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# Collective spin 1 singlet phase in high-pressure oxygen

Edited by David Vanderbilt, Rutgers, The State University of New Jersey, Piscataway, NJ, and approved June 13, 2014 (received for review March 11, 2014)

## Significance

Among the elemental diatomic molecules, O_{2} is the only one carrying a spin 1 magnetic moment. In the high pressure phases of oxygen the magnetic moment conspires with intermolecular forces to generate a rich phase diagram. Whereas, up to 80,000 atmospheres the moment persists, at pressures between 80,000 and 200,000 atmospheres, molecular magnetism apparently disappears, however with a number of unexplained vibrational and optical anomalies. Through a fully quantum treatment of the electronic states of the dense crystalline state we find that in this pressure range oxygen still retains a spin moment in an unconventional and rare state of matter dominated by the quantum fluctuations. This state, a special case of so-called spin liquids, explains most of the observed anomalies.

## Abstract

Oxygen, one of the most common and important elements in nature, has an exceedingly well-explored phase diagram under pressure, up to and beyond 100 GPa. At low temperatures, the low-pressure antiferromagnetic phases below 8 GPa where O_{2} molecules have spin S = 1 are followed by the broad apparently nonmagnetic ε phase from about 8 to 96 GPa. In this phase, which is our focus, molecules group structurally together to form quartets while switching, as believed by most, to spin S = 0. Here we present theoretical results strongly connecting with existing vibrational and optical evidence, showing that this is true only above 20 GPa, whereas the S = 1 molecular state survives up to about 20 GPa. The ε phase thus breaks up into two: a spinless ε_{0} (20−96 GPa), and another ε_{1} (8−20 GPa) where the molecules have S = 1 but possess only short-range antiferromagnetic correlations. A local spin liquid-like singlet ground state akin to some earlier proposals, and whose optical signature we identify in existing data, is proposed for this phase. Our proposed phase diagram thus has a first-order phase transition just above 20 GPa, extending at finite temperature and most likely terminating into a crossover with a critical point near 30 GPa and 200 K.

Molecular systems display at high pressure a horn of plenty of intriguing phases. That is especially true of molecular oxygen, whose diatomic molecule survives unbroken up to at least 133 GPa (1), and where the original spin S = 1 of the gas phase plays an important role. In the phase diagram of O_{2} (Fig. 1), we focus on the wide ε-O_{2} phase between 8 and 96 GPa, a phase which has long intrigued the community (2). Unlike the two bordering phases, δ-O_{2}, an antiferromagnetic (AF) S = 1 correlated insulator at lower pressure, and ζ-O_{2}, a regular and superconducting nonmagnetic (NM) metal at higher pressure, ε-O_{2} is an insulator of more complex nature. Structurally, high-pressure X-ray diffraction (3, 4) revealed in the last decade that at the δ−ε transition at P ∼ 8 GPa, the close-packed O_{2} planes undergo a large distortion giving rise to molecular O_{8} “quartets” (Fig. 1, *Inset*). Spin-polarized neutron diffraction showed that simultaneously there is a collapse of long-range AF Néel order at the δ−ε transition (5). That observation unfortunately did not provide conclusive information about the nature of the ground state in ε-O_{2} and in particular about any further role played in ε-O_{2} by the spin of individual molecules, if any. It has been tempting to imagine that the O_{2} molecular magnetic state could simply collapse from S = 1 to S = 0 at the δ−ε transition. In support of this idea, it can be noted that the metallic state band structure of a hypothetical undistorted nonmagnetic O_{2} (6) is prone to turn spontaneously insulating through a Peierls type distortion, for example dimerizing (7, 8) or tetramerizing (9) the molecules. Further density functional theory (DFT) calculations strengthened that picture, showing that the quartet distorted geometry (10) drives the undistorted metal to a band insulator. Moreover, DFT calculations showed that this state exhibits O_{2} vibrations whose frequency and pressure evolution are, between 20 and 96 GPa, in good agreement with infrared (IR) and Raman data (11), as will be shown in *First-Principles Calculations and Results*.

An alternative, much more speculative picture insisting instead on a correlated state where O_{2} retains its S = 1 gas phase spin, and where the quartet of AF coupled sites yields a singlet ground state as the basic O_{8} block of ε-O_{2} (12), provided no such successful predictions and had limited following so far—see, however, ref. 13.

Let us provide first some background before embarking on theoretical calculations. Several experimental authors (7, 14⇓⇓–17) noted that the behavior of O_{2} in the 8- to 20-GPa pressure range is in many ways anomalous compared with that above 20 GPa (18). Focusing on O_{2} vibrations, the good agreement found by Anh Pham et al. (11) between the band insulator calculations and the measured IR and Raman frequencies is limited to above *P* > 20 GPa. Below this pressure, the IR mode reverts nonmonotonically upward approaching the high-frequency Raman mode at the δ-O_{2} phase boundary of 8 GPa, rather than dropping steadily as predicted by the band insulator. Subtly, but not less significantly, the O_{2} Raman data show that 20 GPa marks a delicate but definite breaking point with a lower rate of decrease of the mode frequency with decreasing pressure. Both IR and Raman elements hint at a possible switch of individual O_{2} molecules from S = 0 to S = 1 upon decreasing pressure near 20 GPa. In particular, the frequency gap between the IR mode, where nearest neighbor O_{2} vibrate out of phase, and the Raman mode where they vibrate in phase is proportional to the IR effective charge, connected with electron current “pumping” between an instantaneously extended molecule (which attracts electrons) and a neighboring compressed one (which expels them) in the IR mode. This electron current is absent in the Raman mode, where neighboring molecules vibrate in phase. If the molecules have spin and correlations are strong, the electron hopping is reduced and the current magnitude, proportional to the IR effective charge, must drop compared with the nonmagnetic band state. Thus, the onset of molecular spin upon decreasing pressure should be accompanied by a collapse of the IR intensity and of the IR−Raman splitting, as is indeed observed (see *First-Principles Calculations and Results*). With this background, experimental facts suggest dividing the vast e-O_{2} phase into two phases. The first, e0-O_{2}, is a higher-pressure phase between 20 and 96 GPa where there is no molecular spin and whose physics, including lattice vibrations, is well described as a band insulator whose gap is due to the Peierls-like quartet distortion. The second phase, e1-O_{2}, is a lower-pressure phase between 8 and 20 GPa where that picture fails, and molecular spin probably resurrects signaling that strong correlations coexist within the quartet distortion of the molecules. The long-range Néel order typical of the lower-pressure undistorted phases is absent here, suggesting that it might be replaced by some kind of correlated singlet state such as that of ref. 12. However, strong and verifiable predictions about the same measured vibrational and optical that are well described by the band insulator in ε_{0}-O_{2} have not been advanced for this type of state.

Our work has the following aims: (*i*) Provide first-principles quantitative calculations of electronic and vibrational properties of same quality for both ε_{0}-O_{2} and ε_{1}-O_{2}, including Raman and IR vibrational frequencies and intensities that could be compared with experiment in both regimes. The approach should allow, even if at the approximate mean field level, for the presence of molecular spin whenever that should lower the total enthalpy. (*ii*) Describe the nature, stability, and optical properties of the resulting model S = 1 collective singlet state description of ε_{1}-O_{2}, extending that in literature (12, 13), particularly including quantum fluctuations. (*iii*) Predict the new phase diagram displaying a first-order ε_{1}–ε_{0} low temperature phase transition near 20 GPa, and a novel phase line predicting at high temperature a new critical point roughly at 30 GPa and 200 K inside the broad ε-O_{2} phase.

## First-Principles Calculations and Results

Our starting point is a density-functional theory (DFT) electronic structure calculation with full structural optimization for the whole ε-O_{2} pressure range. We used spin-polarized, self-interaction corrected calculations (DFT+*U*) (19⇓–21) as implemented in Quantum-ESPRESSO (22). By allowing for static magnetic polarization, we permit the possible presence of molecular spin to emerge—of course at the price of assuming it to exist in static and thus symmetry-breaking form. The generalized gradient approximation (GGA) exchange-correlation functional was used in the version of Perdew et al. (23), and DFT+*U* calculations were carried out in the simplified version of Dudarev et al. (24) as implemented in the Quantum-ESPRESSO code (25). The inclusion of a Hubbard *U* for oxygen p-states is called for to provide cancellation of self-interactions still present in simple GGA DFT. Whereas for the uncorrected choice *U* = 0 the calculated band gap is unrealistically small and spin polarization does not arise (10, 11), calculations in a reasonable range of values of the parameter *U* (0.8, 1.0, 1.5 and 2.0 eV) yield antiferromagnetism below 17.6, 20.0, 25.0, and 30.0 GPa, respectively. We selected *U* = 1.0 eV as the value yielding the transition pressure that fits more reasonably the experimental findings to be described below in the vibrational spectra (7, 14, 16). Their frequency and intensity behavior does not change with the value of *U*. We note that this method to reduce self-interactions is convenient but by no means unique, and other possibilities such as hybrid functional approximations could have been adopted. A 4 × 4 × 4 Monkhorst−Pack k-point mesh (26) Brillouin zone sampling was used throughout. The crystal structure (C2/m) and the initial atomic positions were taken from the experimental data (4). Lattice and internal parameters were then fully optimized at different pressures until forces were smaller than 10^{−5} a.u. The vibrational spectra were calculated using the fropho code that calculates phonons based on the Parlinski−Li−Kawazoe method (27). Infrared intensities were calculated by density functional perturbation theory (28) by single-point DFT calculations with the fully DFT+*U* relaxed crystal structures.

The optimized structure has O_{2} molecules forming quartets in each plane, quite close to the experimental ones (4). A clear AF state prevails below 20 GPa, as shown by the enthalpy difference of Fig. 2 between the NM and the AF state, where molecules are simultaneously quartet distorted and antiferromagnetically spin polarized. As anticipated, the predicted low-pressure resurgence of molecular spin comes with an unrealistic AF static long-range order, which is absent in experiment (5). How this artificial mean field symmetry breaking can be removed by quantum fluctuations will be described later. Ignoring that for the moment, and assuming the mean field to yield as usual a reasonable total energy, we can exploit our first-principles calculations to obtain a prediction for the change of other properties brought about by the instantaneous presence of molecular spin. We calculated, with the DFT+*U* approach, the vibrational spectra of both the nonmagnetic and magnetic states of ε-O_{2}. As is shown in Fig. 3*B*, the onset of spin breaks the monotonic drop of both Raman and IR mode frequencies with decreasing pressure, which was predicted for the nonmagnetic state by previous calculations (11) but which did not agree with experiment below 20 GPa. Direct comparison of Fig. 3*B* with Fig. 3*A* shows now a better agreement, confirming that both the nonmonotonic rise of the IR mode and the slight stiffening of the Raman mode are spin related. Many other phonon modes are also influenced by the onset of spin.

In Fig. 4, our calculated evolution of the main far infrared vibrational mode (see Fig. 4*B*) is shown to drop below 20 GPa in agreement with experimental data (see Fig. 4*A*). This evolution of both high- and low-frequency IR modes upon lowering pressure below 20 GPa goes together with a corresponding change—in fact a decrease—of the mode effective charge. Fig. 5 shows that the dramatic drop of the IR intensity observed in this regime (see Fig. 5, *Inset*), so far unexplained, is now well accounted for by the onset of molecular spin.

We already hinted at the main physical reasons why spin causes all these changes in the vibrational spectrum. The first element is that spin arises in connection with strong electron correlations, which characterize all lower-pressure phases including δ-O_{2}. In the strongly correlated state, O_{2} molecules reduce their mutual electron hopping, and tend to revert toward their gas phase state, which has spin 1, with shorter bond length and about 71 cm^{−1} higher vibrational frequency (29). That explains why the highest (intramolecular) Raman mode reduces its softening rate below 20 GPa (see Fig. 3). The reduced IR effective charge, and the shrinking of the frequency gap between the out-of-phase IR and the in-phase Raman modes as calculated and observed, implies a reduced intermolecular electron hopping in the AF correlated state.

For our subsequent understanding of the magnetic state below 20 GPa, it is also important to estimate the effective Heisenberg exchange couplings among O_{2} molecules. Calling *J*_{1}, *J*_{2}, *J*_{3}, and *J*_{4} the exchange values between the first, second, third, and fourth neighbors (see Fig. 6), we carried out constrained spin polarized DFT+*U* calculations, based on the experimental structure at *P* = 11.4 GPa (3) and a variety of six different AF configurations. Fitting these results, we obtained *J*_{1} = 170.0 ± 30 meV, *J*_{2} = 35.5 ± 2 meV, *J*_{3} = 10.5 ± 2 meV, and *J*_{4} = 14.4 ± 4 meV. On account of the mean-field nature of the DFT calculations, these values are probably somewhat larger values than real and should be considered upper bounds.

## Lattice Singlet Model and Quantum Fluctuations of S = 1 Spins

### S = 1 Lattice Model.

*First-Principles Calculations and Results* showed that the resurgence of molecular spin below 20 GPa could simultaneously explain several observed O_{2} vibrational anomalies. However, the mean-field long-range AF order obtained by DFT disagreed with the lack of long-range spin order found experimentally (5). A state where molecular spins are present without *T* = 0 long-range order would constitute a kind of spin 1 liquid. In this section, we discuss extending the picture earlier proposed by Gomonay and Loktev (12) and more recently pursued by Bartolomei et al. (13) consisting of an overall singlet state of an isolated quartet of *S* = 1 molecules.

The quantum mechanical competition between AF long-range order and an overall singlet state must be pursued in the whole ε_{1}-O_{2} lattice. For that purpose, we simplify the system as a 2D square lattice model made of plaquettes (quartets) of *S* = 1 Heisenberg sites. Each site, representing an O_{2} molecule, is AF coupled to nearest neighbors within the same plaquette by AF exchange couplings *J*_{1} > 0, and to nearest neighbors in the next plaquettes by *J*_{2} < *J*_{1}; see Fig. 7. Two different states compete: the Neél AF configuration, as obtained by DFT and which breaks spin SU(2) symmetry, and a singlet, NM state that is akin to a collection of independent plaquettes, each in its singlet ground state. The singlet ground state of an isolated plaquette of energy *E*_{0} = −6*J*_{1} is obtained by coupling second neighbor sites 1 and 3 to *S*_{13} = *S*_{1} + *S*_{3} = 2, coupling sites 2 and 4 to *S*_{24} = *S*_{2} + *S*_{4} = 2 (see Fig. 7), and then coupling *S*_{13} and *S*_{24} to a total singlet *S* = *S*_{13} + *S*_{24} = 0. The energy per site of an independent collection of plaquettes thus is (note that the number of plaquettes *N*_{□} is one-quarter the number of sites)*J*_{2} < *J*_{1}/2, the nonmagnetic collection of independent plaquettes will be lower in energy than the Neél configuration. If we use *J*_{2} ≃ 35 meV and *J*_{1} ≃ 170 meV > 2 *J*_{2}, we may conclude that the actual ground state is a collection of independent nonmagnetic plaquettes. This state is akin to that proposed in ref. 12. We observe further that the next-nearest-neighbor exchange *J*_{3} > 0 frustrates and penalizes the Neél configuration more than the nonmagnetic one, leading to the energy balance within each plaquette from *E*_{□} − *E*_{Neél} = −*J*_{1}/2 + *J*_{2} → −*J*_{1}/2 + *J*_{2} − *J*_{3}/2. Moreover, the interplane exchange *J*_{4} gives no contribution to the classical energy, since its effects cancel out as can be seen in Fig. 6. Since both *J*_{3} and *J*_{4} are anyhow smaller than *J*_{2}, we shall not consider them in the following analysis.

The above treatment, however, is still crude, as it does not take into account quantum fluctuations. To evaluate their impact, we assess the stability of the nonmagnetic state against quantum fluctuations. Elementary quantum fluctuations are built by combining into an overall singlet two separate spin fluctuations in neighboring plaquettes. The first excited state of the isolated plaquette is still obtained by *S*_{13} = 2 and *S*_{24} = 2, now coupled into a total spin *S* = *S*_{13} + *S*_{24} = 1, at energy *J*_{1} above the ground state. Let us denote as *S*_{z} = *M* = −1, …, +1. If we consider two nearest neighboring plaquettes, identified by the positions **R** and **R**′, application of the exchange *J*_{2} to the state in which both plaquettes are in the ground state,*J*_{2}/4 when they are nearest neighbors. This attraction, however, cannot overcome the hard-core constraint, since the two triplets cannot reside on the same plaquette. The problem is thus equivalent to two hard-core bosons, each costing an energy *J*_{1} and able to hop between nearest neighbor plaquettes with an amplitude −*J*_{2}. Despite the weak nearest neighbor attraction −*J*_{2}/4, their lowest energy state is unbound and has energy 2*J*_{1} − 8*J*_{2}. The overall singlet, spin liquid state will be stable so long as this excitation gap is positive, whereas antiferromagnetism will prevail if it is zero or negative. Based on this result, a better estimate for the stability of the overall singlet is the condition 2*J*_{1} ≳ 8*J*_{2}, which leads to *J*_{2} < *J*_{1}/4. The estimated values of the exchange couplings satisfy this inequality, confirming the stability of an overall singlet state, even once quantum fluctuations are included. The spin liquid singlet state, a lattice of plaquettes each made of four antiferromagnetically correlated S = 1 sites, with weaker but nonzero interplaquette correlations, represents in conclusion our best model for ε_{1}-O_{2} below 20 GPa.

### Infrared Spectrum in the Lattice Singlet Model.

One burning question is, at this point, what evidence can one identify proving the existence of a nonzero molecular spin in low pressure ε-O_{2}, despite its lack of magnetic long-range order? There is, in fact, at least one such piece of evidence, long published but not interpreted yet, and it is optical. Near-infrared spectroscopy across the δ-ε transition shows the excitation of a single O_{2} molecule from its lowest-energy_{2}, is allowed in a lattice of molecules and shows up in high-pressure optical absorption (15). The peak corresponding to*S* = 0 state) is clearly visible in the δ phase, at an energy ∼ 8–9.000 cm^{−1}, a frequency comparable to that of the isolated molecule. When the ε phase sets in above 8 GPa, the^{−1} and very considerably broadened, to the extent that its vibrational satellites are not anymore distinguishable (15). That optical observation can now be explained.

In the *S* = 1 Heisenberg model representation, the^{−1} in the δ phase at much lower pressure (15), where the absorption peak is close to the molecular excitation energy, and its broadening, attributable to shake-up of spin waves, is small. As pressure rises, the exchange coupling *J*_{1} increases rapidly, as signaled by the blue shift and broadening of this transition visible within the δ phase (15). In the O_{2} quartet singlet state, a vacancy costs roughly the energy difference between the ground state of the three surviving O_{2} molecules and the initial four-molecule state. This difference is readily found to be 3*J*_{1}. Moreover, since the three-molecule state has *S* = 1, while the initial four-molecule state was a singlet, an itinerant spin-1 excitation must be created in addition to the vacancy, costing an additional energy ω ∈ [*J*_{1} − 4*J*_{2}, *J*_{1} + 4*J*_{2}] (see Fig. 8, *Inset*). We therefore predict the absorption line to undergo at the δ → ε transition a sudden blue shift of ≲ 4(*J*_{1} − *J*_{2}) with a large broadening ∼ 8*J*_{2} due to interplaquette exchange. With the calculated exchange values at 11.4 GPa, that means ≃ 0.54 eV = 4,355 cm^{−1} and ≃ 0.28 eV = 2,258 cm^{−1}, respectively, values that are reasonably close to the experimental ones (15) (see Fig. 8). In conclusion, the abrupt change of optical absorption is explained by the onset of a correlated singlet state at the δ → ε transition.

## New Phase Diagram

The ground state of high-pressure ε-O_{2}, correctly described as a Peierls-distorted nonmagnetic band insulator and an overall spin singlet above 20 GPa, turns below 20 GPa into a correlated insulator, where molecules recover their S = 1 spin, but where exchange couplings and quantum fluctuations between spins conspire to yield another, different overall spin singlet ground state. Sharing very similar quartet lattice structures, the two states appear to have the same symmetry, and the question is whether the phase diagram should show a phase transition between the two or not. Assuming same symmetry, there could only be a first-order transition or a smooth crossover. On account of the mechanical coupling which the onset of molecular spin must exert on the overall lattice structure, the likeliest candidate is a first-order transition.

As it turns out, there is in literature, ignored so far by most, rather clear evidence of a low-temperature, first-order phase transition (17), signaled around 25 GPa at 20 K by a small but sharp and sudden 10 cm^{−1} splitting of a low-frequency vibration. We propose that this could signal precisely the first-order line separating the low-pressure phase, say ε_{1}-O_{2}, and the high-pressure one, say ε_{0}-O_{2}. This phase line will extend at finite temperature but, on account of same symmetry, should terminate with a critical point. Because there are underlying spins in ε_{1}-O_{2} but not in ε_{0}-O_{2}, the phase line must turn toward higher pressures as temperature grows, because ε_{1}-O_{2} possesses a high-temperature spin entropy *S* ∼ *ln*3 per molecule whereas ε_{0}-O_{2} does not. Accurate room temperature vibrational data (16) actually indicate a smooth crossover between unsplit and split modes above 30 GPa, a pressure definitely higher than the 20 K sharp transition near 25 GPa. That confirms our suggestion, and supports the prediction of a critical point below 300 K and near 30 GPa. Our new proposed phase diagram is therefore summarized in Fig. 9.

## Conclusions

A fresh ab initio study connects with existing vibrational evidence to indicate that molecular spin plays an important role in the lower-pressure part of the ε-O_{2} oxygen phase diagram. Specifically, we propose replacing the single broad ε-O_{2} phase from 8 to 96 GPa with two phases ε_{1}-O_{2} and ε_{0}-O_{2}—the first a local singlet spin 1 liquid, the second a regular, Peierls band insulator—separated by a first-order phase transition near 20 GPa. The predicted phase line should evolve with temperature, terminating with a novel critical point, probably near 30 GPa and 200 K. The high-temperature region below 30 GPa must be characterized by thermally fluctuating S = 1 spins, whose presence should be directly detectable by magnetic susceptibility measurements in the 8- to 30-GPa pressure range. At low temperatures, the wealth of low-energy spin excitations present in ε_{1}-O_{2} but absent in ε_{0}-O_{2} should in addition give rise to very new energy dissipation channels and processes in the former phase.

## Acknowledgments

The authors would like to thank G. Baskaran, Otto Gonzales, Sadhana Chalise, Carlos Pinilla, and Nicola Seriani for fruitful discussions. This work, and in particular Y.C.’s position, was partly sponsored by European Research Council Advanced Grant 320796 – MODPHYSFRICT. Contracts PRIN/ COFIN 2010LLKJBX 004 and 2010LLKJBX 007, European Union-Japan Project LEMSUPER, and Sinergia CRSII2136287/1 are also acknowledged.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: tosatti{at}sissa.it.

Author contributions: E.T. designed research; Y.C., M.F., S.S., and E.T. performed research; Y.C., M.F., S.S., and E.T. analyzed data; and Y.C., M.F., S.S., and E.T. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

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