# Exciton nature of plasma phase transition
in warm dense fluid hydrogen

I. D. Fedorov<EMAIL_ADDRESS>Joint Institute for High Temperatures of the
Russian Academy of Sciences, Izhorskaya 13 Building 2, Moscow 125412, Russian
Federation  Moscow Institute of Physics and Technologies (National Research
University), Institutskij pereulok 9, Dolgoprudny Moscow region 141700,
Russian Federation  V. V. Stegailov<EMAIL_ADDRESS>Joint Institute for
High Temperatures of the Russian Academy of Sciences, Izhorskaya 13 Building
2, Moscow 125412, Russian Federation  Moscow Institute of Physics and
Technologies (National Research University), Institutskij pereulok 9,
Dolgoprudny Moscow region 141700, Russian Federation  HSE University,
Myasnitskaya ulitsa 20, Moscow 101000 Russian Federation

###### Abstract

The transition of a warm dense fluid hydrogen from insulator to a conducting
state at pressures of the order of 20–300 GPa and temperatures of 1000–5000 K
has been the subject of active scientific research over the past few decades.
The use of various experimental methods has not yet led to obtaining reliable
consistent results, and despite numerous attempts, theoretical methods have
not yet been able to explain the existing discrepancies, as well as the
microscopic nature of the mechanism for the transition of hydrogen fluid to a
conducting state. In this work, we have discovered a key mechanism of the
transition associated with formation and dissociation of exciton pairs, which
allows to explain several stages of the transition of hydrogen from molecular
state to plasma. This mechanism is able to give quantitative description of
several experimental results as well as to resolve the discrepancies between
different experimental studies.

Introduction. The behavior of hydrogen under conditions of extreme compression
(100-400 GPa) and high temperatures (1000-4000 K) has been studied for quite a
long time McMahon  _et al._ (2012); Utyuzh and Mikheyenkov (2017); Helled  _et
al._ (2020); Silvera and Dias (2021); Norman and Saitov (2021) but its
theoretical description still poses many unresolved questions Liu  _et al._
(2020); Röpke  _et al._ (2021); Ishkhanyan and Krainov (2021). One of these
problems is the transition of compressed hydrogen in both solid and fluid
states into a conducting state (metallization of solid hydrogen was predicted
a long time ago Wigner and Huntington (1935)). Over the past few decades, a
large number of experimental works have been carried out, both using shock
compression Fortov  _et al._ (2007); Knudson  _et al._ (2015); Mochalov  _et
al._ (2017); Celliers  _et al._ (2018) and using pulsed heating in diamond
anvil cells Loubeyre  _et al._ (2004); Dzyabura  _et al._ (2013); Ohta  _et
al._ (2015); McWilliams  _et al._ (2016); Zaghoo  _et al._ (2018); Houtput
_et al._ (2019); Jiang  _et al._ (2020). At the same time, there is still no
unambiguous theory that explains the experimental results as well as multiple
differences in the observations of different groups.

Initially, the transition of fluid hydrogen from the molecular state to the
atomic state was considered from the point of view of chemical models Norman
and Starostin (1968); Biberman and Norman (1969); Lebowitz and Lieb (1969);
Norman and Starostin (1970, 1970); Ebeling (1971); Kraeft  _et al._ (1986);
Saumon and Chabrier (1991); Reinholz  _et al._ (1995); Ebeling and Norman
(2003); Norman (2006); Khomkin and Shumikhin (2013); Starostin  _et al._
(2016); Ebeling  _et al._ (2017). The concept of plasma phase transition was
introduced Norman and Starostin (1968); Filinov and Norman (1975); Norman and
Saitov (2019). Later, the insulator-to-metal transition concept has become
widespread Redmer and Holst (2010). The most common ab initio approaches are
quantum molecular dynamics (QMD) based on density functional theory and
quantum Monte Carlo methods (see Scandolo (2003); Tamblyn and Bonev (2010);
Morales  _et al._ (2013); Mazzola and Sorella (2017); Knudson and Desjarlais
(2017); Knudson  _et al._ (2018); Karasiev  _et al._ (2018); Geng  _et al._
(2019); Norman  _et al._ (2019); Ramakrishna  _et al._ (2020) and Delaney  _et
al._ (2006); Tubman  _et al._ (2015); Morales  _et al._ (2010); Pierleoni  _et
al._ (2016); Mazzola and Sorella (2015); Mazzola  _et al._ (2018); Rillo  _et
al._ (2019); Gorelov  _et al._ (2020); Tian  _et al._ (2020); Ruggeri  _et
al._ (2020); van de Bund  _et al._ (2021). At the same time, the plephora of
methods cannot fully explain the key properties of the transition, such as a
large isotope effect Zaghoo  _et al._ (2018), a large latent heat of
transition Houtput  _et al._ (2019), as well as differences in the moments of
the detected increase in absorption and reflectivity Rillo  _et al._ (2019).

In this work, we focus on two previously largely neglected phenomena: on the
energy transfer from ions to electrons that is the driving force of thermal
excitations leading to plasma phase transition and on the initial localized
elementary processes of conducting phase formation. For this purpose, we
deploy the methods within the density functional theory (DFT) that give the
possibility to capture the corresponding non-abiabatic and non-equilibrium
processes (in contrast to the finite temperature (FT) DFT method that is
nowadays used routinely for warm dense matter calculations, e.g. see Morales
_et al._ (2013); Knudson and Desjarlais (2017); Knudson  _et al._ (2018);
Karasiev  _et al._ (2018); Geng  _et al._ (2019); Norman  _et al._ (2019);
Ramakrishna  _et al._ (2020)).

Our previous study of the dynamics of individual electronic excitations
Fedorov  _et al._ (2020) showed the possibility of the appearance of an
localized excitons during the transition. The lifetimes of such excitons
turned out to be on the order of 10 fs, which is much larger than a typical
time step of ab initio molecular dynamics. This fact casts doubt on the
applicability of FT DFT, where the instantaneous nature of electronic
transitions and thermalisation is assumed. In our further study Fedorov and
Stegailov (2021), we showed the dissociation of such bound electron-hole
($e$-$h$) pairs. Excitons in liquids are not unusual, e.g. these states were
discovered in Xe Beaglehole (1965); Asaf and Steinberger (1971). In this work,
we extend our preliminary studies to the mechanism of the plasma phase
transition in warm dense hydrogen and compare the results with experimental
data.

Model and calculation details. For modelling of the first singlet excited
electronic state (S1) of the system within DFT, we use the Restricted Open-
Shell Kohn-Sham (ROKS) method Frank  _et al._ (1998); Doltsinis and Marx
(2002). ROKS uses the orbital representation of the many-electron wave
function and therefore allows the analysis of singly occupied molecular
orbitals (SOMOs) by calculation of the corresponding Wannier centers (the
SOMO-1 orbital with lower energy is interpreted as a hole ($h$) and the higher
SOMO-2 orbital is interpreted as an electron ($e$) Schwermann and Doltsinis
(2020)). Together they describe an $e$-$h$ pair or an exciton.

Figure 1: The system of 480 hydrogen atoms (0.8 g/cc, 1400 K). A gradient
color line shows the 30 fs trajectory of the Wannier center of SOMO-2 ($e$)
when the simulation box is centered each time under periodic boundary
conditions with respect to the Wannier center of SOMO-1 ($h$). (see more in
Supplementary Materials).

The calculations are performed using CPMD CPM  with the BLYP exchange-
correlation functional. The QMD step is 4 a.u. 480 hydrogen atoms is
considered in the density range 0.6-1.0 g/cc at temperatures 600-3000 K
corresponding to pressures 60-240 GPa. A plane wave basis with a cutoff of 70
Ry is used. The summation over the Brillouin zone is replaced by the value at
the $\Gamma$-point. The initial configurations for each density and
temperature are obtained by Car-Parrinello MD in the electron ground state
(S0) using the Nose-Hoover thermostat. Then, Born-Oppenheimer MD in S1 without
thermostat is carried out within ROKS at times of at least 1 ps with the
derivation of Wannier centers at each MD step (see Fig. 1).

Due to the technical difficulties of pressure calculations in CPMD, we use
VASP Kresse and Furthmüller (1996) for pressure calculations in the ground
state (S0) (see Supplementary Materials).

QMD results. Previously Fedorov  _et al._ (2020), we have shown the mechanism
of spontaneous non-adiabatic energy transfer from molecular kinetic energy to
excitons. The probability of the formation of such excitons increases with
temperature. In this work, we analyze the properties of a model of a single
exciton at different temperatures and densities and determine the conditions
of exciton dissociation when energy transfer from ions to electrons becomes
irreversible leading to the plasma phase transition.

As can be seen from the visualization of the system in Fig. 1, there are
protons not bound to any Wannier center near the considered $e$-$h$ pair (the
SOMO-1 and SOMO-2 centers, respectively). This is a regular picture (see the
animation in Supplementary Materials). It is worth noting that such “free”
protons collide quite often with hydrogen molecules and replace their protons.
Moreover, as can be seen from the displayed trajectory of the electron
relative to the hole (Fig. 1), most of the time the $e$-$h$ distance is quite
small. The analysis of such distances via the $e$-$h$ radial distribution
function (RDF) (Fig. 2) shows that, up to certain temperatures, the initial
parts of RDFs correspond to the $g(r)\sim\exp(-r/a)$ dependence that is a
well-known behavior of charges in a Coulomb field. The tails of RDFs deviate
from exponential dependence due to the fluid state of the system and high
temperatures, which undoubtedly affect the electron and hole mobility.

Figure 2: RDFs for the Wannier centers of SOMO-1 ($h$) and SOMO-2 ($e$)
orbitals for 0.8 g/cc (other densities are presented in Supplementary
Materials). The red thick line reflects the $\exp(-r/a)$ distribution that
holds at short distances and lower temperatures.

The red line in Fig. 3 shows the dependence of the height of the RDF maximum
$g_{max}=g(r=0)$ on temperature (determined by the $\exp(-r/a)$ approximation
of RDFs at short distances). One can notice a sharp drop with increasing
temperature that indicates the beginning of the dissociation of excitons. Such
a dissociation leads to the fact that the reverse transitions to the ground
electronic state S0 become unlikely, due to the spatial separation of the
$e$-$h$ pair.

Another important characteristic of an exciton is the distance at which RDF
crosses the value of 1 (the green line in Fig. 1). This distance corresponds
to the typical size of a $e$-$h$ pair at a given temperature (the size effect
was checked with a system of 240 H atoms, see Supplementary Materials). Using
this distance as a threshold, we obtain the average waiting time of
dissociation during which a $e$-$h$ pair is in a bound state $\tau_{dis}$ (see
the blue line in Fig. 3). The obtained dependence of lifetime $\tau_{dis}$
correlates quite well with the behavior of $g_{max}$ and demonstrates a sharp
decrease with increasing temperature as well.

The lower threshold for the values of exciton lifetimes that determines the
irreversible dissociation of the $e$-$h$ pair is the average vibrational
period of H2 molecules $\tau_{vib}\sim 7-9$ fs. The temperatures when exciton
lifetime becomes similar or lower than the vibrational period of hydrogen
molecules are the temperatures when the energy transfer from ions to
electronic excitations becomes irreversible. We consider this process as the
first stage of the plasma phase transition (see below).

The analysis of the averaged molecular composition at a given temperature is
carried out by sorting all the positions of protons along the equilibrium S0
QMD trajectory into H2 pairs with distances less than 1.0 Å living longer than
10 fs. The data in Fig. 3 (the purple circles) show that at the exciton
dissociation temperatures (marked by the blue bar) the system has molecular
composition.

The $dE=E(S1)-E(S0)$ values averaged over 250 fs along equilibrium QMD
trajectories are displayed in Fig. 3 as green squares. As can be seen from the
graph, at exciton dissociation temperatures, the values of $dE$ is not lower
than 1 eV. Therefore, we can conclude that the fluid H2 is not metallic at the
beginning of transition.

Figure 3: The upper part (the data from S1 MD): the temperature dependence of
the RDF maxima (red line with circles) and the lifetimes of bound excitons
(blue line with squares). The lower part (the data from S0 MD): the
concentration of molecular hydrogen (purple circles) and the S0$\to$S1
vertical excitation energy (green squares). All data are given for a density
of 0.8 g/cc (the data for 0.6 and 1.0 g/cc are in Supplementary Materials).
The temperature range of exciton dissociation is shown by a blue bar
($\tau_{dis}\sim 7-9$ fs). Figure 4: The scheme of the transition mechanism
proposed. Non-adiabatic transitions (I) have been considered previously
Fedorov  _et al._ (2020). Dissociation of Frenkel excitons is modelled in this
work. Later stages are hypothesised and correlated with experimental data.

Transition mechanism. Thus, we propose the following mechanism for the
transition of a dense fluid molecular hydrogen to plasma at heating (Fig. 4).

I. (formation of short-lived Frenkel excitons via non-adiabatic energy
transfer). At the beginning, we have fluid molecular hydrogen with molecules
moving on the potential energy surface of the ground state (S0). Then, as the
temperature rises, the number of non-adiabatic transitions of electron
subsystem from S0 to S1 increases. Each transition is a result of the atomic
kinetic energy transfer to an electronic excitation and leads to the formation
of an exciton. We assume that in experiments independent excitons can form
simultaneously in different parts of the macroscopic fluid H2 sample. The
probability of such non-adiabatic excitations depends on atomic velocities
that contributes to a large H/D isotope effect for this transition Zaghoo  _et
al._ (2018) (as discussed in Fedorov  _et al._ (2020)). The average $e$-$h$
distance in these excitons is about the size of an H2 molecule that is why we
can call them Frenkel excitons. Most $e$-$h$ pairs formed recombine
spontaneously via internal conversion S1$\to$S0 after $\tau_{vib}$ Fedorov
_et al._ (2020).

II. (formation of Wannier-Mott excitons). At higher temperatures, dissociation
of Frenkel excitons becomes very fast with their average lifetime smaller than
the average vibrational period of H2 molecules $\tau_{dis}\lesssim\tau_{vib}$.
Each dissociated Frenkel exciton becomes less delocalized (we see no
pronounces maximum on $g(r)$) but still bound to protons via electrostatic
interactions. This stage is currently beyond our capabilities for direct QMD
calculations. That is why we describe this stage and the next stages only
qualitatevly. Such dissociated Frenkel excitons can be called Wannier-Mott
excitons. This exciton dissociation leads to the irreversible energy transfer
from kinetic energy of molecules to electronic excitations. Each dissociation
event consumes about $dE\sim 1-2$ eV, which are taken from the kinetic energy
of the ions. It explains the “latent heat of transition” of the order of 1
eV/atom that comes from the analysis of experimental data Houtput  _et al._
(2019) (in contrast to the hydrogen dissociation model that predicts this
value to be about 0.05 eV/atom only Morales  _et al._ (2010). This exciton
dissociation leads to a plateau in the dependence of the peak temperature of
fluid H2 on the absorbed laser energy observed in many works Dzyabura  _et
al._ (2013); Ohta  _et al._ (2015); McWilliams  _et al._ (2016); Zaghoo  _et
al._ (2016); Zaghoo and Silvera (2017); Jiang  _et al._ (2020) (see the red
points in Fig. 5). Formation of many Wannier-Mott excitons in a macroscopic
fluid H2 sample explains the rise of absorption of radiation by fluid H2
Zaghoo  _et al._ (2016); Knudson  _et al._ (2015); Celliers  _et al._ (2018)
(see the green points in Fig. 5). The threshold exciton dissociation
temperatures (when $\tau_{dis}\approx\tau_{vib}$) depend not on the kinetic
energy of molecules but on their velocities (in contrast to molecular
dissociation that depends on energy of molecular vibrations). That is why the
threshold temperatures for D are expected to be about 2 times higher than for
H. We can consider this mechanism as another contribution to the detected
large H/D isotope effect for this transition Zaghoo  _et al._ (2018).

III. (formation of reflective clusters from Wannier-Mott excitons). The next
stage of the plasma phase transition is the formation of localized plasma-like
clusters from single Wannier-Mott excitons. As soon as the average size of
such clusters becomes comparable with the wavelength of probing radiation, the
rise of optical reflectivity by fluid H2 is detected Loubeyre  _et al._ (2004,
2012); Zaghoo  _et al._ (2016, 2018); Jiang  _et al._ (2020); Knudson  _et
al._ (2015); Celliers  _et al._ (2018) (see the blue points in Fig. 5). The
electrons in such plasma clusters is not absolutely free but bound to the
corresponding clusters (similarly to the collective excitations discussed in
Lankin and Norman (2009)). In the single-particle electronic energy spectrum
for Kohn-Sham orbitals, their energies are lower than the continuum level but
could be higher than the SOMO-2 energy level.

IV. (formation of plasma). Eventually, the plasma is formed that leads to the
observation of DC conductivity Weir  _et al._ (1996); Fortov  _et al._ (2003,
2007) (see the purple points in Fig. 5).

Discussion of experiments. Fig. 5 shows the T-P diagram of hydrogen with the
regions of plasma, fluid and solid hydrogen. Various transient processes
observed experimentally are assigned to the plasma phase transition
considered. The (P,T) points on Fig. 5 show the threshold conditions
interpreted as the plasma phase transition points. Experimental studies can be
divided into shock wave (SW) and diamond anvil cell (DAC) experiments.

According to the transition mechanism proposed in this work, during heating of
fluid molecular hydrogen we can distinguish the following phenomena detected
by various techniques: (1) a temperature plateau (shown as red dots) that is
explained by non-adiabatic excitations of a Frenkel excitons and their
irreversible dissociation that transfers $\sim$1-2 eV per event from kinetic
energy of molecules to electronic excitations, (2) the rise of optical
absorption (green dots) that is explained by formation of separated Wannier-
Mott excitons that can absorb light but do not reflect, (3) the rise of
optical reflection (blue dots) that is explained by the formation of excition
(plasma-like) clusters with the size $d>\lambda/2$, where $\lambda$ is the
probing light wavelength, and (4) an increase in DC conductivity (purple dots)
due to plasma formation.

Comparison of our QMD results with experimental data shows that our (P,T)
points obtained with BLYP xc-functional and ROKS as the thresholds of Frenkel
excitons dissociation agree quite well with the DAC results of Ohta et al.
Ohta  _et al._ (2015) and Zaghoo et al. Zaghoo  _et al._ (2016); Zaghoo and
Silvera (2017). On the one side, BLYP and ROKS should be considered as quite
generic approximations to the electronic structure of the dense molecular
hydrogen in the first excited state (the dependence of the results on the
choice of xc-functional was shown previously Knudson and Desjarlais (2017)).
On another side, such simple approximations as BLYP and ROKS could be indeed
quite accurate for such a simple system as dense molecular hydrogen (no strong
correlations and no pronounced dispersion interactions). Therefore, good
agreement with experiments could reflect the fact that in DAC experiments the
threshold (P,T) states correspond to the Frenkel exciton dissociation
conditions. Relatively slow heating in DAC experiments Zaghoo  _et al._
(2016); Zaghoo and Silvera (2017) within the framework of the transition
mechanism proposed can explain why the temperature plateau, the onset of
absorption and the rise of reflectivity are not separated in time. Indeed,
reflective plasma clusters can form on the microsecond timescale of these DAC
experiments.

The corresponding SW experiments are about one order of magnitude faster than
the DAC experiments (see Supplementary Materials). The difference in the
moments when absorption and reflectivity in SW studies are detected can be
explained by the kinetics of plasma-like clusters formation. Before the
reflective clusters of the size $d>\lambda/2$ are formed, the system has been
already driven to higher temperatures and pressures. Different kinetics of
plasma-like clusters formation can explain different (P,T) threshold values
when the rise of reflectivity is observed in different experiments (e.g.
Z-machine Knudson  _et al._ (2015) vs NIF Celliers  _et al._ (2018)). In SW
experiments temperatures are not measured directly but calculated using an
equation of state (EOS) that bring another source of possible inaccuracies. In
some SW experiments Fortov  _et al._ (2007); Mochalov  _et al._ (2017), a
density jump is observed during compression (Fig. 5, brown dots). It is
noteworthy that one of EOS variants places this jump near the DAC results and
near our QMD results for exciton dissociation. Moreover, some of the high-
temperature points can explained by the fact that they were observed in DAC on
cooling Jiang  _et al._ (2020), when the kinetics of plasma recombination is
expected to differ strongly from the kinetics of plasma formation on heating
(for example, in SW studies Knudson  _et al._ (2015) a hysteresis of optical
reflection was observed).

Figure 5: T-P phase diagram. The grey area between two dashed lines with black
circles indicates that the Frenkel exciton lifetime $\tau_{dis}$ becomes of
the order of the oscillation period of the hydrogen molecule ($\tau_{vib}=7–9$
fs). The ten-fold decrease of $e$-$h$ RDF $g_{max}$ is shown by the dotted
line. The experimental marks attributed to the plasma phase transition are
shown: the temperature plateau (red points), the increase in optical
absorption (green points), the increase in optical reflectivity (blue points),
the increase in DC conductivity (purple points). The filled symbols correspond
to H, the open symbols correspond to D. Larger symbols correspond to
experiments with higher heating/compression rates (detailed descriptions of
the points with brief descriptions of the experimental works are presented in
Supplementary Materials).

Conclusions. The new mechanism of plasma phase transition in warm dense fluid
hydrogen has been proposed based on the analysis of formation and growth of
localized electronic excitations during heating. Kinetics of these processes
explains both the existing discrepancies in experimental data and the key
experimental observations: the temperature plateaus and the large detected
“latent heat of transition”, the large isotope effect, the variations in the
moments of absorption and reflectivity rise. The key threshold process of the
proposed mechanism is the dissociation of excitons. QMD calculations with the
BLYP exchange-correlation functional and the ROKS method of S1 electron
dynamics give the threshold temperatures and pressures of the transition that
agree quite well with the appropriate experimental data.

###### Acknowledgements.

We thank Nikos Doltsinis and Genri Norman for useful discussions. This
research was supported by The Ministry of Science and Higher Education of the
Russian Federation (Agreement with Joint Institute for High Temperatures RAS
No. 075-15-2020-785 dated September 23, 2020). The authors acknowledge the
Supercomputer Centre of JIHT RAS and the Supercomputer Centre of MIPT. This
research was supported in part through computational resources of HPC
facilities at NRU HSE.

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correspond to real-world counterparts. Our attempt to add the sliding window
strategy to the number of selected non-terminals and manually adjust
probabilities of expansion rules only partially solves this problem. Without
having the precise input specification in advance, it is not possible to
initialize the generator to produce samples that fully cover the real-world
input space. This problem can be resolved by first mining corresponding
expansion probabilities from a few real-world samples and using them to fine-
tune the model on additionally synthesized samples (Soremekun et al., 2022).
Such samples will serve as guidance for the generator to focus on the
uncovered areas of the input space.

Leveraging program feedback.:

We estimate that the model’s performance can be further improved by utilizing
the feedback from the PUT, when available. In contrast to fundamental LLMs,
Modelizer framework allows relatively cheap continuous re-training of the
learned models. If the specifications of the target system have
insignificantly changed, only affecting the system behavior, but not extending
the population of the input-output vocabularies with new elements, the learned
models can be fine-tuned with a few samples collected from system executions.
In Section 2.4, we have represented a scenario when the PUT serves as an
oracle to validate the model’s predictions. Once the model prediction does not
align with execution results, the collected feedback can be passed to the
model as additional training data to adjust the model’s parameters. The model
has to be repeatedly re-evaluated and fine-tuned till the model’s predictions
match the PUT’s execution results. We plan to study the capabilities of
Modelizer and respective learned models to serve as an oracle for automatic
detecting of the behavior deviation cases between various revisions and
implementations of the same protocol. Also, we hope to explore the possibility
of forging feedback-driven tokenizers, which will be able to automatically
adjust the tokenization granularity and possibly mine the input-output
grammars from black-box systems. They will rely not only on the feedback from
the PUT but also leverage the feedback from the model evaluation.

1-bit models.:

Lastly, we are excited by the idea of replacing 16-bit parameter weights with
a 1-bit representation (Ma et al., 2024), which significantly reduces the
memory requirements, and thus allows training more cost-effective models. We
plan to integrate this feature into Modelizer soon.

Modelizer and all experimental data are available at

https://github.com/2ral/Modelizer

Acknowledgments. We appreciate the thorough review and critical insights
provided by our colleagues, which significantly improved the quality of this
manuscript.
This work is funded by the European Union (ERC S3, 101093186). Views and
opinions expressed are however those of the author(s) only and do not
necessarily reflect those of the European Union or the European Research
Council. Neither the European Union nor the granting authority can be held
responsible for them.

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