# Intensity-resolved measurement of above-threshold ionization of Ar-H2O

Adrian Platz  Institute for Optics and Quantum Electronics, Universität Jena,
D-07743 Jena, Germany  Sebastian Hell  Institute for Optics and Quantum
Electronics, Universität Jena, D-07743 Jena, Germany  Yinyu Zhang  Institute
for Optics and Quantum Electronics, Universität Jena, D-07743 Jena, Germany
Bo Ying  Institute for Optics and Quantum Electronics, Universität Jena,
D-07743 Jena, Germany  Helmholtz Institute Jena, D-07743 Jena, Germany
Gerhard G. Paulus  Institute for Optics and Quantum Electronics, Universität
Jena, D-07743 Jena, Germany  Helmholtz Institute Jena, D-07743 Jena, Germany
Matthias Kübel  Institute for Optics and Quantum Electronics, Universität
Jena, D-07743 Jena, Germany  Helmholtz Institute Jena, D-07743 Jena, Germany
<EMAIL_ADDRESS>
###### Abstract

Above-treshold ionization (ATI) by femtosecond laser pulses centered at 515 nm
is studied for a gas mixture containing the Van-der-Waals complex Ar-H2O. By
detecting photoions and -electrons in coincidence, the ATI spectra for Ar,
Ar2, H2O, and Ar-H2O are discerned and measured simultaneously. Using an
intensity-scanning technique, we observe the red-shift of the ATI spectra as a
function of the laser intensity. The intensity-dependent shift of the ATI peak
positions observed for Ar-H2O and H2O match but significantly differ from the
ones measured for Ar and Ar2. This indicates that the photoelectron is emitted
from the H2O site of the complex and the vertical ionization potential of
Ar-H2O is determined as $(12.4\pm 0.1)$ eV. For resacttered electrons,
however, an enhancement of high-order ATI is observed for Ar-H2O, as compared
to H2O, suggesting that the relatively large Ar atom acts as a scattering
center, which influences the ionization dynamics.

††preprint: APS/123-QED

##  I Introduction

Rare-gas compounds exhibit rich light-matter interactions, and serve as model
systems for electron dynamics in weakly bound molecules. A prominent example
is interatomic Coulombic decay, which has been first observed in Ne2 and
studied using extreme ultraviolet radiation [1, 2]. Also in intense optical
fields, rare gas dimers have attracted significant interest [3, 4, 5] but only
few studies have explored light-matter interaction in more complex rare-gas
compounds [6]. A notable example for such complex target is Ar-H2O, which has
been extensively studied in the past using infrared vibrational spectroscopy
[7, 8, 9, 10, 11]. One focus of attention has been the location of the energy
minimum in the Ar-H2O intermolecular potential energy surface. After initially
contradicting results, studies have converged to the finding that the argon
atom resides in the plane of the water molecule, at an angle close to one of
the hydrogen atoms but at roughly four times the distance from the oxygen atom
[7, 9, 12, 13]. Despite this progress, ionization experiments of Ar-H2O have
been scarce. Conseqeuently, little is known about the structure of the Ar-H2O
cation.

Structural and dynamical information from atoms and molecules has been
successfully infered from above-threshold ionization (ATI) spectra, for
example using laser-induced electron diffraction [14, 15, 16], or
photoelectron momentum imaging [17, 4, 18]. For recent reviews see [19, 20].
Moreover, spectroscopy of electronic states in polyatomic molecules has been
demonstrated in channel-resolved ATI experiments [21, 22]. However, accurately
measuring the ionization potential using ATI is challenging since the ATI peak
positions sensitively depend on the laser intensity. The position of the S-th
ATI peak can be described by (atomic units are used if not otherwise stated)

$E_{S}=(N+S)\omega-E_{\mathrm{IP}}-\Delta E(I),$ (1)

where $\omega$ is the photon energy, $E_{\mathrm{IP}}$ is the ionization
potential of the target atom or molecule, $N=\lceil
E_{\mathrm{IP}}/\omega\rceil$ is the minimum number of photons required to
overcome $E_{\mathrm{IP}}$, and $\Delta E(I)$ is an intensity-dependent peak
shift. The latter is often approximated by the ponderomotive potential
$U_{P}=\frac{I}{4\omega^{2}}$, representing the AC Stark shift of the
ionization continuum. In addition, the ground state exhibits an intensity-
dependent level shift, as well. In the absence of resonances and for not too
intense fields, the AC Stark shift can be reasonably well approximated by the
DC Stark shift, calculated by second-order perturbation theory [23]. It is
given in terms of the peak intensity $I=\mathcal{E}_{0}^{2}$, where
$\mathcal{E}_{0}$ is the peak value of the electric field, and the ground
state polarizability $\alpha$ of the respective atom or molecule, by

$\delta E=-\alpha I/4,$ (2)

and is usually negative. Hence, the ground state stark shift usually increases
the peak shift $\Delta E(I)$. Note that this does not hold in the presence of
resonances. For example, Nicklich, _et al._ observed an intensity-dependent
blue-shift in the multiphoton ionization of Cs by visible wavelengths [24].
The intensity-dependent peak shift of ATI has been observed in various
experiments, which aimed at reducing the focal volume averaging effect [25,
26, 27], or identifying the role of resonances [27, 28]. In many of these
experiments, the Stark shift of the ground state has been neglected.

Here, we present intensity-resolved measurements of ATI of Ar, Ar2, H2O, and
Ar-H2O by 515 nm femtosecond laser pulses. The data allows us to extract the
unkown ionization potential of Ar-H2O and infer that the electron is emitted
from the H2O site of the compound. In contrast to this, we show that elastic
electron rescattering also occurs on the argon atom, whose presence
significantly enhances the elastic scattering cross-section of Ar-H2O with
respect to H2O.

##  II Experimental approach

In our experiment, the gas sample containing Ar-H2O is obtained by bubbling
argon at a pressure of 3 bar through water and co-expanding the mixture
through a 30-$\mu$m-nozzle into a high-vacuum vessel. The nozzle was heated to
50°C to avoid clogging by condensed water. The gas jet subsequently passes a
150-$\mu$m-skimmer and two 1-mm apertures for differential pumping before it
enters a Cold Target Recoil Ion Momentum Specrometer (COLTRIMS), where it
intersects the laser focus at a total distance of
$\approx$1.5\text{\,}\mathrm{m}$$ behind the nozzle.

Figure 1: (a) Schematic of the experiment for intensity-resolved coincidence
measurements of above-threshold ionization. The power of the femtosecond laser
pulses at 1030 nm with are controlled using a motorized half-wave plate
followed by a polarizer. The laser frequency is doubled in a BBO, and the two
colours are separated. The laser power at 515 nm is measured by a photodiode
and a boxcar integrator. In the data acquisition computer, the measured laser
power is correlated with the ion and electron momenta detected in the
COLTRIMS. (b) Recorded photodiode signal as a function of the laser power, as
measured with a power meter before (calibration 1) and after (calibration 2)
conducting the main experiment. (c) Ion time-of-flight spectra recorded in the
COLTRIMS, with various ionic species marked.

Figure 1(a) shows a schematic of the experiment. Femtosecond laser pulses
centered at 1030 nm are obtained from a commercial Yb-doped fiber laser
(Active Fiber Systems) operated at 50 kHz repetition rate. The pulses are
compressed to 36 fs using an argon-filled hollow-core fiber and suitable
chirped mirrors. The laser power is controlled using a motorized half-wave
plate and a broadband thin-film polarizer. The transmitted light is frequency-
doubled using a 300-$\mu$m thick Beta Barium Borate (BBO) crystal and the
fundamental infrared beam is subsequently discarded. A small fraction of the
visible beam is reflected off a wedge and impinges on a photodiode, operated
well below saturation. The photodiode signal is further processed using a
boxcar integrator to yield a measurement of the pulse energy. Importantly, the
photodiode signal is proportional to the laser power recorded with a power
meter, as specifically shown for the range of 200 mW to 600 mW in Fig. 1(b).
Since the second harmonic generation was operated well below saturation, we
assume that the pulse duration and beam profiles remains unchanged when the
laser power is varied using the motorized half-wave plate. Thus, the
photodiode signal is proportional to the intensity in the laser focus. The
main beam is focused (7.5 cm focal length) into the center of a Cold Target
Recoil Ion Momentum Spectrometer (COLTRIMS) [29] where it intersects a cold
gas jet containing the argon-water gas mixture. The momenta of ions and
electrons produced in the laser focus are measured in coincidence using time
and position sensitive detectors on either side of the COLTRIMS. Figure 1(c)
shows the ion time-of-flight spectrum, which permits the distinction of
various ionic species by their time-of-flight, and the accurate measurement of
the recoil momentum along the laser polarization. Momentum conservation
between ion and photoelectron allows us reliably assign photoelectrons to the
ion of the species from which they originate. A total of $6.4\times 10^{4}$
counts are detected for the Ar-H2O+ \+ e- and $3.2\times 10^{7}$ counts for
the Ar+ \+ e- coincidence channels.

Besides ion-electron coincidences, coincident events of two ionic species,
notably, Ar+ \+ H2O+ Coulomb explosion events are observed. The kinetic energy
release of this break-up channel has a mean value of 3.820 eV and a width
(standard deviation) of 0.23 eV. Assuming instantaneous double ionization and
subsequent propagation of a $1/r$ repulsive potential, this corresponds to a
bond distance between Ar and H2O of $(3.77\pm 0.23)\,\AA$, in reasonable
agreement with but slightly longer than the predicted values of $3.636\,\AA$
[7] or $3.603\,\AA$ [9]. The experimental result might be systematically
shifted towards larger values due to the nonlinear ionization rate, especially
in the second ionization step, for which the ionization potential is expected
to strongly decrease with increasing internuclear distance. Moreover, we note
that the measured KER results from the effective distance of the two charges
rather than the center-of-mass distance between Ar and H2O.

Figure 2: Measured Energy distributions of photoelectrons detected in
coincidence with Ar+ (black), H2O+s (dashed blue) and Ar-H2O+ (green). Each
curve is normalized to its maximum (851059 counts for Ar, 22822 for H2O and
1036 for Ar-H2O). Missing data points are due to a lack of statistics. The
upper curves represent all data acquired in the experiment with a peak
intensity of $\approx 1.0\times 10^{14}\,\mathrm{W/cm^{2}}$, the lower ones
were recorded at $\approx 0.5\times 10^{14}\,\mathrm{W/cm^{2}}$. The data sets
with different intensity values are vertically displaced for visual
convenience. The inset shows the low-energy range of the ATI spectra.

##  III Results and Discussion

###  III.1 Intensity-dependent ATI spectra

Figure 2 shows photoelectron energy spectra for ATI of Ar, H2O, and Ar-H2O.
The spectra exhibit a typical shape with a rapid drop over the first few ATI
orders, followed by the rescattering plateau [30] that cuts off around 30 eV
(20 eV) for the upper (lower) curve. Remarkably, the signal in the plateau
region is approximately two times higher for Ar and Ar-H2O than for H2O. We
will return to this observation in the context of the photoelectron momentum
distributions presented in Fig. 5 below. The photoelectron spectra are
modulated by the ATI peaks, which are spaced by the photon energy of 2.4 eV,
as expected from equation 1. Furthermore, the ATI combs recorded for different
targets are shifted with respect to each other, due to the difference in the
ionization potentials. A double peak structure is observed for Ar at higher
intensity, which reduces the contrast between ATI peaks and suggests the
involvement of resonances. Comparing the ATI combs recorded at higher
intensity (upper curves) to the ones recorded at lower intensity (lower
curves) reveals the intensity-dependent peak shift. Importantly, a close
examination of the peak positions (see inset of Fig. 2 shows that the offset
between the peak positions observed for Ar and Ar-H2O differ for the two
intensity values. Hence, the ionization potential of Ar-H2O cannot be
retrieved simply by comparing the respective ATI peak positions to the ones
measured for Ar. The ATI peaks measured for H2O and Ar-H2O, however, are close
to each other for both intensity values. Whether the offset between the ATI
peaks for the two targets allow us to extract the ionization potential of
Ar-H2O will be tested using an intensity scan.

The intensity scan is carried out by scanning the half-wave plate while the
laser power transmitted trhough the polarizer is tracked by the photodiode, as
described above. The measured power-dependent ATI spectra for Ar are presented
in Fig. 3(a). The ATI spectra exhibit a pronounced intensity dependence: while
the cut-off energy increases proportionally to the laser power, the individual
ATI peaks are red-shifted and also broadened. This behaviour is an impressive
manifestation of the focal volume effect: with increasing peak intensity, more
and more intensity values contribute to the ATI yield. Since different
intensity values corresponds to different peak positions, the peaks in the
observed ATI spectra are broadened and eventually washed out. In addition,
Freeman (multi-photon) resonances [31, 32] lead to horizontal features in the
intensity-dependent ATI spectrum (see inset of Fig. 3), and are likely
responsible for the double peak structure observed for Ar in Fig. 2 at higher
intensity.

The intensity-resolved ATI spectra allow us to find the proportionality
constant between measured laser power and peak intensity. This is achieved by
comparison of the experimental results to numerical results displayed in Fig.
3(b). These were obtained by solving the one-dimensional time-dependent
Schrödinger equation (TDSE) using the split-operator method. The Hamiltonian
is given by $H(x,t)=\frac{Z}{|x|+\alpha}+\mathcal{E}(t)\cdot x$. The softcore
parameters $Z$ and $\alpha$ where tuned such that the energies of the ground
state and first-excited state of the model atom match those of real argon. The
laser field $\mathcal{E}(t)=\mathcal{E}_{0}(t)\cos{\omega t}$ has a Gaussian
pulse envelope $\mathcal{E}_{0}(t)$ with a full-width at half-maximum of
$40\text{\,}\mathrm{f}\mathrm{s}$ and a carrier frequency of
$\omega=$0.088\text{\,}\mathrm{a}\,\mathrm{u}\,$$ The calculated photoelectron
spectra were integrated over a Gaussian focal volume [33] and agree well with
the measured ones. The proportionality constant between peak intensity and
laser power in the experimental data is determined by minimizing the standard
deviation between the positions of ATI peaks of orders 1 to 10 in the measured
and calculated ATI spectra. We find best agreement for $(6.40\pm
0.30)\,\frac{\mathrm{mW}}{\mathrm{TW/cm^{2}}}$. The uncertainty indicates the
range in which the standard deviation increases by no more than 10%, see inset
of Fig. 3(b). Using this conversion, we confirm that the high-energy cut-off
of the spectra can be adequately described by the extended cut-off law
proposed in Ref. [34]: $E_{\mathrm{max}}=10\,U_{P}+0.538E_{\mathrm{IP}}$, as
indicated by the red lines in Fig. 3.

Figure 3: Intensity-dependent ATI spectra for ionization of Ar by 515 nm
light. (a) Experimental results obtained by integrating the measured
photoelectron momentum distributions over the full solid angle. The red solid
line indicates the cut-off energy of the photoelectron spectra and is
described by the expression
$0.0391\,\mathrm{eV}/\mathrm{mW}+8.5\,\mathrm{eV}$. (b) Numerical results
obtained by solving the one dimensional TDSE. The red line is described by the
expression $25.0\,\mathrm{eV}/(10^{14}\,\mathrm{W/cm^{2}})+8.5\,\mathrm{eV}$
indicating the high-energy cut-off. The inset shows the standard deviation
$\sigma$ as a function of the proportionality constant between peak intensity
and laser power.

###  III.2 Peak shifts and ionization potential of Ar-H2O

In order to quantitatively analyze the intensity-dependent ATI spectra for all
targets, we extract the ATI peak positions from the measured photoelectron
spectra as follows. The photoelectron energy distribution $Y(E)$ around the
first ATI peak is fitted by a Gaussian,

$Y(E)=A\exp{\left(-\frac{(E-E_{C})^{2}}{\text{w}^{2}}\right)}+Y_{0}.$ (3)

We note that, especially for high intensities, the shape of the ATI peaks
deviates significantly from a Gaussian shape. Nevertheless, we obtain good
agreement for the peak positions, see figure reffigs:consistency, which shows
the measured intensity-dependent spectra along with the retrieved centroids of
the ATI peaks measured at each laser power and target. The procedure is
repeated for 25 different values of the measured laser power, which yields the
intensity-dependence of the ATI peak positions displayed in Fig. 4.

Figure 4: Intensity-dependent ATI peak positions for various targets. Shown is
the measured offset energy of the ATI comb for Ar, Ar2, H2O and Ar-H2O, as
function of the peak intensity. The colored solid lines represent linear fit
functions to the data. The peak positions extracted for the simulated data is
also shown (black). Around $7\cdot 10^{13}\,\mathrm{W/cm^{2}}$, no accurate
fit could be obtained due to strong modulations in the calculated
photoelectron spectra.

Figure 4 shows the offsets $E_{0}=E_{C}-2.4\,\mathrm{eV}$ of the ATI comb,
retrieved by the fitting procedure described above, as a function of the
calibrated laser intensity. Indeed, the ATI peaks measured for Ar and Ar-H2O
exhibit significantly different shifts as a function of intensity. In
contrast, the intensity-dependence of the ATI peak positions measured for Ar
and Ar2 are nearly identical. Indeed, the ionization potentials of Ar
($15.76\,\mathrm{eV}$) and Ar2 ($15.65\,\mathrm{eV}$) are very close and their
ATI peaks exhibit only a small offset with a mean of $(0.08\pm
0.01)\,\mathrm{eV}$, in good agreement with the difference in ionization
potential. Analogously, the energy offset between the ATI peaks measured for
H2O and Ar-H2O remains essentially constant at $0.26\,\mathrm{eV}$ within a
margin of $\pm 0.06\,\mathrm{eV}$ over the entire intensity range studied in
the present experiment. Therefore, it is reasonable to assume that the peak
offset corresponds to the difference in ionization potential (cf. equation 1).
With the known ionization potential of H2O of $12.6\,\mathrm{eV}$, we find a
value of $(12.4\pm 0.1)\,\mathrm{eV}$ for Ar-H2O, where the uncertainty
corresponds to the statistical error of the differences in the measured peak
positions. We point out that this deviation is independent of the intensity
determination conducted above.

We will now further investigate the differences in the intensity-dependence of
the ATI peak positions recorded for different species. Clearly, the red-shift
of the ATI peaks (cf. equation 1) is target-dependent. Specifically, the
linear fit functions, applied over the entire intensity range of the
experiment, yield a slope of $-2.2\,\mathrm{eV}/(10^{14}\,\mathrm{W/cm^{2}})$
for Ar and Ar2. These values are in reasonable agreement with the
ponderomotive shift of $-2.5\,\mathrm{eV}/(10^{14}\,\mathrm{W/cm^{2}})$ for
515-nm light and the difference is consistent with the effect of focal volume
averaging. The TDSE results for Ar exhibits some deviations from linearity,
likely due to the effect of resonances. Up to an intensity of approximately
$1.0\cdot 10^{14}\,\mathrm{W/cm^{2}}$, the intensity-dependent peak shift of
the TDSE results agrees with the experimental results, however, the curve is
blue shifted by approximately 0.2 eV. Such deviation is possible despite the
intensity calibration because the fitting performed above is carried out for
the positions of ATI orders 1 to 10, while in Fig. 4, only ATI order 1 is
analyzed. Above $1.0\cdot 10^{14}\,\mathrm{W/cm^{2}}$, the measured and
calculated peak positions deviate, potentially because the influence of
resonances is exaggerated in the one-dimensional TDSE calculations. This
observation emphasizes that the linear relationship between ATI peak position
and intensity is only applicable in the absence of resonances.

The ATI peak shift measured for Ar-H2O and H2O is only
$-1.5\,\mathrm{eV}/(10^{14}\,\mathrm{W/cm^{2}})$, significantly less than the
ponderomotive shift of $-2.5\,\mathrm{eV}/(10^{14}\,\mathrm{W/cm^{2}})$ for
515-nm light. A first candidate to explain the observed differences between Ar
and Ar2 on one hand and H2O and Ar-H2O on the other hand is the AC stark shift
of the ground state. However, the ground state polarizabilities of argon
($\alpha_{\textrm{Ar}}=$11.07\text{\,}\mathrm{a}_{\mathrm{0}}^{3}$$) and water
($\alpha_{\textrm{H2O}}=$10.74\text{\,}\mathrm{a}_{\mathrm{0}}^{3}$$) [35, 36,
37] are very similar and both lead to a Stark shift of the ground state energy
by $-0.2\,\mathrm{eV}/10^{14}\,\,\mathrm{W/cm^{2}}$. Hence, the difference of
ground state polarizabilities for argon and water is too small to explain the
observed difference in the intensity-dependent ATI peak shifts of the two
targets. Furthermore, as the different shift is observed over the entire
intensity range, resonances are unlikely to be responsible for the observed
differences. We propose that the smaller red shift of the ATI peaks observed
for H2O and Ar-H2O results from the permanent dipole [23]. Its effect may
become measurable due to the orientation-dependence of the ionization
probability of water [38]. However, a quantitative calculation of the
intensity-dependent ATI peak shift of water is outside the scope of the
present paper.

###  III.3 Rescattering from Ar-H2O

In this section, we return to the observation of enhanced backscattering from
Ar-H2O with respect to H2O. Figure 5 presents the photoelectron momentum
distributions measured for Ar-H2O (upper half), H2O (bottom left quadrant) and
Ar (bottom right quadrant) corresponding to the ATI spectra presented in Fig.
2 at high intensity. The momentum distribution measured for Ar2 is compared to
the one for Ar in Fig. 7. The central, low-energy, part of the momentum
distributions for all three targets are governed by direct electron emission
and full ATI rings are observed. At energies significantly larger than
$2\,U_{\mathrm{P}}$, corresponding to $p\approx$0.6\text{\,}\,\mathrm{a.u.}$$,
re-scattered electrons dominate and the angular distributions are
significantly narrower such that only segments of the ATI rings are observed.

We first compare Ar-H2O to H2O and observe that the low-energy part of the
momentum distribution is nearly identical for the two targets. The slightly
lower ionization potential of Ar-H2O leads to slightly larger radii of the ATI
rings. In the high-energy part along the laser polarization, the signal
observed for Ar-H2O is slightly larger than for H2O. This is evident for the
highest ATI peaks but also the signal in the intermediate ones is higher by a
a factor of $\approx 2$, see Fig. 2. This enhancement suggests a larger
contribution from backscattering in the case of Ar-H2O. Compared to Ar (bottom
right quadrant), however, the signal remains slightly lower. The stronger
signal observed for Ar might be a consequence of focal volume averaging, which
leads to higher effective intensity for Ar, and therefore higher signal at
larger electron momenta. Our results suggest that in Ar-H2O, the relatively
large Ar atom acts as a scattering center, which enhances the contribution of
backscattering to the photoelectron yield. Indeed, for the differential
elastic electron scattering cross section around $180^{\circ}$, two to three
times larger values have been reported for Ar [39] than for water [40],
consistent with our results.

Figure 5: (a)Photoelectron momentum distribution recorded for non-dissociative
single-ionization of Ar-H2O (upper half), compared to the ones recorded for
H2O (bottom left), and Ar (bottom right). The ionization yield is integrated
over all intensity values and normalized to 1 by dividing by the number of
detected events for each species. The plot shows the normalized ionization
yield per momentum bin, as a function of the momentum components parallel
$p_{||}=p_{z}$ and perpendicular $p_{\perp}=\sqrt{p_{x}^{2}+p_{y}^{2}}$ to the
laser polarization, taking into account the appropriate Jacobian. The minimum
of the color scale is limited by the low statistics of the Ar-H2O channel.

##  IV Summary and Outlook

In conclusion, we have presented measurements of ATI of Ar-H2O and compared
them to simultaneous measurements of both Ar and H2O. We found that the peak
shift for Ar-H2O matches that for H2O, while the ionization potential is
lowered by some $0.2\,\,\mathrm{eV}$ with respect to H2O. These observations
indicate that, on the one hand, strong-field ionization occurs on the H2O side
of the weakly bound Van-der-Waals molecule. On the other hand, the reduction
of the ionization potential indicates that the molecular ion is more tightly
bound than neutral Ar-H2O. This raises the question about details of the
potential energy landscape of the Ar-H2O cation, for example where the argon
atom resides with respect to the H atoms. A suitable tool to shed light on
this question is laser-driven Coulomb explosion imaging. While the Ar atom has
little effect on the initial step of strong-field ionization, we have further
shown that elastic rescattering in Ar-H2O is enhanced by the presence of the
Ar atom. By extension to inelastic rescattering, this suggests the existence
of a special mechanism for non-sequential double ionization, where a first
electron is emitted from water and its subsequent recollision with the parent
ion leads to ionization of the argon atom. This mechanism could lead to a
peculiar effect: at relatively low intensity and sufficiently long wavelength,
Ar-H2O could be doubly ionized even before single ionization of Ar becomes
efficient.

## Acknowledgements

We thank T. Weber, A. Rose, H. Wöhl, F. Ronneberger, T. Farhadova, J. Yu, S.
Voss and A. Czasch for technical support, in particular when setting up the
COLTRIMS. Fruitful discussions with M. Lesiuk and R. Della Picca are
acknowledged. This work has been funded by the DFG under Project No.
437321733.

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##  Appendix A Intensity-dependent ATI peak positions

Fig. 6 shows experimental results for the four targets along with the
centroids obtained from the fitting procedure described above. The fitting is
performed in a range of $\pm$1\text{\,}\mathrm{e}\mathrm{V}$$ around the
approximate position of the first ATI peak. The accuracy of the results is not
notably improved if the fitting is repeated to ATI peaks of higher order. The
zeroth ATI peak was ignored because of channel closing. In the case of water,
no accurate fit was obtained at the highest intensities used in the
experiment.

(a)

(b)

(c)

(d)

Figure 6: Intensity-dependent ATI spectra of (a) argon, (b) argon dimer, (c)
argon-water and (d) water. The dots mark the centroids of Gaussians fitted to
the experimental data at various intensity values. The statistcal error of the
fits is on the order of the dot size.

##  Appendix B Photoelectron momentum distribution for argon dimers

Figure 7 compares the photoelectron momentum distributions recorded for non-
dissociative single ionization of Ar2 to that for Ar. The momentum
distributions for the two targets are very similar to each other, in agreement
with earlier experiments using 800 nm light [41]. A slight yield enhancement
at minimal electron energy reported for the dimer in Ref. [41] can be
identified.

Figure 7: Same as Fig. 5 but shown the momentum distribution recorded for non-
dissociative single-ionization of Ar2 (upper half) compared to the one
recorded for Ar (bottom half), as in Fig. 5. At
$p_{||}\approx-1.3\,\mathrm{a.u.}$ and $p_{||}\approx 0.8\,\mathrm{a.u.}$, the
detector provides no resolution for the perpendicular momentum.


##  1 Introduction

The statistical distribution of the zeroes of the Riemann zeta function, and
the related family of Dirichlet $L$-functions, qualitatively resemble the
eigenvalue distribution of a random ensemble of unitary matrices[1, 2, 3]. It
is also reminiscent of the distribution of zeroes of partition functions of
statistical models. The latter observation is the motivation to search for a
suitable model in physicists’ approach to the problem—the literature is vast,
however, see e.g., Refs.[4, 5, 6, 7, 8, 9], the review [10] and references
therein. This resemblance may be an important guide as the zeroes of the
partition function of many systems, by the Yang-Lee type theorems [11], all
lie parallel to the imaginary axis (or on the unit circle). These zeroes are
called Yang-Lee zeroes or Fisher zeroes, depending upon whether the partition
function is viewed as a function of the applied external field, e.g., magnetic
field, or of $\beta=1/(k_{B}T)$, the inverse temperature. The arithmetic or
primon gas of Refs. [5, 7, 6] and the number theoretic spin chain of Refs.[8,
9], in particular, proposed interesting models for which the partition
functions are directly related to the Riemann zeta function. In this approach
the non-trivial zeroes of the zeta function are to be identified with the
Yang-Lee or Fisher zeroes.

Motivated by these, we shall propose a statistical model and compute its
partition function. The idea is again to associate the spectrum of an operator
with the Fisher zeroes of the partition function. In addition, however, we
shall study the spectrum of some relevant operators of these models. The
systems that relate to the $L$-functions of our interest can be thought of as
spins in an external magnetic field. Since the spectrum of a Hamiltonian of
this type of spin systems is discrete (the spins being integer/half-integer
valued) this operator is similar to the number operator of an oscillator. A
phase operator that is conjugate to this will be a new ingredient in our
investigation. The construction of a phase operator which is truly canonically
conjugate to the number operator is a subject of long-standing quest that may
not be completely closed yet. Nevertheless, several different ways to define
the phase operator have been proposed, for example, Refs.[12, 13, 14, 15, 16,
17, 18] is a partial list. In particular, we shall investigate two ways of
defining it for the spin models corresponding to the family of $L$-functions.
In the first construction, we follow Ref.[17], where the authors propose an
operator by directly constructing eigenstates of phase for a system with a
discrete spectrum. The second approach is motivated by the proposal in [15].
We shall argue that there are enough hints in these proposals to understand
the correspondence between the spectrum of these operators and the zeroes of
the partition function.

In the following, we shall first review (in Section 2) some of the relevant
arguments and results from the cited references, in the context of a simple
spin system on a one-dimensional lattice. In Section 3, after recalling some
properties of the Riemann zeta function and our earlier work on its relation
to operators on the Hilbert space of complex valued functions on the $p$-adic
number field $\mathbb{Q}_{p}$ [19, 20], we elaborate on a proposal to view it
as a statistical model of spins. In Sections 3.1 and 3.2 we detail two
constructions of the phase operators for the spin model for the Riemann zeta
function, which are then extended to the family of Dirichlet $L$-functions in
Section 4.

##  2 Quantum spins in external field

Spin models in one dimension are among the simplest statistical models, yet
they offer an arena rich enough to experiment, before considering more
complicated systems. The variables are ‘spins’ $s_{n}$ at lattice points
$n\in\mathbb{Z}$ or $n\in\mathbb{N}$ that can take $(2j+1)$ values
$\\{-j,-j+1,\cdots,j-1,j\\}$ in the spin-$j$ representation. In models of
magnetism, these spins interact locally, usually with the nearest neighbours.
In addition, one may turn on an external magnetic field.

Let us digress to recall the properties of a simpler model, the Ising model,
in which the classical spins take one of two possible values $\pm 1$ and the
total Hamiltonian is $H=-J\sum_{n}s_{n}s_{n+1}-B\sum_{n}s_{n}$, where $J$ is
the strength of interaction ($J>0$ being ferromagnetic and anti-ferromagnetic
otherwise) and the second term arises from an interaction with an external
magnetic field $B$. The partition function (in the absence of an external
field) of an Ising system of size $L$ at inverse temperature $\beta$ is

$Z(\beta)\equiv\text{Tr }e^{-\beta
H}=\sum_{\left\\{s_{n}\right\\}}\exp\left(\beta
J\sum_{n=1}^{L-1}s_{n}s_{n+1}\right)$

In this simple case, one may also change variables to
$\sigma_{n}\equiv\sigma_{\langle n-1,n\rangle}=s_{n-1}s_{n}$ associated to the
edges $\langle n-1,n\rangle$ joining nearest neighbours. Evidently,
$\sigma_{n}=\pm 1$ as well. Thus

$\displaystyle Z(\beta)$ $\displaystyle=$ $\displaystyle
2\sum_{\left\\{\sigma_{n}\right\\}}\exp\left(\beta
J\sum_{n=2}^{L}\sigma_{n}\right)$ $\displaystyle=$ $\displaystyle
2\sum_{\sigma_{n}}\left\langle\sigma_{2},\sigma_{3},\cdots\right|\exp\left(\beta
J\sum_{j}S_{j}\right)\left|\sigma_{2},\sigma_{3},\cdots\right\rangle$

where we have defined vectors $|\sigma_{n}\rangle$ in a (two-dimensional)
Hilbert space corresponding to the edge $\langle n-1,n\rangle$ and $S_{n}$s
are spin operators such that
$S_{n}\left|\sigma_{n}\right\rangle=\sigma_{n}\left|\sigma_{n}\right\rangle$.
A generalisation of this model allows the coupling constants $J$ to be
position dependent, so that the Hamiltonian is $H=-\sum_{n}J_{n}\sigma_{n}$
and

$Z(\beta)=2\sum_{\sigma_{n}}\big{\langle}\sigma_{2},\sigma_{3},\cdots\big{|}e^{\beta\sum_{n}J_{n}S_{n}}\big{|}\sigma_{2},\sigma_{3},\cdots\big{\rangle}$

is the canonical partition function of the generalised model at the
temperature $k_{B}T=1/\beta$.

We would like to consider the general case where the spins to be valued in the
spin-$j$ representation of $\mathfrak{su}(2)$. Although we seek a partition
function of the form as above, the general spin case is cannot be realised as
an Ising type model, rather it will be a model of spins in an external local
magnetic field $B_{n}$ at site $n$. It will be useful to think of $S_{n}$ to
be the third component $S_{n3}$ of the $\mathfrak{su}(2)$ spin operators on
the edge/site, the others being $S_{n\pm}$. The vectors
$|\sigma_{2},\sigma_{3},\cdots\rangle=|\sigma_{2}\rangle\otimes|\sigma_{3}\rangle\otimes\cdots$
belong to the product space.The interaction $\sim\mathbf{B}\cdot\mathbf{S}$
between the spin (magnetic moment to be precise) and the external field
(assumed to be along the $z$-direction) described by the Hamiltonian
$H=-\sum_{n}B_{n}\sigma_{n}$ leads to the partition function

$Z(\beta)=\sum_{\sigma_{n}}\big{\langle}\sigma_{2},\sigma_{3},\cdots\big{|}e^{\beta\sum_{n}B_{n}S_{n}}\big{|}\sigma_{2},\sigma_{3},\cdots\big{\rangle}$

at the temperature $k_{B}T=1/\beta$. Our objective is to obtain an identity
for the partition function for this model. To this end, we shall seek an
operator that, in a certain well defined sense, is formally canonically
conjugate to the $z$-component $S_{n3}$ of the spin operator at site $n$.
There are well known difficulties in defining such an operator, however, we
shall see that one needs to make a much weaker demand.

In this context, it is useful to remeber the Schwinger oscillator realisation
of the algebra $\mathfrak{su}(2)$ in terms of a pair of bosonic
creation/anhilation operators $(a_{1}^{\dagger},a_{1},a_{2}^{\dagger},a_{2})$
at each edge, where we drop the edge index for the time being. Then
$S_{+}=a_{1}^{\dagger}a_{2}$, $S_{-}=a_{2}^{\dagger}a_{1}$ and the third
component is the difference of the number operators
$S_{3}=\frac{1}{2}(n_{1}-n_{2})=\frac{1}{2}\left(a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2}\right)$.
One can formally introduce the _phase operator_
$\Phi=\frac{1}{2}(\phi_{1}-\phi_{2})$ such that
$[\phi_{a},n_{b}]=i\delta_{ab}$, however, there are several mathematical
difficulties in defining the above[12, 13]. We will now review an explicit
construction to show how one can still work around this problem.

###  2.1 Phase operator via phase eigenstates

Let us label the eigenstates in the spin-$j$ representation of $S_{3}$ as
$|m\rangle$, for $m=-j,\cdots,j$. One can define an eigenstate of phase as a
unitary transform of these states as

$\displaystyle|\phi_{k}\rangle$
$\displaystyle=\frac{1}{\sqrt{2j+1}}\sum_{m=-j}^{j}e^{-im\phi_{k}B}|m\rangle$
$\displaystyle\text{where, }\;\phi_{k}$ $\displaystyle=\frac{2\pi
k}{B(2j+1)},\qquad k=-j,\cdots,j$ (1)

are the eigenvalues of the phase. The phase eigenstates satisfy

$\langle\phi_{k^{\prime}}|\phi_{k}\rangle=\frac{1}{2j+1}\sum_{m=-j}^{j}e^{-imB(\phi_{k}-\phi_{k^{\prime}})}=\delta_{k,k^{\prime}}$
(2)

and thus provide an orthonormal basis of the Hilbert space of states.

In terms of these, we may define the the ‘phase operator’ through spectral
decomposition as

$\hat{\phi}=\sum_{k=-j}^{j}\phi_{k}\,|\phi_{k}\rangle\langle\phi_{k}|$ (3)

We shall now show that it transforms covariantly when conjugated by $e^{\beta
BS_{3}}$. This works for special values of $\beta$ since, an eigenvalues
$\phi_{k}$ of $\hat{\phi}$, being angle-valued, is only defined modulo
$2\pi/B$. In order to see this, we note that

$e^{-\beta BS_{3}}\hat{\phi}\,e^{\beta
BS_{3}}=\sum_{k=-j}^{j}\frac{\phi_{k}}{2j+1}\sum_{m=-j}^{j}\sum_{m^{\prime}=-j}^{j}e^{-im(\phi_{k}-i\beta)B+im^{\prime}(\phi_{k}-i\beta)B}|m\rangle\langle
m^{\prime}|$

There are two cases to consider. The first is trivial: for $\beta=0$ or any of
its periodic images $i\beta=\frac{2\pi n}{B}$ ($n\in\mathbb{Z}$) in the
complex $\beta$-plane, the RHS is the phase operator $\hat{\phi}$. More
interestingly, if $i\beta$ takes any of the specific discrete values
$\frac{2\pi k^{\prime}}{B(2j+1)}+\frac{2\pi n}{B}$, where
$k^{\prime}=-j,\cdots,j$ (but $k^{\prime}\neq 0$) and $n\in\mathbb{Z}$, i.e.,
$i\beta$ is a difference between the phase eigenvalues (mod ${2\pi}/{B}$),
then $\phi_{k}-i\beta$ is again an allowed eigenvalue of the phase operator
$\pmod{2\pi/B}$. In this case, we can add and subtract $i\beta$ to the
eigenvalue $\phi_{k}$ and use the completeness of basis, to find

$e^{-\beta BS_{3}}\hat{\phi}\,e^{\beta BS_{3}}=\hat{\phi}+i\beta\>\text{ only
for }\,0\neq\beta=-\frac{2\pi ij}{B(2j+1)},\cdots,\frac{2\pi
ij}{B(2j+1)}\>\Big{(}\text{mod }\frac{2\pi}{B}\Big{)}$ (4)

This is called a shift covariance relation [21]. It may also be rewritten as a
commutator

$\big{[}\hat{\phi},e^{\beta BS_{3}}\big{]}=i\beta\,e^{\beta BS_{3}}\>\text{
only for }\,0\neq\beta=-\frac{2\pi ij}{B(2j+1)},\cdots,\frac{2\pi
ij}{B(2j+1)}\>\Big{(}\text{mod }\frac{2\pi}{B}\Big{)}$

i.e., at special values of the inverse temperature.

To summarise, we find that $\hat{\phi}$ in Eq. 3 satisfies shift covariance,
alternatively, though somewhat loosely, it is ‘canonically conjugate’ to
$S_{3}$ only for a special set of an infinite number of imaginary values of
$\beta$, all on the line $\text{Re}\,\beta=0$ as above. At $\beta=0$ (mod
$\frac{2\pi}{B}$), the commutator is trivial111It is also reflected in the
resolvent of the phase operator, as we shall see in the following..

In passing, it is instructive to take the trace of the ‘canonical commutator’.
The left hand side evidently vanishes, since the vector space of states is
finite, namely $(2j+1)$, dimensional. On the right hand side, the trace of
$e^{-\beta H}$, the partition function which vanishes, being a sum over the
roots of unity. Thus the values of $\beta$ for which Eq. 4 is valid must also
satisfy the condition $\text{Tr }e^{-\beta H}=0$. This means that mod
${2\pi}/{B}$, these values of $i\beta\neq 0$ for which the partition function
has a zero are same as that of the eigenvalues of $\hat{\phi}$.

The resolvent of the exponential of the phase operator at a single site (as a
function of $z=e^{i\phi}$) is

$\hat{R}[\hat{\phi}](\phi)=\left(1-e^{-i\phi}e^{i\hat{\phi}}\right)^{-1}$

and its trace is

$\mathrm{Tr}\left(\hat{R}[\hat{\phi}](\phi)\right)=\sum_{k=-j}^{j}\Big{\langle}\phi_{k}\Big{|}\frac{1}{1-e^{i(\hat{\phi}-\phi)}}\Big{|}\phi_{k}\Big{\rangle}\\\
=\sum_{k=-j}^{j}\frac{1}{1-e^{i(\phi_{k}-\phi)}}$

On the other hand, the partition function at a single site
$Z_{1}(\beta)=\text{Tr}\,e^{-\beta H}=\sum_{m}e^{\beta Bm}$ vanishes at
special values of the inverse temperature $\beta=\frac{2\pi mi}{B(2j+1)}$ (mod
$\frac{2\pi}{B}$) where $m\in\\{-j,\cdots,j\\}$ but $m\neq 0$. These zeroes of
the partition function in the complex $\beta$-plane are called Fisher zeroes.
At precisely these values, the resolvent function develops poles.

##  3 The case of Riemann zeta function

Before we get to our main goal to interpret the Riemann zeta function as a
partition function, let us briefly recall some of its relevant properties.
Originally defined by the analytical continuation of the series

$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}=\prod_{p\,\in\,{\mathrm{primes}}}\frac{1}{\left(1-p^{-s}\right)},\qquad\mathrm{Re}(s)>1$
(5)

to the complex $s$-plane by Riemann, the zeta function has a set of equally
spaced zeroes at negative even integers $-2n$, $n\in\mathbb{Z}$ called its
trivial zeroes. More interestingly, it has another infinite set of zeroes,
which, according to the _Riemann hypothesis_ lie on the _critical line_
${\mathrm{Re}}(s)=\frac{1}{2}$. The related Riemann $\xi$-function (sometimes
called the _symmetric zeta-function_) and the adelic zeta function share only
the latter (non-trivial) zeroes with Eq. 5 (i.e., the set of trivial zeroes
are absent in the following functions)

$\xi(s)=\frac{1}{2}s(s-1)\zeta_{\mathbb{A}}(s)=\frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)\zeta(s)$
(6)

both of which satisfy the reflection identity $\xi(s)=\xi(1-s)$, respectively,
$\zeta_{\mathbb{A}}(s)=\zeta_{\mathbb{A}}(1-s)$, derived from a similar
identity for the original zeta function. The former is a holomorphic function
while the latter, $\zeta_{\mathbb{A}}(s)$, is meromorphic.

The non-trivial zeroes of $\zeta(s)$ (which are the only zeroes of $\xi(s)$
and $\zeta_{\mathbb{A}}(s)$) conjecturally on the critical line, seem to occur
randomly, although they are found to be correlated in the same way as the
eigenvalues of a Gaussian ensemble of $N\times N$ hermitian or unitary
matrices in the limit $N\to\infty$ [1, 2, 3]. Starting from Hilbert and Pólya,
it has long been thought that these zeroes correspond to the eigenvalues of an
operator, that is self-adjoint in an appropriately defined sense. A direct
analysis of the spectrum of the purported operator may lead to a proof of the
Riemann hypothesis. Despite many ingenious efforts, an operator has not yet
been found. In Ref.[19], in a larger collaboration, we attempted to find a
suitable operator by _assuming the validity of the hypothesis_ , specifically,
by assuming that the zeroes _are_ the eigenvalues of a unitary matrix model
(UMM). We found that the partition function can be expressed as the _trace_ of
an operator on the Hilbert space of complex valued locally constant Bruhat-
Schwarz functions supported on a compact subset $p^{-1}\mathbb{Z}_{p}$ of the
$p$-adic field $\mathbb{Q}_{p}$. This was achieved in two steps. First a UMM
was constructed for each prime $p$ corresponding to the Euler product form in
Eq. 5. These (as well as a UMM for the trivial zeroes) were combined to define
the random matrix model. In this paper, we shall use some of the technology
that were useful in [19], however, our goal will be different.

We begin by expanding the prime factors in the Euler product form of the zeta
function

$\zeta(s)=\prod_{p\,\in\,{\mathrm{primes}}}\frac{1}{\left(1-p^{-s}\right)}=\prod_{p\,\in\,{\mathrm{primes}}}\sum_{n^{(p)}=0}^{\infty}p^{-sn_{(p)}},\qquad{\mathrm{Re}}(s)>1$
(7)

The factor

$\zeta_{p}(s)=\frac{1}{\left(1-p^{-s}\right)}\>\>{\text{for a fixed prime }}p$
(8)

is sometimes called the _local_ zeta function at $p$. It can be thought of as
a complex valued function on the field $\mathbb{Q}_{p}$ (of $p$-adic numbers).
The prefactor $\zeta_{\mathbb{R}}(s)=\pi^{-{s\over 2}}\Gamma\left({s\over
2}\right)$ in Eq. 6 is known as the _local_ zeta functions corresponding to
$\mathbb{R}$ (of real numbers). It is the Mellin transform of the Gaussian
function $e^{-\pi x^{2}}$. In an exactly analogous fashion, $\zeta_{p}(s)$ in
Eq. 8 is the Mellin transform of the equivalent of the Gaussian function (in
the sense of a function that is its own Fourier transform) on
$\mathbb{Q}_{p}$.

We can express the sum in Eq. 7 as the trace of an operator. To this end, let
us recall that the space of (mean-zero) square integrable complex valued
functions on $\mathbb{Q}_{p}$ is spanned by the orthonormal set of Kozyrev
wavelets $\psi^{(p)}_{nml}(\xi)\in\mathbb{C}$ (for $\xi\in\mathbb{Q}_{p}$),
which have compact support in $\mathbb{Q}_{p}$[22]. In $p$ segments (of equal
Haar measure) its values are the $p$-th roots of unity. They are analogous to
the generalised Haar wavelets, with the labels $n$, $m$ and $l$ referring to
scaling, translation and phase rotation. Interestingly, the Kozyrev wavelets
are eigenfunctions of an operator with eigenvalue $p^{\alpha(1-n)}$

$D_{(p)}^{\alpha}\psi_{n,m,l}^{(p)}(\xi)=p^{\alpha(1-n)}\psi_{n,m,l}^{(p)}(\xi)$
(9)

where, the pseudodifferential operators $D_{(p)}^{\alpha}$, called the
_generalised Vladimirov derivatives_ , are defined by the following integral
kernel as

$D^{\alpha}_{(p)}f(\xi)=\frac{1-p^{\alpha}}{1-p^{-\alpha-1}}\,\displaystyle\int_{\mathbb{Q}_{p}}d\xi^{\prime}\,\frac{f(\xi^{\prime})-f(\xi)}{|\xi^{\prime}-\xi|_{p}^{\alpha+1}},\quad\alpha\in{\mathbb{C}}$

They satisfy
$D^{\alpha_{1}}_{(p)}D^{\alpha_{2}}_{(p)}=D^{\alpha_{2}}_{(p)}D^{\alpha_{1}}_{(p)}=D^{\alpha_{1}+\alpha_{2}}_{(p)}$.

Since the roles of translation and phase are not going to be important in what
follows, let us set $m=0$ and $l=1$ and define vectors $|n^{(p)}\rangle$
corresponding to $\psi^{(p)}_{-n+1,0,1}(\xi)$

$\psi^{(p)}_{-n+1,0,1}(\xi)\>\longleftrightarrow\>|n_{(p)}\rangle$ (10)

in the Hilbert space $L^{2}(\mathbb{Q}_{p})$. Then

$\displaystyle D_{(p)}^{\alpha}|n_{(p)}\rangle$
$\displaystyle=p^{n^{(p)}\alpha}|n_{(p)}\rangle$
$\displaystyle\log_{p}D_{(p)}|n_{(p)}\rangle$
$\displaystyle=\displaystyle{\lim_{\alpha\to
0}}\,\frac{D_{(p)}^{\alpha}-1}{\alpha\ln
p}|n_{(p)}\rangle\>=\>n_{(p)}|n_{(p)}\rangle$ (11)

The wavelets, by construction, transform naturally under the affine group of
scaling and translation. However, it was shown in Ref.[23] that the scaling
part of it enhances to a larger SL(2,$\mathbb{R})$ symmetry. In terms of the
raising and lowering operators
$a^{(p)}_{\pm}|n_{(p)}\rangle=|n_{(p)}\\!\pm\\!1\rangle$ the generators of
SL(2,$\mathbb{R}$) are $J^{(p)}_{\pm}=a_{\pm}^{(p)}\log_{p}\\!D_{(p)}$ and
$J_{3}^{(p)}=\log_{p}D_{(p)}$. The algebra of these generators and their
action on the wavelet states are as follows.

$\displaystyle\begin{split}\left[J_{3}^{(p)},J^{(p)}_{\pm}\right]=\pm
J^{(p)}_{\pm},&\qquad\left[J_{+}^{(p)},J^{(p)}_{-}\right]=-2J^{(p)}_{3}\\\
J^{(p)}_{3}|n_{(p)}\rangle=n_{(p)}|n_{(p)}\rangle,&\qquad
J^{(p)}_{\pm}|n_{(p)}\rangle=n_{(p)}|n_{(p)}\\!\pm\\!1\rangle\end{split}$ (12)

We can now write Eq. 7 as

$\displaystyle\zeta(s)$ $\displaystyle=$
$\displaystyle\prod_{p\,\in\,{\mathrm{primes}}}\sum_{n^{(p)}=0}^{\infty}\left\langle
n_{(p)}\right|D_{(p)}^{-s}\left|n_{(p)}\right\rangle$ (13) $\displaystyle=$
$\displaystyle\sum_{{\mathbf{n}}=(n_{(2)},n_{(3)},\cdots)}\\!\\!\\!\\!\left\langle{\mathbf{n}}\right|e^{-s\ln\mathcal{D}}\left|{\mathbf{n}}\right\rangle$

where we have used a shorthand $\ln\mathcal{D}\equiv\sum_{p}\ln
p\,\log_{p}\\!D_{(p)}$, and the vectors $\left|{\mathbf{n}}\right\rangle$
belong to the product of the Hilbert spaces for all primes
$\bigotimes_{p}L^{2}(\mathbb{Q}_{p})$. However, since the sum runs only over
the _positive_ integers (including zero), this subspace is actually
$\bigotimes_{p}L^{2}(p^{-1}\mathbb{Z}_{p})$, spanned by the Bruhat-Schwarz
functions restricted to $p^{-1}\mathbb{Z}_{p}$ due to which the trace is well
defined (see [22, 23, 24] for details on the wavelet functions). This
expression leads us to think of the zeta function as the partition function of
a statistical system, the configurations of which are parametrised by the
integers ${\mathbf{n}}=(n_{(2)},n_{(3)},\cdots)$.

The $\mathfrak{sl}_{2}(\mathbb{R})$ algebra Eq. 12 can be realised in terms of
a pair of oscillators in the Schwinger representation

$J^{(p)}_{3}=\log_{p}D_{(p)}=\frac{1}{2}\left(N_{\mathrm{I}(p)}-N_{\mathrm{II(p)}}\right),\>\,J^{(p)}_{+}=a_{\mathrm{I}(p)}^{\dagger}a_{\mathrm{II}(p)}\>\,\text{
and }\>\,J^{(p)}_{-}=a_{\mathrm{II}(p)}^{\dagger}a_{\mathrm{I}(p)}$ (14)

Formally there is a _phase difference operator_
$\Phi_{(p)}=\left(\Phi^{(p)}_{\mathrm{I}}-\Phi^{(p)}_{\mathrm{II}}\right)$
conjugate to the number difference operator
$N_{(p)}=\frac{1}{2}\left(N_{\mathrm{I}(p)}-N_{\mathrm{II}(p)}\right)$, such
that

$[\Phi_{I(p)}\,,N_{J(p^{\prime})}]=i\delta_{IJ}\delta_{pp^{\prime}},\qquad[\Phi_{(p)},N_{(p^{\prime})}]=i\delta_{pp^{\prime}}$
(15)

In Section 2 we reviewed a construction for the phase operator following [12,
13, 14, 15, 16, 17, 18]. Assuming for the moment that a phase operator with
the desired properties can be constructed, we define the operator
$\frac{1}{\ln p}\,\Phi_{(p)}p^{-N_{(p)}}=\frac{1}{\ln p}\,\Phi_{(p)}e^{-N\ln
p}$ and evaluate the following commutator

$\left[\frac{1}{\ln
p}\,\Phi_{p}\,p^{-N_{(p)}},{p^{\prime}}^{N_{(p^{\prime})}}\right]=i\delta_{pp^{\prime}}$
(16)

using Eq. 15. Thus, the operator $\frac{1}{\ln p}\,\Phi_{(p)}\,{p}^{-N_{(p)}}$
is formally canonically conjugate to $p^{N_{(p)}}=D_{(p)}$.

We would now like to extend it to the large Hilbert space obtained by
combining all primes. Let us first consider all prime numbers up to a fixed
prime $\mathfrak{p}$. The number of such primes is $\pi(\mathfrak{p})$, where
$\pi(x)$ is the prime counting function. We now define

$\mathcal{O}_{\mathfrak{p}}=\frac{1}{\pi(\mathfrak{p})}\sum_{p=2}^{\mathfrak{p}}\frac{1}{\ln
p}\,\Phi_{(p)}\,\mathcal{D}_{\mathfrak{p}}^{-1}\quad\text{ and
}\quad\ln\mathcal{D}_{\mathfrak{p}}={\sum_{p=2}^{\mathfrak{p}}\ln D_{(p)}}$

which are operators in the truncated Hilbert space
$\displaystyle{\bigotimes_{p=2}^{\mathfrak{p}}}L^{2}(p^{-1}\mathbb{Z}_{p})$.
These are canonically conjugate since
$\left[\mathcal{O}_{\mathfrak{p}},\mathcal{D}_{\mathfrak{p}}\right]=i$. Now we
take the limit $\mathfrak{p}\to\infty$ to obtain the canonically conjugate
operators

$\mathcal{O}=\lim_{\mathfrak{p}\to\infty}\mathcal{O}_{\mathfrak{p}},\quad\mathcal{D}=\lim_{\mathfrak{p}\to\infty}\mathcal{D}_{\mathfrak{p}}\quad\text{such
that}\quad\left[\mathcal{O},\mathcal{D}\right]=i$

on the large Hilbert space $\bigotimes_{p}L^{2}(p^{-1}\mathbb{Z}_{p})$. This
limit is analogous to the _thermodynamic limit_ of statistical models, as we
shall see in Section 3.1.

Associated to these operators is the Weyl symmetric product

$\displaystyle\begin{split}\frac{1}{2}\left(\mathcal{D}\mathcal{O}\right.&+\left.\mathcal{O}\mathcal{D}\right)=\lim_{\mathfrak{p}\to\infty}\frac{1}{2}\left(\mathcal{D}_{\mathfrak{p}}\mathcal{O}_{\mathfrak{p}}+\mathcal{O}_{\mathfrak{p}}\mathcal{D}_{\mathfrak{p}}\right)\>=\>\mathcal{O}\mathcal{D}-\frac{i}{2}\\\
&=\lim_{\mathfrak{p}\to\infty}\frac{1}{\pi(\mathfrak{p})}\sum_{p=2}^{\mathfrak{p}}\left(\mathbf{1}\otimes\cdots\otimes
e^{\frac{1}{2}\\!\ln D_{(p)}}\frac{\Phi_{(p)}}{\ln p}e^{-\frac{1}{2}\\!\ln
D_{(p)}}\otimes\mathbf{1}\otimes\cdots\right)\end{split}$ (17)

which is (formally) self-adjoint. In the last line, we have a similarity
transform of the sum of the $\Phi_{(p)}$ operators. As has been emphasised,
e.g. in Ref. [15], the operator canonically conjugate to the number operator
can only be defined up to a similarity transformation. Hence there ought to be
more than one (which could be infinite in number) _total_ phase operators
$\Phi$ canonically conjugate to the _total_ number operator
$\ln\mathcal{D}=\sum_{p}\ln D_{(p)}$. One may follow proposals in the
literature (e.g. [15]) to define $\Phi_{(p)}$, which would result in
$\frac{1}{2}\left(\mathcal{D}_{\mathfrak{p}}\mathcal{O}_{\mathfrak{p}}+\mathcal{O}_{\mathfrak{p}}\mathcal{D}_{\mathfrak{p}}\right)$,
canonically conjugate to $\ln\mathcal{D}_{\mathfrak{p}}$, on the outer product
of a dense subspace of the Hilbert space $L^{2}(p^{-1}\mathbb{Z}_{p})$ at the
$p$-th place. It is worth reiterating that the construction discussed above is
formal. The limit $\mathfrak{p}\to\infty$ is not straightforward. There is a
more convenient way to construct the phase operator over a subspace of the
Hilbert space. We shall attempt to do so in the next two subsections.

###  3.1 Aggregate phase operator for the Riemann zeta function

Let us return to the model of $\mathfrak{su}(2)$ spin in an external field of
Section 2 with the Hamiltonian containing a site dependent magnetic field

$H=-\displaystyle{\sum_{p=2}^{\mathfrak{p}}}B_{p}N_{p}=-\displaystyle{\sum_{p=2}^{\mathfrak{p}}}B_{p}\left(S_{3,p}+j\mathbf{1}\right)$

where we have now chosen an unusual convention222It should, however, be
mentioned that this type of numbering has been used before in [4, 5, 7, 6]. of
using _prime numbers_ $p$ to label the sites, $B_{p}$ are the values of the
magnetic field at site $p$ and we have shifted the zero of the energy for
convenience. The latter amounts to a shift in the spectrum of $S_{p,3}$ by
$S_{p,3}\rightarrow N_{p}=S_{3,p}+j\mathbf{1}$ so that $N_{p}$ takes the
integer value $0,1,\cdots,\mathfrak{n}$.

In this case one can define a phase operator at an individual site, say
$\boldsymbol{\phi}_{p}$ at the $p$-th site, as in Section 2. Each of these
individual operators satisfies the shift covariance relation (or commutator)
for special values of $\beta$

$\displaystyle\big{[}\boldsymbol{\phi}_{p},e^{\beta\sum_{2}^{\mathfrak{p}}B_{p}N_{p}}\big{]}$
$\displaystyle=i\beta e^{\beta\sum_{2}^{\mathfrak{p}}B_{p}N_{p}}$ (18)
$\displaystyle\text{for }\beta=\frac{2\pi
ik}{B_{p}(\mathfrak{n}+1)}\pmod{{2\pi}/{B_{p}}}\>$ $\displaystyle\text{ with
}k=1,\cdots,\mathfrak{n}\text{ and }p=2,\cdots,\mathfrak{p}$

This is valid over the entire Hilbert space, i.e., on an arbitrary state
vector, but only for these special values of $\beta$. Thus there are as many
shift covariant phase operators as the number of sites, and each individual
phase operator is covariant under the specific choices of $\beta$. Moreover,
since each of the Hilbert spaces, labelled by $p$, is finite dimensional, the
trace is a product over traces in each Hilbert space. Hence if we take the
trace of Eq. 18, exactly as in the case of the spin model in Section 2, the
trace of the commutator is zero, therefore,
$\text{Tr}\,e^{\beta\sum_{p}B_{p}N_{p}}=0$. Thus the shift covariance relation
is valid for those values of $\beta$ which also satisfy the zero trace
condition. This relates the zeroes of the partition function to the poles of
the following resolvent operators

$\mathfrak{R}[e^{i\boldsymbol{\phi}_{p}}](\phi)=\left(1-e^{-i\phi}e^{i\boldsymbol{\phi}_{p}}\right)^{-1}$

for all $p$. The trace of the resolvent is

$\sum_{k_{p}=0}^{\mathfrak{n}}\frac{1}{1-e^{-i\phi+i\phi_{k_{p}}}}$

apart from the pole for $k=0$, which yields the trivial commutator.

The similarity between the spin in a magnetic field and the zeta function is
apparent at this stage. (Recall that we have labelled the edges connecting
adjacent sites by the first $\mathfrak{p}$ prime numbers with this objective.)
Indeed, if we choose the local magnetic field $B_{p}=\ln p$, then the
partition function becomes

$Z(\beta)=\prod_{p=2}^{\mathfrak{p}}\bigg{(}\sum_{m_{p}=0}^{\mathfrak{n}}e^{\beta
m_{p}\ln
p}\bigg{)}=\prod_{p=2}^{\mathfrak{p}}\frac{1-p^{\beta(\mathfrak{n}+1)}}{1-p^{\beta}}$

In the thermodynamic limit $\mathfrak{p}\to\infty$, even for finite
$\mathfrak{n}$, the partition function has a simple form in terms of a ratio
of the Riemann zeta functions

$Z(\beta)=\lim_{\mathfrak{p}\to\infty}\,\prod_{p=2}^{\mathfrak{p}}\frac{1-p^{\beta(\mathfrak{n}+1)}}{1-p^{\beta}}=\frac{\zeta(-\beta)}{\zeta\left(-(\mathfrak{n}+1)\beta\right)}$
(19)

Remarkably this has the exact same form as the partition functions of a
$\kappa$-parafermionic primon gas of [5, 7, 6, 10] with
$\kappa=\mathfrak{n}+1$ and $s=-\beta$. It would be interesting to try to
relate the parafermionic variables to the spin degrees of freedom. Notice that
$Z(\beta)$ has zeroes at the non-trivial zeroes of $\zeta(-\beta)$ from the
numerator, as well as at $\beta=-1/(\mathfrak{n}+1)$ from the pole of
$\zeta\left(-(\mathfrak{n}+1)\beta\right)$ from the denominator. The latter is
the only real zero, although it is at an unphysical value of the (inverse)
temperature. However, the trivial zeroes of $\zeta(-\beta)$ are not zeroes of
the partition function. This is due to the fact that at these points, both the
numerator and the denominator have simple zeroes, hence

$\lim_{\beta\to
2n}\frac{\zeta(-\beta)}{\zeta\left(-(\mathfrak{n}+1)\beta\right)}=\text{
finite}$

Thus the nontrivial zeroes are the Fisher zeroes of the spin model in the
complex (inverse temperature) $\beta$-plane. However, since the zeroes of the
Riemann zeta function are believed to be isolated (and since there is no
accumulation point on the real line) these zeroes are not related to any phase
transition. This is consistent as the system of spins in a magnetic field is
not expected to undergo a phase transition. Finally, the partition function
has additional poles from the zeroes of the zeta function in the denominator.

The spectrum of the zeroes of the partition function is then given by

$\sum_{p=2}^{\mathfrak{p}}\sum_{{n_{p}\in\mathbb{Z}}}\sum_{k_{p}=0}^{\mathfrak{n}}\frac{1}{1-e^{i\phi_{k_{p}}+i\frac{2\pi
n_{p}}{\ln p}-i\phi}}-\sum_{p,n_{p}}\frac{1}{1-e^{i\frac{2\pi n_{p}}{\ln
p}-i\phi}}$ (20)

where we have subtracted the pole due to $k=0$. This function may be rewritten
as follows.

$\displaystyle\sum_{p=2}^{\mathfrak{p}}\sum_{n\in\mathbf{Z}}\frac{1}{1-e^{i\left(\frac{2\pi
n}{(\mathfrak{n}+1)\ln
p}-\phi\right)}}-\sum_{p=2}^{\mathfrak{p}}\sum_{n\in\mathbf{Z}}\frac{1}{1-e^{i\left(\frac{2\pi
n}{\ln p}-\phi\right)}}$
$\displaystyle\buildrel\text{sing.}\over{\approx}\sum_{p=2}^{\mathfrak{p}}\sum_{n\in\mathbf{Z}}\frac{-i}{\phi-\frac{2\pi
n}{(\mathfrak{n}+1)\ln
p}}-\sum_{p=2}^{\mathfrak{p}}\sum_{n\in\mathbf{Z}}\frac{-i}{\phi-\frac{2\pi
n}{\ln p}}$
$\displaystyle\approx\sum_{p=2}^{\mathfrak{p}}\frac{d}{d(i\phi)}\ln(1-p^{-(\mathfrak{n}+1)i\phi})-\frac{d}{d(i\phi)}\sum_{p=2}^{\mathfrak{p}}\ln(1-p^{-i\phi})$
$\displaystyle\approx-i\frac{d}{d\phi}\ln\bigg{(}\prod_{p=2}^{\mathfrak{p}}\frac{1-p^{-(\mathfrak{n}+1)i\phi}}{1-p^{-i\phi}}\bigg{)}$

In the above we have used the Mittag-Leffler expansion, assuming analyticity
of the partition function. The expression above, in the limit
$\mathfrak{p}\to\infty$, becomes

$-i\frac{d}{d\phi}\ln\bigg{(}\frac{\zeta(i\phi)}{\zeta\left((\mathfrak{n}+1)i\phi\right)}\bigg{)}$

for $\mathrm{Re}\,(i\phi)>1$.

We will now try to construct a single operator that can be understood as
‘canonically conjugate’ to the Hamiltonian. If we define the total phase
operator as $\boldsymbol{\Phi}=\sum_{p}\boldsymbol{\phi}_{p}$ (which is the
sum of individual phase operators $\boldsymbol{\phi}_{p}$ as defined in
Section 2) it does not, unfortunately, have the desired shift covariance
relation Eq. 4 with the Hamiltonian. This is due to the site dependence of the
magnetic field $B_{p}$, as is apparent from the steps leading to Eq. 4. The
commutator there is obtained only at specific discrete values of $\beta$ which
are integer multiples of $2\pi k^{\prime}/B_{p}(\mathfrak{n}+1)$. Therefore,
unless the magnetic field $B_{p}$ at all the sites are commensurate, which is
certainly not the case for $B_{p}=\ln p$, it is not possible to get the
desired commutator this way.

Instead, we propose to work with an aggregate phase operator
$\boldsymbol{\varphi}$ such that its action on the composite state
$\bigotimes_{p}\big{|}\phi_{p,k_{p}}\big{\rangle}$ is defined to be
$e^{i\hat{\phi}_{p}}$ if the eigenvalue $\phi_{p}\neq 0$, while all other
eigenvalues $\phi_{q\neq p}$ are zero, otherwise it acts as the identity.
Thus, _if two or more of the phases are non-zero_ ,
$e^{i\boldsymbol{\varphi}}=\mathbf{1}$. This may be expressed as

$e^{i\boldsymbol{\varphi}}=\sum_{p=2}^{\mathfrak{p}}e^{i\hat{\phi}_{p}}\prod_{q\neq
p}\delta_{\phi_{q},0}+\frac{2}{n_{\neq 0}(n_{\neq
0}-1)}\sum_{p_{1},p_{2}=1}^{\mathfrak{p}}\prod_{p_{1}\neq
p_{2}}(1-\delta_{\phi_{p_{1}},0})(1-\delta_{\phi_{p_{2}},0})$ (21)

where $n_{\neq 0}=\sum_{p}(1-\delta_{\phi_{p},0})$ is the number of sites
where the phase is non-zero. This is equivalent to projecting on a subspace
$\mathcal{H}^{(1)}$ of the Hilbert space, in which only one, and exactly one,
phase is different from zero333This is analogous, though not exactly
equivalent, to a projection of the Fock space of a quantum field theory of,
say a scalar field, on a subspace with single-particle excitation.. After the
projection, one can use the total phase operator in the subspace
$\boldsymbol{\Phi}|_{\mathcal{H}^{(1)}}=\Pi_{\mathcal{H}^{(1)}}\left(\sum_{p}\Phi_{p}\right)\Pi_{\mathcal{H}^{(1)}}$.
From either point of view, the action of the above is nontrivial on a subspace
of the Hilbert space parametrised by only one of the eigenvalues $\phi_{p}$ at
a time, i.e., on a union of circles $\cup_{p}S^{1}_{(p)}$ while the full
Hilbert space is parametrised by $(S^{1})^{\mathfrak{p}}$. In the complement
of this subspace, it is identity. In this subspace $\mathcal{H}^{(1)}$, we can
follow the steps leading to Eq. 4 to compute the commutator

$\left[\boldsymbol{\varphi},\Pi_{\mathcal{H}^{(1)}}e^{-\beta
H}\Pi_{\mathcal{H}^{(1)}}\right]=i\beta\,\Pi_{\mathcal{H}^{(1)}}e^{-\beta
H}\Pi_{\mathcal{H}^{(1)}}$

which holds in $\mathcal{H}^{(1)}$ for all $\beta=\frac{2\pi
k}{B_{p}(\mathfrak{n}+1)}$ (mod $2\pi/B_{p}$) where $k=1,\cdots,\mathfrak{n}$
and $p=2,\cdots,\mathfrak{p}$. It is worth emphasising that, as in several
examples in quantum theory, the domain of the canonical commutator is not the
entire Hilbert space, but a direct sum of closed orthogonal subspaces[14] of
the type $\mathcal{H}^{(1)}$. In the limit $\mathfrak{p}\to\infty$, one take
the closure of this subspace to obtain a closed subspace of the entire Hilbert
space.

The resolvent function of the exponential of the aggregate phase operator Eq.
21

$\mathfrak{R}[e^{i\boldsymbol{\varphi}}](\phi)=\left(1-e^{-i\phi}e^{i\boldsymbol{\varphi}}\right)^{-1}$
(22)

has the trace

$\displaystyle\mathrm{Tr}\left(\mathfrak{R}[e^{i\boldsymbol{\varphi}}](\phi)\right)$
$\displaystyle=\sum_{p=2}^{\mathfrak{p}}\sum_{k_{1},\cdots,k_{\mathfrak{p}}}\left(\otimes_{i=1}^{\mathfrak{p}}\big{\langle}\phi_{k_{i}}\big{|}\right)e^{-\beta
H}\sum_{n=0}^{\infty}e^{in(\boldsymbol{\varphi}-\phi)}\left(\otimes_{i=1}^{\mathfrak{p}}\big{|}\phi_{k_{i}}\big{\rangle}\right)$
$\displaystyle=\sum_{p=2}^{\mathfrak{p}}\Bigg{(}\sum_{{k_{p}\atop{\text{exactly
one}\atop\phi_{k_{p}}\neq
0}}}\sum_{n=0}e^{in\phi_{k_{p}}}e^{-in\phi}+\sum_{{k_{p}\atop{\text{at least
two}\atop\phi_{k_{p}}\neq 0}}}e^{-in\phi}\Bigg{)}$
$\displaystyle=\,\sum_{p=2}^{\mathfrak{p}}\Bigg{(}\sum_{{k_{p}\atop{\text{exactly
one}\atop\phi_{k_{p}}\neq
0}}}\frac{1}{1-e^{i\phi_{k_{p}}-i\phi}}+\sum_{{k_{p}\atop{\text{at least
two}\atop\phi_{k_{p}}\neq 0}}}\frac{1}{1-e^{-i\phi}}\Bigg{)}$ (23)

Except for the pole at $\phi=0$, this behaviour is in fact identical to that
of the resolvent
$\displaystyle\sum_{p=2}^{\mathfrak{p}}\left(1-e^{-i\phi}e^{i\hat{\phi}_{p}}\right)^{-1}$
in Eq. 20.

Even though we do not require to take the limit $\mathfrak{n}\to\infty$, it is
interesting to note that the phase operator $\boldsymbol{\phi}_{p}$ at the
$p$-th site approaches the phase operator described in [15], which is a
Toeplitz operator [25, 26, 27] in this limit. This has been shown in [17].
This provides a way to understand the relation to the spectrum without a
truncation to a finite $\mathfrak{n}$ case. As shown in [15, 17], each pair of
operators $(\boldsymbol{\phi}_{p},N_{p})$ satisfies the canonical commutation
relation in a subspace $\Omega_{p}$ of the $p$-th Hilbert space
$L^{2}(p^{-1}\mathbb{Z}_{p})$ as follows

$\Omega_{p}=\Big{\\{}|f\rangle_{(p)}=\displaystyle\sum_{n_{p}=0}^{\infty}f_{n_{p}}|n_{p}\rangle_{(p)}:\displaystyle\sum_{n_{p}=0}^{\infty}f_{n_{p}}=0\Big{\\}}$

where, $|n_{p}\rangle_{(p)}$ is an eigenstate of $N_{p}$ corresponding to the
eigenvalue $n_{p}$. Thus, each of the phase operators $\boldsymbol{\phi}_{p}$
satisfies the canonical commutator over a dense subspace of the Hilbert space

$\Big{[}\frac{1}{\ln
p}\,\boldsymbol{\phi}_{p}\,,\sum_{p\in\text{prime}}\\!\ln\mathcal{D}_{(p)}\Big{]}=i,\quad\text{in
}\,\Omega_{p}\,\bigotimes_{p^{\prime}\neq p}\mathcal{H}_{p^{\prime}}$

This is similar to the operator defined earlier, but without the sum
restricted to a finite prime $\mathfrak{p}$ (along with the normalisation
factor $\pi(\mathfrak{p})$ in the denominator). This is due to the fact that
in this case, one gets a contribution from only one of the subspaces (one
prime) at a time. In the next subsection we shall take a similar route to
define another phase operator.

###  3.2 Total phase operator for the Riemann zeta function

Following [15] (see also [17]) we would like to discuss another construction
of the phase operator $\hat{\Phi}$ conjugate to the number operator $\hat{N}$
such that

$\displaystyle\hat{N}|n\rangle$ $\displaystyle=n|n\rangle,\qquad
n=0,1,\cdots,\mathfrak{n}$ $\displaystyle\hat{\Phi}$
$\displaystyle=\sum_{m\neq n}\frac{i\,|m\rangle\langle n|}{m-n}$ (24)

This is a $(\mathfrak{n}+1)\times(\mathfrak{n}+1)$ hermitian Toeplitz matrix
[25, 26, 27]. When applied on a state
$|\mathbf{v}\rangle=\sum_{n}v_{n}|n\rangle$, we find that

$\big{[}\hat{\Phi},\hat{N}\big{]}|\mathbf{v}\rangle=i|\mathbf{v}\rangle\quad\text{if
and only if }\sum_{n=0}^{\mathfrak{n}}v_{n}=0$ (25)

Thus the commutator is valid in a codimension one subspace. For example, we
could choose the $v_{n}$s to be the nontrivial $(\mathfrak{n}+1)$-th roots of
unity. Toeplitz matrices and operators have a long history and have been
studied extensively (see e.g., [27]). Although eigenvalues $k_{0}\leq
k_{1}\leq\cdots\leq k_{\mathfrak{n}}$ and the corresponding eigenvectors
$|k_{m}\rangle$ ($k=0,1,\cdots,\mathfrak{n}$) of the matrix above exist, one
cannot write them explicitly. Moreover, by Szegö’s theorems, the spectrum is
bounded by $\pi$ as $\mathfrak{n}\to\infty$ (so that the matrix size goes to
infinity) and the eigenvalues are distributed uniformly and symmetrically
around zero, as one can also check numerically for small values of
$\mathfrak{n}$.

Coming back to the problem of our interest, in which the Hamiltonian is
$H=\sum_{p}\ln p\,N_{(p)}$, where $N_{(p)}$ is the number operator at the
$p$-th site, which in turn can be expressed in terms of the generalised
Vladimirov derivative. For a natural number $n\in\mathbb{N}$, we use the prime
factorisation444Prime factorisation also plays an important role in the
arithmetic gas models [4, 5, 7, 6]. to associate a vector in
$\otimes_{p}L^{2}(p^{-1}\mathbb{Z}_{p})$ as

$n=\prod_{p}p^{n_{(p)}}\>\longleftrightarrow\>|\mathbf{n}\rangle=\otimes_{p}|n_{(p)}\rangle$
(26)

using the wavelet basis. We emphasise that only a finite number of entries in
the infinite component vector are non-zero integers. Clearly
$|\mathbf{n}\rangle$ is an eigenvector of $H$

$H|\mathbf{n}\rangle=\sum_{p}n_{(p)}\ln p\,|\mathbf{n}\rangle=\ln
n\,|\mathbf{n}\rangle$

Moreover, these states are orthonormal
$\langle\mathbf{n}_{i}|\mathbf{n}_{j}\rangle=\prod_{p}\langle
n^{i}_{(p)}|n^{j}_{(p)}\rangle=\prod_{p}\delta_{n^{i}_{(p)}n^{i}_{(p)}}=\delta_{n_{i}n_{j}}$.

When restricted to a fixed value of $p$, the following definition for the
phase operator

$\frac{i}{\ln p}\sum_{n_{a}^{(p)}\neq
n_{b}^{(p)}}\frac{|\mathbf{n}^{(p)}_{a}\rangle\,\langle\mathbf{n}^{(p)}_{b}|}{(n^{(p)}_{a}-n^{(p)}_{b})}$

on $L^{2}(p^{-1}\mathbb{Z}_{p})$ is natural. This is a Toeplitz matrix,
therefore, it has eigenvectors $|k_{(p)}\rangle$. Let us define the phase
operator on the full space $\otimes_{p}L^{2}(p^{-1}\mathbb{Z}_{p})$
schematically to be of the form

$\Phi_{\text{tot}}\sim\sum_{n_{a}\neq
n_{b}}\frac{i\,|\mathbf{n}_{a}\rangle\langle\mathbf{n}_{b}|}{\ln n_{a}-\ln
n_{b}}=\sum_{\stackrel{{\scriptstyle\text{not
all}}}{{n_{a}^{(p)}=n_{b}^{(p)}}}}\frac{i\left(\otimes_{p_{a}}|\mathbf{n}^{(p_{a})}_{i}\rangle\right)\left(\otimes_{p_{b}}\langle\mathbf{n}^{(p_{b})}_{b}|\right)}{\sum_{p}(n^{(p)}_{a}-n^{(p)}_{b})\ln
p}$

We need to specify the limits of sums over the integers in the above. However,
before we undertake that exercise, we would like to check if $H$ and
$\Phi_{\text{tot}}$ could be a canonically conjugate pair, possibly on a
subspace spanned by vectors of the form in Eq. 26.

To this end, let us now consider a finite linear combination of the form
$|\mathbf{v}\rangle=\sum_{\mathbf{n}}v_{\mathbf{n}}|\mathbf{n}\rangle$, in
which we further require the coefficients to factorise as
$v_{\mathbf{n}}\equiv v_{(n_{(2)},n_{(3)},\cdots)}=\prod_{p}v_{n_{(p)}}$. One
can compute the commutator and verify that on such a state

$\big{[}\Phi_{\text{tot}},H\big{]}|\mathbf{v}\rangle=i|\mathbf{v}\rangle\quad\text{if
and only if }\sum_{n}v_{n}=0$ (27)

where the upper limit of the sum is the maximum integer $n_{\text{max}}$ that
appear in the definition of the vector $|\mathbf{v}\rangle$. Consider all the
vectors $|n\rangle$ that appear in the linear combination in defining
$|\mathbf{v}\rangle$ on which we want to check for the commutator, and the
prime factorisations of the corresponding integers $n$. Let the maximum
$\text{max}_{p}\\{n^{(p)}\\}$ of these be $\mathfrak{n}\in\mathbb{N}$. There
is also a highest prime $\mathfrak{p}$, i.e., above which all
$n^{(p>\mathfrak{p})}=0$ in the factorisations. We can now make the proposal
for the phase operator more precise. It is

$\Phi_{\text{tot}}=\begin{cases}\displaystyle{\sum^{\mathfrak{n}}_{{n_{a}^{(p)},\,n_{b}^{(p)}=0}\atop{\text{not
all
}n_{a}^{(p)}=n_{b}^{(p)}}}\\!\\!\\!\\!\frac{i\left(\otimes_{p_{a}\leq\mathfrak{p}}\big{|}\mathbf{n}^{(p_{a})}_{i}\big{\rangle}\right)\left(\otimes_{p_{b}\leq\mathfrak{p}}\big{\langle}\mathbf{n}^{(p_{b})}_{b}\big{|}\right)}{\sum_{p\leq\mathfrak{p}}(n^{(p)}_{a}-n^{(p)}_{b})\ln
p}}&\mbox{for }n_{a}^{(p)},n_{b}^{(p)}\leq\mathfrak{n}\\\
\displaystyle{\bigotimes_{p>\mathfrak{p}}\big{|}n_{(p)}=0\big{\rangle}\big{\langle}n_{(p)}=0\big{|}}&\mbox{otherwise}\end{cases}$
(28)

and acts on a space spanned by vectors of the form

$|\mathbf{k},\mathfrak{p}\rangle=\left(\otimes_{p\leq\mathfrak{p}}|k_{(p)}\rangle\right)\otimes\left(\otimes_{p^{\prime}>\mathfrak{p}}|n_{(p^{\prime})}=0\rangle\right)$

where at least one $k_{(p)}\neq 0$ for a prime $p\leq\mathfrak{p}$ and for
$p>\mathfrak{p}$, we have chosen the ‘vacuum’ state in the number
representation. In the limit $\mathfrak{p}\to\infty$ (even for finite values
of $\mathfrak{n}$), we expect this to be a well defined Toeplitz operator on
$\otimes_{p}L^{2}(p^{-1}\mathbb{Z}_{p})$. However, we are not able to offer a
rigorous mathematical proof of this assertion.

The phase operator cannot be defined uniquely, it is ambiguous upto a
similarity transform [15]. Given a phase operator $\Phi$, for example, as
defined in Eq. 24, let us consider the operator $\Phi_{\beta}=e^{-\beta N}\Phi
e^{\beta N}$ related by a similarity transform labelled by a parameter
$\beta$. This is would have been a trivial statement had the commutator Eq. 25
been true in the full vector space, however, as we have seen this relation
holds in a subspace of codimension one. It is straightforward to check that
the condition that restricts to the subspace is modified to $\sum_{n}e^{\beta
n}v_{n}=0$ for $\Phi_{\beta}$ to be conjugate to $N$. We may choose
$v_{n}=e^{2\pi im_{1}n/(\mathfrak{n}+1)}$ and $\beta=2\pi
im_{2}/(\mathfrak{n}+1)$ with $m_{1}+m_{2}\neq 0\pmod{\mathfrak{n}+1}$. This
condition is identical to the vanishing of the partition function $Z=\text{Tr
}e^{-\beta H}$ for the Hamiltonian $H=-N$ at these special values of $\beta$.

Now consider $\Phi_{\text{tot},\beta}=e^{-\beta H}\Phi_{\text{tot}}e^{\beta
H}$, the similarity transformation of Eq. 28. The modified condition that
defines the subspace is $\prod_{p}\sum_{n_{(p)}}e^{\beta n_{(p)}\ln
p}v_{n_{(p)}}=0$. If we choose the coefficient
$v_{n_{(p)}}=\chi(p)^{n_{(p)}}$, where $\chi(p)$ is a Dirichlet character (see
Eq. 30), the subspace is defined by the vanishing of

$\prod_{p=2}^{\mathfrak{p}}\sum_{n_{(p)}=0}^{\mathfrak{n}}\left(\chi(p)p^{\beta}\right)^{n_{(p)}}=\prod_{p=2}^{\mathfrak{p}}\frac{1-\chi(p^{\mathfrak{n}+1})p^{\beta(\mathfrak{n}+1)}}{1-\chi(p)p^{\beta}}$
(29)

which, in the limit $\mathfrak{p}\to\infty$ is a ratio of Riemann zeta or
Dirichlet $L$-functions, depending whether the character is trivial or not, as
in Eqs. 19 and 38, respectively. Thus the subspace in which the phase operator
Eq. 28, or its similarity transform, is canonically conjugate to the
Hamiltonian is defined by the vanishing of the Riemann zeta function (at
special values of the inverse temperature $\beta\neq 0$). We have previously
encountered this in Eq. 19 with the aggregate phase operator defined in
Section 3.1. As we see, different choices for the coefficients relate to the
vanishing of Dirichlet $L$-functions, to which we shall now turn our
attention.

##  4 Extension to the Dirichlet $L$-functions

The Riemann zeta function belongs to a family of functions, called the
Dirichlet $L$-functions, that are defined as the analytic continuation of the
Dirichlet series

$L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}=\prod_{p\,\in\,{\mathrm{primes}}}\frac{1}{1-\chi(p)p^{-s}},\qquad\mathrm{Re}(s)>1$
(30)

to the complex $s$-plane. In the above, $\chi_{(}n)$, called the Dirichlet
character, is a homomorphism from the multiplicative group
$G(k)=\left(\mathbb{Z}/k\mathbb{Z}\right)^{*}$ of _invertible_ elements of
$\mathbb{Z}/k\mathbb{Z}$ to $\mathbb{C}^{*}$, which is then extended as a
character for all $\mathbb{Z}$ by setting $\chi(m)=0$ for all $m$ which are
zero $\pmod{k}$ [28]. A Dirichlet character so defined satisfy the following
properties

  1. 1.

For all $m_{1},m_{2}\in\mathbb{Z}$, $\chi(m_{1}m_{2})=\chi(m_{1})\chi(m_{2})$

  2. 2.

$\chi(m)\neq 0$ if and only if $m$ is relatively prime to $k$

  3. 3.

$\chi(m_{1})=\chi(m_{2})$ if $m_{1}\equiv m_{2}$ $\pmod{k}$

Therefore, $\chi$ is a multiplicative character, defined modulo $k$, on the
set of integers. It is this multiplicative property that justifies the sum to
be written as an infinite product in Eq. 30.

There is a trivial character that assigns the value 1 to all integers,
including 0. (This may be taken to correspond to $k=1$.) The Riemann zeta
function corresponds to the choice of the trivial character. In all other
cases, only those integers (respectively, primes), the Dirichlet characters of
which are not zero, contribute to the sum (respectively, the product). This of
course depends on the periodicity $k$ of the character. Therefore, the product
restricts to primes that do not divide $k$

$\prod_{p}\frac{1}{1-\chi_{k}(p)p^{-s}}=\prod_{(p,k)=1}\frac{1}{1-\chi_{k}(p)p^{-s}}=\prod_{(p,k)=1}\sum_{n_{p}=0}^{\infty}\chi_{k}(p)^{n_{p}}p^{-n_{p}s}$

With this understanding, one can define the _inverse_ $\chi^{-1}$ by
restricting to the relevant set of primes. For these primes $p\nmid k$, the
Dirichlet character satisfies $\chi_{k}^{-1}(p)=\chi_{k}^{*}(p)$. (Formally,
for the others, we may take $\chi$ as well as $\chi^{-1}$ to be zero.)

Everything we discussed in the context of the Riemann zeta function in Section
3, including all the caveats, apply to the Dirichlet $L$-functions, with
obvious modifications at appropriate places. The role of the generalised
Vladimidrov derivative, acting on complex valued functions on the $p$-adic
numbers $\mathbb{Q}_{p}$ is played by the generalised Vladimirov derivative
_twisted by the character $\chi$_ [20], be denoted by $D_{(p)\mathfrak{x}}$.
The Kozyrev wavelets are eigenfunctions of these operators for all $\chi$. The
eigenvalues, however, are different and involve the Dirichlet character as
follows

$D_{(p)\mathfrak{x}}\psi_{1-n,m,j}^{(p)}(\xi)=\chi_{k}(p^{n})p^{n}\,\psi_{1-n,m,j}^{(p)}(\xi)$
(31)

We refer to [20] for details of the construction and other properties of these
operators.

The above equation and Eq. 9 lead to the conclusion that $D$ and
$D_{\mathfrak{x}}$ are simultaneously diagonalisable, hence the Kozyrev
wavelets are also eigenfunctions of the _unitary_ operator
$U_{\mathfrak{x}}=D_{\mathfrak{x}}D^{-1}$

$U_{(p)\mathfrak{x}}\psi_{1-n,m,j}^{(p)}(\xi)=D_{(p)\mathfrak{x}}D^{-1}_{(p)}\psi_{1-n,m,j}^{(p)}(\xi)=\chi_{k}(p^{n})\psi_{1-n,m,j}^{(p)}(\xi)$
(32)

We can define its inverse
$U_{(p)\mathfrak{x}}^{-1}=U_{(p)\mathfrak{x}}^{\dagger}=D_{(p)}D^{-1}_{(p)\mathfrak{x}^{*}}$
for those $k$ which do not contain $p$ in its factorisation (otherwise it is
the identity operator). Conversely, when we consider all primes, for a given
$k$, we need to restricted to the set of primes that do not divide $k$, i.e.,
with the formal extension of the inverse given after LABEL:EulerL.

As in the case of the Riemann zeta function, we can combine all the prime
factors to write $L(s,\chi_{k})$ as a trace

$L(s,\chi)=\sum_{{\mathbf{n}}=(n^{(2)},n^{(3)},\cdots)}\\!\\!\\!\\!\big{\langle}{\mathbf{n}}\big{|}\,\mathcal{U}_{\mathfrak{x}}e^{-s\ln\mathcal{D}}\,\big{|}\mathbf{n}\big{\rangle}$

where $\mathcal{U}_{\mathfrak{x}}=\otimes_{p}U_{(p)\mathfrak{x}}$. If we want
to interpret this as the partition function of a statistical mechanical model,
the Hamiltonian is such that

$e^{-\beta
H_{\mathfrak{x}}}\>\longleftrightarrow\>\mathcal{U}_{\mathfrak{x}}e^{-s\ln\mathcal{D}}=\mathcal{D}^{-1}\mathcal{D}_{\mathfrak{x}}\mathcal{D}^{-s}=\mathcal{D}_{\mathfrak{x}}\mathcal{D}^{-s-1}$

which reduces to $e^{-\beta H}\sim\mathcal{D}^{-s}$ corresponding to the
Riemann zeta function in Eq. 13, upto a phase. Now since a non-zero
$\chi_{k}(p)=e^{i\omega_{p}}$ is a root of unity, we can define a new phase
state555This is done by truncating the spectrum to relate with the previous
case.

$|\phi_{k_{p}}^{(\mathfrak{x})}\rangle=\frac{1}{\sqrt{\mathfrak{n}+1}}\sum_{n_{p}=0}^{\mathfrak{n}}e^{-in_{p}(\phi_{k_{p}}\ln
p+\omega_{p})}|n_{p}\rangle$ (33)

which provide an orthonormal set

$\big{\langle}\phi_{k^{\prime}_{p}}^{(\mathfrak{x})}\big{|}\phi_{k_{p}}^{(\mathfrak{x})}\big{\rangle}=\delta_{k_{p},k^{\prime}_{p}}$

One may construct a phase operator

$\hat{\phi}_{(p)\mathfrak{x}}=\sum_{k_{p}}\left(\phi_{k_{p}}+\frac{\omega_{p}}{\ln
p}\right)\big{|}\phi_{k_{p}}^{(\mathfrak{x})}\big{\rangle}\big{\langle}\phi_{k_{p}}^{(\mathfrak{x})}\big{|}\;\equiv\>\sum_{k_{p}}\phi_{k_{p}}^{(\mathfrak{x})}\big{|}\phi_{k_{p}}^{(\mathfrak{x})}\big{\rangle}\big{\langle}\phi_{k_{p}}^{(\mathfrak{x})}\big{|}$
(34)

using the eigenvalues and eigenstates as before.

Now we define the operator $U_{(p)}e^{\beta\ln pN_{p}}$ such that

$U_{(p)}e^{\beta\ln pN_{p}}|n_{p}\rangle=e^{in_{p}\omega_{p}}e^{\beta n_{p}\ln
p}|n_{p}\rangle$ (35)

It follows that

$\displaystyle\hat{\phi}_{(p)\mathfrak{x}}U_{(p)}e^{\beta\ln p\,N_{p}}$
$\displaystyle=U_{(p)}e^{\beta\ln
p\,N_{p}}\,\sum_{k_{p}}\left(\phi_{k_{p}}^{(\mathfrak{x})}+\frac{\omega_{p}}{\ln
p}-i\beta\right)\Big{|}\phi_{k_{p}}^{(\mathfrak{x})}+\frac{\omega_{p}}{\ln
p}-i\beta\Big{\rangle}\Big{\langle}\phi_{k_{p}}^{(\mathfrak{x})}+\frac{\omega_{p}}{\ln
p}-i\beta\Big{|}$ $\displaystyle{}\quad+U_{(p)}e^{\beta\ln
p\,N_{p}}\left(i\beta-\frac{\omega_{p}}{\ln
p}\right)\sum_{k_{p}}\Big{|}\phi_{k_{p}}^{(\mathfrak{x})}+\frac{\omega_{p}}{\ln
p}-i\beta\Big{\rangle}\Big{\langle}\phi_{k_{p}}^{(\mathfrak{x})}+\frac{\omega_{p}}{\ln
p}-i\beta\Big{|}$

This relation can be obtained by the same method used in the earlier sections.
If $i\beta$ takes any of the values $\frac{2\pi k_{p}}{(\mathfrak{n}+1)\ln
p}+\frac{\omega_{p}}{\ln p}$ then in the first term above, one gets the phase
operator Eq. 34, since $\phi_{k_{p}}$ is defined modulo $\frac{2\pi}{\ln p}$.
Hence, as in the case of the Riemann zeta function,

$\Big{[}\hat{\phi}_{(p)\mathfrak{x}},U_{(p)\mathfrak{x}}e^{\beta\ln
p\,N_{p}}\Big{]}=\left(i\beta-\frac{\omega_{p}}{\ln
p}\right)U_{(p)\mathfrak{x}}e^{\beta\ln p\,N_{p}}$ (36)

The definition of the (exponential of the) resolvent is completely analogous
to the case of the Riemann zeta function Eq. 22 — one only needs to substitute
$\hat{\phi}_{p}\to\hat{\phi}_{(p)\mathfrak{x}}$, resulting in the trace

$\sum_{p=2}^{\mathfrak{p}}\Bigg{(}\sum_{{k_{p}\atop{\text{exactly
one}\atop\phi_{k_{p}}\neq
0}}}\frac{1}{1-e^{i\left(\phi_{k_{p}}+\frac{\omega_{p}}{\ln
p}\right)-i\phi}}\;+\sum_{{k_{p}\atop{\text{at least two}\atop\phi_{k_{p}}\neq
0}}}\frac{1}{1-e^{\frac{i\omega_{p}}{\ln p}-i\phi}}\Bigg{)}$

in place of Eq. 23. Once again the poles (apart from that at $\phi=0$)
coincide with the zeroes of the partition function, which is666The unitary
opeartor $\mathcal{U}_{\mathfrak{x},\mathfrak{p}}$ is the product of the
corresponding operators at all sites
$\mathcal{U}_{\mathfrak{x},\mathfrak{p}}=\displaystyle{\prod_{p=1}^{\mathfrak{p}}}U_{(p)\mathfrak{x}}$.

$Z(\beta)=\mathrm{Tr}\left(\mathcal{U}_{\mathfrak{x},\mathfrak{p}}\exp\Big{(}\beta\sum_{p=2}^{\mathfrak{p}}\ln
p\,N_{p}\Big{)}\right)=\prod_{p=2}^{\mathfrak{p}}\frac{1-\chi^{\mathfrak{n}+1}(p)p^{\beta(\mathfrak{n}+1)}}{1-\chi(p)p^{\beta}}$
(37)

where we have used the fact that
$\chi(p^{\mathfrak{n}+1})(p)=\chi^{\mathfrak{n}+1}(p)$ is again a character
with the same periodicity. In the thermodynamic limit $\mathfrak{p}\to\infty$
we get the following ratio of the Dirichlet $L$-functions

$Z(\beta)=\frac{L(-\beta,\chi_{k})}{L(-(\mathfrak{n}+1)\beta,\chi^{\mathfrak{n}+1}_{k})}$
(38)

In the special case where $\mathfrak{n}+1$ is the Euler totient function
$\varphi(k)$ or its integer multiple,
$\chi(p^{\varphi(k)})=\left(\chi(p)\right)^{\varphi(k)}=\chi_{k,0}(p)$, the
principal character, which is 1 if $(p,k)=1$ and 0 otherwise. Except for the
trivial zeroes, the script of the discussions above is very similar to what we
argued for the Riemann zeta function.

In summary, we have proposed to view the Riemann zeta and the Dirichlet
$L$-functions as the partition functions (upto multiplication by a function
that plays no essential role) of _quantum spins_ in magnetic fields, the
values of which depend on the site. We have argued how to make sense of the
phase operator (upto a similarity transformation). The zeroes of the partition
function coincide with the poles of the resolvent function of the exponential
of the aggregate or total phase operators, as discussed in Sections 3.1 and 4.
A different approach to the phase operator was discussed in Section 3.2. Its
relation to the partition function, via similarity transforms, seems to relate
the zeta function and the $L$-functions in the same framework.

Acknowledgements: We thank Surajit Sarkar for collaboration at initial stages
of this work. It is a pleasure to acknowledge useful discussions with Rajendra
Bhatia and Ved Prakash Gupta. We would like to thank Toni Bourama and Wilson
Zùñiga-Galindo for the invitation to write this article.

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