# First Principles Theory of the Pressure Induced Invar Effect in FeNi Alloys

Amanda Ehn, Björn Alling, Igor A. Abrikosov  Department of Physics, Chemistry,
and Biology (IFM), Linköping University, SE-58183 Linköping, Sweden

###### Abstract

The Fe0.64Ni0.36 alloy exhibits an anomalously low thermal expansion at
ambient conditions, an effect that is known as the invar effect. Other
FexNi1-x alloys do not exhibit this effect at ambient conditions but upon
application of pressure even Ni-rich compositions show low thermal expansion,
thus called the pressure induced invar effect. We investigate the pressure
induced invar effect for FexNi1-x for x = 0.64, 0.50, 0.25 by performing a
large set of supercell calculations, taking into account noncollinear magnetic
states. We observe anomalies in the equation of states for the three
compositions. The anomalies coincide with magnetic transitions from a
ferromagnetic state at high volumes to a complex magnetic state at lower
volumes. Our results can be interpreted in the model of noncollinear magnetism
which relates the invar effect to increasing contribution of magnetic entropy
with pressure.

##  I Introduction

In 1897 Ch.E. Guillaume discovered that the face centred cubic (fcc)
Fe0.64Ni0.36 alloy exhibits an anomalously low thermal expansion for a wide
range of temperatures, an effect that is now called the invar effect, and the
specific FeNi composition is now called invar. [1] The effect has been shown
to exist in other alloys as well, such as Fe-Pt and Fe-Pd, among others.
Besides the anomalously low thermal expansion, other properties such as the
magnetisation, elastic constants, and spontaneous volume magnetostriction show
anomalous behaviour. [2, 3] The effect is utilised in industry for precision
tools and instruments.

It is agreed that an explanation of the invar effect involves a negative
contribution to the thermal expansion from the alloys’ magnetic properties
which compensates for the positive contribution from lattice vibrations. [4]
Since the discovery of the invar effect several theories have been suggested
to explain it in detail but the topic remains controversial up to date.

Perhaps the most well-known theory to describe the effect was proposed by
Weiss, the so-called 2$\gamma$-state model. [5] In this model the Fe-atoms can
occupy two distinct states: a high spin (HS), high volume and a low spin (LS),
low volume state. The HS state has a slightly lower equilibrium energy but a
transition from the HS state to the LS state can be thermally induced. At the
transition the LS state population is gradually increased while the HS state
population decreases. The transition from the high volume state to the low
volume state compensates for the unusual thermal expansion. This theory has
been supported by several experiments and earlier ab initio calculations [6,
7, 8].

More recently van Schilfgaarde et al. [9] proposed a new model to explain the
anomaly. In their model the anomaly is caused by a continuous transition from
a HS ferromagnetic (FM) high volume state to an increasingly complex
noncollinear (NC) magnetic state at low volume. The net magnetisation of the
system decreases as individual iron spins are canted away from the global
magnetisation direction. This noncollinearity results in an anomalous binding
energy curve which in a simple model is related to the thermal expansion
coefficient via the bulk modulus and Grüneisen constant. With the new theory
came a newfound interest in the anomaly and several experiments were carried
out to investigate its validity. A number of experiments have found results
that are consistent with the noncollinear magnetism model [10, 11, 12], but
other studies have found disagreements with the theory. [13, 7]

In one experiment stimulated by Ref. [9] the authors found that the invar
effect exists also in FexNi1-x alloys with high nickel concentrations upon
applying pressure to the material, thus called the pressure induced invar
effect. [14] According to the theory proposed in Ref. [9] this behaviour could
be present in all FeNi alloys as the noncollinearity in the magnetic state is
induced by pressure. The anomalous behaviour is related to the transition from
the high volume FM to low volume NC state. By applying pressure to Ni-rich
FexNi1-x alloys the magnetic moments can be induced to form noncollinear
magnetic configurations. This pressure induced invar effect has since been
reproduced in further experiments. [10]

To the best of our knowledge the pressure induced invar effect has not been
investigated in detail using ab initio methods. We set out to investigate the
pressure induced invar effect and the relations between pressure, volume, and
magnetic configurations in disordered FexNi1-x for x = 0.25, 0.50, 0.64 using
ab initio theory. We perform calculations on both noncollinear and collinear
magnetic states for each composition and volume. We observe a magnetic
transition from a ferromagnetic state at high volumes to a complex magnetic
state at lower volumes for all compositions. At the corresponding volumes
anomalies are observed in the equation of states, although for the most Ni-
rich composition this anomaly is very weak.

##  II Method

###  II.1 Computational details

The calculations are carried out by the Vienna ab initio simulation package
(VASP), using the 5.4.4 version. [15, 16, 17] The projector augmented wave
(PAW) method is chosen. [18] To obtain the 64 atoms supercells we use the
special quasi-random structures (SQSs) method. [19] The three supercells have
compositions Fe0.25Ni0.75, Fe0.5Ni0.5, Fe0.64Ni0.36, and for the exact number
of ion-types see Table 1 which also contain information about experimental
equilibrium lattice parameter and atomic volume.

In order to find the ground state and the corresponding magnetic structure the
calculations are initialised with ideal positions of the atoms in a fcc
lattice where the ions are allowed to relax around their initial positions.
The magnetic moments for the Fe-atoms are initialised with random directions
while the Ni-atoms have spins initialised in one direction. During the
calculations both the ionic positions and magnetic moments gradually relax
toward their most preferred states. Magnetic moments are free to change
magnitude and rotate in a noncollinear fashion. To avoid that the calculations
are trapped in a local energy minimum far away from the global minimum several
calculations are carried out for each composition and volume. For each of
these the initialised magnetic structure is a unique random magnetic
configuration for Fe moments. In the following sections the lowest energy
results for each composition and volume are presented.

Both NC and collinear magnetism are investigated. In the NC calculations the
magnetic moments can evolve in all three dimensions and are thus free to
rotate during the optimisation steps. In the collinear case the moments are
restricted to a spin ’up’ or spin ’down’ state. For the collinear calculations
the magnetic structure is always initialised in a ferromagnetic state. The
magnitudes of the local moments are always free to evolve during the self-
consistent cycle.

Calculations are carried out with a 3x3x3 k-point grid generated with the
Monkhorst-Pack scheme [20]. The Methfessel-Paxton method was used with a
smearing of 0.1 eV [21]. The energy cut-off is set to 500 eV. The convergence
criteria for energy was set to 10-5 and for forces 10-4. Throughout the work
spin-orbit coupling was not taken into account.

Table 1: Experimental lattice parameters and supercell composition information. Column 1 gives the compsition. Column 2 and 3 display the experimental equilibrium lattice parameters (from Refs. [22, 14]) and corresponding atomic volume for the three compositions. Column 4 and 5 display the number of Fe and Ni-atoms in the supercells, respectively. Composition | a0 (Å) | Atomic volume (Å3) | #Fe | #Ni
---|---|---|---|---
Fe0.25Ni0.75 | 3.545 | 11.14 | 16 | 48
Fe0.5Ni0.5 | 3.578 | 11.45 | 32 | 32
Fe0.64Ni0.36 | 3.5957 | 11.62 | 41 | 23

###  II.2 Choice of exchange-correlation functional

The choice of the functional that describe exchange-correlation effects in
density functional theory (DFT) requires some discussion. One advantage of the
generalized gradient approximation (GGA) compared to the local-density
approximation (LDA) functional is that it typically more accurately reproduces
the equilibrium volume for 3d metals. At the same time the GGA overestimates
the magnetic moments of the 3d metals and as a result also the magnetic
energy. [23, 24] Of course, when considering Fe the GGA functionals succeed in
reproducing the ground state body centred cubic (bcc) crystal structure of Fe
while the LDA generally does not, as discussed in Ref. [25]. In the same paper
it is argued that the reason the stabilisation of the bcc ground state of Fe
fails in the LDA is due to a weakening of the magnetic contribution at the
equilibrium volume. Meanwhile the GGA corrects the volume and the magnetic
contribution, thus savouring the bcc Fe structure. In addition the choice of
parameterisation of the GGA functional can produce different results for fcc
Fe, as found in Ref. [26] where different GGA functionals lead to different
potential energy landscapes for different magnetic states.

Previous studies on the magnetic ground state of the invar alloy done with the
LDA and GGA functionals yield different magnetic structures in the ground
state. GGA functionals lead to a ferromagnetic ordering in the ground state
for the invar alloy. [27] In contrast, theoretical studies with the LDA
functional have resulted in NC magnetic orderings in the ground state. [9]
Thus the ground state magnetic structure depend on the choice of functional.
For this study the LDA functional was chosen as it gives better magnetic
properties at a fixed volume [25] while GGA overstabilises a ferromagnetic
state. To address the issue of both functionals finding the wrong equilibrium
volume we plot our results in relative change in pressure and volume. To look
for trends across functionals the composition x = 0.5 was additionally
investigated with the PBE and PW91 functionals. Results from calculations
performed with the PBE and PW91 GGA functionals are presented in the
Supplementary Material [28].

##  III Results

###  III.1 Evolution of magnetic structure

In this section the evolution of magnetic structure for the three studied
compositions are presented and analysed.

####  III.1.1 Fe0.64Ni0.36

Figure 1: Visualisation of magnetic structures of Fe0.64Ni0.36 at three atomic
volumes, from top to bottom: 12.46, 11.57, 11.18 Å3. Magnetic transition
begins at 11.57 Å3. Gold and silver spheres represent Fe and Ni, respectively.

A visualisation of the magnetic evolution can be seen in Fig. 1, where the
magnetic structure is shown at three volumes. The transition from a FM
ordering to an increasingly complex structure is seen as the volume is
reduced. The transition from a FM state to a noncollinear complex state is
continuous and begins at atomic volume 11.57 Å3, which is 0.5% smaller than
experimental equilibrium atomic volume 11.62 Å3. These results are in good
agreement with the work by van Schilfgaarde et al. [9] that found a similar
transition in the magnetic ordering for the invar composition. For the
calculations where the moments are restricted to collinear alignments, a
transition with spin-flips begin at 10.95 Å3. Note that exact values for the
transition could be affected by the computational details, e.g. the size of
the supercell. However, good agreement between our simulations and those
reported in Ref [9] confirm the reliability of the existence of the transition
itself.

Figure 2: Histograms of spin-pair correlation functions for Fe0.64Ni0.36
between neighbouring Fe-spins (panel a) and Ni-spins (panel b) at several
volumes (in Å3). $\hat{s}_{i}\cdot\hat{s}_{j}=cos(\theta)$. The right y-axis
gives the atomic volumes in units Å3. Inserted plot shows the average magnetic
moment size for noncollinear and collinear calculations in filled in blue
circles and red crosses, respectively. Dashed line is experimental equilibrium
volume.

In Fig. 2 the spin-pair correlation functions for Fe-Fe and Ni-Ni nearest
neighbours (NNs) are shown. For the Fe-Fe NNs, we see a transition from a
nearly ferromagnetic (FM) state at large volume to a complex state and
eventually to a nearly antiferromagnetic (AFM) state. It can be noted that the
average Fe moment size is very small at the lowest calculated volume (9.0 Å3):
in Fig. 3 individual moments have values $\sim$ 0.25 $\mu_{B}$. For the Ni-Ni
NNs spins remain in the FM state longer upon a reduction of volume, compared
to the Fe-Fe pairs, until they too undergo a transition into a complex state,
i.e. spin-flips.

When comparing the average magnetic moments for Fe and Ni in the NC and
collinear calculations they behave differently. If we start from large volumes
and follow the trend in decreasing volume the Fe-atoms show a continuous
reduction of the net moment after the magnetic transition from the FM state
begins. Once the transition starts the magnetic moment size declines nearly
linearly from 2.2 $\mu_{B}$ at 11.57 Å3, 1.5 $\mu_{B}$ at 10.72 Å3, and 0.8
$\mu_{B}$ at 9.83 Å3. The magnetic transition occurs at higher atomic volumes
in noncollinear calculations compared to collinear calculations, which can be
observed in the inserts in Fig. 2. In the collinear calculations the
transition start at a smaller volume, and once it starts the average magnetic
moment size decreases faster.

In Fig. 3 we show individual magnetic moments sizes from NC calculations,
where we observe a sharp drop in the local moment size for some atoms, while
for other atoms the local moments decrease is less sharp.

Figure 3:  Absolute values of local magnetic moments for individual atoms in
Fe0.64Ni0.36, both Fe and Ni, as a function of atomic volume Å3. Dashed and
dotted lines are experimental equilibrium volume and transition volume,
respectively.

####  III.1.2 Fe0.5Ni0.5

A visualisation of the magnetic structure and its evolution can be seen in
Fig. 4, where the magnetic structure is shown for three volumes. As the volume
is reduced, during the initial stage of the transition the magnetic moments
are tilted away from, or in some cases nearly antiparallel, to the global
magnetisation direction. Further reduction of the volume increases the
complexity in the magnetic structure.

Figure 4: Visualisation of magnetic structures of Fe0.5Ni0.5 at three atomic
volumes, from top to bottom: 11.18, 10.95, and 10.54 Å3. Magnetic transition
begins at 10.95 Å3. Gold and silver spheres represent Fe and Ni, respectively.

Similar to the invar composition, we can see that when looking at the spin-
pair correlation functions for Fe0.5Ni0.5 the magnetic state at the largest
volume is FM and the transition shows up as a complex state at lower volumes,
as is seen in Fig. 5. The spin-pair correlation functions behave much as for
the invar composition: the Fe-NNs pairs are all parallel at large volumes, but
exhibit increasing complexity as the volume reduces until it eventually ends
up in an AFM state. For the Ni-Ni spin-pair correlation functions the
behaviour is similar to that of the invar composition with deviations from
parallel pairs occurring at lower volumes than for the Fe-Fe spin-pairs.

Figure 5: Spin-pair correlation functions for Fe0.5Ni0.5 between neighbouring
Fe-spins (panel a) and Ni-spins (panel b) at several volumes (in Å3).
$\hat{s}_{i}\cdot\hat{s}_{j}=cos(\theta)$. The right y-axis gives the atomic
volumes in units Å3. Inserted plot shows the average magnetic moment size for
noncollinear and collinear calculations in filled in blue circles and red
crosses, respectively. Dashed line is experimental equilibrium volume.

Comparing the result from NC and collinear calculations we observe that the
magnetic transition from a FM to a complex state occurs at higher volumes for
the NC case, as can be seen in the average magnetisation for Fe and Ni atoms,
insets in Fig. 5. The transition occurs for the NC case at atomic volume 10.95
Å3 at which point the average magnetisation for the Fe-moments is reduced.
Thus the transition occurs at an atomic volume that is 4.4 % lower than the
experimental equilibrium volume. For the collinear case, the transition begins
at 10.36 Å3, a volume that is 9.6 % lower than the experimental equilibrium
volume. The behaviour at and after the transition differs between the two
cases. For the NC case the reduction in moment size is nearly linear, but for
the collinear case the decline in average moment size is very rapid. Thus it
is observed that the Fe magnetic moments can, on average, retain larger sizes
in a complex noncollinear state than when restricted to spin ’up’ and ’down’
states.

For individual atoms’ local magnetic moments there are some that decline
faster in size compared to the average size, as seen in Fig. 6. The transition
begins at 10.95 Å3, and here a number of Fe-atoms’ magnetic moments decline
quickly.

Figure 6:  Magnetic moment sizes for all individual atoms in Fe0.5Ni0.5, both
Fe and Ni, as a function of atomic volume Å3. Dashed and dotted lines are
experimental equilibrium volume and transition volume, respectively.

####  III.1.3 Fe0.25Ni0.75

In contrast to the two other compositions Fe0.25Ni0.75 exhibits a smoother
variation in average magnetisation and individual local moments. The results
from collinear and NC calculations are very close in average magnetisation and
local moments.

Visualisation of the magnetic evolution can be seen in Fig. 7 where the
magnetic structure is shown for three volumes. At high volumes there is a FM
ordering, but upon compression we observe the transition to NC alignments, at
least for some atoms. Further reduction in volume increases the complexity in
the magnetic structure, though the effect is less pronounced than at higher
concentrations of Fe.

Figure 7: Visualisation of the magnetic structures of Fe0.25Ni0.75 at three
atomic volumes, from top to bottom: 10.54, 9.65, 9.31 Å3. Magnetic transition
begins at 9.65 Å3. Gold and silver spheres represent Fe and Ni, respectively.

The spin-pair correlation functions for Fe NNs confirm the NC magnetic
structure upon the magnetic transition: the two largest volumes have a FM
ordering while the intermediate volume, 9.7 Å3, has magnetic moments where
some pairs are tilted away from each other but still close to the global
magnetisation direction. At the lower volumes the pairwise alignments are
nearly randomly distributed, as is seen in Fig. 8a. Meanwhile the Ni NNs pairs
show almost no change in pairwise alignment between the largest and smallest
volume, staying parallel at each volume. While the magnetic transition from a
ferromagnetic ordering to a complex ordering occurs it happens at atomic
volume 9.65 Å3, 13.3 % lower than the experimental equilibrium volume. This
can be compared to the two other compositions where corresponding numbers are
0.5 % and 4.4 % smaller than the experimental equilibrium volume for
Fe0.64Ni0.36 and Fe0.5Ni0.5, respectively. In calculations with strictly
collinear moments the transition with spin-flips begins at 9.15 Å3.

The moment sizes for individual atoms are seen in Fig. 9. Interestingly, there
are no sharp drops in the magnitudes of local moments, in contrast to the two
previous compositions. The magnetic moments slowly decline until volumes of
about 9.5 Å3 at which point some atoms decline quicker as a result of the
magnetic transition.

Figure 8: Spin-pair correlation functions for Fe0.25Ni0.75 between
neighbouring Fe-spins (panel a) and Ni-spins (panel b) at several volumes (in
Å3). $\hat{s}_{i}\cdot\hat{s}_{j}=cos(\theta)$. The right y-axis gives the
atomic volumes in units Å3. Inserted plot shows the average magnetic moment
size for noncollinear and collinear calculations in filled in blue circles and
red crosses, respectively. Dashed line is experimental equilibrium volume.
Figure 9:  Magnetic moment sizes for all individual atoms in Fe0.25Ni0.75,
both Fe and Ni, as a function of atomic volume Å3. Dashed and dotted lines are
experimental equilibrium volume and transition volume, respectively.

###  III.2 Enthalpy

The magnetic transitions above are reflected in the enthalpy curves when
comparing NC and collinear calculations. This can be seen in Fig. 10 where the
difference between the NC and collinear calculations are plotted. Note that
$\Delta H$ before the magnetic transition from a FM state to a complex state
sets in is within the error bars and most likely caused by the different
settings in the calculations. Once the transition begins there are noticeable
enthalpy differences for all compositions. The differences become smaller at
higher pressures (lower volumes) when the collinear cases exhibit spin-flips.
The difference is relatively small though, less than 1 meV per atom, and are
thus within the error bars. The difference is likely caused by the different
settings in the calculations.

It should be noted that the enthalpy difference between NC and collinear
calculations should be 0 for the FM solutions. However, we see a nearly
constant shift for all compositions and functionals leading to a negative
difference for FM solutions. The settings in the calculations are identical
apart from allowing noncollinear and collinear magnetic moments, so this seems
to be result from the technical settings in VASP.

Figure 10:  Enthalpy differences between noncollinear and collinear
calculations for all three compositions. Dashed lines represent the magnetic
transitions in NC calculations.

###  III.3 Equation of state

Next we present the equation of states for the three compositions and compare
them with our collinear calculations and the experimental results from Ref.
[14]. The results are shown in Figs. 11-13. For all compositions from the NC
calculations the anomaly appears at volumes corresponding to the onset of the
magnetic transition from a FM state to a complex state, although for
Fe0.25Ni0.75 the anomaly is extremely weak. As both NC and collinear
calculation reproduce this magnetic transition, albeit at different volumes,
the anomalies in the equation of state (EOS) appear at separate volumes. The
Birch-Murnaghan equation of state is fitted to the ferromagnetic results in
each composition (i.e. to datapoints corresponding to volumes higher than the
magnetic transition volume) and shown as a dotted line. For easier comparison
with results in previous sections, Table 2 relates the volumes and magnetic
transition volumes to experimental equilibrium volumes in the studied Fe-Ni
alloys.

Table 2:  Magnetic transition volumes from calculations and $V/V_{0}^{exp}$ for the studied compositions of fcc Fe-Ni alloys. Composition |  $V_{FM\rightarrow NC}$ (Å3) | $V_{FM\rightarrow NC}/V_{0}^{exp}$
---|---|---
Fe0.64Ni0.36 | 11.57 | 0.996
Fe0.5Ni0.5 | 10.95 | 0.956
Fe0.25Ni0.75 | 9.65 | 0.866
Figure 11:  Equation of state for Fe0.64Ni0.36 from NC calculations, collinear
calculations, and experiment from Ref. [14]. The dashed line shows the Birch-
Murnaghan equation of state fitted to the ferromagnetic datapoints. Figure 12:
Equation of state for Fe0.5Ni0.5 from NC calculations, collinear calculations,
and experiment on Fe0.55Ni0.45 from Ref. [14]. The dashed line shows the
Birch-Murnaghan equation of state fitted to the ferromagnetic datapoints.
Figure 13:  Equation of state for Fe0.25Ni0.75 from NC calculations, collinear
calculations, and experiment on Fe0.2Ni0.8 from Ref. [14]. The dashed line
shows the Birch-Murnaghan equation of state fitted to the ferromagnetic
datapoints.

##  IV Discussion

As was pointed out earlier [25, 26] and additionally underlined in Sec. II.B
the existing approximations for the exchange-correlation functionals within
DFT might not be sufficient for a quantitatively accurate and detailed
description of Fe-Ni invar alloys. Indeed, comparing results calculated for
the fcc Fe0.5Ni0.5 alloy (Sec. III) with GGA calculations (see Supplementary
Material and more detailed discussion below) one sees their significant
dependence on the chosen computational scheme. However, many essential
features of the concentration and pressure dependences of magnetic,
thermodynamic and structural properties calculated for the fcc Fe-Ni system in
this work are general, and therefore we start our discussion with the
observations that do not depend on the details of the methodology.

Most importantly, our calculations clearly demonstrate the existence of a
magnetic transition from the ferromagnetic state at large volumes to
increasingly NC configurations upon volume decrease. This effect is observed
for all the alloy compositions considered in this study, not just at invar
composition. However, in Fe0.64Ni0.36 invar the magnetic transition occurs in
an immediate vicinity of the experimental equilibrium volume (at the LDA level
of theory), while in alloys with higher Ni concentrations it is induced by
pressure, in agreement with experimental observations of the pressure induced
invar effect. [14] The transition is initiated at Fe atoms, implying that the
Fe atoms’ magnetic moments are driving the transition, and thus they are
responsible for the anomalous behaviour of properties of the alloys. Upon the
transition from the FM to the NC state both longitudinal and transverse
degrees of freedom are affected: Fe local moments shrink Figs. 3, 6, 9) and
become canted (Figs. 1, 4, 7). On the other hand, local moments at Ni preserve
their FM alignment to much lower volumes in comparison to Fe (Figs. 2, 5, 8).
This is so, because the Ni moments are known to be more itinerant, and
therefore they are oriented along the net magnetisation direction of the
system. In fact, this observation agrees with a recent experimental
observation of very high stability of the ferromagnetic state in fcc Ni, with
the Curie temperature above the room temperature at least up to 260 GPa. [29]

Comparison of the enthalpies between NC and collinear calculations (Fig. 10)
clearly favour the NC model of the pressure induced magnetic phase transition.
Interestingly, for all the studied compositions our calculations predict that
upon further increase of pressure the NC state evolves towards the LS nearly
AFM state at the lowest volumes considered in this study (see Figs. 2, 5, 8).
Though the stability of the LS AFM states with respect to more NC states at
the highest considered pressure is at the limit of the accuracy of our
calculations (see Supplementary Material Figs. 1-3 and Tables I-III), the fact
that the ”starting” and the ”end” points of the magnetic transition are the HS
FM and LS AFM states, respectively, is seen at all the compositions. Moreover,
it does not depend on the approximation for the DFT exchange-correlation
functional (see Supplementary Material Figs. 4 and 5, and Tables IV and V).
This observation provides a link between the classic Weiss 2$\gamma$-state
model of the invar effect [5] and the more recent NC picture [9], confirmed
also in this study.

It is clearly seen that the magnetic phase transitions lead to well defined
peculiarities at the EOS at all the considered compositions, Figs. 11-13, in
qualitative agreement with experiment. [14] A discussion of the quantitative
agreement is more challenging due to the limited accuracy of the existing
exchange-correlation functionals. Let us reiterate an important point from
Sec. IIB: to reduce the uncertainty due to computational details, we plot the
data in Figs. 11-13 as $V/V_{0}^{exp}$ vs $P-P(V_{0}^{exp})$, where the latter
pressure, $P(V_{0}^{exp}$), is the calculated pressure at the experimental
volume (obviously, it is 0 for the experimental data).

Starting with simulations for Fe0.64Ni0.36 invar composition in the LDA, the
NC calculations predict that the deviation from the EOS fitted to the FM data
(dotted line in Fig. 11) occurs when the magnetic transition begins at volume
11.57 Å3, corresponding to $V/V_{0}^{exp}=0.995$. A similar deviation is
observed in the collinear calculations when the magnetic moments begin to
flip, at volume 10.95 Å3 ($V/V_{0}^{exp}=0.942$). As has been pointed out
already in Ref. [14], the peculiarity at the experimental curve is hardly
visible, because the FM branch of the EOS corresponds to a negative pressure,
and therefore cannot be measured. However, we note good agreement between the
curvature of the experimental calculated EOS above the transition, and a
significant deviation of the curvature from the extrapolated EOS of the HS FM
state. In fact, it is reasonable to assume that experimentally the
Fe0.64Ni0.36 invar alloy is in the very vicinity of the magnetic phase
transition and is nearly ferromagnetic at its equilibrium volume. This is
supported by a high experimental value of its net magnetisation, which is
(almost) at the Slater-Pauling curve when measured at temperature T=77K [30],
as well as by an observation from neutron scattering experiments with
polarisation analysis that the magnetic moments must be collinear on atomic
length scales at ambient pressure [12]. With this assumption, we argue that
the position of the peculiarity at the EOS calculated in the NC model using
LDA is in good agreement with experiment.

In this respect, the simulations for the equiatomic alloy are more
informative, as both theory and experiment capture the HS FM state at low
positive pressure, and therefore see the whole transition. From the experiment
one sees that there is an anomaly at the curve between 5 and 9 GPa. Pronounced
anomalies are also observed in our ab inito results. In the NC calculations
the deviation occurs when the magnetic transitions begins at atomic volume
10.95 Å3 ($V/V_{0}^{exp}=0.956$). A similar deviation is observed in the
collinear calculations when the magnetic moments begin to flip, but at
significantly lower atomic volume 10.36 Å3 ($V/V_{0}^{exp}=0.904$ ). In fact,
the equation of state for the NC calculations locates the EOS anomaly at
relative volumes and pressure that are in very good agreement with the
experiment, though the experimental results are for a slightly lower Ni-
concentration, $Fe_{0.55}Ni_{0.45}$. While position of the anomaly and the
curvatures of the both HS FM branch above the transition and the LS AFM branch
below $P-P(V_{0}^{exp})$ 10 GPa are well reproduced, the volume collapse upon
the transition is significantly overestimated. In fact, similar effect is seen
in LDA calculations for Fe0.64Ni0.36 invar alloy following the assumption that
it is (nearly) ferromagnetic at equilibrium volume, as discussed above.

To investigate sensitivity of the volume collapse to computational details,
the evolution of magnetic structure and the EOS of the fcc Fe0.5Ni0.5 alloy
were simulated using two different forms of the GGA, the PBE and PW91. The
results are presented in Figs. 6-15 in the Supplementary Material.
Qualitatively, they are quite similar to the LDA results. Looking at the EOS
calculated using GGA, Figs. 9 and 14, we also observe the peculiarities
induced by the magnetic phase transition. While $V/V_{0}^{exp}$ at the
transition from FM to NC state is significantly lower than experimental value,
especially for the PBE functional form of the GGA, the magnitude of the volume
collapse upon the transition is very well reproduced, in contrast to the LDA
EOS. This observation strengthens our confidence in relating experimentally
observed peculiarity of the EOS to the pressure induced FM-to-NC transition of
the magnetic structure of the alloy.

Moreover, we would like to point out that the original interpretation of the
experimental results in Ref. [14] was that the anomaly at each composition
should be interpreted as a ’bump’ on the (conventional) P-V curves. Our
results reproduce anomalies in both noncollinear and collinear calculations
but offer an alternative interpretation. We believe that the EOS should be
considered as consisting of the two branches, corresponding to the HS FM state
at high volumes and LS AFM (probably followed by the non-magnetic) state at
low volumes smoothly connected to each other in the (relatively narrow)
transition region. Such an interpretation also provides a link between the
2$\gamma$-state model of Weiss [5] and the NC picture. [9]

In addition, we observe differences in the magnetic behaviour of different Fe
atoms, Figs. 3, 6, 9. These differences are due to different chemical local
environments and connects to the main focus of the theories of F. Liot that
observed Fe spin-flips in Fe-rich local environments in collinear
calculations. [27, 26] However, our noncollinear calculations show that many
of the Fe moments do participate in noncollinear rotations. Typically in Fe-
rich local environments but not only those in the most Fe-rich environments.
Further investigations, using larger supercells, are needed to fully quantify
the precise local environment where magnetic transitions away from the FM
state happens at the largest volume. Thus our theory also provides a link
between previous works focusing mainly on the global composition, [31, 32, 33,
34, 35, 36, 37] and those with full focus on the local environment effects.
[27, 26] Furthermore, it may be useful for a thermodynamic description of Fe-
Ni invar alloys, e.g. using methods of computational thermodynamics, like
CALPHAD.

Turning now to a comparison between the experimental EOS for the Ni-rich fcc
Fe0.20Ni0.80 alloy and the calculated EOS for the Fe0.25Ni0.75 alloy shown in
Fig. 13, one sees for the former the behaviour of the P-V curve is anomalous
between 9 and 14 GPa. In contrast to alloys with lower Ni concentrations, one
does not see pronounced anomalies in the ab inito results. However, according
to our calculations the magnetic transitions in NC calculations begins at
volume 9.65 Å3 ($V/V_{0}^{exp}=0.866$), and one sees that at the transition
the calculated EOS starts to deviate significantly from the curve obtained by
the extrapolation of the FM EOS (the dotted line in Fig. 13). The transition
volume is clearly underestimated for the Fe0.25Ni0.75 alloy. Note however,
that the spin flips in the collinear calculations start to occur at even lower
volume 9.15 Å3 ($V/V_{0}^{exp}=0.821$ ). On the other hand, the volume
collapse is reproduced much better in comparison to simulations for the alloys
with lower Ni concentration already at the LDA level of the DFT. One possible
reason for the underestimation of the transition volume in Ni-rich alloys is
that the transition is initiated at Fe atoms in Fe-rich chemical environments,
as discussed above. For the Ni-rich alloys they could not be well represented
in our relatively small supercell. On the contrary, they are present in real
macroscopic alloys, and act as nuclei that initiate the transition, most
probably at higher volumes.

Summarising the discussion of the EOS, we conclude that our results provide
convincing explanation of the pressure induced peculiarities observed at the
EOS of fcc Fe-Ni alloys in experiment [14] and identify it with the FM-to-NC
magnetic transitions upon the compression of the alloys. In its own turn, the
experiment unambiguously couples the EOS peculiarities to the very low thermal
expansion measured for the alloys at the pressures corresponding to the
peculiarities (see Fig. 4 of Ref. [14]). In this paper we do not aim to
calculate the thermal expansion explicitly, as this requires highly time-
consuming ab initio molecular dynamics simulations coupled to the simulations
of the spin dynamics, for instance using the methodology of [38]. Such
simulations are beyond the scope of the present study and would be reported
elsewhere. However, a qualitative picture could be captured based on simple
thermodynamic arguments put forward by Abrikosov et al. in [26]. Let us
connect the temperature dependence of volume V to the pressure dependence of
entropy S via Maxwell relations:

$\left(\frac{\partial{V}}{\partial{T}}\right)\biggr{\rvert}_{P}=-\left(\frac{\partial{S}}{\partial{P}}\right)\biggr{\rvert}_{T}$
(1)

Clearly, the increasing complexity and frustration of the alloys magnetic
structure, both in longitudinal and transverse degrees of freedom, which is in
fact maximal in the transition region (Figs. 1, 3, 4, 6, 7, 9) increases
entropy making the magnetic contribution to the term in the right-hand size of
Eq. (1) positive. The corresponding contribution to the temperature dependent
volume expansion in the left-hand side of Eq. (1) is therefore negative,
compensating for the positive contribution due to thermal lattice disorder. In
fact, recent calculations by Heine et al. [39] have demonstrated that for the
fcc Fe0.65Ni0.35 invar alloy at ambient pressure the contributions due to the
magnetic and vibrational disorder cancel each other exactly. Our study
underlines that the magnetic complexity at the HS FM-to-NC transition
increases greatly in comparison to non-invar compositions and pressures.

##  V Conclusions

We have carried out a large set of supercell calculations to investigate the
pressure induced invar effect in the fcc FexNi1-x alloys for three
compositions of Fe: 64, 50 and 25 at%. A transition from the high-spin
ferromagnetic ordering at large volumes to a complex non-collinear state at
lower volumes is detected in all three alloys considered in our study. At the
transition volumes we observe significant anomalies in the equation of state.
These results can be interpreted within the model of noncollinear magnetism
for the invar effect, in which the increasing magnetic complexity in the
region of the HS FM-to-NC transition leads to a positive sign of the pressure
dependence of the magnetic entropy, which in its own turn is coupled to
anomalously large negative magnetic contribution to the temperature dependence
of volume of invar alloys via Maxwell relation, Eq. (1). For the invar
composition the HS FM-to-NC transition occurs at ambient conditions, but for
the more Ni-rich compositions it can be induced by external pressure. Thus our
first-principles theory provides convincing qualitative explanation for the
pressure induced invar effect, discovered in Ref. [14].

## Data availability

All data can be found at
https://data.openmaterialsdb.se/pressure_induced_invar_effect.

## Acknowledgements

We gratefully acknowledge support from the Knut and Alice Wallenberg
Foundation (Wallenberg Scholar grant no. KAW-2018.0194), the Swedish
Government Strategic Research Area in Materials Science on Functional
Materials at Linköping University (Faculty Grant SFO-Mat-LiU No. 2009-00971)
and the Swedish Research Council (VR) Grant No. 2019-05600. B.A. acknowledges
financial support from the Swedish Research Council (VR) through Grant No.
2019-05403, as well as support from the Swedish Foundation for Strategic
Research (SSF) through the Future Research Leaders 6 program, FFL 15-0290. The
computations were enabled by resources provided by the Swedish National
Infrastructure for Computing (SNIC) located at National Super Computer Centre
(NSC) in Linköping, partially funded by the Swedish Research Council through
Grant Agreement No. 2018-05973.

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$\displaystyle\operatorname{\mathbb{D}}^{2}$
$\displaystyle={\sum_{i}e_{i}^{2}\otimes
X_{i}^{2}}+\sum_{i<j}\left(e_{i}e_{j}\otimes X_{i}X_{j}+e_{j}e_{i}\otimes
X_{j}X_{i}\right)$
$\displaystyle=-1\otimes\left(\Omega_{\mathfrak{g}}-\Omega_{\mathfrak{k}}\right)+\sum_{i<j}e_{i}e_{j}\otimes{[X_{i},X_{j}]},$

where $\Omega_{\mathfrak{g}}=\Omega$ is the Casimir operator in
$U\mathfrak{g}$. Since the Cartan involution is a Lie algebra automorphism,
$[X_{i},X_{j}]\in\mathfrak{k}$; furthermore, $\Omega_{\mathfrak{k}}$ acts by 0
on $G/K$. Thus
$\operatorname{\mathbb{D}}^{2}=-1\otimes\Omega_{\mathfrak{g}}=-\Delta$ on
$G/K$. ∎

###  4.2. A canonical confining potential

Next, we motivate the choice of the simplest automorphic Hamiltonian.

######  Theorem 4.2 (S.).

Let $G$ be a semi-simple Lie group with invariant Dirac operator
$\operatorname{\mathbb{D}}$. If $\beta$ acts by central scalars and
$\Delta\beta\in\operatorname{\mathbb{R}}$, then the Hamiltonian
$S=-\Delta-(\operatorname{\mathbb{D}}\beta)^{2}$ has ground state
$f=e^{-\beta}$ and bottom eigenvalue $\Delta\beta$.

###### Proof.

By design, $S$ factors into raising and lowering operators

$S=-\Delta+q=(\operatorname{\mathbb{D}}-\operatorname{\mathbb{D}}\beta)(\operatorname{\mathbb{D}}+\operatorname{\mathbb{D}}\beta)+[\operatorname{\mathbb{D}}\beta,\operatorname{\mathbb{D}}]=RL+[\operatorname{\mathbb{D}}\beta,\operatorname{\mathbb{D}}],$

and Leibniz’ rule simplifies the commutator

$[\operatorname{\mathbb{D}}\beta,\operatorname{\mathbb{D}}]f=\operatorname{\mathbb{D}}f\cdot\operatorname{\mathbb{D}}\beta-\operatorname{\mathbb{D}}f\cdot\operatorname{\mathbb{D}}\beta-f\cdot\operatorname{\mathbb{D}}^{2}\beta=f\cdot\Delta\beta.$

Since the commutator
$[\operatorname{\mathbb{D}}\beta,\operatorname{\mathbb{D}}]=\Delta\beta$ is a
real scalar, it is also the lowest possible eigenvalue of $S$ by mutual
adjointness of the raising and lowering operators; furthermore, $S$ has ground
state $f=e^{-\beta}$, since $Lf=0$. ∎

###  4.3. Kronecker’s First Limit Formula

Largely following the discussion in [Ter73], [Sie61], [DIT18], and [LM15], we
recall Kronecker’s first limit formula and several immediately relevant
generalizations. [Sie61] elucidates the significance of Kronecker limit
formulas in number theory, and [Wei76] traces their historical development.

Introduced in [Kro68], Kronecker’s first limit formula is a two-dimensional
analogue of the limit formula for Riemann’s $\zeta(s)$:

$\lim_{s\to 1}\left(\zeta(s)-\frac{1}{s-1}\right)=\gamma_{o},$

where $\gamma_{o}$ is the Euler-Mascheroni constant. For $z\in\mathfrak{H}$,
let $\displaystyle Q(u,v)={\frac{\text{Im }z}{|uz+v|^{2}}}$, and define

$\zeta_{Q}(s)=\sum_{0\neq(m,n)\in\operatorname{\mathbb{Z}}^{2}}Q(m,n)^{-s}={2}\zeta(2s)E_{s}(z),\hskip
14.22636pt\text{Re }s>1.$

Kronecker’s first limit formula is

$\lim_{s\to 1}\left(\zeta_{Q}(s)-\frac{\pi}{s-1}\right)=2\pi(\gamma_{o}-\log
2-\log(\sqrt{y}|\eta(z)|^{2})),$

where $\eta(z)$ is Dedekind’s eta function. There are many proofs of this
result, including Siegel’s proof by Poisson summation [Sie61]*pp. 4-13. In his
1903 paper, Epstein obtained an analogous formula [Eps03]*pp. 644 for his
namesake zeta function

$Z_{r}(Q,s)=\sum_{0\neq v\in\operatorname{\mathbb{Z}}^{r}}(v\hskip
2.84526ptQ\hskip 2.84526ptv^{\top})^{-s},\hskip
14.22636pt\text{Re}(s)>\frac{r}{2},$

for a positive-definite matrix $Q\in GL_{r}(\operatorname{\mathbb{R}})$ with
Iwasawa decompositions

$Q=\begin{pmatrix}1_{\ell}&*\\\ 0&1_{r-\ell}\end{pmatrix}\begin{pmatrix}A&0\\\
0&D\end{pmatrix}\cdot k$

with $A\in GL_{\ell}(\operatorname{\mathbb{R}})$ and $D\in
GL_{r-\ell}(\operatorname{\mathbb{R}})$. As proven by Epstein for $\ell=1$ and
for $1\leq\ell<r$ in [Ter73]*Theorem 4,

$\lim_{s\to
r/2}\left(Z_{r}(Q,s)-\frac{\pi^{r/2}}{\sqrt{|Q|}\Gamma\left(\frac{r}{2}\right)(s-\frac{r}{2})}\right)=2Z^{*}_{r/2}(Q),$

where the factor of $2$ on the right-hand side of the equality comes from our
normalization of $Z_{r}(Q,s)$,

$\displaystyle Z^{*}_{r/2}(Q)$
$\displaystyle=Z_{r-\ell}\left(D,\frac{r}{2}\right)+\frac{\pi^{\frac{r-\ell}{2}}\Gamma\left(\frac{\ell}{2}\right)}{\sqrt{\det
D}\cdot\Gamma\left(\frac{r}{2}\right)}\cdot Z^{*}_{\ell/2}(A)$
$\displaystyle+\frac{\pi^{\frac{r}{2}}}{\Gamma\left(\frac{r}{2}\right)}\cdot
H_{\ell,r-\ell}\left(Q,\frac{r}{2}\right)+\frac{\pi^{\frac{r}{2}}}{2\sqrt{\det
Q}\cdot\Gamma\left(\frac{r}{2}\right)}\cdot\left(\frac{\Gamma^{\prime}\left(\frac{\ell}{2}\right)}{\Gamma\left(\frac{\ell}{2}\right)}-\frac{\Gamma^{\prime}\left(\frac{r}{2}\right)}{\Gamma\left(\frac{r}{2}\right)}\right),$
$H_{\ell,r-\ell}\left(Q,\frac{r}{2}\right)=\sum_{\begin{subarray}{c}0\neq
u\in\operatorname{\mathbb{Z}}^{\ell}\\\ 0\neq
v\in\operatorname{\mathbb{Z}}^{r-\ell}\end{subarray}}\frac{\exp^{(2\pi i\cdot
vQu^{\top})}}{\sqrt{\det
D}}\left(\frac{vD^{-1}v^{\top}}{uAu^{\top}}\right)^{\frac{\ell}{4}}\\\
K_{\frac{\ell}{2}}\left(2\pi\left(uAu^{\top}vD^{-1}v^{\top}\right)^{1/2}\right),$

and $K_{\nu}$ is the modified Bessel function of the second kind defined by

$K_{\nu}(z)=\frac{1}{2}\int_{0}^{\infty}\exp\left(-z(u+1/u)/2\right)u^{\nu-1}du,\hskip
14.22636pt\text{for }|\arg z|<\pi/2.$

Terras’ proof generalizes the proof of [BG64]*Theorem 1 for the case of binary
quadratic forms.

We now discuss Kronecker limit formulas for Dedekind zeta functions of an
ideal class $\mathfrak{a}$ of a number field $K$, defined by

$\zeta_{K}(s,\mathfrak{a})=\sum_{\mathfrak{b}\sim\mathfrak{a}}(N{\mathfrak{b}})^{-s},\hskip
14.22636pt(\text{Re }s>1).$

Using the classical identity of Dirichlet/Hecke relating
$\zeta_{K}(s,\mathfrak{a})$ for $K=\operatorname{\mathbb{Q}}(\sqrt{-D})$ an
imaginary quadratic field to the value of $E_{s}(z)$ at the Heegner point
$\tau_{\mathfrak{a}}$ (see [Hec17a] or e.g. [Gar14], [Tem11]),

$E_{s}(\tau_{\mathfrak{a}})=\frac{\\#K^{\times}}{4}\left(\frac{\sqrt{|-D|}}{2}\right)^{s}\frac{\zeta_{K}(s,\mathfrak{a})}{\zeta(2s)},$

Kronecker’s first limit formula yields a limit formula for
$\zeta_{K}(s,\mathfrak{a})$ of imaginary quadratic field. In [Hec17b], Hecke
proved a similar limit formula for real quadratic fields by relating
$\zeta_{K}(s,\mathfrak{a})$ to an integral of $E_{s}(z)$ over a Heegner cycle
in $\mathfrak{H}$. Decades later, [BG84] proved the limit formula for real
cubic fields using an identity relating the integral of a minimal parabolic
Eisenstein series on $SL_{3}(\operatorname{\mathbb{R}})$ over a Heegner cycle
to the Rankin-Selberg integral of a Hilbert modular Eisenstein series, and
[Efr92] proved a similar result for non-real cubic fields by using a maximal
parabolic Eisenstein series. More recently, [LM15] generalize Efrat’s method
to identify an analogous limit formula for $\zeta_{K}(s,\mathfrak{a})$ of a
wide ideal class $\mathfrak{a}$ of a totally real number field $K$ of degree
$r\geq 2$. Their proof closely follows [Efr92], [Fri87], [Gol15], and [Ter73].

###  4.4. Sufficient growth of the confining potential

We now give a sufficient condition for a Hamiltonian $S=-\Delta+q$ to have
purely discrete spectrum on $L^{2}(\Gamma\backslash G/K)$ and characterize a
nuclear Fréchet Schwartz space. We largely follow the discussion in [Gar13].
Let $V^{n}$ be the Hilbert-space completions of automorphic test functions
$C_{c}^{\infty}(\Gamma\backslash G/K)$ under the Sobolev-like norms

$|f|^{2}_{V^{n}}=\langle S^{n}f,f\rangle_{L^{2}}.$

The confining potential $q$ has “sufficient growth” when the $V^{n}$ have
Hilbert-Schmidt transition maps. This is convenient because, as discussed in
Section 2.7, the projective limit of Hilbert spaces with Hilbert-Schmidt
transition maps is nuclear Fréchet. Nuclearity of $V^{\infty}=\lim_{n}V^{n}$
guarantees existence of a genuine tensor product and, therefore, gives
Schwartz’ kernel theorem, by the Cartan-Eilenberg adjunction. As one might
imagine, nuclearity requires more than compactness of the inclusion, and more
growth of the potential is required as
$\dim_{\operatorname{\mathbb{R}}}(\Gamma\backslash G/K)$ increases.

[Gar13] suffices for the simplest case:

######  Theorem 4.3.

Let $\Gamma\backslash G/K=SL_{2}(\operatorname{\mathbb{Z}})\backslash
SL_{2}(\operatorname{\mathbb{R}})/SO(2,\operatorname{\mathbb{R}})$, and let
$V^{1}$ be the Hilbert-space completion of $C^{\infty}_{c}(\Gamma\backslash
G/K)$ under the Sobolev-like norm

$|f|^{2}_{V^{1}}=\langle Sf,f\rangle_{L^{2}}.$

The inclusion $V^{1}\to L^{2}(\Gamma\backslash G/K)$ is compact for any
$q(x+iy)\gg y^{\varepsilon}$ for $\varepsilon>0$.

###### Proof.

The proof is an easier variant of the compactness argument in [LP76]*pp.
204-206. By the total boundedness criterion for relative compactness, it
suffices to show that the image of the unit $V^{1}$-ball $B$ is totally
bounded in $L^{2}(\Gamma\backslash G/K)$. That is, we want to show that given
$\varepsilon>0$, the image of $B$ in $L^{2}(\Gamma\backslash G/K)$ can be
covered by finitely-many balls of radius $\varepsilon$. The usual Rellich-
Kondrachev lemma on compact Riemannian manifolds asserts compactness of the
standard Sobolev spaces $H^{k+1}(\operatorname{\mathbb{T}}^{n})\to
H^{k}(\operatorname{\mathbb{T}}^{n})$, so the issue reduces to an estimate on
the tail of $\Gamma\backslash G/K$:

Given $c\geq 1$, let $Y_{o}$ be the image of $\frac{\sqrt{3}}{2}\leq y\leq
c+1$ in $\Gamma\backslash G/K$ and $Y_{\infty}$ the image of $y\geq c$. We may
cover $Y_{\infty}$ by one large open set $U_{\infty}$ and $Y_{o}$ by small
coordinate patches $U_{i}$. By compactness of $Y_{o}$, there exists a finite
sub-cover of $Y_{o}$. Choose a smooth partition of unity $\\{\varphi_{i}\\}$
subordinate to the finite sub-cover and $U_{\infty}$ , and let
$\varphi_{\infty}$ be a smooth function identically 1 on the image of $y\geq
c+1$. On $Y_{o}$, we may write $f\in V^{1}$ as a finite sum of functions
$\varphi_{i}\cdot f$ with compact support on the $U_{i}$ identified with
$\operatorname{\mathbb{T}}^{2}$, and lying in
$H^{1}(\operatorname{\mathbb{T}}^{2})$ since

$|\varphi_{i}f|^{2}_{V^{1}}=\langle(-\Delta+q)\varphi_{i}f,\varphi_{i}f\rangle\ll_{i}\langle(-\Delta^{E}+1)\varphi_{i}f,\varphi_{i}f\rangle=|\varphi_{i}f|^{2}_{H^{1}(\operatorname{\mathbb{T}}^{2})},$

for $\Delta^{E}$ the Euclidean Laplacian on $\operatorname{\mathbb{T}}^{2}$.
By the usual Rellich lemma, each copy of the inclusion map
$H^{1}(\operatorname{\mathbb{T}}^{2})\to L^{2}(\operatorname{\mathbb{T}}^{2})$
is compact, so $\varphi_{i}\cdot B$ is totally bounded in
$L^{2}(\Gamma\backslash G/K)$.

It remains to prove that given $\delta>0$, the cut-off $c$ can be made
sufficiently large such that the image of the smoothed tail
$\varphi_{\infty}\cdot B$ lies in a single ball of radius $\delta$. Since
$0\leq\varphi_{\infty}\leq 1$, it suffices to show

$\lim_{c\to\infty}\int_{y>c}|f(z)|^{2}\frac{dxdy}{y^{2}}\to 0\hskip
28.45274pt(\text{uniformly for }|f|_{V^{1}}\leq 1).$

Since $q\gg y^{\varepsilon}$ in $Y_{\infty}$,

$\displaystyle\int_{y>c}|f(z)|^{2}\frac{dxdy}{y^{2}}\leq
c^{-\varepsilon}\int_{y>c}|f(z)|^{2}y^{\varepsilon}\frac{dxdy}{y^{2}}\leq
c^{-\varepsilon}|f|^{2}_{V^{1}}\leq c^{-\varepsilon}\to 0$

as $c\to\infty$. Thus, the image of $B$ in $L^{2}(\Gamma\backslash G/K)$ is
totally bounded, and by the total boundedness criterion for relative
compactness, the inclusion $V^{1}\to L^{2}(\Gamma\backslash G/K)$ is compact.
∎

Next, we identify a growth condition such that the transition maps $V^{n}\to
V^{n-2}$ are Hilbert-Schmidt. Certainly, the transition maps are Hilbert-
Schmidt if $-\Delta+q$ has Hilbert-Schmidt resolvent: for $\\{u_{i}\\}$ an
orthonormal basis of eigenfunctions for $L^{2}(\Gamma\backslash G/K)$, with
positive eigenvalues $\lambda_{i}$, the vectors $u_{i}/\lambda_{i}^{n/2}$ form
an orthonormal basis for $V^{n}$. With respect to these bases, the inclusions
$V^{n}\to V^{n-2}$ are multiplication maps

$\sum_{i}c_{i}\frac{u_{i}}{\lambda_{i}^{n/2}}\mapsto\sum_{i}c_{i}\cdot\lambda_{i}^{-1}\frac{u_{i}}{\lambda_{i}^{(n-2)/2}},$

and such a map is Hilbert-Schmidt if and only if
$\sum_{i}(\lambda_{i}^{-1})^{2}<\infty$. The resolvent of the Friedrichs
extension of $-\Delta+q$ has eigenvalues $\lambda_{i}^{-1}$, and the Hilbert-
Schmidt property is the same equality.

As in [Gar13], we can identify a growth condition for Hilbert-Schmidt
resolvent in the simplest case:

######  Theorem 4.4.

Let $\Gamma\backslash G/K=SL_{2}(\operatorname{\mathbb{Z}})\backslash
SL_{2}(\operatorname{\mathbb{R}})/SO(2,\operatorname{\mathbb{R}})$, and let
$V^{n}$ be the Hilbert-space completions of $C^{\infty}_{c}(\Gamma\backslash
G/K)$ under the Sobolev-like norm

$|f|^{2}_{V^{n}}=\langle S^{n}f,f\rangle_{L^{2}}.$

The Hamiltonian $-\Delta+q$ has Hilbert-Schmidt resolvent when $q(x+iy)\gg
y^{2}$.

###### Proof.

As in the proof of compactness of the inclusion $V^{1}\to
L^{2}(\Gamma\backslash G/K)$, the fact that
$H^{{k+2}}(\operatorname{\mathbb{T}}^{2})\to
H^{{k}}(\operatorname{\mathbb{T}}^{2})$ is Hilbert-Schmidt reduces the
discussion to the non-compact but geometrically simpler part of
$\Gamma\backslash G/K$.

Specifically, it suffices to consider the restriction $T$ of $S=-\Delta+y^{2}$
to test functions on the tapering cylinder
$X=\operatorname{\mathbb{T}}^{1}\times[1,\infty)$ with measure
$\frac{dxdy}{y^{2}}$. Thus, the domain of $T$ includes test functions on $X$
vanishing to infinite order on the boundary $\partial
X=\operatorname{\mathbb{T}}^{1}\times\\{1\\}$. Let $\tilde{T}$ be the
Friedrichs self-adjoint extension of $T$.

The circle group $\operatorname{\mathbb{T}}^{1}$ acts on $X$ and commutes with
$T$ and $\tilde{T}$. Thus, $L^{2}(X)$ decomposes orthogonally into components
indexed by characters $\psi_{n}(x)=e^{inx}$ of
$\operatorname{\mathbb{T}}^{1}$. The differential equation for the $n^{th}$
component of a fundamental solution $u_{a}=u_{a}(y)$ at $y=a$ is given by

$\delta_{a}=(-\Delta+y^{2})(e^{inx}u_{a}(y))=y^{2}\left(n^{2}-\frac{\partial^{2}}{\partial
y^{2}}+{1}\right)u_{a}(y)=y^{2}\left(-u_{a}^{\prime\prime}+(n^{2}+{1})u_{a}\right)$

and simplifies to a (nonhomogeneous) constant-coefficient equation

$\frac{\delta_{a}}{{a}^{2}}=-u_{a}^{\prime\prime}+(n^{2}+{1})u_{a},\hskip
14.22636pt(\text{with }a>1).$

As usual, we attempt to piece together $u_{a}$ from solutions $e^{\pm cy}$ to
the associated homogeneous equation $-u_{a}^{\prime\prime}+(n^{2}+{1})u=0$,
where $c=\sqrt{n^{2}+{1}}\geq 1$. Since $u_{a}(y)$ must have moderate growth
enough as $y\to+\infty$ such that it is in $L^{2}(X)$ with measure
$\frac{dxdy}{y^{2}}$, go to zero as $y\to 1^{+}$, be continuous, and be non-
smooth enough at $y=a$ to produce the required multiple of $\delta_{a}$,
$u_{a}$ must be of the form

$u_{a}(y)=\begin{cases}A_{a}e^{cy}+B_{a}e^{-cy}\hskip 14.22636pt(\text{for
}1<y<a)\\\ C_{a}e^{-cy}\hskip 56.9055pt(\text{for }a<y)\end{cases}$

for some constants $A_{a},B_{a},C_{a}$ satisfying the following conditions:

$\begin{cases}A_{a}e^{c}+B_{a}e^{-c}=0\hskip 136.57323pt(\text{vanishing at
}y\to 1^{+})\\\ A_{a}e^{ca}+B_{a}e^{-ca}=C_{a}e^{-ca}\hskip
99.58464pt(\text{continuity at }y=a)\\\
-cC_{a}e^{-ca}-(cA_{a}e^{ca}-cB_{a}e^{-ca})=\frac{1}{a^{2}}\hskip
39.83385pt(\text{change of slope by }\frac{1}{a^{2}}\text{ at
}y=a)\end{cases}$

From the first equation, $B_{a}=-e^{2c}\cdot A_{a}$ such that the system
becomes

$\begin{cases}A_{a}(e^{ca}-e^{2c}e^{-ca})=C_{a}e^{-ca}\hskip
88.20354pt(\text{continuity at }y=a)\\\
-cC_{a}e^{-ca}-cA_{a}(e^{ca}+e^{2c}e^{-ca})=\frac{1}{a^{2}}\hskip
42.67912pt(\text{change of slope by }\frac{1}{a^{2}}\text{ at
}y=a).\end{cases}$

Substituting $C_{a}=A_{a}\cdot(e^{2ca}-e^{2c})$ from the first equation into
the second yields the condition

$\displaystyle\frac{1}{a^{2}}$
$\displaystyle=A_{a}\cdot\left(-c(e^{2ca}-e^{2c})e^{-ca}-c(e^{ca}+e^{2c}e^{-ca}\right)=A_{a}\cdot(-2ce^{ca}).$

Rearranging, we have $A_{a}=-e^{-ca}/2ca^{2}$ such that

$\displaystyle B_{a}$ $\displaystyle=-e^{2c}\cdot A_{a}=e^{2c-ca}/2ca^{2}$
$\displaystyle C_{a}$
$\displaystyle=A_{a}\cdot(e^{2ca}-e^{2c})=(e^{2c}e^{-ca}-e^{ca})/2ca^{2}.$

Thus,

$u_{a}(y)=\begin{dcases}\frac{e^{2c}e^{-cy}-e^{cy}}{2ca^{2}}\cdot
e^{-ca}\hskip 14.22636pt(\text{for }1<y<a)\\\
\frac{e^{2c}e^{-ca}-e^{ca}}{2ca^{2}}\cdot e^{-cy}\hskip 14.22636pt(\text{for
}a<y).\end{dcases}$

Since $(c^{2}-\frac{\partial^{2}}{\partial y^{2}})u=f$ is solved by

$u(y)=\int_{1}^{\infty}a^{2}\cdot u_{a}(y)f(a)da,$

the symmetry of $-\Delta+q$ with respect to the measure $dy/y^{2}$ implies
that $a^{2}\cdot u_{a}(y)$ is symmetric in $y$ and $a$; furthermore, the
integral kernel for the resolvent is

$a^{2}\cdot u_{a}(y)=\begin{dcases}\frac{e^{2c}e^{-cy}-e^{cy}}{2c}\cdot
e^{-ca}\hskip 14.22636pt(\text{for }1<y<a)\\\
\frac{e^{2c}e^{-ca}-e^{ca}}{2c}\cdot e^{-cy}\hskip 14.22636pt(\text{for
}a<y).\end{dcases}$

The resolvent being Hilbert-Schmidt is equivalent to

$\int_{1}^{\infty}\int_{1}^{\infty}|a^{2}\cdot u_{a}(y)|^{2}\hskip
2.84526pt\frac{da}{a^{2}}\hskip 2.84526pt\frac{dy}{y^{2}}<\infty.$

By symmetry, it suffices to integrate over $1<y<a<\infty$, and

$\displaystyle\iint_{1<y<a}|a^{2}\cdot u_{a}(y)|^{2}\hskip
2.84526pt\frac{da}{a^{2}}\hskip
2.84526pt\frac{dy}{y^{2}}=\iint_{1<y<a}\frac{|e^{4c}e^{-2cy}-2e^{2c}+e^{2cy}|\cdot
e^{-2ca}}{4c^{2}}\hskip 2.84526pt\frac{da}{a^{2}}\hskip
2.84526pt\frac{dy}{y^{2}}$
$\displaystyle\ll\frac{1}{n^{2}+1}\iint_{1<y<a}(e^{4c}e^{-2cy}+2e^{2c}+e^{2cy})\cdot
e^{-2ca}\hskip 2.84526pt\frac{da}{a^{2}}\hskip 2.84526pt\frac{dy}{y^{2}}\ $

Replacing $y,a$ by $y+1,a+1$, the integral becomes

$\displaystyle\iint_{0<y<a}(e^{-2cy}+2+e^{2cy})$ $\displaystyle\cdot
e^{-2ca}\hskip 2.84526pt\frac{da}{(a+1)^{2}}\hskip
2.84526pt\frac{dy}{(y+1)^{2}}\ll\iint_{0<y<a}\frac{da}{(a+1)^{2}}\hskip
2.84526pt\frac{dy}{(y+1)^{2}}$
$\displaystyle\ll\int_{0}^{\infty}\frac{da}{(a+1)^{2}}\cdot\int_{0}^{\infty}\frac{dy}{(y+1)^{2}}<\infty$

Since the $L^{2}$-norm of the $n^{th}$ component of the integral kernel is
uniformly bounded by a constant multiple of $1/(n^{2}+1)$, the sum over
$n\in\operatorname{\mathbb{Z}}$ is finite. That is, the resolvent is Hilbert-
Schmidt. ∎

One should identify an analogous growth condition for
$G=SL_{r}(\operatorname{\mathbb{R}})$.

###  4.5. The simplest automorphic Hamiltonian

The previous discussion leads to the following results.

######  Theorem 4.5 (S.).

Let $\Gamma\backslash G/K=SL_{2}(\operatorname{\mathbb{Z}})\backslash
SL_{2}(\operatorname{\mathbb{R}})/SO(2,\operatorname{\mathbb{R}})$. For
$\operatorname{\mathbb{D}}$ the invariant Dirac operator and $E_{1}^{*}$ the
zero-order term of the Laurent expansion of the simplest (degenerate)
Eisenstein series $E_{s}$ at $s=1$ with eigenvalue $\lambda_{s}=2s(s-1)$, the
Friedrichs extension of the Hamiltonian
$S=-\Delta-(\operatorname{\mathbb{D}}E_{1}^{*})^{2}$ has purely discrete
spectrum on $L^{2}(\Gamma\backslash G/K)$, ground state $f=e^{-E_{1}^{*}}$,
and lowest eigenvalue $\lambda_{o}={6/\pi}$. Furthermore, $S$ characterizes a
nuclear Fréchet Schwartz space $V^{\infty}=\lim_{n}V^{n}$, for $V^{n}$ the
completion of $L^{2}(\Gamma\backslash G/K)$ with respect to the Sobolev norm

$|f|_{V^{n}}=\langle S^{n}f,f\rangle_{L^{2}}.$

###### Proof.

For $x+iy\in\mathfrak{H}$ and $j,k$ the usual Hamiltonian quaternions,
$\operatorname{\mathbb{D}}=y(j\partial_{x}+k\partial_{y})$ with

$\operatorname{\mathbb{D}}^{2}=y^{2}\left(j^{2}\partial_{x}^{2}+(jk+kj)\partial_{x}\partial_{y}+k^{2}\partial_{y}^{2}\right)=-y^{2}\left(\partial_{x}^{2}+\partial_{y}^{2}\right)=-\Delta.$

The Laurent expansion of $E_{s}$ about $s=1$ is

$E_{s}(z)=\frac{3/\pi}{(s-1)}+E_{1}^{*}(z)+O(s-1),$

with $\displaystyle E_{1}^{*}(z)=\frac{6}{\pi}\left(\gamma-\log
2-\log\left(y^{1/2}|\eta(z)|^{2}\right)\right)$ given by Kronecker’s first
limit formula.

In what follows, we view $s\to E_{s}$ as a meromorphic function-valued
function of $s$. By the (vector-valued) Laurent expansion of $s\to E_{s}$,

$\displaystyle 0$
$\displaystyle=\frac{d}{ds}[(\Delta-\lambda_{s})(s-1)E_{s}]=(\Delta-\lambda_{s})\frac{d}{ds}[(s-1)E_{s}]-2(2s-1)(s-1)E_{s}$
$\displaystyle=(\Delta-\lambda_{s})\frac{d}{ds}[(s-1)E_{s}]-2(2s-1)[3/\pi+(s-1)E_{1}^{*}+O(s-1)].$

Evaluating the expression at $s=1$ yields $\Delta
E_{1}^{*}={6}/\pi\in\operatorname{\mathbb{R}}$.

We claim the confining potential $q=-(\operatorname{\mathbb{D}}E_{1}^{*})^{2}$
has sufficient growth (see Section 4.4). By Theorem 4.4, we must show $q\gg
y^{2}$ as $y\to+\infty$. Let $\mathfrak{g}$ be the Lie algebra of $G$ and
$\mathfrak{z}$ the center of the universal enveloping algebra $U\mathfrak{g}$,
generated by Casimir. Since $\Delta E_{1}^{*}\in\operatorname{\mathbb{R}}$ is
an eigenfunction of the Laplacian, $E_{1}^{*}$ and its derivatives are
$\mathfrak{z}$-finite of moderate growth. By the theory of the constant term
[Gar18]*Theorem 8.1.1, it suffices to show
$-c_{P}(\operatorname{\mathbb{D}}E_{1}^{*})^{2}\gg y^{2}$, where

$\text{constant term of }f=c_{P}f=\int_{(N\cap\Gamma)\backslash N}f(nx)dn.$

By left $G$-invariance and Gelfand-Pettis corollaries (see [Gro53] and
[Gar18]), $\operatorname{\mathbb{D}}$ commutes with evaluation of the constant
term $c_{P}$ such that

$c_{P}(\operatorname{\mathbb{D}}E_{s})=\operatorname{\mathbb{D}}(c_{P}E_{s})=\operatorname{\mathbb{D}}\left(y^{s}+\frac{\xi(2s-1)}{\xi(2s)}\hskip
2.84526pty^{1-s}\right)=k\left(sy^{s}+\frac{\xi(2s-1)(1-s)}{\xi(2s)}\hskip
2.84526pty^{1-s}\right).$

Evaluating the Laurent expansion of $E_{s}$ at $s=1$ for large $y$,

$c_{P}(\operatorname{\mathbb{D}}E_{1}^{*})=k\left(y-\frac{\text{Res}_{s=1}\xi(s)}{\xi(2)}\right)=k\left(y-\frac{3}{2\pi}\right)\gg
ky,$

so

$-c_{P}(\operatorname{\mathbb{D}}E_{1}^{*})^{2}\gg-(ky)^{2}=y^{2}.$

Since $q=-(\operatorname{\mathbb{D}}E_{1}^{*})^{2}$ has sufficient growth, the
Friedrichs extension of $S=-\Delta+q$ has purely discrete spectrum on
$L^{2}(\operatorname{\Gamma\backslash G/K})$, and for $V^{n}$ the completion
of automorphic test functions with respect to the Sobolev norm

$|f|_{V^{n}}=\langle S^{n}f,f\rangle_{L^{2}},$

the projective limit $V^{\infty}=\lim_{n}V^{n}$ is nuclear Fréchet. $S$ also
factors into mutually adjoint raising and lowering operators,
$R=(\operatorname{\mathbb{D}}-\operatorname{\mathbb{D}}E_{1}^{*})$ and
$L=(\operatorname{\mathbb{D}}+\operatorname{\mathbb{D}}E_{1}^{*})$: for $f$ in
$L^{2}(\operatorname{\Gamma\backslash G/K})$,

$Sf=(\operatorname{\mathbb{D}}-\operatorname{\mathbb{D}}E_{1}^{*})(\operatorname{\mathbb{D}}+\operatorname{\mathbb{D}}E_{1}^{*})f+[\operatorname{\mathbb{D}}E_{1}^{*},\operatorname{\mathbb{D}}]f=RLf+[\operatorname{\mathbb{D}}E_{1}^{*},\operatorname{\mathbb{D}}]f.$

By Leibniz’ rule, the commutator is $\Delta E_{1}^{*}=6/\pi$, and since
$f=e^{-E_{1}^{*}}\in\ker L$,

$Sf=RLf+[\operatorname{\mathbb{D}}E_{1}^{*},\operatorname{\mathbb{D}}]f=RLf+\Delta
E_{1}^{*}\hskip 2.84526ptf=\Delta E_{1}^{*}f.$

Thus, $f=e^{-E_{1}^{*}}$ is the ground state of $\tilde{S}$ with eigenvalue
$\Delta E_{1}^{*}=6/\pi$. ∎

######  Theorem 4.6 (S.).

Let $\Gamma\backslash G/K=SL_{r}(\operatorname{\mathbb{Z}})\backslash
SL_{r}(\operatorname{\mathbb{R}})/SO(r,\operatorname{\mathbb{R}})$ and
$P=P^{r-1,1}$ the standard maximal proper parabolic subgroup of $G$. For
$\operatorname{\mathbb{D}}$ the invariant Dirac operator, $\varphi^{P}_{s}$
the spherical vector in the principal series representation of $G$,
$E^{P}_{s,\varphi}$ the corresponding degenerate Eisenstein series with
eigenvalue $\lambda_{s}=r(r-1)s(s-1)$, and $E_{1}^{*}$ the zero-order term of
the Laurent expansion of $E^{P}_{s,\varphi}$ at $s=1$, the Hamiltonian
$S=-\Delta-(\operatorname{\mathbb{D}}E_{1}^{*})^{2}$ has ground state
$f=e^{-E_{1}^{*}}$ and lowest eigenvalue

$\displaystyle\lambda_{o}=\frac{r(r-1)\pi^{r/2}}{2\Gamma(r/2)\zeta(r)}.$

Granting sufficient growth of the confining potential, the Friedrichs
extension $\tilde{S}$ has purely discrete spectrum on $L^{2}(\Gamma\backslash
G/K)$ and characterizes a nuclear Fréchet Schwartz space
$V^{\infty}=\lim_{n}V^{n}$, for $V^{n}$ the completion of
$L^{2}(\Gamma\backslash G/K)$ with respect to the Sobolev norm

$|f|_{V^{n}}=\langle S^{n}f,f\rangle_{L^{2}}.$

###### Proof.

By design, the proof is similar to the $SL_{2}(\operatorname{\mathbb{R}})$
case. The Laurent expansion of $E^{P}_{s}(g)$ about $s=1$ is

$E^{P}_{s,\varphi}(g)=\frac{R}{(s-1)}+E_{1}^{*}(g)+O(s-1),$

with $\displaystyle R=\frac{\pi^{r/2}}{2\Gamma(r/2)\zeta(r)}$ the residue of
$E_{s}^{P}(g)$ at $s=1$ and $E_{1}^{*}(g)$ given by a generalization of
Kronecker’s first limit formula.

In what follows, we view $s\to E^{P}_{s,\varphi}$ as a meromorphic function-
valued function of $s$. By the (vector-valued) Laurent expansion of $s\to
E^{P}_{s,\varphi}$,

$\displaystyle 0$
$\displaystyle=\frac{d}{ds}[(\Delta-\lambda_{s})\left(s-1\right)E^{P}_{s,\varphi}]=(\Delta-\lambda_{s})\frac{d}{ds}[\left(s-1\right)E^{P}_{s,\varphi}]-r(r-1)(2s-1)(s-1)E^{P}_{s,\varphi}$
$\displaystyle=(\Delta-\lambda_{s})\frac{d}{ds}[(s-1)E^{P}_{s,\varphi}]-r(r-1)(2s-1)[R+(s-1)E_{1}^{*}+O(s-1)].$

Evaluating the expression at $s=1$ yields $\Delta
E_{1}^{*}=r(r-1)R=\lambda_{o}$.

Granting that $q=-(\operatorname{\mathbb{D}}E_{1}^{*})^{2}$ has sufficient
growth, the Friedrichs extension of $S=-\Delta+q$ has purely discrete spectrum
on $L^{2}(\operatorname{\Gamma\backslash G/K})$ and for $V^{n}$ the completion
of automorphic test functions with respect to the Sobolev norm

$|f|_{V^{n}}=\langle S^{n}f,f\rangle_{L^{2}},$

the projective limit $V^{\infty}=\lim_{n}V^{n}$ is nuclear Fréchet.

$S$ also factors into mutually adjoint raising and lowering operators,
$R=(\operatorname{\mathbb{D}}-\operatorname{\mathbb{D}}E_{1}^{*})$ and
$L=(\operatorname{\mathbb{D}}+\operatorname{\mathbb{D}}E_{1}^{*})$: for $f$ in
$L^{2}(\operatorname{\Gamma\backslash G/K})$,

$Sf=(\operatorname{\mathbb{D}}-\operatorname{\mathbb{D}}E_{1}^{*})(\operatorname{\mathbb{D}}+\operatorname{\mathbb{D}}E_{1}^{*})f+[\operatorname{\mathbb{D}}E_{1}^{*},\operatorname{\mathbb{D}}]f=RLf+[\operatorname{\mathbb{D}}E_{1}^{*},\operatorname{\mathbb{D}}]f.$

By Leibniz’ rule, the commutator is $\Delta E_{1}^{*}=\lambda_{o}$. Since
$f=e^{-E_{1}^{*}}\in\ker L$,

$Sf=RLf+[\operatorname{\mathbb{D}}E_{1}^{*},\operatorname{\mathbb{D}}]f=RLf+\Delta
E_{1}^{*}f=\Delta E_{1}^{*}f.$

That is, $f=e^{-E_{1}^{*}}$ is the ground state of $\tilde{S}$ with bottom
eigenvalue
$\displaystyle\lambda_{o}=\frac{r(r-1)\pi^{r/2}}{2\Gamma(r/2)\zeta(r)}$. ∎

## References