−750 picokelvin: below zero kelvin, and hotter than the Sun

Part IA minus sign that means hotter

Chapter 1Minus seven hundred and fifty picokelvin

On 3 May 1993, Pertti Hakonen, R. T. Vuorinen and J. E. Martikainen of the Low Temperature Laboratory at the Helsinki University of Technology, now part of Aalto University, published measurements on the nuclear spins of polycrystalline rhodium foils. On the positive side of the scale they reached 280 picokelvin. On the negative side they reached minus seven hundred and fifty picokelvin: −750 pK. Their abstract called the pair "record-low and -high nuclear spin temperatures".¹ A picokelvin is a trillionth of a kelvin.

What sat at those temperatures was not the whole metal. A rhodium nucleus is a tiny magnet, and in a magnetic field it can point along the field or against it. The experiment cooled the collection of these nuclear magnets, the nuclear spins, while the electrons and the crystal lattice of the same metal kept a temperature of their own, positive and far above the picokelvin range (Chapter 8). The laboratory describes its apparatus as a cascade: a dilution refrigerator at about 3 mK, a first nuclear cooling stage at about 50 µK, and finally the sample itself, whose nuclear spins reach the picokelvin range.² Both 1993 values were reached by adiabatic demagnetisation, starting from nuclear polarisations of 83% for the positive run and −60% for the negative one. In the negative run, the net magnetisation of the nuclei had been made to point against the field before the field was lowered.¹

Absolute zero, 0 K, is the bottom of the temperature scale. No refrigerator can reach it, and nothing can be colder. So what can −750 pK mean? Not "colder than nothing". The laboratory's own explanation is that the positive and negative sides of the scale are connected through infinite temperature, not through zero.² A negative temperature is reached by going up, through infinitely hot, and not down through absolute zero.

The practical meaning is this. Put the −750 pK spins in thermal contact with anything at a positive temperature, and heat flows out of the spins. That includes the centre of the Sun, at 1.571 × 10⁷ K,³ and it includes the Planck temperature, about 1.4 × 10³² K. Every negative temperature is hotter than every positive one.

A year after the rhodium paper, Hakonen and Olli Lounasmaa described the pair in Science as "the lowest and 'highest' temperatures ever produced on this scale and measured", with the quotation marks around "highest" their own.⁴ This page makes no such claim for −750 pK. In 2011 a team at MIT reported spin temperatures of ±50 pK in an ultracold gas, and Chapter 4 shows that −50 pK is further along the hot side of the scale than −750 pK.⁵ The claim that survives every caveat, including a live dispute about definitions (Chapter 12), is the one in this page's title: −750 pK is hotter than the Sun.

The rest of the page builds up why. Temperature is not "how fast the atoms jiggle". It is defined by which way heat flows, and the quantity that decides that is a slope: how fast a system's entropy grows as you add energy. For almost everything that slope is positive. For a collection of magnets with a ceiling on their energy, it can turn negative.

Chapter 2What a thermometer ranks

Start with what "hotter" means in practice. When two bodies touch and heat flows from A to B, A is the hotter one. If A and B are each in equilibrium with a third body C, they are in equilibrium with each other: this is the zeroth law of thermodynamics, and it is what makes "equally hot" a consistent label, so that bodies can be ranked. A thermometer is just a small body that comes to equilibrium with whatever it touches and displays a number for where it ended up.

Statistical mechanics turns the ranking into a definition. A body with energy E can be arranged microscopically in many ways, and its entropy S counts them: S = k_B ln W for W equally likely arrangements. Its temperature is defined by how fast that entropy rises as energy is added:

1/T = ∂S/∂E, β = 1/(k_BT) = (1/k_B) ∂S/∂EThe derivative is taken with the volume, the particle number and any applied field held fixed. k_B = 1.380649 × 10⁻²³ J/K exactly. β is sometimes called the coldness; it has units of inverse energy.

Why this definition? Let a small amount of energy dE pass from body A to body B. A's entropy falls by dE/T_A and B's rises by dE/T_B, so the total changes by

dS = (1/T_B − 1/T_A) dE = k_B(β_B − β_A) dEFor two bodies that exchange energy with each other and nothing else, the second law says this cannot be negative.

The second law requires dS ≥ 0, so energy flows into the body with the larger β. Heat flows from small β to large β, and stops when the two are equal. That one sentence is the whole of this page. It says nothing about speed, motion or kinetic energy.

For ordinary matter β is positive. It is very large near absolute zero and shrinks towards zero as a body gets hotter. Temperature, T = 1/(k_Bβ), is its reciprocal, chosen historically so that a bigger number means hotter, and the choice works perfectly as long as β stays positive. The familiar rule that the atoms of an ideal gas carry an average kinetic energy of (3/2)k_BT is a consequence of this definition for one kind of system, not the definition itself.

This site has used the definition before without stopping on it. Nº 3 wrote the first law as dS = dE/T to work out how much entropy a cosmic horizon loses when an object appears inside it. This page asks what happens when the slope dS/dE turns negative. Simon Braun and colleagues, who made a negative temperature in moving atoms in 2013 (Chapter 10), put the answer in one line: the jump of T through infinity is a consequence of the historic definition, and "a continuous and monotonically increasing temperature scale" is −β.⁶

Chapter 3Counting states

Take N spins in a magnetic field. Each one points either along the field, with energy 0, or against it, with energy ε; for a nucleus with magnetic moment μ in a field B, ε = 2μB. If n spins point against the field (call them "up"), the energy is E = nε, and the number of arrangements is a binomial coefficient:

W = C(N, n) = N! / [n! (N − n)!], S = k_B ln WThe spins are treated as independent: their interactions with each other are neglected (Chapter 9 brings them back).

For ten spins, with n = 0, 1, 2, … 10 against the field, the numbers of arrangements are 1, 10, 45, 120, 210, 252, 210, 120, 45, 10 and 1: 1,024 in all. The entropy rises from zero, peaks at five spins up, and falls back to zero at ten. On the left of the peak, adding one quantum of energy ε adds arrangements, so the slope is positive and so is T. At the peak the slope is zero, so T is infinite. On the right, adding energy removes arrangements: the slope is negative, and so is T.

Figure 1 draws this hill for 10, 100 or 10²⁰ spins, with a tangent line whose slope is proportional to 1/T. Drag the energy up past the middle and watch two readouts at once: the temperature, and the coldness strip under the chart.

Figure 1🎲 Try it

Count the ways

Drag the energy up

Spins flipped against the field

Helsinki, 1993, before demagnetising:

Ways to arrange

Entropy

Coldness β × ε

Temperature kT/ε

N two-level spins in a magnetic field, each either along the field (energy 0) or against it (energy ε). The curve is the Boltzmann entropy S = k ln W, where W = C(N, n) counts the arrangements with n spins against the field: exact for 10 and 100 spins; for 10²⁰ spins the per-spin limit s = −x ln x − (1 − x) ln(1 − x), with x = n/N. The tangent's slope is proportional to 1/T = ∂S/∂E (the axes are in bits and in spins flipped, so the slope is βε/ln 2). Readouts give β = 1/kT in units of 1/ε (a central difference for 10 and 100 spins, ln((1 − x)/x) for the smooth curve) and kT/ε. The spins are independent; interactions are neglected. Because energies are in units of ε, the figure applies to any two-level spin system, rhodium nuclei in a field included. This page's model.Each arrow is a tiny magnet; pointing up, against the field, costs energy. Drag the slider to add energy. The number of ways to arrange the arrows rises, peaks at halfway, then falls. Temperature is set by the slope of that hill: watch it leap from plus infinity to minus infinity at the top, while the coldness just glides through zero.

The contrast is the point. Going from four spins up to five to six, the temperature kT/ε jumps from +2.70 to ±∞ to −2.70, while the coldness βε moves calmly from +0.371 through 0 to −0.371. The jump belongs to the reciprocal, not to the spins. Nothing happens to them at the top of the hill; only the label T = 1/(k_Bβ) blows up.

For many spins, write x = n/N for the fraction against the field. The entropy per spin is s = −x ln x − (1 − x) ln(1 − x) (in units of k_B), and its slope gives

βε = ln[(1 − x)/x] ⇔ x = 1/(1 + e^βε)The second form is the Boltzmann distribution for a two-level system, and it holds for either sign of β: β > 0 gives x < ½; β < 0 gives x > ½, more spins in the upper level than the lower, a population inversion.

At a negative temperature the Boltzmann factor e^−E/k_BT grows with energy, so higher levels are more populated than lower ones. That is only possible because there is a top level: for a system with no ceiling, a weight that grows with energy could never add up to a finite total. Norman Ramsey worked out the thermodynamics of such inverted populations in 1956.⁷ Chapter 6 returns to the ceiling.

At the top of the hill, x = ½ and β = 0: each spin is as likely to point one way as the other. For many spins the entropy is then k_B ln 2 per spin: one bit per spin. That is the same k ln 2 that Landauer's principle charges, as heat kT ln 2, for erasing one bit (Nº 2). An infinitely hot collection of spins is a memory full of random bits.

Putting the pieces together gives the order of hotness:

0⁺ K < 1 K < 300 K < 10⁶ K < +∞ = −∞ < −10⁶ K < −300 K < −1 K < 0⁻ KColdest on the left, hottest on the right. 0⁺ K is every spin along the field; 0⁻ K is every spin against it, the most energetic state the spins have. Neither end can be reached.

Absolute zero is never crossed. The negative side is reached the long way round, through ±∞.

Chapter 4The coldness ruler

The order of hotness shows what is wrong with the kelvin number as a ranking: it puts the negative temperatures below zero, where they look colder than everything, when they are hotter than everything. The quantity that ranks correctly is β, or −β if you want "up" to mean hotter, because heat always flows from smaller β to larger. On a β ruler the negative temperatures sit beyond β = 0, on the far side of infinitely hot, and ±∞ is a single point in the middle.

Figure 2 places seventeen temperatures twice: on the left by their kelvin number, on the right by β. Press the button to re-rank them by heat flow, or step through them with the slider; the line under the figure gives each one's source.

Figure 2📏 Try it

The thermometer and the coldness ruler

Tap a temperature

● measured or standard · ○ this page's estimate, or a label

Step through them, coldest to hottest

Temperature

Coldness β (per eV) (bigger = colder)

Hotter than how many of the other 16

Seventeen temperatures placed twice. Left: the kelvin number on a logarithmic scale each side of zero, the order a naive thermometer suggests. Right: the coldness β = 1/k_BT on an asinh scale, u = asinh(β/β₀) with β₀ = 10⁻³⁰ eV⁻¹, logarithmic in |β| far from zero and linear through β = 0, so T = ±∞ sits in the middle; heat flows down this scale. Filled markers: measured or standard values (Hakonen et al. 1993; Aalto; Medley et al. 2011, an effective spin temperature from a fit; NASA; Nº 3 and Nº 4). Rings: this page's arithmetic or estimates (the Hawking and Planck temperatures, Purcell and Pound's spins, Braun et al.'s −2.2 J/k_B converted with J/h ≈ 470 Hz, an upper bound on |T|) and labels that are not equilibrium temperatures (relic neutrinos; a He-Ne line with 10% inversion).Each dot is a temperature. On the left, the plain kelvin number: there, minus 750 picokelvin sits below zero, as if it were colder than everything with a plus sign. On the right, the same dots ranked by which way heat flows, hotter at the top. Every negative temperature lands above the Sun and above the Planck temperature. Press the button to watch them move, or step through them one by one.

Every line from a negative temperature runs up, every line from a positive one runs down, and the two families cross. On the β side, −750 pK sits above the Sun's core and above the Planck temperature. The Planck temperature, 1.4 × 10³² K, is a natural unit built from ħ, c, G and k_B, not a known ceiling, and it still sits on the cold side of β = 0, at β ≈ 8 × 10⁻²⁹ per eV (this page's arithmetic). The laser line at about −240,000 K looks, by its kelvin number, like the coldest entry on the list; on the β ruler it is the least hot of the negatives, and it is not a true temperature at all (Chapter 11).

Among negative temperatures, the one closer to zero is hotter. −750 pK has β = −1.5 × 10¹³ per eV; −50 pK has β = −2.3 × 10¹⁴ per eV, fifteen times further along the hot side (this page's arithmetic). The two numbers describe different things. Patrick Medley, Wolfgang Ketterle and colleagues at MIT prepared "isolated spin distributions" in an ultracold rubidium gas at spin temperatures of ±50 pK: an effective temperature describing how the atoms' spin states were distributed, not how the atoms moved.⁵ The Helsinki value is the nuclear spin temperature of a macroscopic piece of metal.¹ Neither supports a single "hottest ever" claim, and this page makes none.

On the β ruler, the step from +280 pK across infinity to −750 pK is 5.7 × 10¹³ per eV long. The step from room temperature, taken as 293 K, to the surface of the Sun, 5,772 K,³ is 37.6 per eV. On the scale that decides heat flow, the gap between the Helsinki team's two records, which spans infinity, is about 1.5 trillion times longer than the step from a room to the Sun's surface (this page's arithmetic).

The positive side of the same ruler has a later Helsinki entry: the nuclear spins of a piece of rhodium of about two grams, cooled to 100 pK. The laboratory's page, last updated on 24 January 2024, says it still holds the low-temperature record for macroscopic condensed-matter systems.²

Chapter 5The touch test

A definition of "hotter" has to pass the test of contact. Take two boxes of independent spins with the same level spacing ε, and let them exchange energy with each other and nothing else. Energy is conserved, so the total number of spins pointing against the field is fixed. The most probable final state shares those spins so that both boxes end with the same fraction up, which is the same β. Until they get there, heat flows from the box with the smaller β to the box with the larger β.

Temperatures do not average; energies do. A classic textbook example puts two equal spin systems, one at −300 K and one at +300 K, in contact. They do not end at 0 K, the naive "average". They end with exactly half the spins up in both, which is T = ±∞. The same happens to two boxes of 1,000 spins with 70% and 30% up, at kT/ε = −1.18 and +1.18.

Figure 3 lets you set two boxes, their sizes and their fractions up, and then press Touch. Try the presets: equal and opposite temperatures; a small negative box against a big positive one; and the reverse. The switch labelled Gibbs belongs to Chapter 12.

Figure 3🤝 Try it

The touch test

Set two boxes, then touch

Box A: spins up

Size

Box B: spins up

Size

A's temperature kT/ε

B's temperature kT/ε

Heat flows

After touching

Touching multiplies the ways byHeat flows whichever way makes this number bigger.

Two systems of N independent two-level spins with the same spacing ε exchange energy and nothing else. Temperatures are Boltzmann's, from the exact counts (central difference of ln C(N, n)), or, with the switch, Gibbs's, from the number of states at or below each energy, Ω(n) = Σ_k≤n C(N, k). The touch ends in the average shared state, equal fractions up (to the nearest whole spin), which conserves energy and lies within one spin of the most probable split; once touching, the two boxes are one system of N_A + N_B spins: "After touching" gives its temperature from the joint count, and the A and B tiles still give each box's own. Because the counts are exact, boxes of different sizes at the same fraction up can read slightly different temperatures, by less than the change one quantum makes to the smaller box's. Boltzmann's temperatures always predict the direction of flow; in the 'Gibbs ranks B hotter' preset, Gibbs temperatures of about 1.0 × 10⁵ and 1.1 × 10⁵ ε/k label B the hotter (computed with the forward difference ln Ω(n + 1) − ln Ω(n); other rules change the numbers but not the order), yet energy flows from A to B, and after the touch the two boxes' own Gibbs temperatures, about 42 and 4.3 × 10⁶ ε/k, differ some 100,000-fold although the boxes are in equilibrium (the zeroth-law objection of Frenkel and Warren, 2015). A real negative-temperature system touching an ordinary body, which has no energy ceiling, always ends at a positive temperature. "Touching multiplies the ways by" is C(N_A, a) C(N_B, b) / [C(N_A, n_A) C(N_B, n_B)] = e^ΔS/k, from the starting split (n_A, n_B) to the final one (a, b); it exceeds 1 whenever more than one quantum moves (a single quantum, rounded up from a smaller average, can lower it slightly). This page's model.Two boxes of tiny magnets. Set how many point up in each, and how big each box is, then press Touch to let them swap energy. Heat always flows out of the negative box into the positive one. Two equal boxes at minus and plus the same temperature don't end at zero: they end at infinity. The Gibbs switch shows a rival definition of temperature (chapter 11); try 'Gibbs ranks B hotter'.

Three things to take from it. Heat always leaves the negative box for the positive one. Whether the pair ends at a positive or a negative temperature depends on whether, taken together, more than half of all the spins point against the field, so the sizes of the boxes matter as much as their temperatures. And no touch ever passes through 0 K.

Real negative-temperature spins are never alone: they sit in a metal or a crystal. An ordinary body, a gas or a crystal lattice, has no energy ceiling. It can always absorb more energy and stay at a positive temperature. So a negative-temperature system that touches an ordinary body gives heat until both share one positive temperature: the pair always ends on the positive side (this page's model). That is how the nuclear-spin negative temperatures in this page ended (Chapters 7 and 8).

Part IIGetting past infinity

Chapter 6Why you need a ceiling

A free gas, a liquid or the vibrating lattice of a solid cannot reach a negative temperature, because it always has more room at higher energy. Its atoms can always move faster or vibrate harder, so the number of available states keeps growing as energy is added. The slope ∂S/∂E stays positive at every energy, and so does T.

A negative slope needs the states to run out at high energy. The simplest way is a ceiling: a highest energy above which there are no states. Then, past the energy where the states are most crowded (exactly halfway up, for independent spins), the entropy must fall. Three kinds of limit appear in this page:

spins in a magnetic field, whose top state is every spin against the field (Chapters 7 and 8);
atoms confined to the lowest band of an optical lattice, whose kinetic energy cannot exceed the top of the band (Chapter 10);
point vortices in a two-dimensional fluid in a bounded container. Their energy has no ceiling (it grows without limit as two vortices turning the same way approach each other), but the walls make the space of possible arrangements finite, so the count must still peak and then fall (Chapter 11).

Ramsey's 1956 paper set out when such a system really has a negative temperature. In the usual summary of his conditions, paraphrased here: the elements of the system must be in thermal equilibrium among themselves, so that a temperature exists at all; their allowed energies must have an upper bound; and the system must be thermally isolated from systems that lack these properties, or at least reach equilibrium internally much faster than it leaks energy to them.⁷

The third condition is where every real experiment lives. The bounded part always sits inside an unbounded one: nuclear spins inside a crystal, the atoms' motion inside a trap. The negative temperature lasts while the two are weakly coupled, and ends as heat leaks into the ordinary part (Chapter 5).

That makes a negative temperature sound fragile, but in isolation it is not. At a given energy, the inverted state is the most probable arrangement: a genuine equilibrium, not an excitation waiting to collapse. It decays only through its coupling to ordinary matter, and Chapter 10 describes a gas that was more stable at negative temperature than it would have been at a positive one.

Chapter 7Minus room temperature, in a crystal

The first laboratory negative temperature was reported by Edward Purcell and Robert Pound in a paper titled "A Nuclear Spin System at Negative Temperature", published on 15 January 1951.⁸ Their sample was a crystal of lithium fluoride, whose nuclear spins exchange energy with the crystal lattice only slowly. They magnetised the spins in a field, then reversed the field faster than the spins could follow. The spins were left pointing against the new field: more of them in the upper level than in the lower, an inverted population holding more energy than any positive temperature allows. The state lasted long enough to be measured before the lattice pulled the spins back towards ordinary equilibrium.

How hot were they? Before the reversal the spins were in equilibrium with a crystal at room temperature, about +300 K, so each higher level held slightly fewer spins than the one below, by the Boltzmann factor for 300 K. Reversing the field turns the ladder of levels upside down without changing how many spins sit on each rung, so the same population ratio now corresponds to a coldness of the opposite sign. If no polarisation was lost and the spins are compared in a field of the same strength, their temperature was about −300 K: roughly minus room temperature (this page's estimate). Any loss of polarisation moves the temperature further from zero, to −1,000 K for example, which is less hot. This page could not verify the fields and times in the original paper, so the estimate rests on that stated assumption.

The twist is that the first negative temperature was made in a crystal at room temperature, not in an ultracold laboratory. Spins at about −300 K sat in a lattice at about +300 K: the same number with opposite signs, and the spins were the hotter.

Was it really a temperature? The sceptics of Chapter 12 say no. In 2017 Eitan Abraham and Oliver Penrose re-analysed the thermodynamics and concluded that negative absolute temperatures "did occur in the experiments designed to produce them".⁹

Chapter 8Two temperatures in one metal

In a metal at microkelvin temperatures, the nuclear spins exchange energy with one another much faster than with the conduction electrons. So the spins reach equilibrium among themselves and have a temperature of their own, which can be utterly different from that of the electrons and the lattice of the same piece of metal. That is how a sample held at a positive temperature in a cryostat can contain nuclear spins at −750 pK.

The tool is adiabatic nuclear demagnetisation. Polarise the spins in a large field while they touch a cold stage, isolate them, and lower the field slowly. For ideal independent spins the entropy depends only on the ratio B/T, so at constant entropy the temperature falls in proportion to the field. Real spins depart from this at the lowest fields, where their interactions with each other matter. The same rule works on the negative side: lower the field on an inverted spin system and its temperature shrinks in proportion too, from a negative value towards 0⁻, the hot end of the scale. The same step that cools a positive spin system makes a negative one hotter (this page's explanation from the ideal-spin model).

That is the route the Helsinki team took: polarisations of 83% and −60% prepared in a larger field, then demagnetisation to 280 pK and −750 pK.¹ The laboratory says the spins are "tricked" to negative temperatures by a suitable manipulation of the external field, and that the result seeks equilibrium by giving energy to its surroundings.² That is what ended each negative run: the −750 pK spins slowly gave heat to the electrons and the lattice, which were at a positive temperature. Heat flowing from −750 pK into a metal at a positive temperature is this page's headline, happening inside a cryostat.

Silver came first. In 1990 Hakonen, S. Yin and Lounasmaa took silver's nuclear spins down to 0.8 nK and, on the negative side, as far as −4.3 nK, and found that antiferromagnetic alignment at positive temperatures turns into ferromagnetic orientation at negative ones.¹⁰ In 1992 the group reported ferromagnetic order in silver's nuclear spins at negative temperatures hotter than −1.9 nK, that is, closer to zero on the negative side, an order caused, in their words, by the antiferromagnetic Ruderman–Kittel interaction.¹¹ The same interaction produces opposite kinds of order on the two sides of infinity; Chapter 9 explains why.

Rhodium's nuclear spins behaved differently. On the positive side they showed an antiferromagnetic tendency, and on the negative side the group found a crossover from a ferromagnetic to an antiferromagnetic tendency at a few nanokelvin, around −5 to −6 nK in its accounts.^1,12 Hakonen and Lounasmaa summarised it a year later: in rhodium the antiferromagnetic state is preferred at both signs of temperature.⁴ That does not break the mirror rule of Chapter 9. The group's explanation is rhodium's face-centred cubic lattice with an appreciable interaction between next-nearest neighbours, which makes both the lowest-energy arrangement, favoured at positive temperature, and the highest-energy one, favoured at negative temperature, antiferromagnetic.¹² Even at 100 pK, in the later experiment, the rhodium moments did not freeze into an ordered state.² In 1997 A. S. Oja and Lounasmaa reviewed nuclear magnetic ordering in simple metals at both signs of temperature.¹³

Chapter 9Energy turned upside down

There is a compact way to see what a negative temperature does. In equilibrium, a state of energy E has a probability proportional to e^−βE. If β is negative, write it as −|β|:

e^−βH = e^{−|β|(−H)} (β < 0)H is the energy function, the Hamiltonian. A system at temperature −|T| has the same equilibrium probabilities as the same system with its energy turned upside down, at temperature +|T|.

At a negative temperature, the highest-energy states play the role of ground states. Suppose neighbouring spins lower their energy by pointing opposite ways, an antiferromagnetic coupling. Turn the energy upside down and pointing the same way becomes the favoured arrangement. At a positive temperature such a magnet orders antiferromagnetically; at a negative temperature it orders ferromagnetically. That is what silver's nuclear spins did: Hakonen and Lounasmaa described the two orders as "caused by the same interactions", depending only on the sign of the temperature.^4,13

Figure 4 is a cartoon of the sign flip: a chain of sixteen spins whose neighbours prefer to point opposite ways. Slide the coldness from positive, through zero, to negative, then switch the rule to check the mirror.

Figure 4🙃 Try it

Turn the energy upside down

Slide the coldness through zero

Coldness β × J

+3, colder0−3, hotter

Temperature kT/J

Neighbours alike

The chain looks

A cartoon of the sign flip: an open chain of sixteen Ising spins with nearest-neighbour coupling, E = ±J Σ s_is_i+1, sampled exactly at coldness βJ (fixed random numbers, so the sample changes bond by bond). The fraction of alike neighbours is 1/(1 + e^±2βJ). Because e^−βH = e^{−|β|(−H)} for β < 0, the chain at negative temperature looks like the opposite rule at positive temperature. Silver's nuclear spins order antiferromagnetically at positive and ferromagnetically at negative nanokelvin temperatures from the same interactions (Hakonen et al. 1992; Oja and Lounasmaa 1997); that is a three-dimensional magnet, whereas a one-dimensional chain never orders. This page's model.A row of tiny magnets that like to point opposite ways to their neighbours. Slide the coldness from positive, through zero, to negative. At negative temperature the very same magnets line up, because a negative temperature turns the energy upside down. Silver's nuclei did exactly this.

At βJ = +1.5, about 5% of neighbouring pairs are alike; at βJ = −1.5, about 95% are (this page's arithmetic from the exact chain result, a fraction 1/(1 + e^2βJ) of alike pairs). The fraction passes smoothly through 50% at βJ = 0, just as β passed smoothly through zero in Figure 1, and the chain at negative coldness looks exactly like the opposite rule at positive coldness. One caveat: a one-dimensional chain never orders at any nonzero temperature, so the figure shows only how neighbours are correlated. Silver's nuclear spins form a three-dimensional magnet, which does order.

The same mirror turns attraction into repulsion. A gas of atoms that attract each other behaves, at negative temperature, like a gas whose atoms repel. That is the subject of the next chapter.

Chapter 102013: moving atoms past infinity

Spins have a natural ceiling. Motion does not: an atom can always move faster. Until 2013, as the team that changed this put it, negative temperatures had been "demonstrated in localized systems with finite, discrete spectra".⁶ Ideas for doing it with motion already existed. In 2005 A. P. Mosk proposed negative kinetic temperatures for atoms in optical lattices,¹⁴ and in 2010 Akos Rapp, Stephan Mandt and Achim Rosch showed that reversing the confining potential could create stable condensates of attractive bosons at negative temperature.¹⁵

On 4 January 2013, Simon Braun, Ulrich Schneider, Immanuel Bloch and colleagues at LMU Munich and the Max Planck Institute of Quantum Optics published the experiment in Science.⁶ The trick is to put a ceiling on every kind of energy the atoms have:

kinetic energy: in an optical lattice, an atom in the lowest band cannot have more kinetic energy than the top of that band;
interaction energy: bounded above only if the atoms attract each other;
potential energy: bounded above only if the trap is turned into an anti-trap, a hill instead of a bowl.

The recipe followed. A Bose–Einstein condensate of about 1.1 × 10⁵ potassium-39 atoms was loaded into a deep three-dimensional lattice, where it became a Mott insulator, with the atoms frozen one per site. While they were frozen, a Feshbach resonance switched their interactions from repulsive to attractive, and the trap was turned into an anti-trap. Then the horizontal lattice was lowered to 6.1 recoil energies, so the atoms could move again. They piled up at the top of the kinetic-energy band: the quasi-momentum distribution developed sharp peaks at the upper band edge, the mirror image of an ordinary condensate sitting at the bottom.⁶

The number needs care. Fitting the quasi-momentum distribution gave T = −2.2 J/k_B, where J is the energy scale for an atom tunnelling to a neighbouring site, and the paper says that its fitted temperatures "only represent upper bounds for the absolute values" of T. The paper gives no value in kelvin. With J/h ≈ 470 Hz for that lattice (this page's band calculation; the paper's own effective mass gives 476 Hz), −2.2 J/k_B is about −50 nK, and the true value is at least that close to zero (this page's conversion).⁶ The institute's German press release spoke of "minus a few billionths of a kelvin" and called the atoms the hottest systems in the world; neither phrase is a result from the paper.¹⁶

On the coldness ruler, the 2013 atoms (β ≈ −2.3 × 10¹¹ per eV at the fitted value) are less hot than the 1993 rhodium spins (−1.5 × 10¹³ per eV). The bound would have to overstate |T| by a factor of more than 60 for the atoms to be the hotter (this page's arithmetic). The 2013 result is the newer and harder achievement, not the hotter one. The MIT ±50 pK of Chapter 4 belonged to the atoms' spins; Munich's feat was a negative temperature in their motion.

A negative temperature also implies a negative pressure. The first law reads dS = dE/T + (P/T) dV, and in equilibrium the entropy cannot fall when the volume grows, so P/T ≥ 0: a gas at negative temperature must have negative pressure. The attractive gas, which would collapse at a positive temperature, was "stable against collapse for arbitrary atom numbers" at a negative one.⁶ The paper notes that negative pressure is also what dark energy needs in cosmology; that is an analogy, not a model of dark energy. In the paper's preprint, the coherence of the negative-temperature state lasted more than 600 ms for the best anti-trapping potential.⁶

Part IIIWhat the minus sign means

Chapter 11Whirlpools, light and labels

The same mathematics turns up wherever the number of states is limited, by a ceiling on the energy or by walls, but each case has a different status, and it matters which.

Whirlpools. In 1949 Lars Onsager treated a two-dimensional flow as a gas of point vortices in a bounded region.¹⁷ A bounded region means bounded arrangements, so the entropy has a maximum, and above it the temperature is negative: vortices of the same sign then clump into large clusters. Onsager proposed this as the reason two-dimensional turbulence builds large, long-lived vortices. Seventy years later two teams saw it in thin superfluids. At the University of Queensland, Guillaume Gauthier and colleagues made vortex clusters in a rubidium condensate in an elliptical trap that persisted "far from global equilibrium", in a regime "at negative absolute temperatures".¹⁸ At Monash University, Shaun Johnstone and colleagues dragged a grid through a flattened condensate and watched the vortex mix evolve to steady states characterised by negative temperatures.¹⁹ Same mathematics as the spins; the status was different, because neither vortex gas was in global equilibrium. That gap has since closed. In 2022 the Queensland group, with Matthew Reeves, Tyler Neely and colleagues, stirred vortices into a condensate and let them relax. The long-time patterns matched the equilibrium theory at positive, infinite and negative absolute temperatures, and at the highest energies the cluster broke the symmetry of its container and settled off-centre, an equilibrium phase at negative absolute temperature.³⁹ As with the nuclear spins, this is the temperature of one part of the system, the vortices, inside a condensate that stays at a positive temperature.

Light. A multimode optical fibre guides only a finite number of modes, and the modes play the role of energy levels, so light sharing its power among them has a ceiling. In 2023 K. Baudin and colleagues saw light in such fibres settle into negative-temperature equilibrium states, with high-order modes more populated than low-order ones, in agreement with theory "without free parameters".²⁰ The same year, A. L. Marques Muniz and colleagues in Jena and Central Florida showed light pulses in optical fibres following the rules of thermodynamics at negative optical temperature;²¹ the Jena press release was headlined "Hotter than infinity".²² The optical temperature describes how light's power is shared among the modes. The glass stays at room temperature. A May 2025 preprint by Mario Ferraro and colleagues reports that in a few-mode fibre, raising the laser power inverts the mode populations until the optical temperature flips sign.²³

Lasers and masers. A pumped laser medium has more atoms in the upper laser level than in the lower one. Read through the Boltzmann formula, that one pair of levels has a negative excitation temperature:

T_ex = −hν / [k_B ln(N₂/N₁)]N₂ and N₁: populations per state of the upper and lower level; ν: the frequency of the transition between them. N₂ > N₁ makes T_ex negative.

But a laser is driven, constantly pumped; its other levels are not inverted, and it has no single temperature. It fails Ramsey's first condition. Its negative excitation temperature is a label for one pair of levels, not a thermometer reading, even though reviews list lasers among the systems associated with negative temperature.²⁴ Interstellar and circumstellar masers are natural population inversions of the same kind,²⁵ so the same caveat applies to them.

Label	Value	What it is
Helium-neon red line (632.8 nm), 10% more atoms per state up than down	about −240,000 K	excitation temperature of one inverted pair of levels
The same line, 1% inversion	about −2.3 million K	a smaller inversion gives a larger number, closer to ±∞
OH maser line (1,667 MHz), 1% inversion	about −8 K	the same inversion at a radio frequency gives a small number
Maser brightness temperatures²⁵	10¹⁵ K and more	the temperature a blackbody would need to be as bright
Relic neutrinos (Nº 4)	1.95 K	a parameter of a momentum distribution

The excitation temperatures are this page's arithmetic from the formula above. None of the five is the temperature of a system in equilibrium.

As Nº 4 put it for the relic neutrinos, a temperature can be "a label, not a thermometer reading". These labels even come out negative, and the further their number is from zero, the closer the two populations are to equal.

Does nature make negative temperatures outside the laboratory? Masers are natural inversions, but driven ones. We found no natural system known to meet all three of Ramsey's conditions. That is an absence of evidence, not a proof.

Chapter 12The fight over a word

Not every physicist accepts that these are temperatures, and the disagreement is about which entropy to use. Boltzmann's entropy, used throughout this page, counts the arrangements at the energy E. Gibbs's "volume" entropy counts every arrangement with energy E or less:

S_B = k_B ln ω(E), S_G = k_B ln Ω(E), Ω(E) = Σ_{E′ ≤ E} ω(E′)ω is the number of arrangements at energy E (W in Chapter 3). Ω can only grow as E grows, so the Gibbs temperature is never negative.

For ordinary matter almost all the states below E lie just below E, so the two entropies give the same temperature. They part ways only where ω turns over, which is exactly this page's territory. Below half-full they agree: at 30% of the spins up, kT/ε = 1.18 under either definition (this page's arithmetic). Above half-full, the Boltzmann temperature is negative and the Gibbs temperature stays positive.

In a paper published online on 8 December 2013, Jörn Dunkel and Stefan Hilbert argued that Gibbs's entropy is the consistent one; its title is "Consistent thermostatistics forbids negative absolute temperatures". Their preprint called the Boltzmann definition "popular yet inconsistent", and they commented on heat engines with efficiencies greater than one.²⁶ With Peter Hänggi they restated the case in 2016: only the volume entropy satisfies the traditional form of the laws of thermodynamics exactly, for a broad class of systems that includes all standard classical Hamiltonian systems, regardless of their size, so negative absolute temperatures and Carnot efficiencies above one are not achievable within a standard framework. They noted open mathematical problems for systems with discrete energy levels, such as this page's spins.²⁷

The replies came quickly. Daan Frenkel and Patrick Warren argued in 2015 that the Gibbs entropy fails a basic requirement, that two bodies in thermal equilibrium should be at the same temperature, and that negative temperatures are "inevitable" in systems with bounded spectra.²⁸ Robert Swendsen and Jian-Sheng Wang concluded that "negative temperature is a valid extension of thermodynamics".²⁹ Pierfrancesco Buonsante, Roberto Franzosi and Augusto Smerzi argued that for systems whose statistical ensembles are equivalent, Boltzmann's is the correct entropy.³⁰ Abraham and Penrose defended the Purcell and Pound interpretation (Chapter 7).⁹ The 2021 review by Marco Baldovin, Stefano Iubini, Roberto Livi and Angelo Vulpiani in Physics Reports concluded that negative temperatures "are consistent with equilibrium as well as with non-equilibrium thermodynamics".²⁴ New proposals keep arriving, such as a 2024 paper that counts states in a relative energy window and, in its title's words, "allows negative absolute temperatures".³¹

As of 24 October 2026 there is no consensus statement. The most recent comprehensive review we found accepts negative temperatures; since 2013, a school led by Dunkel and Hilbert has called them an artefact of the definition.

Both sides accept the same measurements, and both agree on which way heat flows: out of inverted spins into anything ordinary. Under Boltzmann's definition, that is because their temperature is negative. Under Gibbs's, their temperature is astronomically large and positive, which ranks them above ordinary matter all the same. For 10²⁰ spins with 60% of them pointing against the field, the Gibbs value is

kT_G/ε ≈ Ω/ω ≈ e^{N[ln 2 − s(x)]} ≈ 10^{8.7 × 10¹⁷}For x > ½ and large N; s(x) is the entropy per spin of Chapter 3. This page's arithmetic.

That is a number with nearly a billion billion digits. "Hotter than the Sun" holds under either definition. The dispute is mainly over the word "temperature", not over which way heat goes, though the Boltzmann side argues that experiments already favour it. Hakonen and Lounasmaa took silver's switch to ferromagnetic order at negative temperature (Chapters 8 and 9) to show that negative temperatures are "real, not fictitious, quantities",⁴ and Abraham and Penrose argue that the volume entropy gives predictions inconsistent with the nuclear-spin experiments.⁹

The sharpest form of the disagreement is the zeroth-law test, and Figure 3 can run it. Choose the preset "Gibbs ranks B hotter": box A holds 100 spins with 70 up, box B 1,000 spins with 562 up. With the finite-difference rule the figure uses, ε/kT_G = ln Ω(n + 1) − ln Ω(n), their Gibbs temperatures are kT_G/ε = 102,139 and 112,536, B the higher by about 10%. For boxes this small the numbers depend on the rule: the formula Ω/ω above gives about 43,000 and 88,000, but every rule makes B the higher. Yet when they touch, energy flows from A to B, until 57.45% of all the spins point up, A with 57 of its 100 and B with 575 of its 1,000 (this page's arithmetic). Their Boltzmann temperatures, −1.19 and −4.02, predict the flow correctly: A, closer to 0⁻, is the hotter. And once they have settled, in equilibrium with each other, their own Gibbs temperatures are kT_G/ε ≈ 42 and 4.3 million with the figure's rule (about 31 and 3.1 million with Ω/ω), some 100,000-fold apart, while their Boltzmann temperatures, −3.58 and −3.31, agree as closely as whole spins allow (this page's arithmetic). Two bodies in equilibrium with very different temperatures: that is Frenkel and Warren's objection in miniature; the Gibbs side answers that exact consistency of the thermodynamic laws at every size matters more.^28,27

Chapter 13Efficiency above one

A reversible Carnot engine takes heat Q_h from a hot reservoir, delivers work W, and gives heat Q_c to a cold one. Reversibility means the entropy taken from one reservoir equals the entropy given to the other, Q_h/T_h = Q_c/T_c, and the efficiency is

η = W/Q_h = 1 − T_c/T_h = 1 − β_h/β_cA reversible engine between reservoirs at T_h and T_c; η counts work per unit of heat taken from the hot side.

Written in coldness, η is a straight line in the hot reservoir's β. As the hot reservoir moves along the coldness ruler from T_c through +∞ into negative temperatures, η rises smoothly from 0 through 1 and beyond.

Why beyond? When T_h is negative, taking heat out of the hot reservoir raises its entropy, because its entropy falls as its energy rises. The engine can then also take heat from the cold reservoir, which lowers that reservoir's entropy by the same amount, and turn both into work:

W = Q_h + |Q_c|, η = 1 + T_c/|T_h| > 1 (T_h < 0)Here the heat |Q_c| = Q_hT_c/|T_h| flows out of the cold reservoir.

With T_h = −300 K and T_c = +300 K, η = 2: one joule from the hot side and one joule from the cold side become two joules of work. With T_h = −1 K, η = 301 (this page's arithmetic). Braun and colleagues listed Carnot engines "above unity efficiency" among the counterintuitive effects that negative temperatures allow.⁶

Figure 5 draws the energy flows per joule taken from the hot tank, with the cold tank at 300 K. Slide the hot tank from 300 K through infinity to negative temperatures; then switch to the spin engine.

Figure 5⚙️ Try it

An engine with its hot tank past infinity

Slide the hot tank past ±∞

How hot is the hot tank?

300 K±∞−300 K−100 K

Hot tank

Heat from the cold tank

Work out

Efficiency

Ideal engine: a reversible Carnot engine between a hot reservoir at T_h and a cold one at T_c = 300 K, per joule of heat from the hot side; η = 1 − T_c/T_h = 1 − β_h/β_c, a straight line in the hot side's coldness. For T_h < 0 the engine also draws heat from the cold reservoir, so W = Q_h + |Q_c| and η > 1; energy is conserved, and preparing the inverted reservoir costs work that the figure does not count. Spin engine: a spin-1/2 quantum Otto cycle with gap ε₁ = 10 kT_c at the hot bath and ε₂ = ε₁/2 at the cold bath (this page's model); η = 1 − ε₂/ε₁ = 50% whatever the bath, and the work per cycle, (ε₁ − ε₂)(p₁ − p₂), roughly doubles from an infinitely hot bath to a strongly inverted one. Experiments (de Assis et al. 2019; Nettersheim et al. 2022, up to 30% more power) ran quantum Otto cycles with other working media and settings.An engine takes heat from a hot tank, turns some of it into work, and normally dumps the rest into a cold tank. Slide the hot tank hotter, through infinity and into negative temperatures. Past infinity the cold tank's arrow turns round: it gives heat too, and the 'efficiency' goes over 100%. Nothing is free: the energy still adds up, and making the hot tank costs at least as much work as you get back.

The cold tank's arrow shrinks to nothing as the hot tank reaches ±∞, then turns round and grows. Nothing else special happens at infinity: the efficiency just keeps climbing in a straight line.

None of this is free energy. The efficiency counts only the hot side's heat, and energy is conserved. And the inverted reservoir has to be made. Suppose it is made from matter at the cold bath's temperature and ends there after the engine has drained it. Then the whole process, preparation included, exchanges heat with a single bath, and Kelvin's statement of the second law forbids a cycle that turns heat from a single bath into net work. You get back at most the work you spent flipping the spins (this page's argument). The Gibbs camp goes further and calls efficiencies above one an artefact of the definition.^26,27

Ramsey drew a stranger consequence in 1956. Kelvin's statement says that no cycle can turn heat from a single reservoir entirely into work, and it holds for reservoirs at positive temperature. For a reservoir at a negative temperature it has to be amended: heat drawn from such a reservoir can be turned entirely into work, with no cold reservoir at all, and what becomes impossible instead is turning work entirely into heat inside it.^7,24 In 2018 Henning Struchtrup offered a third reading of the fight in Chapter 12: an inverted spin system is an unstable state with no temperature at all, positive or negative, and the work spent flipping its spins is stored in it, as in a battery or a spring.³²

Real experiments use quantum Otto engines rather than Carnot engines. In an Otto cycle the working medium touches one bath at a time, and in the ideal case its efficiency is fixed by the cycle's design, not by the baths. In 2019 Rogério de Assis and colleagues ran an NMR engine with one reservoir at an effective negative spin temperature and found its efficiency could exceed that of the same engine with both reservoirs positive, but beat the standard Otto efficiency only when run in finite time.³³ In 2022 Jens Nettersheim, Artur Widera and colleagues ran an engine on single caesium atoms in a rubidium bath and reported up to 30% more power at negative effective spin temperature than at positive temperatures, "including even the case of infinite temperature".³⁴ On the theory side, Alberto Brollo, Adolfo del Campo and Alvise Bastianello found in November 2025 that extra conservation laws reduce the efficiency of such engines at positive temperatures but enhance it at negative ones.³⁵

The figure's spin mode is a cartoon of that kind of engine: its efficiency is fixed at 50% by design, and a more inverted bath fills the upper level more, so each cycle does up to about twice the work it would with an infinitely hot bath. That factor is this page's model, not a measurement. The case with both reservoirs at negative temperatures has its own bookkeeping, analysed by P. T. Landsberg in 1977, and this page leaves it out.³⁶

Chapter 14Temperature is a slope

One more oddity is often confused with this one. A black hole has a positive temperature, the Hawking temperature T_H = ħc³/(8πGMk_B), about 6.2 × 10⁻⁸ K for one solar mass (this page's arithmetic),³⁷ and it gets hotter as it loses energy: a negative heat capacity. Two-level spins do the opposite. Their heat capacity, C/(Nk_B) = (βε)²e^βε/(1 + e^βε)², is positive for either sign of β; it is 0.42 at βε = ±2. A black hole has a positive temperature and a negative heat capacity; inverted spins have a negative temperature and a positive heat capacity.

Five things the minus sign does not mean:

Below absolute zero: Negative temperatures lie beyond infinity. Absolute zero stays the unreachable floor of the positive side.
Very cold: Among negative temperatures, the closer to zero, the hotter: −50 pK is hotter than −750 pK (Chapter 4).
The hottest ever: Every negative temperature is hotter than the Sun, but different negative temperatures measure different things, and no single one is safely the hottest (Chapters 4 and 10).
A free lunch: An efficiency above one counts only one reservoir's heat, and making the inverted reservoir costs at least the work it gives back (Chapter 13).
A laser's temperature: An inverted pair of levels in a driven medium has a negative label, not a temperature (Chapter 11).

Where things stand, as of 24 October 2026: the negative temperatures of matter in equilibrium that we found are the nuclear-spin and cold-atom experiments in this page, the 2022 vortex gas of Chapter 11, and a programme by Ulrich Schneider's group in Cambridge (Schneider was one of the authors of the 2013 Munich experiment) that prepares potassium-39 atoms at negative absolute temperatures in triangular and kagome optical lattices, where the energy turned upside down (Chapter 9) reverses the sign of the tunnelling and makes the lattice frustrated. The group has described this work in conference abstracts from 2023 to 2025,^40,42 in two 2024 Cambridge PhD theses^41,43 and, in September 2025, in a first paper: a preprint by Mehedi Hasan and colleagues that used a negative absolute temperature to hold the atoms at the frustrated top of a triangular lattice's band. There the superfluid gave way to a Mott insulator at interactions, relative to tunnelling, about 2.7 times weaker than in the same lattice at positive temperature, where it is unfrustrated.⁴⁴ On 24 October 2026 neither arXiv nor the group's publication list gave that preprint a journal reference, and a search found no journal paper from 2025 or 2026 reporting a new negative temperature in matter. That preprint is the newest negative-temperature experiment in matter we found; the newest in light is the May 2025 few-mode-fibre preprint of Chapter 11. The theory is active: the 2025 engine result of Chapter 13, and a 2026 Physical Review A paper that predicts light in waveguides condensing at negative temperatures.³⁸ The dispute over the word is open.

The thread through all of it is one rule: heat flows from small β to large β. The temperature T = 1/(k_Bβ) is a convenient label for ordinary matter, defined so that bigger means hotter, and it misbehaves only when the slope turns over. Nº 3 used that slope to price a fluctuation against a cosmic horizon; Nº 4 warned that a temperature can be a label; Nº 2's bit of information is the entropy of a spin at the top of the hill. And a piece of rhodium near Helsinki held, for a while, nuclear spins that would have given heat to the Sun.

Temperature is a slope.

Glossary

Nuclear spin: The tiny magnet carried by an atomic nucleus. In a magnetic field, the magnet of a spin-½ nucleus can point along the field (lower energy) or against it (higher energy).
Entropy, S: Boltzmann's constant times the logarithm of the number of microscopic arrangements. One bit is k_B ln 2.
Coldness, β: 1/(k_BT), the slope of entropy against energy divided by k_B. Heat flows from small β to large β. β = 0 is T = ±∞.
Population inversion: More particles in an upper energy level than in a lower one. In equilibrium, it means a negative temperature.
Boltzmann and Gibbs entropy: Two ways to count states: those at a given energy (Boltzmann) or all those at or below it (Gibbs). Only the first allows negative temperatures.
Excitation temperature: The temperature that would give one pair of levels its observed population ratio. A label, not a thermometer reading, unless the system is in equilibrium.
Adiabatic demagnetisation: Lowering the magnetic field on thermally isolated spins. It cools spins at positive temperature and makes spins at negative temperature hotter.
Optical lattice: A pattern of laser light that holds atoms in a regular array, like eggs in a box.
Bose–Einstein condensate: A gas of bosons cold enough that a large fraction of them share one quantum state.
Mott insulator: A state of atoms in a lattice in which repulsion pins a fixed whole number of atoms, here one, to each site.
Carnot efficiency: The efficiency of an ideal reversible engine between two reservoirs, 1 − T_c/T_h.
Heat capacity: The energy needed to raise a temperature by one kelvin. Negative for black holes, positive for spins on both sides of infinity.
Picokelvin, nanokelvin: 10⁻¹² K and 10⁻⁹ K.

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−750 picokelvin

Below zero?

What "hotter" really means

Count the ways

A better thermometer

The touch test

A ceiling, and the first flip

Two temperatures in one metal

Upside down

Atoms past infinity

Whirlpools, light and labels

The fight over a word

Efficiency over 100%

Temperature is a slope

Minus seven hundred and fifty picokelvin