque and J. W. Stout in 1936, from 15 K upward, left about 0.82 ± 0.05 cal/(K mol) of entropy unaccounted for at the bottom, in agreement with Pauling.16 Nagle's refined count agrees with that measurement even better than Pauling's.17
Entropy is the logarithm of a count of the microscopic arrangements that look the same from outside (Nº 3 puts a price on it), and ice is the classic example of such a count. The count is absurd: a 1 g ice cube can arrange its protons in about 1.53.3 × 1022 ways, a number with about 5.9 × 1021 digits (this page's arithmetic). For the flat version of the rule used in Figure 1, Elliott Lieb found the exact count in 1967: (4/3)3/2 ≈ 1.540 per crossing, a few per cent above Pauling's 1.5.18 Figure 1's "Water ice" view draws the puzzle as oxygens and protons.
Does this break the third law of thermodynamics? No. The third law is about equilibrium, and ice's leftover entropy belongs to disorder that is frozen in because the protons can no longer rearrange. Given a way to move, the protons do order: ice doped with a trace of potassium hydroxide, which lets them rearrange, turns into proton-ordered ice XI below about 72 K (Tajima, Matsuo and Suga, 1982). Rainer Feistel has argued that such frozen states need the different kinds of entropy, those of Clausius, Boltzmann and Pauling, kept apart.19 The safe statement is "leftover entropy at the lowest temperatures reached".
Chapter 4The same rule in a magnet
Now replace the oxygens by the centres of tetrahedra and the hydrogens by magnetic moments. In the pyrochlore oxides Dy₂Ti₂O₇ and Ho₂Ti₂O₇, the rare-earth ions sit on a network of corner-sharing tetrahedra: each ion belongs to two tetrahedra, and each tetrahedron has four ions. The electric field of the surrounding oxygen atoms forces each ion's moment, about 10 μB, to lie along the line joining the centres of its two tetrahedra. So the moment can only point into one tetrahedron and out of the other, or the reverse. It is an Ising variable: in or out.3,20
The four moments of a tetrahedron have 24 = 16 arrangements. Six are two in, two out. Eight are three in and one out, or the reverse: a single rule-breaker. Two are all in or all out: a double one. If the moments pointed at random, 10 of every 16 tetrahedra, 62.5%, would break the rule. In spin ice at 1 K only about one in 18 does (this page's model, Chapter 10), and far fewer at lower temperatures. The map to water ice is exact at the level of the rule:
Water ice
Spin ice
oxygen atom
centre of a tetrahedron
hydrogen bond
magnetic moment
proton close to this oxygen
moment pointing into this tetrahedron
two near, two far
two in, two out
Pauling's (3/2)N for N molecules
(3/2)N/2 for N moments
The last row follows from the same count: 2N arrangements of N moments, times 6/16 for each of the N/2 tetrahedra, gives (3/2)N/2, a leftover entropy of (R/2) ln(3/2) ≈ 1.69 J/(K mol) of moments.2
The idea is older than the materials. In 1956 Philip Anderson, studying how magnetic ions order in ferrites, analysed Ising spins on this lattice with an antiferromagnetic coupling and found a macroscopic number of ground states, a problem that maps onto Pauling's ice.21,20 For decades it was a model without a material. In 1997 Mark Harris, Steven Bramwell and colleagues found that the ferromagnetic pyrochlore Ho₂Ti₂O₇ does not order as a ferromagnet should, explained it by the local axes of its moments, and named the state spin ice.3 Two years later Arthur Ramirez and colleagues measured the heat Dy₂Ti₂O₇ absorbs as it warms from below a kelvin and found its entropy short by close to Pauling's value: ice's leftover entropy, in a magnet.22
That measurement has been argued over. In 2013 Pomaranski and colleagues, equilibrating Dy₂Ti₂O₇ over much longer times, reported that the residual entropy was absent.23 In 2018 Giblin and colleagues re-examined the question and found no evidence for a loss of the ice-rule correlations or of the entropy at long times.24 In March 2026 Sergey Syzranov and Arthur Ramirez proposed a simpler thermodynamic test, which relies on a Maxwell relation rather than on integrating the heat capacity over a wide range (a preprint).25 So, again, the safe statement is "leftover entropy at the lowest temperatures reached", with the long-time question still open.
The analogy has limits. The "protons" are moments; the energy that enforces the rule is a few kelvin rather than thousands; and the moments interact over long distances through their dipolar fields, which ought to spoil a rule that only counts nearest neighbours. Chapter 5 shows why it does not. Spin ice has since become a field of its own: in 2020 Bramwell and Harris counted "well over five thousand papers" since its discovery.26
Part IIIBreaking the rule
Chapter 5Every magnet is two charges
A magnetic dipole can be pictured as two opposite magnetic charges a short distance apart. CMS chose the distance cleverly. They stretched each atomic moment into a dumbbell long enough to reach from the centre of one tetrahedron to the centre of the next, with charges +μ/ad and −μ/ad at its two ends, so that the dipole moment stays the same.2 All the charges now sit on the tetrahedron centres, which form a diamond lattice with spacing ad = √(3/2) a, where a is the distance between neighbouring ions.
The dipolar energy of the whole crystal then becomes a Coulomb energy between the net charges of the tetrahedra, plus corrections that fall off at least as fast as 1/r5.2 Two in, two out means two + ends and two − ends meet in every tetrahedron: every tetrahedron is neutral. That is why the long-range dipolar force does not spoil the ice rule. A crystal of neutral tetrahedra has, to this accuracy, the same energy in every ice-rule arrangement, so it keeps Pauling's entropy. Not quite exactly: the small corrections favour one ordered arrangement, and simulations predict that a crystal given unlimited time would order at about 0.2 K (Melko, den Hertog and Gingras, 2001). Real crystals freeze long before they could find it, which is one reason the long-time entropy measurements of Chapter 4 matter. Ryzhkin had already treated the ice-rule defects as magnetic charges in a 2005 theory of how these crystals relax;27 the dumbbell picture showed that they interact as Coulomb charges and are free to separate.
Flip one moment. One tetrahedron is now three in, one out: three + ends and one − end, a net charge of +2μ/ad. Its neighbour is one in, three out, with −2μ/ad. That is a monopole and an antimonopole at contact. With μ = 10 μB and a = 3.54 Å, the values CMS used:
q = 2μ/ad = 4.61 μB/Å = 4.28 × 10−13 A·m qD = h/(μ0e) = 35,487 μB/Å, qD/q = 7,693This page's arithmetic, with ad = 4.336 Å and 1 μB/Å = 9.274 × 10−14 A·m. CMS (2011) quote q ≃ 4.6 μB/Å ≃ 4.28 × 10−13 J/(T m).
The energy of a pair at contact sets the scale for everything that follows:4
Enn = μ0q2/(4πadkB) ≈ 3.06 KThe Coulomb attraction of a monopole and an antimonopole on neighbouring tetrahedra, in kelvin. A doubly charged tetrahedron (all in or all out) costs about four times as much as a single one, so doubles are rare at low temperature.
Figure 2 shows two corner-sharing tetrahedra with the shared moment already flipped. Switch between magnets and dumbbells, flip the shared moment back and forth, and squeeze the crystal with the spacing slider. Watch which readouts move.
Figure 2🧲 Try it
Where one eight-thousandth comes from
Flip, turn, squeeze
Each pole carries
Share of Dirac's charge
Pull when touching
Two corner-sharing tetrahedra of the pyrochlore lattice, with each atomic moment (about 10 μB for Dy³⁺ and Ho³⁺) along the line joining the centres of the two tetrahedra it belongs to. In the dumbbell model (Castelnovo, Moessner and Sondhi 2008) each moment becomes charges ±μ/ad at those centres, ad = √(3/2) a being the distance between tetrahedron centres; the beads are drawn a little off the centres so that all four at each centre can be seen. Two in, two out makes every tetrahedron neutral; flipping the shared moment leaves charges ±2μ/ad = ±4.61 μB/Å (4.28 × 10−13 A·m), and the Dirac charge h/(μ0e) is 35,487 μB/Å, so the ratio is about 1/7,700 (this page's arithmetic with their inputs; the paper rounds it to 1/8000). 'Pull when touching' is the Coulomb energy at one spacing, μ0q2/(4πad), in kelvin (3.06 K, CMS 2011). The slider ranges go beyond real materials; that pressure would tune the charge is a theoretical remark (CMS 2008), not yet a measurement.Two pyramids of atomic magnets that share one corner. It opens as dumbbells: each magnet becomes a + blob and a − blob sitting in the middles of the pyramids (tap 'Magnets' to see the arrows). When the rule holds, each pyramid's blobs cancel. Flip the shared magnet and one pyramid is left with two spare + blobs, the other with two spare −: a north and a south pole. Squeeze the crystal and watch the charge change. Dirac's charge never could.
The charge appears only where the rule breaks, and its size is a property of the material. A 1% squeeze of the spacing would raise it by about 1% (this page's arithmetic). CMS remarked that the charge "can even be tuned continuously by applying pressure";2 spin ice has been squeezed in the lab (neutron measurements on holmium titanate under uniaxial pressure, Edberg and colleagues, 2019), but we found no experiment that has measured a change in the charge, so "would" is the right word. Dirac's charge, fixed by h and e, could never move.
Some numbers to calibrate how strong these charges are (this page's arithmetic):
The force between two spin-ice monopoles is 12,600 times weaker than between two electrons at the same distance; between two Dirac monopoles it would be 4,695 times stronger. Between the two kinds of monopole lies a factor of 59 million, 7,6932.
One tetrahedron spacing away (4.3 Å), a monopole's field is about 0.23 T, several thousand times the Earth's field of 25–65 µT. At 10 nm it has fallen to 0.4 mT.
The attraction falls to a tenth of its contact value, 0.3 K, at ten spacings, about 4 nm.
Chapter 6A string that costs nothing
Return to the pair at contact. Flip one of the three moments that point into the north pole's tetrahedron. That tetrahedron is back to two in, two out, and the tetrahedron at the other end of the flipped moment becomes three in, one out. The pole has hopped one step. Repeat, and the two poles separate along a chain of flipped moments.
The chain costs nothing, because every tetrahedron along it still obeys the rule and every ice-rule arrangement has, to a very good approximation, the same energy. The only cost is the Coulomb attraction between the ends:
E(r) = −μ0q2/(4πr)The energy of the pair relative to infinite separation. It stays finite as the pair is pulled apart: separating it completely costs only about 3 K more than the pair at contact. The poles are deconfined.
Compare an ordinary ferromagnet. There, a line of flipped spins costs energy along its whole length, so the cost grows with the separation and the two ends stay bound: confined. In spin ice the low-temperature state is already a soup of all possible strings, so one more string is free. That is what "deconfined" means, and it is why the ends act as independent particles.
What kind of charge are they? Inside matter there are two magnetic fields. B turns compass needles and pushes on moving charges. H is defined through B = μ0(H + M), where M is the magnetisation, so
∇·B = 0, ∇·H = −∇·MWherever the magnetisation converges or diverges, H has a source or a sink, with density −∇·M. The poles at the ends of a bar magnet are sources of this kind.
A spin-ice monopole is a point where the moments converge (three in) or diverge (three out): a point source of H. B has no source there. In CMS's words, "a compensating flux travels along the (unquantised) 'Dirac string'" of flipped moments.2 So the trail is a Dirac string in the literal sense: a thin tube that carries the flux of B from one pole back to the other, so that the field lines of B never end. Maxwell's equations hold. What is unusual is not a broken law but a magnetisation that has organised itself into mobile, point-like poles.
How do the poles move? Not like billiard balls. Jaubert and Holdsworth put it this way: "the charges have no mass and therefore have diffusive, rather than Newtonian dynamics".28 A pole moves only when one moment flips, and at low temperature the flip is a quantum process. Bovo and colleagues found that monopoles in Dy₂Ti₂O₇ hop 2,430 times a second, at a rate that does not change with temperature over the range they measured, from about 2 K up: the signature of a moment tunnelling through its energy barrier rather than climbing over it.29 This page uses the same rate at lower temperatures, which is an extrapolation.
D = ν0ad2/6 ≈ 7.6 × 10−17 m2/s, √(6Dt) ≈ 21 nm after one secondThis page's arithmetic for a free random walk, with ν0 = 2.43 × 103 s−1 (Bovo et al. 2013) and ad = 4.34 Å: 1.3 µm in an hour, 6.3 µm in a day, for a pole that walked freely that long. None does: at 1 K the poles are about 1.3 nm apart (Chapter 10), a distance a free walker covers in a few milliseconds, and the real walk is cramped (Chapter 11). If every hop went the same way, a pole would cover about 1 µm per second.
The walk is also constrained. A north pole can move only by flipping one of its three "in" moments, so at most three of its four directions are open, and north and south poles travel the network of strings in opposite senses.28 A pole that retraces its steps erases its string. A pole that meets an antipole by another route annihilates it and leaves a closed loop of flipped moments: a new ice-rule state, so the crystal remembers. Figure 1 shows each of these in the flat cartoon. Christopher Henley's review calls the whole low-temperature state a Coulomb phase, in which the defects "behave as effective charges with Coulomb interactions".30
Chapter 7Dirac's argument, run backwards
Now the centre of the page. Dirac's charge is quantised because his string must be invisible: the phase condition of Chapter 2 is the price of hiding it. CMS make the point directly: the quantisation "follows from the monopole being attached to a Dirac string which has to be unobservable".2
In spin ice the string is a chain of real flipped moments, and it is observable. Neutrons see the statistics of such strings (Chapter 9), and in arrays of nanomagnets, spin ice's man-made cousins, strings have been imaged one by one. With a visible string, nothing imposes the phase condition, so nothing quantises the charge. The charge is whatever 2μ/ad is for the crystal at hand: about 1/7,700 of Dirac's for these two materials with CMS's inputs, different in another material, and different again under pressure. CMS's 2012 review lists "observable Dirac strings, and irrational charges" among the things spin ice offers.31
Dirac's logic is not broken; it runs in reverse. Invisible string, fixed charge. Visible string, any charge. Seen this way, the fraction is not a curiosity. It is the clearest sign that these are not Dirac's monopoles.
Nor can they leave. A spin-ice monopole is a configuration of the moments, in the way a wave is a configuration of water or a hole is a missing electron in a semiconductor. Take the crystal away and nothing is left to carry the charge.
Dirac's monopole
Spin-ice monopole
Source of
B (and H)
H only; the flux of B returns along the string
String
invisible, by construction
a real chain of flipped moments, seen by neutrons
Charge
fixed: nh/(μ0e)
set by the crystal: 2μ/ad, about 1/7,700 of Dirac's
Mass
not fixed by the argument; grand unified versions as heavy as a pollen grain (Nº 0)
no inertia in the standard picture: diffusive hops
Where
anywhere in the universe
only inside the crystal
Found
no, as of 24 October 2026
yes, as emergent quasiparticles
Chapter 8Fractional charges in your freezer
Water ice had fractional charges all along. It has two kinds of rule-breaker. Ionic defects, H₃O⁺ and OH⁻, break the rule of two hydrogens per oxygen. Bjerrum defects break the rule of one hydrogen per bond: a D defect has two protons on one bond, an L defect has none.32
A moving defect turns water molecules as it passes, so the charge it carries along is less than e. In Jaccard's theory of the electrical properties of ice, ionic defects carry an effective charge e± ≈ 0.62 e and Bjerrum defects eDL ≈ 0.38 e, and the two add up to exactly e (values from Petrenko and Whitworth, as used by De Poorter).33,32 An ab initio calculation of 1983 put the Bjerrum share at 0.36 ± 0.03 e.34 Put one extra proton into perfect ice and it splits into an H₃O⁺ ion and a D defect that wander off independently, one carrying about 0.62 e, the other about 0.38 e.
e± + eDL = eThe effective charges are defined through the polarisation the defects leave behind as they move. Nobody isolates a particle of 0.62 e: the fraction is real inside the crystal and meaningless outside it.
Fractional charges of this kind are not unique to ice. In the fractional quantum Hall effect, a thin sheet of electrons in a strong magnetic field has quasiparticles of charge e/3, and in 1997 two groups measured that charge from the shot noise of the current: the size of the lumps in which charge arrives.35,36 Chapter 11 listens to spin ice's noise too, though there it was used to learn how the charges move, not how big they are.
Fennell and colleagues described the spin-ice monopoles as "analogous to ice's mobile ionic defects".6 The analogy is close. Spin ice has only the ionic kind, because every bond carries exactly one moment; there is no bond with two or with none. The spin-ice monopole is the magnetic twin of H₃O⁺ and OH⁻, and its fraction of Dirac's charge plays the part of ice's 0.62 e. Figure 1's "Water ice" view relabels the puzzle this way: a crossing with three near protons is an H₃O⁺ ion, one with a single near proton an OH⁻.
Part IVSeeing, hearing, and a universe inside
Chapter 9What the neutrons saw
Neutrons carry a magnetic moment, so they scatter off magnetic moments, and the pattern of scattered intensity measures how the moments are correlated. What would the ice rule look like? Coarse-grain the moments into a smooth field. Two in, two out at every tetrahedron means the field has zero divergence everywhere: it behaves like a magnetic field with no sources. A random field with that single constraint has dipolar correlations at long distance, and in reciprocal space its spectrum is the transverse projector:37,30
Sab(q) ∝ δab − qaqb/q2a and b label components of the moment field. The component along q is missing. Near each reciprocal-lattice point the intensity of one component is a bow tie: bright along one direction, dark across it, pinched to a point at the centre.
These bow ties are called pinch points. Monopoles break the constraint, so the rule holds only over a finite length ξ, and the bow ties blunt. Fennell and colleagues described the softened pattern near a pinch point as
S ∝ (ql2 + ξ−2)/(q2 + ξ−2)ql is the component of q along the bright direction, measured from the pinch point. Along the dark direction the intensity no longer falls to zero at the centre: a region about 1/ξ wide around the pinch point fills in. Random moments give a flat pattern.
In Ho₂Ti₂O₇ at 1.7 K, Fennell and colleagues used polarised neutrons: they sorted the scattered neutrons by whether their spin had flipped. The pinch points appeared only in the spin-flip channel. In the other channel the scattering is strong near the zone centre, and adding the two hides the bow ties. They found an ice-rule correlation length of 182 ± 65 Å at 1.7 K, about 18 nm or some 40 tetrahedron spacings (this page's arithmetic), growing to 234 ± 67 Å at 1.3 K.6 They read ξ as the maximum length of the Dirac strings, and found it growing on cooling roughly as eE/T, with an energy E = 1.7 ± 0.1 K, close to the effective exchange energy; at low temperature, they concluded, the monopoles are free over at least micrometres. Figure 3 shows the same physics for the flat cartoon of Figure 1: change the share of broken crossings, move the cut lines, and switch between one channel and both.
Figure 3🔦 Try it
The fingerprint of a field with no sources
Add monopoles
Neutrons
no scatteringstrong
Monopoles
Ice rule holds over
Bow tie
The scattered intensity of one arrow component, Sxx(q), for the flat square-ice cartoon of Figure 1, over two Brillouin zones in each direction (q in reciprocal lattice units): Sxx ∝ (sy2 + κ2)/(sx2 + sy2 + κ2) with s = sin(πq) (so the pinch points sit at whole numbers), normalised to a mean of 1, the lattice form of the transverse projector δab − qaqb/q2 that a divergence-free field produces (Isakov et al. 2004; Henley 2010). κ is set from the monopole share (every crossing that breaks the rule: random arrows give 50% single and 12.5% double, 62.5% in all) by fits to this page's Monte Carlo of square ice (seed 14 on a 24 × 24 lattice; the fitted points are in the page's source); the ice rule then holds over about 1/(2κ) lattice spacings of the cartoon (its lengths are not the crystal's). Next to the map (below it on a phone): the intensity along the two cut lines. 'All together' adds the other component, which in this 2D model fills the bow tie in completely (with monopoles present, a soft round spot remains: the monopoles' own scattering); in the real 3D crystal the non-spin-flip channel is strong near the zone centre and hides it, which is why Fennell et al. used polarised neutrons and saw pinch points only in the spin-flip channel, with an ice-rule correlation length of 182 ± 65 Å at 1.7 K (Ho₂Ti₂O₇, 2009).What a beam of neutrons would see bouncing off the puzzle crystal. When every crossing keeps the rule, the pattern has sharp bow ties: that's the fingerprint of a magnetic field whose lines never start or stop. Add monopoles and the bow ties blur. Count every neutron, whatever happened to its spin, and in this cartoon the bow ties disappear completely; in the real crystal the other neutrons drown them out, which is why the 2009 experiment had to sort them.
The bow tie is as close as one gets to seeing "no sources" directly. Monopoles set a length ξ beyond which the rule is lost, and the bow ties blur over a region about 1/ξ wide. The 'All together' switch shows why the experiment needed polarised neutrons: in the cartoon the second component fills the bow tie in completely, and in the real crystal it hides it.
In the same issue, dated 16 October 2009 (both papers had appeared online that September), Science published a second experiment. Morris and colleagues put Dy₂Ti₂O₇ in a magnetic field along the [001] direction, close to saturation, where nearly all moments are pinned. The only excitations left are strings of flipped moments running roughly along the field, which behave like random walks with distance along the field playing the part of time. Instead of pinch points the neutrons showed cone-shaped scattering from strings of order 50 sites long at 0.7 K, and when the field was tilted by about 10° the cones collapsed into sheets, as tilted random walks should. The heat capacity was fitted by the Debye–Hückel theory of a gas of monopoles (Chapter 10). They measured above 0.6 K, because below that the sample froze.5
Neither experiment sees a single monopole. Neutrons measure correlations: the pattern the ice rule makes, and the statistics of the strings. Single monopoles and strings have been seen one by one in artificial spin ice, arrays of nanoscale magnets made by lithography: monopole defects in a connected array (Ladak and colleagues, 2010), monopoles with their Dirac strings in a kagome array (Mengotti and colleagues, 2011), and their motion in real time in arrays that fluctuate thermally (Farhan and colleagues, 2019).38,39,40 In a "qubit spin ice" built from superconducting qubits, flipping a spin at the boundary created a monopole inside.41 These are cousins, not spin ice itself. In artificial square ice of the kind Wang and colleagues introduced in 2006, the six two-in-two-out arrangements of a crossing are not equivalent, so its lowest-energy state is ordered rather than ice-like;42 that is why Figures 1 and 3 call the flat lattice a cartoon. A square array built so that the six crossings cost the same does reach the ice-like state, with pinch points and freely moving monopoles (Perrin and colleagues, 2016).
Chapter 10A gas of magnetic charges
At any temperature above zero some moments are flipped against the rule, so the crystal holds a gas of monopoles. Without interactions their density per tetrahedron would follow a Boltzmann law,
ρ = x/(1 + x), x = 2 e−Δ/TΔ is the cost of creating an isolated monopole, in kelvin: 4.35 K in Dy₂Ti₂O₇ and 5.79 K in Ho₂Ti₂O₇ (CMS 2011).
But each monopole attracts its opposites and repels its own kind, so it gathers a screening cloud that lowers its energy and raises the density. CMS treated the gas with the Debye–Hückel theory of electrolytes. Around 1 K, they found, screening can more than double the density; in this page's model it does so between about 0.65 and 1.2 K. At lower temperatures the monopoles are too sparse to screen one another and the gain fades: a factor of 1.5 at 0.5 K, only 1.05 at 0.3 K. CMS also note that the theory is "near an 'internal' limit of validity" throughout this range.4 This page's model uses their theory for Dy₂Ti₂O₇, with Δ = 4.35 K and Enn = 3.06 K:
Temperature
Monopoles per tetrahedron
One in
Spacing
Per cm3
Relaxation time (model)
0.3 K
1.1 × 10−6
950,000
49 nm
8 × 1015
28 s
0.5 K
5.1 × 10−4
2,000
6.3 nm
4 × 1018
98 ms
1.0 K
0.056
18
1.3 nm
4 × 1020
1.8 ms
2.0 K
0.27
3.7
0.77 nm
2 × 1021
0.73 ms
This page's model: Debye–Hückel theory of CMS (2011) for Dy₂Ti₂O₇. Spacing is the cube root of the volume per monopole; "per cm3" uses 8.0 × 1021 tetrahedra per cm3 from a = 3.54 Å. The relaxation time is the model described below. Below about 0.5 K real samples do not reach these equilibrium values. Above about 1 K the theory's screening length is shorter than a lattice spacing, so CMS advise reading it with caution; above about 2 K the monopole picture itself stops being useful.
Figure 4 puts the table on a slider. Cool the slab and count the poles, switch the pull between them off to see how much the screening adds, and watch the shaded bands at both ends of the temperature axis.
Figure 4🌡️ Try it
Cool it down and count the poles
Drag the temperature
north polesouth polewith their pullwithout it
Pull between poles
Polesof the tetrahedraof the pyramids
In 1 cm3 of crystalIn a 1 cm cube
Typical spacing
Magnet settles in
This page's model of the monopole gas in Dy₂Ti₂O₇: the density per tetrahedron from Debye–Hückel theory (Castelnovo, Moessner and Sondhi 2011; Δ = 4.35 K, Enn = 3.06 K), with the dashed curve and the 'Off' setting giving the density without the Coulomb attraction, 2e−Δ/T/(1 + 2e−Δ/T). The slab is a schematic window of 240 tetrahedra; the number marked is the density times 240, rounded to an even number so that norths and souths come in pairs. Spacing: the cube root of the volume per monopole. 'In 1 cm3': the density times the 8.0 × 1021 tetrahedra in a cubic centimetre, as in the table. 'Magnet settles in': the relaxation time τ = V0kBT/(μ0Dq2ρ) of Ryzhkin's theory with the hop rate Bovo et al. (2013) measured above about 2 K, used here at lower temperatures too; it ignores the constrained, fractal motion of Chapter 11 and so underestimates the slowdown at low temperature. Shaded: below about 0.5 K real samples fall out of equilibrium (Morris et al. 2009; CMS 2011); above about 2 K the gas is too dense for the monopole picture; Debye–Hückel theory is already rough above about 1 K, where its screening length falls below a lattice spacing (CMS 2011). Ho₂Ti₂O₇ (Δ = 5.79 K) holds fewer monopoles at every temperature.Each mark in the slab is a pole: filled dots for north, rings for south. Drag the temperature down and watch them disappear: at 1 degree above absolute zero about 1 pyramid in 18 holds one, at 0.3 degrees about 1 in a million. With fewer poles the magnet takes longer to settle. Below about half a degree the crystal freezes before it can settle; above 2 degrees there are so many that the idea stops helping.
From 1 K to 0.5 K the number of monopoles falls about a hundredfold, and from 1 K to 0.3 K about fifty-thousandfold (this page's model). The window in which the monopole picture is useful is narrow. Below about 0.5–0.6 K real crystals fall out of equilibrium; above about 2 K a quarter or more of the tetrahedra are broken, and a dilute gas is the wrong picture.5,4
Within the window, the gas explains a lot. The heat capacity of Dy₂Ti₂O₇ at low temperature follows it: Morris and colleagues fitted theirs with Debye–Hückel theory.5 The magnetisation relaxes through currents of monopoles, so the relaxation rate is proportional to their density times their mobility. As the density collapses on cooling, the relaxation time grows from milliseconds towards seconds and beyond.27,28,29 This page's version of that model is
τ = V0kBT/(μ0Dq2ρ)Ryzhkin's relaxation time, with V0 the volume per tetrahedron and the diffusion constant D from the measured hop rate (Bovo et al. 2013). It gives 1.8 ms at 1 K and 98 ms at 0.5 K. It ignores the constrained, fractal motion of Chapter 11, so it underestimates the slowdown at low temperature.
Two experiments show the gas behaving like an electrolyte. Giblin and colleagues applied a magnetic-field pulse to a Dy₂Ti₂O₇ crystal at 0.36 K and created a relaxing magnetic current that lasted for several minutes, measured through the voltage it induced in a coil.43 And with the field along the [111] direction, Sakakibara and colleagues found in 2003 that a jump in the magnetisation near 0.9 T becomes a first-order, liquid–gas-type transition below about 0.36 K.44 CMS explained it as the monopole gas condensing into a monopole liquid. Their simulation puts the critical end point at 0.57 ± 0.06 K and 0.86 ± 0.03 T. The field agrees with experiment within a few per cent; the measured temperature is about a third lower, which they attribute to small farther-neighbour interactions.2
One measurement is best told as contested. In 2009 Bramwell and colleagues used muon spin rotation to follow how a field changes the number of free monopoles. Through Onsager's theory of the field-driven dissociation of ion pairs in weak electrolytes (the second Wien effect) they extracted a monopole charge of about 5 μB/Å, in agreement with theory, and called the monopole currents "magnetricity".45 In 2011 Dunsiger and colleagues reported new muon experiments and concluded that such an effect "is not observable" with that method; they saw instead spin fluctuations that become independent of temperature at low temperatures.46 Bramwell and Giblin replied that the criticism did not warrant any revision of their conclusions, in a comment posted on arXiv.47 In 2012 Stephen Blundell showed that what such weak-field muon experiments see can come from the macroscopic stray field of the magnetised sample, rather than from the monopoles directly (Phys. Rev. Lett. 108, 147601). We found no resolution of the charge measurement in print. The Wien effect in spin ice has since been developed in theory,48,49 and monopole currents have been measured directly with coils and SQUIDs.43,50 The charge itself does not rest on the muon measurement: it rests on the dumbbell calculation and on the neutron, thermal and noise evidence.
Chapter 11Listening to monopoles
In 2008 CMS proposed detecting spin-ice monopoles "in an experiment modelled after the celebrated Stanford magnetic monopole search":2 Cabrera's superconducting loop, the one that recorded the Valentine's Day event.9 A Dirac monopole passing through such a loop changes the flux through it by h/e, a clean step. A spin-ice monopole carries μ0q = 5.4 × 10−19 Wb, 2.6 × 10−4 of a superconducting flux quantum (this page's arithmetic). It would take 7,693 of them to make one Dirac step, and between 1.2 and 2 K a cubic centimetre holds of order 1021 of them (this page's model). So there are no steps to see. There is only noise.
In 2019 Ritika Dusad and colleagues put a Dy₂Ti₂O₇ crystal inside the pick-up coil of a SQUID and recorded its magnetisation noise between 1 Hz and 2.5 kHz at temperatures from 1.2 to 4 K. Pairs of monopoles are born and recombine, and their number and motion fluctuate; the flux through the coil fluctuates with them. The spectrum was flat at low frequency and fell off above a corner frequency of order 1/(2πτ), with τ in the millisecond range, so the amplified noise is audible.7 They fitted it with
S(f)/S(0) = 1/[1 + (2πfτ)b], b ≈ 1.2 to 1.5Free, independent monopoles would give a Lorentzian, b = 2. In this page's model τ ≈ 1.2 ms at 1.2 K, a corner at 1/(2πτ) ≈ 137 Hz.
The slope was wrong. Monopoles diffusing freely and independently give b = 2; the measured exponents lay between 1.2 and 1.5.7 Figure 5 compares the two shapes. Drag the slope and the frequency cursor, and press Listen to hear this page's synthesis of the spectrum you have chosen; the sound is made from the formula, not recorded.
Figure 5🎧 Try it
The hiss of a monopole gas
Listen, then compare
free polesthis slopeband measured in 2019
3 seconds of sound
Sound plays only when you press Listen.
Frequency
Free poles
This slope
Louder byCompared with free poles
Magnetisation noise spectra S(f)/S(0) = 1/(1 + (2πfτ)b) (the form fitted by Dusad et al. 2019), with τ = 1.17 ms from this page's model at 1.2 K (corner 1/(2πτ) = 137 Hz; Dusad et al. measured τ in the millisecond range between 1.2 and 4 K over 1 Hz–2.5 kHz). Free, uncorrelated monopoles give a Lorentzian, b = 2 (dashed); the measured exponents lie between 1.2 and 1.5. The readouts give each spectrum's power at the cursor, as a share of its low-frequency level. Hallén et al. (2022) traced the difference to motion on a fractal: local transverse fields freeze a further quarter of the spins, so within about 14 hops a monopole can reach only about 130 of the roughly 2,000 sites an unconstrained walker could. Each monopole carries 2.6 × 10−4 flux quanta (this page's arithmetic), so the SQUID records the noise of very many, not single steps. The sound is this page's synthesis of the plotted spectrum, not a recording.The noise a crystal full of poles makes in a super-sensitive magnetic detector. The dashed line is what free, independent poles would do; the solid line falls off with the slope measured in 2019. Press Listen to hear it (it's made from the plotted shape, not recorded). Drag the slope and the cursor: the real poles keep more high-pitched hiss, which a 2022 study traced to the tangled, branching paths they must walk.
At ten times the corner frequency the measured shape is about three times louder than free monopoles would be (for b = 1.5) or six times (for b = 1.2); at a hundred times the corner, beyond the band measured in 2019, the fitted form would give about ten and forty times (this page's arithmetic). The real monopoles keep more of the high-pitched hiss.
Why? Samarakoon and colleagues argued in 2022 that the landscape the monopoles move through is not perfectly flat.51 The same year Jonathan Hallén and colleagues gave a theoretical explanation that needs no disorder: it works in a perfectly clean crystal. Besides the ice rule, local transverse fields set by the neighbouring moments, which control how fast each moment can tunnel (Tomasello and colleagues, 2019), effectively block "another quarter of the spins". A monopole is left with between zero and three possible next steps, two on average, and the set of places it can reach sits close to a percolation threshold: a fractal. The number of sites it can reach grows with the number of hops n only as n1.85, so that up to about 14 hops a monopole can reach only about 130 of the 2,071 sites that Hallén et al. count for an unconstrained walker. Motion on that fractal gives an exponent close to 1.5, and it also explains why the relaxation time grows so fast on cooling.52
Later work refines the picture. Hsu and colleagues found two distinct transport processes in monopole fluids, separated near 600 µs.50 In 2025 Huster Zapke and Holdsworth argued from simulations that, over long times, most of the correlated noise comes from the strings rather than from the monopoles themselves.53 Either way the monopoles are real quasiparticles, and their walk is shaped by the strings they leave.
Chapter 12A universe with its own light
Everything so far is classical: moments flip one at a time, by heat or by tunnelling, and the monopoles hop. Suppose instead that the moments can tunnel collectively between whole ice-rule arrangements, for example by reversing the six moments around a hexagon of the lattice at once, with an energy scale g of order a micro-electronvolt. The ice-rule states then mix into one quantum state, and theory says that a quantum electrodynamics of its own emerges, described by a compact U(1) gauge theory (Michael Hermele, Matthew Fisher and Leon Balents, 2004, for this lattice, building on three-dimensional models of 2002–03 by Olexei Motrunich and T. Senthil and by Roderich Moessner and Shivaji Sondhi).54 It has:
Photons: transverse waves of the moment field, with frequency proportional to wavenumber at long wavelength. Like the phonons of a solid or blackbody radiation, they add a heat capacity proportional to T3. Benton, Sikora and Shannon worked out how experiments would see this light (2012).55
Electric charges: the "spinons", which are the ice-rule defects. A three-in tetrahedron, a magnetic monopole to us, is an electric charge in the crystal's own electrodynamics.56
Magnetic monopoles of the emergent theory ("visons"), and for these Dirac's rule holds: their charge is e/(2α) in the theory's own units.56
How different are its constants? Salvatore Pace, Siddhardh Morampudi, Roderich Moessner and Chris Laumann estimated them in 2021. The emergent fine-structure constant is "more than an order of magnitude" larger than our 1/137: about 1/10 for typical parameters, and tunable from zero up to about 0.2, close to the strongest coupling at which the emergent light is believed to survive before the theory confines. The emergent light is slow: about 1 m/s for a cubic lattice constant of about 10 Å and g ~ 1 µeV.56 With their formula, c ≈ 0.51 ag/ħ, g = 1 µeV gives 0.77 m/s, slower than a person walking (about 1.4 m/s) and some 390 million times slower than our light (this page's arithmetic). Matter in that universe could outrun its light and emit Cherenkov radiation. These are theory estimates for typical parameters, not measured values; as of 24 October 2026 no experiment has pinned either down in a real material.
Figure 6 races our light, the crystal's light and a walker across an imaginary 10 m crystal (real crystals are millimetres across, which the emergent light would cross in a few milliseconds). Turn up the tunnelling energy and the crystal's light catches up; change α and watch the emergent monopole charge move the other way.
Figure 6🏁 Try it
A universe with its own light
Race the light
Its fine-structure constant
Speed of lightours: 299,792,458 m/s
Fine-structure constantours: 1/137
Smallest magnetic chargeours: 68.5 e
Over 10 metresyou winlight winsa dead heat
Order-of-magnitude constants of the emergent electrodynamics of quantum spin ice from Pace, Morampudi, Moessner and Laumann (2021): emergent light speed c ≈ 0.51 ag/ħ for the cubic lattice constant a = 10 Å and ring-exchange energy g (Pace et al. quote about 1 m/s for g ~ 1 µeV), fine-structure constant about 1/10 (tunable from zero up to about 0.2, beyond which the emergent theory is believed to confine; the chips show 0.06, 1/10 and 0.2, values from their study), and the emergent monopole charge e/(2α) required by Dirac quantisation inside the emergent theory. The race is over an imaginary 10 m crystal (real crystals are millimetres across), with time on screen slowed so that the walker, at 1.4 m/s, always takes 4 s; the two speeds keep their true ratio. These are theory estimates, not measurements. Candidate materials: Ce₂Zr₂O₇ shows neutron and thermodynamic evidence for emergent photons and spinons (2025; field-tuned spectroscopy, June 2026), and Ce₂Sn₂O₇ neutron evidence for spinons coupled to an emergent gauge field (2025; the result depends on how the samples are made). Decisive confirmation is still described as outstanding (Naik et al., June 2026).If the crystal's magnets can tunnel quantum mechanically, theory says it becomes a tiny universe with its own light. Race it against a person walking across an imaginary 10-metre crystal: at typical settings, you win. Turn up the tunnelling and light catches up. The numbers are theory, and experiments are getting close but haven't settled it yet.
Look at the readout for the smallest magnetic charge. In our universe the smallest Dirac charge is 68.5 e, because α is 1/137. In the crystal's universe, with α near 1/10, it is about 5 e. Dirac's rule, which the classical crystal escapes because its string is visible, returns inside the quantum crystal, for that crystal's own monopoles.
Is it real? The leading candidates are cerium pyrochlores. In Ce₂Zr₂O₇, Bin Gao and colleagues reported signatures of a three-dimensional quantum spin liquid in 2019,57 and in 2025 neutron-scattering and thermodynamic evidence for emergent photons and fractionalised spinons, with excitations near zero energy at 50 mK.58 In Ce₂Sn₂O₇, Victor Porée and colleagues reported evidence for fractional matter coupled to an emergent gauge field (2025),59 although samples made by another method behave differently and point to an ordered ground state, so that material is still argued over.60 In June 2026 Gao and colleagues applied a field of about 0.15 T along [111], which suppresses the photon signal and leaves the spinon continuum, and so separated the two in Ce₂Zr₂O₇; they read the result as strong evidence for the predicted "π-flux" quantum spin ice.61 A 2025 review calls Ce₂Zr₂O₇ "an excellent QSI ground-state candidate".62
The photon is hard to catch: it lives at frequencies below about a gigahertz and temperatures below about half a kelvin, with vanishing weight at zero momentum. A theory paper of June 2026 proposes listening for its stray magnetic noise with solid-state defect magnetometers, and it opens by saying that decisive experimental confirmation "remains an outstanding challenge".63 The status, then: strong evidence in 2025 and 2026; decisive confirmation still outstanding as of 24 October 2026.
So the story ends where it began. Dirac's monopole is still unfound. Meanwhile a crystal a few millimetres across hosts north and south charges that obey Coulomb's law, trail a string that experiments can detect, and carry a charge that pressure could tune. A quantum cousin of that crystal may hold a whole electrodynamics, with its own light and its own Dirac monopoles.
Chapter 13Glossary and sources
Spin ice
A magnetic crystal, such as Dy₂Ti₂O₇ or Ho₂Ti₂O₇, whose atomic moments obey the ice rule at low temperature.
Pyrochlore lattice
A three-dimensional network of tetrahedra that share corners; each corner belongs to two tetrahedra.
Ice rule (two in, two out)
At every tetrahedron, two moments point in and two point out; in water ice, every oxygen has two near hydrogens and two far ones.
Magnetic monopole (Dirac)
A hypothetical particle carrying an isolated magnetic charge. Dirac showed in 1931 that its charge would be a multiple of h/(μ0e), and that one would make every electric charge a multiple of a unit. None has been found.
Emergent quasiparticle
Something that behaves like a particle but is a collective pattern of many atoms, and exists only inside the material.
Dirac string
In Dirac's theory, an invisible line carrying a monopole's flux; in spin ice, the visible chain of flipped moments joining two monopoles.
B and H
The two magnetic fields inside matter, related by B = μ0(H + M). B never has sources; H has sources wherever the magnetisation M converges or diverges.
Bohr magneton, μB/Å
The natural unit of an atomic magnetic moment (9.274 × 10−24 J/T); spin-ice papers give magnetic charge in Bohr magnetons per ångström, 1 μB/Å = 9.274 × 10−14 A·m.
Flux quantum
h/2e ≈ 2.07 × 10−15 Wb, the unit of magnetic flux in a superconducting loop. A Dirac monopole carries two; a spin-ice monopole 2.6 × 10−4.
Deconfinement
The state in which two charges can be separated to any distance at finite cost, because the string between them costs no energy.
Coulomb phase
A state whose local rule makes the coarse-grained moments a field with no sources, with dipolar correlations and charges that interact by Coulomb's law.
Pinch point
The bow-tie pattern that a field with no sources produces in neutron scattering.
Debye–Hückel theory
The theory of a dilute gas of charges in which each charge is screened by a cloud of opposite ones; first made for electrolytes.
Relaxation time
How long a magnet takes to settle after a change, here set by how many monopoles there are and how fast they move.
SQUID
A superconducting quantum interference device, an extremely sensitive detector of magnetic flux.
Residual (Pauling) entropy
Entropy left over at the lowest temperatures reached because many arrangements obey the same rule.
Ionic and Bjerrum defects
Rule-breakers in water ice: H₃O⁺ and OH⁻ break the two-near rule, D and L defects the one-hydrogen-per-bond rule.
Quantum spin ice
A spin ice in which quantum tunnelling between ice-rule arrangements creates an emergent electrodynamics; strong candidates, not yet established.
Spinon and vison
The electric charges and the magnetic monopoles of quantum spin ice's emergent electrodynamics.
Fine-structure constant, α
How strongly charges couple to light: 1/137 in our universe, about 1/10 in theory for quantum spin ice.
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