# Carbon-contaminated topological defects in hBN: a potential new class of
single photon emitters

Rohit Babar1,2,∗ Igor A. Abrikosov3 Gergely Barcza1,2 Viktor Ivády2,3,4,∗

###### Abstract

Topological defects, such as Stone-Wales defects and grain boundaries, are
common in 2D materials. In this exploratory study, we investigate the
intricate interplay of carbon contamination and topological defects revealing
a new class of color centers in hexagonal boron nitride. We demonstrate that
carbon contamination and strain can both stabilize Stone-Wales configurations
with desirable optical properties. Inspired by these results, we further
demonstrate that carbon atoms at grain boundaries can resolve energetic B-B
and N-N bonds and give rise to highly favorable atomic structures potentially
leading to the accumulation of carbon contamination at grain boundaries. We
identify carbon-contaminated topological defects that give rise to color
centers emitting in the visible spectral range with short radiative lifetime
and high Debye-Waller factors. Our discoveries shed light on a new class of
defects and pave the way towards the identification of color centers and
single photon emitters in hBN.

Wigner Research Centre for Physics, PO Box 49, H-1525, Budapest, Hungary

MTA–ELTE Lendület ”Momentum” NewQubit Research Group, Pázmány Péter, Sétány
1/A, 1117 Budapest, Hungary

Department of Physics, Chemistry and Biology, Linköping University, SE-581 83
Linköping, Sweden

Department of Physics of Complex Systems, Eötvös Loránd University, Egyetem
tér 1-3, H-1053 Budapest, Hungary

email<EMAIL_ADDRESS><EMAIL_ADDRESS>
August 28, 2024

##  1 Introduction

Wide-bandgap layered semiconductors have recently emerged as a capable
platform for developing integrated photonic devices. [1, 2] Similar to
conventional bulk semiconductors, atomic-scale structural defects in these
materials can implement single photon sources with desirable optical
properties. [3] While layered materials may contain defects and impurities in
high concentrations, exfoliation of the layered structure provides an ultimate
method to control the number of defects in the sample. Thin flakes may contain
emitters in such a low number that individual emitters may be resolved by
confocal microscopy.[4] Furthermore, due to the absence of internal reflection
in thin samples, the photons emitted by the color centers may be collected
with near unity efficiency.[5, 6, 7] These attributes make layered
semiconductors a desirable platform for integrated photonics.

Hexagonal boron nitride (hBN) is famous for its applicable single photon
sources[8, 9] emitting in a broad spectral range starting at the near-infrared
region[10], involving the visible spectrum[11, 12], and finishing at the UV-B
ultraviolet spectral region[13]. It is generally accepted nowadays that these
emitters are related to structural defects of the lattice forming deep levels
in the 6.1 eV band gap of hBN. The emitters that have been reported in the
literature possess a wide variety of properties, some of them exhibit a bright
emission in the zero-phonon line (ZPL)[8], while others emit in a broad
spectral range.

Despite the numerous experimental and theoretical studies[9], the microscopic
structure of most of the emitters is yet to be revealed. A plethora of
native[14, 15, 16] and impurity-related defects[17, 18, 19, 20, 21, 22, 23,
24, 25] has been computationally studied in an attempt to account for the
experiential observations. Except for a few examples[14, 25], no consensus has
been achieved so far regarding the microscopic origin of the emitters. This
raises the question of whether all the relevant defect structures have been
taken into consideration to unambiguously identify the unknown emitters in
hBN.

It is known from studies on graphene that the formation of topological
defects, such as the Stone-Wales (SW) defects[26] and grain boundaries[27] in
the honeycomb lattice of graphene is common. The formation of SW defect in hBN
is, however, less favorable, due to the polarized nature of the B-N bonds.[28]
Indeed, the formation energy of the 5-7 ring SW defect is as high as 7.2 eV
[29, 30, 31] in hBN compared to the 5.1 eV of graphene SW defect [32]. These
results suggest that topological SW defects are less common in hBN. It is
worth mentioning that high-energy irradiation may provide sufficient energy to
the lattice to form various SW defects, see Ref. [33].

The formation of grain boundaries, featuring various types of topological
defects, is governed not only by thermodynamics but also by the conditions of
the growth. In particular, simultaneous nucleation of hBN seeds on the
substrate may lead to hBN grains of different orientations, reversed edges,
and various relative shifts deviating from the periodicity of the hBN lattice.
The interfaces of such grains give rise to various grain boundaries, which
form topological line defects [34, 35]. Grain boundaries in hBN have been
observed by transmission electron (TEM), scanning tunneling (STM) and atomic
force microscopy (AFM)[33, 36, 37, 38, 39, 40, 41, 42] and studied
computationally by density functional theory (DFT).[43, 44, 45] Recently,
quantum emitters from grain boundaries in hBN have been reported.[46, 47, 48,
49, 50, 51] Bright emitters along lines in hBN have been speculated to be
related to SW defects.[46] Furthermore, cathodoluminescence (CL) study reveals
5.46 and 5.63 eV emission lines (D series) localized at extended defects in
hBN.[52, 53] Their origin is attributed to excitons bound to grain boundary,
dislocations [53, 54] or stacking faults[55, 56]. Recently, the presence of PL
and CL emission around 2.3 eV is proposed to originate from an array of
dislocations.[57]

In this article, we show that the formation energy of topological defects can
be reduced by incorporating carbon contamination into the structure. To
demonstrate this, we study a set of carbon-containing Stone-Wales defects
(SW-C), with 5-7 rings and 1-4 carbon contaminants in different charge states.
In addition, we study 10 different types of grain boundaries and numerous
carbon dimer configurations adjacent to them. We show that the formation
energy of SW-C defects and carbon dimers at certain positions of the grain
boundary is significantly reduced. We also show that applying tensile strain
to the hBN sample further favors the formation of SW-C defect. Furthermore, we
study the energy balance of different grain boundary-adatom configurations and
conclude that mobile carbon atoms may accumulate at the grain boundaries and
form stable carbon-contaminated line defects. Finally, we show that the SW-C
defects as well as the favorable carbon dimers at the grain boundary can give
rise to color centers with short excited state lifetime and high Debye-Waller
factor in the visible spectral region.

The rest of the paper is organized as follows. In section II. we provide the
details of the methods used in our first-principles study. Section III.
presents and discusses our computational results. Finally, section IV.
summarizes and concludes the work.

##  2 Methods

Density functional theory (DFT) calculations for single-layer hBN models are
performed using the Vienna Ab-initio Simulation Package (VASP)[58], which
utilizes the projector-augmented wave method[59] and a plane-wave basis set.
The plane-wave and kinetic energy cutoff are set to 450 and 900 eV,
respectively. Exchange-correlation effects of the many-electron system are
described by the generalized gradient approximation of Perdew, Burke, and
Ernzerhof (PBE)[60] and also the screened hybrid functional of Heyd-Scuseria-
Ernzerhof (HSE06)[61] with a modified exact exchange fraction of 0.32 and
screening parameter of 0.2 Å-1. The PBE functional is used in the study of
defect formation energies and grain boundary electronic structures, while the
HSE06 functional is used to obtain ZPL energies and the electronic structure
of point defects. The van der Waals interaction is included using the
Grimme-D3 method[62].

In our study, we consider a 128-atom hexagonal supercell for the SW-C defect
calculations including a vacuum of 18 Å between the periodic hBN layers. The
grain boundaries are modeled in orthorhombic supercells of dimensions 18-20 Å
along with the grain boundary and varying lengths of 30-80 Å in the
perpendicular direction. Following the coincidence site lattice theory,[63]
the periodicity of supercell is retained by including a single, parallel, or
antiparallel grain boundary. The electronic structure of carbon dimer
containing grain boundaries is also verified in hydrogen-terminated ribbon
models for selected cases. The reciprocal space is sampled with
$\Gamma$-point. Atomic positions are relaxed until the forces are less than
0.01 eV/Å. The bond rotation barrier energies are calculated with PBE
functional using the climbing image nudged elastic band method [64, 65].

The excited state properties are obtained using a combination of electronic
and vibrational calculation setups. The ZPL energy is calculated as the energy
difference between ground and excited states, where the excited state
configuration is found using $\Delta$-SCF method[66]. The photoluminescence
spectra and the Debye-Waller factor are calculated with the PyPhotonics
package[67]. Here, the phonon modes are calculated from Phonopy code[68, 69]
using the finite-displacement method. The grain boundary supercell dimensions
are reduced to manage the high computational cost of phonon calculations.

The radiative lifetime is calculated as[70]

$\begin{split}\tau=\frac{3\pi\epsilon_{0}c^{3}{\hbar}^{4}}{nE_{\text{ZPL}}^{3}\mu^{2}}\text{,}\end{split}$
(1)

where $\epsilon_{0}$ is the vacuum permittivity, $c$ is the speed of light,
$n$ is the refractive index of hBN and $\mu$ is the transition dipole moment
of the electronic states corresponding to the ZPL.

##  3 Results and discussions

###  3.1 Formation energy of carbon-containing Stone-Wales defects in hBN

In Fig. 1 we depict the considered Stone-Wales and carbon-containing Stone-
Wales configurations. In a pristine hBN layer, the 5-7 SW defect can be
obtained by ”rotating” a first neighbor B-N pair by 90∘, see Fig. 1(a). The
numbers 5 and 7 indicate the number of vertices of polygons formed by the SW
defect. In addition to the 5-7 SW defect, we investigate various SW-C
structures obtained from carbon substitutional defects by rotating either a
C-N, a C-B, or a C-C bond by 90∘. In Fig. 1(b), we depict the resulting
configurations including either one, two, three, or four carbon atoms. We note
that not all the possible configurations are depicted in Fig. 1. We focus on
those configurations where the carbon atoms are connected and centered around
the shared atoms of the 5 and 7 rings, see Fig. 1. In general, we find these
configurations to be energetically the most favorable ones, see related
results later and in the Supplementary Information. Furthermore, we note that
the SW-C${}_{\text{B}}$, SW-C${}_{\text{N}}$, and SW-C${}_{\text{2}}$
configurations tend to buckle the 2D hBN flake at the defect site. Such out-
of-plane relaxations are not observed for other SW configurations containing
more than two carbon atoms.

Figure 1: Stone-Wales defect in pure and carbon doped hBN. (a) The ideal
honeycomb lattice of hBN transforms to SW defect through a bond rotation
indicated by the curved arrows. (b) SW-C defect counterparts for the most
prevalent carbon clusters in hBN. Green, gray, and brown spheres denote boron,
nitrogen, and carbon atoms respectively.

Next, we compute $F_{\text{SW-C-cluster}}^{\text{rel}}$, the relative
formation energy of the SW-C defects compared to the formation energy of the
corresponding carbon substitutional defect as

$\begin{split}F_{\text{SW-C-cluster}}^{\text{rel}}\left(q\right)=\Delta
E_{\text{SW-C-cluster}}^{\text{tot}}\left(q\right)=\\\
E^{\text{tot}}_{\text{SW-C-
cluster}}\left(q\right)-E^{\text{tot}}_{\text{C-cluster}}\left(q\right)\end{split}$
(2)

where $E^{\text{tot}}_{\text{SW-C-cluster}}\left(q\right)$ and
$E^{\text{tot}}_{\text{C-cluster}}\left(q\right)$ are the total energies of
defective supercells in charge state $q$. The total energy difference is equal
to the difference of the formation energy [71] of the defects assuming the
following requirements are fulfilled. a) The structures contain the same
number of atoms. b) The finite size corrections for the total energy of the
SW-C-cluster and the C-cluster are approximately equal and thus they cancel
out. c) There exists a Fermi energy interval in the band gap of hBN, where
both defects possess the same charge state. The equality holds only in this
interval. Condition a) is trivially fulfilled in our simulations, while
condition b) should be ensured by the large models considered in our
calculations. Condition c) is expected to be fulfilled when the Fermi energy
is in the middle of the bandgap, see discussions of the electronic structure
later. Therefore, we compute the relative formation energy of the SW-C defects
according to Eq. (2).

Figure 2: Formation energy differences of SW-C defects and corresponding
carbon substitutions complexes for different charge states, negative (amber
columns), neutral (light blue columns), and positive (green columns).

As shown in Fig. 2, the relative formation energy of planar SW-C structures
reduces significantly by replacing 1, 2, 3, or 4 boron and nitrogen atoms of
the SW defect. In the neutral charge state, replacing one boron (nitrogen)
with a carbon atom results in a 33 % (27 %) reduction of the relative
formation energy of the SW configurations. Replacing the boron and nitrogen
atoms in between the 5 and 7 atoms rings results in an even larger, 49 %
reduction of the relative formation energy of the SW configurations. Including
three carbon atoms in the SW configuration does not decrease the formation
energy further compared to the SW-C2 configuration. Finally, including 4
carbon atoms in the SW structure considerably lowers the energy differences
compared to the SW-C2 configuration.

To explain our observations, we start with a discussion of the stability of
the SW configuration. In the case of the hBN-SW defect, B-B and N-N bonds are
formed, which are energetically unfavorable. Incorporating carbon impurities
in the structure relaxes the frustrated bonds through the replacement of B-B
and N-N bonds with C-N and C-B bonds. The energy gain for replacing the B and
N atoms shared by the 5 and 7 rings is 2.3 eV and 1.9 eV, respectively. The
carbon substitution at the next neighbor site that resolves only one-half of
the frustrated bond is less favorable in case of SW-C${}_{\text{B}}$ while
such an arrangement is preferred for SW-C${}_{\text{N}}$. When both middle
atoms are replaced by carbon, the energy gain is approximately the sum of
these two contributions, i.e. 3.6 eV. Two carbon atoms can resolve both
energetically unfavorable bonds of the hBN SW defect. The inclusion of an
additional carbon atom does not lower the formation energy further, see Fig.
2. The inclusion of four carbon atoms in the SW defect reduces the formation
energy further.[72] The inclusion of disjoint clusters of carbon atoms at the
periphery of the SW defect does not significantly affect the formation energy,
thus the configurations depicted in Fig. 1 can be considered the most
favorable ones for a given stoichiometry.

Considering the charge state dependence of the relative formation energy of
the SW-C defects, we observe a clear pattern for an odd number of carbon
atoms. When more boron atoms are substituted by carbon than nitrogen, i.e.
SW-C${}_{\text{B}}$ and SW-C${}_{\text{2}}$C${}_{\text{B}}$, the negative
charge further stabilizes the SW configuration, while the positive charges
increase the energy of the SW configuration, see Fig. 2. The opposite is true
for the SW-C defects where more N atoms are replaced by carbon atoms than
boron atoms. For an even number of carbon atoms, no clear relationship between
the charge state and the relative formation energy is observed.

Figure 3: Relative formation energies for SW-C defects under tensile strain.
The SW-C defects are favorable for 4-7 % strain, applied parallel to the SW
bond. Here, $\Delta$E refers to the energy difference between the substituted
carbon cluster and corresponding SW-C configurations. A negative value denotes
the favorable formation of SW-C defects.

The SW configuration introduces highly anisotropic strain in the hBN lattice
which can help to relax external uniaxial stress. Consequently, the formation
of SW configurations is favorable under tension.[26] Therefore, we investigate
the effect of strain on the formation energy of SW, SW-C2,
SW-C${}_{\text{(B)3}}$C${}_{\text{N}}$, and
SW-C${}_{\text{B}}$C${}_{\text{(N)3}}$ configurations. As can be seen in Fig.
3, a strain applied along the axis passing through the shared atoms between
the 5 and 7-atom rings further reduces the relative formation energy of the
SW-C defects. Due to the lower relative formation energy of the SW-C defects
than the hBN SW defect, the SW-C defect may become more favorable than the
corresponding non-topological carbon cluster for an applied strain as large as
4-7%. We observe that the lower the relative formation energy of the SW-C
defects the lower the strain needed to make the SW-C configuration favorable.
Therefore, we anticipate that defected regions in hBN with a high
concentration of carbon atoms may host Stone-Wales structures.

###  3.2 Formation energy of carbon dimers at grain boundaries

Figure 4: (a) Grain boundaries and unique sites for carbon substitution
considered in our study. (b) Favorable sites for carbon dimer at the grain
boundary. The gray dashed line denotes the reference energy for the carbon
dimer away from the respective grain boundary.

The formation of isolated carbon-containing Stone-Wales defects may still be
hindered by the relatively low yet positive formation energy of the
structures. However, the growth of hBN has the potential to give rise to grain
boundaries, inevitably leading to the formation of Stone-Wales-like defects.
In hBN, such grain boundaries comprise of line defects that incorporate
different types of B-N rings, most often 4,5,6,7, and 8-atom rings, in a
certain sequence. Details of the line defect formed depend on the relative
orientation and position of the hBN grains interfacing at the grain boundary.
In this section, we investigate the potential of carbon contamination to
stabilize high energy bonds at grain boundaries, similar to the case of
isolated Stone-Wales defects.

We consider 10 different boundaries as depicted in Fig. 4(a). Most of these
grain boundaries have been observed experimentally either in isolated or
combined form [33, 36, 37, 39, 41] and studied computationally [43, 44, 73].
To distinguish different grain boundaries, we introduce the following
notation: integers $i$, $j$, and $k$ separated by vertical lines, such as
$i|j|k$, define the number of atoms forming the B-N rings at the grain
boundary, while letter pairs in parentheses characterize the bonding (B-B,
B-N, and N-N) at the interface. The examined grain boundaries can be
categorized into four distinct groups based on the sequence of the different
B-N rings, i.e., there are two $4|8$, three $5|5|8$, two $6|6$, and three
$5|7$ type grain boundaries. To accommodate these line defects in periodic
supercells, we use a set of single-layer periodic models and nano-ribbons
terminated with hydrogen atoms at the edges, see Methods for details. It is
worth mentioning that the formation energy of the different types of grain
boundaries can differ significantly, however, their appearance is governed not
only by thermodynamics but also by the nucleation and growth of the BN nano-
flakes. Earlier theoretical studies calculate the grain boundary formation
energies between 0.2-1.4 eV/Å depending on the bonding and chemical
potential.[43, 73, 45, 39] High formation energy boundaries can also be
observed in experiments.[39]

In the following, we consider one type of carbon defect at the grain
boundaries, namely the highly stable carbon dimer.[17, 72] Within each grain
boundary type, we identify a non-exhaustive set of atomic sites close to the
boundary and number them from 1 to 12, see Fig. 4(a). In addition, we consider
an adjacent B-N pair far away from the boundary, referred to as far site
hereinafter (not shown in Fig. 4(a)). We introduce carbon dimers by replacing
adjacent B and N atoms at the numbered sites as well as at the far site. The
carbon dimers are referred to as Cx-Cy, where $x$ and $y$ site indexes are
specified in Fig. 4(a). B-B and N-N first neighbor pairs are not considered in
our study to avoid dependence of the relative formation energy on the chemical
potential of boron, carbon, and nitrogen, which may overwhelm the trends we
are interested in. This way we directly compare the formation energy of the
near-site dimers to the far-site dimer. The far-site carbon dimer is at least
12 Å away from the grain boundary, where the dimer experiences a bulk-like
environment. Due to the large number of models and defect configurations
considered, we use the computationally affordable PBE[74] exchange-correlation
functional in the study of the grain boundaries.

Fig. 4(b) depicts the relative formation energy of various carbon dimers at
different types of grain boundaries compared to the formation energy of the
far-site carbon dimer, which serves as the reference for the energy scale.
Negative (positive) values mean favorable (unfavorable) configurations
compared to the far-site carbon dimer. Our calculations identify several
favorable carbon dimer sites at the grain boundaries. Taking a closer look, we
can observe a similar trend as for the isolated 5-7 Stone-Wales defects, i.e.
replacement of unfavorable B-B and N-N bonds by C-B, C-N, or C-C bond gives
rise to a considerable energy gain. In particular, the C2-C3 and C1-C4 and the
C1-C2 carbon dimers at the $4|8$ (BB,NN) boundary resolve one and two high
energetic bonds in the 4-atom ring and lower the formation energy by $\sim$1.5
eV and 3.1 eV, respectively. A pair of carbon substitutional defects in the
8-ring is less favorable. In contrast, there are no high energy B-B and N-N
bounds at the $4|8$ (BN) boundary, thus the formation energies of boundary-
carbon dimers are either comparable to or slightly higher than the far-site
carbon dimer. Considering the example of $5|5|8$ boundaries, we always find
high energetic bonds, thus we can identify favorable carbon pairs. For
instance, the replacement of any of the B and N atoms at lattice sites 1, 3,
6, and 9 for the $5|5|8$(BN) grain boundary gives rise to a stable
substitutional carbon atom where at least one of the frustrated bonds[28] are
relaxed. In addition to the high-energy-bond-relaxation mechanism, we observe
an additional effect that influences the formation energy. The 5-ring
environment with more N atoms is preferred. Similarly, sites 1 and 2 are
preferred in $5|5|8$ with B-B or N-N bonds at the boundary. In the case of
$6|6$ grain boundaries, we identify only one dimer site at the junction, which
is energetically favorable compared to the far-site dimer, see Fig. 4.
Finally, the complex $5|7$ grain boundary also shows numerous favorable carbon
dimer sites. The energetically most beneficial configurations are those that
resolve either a B-B or N-N bound. The energy difference between the most and
least favored site in $5|5|8$ and $5|7$ grain boundaries are comparable to the
trends observed for the SW-C${}_{\text{2}}$ (see Supplementary Fig. S1) Note
that there are also non-stoichiometric carbon dimer configurations when the
carbon atoms substitute either an adjacent pair of B atoms or an adjacent pair
of N atoms. Since iso-element pairs are unfavorable in BN, carbon
substitutions at these sites could also be highly favorable, however, the
formation energy would depend on the chemical potential of the involved
elements.

Our results clearly show that there are preferred carbon dimer configurations
at grain boundaries, where the most favorable ones are resolving two high-
energy B-B or N-N bonds. The formation energy of the most favorable carbon
dimer configurations in our study is lower by $\sim$3 eV than the bulk-like
configuration. Taking into consideration the 2.2 eV formation energy (obtained
with HSE(0.4) functional) of the carbon dimer in bulk hBN [17], the formation
energy of the most favorable carbon dimers at the grain boundary becomes
negative. This is only possible by lowering the formation energy of the grain
boundary itself. Our results in agreement with Refs. [44, 73] show that carbon
atoms can further stabilize grain boundaries by replacing high-energy bonds.

###  3.3 Kinetic stability and dynamical processes of SW-C defect formation

In order to assess the stability of the SW-C defects, we study the height of
the energy barrier between the substitutional carbon cluster and the
corresponding SW-C configuration. In thermal equilibrium, the probability of
the stable and the metastable configurations is determined by the formation
energy difference of the configurations. At low temperatures, when
$k_{B}T<<E_{b}$, where $E_{b}$ is the reaction barrier between the two
configurations, the atomic configuration may stuck in a higher energy
configuration.

Figure 5: Barrier energy for transformation of substituted C2 defect to SW-C2.

Fig. 5 depicts the reaction pathway and corresponding energy for transforming
a C2 cluster into a SW-C2 configuration. We considered both strained and
unstrained hBN sheets. The energy barrier of the C2 $\rightarrow$ SW-C2
process is found to be 6.3 eV, while the barrier for the reversed process is
2.7 eV. Considering the intermediate steps of the transition, the highest
energy point is associated with the 56∘ rotated and buckled configuration of
the carbon dimer. We conclude that both the carbon clusters and the SW-C
defect are kinetically stable under ambient temperatures, i.e. no transition
is expected between the stable and metastable configurations.

These findings are not in favor of the formation of SW and SW-C defects. On
the other hand, as demonstrated for graphene, there are alternative mechanisms
that can give rise to SW and SW-C defects in hBN. The recombination of
divacancy and a mobile carbon adatom can yield a SW-like defect in graphene
[75], while a nitrogen-containing vacancy can lower the barrier to form SW-N
in graphene[76]. Here, the sliding motion of lattice-carbon atom facilitates
the absorption of migrating atoms. In addition, irradiation may transfer
sufficient amount of energy to the lattice to form SW and SW-C configurations.
Such transition may be more frequent for C clusters than for pristine hBN, due
to the lower formation energy and lower energy barrier of the former. Taking
into account the positive effect of tensile strain on the formation SW-C
defects, such defects may form in higher concentration at strained areas under
irradiation.

According to a recent theoretical work in Ref. [15], carbon and boron
interlayer interstitial atoms are mobile at room temperature. Therefore, they
either diffuse out from the layers or recombine with another defect in
lattice. We study the energetics of various steps of mobile carbon atom-grain
boundary recombination. As an example, we consider the case of $4|8$ (BB,NN)
grain boundary and two carbon adatoms.

Figure 6: Energetics for carbon dimer formation at the 4$|$8(BB,NN) grain
boundary. Colored lines with labels indicate the energy of different
configurations of a system of a single-layer grain boundary and two adatoms.
The number of atoms and electrons are the same in all configurations, thus the
energies are directly comparable. Ca, Ba, and Na label different adatoms,
while $4|8$(BB,NN), $4|8$(C1B,NN), $4|8$(BB,C2N), and $4|8$(C1B,C2N) label
grain boundary configurations with no carbon contamination, one carbon atom
substituting the boron atom at site 1, one carbon atom substituting the
nitrogen atom at site 2, and two carbon atom substituting both the boron atom
at site 1 and the nitrogen atom at site 2, respectively. The adatoms are
placed far away from the grain boundary in all cases. We note that the
energies may be lowered depending on the charge states[77].

Fig. 6 shows the local minima of the configuration energy landscape of a $4|8$
(BB,NN) grain boundary with two carbon adatoms. Going from left to right, the
carbon adatoms replace the boron and nitrogen atoms one by one and on the
right reach an energetically favorable configuration where the two carbon
atoms are incorporated into the hBN layer and a boron and a nitrogen atom are
kicked out. Substituting the boron atom at the grain boundary is energetically
unfavourable, however, the substitution of the nitrogen atom and creation of
the nitrogen adatom is favoured. Importantly, substituting the adjacent B-N
pair at the boundary with carbons, C1-C2 configurations, and creating both
boron and nitrogen adatoms is energetically highly beneficial for the system.
Therefore, we conclude that the incorporation of carbon atoms at the grain
boundaries, even at the cost of creating nitrogen and boron atoms is
preferential, which may lead to the accumulation of carbon atoms and the
formation of carbon-contaminated topological defects at the grain boundaries.

In the case of hBN, the substitution of B-N with C-C pair is observed to be
more favorable than incorporating a single C atom[78]. DFT study shows the
absorption of carbon pair in 5$|$7 grain boundaries can aid the migration and
bond rotation in defected areas[79].

###  3.4 Electronic structure and optical properties

Figure 7: Kohn-Sham electronic structure of various neutral SW-C defects in
hBN. Red and light blue lines depict spin-up and spin-down Kohn-Sham energy
levels between the valence band and the conduction band (gray bars). The
occupation of the defect states is indicated by filled and empty spheres. The
electronic structure is obtained by using HSE(0.32) exchange-correlation
functional.

In the previous sections, we analyzed the formation energy, stability, and
alternatives of formation of carbon-contaminated SW configurations and grain
boundaries. The appearance of the SW-C defects and related configurations may
play a role in defining the electrical, optical, and magnetic properties of
hBN samples. Considering magnetism, we either find singlet, spin-0, or
doublet, spin-1/2, ground state for the SW-C defects and carbon-containing
grain boundaries depending on the parity of the total number of electrons, see
Fig. 7 and Table 1. For carbon dimers in the SW-C and grain boundary
configurations, we obtain a spin-singlet ground state due to the even number
of electrons in the neutral charge state, see Figs. 7 and 8. Since none of the
studied configurations possess a high-spin ground state, we anticipate that
these defects cannot implement spin qubits in hBN. On the other hand, we find
both occupied and unoccupied defect states in the bandgap of hBN for the SW-C
defects, see Fig. 7, as well as for the energetically favorable carbon-
contaminated grain boundary configurations, see Fig. 8.

Figure 8: PBE Kohn-Sham electronic structure of different highly stable grain
boundary carbon dimer configurations in the neutral charge state in hBN.
Colored lines with filled and empty circles indicate occupied and unoccupied
Kohn-Sham energy levels in the bandgap of hBN. The valance band maximum and
the conduction band minimum of pristine hBN are depicted by gray bands. Black
lines denote electronic states localized on carbon atoms. The colored (amber
and light blue) bars indicate the position of the grain boundary states when
no carbon contamination is included in the model. The electronic structures
are obtained by using the PBE exchange-correlation functional, which
underestimates the energy gaps. Nevertheless, the PBE results are
qualitatively valid.

In the neutral charge state, most of the configurations exhibit occupied
defect states in the lower half of the bandgap and unoccupied states in the
upper half of the bandgap. Similar trends are observed in C-SW defects in hBN
and nanotubes [80, 81, 31]. We anticipate from these observations that the
neutral charge state of most of the SW-C and carbon dimer containing grain
boundary defect configurations is stable when the Fermi energy is in the
middle of the band gap of hBN.

It is worth noting that the bandgap of grain boundary models is significantly
reduced compared to pristine hBN, due to the quasi-localized grain boundary
states. The 4.8 eV PBE band-gap of hBN reduces to 2.6-4.2 eV, in particular
4$|$8(BB,NN): 2.8 eV, 4$|$8(BN): 4.2 eV; $5|5|8$(BB): 2.6 eV; $5|5|8$(NN): 3.4
eV; $5|5|8$(BN): 2.7 eV; $6|6$(BB): 2.9 eV; $6|6$(NN): 2.9 eV; $5|7$(BB): 3.8
eV; $5|7$(NN): 3.9 eV; $5|7$(BB,NN): 3.1 eV, see colored shaded areas in Fig.
8. Since both the conduction and the valence band edges are affected by the
grain boundary states, grain boundaries can act as charge carrier traps.
Furthermore, we notice in the calculations that the energy of the grain
boundary states is often altered by the carbon dimer, which indicates strong
hybridization between the states of the line and point defects.

In Figs. 7 and 8, all the electronic structures exhibit at least one occupied
as well as at least one unoccupied defect state in the band gap of either hBN
or the considered grain boundary configuration. These results suggest that the
SW-C configurations and carbon dimers at the grain boundary can give rise to
color centers in the visible spectral range. In low defect concentrations,
they may behave like single photon emitters in hBN.

Table 1:  Ground state spin, ZPL energy, excitation lifetime of the considered SW-C defects in hBN. The ZPLs are obtained on HSE(0.32) level of theory. Defect | Spin | ZPL (eV) |  $\tau$ (ns) | DW
---|---|---|---|---
SW-C${}_{\text{B}}$ | 1/2 | 1.96 | 59.1 |
SW-C${}_{\text{N}}$ | 1/2 | 1.17 | 1.6 $\times 10^{3}$ |
SW-C${}_{\text{2}}$ | 0 | 3.30 | 0.7 | 0.18
SW-C${}_{\text{2}}$C${}_{\text{B}}$ | 1/2 | 2.12 | 16.8 |
SW-C${}_{\text{2}}$C${}_{\text{N}}$ | 1/2 | 2.74 | 9.9 |
SW-C${}_{\text{(B)3}}$C${}_{\text{N}}$ | 0 | 2.03 | 37.0 | 0.09
SW-C${}_{\text{B}}$C${}_{\text{(N)3}}$ | 0 | 1.92 | 37.2 | 0.15

To further investigate this possibility, we first calculate the zero-phonon
line (ZPL) energy of different SW-C configurations in hBN using HSE(0.32)
functional, which are summarized in Table 1. We find that the SW-C2 defect
emits at the highest energy, around 3.3 eV, while the most favorable four
carbon-containing configurations are around 2 eV. The estimated error margin
of the ZPL values is expected about 0.1-0.2 eV in our calculations. In order
to estimate the brightness of the SW-C color centers, we calculate the
radiative lifetime of the SW-C${}_{\text{2}}$,
SW-C${}_{\text{(B)3}}$C${}_{\text{N}}$, and
SW-C${}_{\text{(B)}}$C${}_{\text{(N)3}}$ configurations. We respectively
obtain 0.7 ns, 37.0 ns, and 37.2 ns for the optical lifetime of these defects.
Due to the relatively large ZPL and the presumably absent alternative spin
shelving states, we expect high quantum efficiency and bright visible emission
for these defects.

Figure 9:  Photoluminescence spectra and phonon energy spectral function for
select SW-C defects. Upper panel denotes the normalized PL spectra for
SW-C${}_{\text{2}}$ (orange), SW-C${}_{\text{(B)3}}$C${}_{\text{N}}$ (green)
and SW-C${}_{\text{(B)}}$C${}_{\text{(N)3}}$ (blue) defects while the lower
panel shows the spectral function S for phonon modes contributing to the PL
spectra. The unequal carbon bonds of SW-C${}_{\text{(B)3}}$C${}_{\text{N}}$
result in two phonon sidebands at 171 and 210 meV.

Accordingly, we determine their PL spectra, depicted in Fig. 9. As can be
seen, the C-C phonon modes are similar in nature to the carbon substitutional
defects albeit with energy shifts corresponding to the altered bond strengths
in SW configuration. The Debye-Waller factors for SW-C${}_{\text{2}}$ and
SW-C${}_{\text{(B)}}$C${}_{\text{(N)3}}$ are found to be comparable to the
values predicted for hBN SW and carbon substitutional defects.[17, 16, 82]

Table 2:  ZPL energy, excitation lifetime of the highly stable carbon dimers at $4|8$ grain boundaries in hBN. The ZPLs are obtained on HSE(0.32) level of theory. Grain boundary | Configuration | ZPL (eV) |  $\tau$ (ns) | DW
---|---|---|---|---
$4|8$(BB,NN) | C1-C2 | 2.14 | 22.3 | 0.02
$4|8$(BN) | C3-C4 | 3.47 | 15.3 | 0.24

Figure 10:  Photoluminescence spectra and phonon energy spectral function for
carbon dimers at grain boundaries $4|8$(BB,NN) (pink) and $4|8$(BN) (blue).

We also explore the optical properties of grain boundaries which largely
depend on the local bonding at the boundary. Owing to computational demand, we
consider only a few highly favorable carbon dimer-containing grain boundary
configurations as illustration, see Table 2. We find that the studied
configurations emit in a similar spectral range as the SW-C defects. The
corresponding radiative lifetime for the most stable carbon dimers in
$4|8$(BB,NN) and $4|8$(BN) is 22.3 ns and 15.3 ns respectively. As shown in
Fig. 10 and in Table 2, the C1-C2 carbon dimer in $4|8$(BB,NN) grain boundary
emits in the visible range with a predominant phonon contribution to the PL
spectra. In contrast, the UV active C3-C4 defect hosted in the $4|8$(BN) grain
boundary exhibits a significant Debye-Waller factor (0.24). Therefore, we
expect other grain boundaries to host bright color centers as well. Point
defects may act as a recombination center for excitons and separated electron
and hole pairs captured by the quasi-localized states of the grain boundary.
Realizing charge carrier injection in such structures may lead to
electroluminescence.[83, 84]

From the computed optical properties of the SW-C defects and carbon-
contaminated grain boundaries, we conclude that some of these defects are
particularly interesting for quantum photonic applications due to their short
lifetime and dominating ZPL emission.

##  4 Summary

In this article, we demonstrated that carbon contamination can play an
important role in stabilizing topological defects, such as Stone-Wales defects
and grain boundaries in hBN. Carbon Stone-Wales may appear in increased
concentration in strained areas of irradiated hBN, while carbon impurities may
accumulate at grain boundaries and form a series of highly stable carbon-
containing defect structures. We also showed that the studied defects can give
rise to color centers in hBN emitting in the visible spectral range with
desirable optical properties.

The presented novel class of defects not only broadens our understanding of
hBN’s defect landscape but also points to a potential avenue for identifying
previously unknown hBN color centers. Ultimately, our findings may pave the
way toward the controlled fabrication of visible emitters in hBN, opening new
possibilities for applications in quantum technologies and optoelectronics.

##  Data availability

The main data supporting the findings of this study are available within the
paper and its Supplementary Information. Further numerical data are available
from the authors upon reasonable request.

## Acknowledgments

This research was supported by the National Research, Development, and
Innovation Office of Hungary within the Quantum Information National
Laboratory of Hungary (Grant No. 2022-2.1.1-NL-2022-00004) and within grants
FK 135496 and FK 145395. V.I. also appreciates the support from the Knut and
Alice Wallenberg Foundation through WBSQD2 project (Grant No. 2018.0071). The
calculations were performed on resources provided by the Swedish National
Infrastructure for Computing (SNIC) at the National Supercomputer Centre
(NSC). We acknowledge KIFÜ for awarding us access to computational resources
based in Hungary.

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Variance of voltages in a lattice Coulomb gas

Diana Conache111Technische Universität München, Zentrum Mathematik, Bereich
M5, D-85747 Garching bei München, Germany. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>Markus Heydenreich22footnotemark: 2 Franz
Merkl222Mathematical Institute, Ludwig-Maximilians-Universität München,
Theresienstr. 39, D-80333 Munich, Germany. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>Silke W.W. Rolles11footnotemark: 1

###### Abstract

We study the behavior of the variance of the difference of energies for
putting an additional electric unit charge at two different locations in the
two-dimensional lattice Coulomb gas in the high-temperature regime. For this,
we exploit the duality between this model and a discrete Gaussian model. Our
estimates follow from a spontaneous symmetry breaking in the latter model.
Keywords: Spontaneous symmetry breaking, Poisson summation, lattice Coulomb
gas.
MSC 2020: Primary 82B20, Secondary 60K35.

##  1 Introduction

The Poisson summation formula is by now a standard tool in Statistical
Mechanics for proving duality between models in the high and low-temperature
regime. It was first used by McKean [14] to prove the Kramers-Wannier duality
for the infinite two-dimensional Ising Model. Further results for other models
followed. Gruber and Hintermann [9] showed that for arbitrary lattice spin
systems one can relate the high and low temperature expansions by means of the
Poisson summation formula. Several overview papers of the results on duality
appeared in the beginning of the 1980’s both from a mathematical perspective
[13] and from a physics one [15].

In this note, we make use of the Poisson summation formula to show the duality
between a two-dimensional discrete Gaussian model with pinning at the origin
at low temperature and a lattice Coulomb gas at high temperature. Such lattice
Coulomb gas representations were previously treated in [10]. Expected squared
height differences in the discrete Gaussian model correspond by duality to
variances of voltages in the lattice Coulomb gas. These voltages are highly
non-local observables in terms of the charge distribution, so that known
results on Debye screening [2, 3, 4] do not apply to them. In Corollary 7 we
give a proof of the spontaneous breaking of the internal ${\mathbb{Z}}^{1}$
symmetry in the discrete Gaussian model via a Peierls argument on the torus,
taking also care of contours with non-zero homology. This comes in contrast to
a continuous version of the model, where the symmetry cannot be broken, even
for more general models due to the Mermin-Wagner theorem. We then employ this
symmetry breaking to show that the variance of the energy cost for
transporting a unit test charge between two given sites $i$ and $j$ of a
lattice, i.e. the variance of the voltage between $i$ and $j$, is
asymptotically proportional to $(\beta^{*})^{-1}\log|i-j|$ for small inverse
temperature $\beta^{*}$. We make this precise in Theorem 8 below.

The duality between the discrete Gaussian model and the lattice Coulomb gas
was also discussed by Fröhlich and Spencer in [6]; see also earlier work on
the Kosterlitz-Thouless transition by [11, 17]. A discussion on the connection
to the results in [6] is included in the last section. There we also contrast
our results to Debye screening proven in three space dimensions by Brydges and
Federbush [2, 3].

##  2 The two models

###  2.1 The discrete Gaussian model

Let $\Lambda\Subset{\mathbb{Z}}^{2}$ be a finite square box with periodic
boundary conditions containing the origin and let $\mathcal{E}_{\Lambda}$
denote the set of undirected edges between nearest-neighbor points in
$\Lambda$. Consider the space of integer-valued configurations which are
pinned at the origin:

$\displaystyle\Omega_{\Lambda}=\\{x\in{\mathbb{Z}}^{\Lambda}:x_{0}=0\\}.$ (1)

We consider the Hamiltonian

$\displaystyle
H_{\Lambda}(x)=\sum_{\\{i,j\\}\in\mathcal{E}_{\Lambda}}(x_{i}-x_{j})^{2},\quad
x\in\Omega_{\Lambda}.$ (2)

Using the lattice Laplacian $\Delta=(\Delta_{ij})_{i,j\in\Lambda}$,

$\displaystyle\Delta_{ij}=\left\\{\begin{array}[]{ll}1&\text{if }i\sim j,\\\
-4&\text{if }i=j,\\\ 0&\text{else, }\end{array}\right.$ (6)

we can rewrite it as

$\displaystyle H_{\Lambda}(x)=-\sum_{i,j\in\Lambda}x_{i}\Delta_{ij}x_{j},\quad
x\in\Omega_{\Lambda}.$ (7)

The partition sum and the corresponding Gibbs measure of the discrete Gaussian
model with pinning at the origin at given inverse temperature $\beta>0$ are
given by

$\displaystyle
Z_{\Lambda,\beta}=\sum_{x\in\Omega_{\Lambda}}{\mathrm{e}}^{-\beta
H_{\Lambda}(x)}\quad\text{ and }\quad
P_{\Lambda,\beta}=\frac{1}{Z_{\Lambda,\beta}}\sum_{x\in\Omega_{\Lambda}}{\mathrm{e}}^{-\beta
H_{\Lambda}(x)}\delta_{x},$ (8)

respectively; here $\delta_{x}$ denotes the Dirac measure in $x$.

###  2.2 The lattice Coulomb gas

Consider the Fourier Transform of a given integrable function $f$ on
${\mathbb{R}}^{n}$:

$\displaystyle\hat{f}(k)=\int_{{\mathbb{R}}^{n}}{\mathrm{e}}^{-{\mathrm{i}}k\cdot
x}f(x){\mathrm{d}}x,\qquad k\in{\mathbb{R}}^{n}.$ (9)

#### Poisson Summation Formula.

We recall a well-known duality result from Fourier Analysis.

######  Fact 1 (Poisson Summation Formula, [8, Theorem 3.2.8]).

Let $n\in{\mathbb{N}}$ and let $f$ be a continuous function on
${\mathbb{R}}^{n}$ which satisfies for some $C,\delta>0$ and for all
$x\in{\mathbb{R}}^{n}$

$\displaystyle|f(x)|\leq C(1+|x|)^{-n-\delta},$ (10)

and whose Fourier transform $\hat{f}$ restricted to $2\pi{\mathbb{Z}}^{n}$
satisfies

$\displaystyle\sum_{k\in 2\pi{\mathbb{Z}}^{n}}|\hat{f}(k)|<\infty.$ (11)

Then for all $y\in{\mathbb{R}}^{n}$ we have

$\displaystyle\sum_{k\in
2\pi{\mathbb{Z}}^{n}}\hat{f}(k){\mathrm{e}}^{{\mathrm{i}}k\cdot
y}=\sum_{x\in{\mathbb{Z}}^{n}}f(x+y),$ (12)

and in particular

$\displaystyle\sum_{k\in
2\pi{\mathbb{Z}}^{n}}\hat{f}(k)=\sum_{x\in{\mathbb{Z}}^{n}}f(x).$ (13)

In order to show the duality between the discrete Gaussian model and the
lattice Coulomb gas, we make use of the Poisson summation formula by applying
it to the Boltzmann factor of the first model. Note that, for a positive
definite matrix $A\in{\mathbb{R}}^{n\times n}$ and $g(x)=\exp(-x^{t}Ax)$, one
has

$\displaystyle\hat{g}(k)=\frac{\pi^{\frac{n}{2}}}{\sqrt{\det
A}}{\mathrm{e}}^{-\frac{1}{4}k^{t}A^{-1}k}.$ (14)

Since $g$ satisfies the hypotheses of Fact 1, formulas (12) - (13) hold for
$g$.

#### Green’s function.

Let $P_{i}$ be the law of a simple random walk
$(X_{t})_{t\in{\mathbb{N}}_{0}}$ on $\Lambda$ started at $i\in\Lambda$. For
$i,j\in\Lambda$, we define the Green’s function

$\displaystyle G_{ij}=\lim_{\lambda\downarrow
0}\sum_{t\in{\mathbb{N}}_{0}}\left(P_{i}(X_{t}=j)-\frac{1}{|\Lambda|}\right){\mathrm{e}}^{-\lambda
t}.$ (15)

The limit exists and is finite. To see this, combine odd and even times in
pairs of two and use that $P_{i}(X_{2t}=j)+P_{i}(X_{2t+1}=j)$ converges
exponentially fast to $2/|\Lambda|$ as $t\to\infty$.

######  Lemma 2.

For $j,k\in\Lambda$, we have

$\displaystyle-\frac{1}{4}\sum_{i\in\Lambda}\Delta_{ki}G_{ij}=\delta_{kj}-\frac{1}{|\Lambda|}.$
(16)

###### Proof.

For all $k,i\in\Lambda$, we have

$\displaystyle\Delta_{ki}=4(P_{k}(X_{1}=i)-\delta_{ki}).$ (17)

Therefore, for $j,k\in\Lambda$, it follows

$\displaystyle-\frac{1}{4}\sum_{i\in\Lambda}\Delta_{ki}G_{ij}=-\sum_{i\in\Lambda}(P_{k}(X_{1}=i)-\delta_{ki})G_{ij}=G_{kj}-\sum_{i\in\Lambda}P_{k}(X_{1}=i)G_{ij}$
$\displaystyle=$ $\displaystyle\lim_{\lambda\downarrow
0}\sum_{t\in{\mathbb{N}}_{0}}\left(P_{k}(X_{t}=j)-\frac{1}{|\Lambda|}\right){\mathrm{e}}^{-\lambda
t}-\sum_{i\in\Lambda}P_{k}(X_{1}=i)\lim_{\lambda\downarrow
0}\sum_{t\in{\mathbb{N}}_{0}}\left(P_{i}(X_{t}=j)-\frac{1}{|\Lambda|}\right){\mathrm{e}}^{-\lambda
t}$ $\displaystyle=$ $\displaystyle\lim_{\lambda\downarrow
0}\sum_{t\in{\mathbb{N}}_{0}}\left(P_{k}(X_{t}=j)-\frac{1}{|\Lambda|}\right){\mathrm{e}}^{-\lambda
t}-\lim_{\lambda\downarrow
0}\sum_{t\in{\mathbb{N}}_{0}}\left(P_{k}(X_{t+1}=j)-\frac{1}{|\Lambda|}\right){\mathrm{e}}^{-\lambda
t}{\mathrm{e}}^{-\lambda}$ $\displaystyle=$ $\displaystyle
P_{k}(X_{0}=j)-\frac{1}{|\Lambda|}=\delta_{kj}-\frac{1}{|\Lambda|}.$ (18)

For the third line we used the Markov property and introduced an extra factor
${\mathrm{e}}^{-\lambda}$, which converges to $1$ as $\lambda\downarrow 0$. ∎

The following identity also has a continuous analogue for the Green’s function
of a randomly shifted Gaussian Free Field. More precisely, see Remark 8.20 and
Exercise 8.7 in [5] for the continuum Gaussian Free Field and Exercise 1.4. in
[1] for the discrete Gaussian Free Field.

######  Lemma 3.

For $i,j\in\Lambda\setminus\\{0\\}$, we have that the inverse of the matrix
$\Delta_{0^{c}0^{c}}=(\Delta_{ij})_{i,j\in\Lambda\setminus\\{0\\}}$ is given
by

$\displaystyle-4((\Delta_{0^{c}0^{c}})^{-1})_{ij}=G_{ij}-G_{i0}-G_{0j}+G_{00}.$
(19)

###### Proof.

Since

$\displaystyle\sum_{i\in\Lambda}\Delta_{ki}=0$ (20)

for all $k\in\Lambda$, we calculate for $j,k\in\Lambda\setminus\\{0\\}$, using
Lemma 2 in the third step,

$\displaystyle\sum_{i\in\Lambda\setminus\\{0\\}}\Delta_{ki}(G_{ij}-G_{i0}-G_{0j}+G_{00})=\sum_{i\in\Lambda}\Delta_{ki}(G_{ij}-G_{i0}-G_{0j}+G_{00})$
$\displaystyle=$
$\displaystyle\sum_{i\in\Lambda}\Delta_{ki}(G_{ij}-G_{i0})=-4\left(\delta_{kj}-\frac{1}{|\Lambda|}\right)+4\left(\delta_{k0}-\frac{1}{|\Lambda|}\right)=-4\delta_{kj}.$
(21)

∎

#### Duality between the discrete Gaussian model and the lattice Coulomb gas.

The matrix $\Delta_{0^{c}0^{c}}=(\Delta_{ij})_{i,j\in\Lambda\setminus\\{0\\}}$
is negative definite. Applying the Poisson summation formula (13) to
$g(x)=\exp(-x^{t}Ax)$ with $n=|\Lambda|-1$ and $A=-\beta\Delta_{0^{c}0^{c}}$,
we connect the partition sum $Z_{\Lambda,\beta}$ of the discrete Gaussian
model defined in (8) with the partition sum

$Z^{*}_{\Lambda,\beta^{*}}:=\sum_{k\in
2\pi{\mathbb{Z}}^{\Lambda\setminus\\{0\\}}}{\mathrm{e}}^{\beta^{*}k^{t}(\Delta_{0^{c}0^{c}})^{-1}k}$
(22)

of the lattice Coulomb gas at inverse temperature $\beta^{*}:=(4\beta)^{-1}$
by the following duality equation

$\displaystyle
Z_{\Lambda,\beta}=\sum_{x\in{\mathbb{Z}}^{\Lambda\setminus\\{0\\}}}{\mathrm{e}}^{-\beta
x^{t}(-\Delta_{0^{c}0^{c}})x}=\left(\frac{\pi}{\beta}\right)^{\frac{|\Lambda|-1}{2}}\det(-\Delta_{0^{c}0^{c}})^{-\frac{1}{2}}Z^{*}_{\Lambda,\beta^{*}}.$
(23)

Let
$\mathcal{K}_{\Lambda}:=\\{k\in{\mathbb{R}}^{\Lambda}:\,\sum_{i\in\Lambda}k_{i}=0\\}$,
which we view as dual to
$\Omega_{\Lambda}^{\mathbb{R}}:=\\{x\in{\mathbb{R}}^{\Lambda}:x_{0}=0\\}$.
Define
$\Omega_{\Lambda}^{*}:=2\pi{\mathbb{Z}}^{\Lambda}\cap\mathcal{K}_{\Lambda}$.
The Gibbs measure of the lattice Coulomb gas on $\Omega_{\Lambda}^{*}$ is
given by

$P_{\Lambda,\beta^{*}}^{*}:=\frac{1}{Z_{\Lambda,\beta^{*}}^{*}}\sum_{k\in
2\pi{\mathbb{Z}}^{\Lambda\setminus\\{0\\}}}{\mathrm{e}}^{\beta^{*}k^{t}(\Delta_{0^{c}0^{c}})^{-1}k}\delta_{k}.$
(24)

######  Corollary 4 (The dual lattice Coulomb gas in terms of the Green’s
function).

For all $k\in\mathcal{K}_{\Lambda}$, writing
$k_{0^{c}}=(k_{i})_{i\in\Lambda\setminus\\{0\\}}$, one has

$\displaystyle-4k_{0^{c}}^{t}(\Delta_{0^{c}0^{c}})^{-1}k_{0^{c}}=k^{t}Gk.$
(25)

In particular,

$\displaystyle
Z^{*}_{\Lambda,\beta^{*}}=\sum_{k\in\Omega_{\Lambda}^{*}}{\mathrm{e}}^{-\frac{\beta^{*}}{4}k^{t}Gk}\quad\text{
and }\quad
P_{\Lambda,\beta^{*}}^{*}=\frac{1}{Z_{\Lambda,\beta^{*}}^{*}}\sum_{k\in\Omega_{\Lambda}^{*}}{\mathrm{e}}^{-\frac{\beta^{*}}{4}k^{t}Gk}\delta_{k}.$
(26)

###### Proof.

Let $\mathbf{1}=(1)_{i\in\Lambda\setminus\\{0\\}}$ denote the column vector
consisting of ones. For $k\in\mathcal{K}_{\Lambda}$ we have by Lemma 3

$\displaystyle k^{t}Gk$
$\displaystyle=k_{0^{c}}^{t}G_{0^{c}0^{c}}k_{0^{c}}+k_{0^{c}}^{t}G_{0^{c}0}k_{0}+k_{0}G_{00^{c}}k_{0^{c}}+k_{0}G_{00}k_{0}$
$\displaystyle=k_{0^{c}}^{t}(G_{0^{c}0^{c}}-G_{0^{c}0}\mathbf{1}^{t}-\mathbf{1}G_{00^{c}}+\mathbf{1}G_{00}\mathbf{1}^{t})k_{0^{c}}=-4k_{0^{c}}^{t}(\Delta_{0^{c}0^{c}})^{-1}k_{0^{c}}.$
(27)

Substituting this in (22) and (24), the second claim also follows. ∎

##  3 Peierls argument and breaking of symmetry in the discrete Gaussian model

The following results follow the lines of Section 6.3 in [7], where a thorough
discussion regarding the symmetries and the ground states of the discrete
Gaussian model (without pinning) based on [16] is given. While Peierls
arguments on planar domains are classical, the treatment of non-zero
homologies are less common.

We construct a contour model as follows. Let $\Lambda^{*}:=\Lambda+(1/2,1/2)$
be the dual box with periodic boundary conditions and
$\mathcal{E}_{\Lambda^{*}}$ the set of edges between nearest-neighbor points
in $\Lambda^{*}$. Let $x\in\Omega_{\Lambda}$ and
$\\{i,j\\}\in\mathcal{E}_{\Lambda}$. There exists a unique
$\\{i,j\\}^{*}\in\mathcal{E}_{\Lambda^{*}}$ intersecting $\\{i,j\\}$. Then, if
$x_{i}-x_{j}=n\in{\mathbb{N}}$, we draw $n$ distinct arrows on $\\{i,j\\}^{*}$
such that when one looks in the direction of the arrows, the vertex $i$ is to
the left and the vertex $j$ to the right. In other words, the larger value is
to the left of the arrow and the smaller one to the right. For a dual edge $e$
directed from dual vertex $e_{-}$ to $e_{+}$, we denote its direction vector
by $\vec{e}:=e_{+}-e_{-}$.

Take now two different vertices $i,j\in\Lambda$ and fix a configuration $x$ in
the set $\Omega_{i,j}:=\\{x\in\Omega_{\Lambda}:x_{i}>x_{j}\\}$. Consider the
set $M_{i}:=\\{k\in\Lambda:x_{k}\geq x_{i}\\}$. Let $C_{i}$ be the connected
component of $M_{i}$ containing $i$. Let $\partial C_{i}$ be its boundary,
viewed as a set of directed edges in the dual lattice, where the direction
follows the same rule as for the arrows described above. The connected
components of $\partial C_{i}$ consist of closed contours; at the intersection
of four dual edges belonging to contours we use the North-West/South-East
deformation rule (see e.g. Fig. 3.11 in [5]) to generate self-avoiding
contours.

Zero or two of the connected components of $\partial C_{i}$ may wind around
the torus (let’s call them $\gamma_{-}$ and $\gamma_{+}$, if they exist), but
all other contours $\gamma_{1},\ldots,\gamma_{m}$ do not wind around it. One
can see this as follows. Winding connected components $\gamma$ are
characterized by their period $p(\gamma):=\sum_{e\in\gamma}\vec{e}$ being non-
zero. The period characterizes the homology class of $\gamma$. Due to the
periodic boundary conditions, for any set of vertices $C\subseteq\Lambda$, the
balance equation $\sum_{e\in\partial C}\vec{e}=0$ holds. This shows that there
are either zero or at least two winding connected components of $\partial
C_{i}$. The period $p(\gamma)$ is the asymptotic direction of any lifting
$\hat{\gamma}$ of $\gamma$ to a two-sided infinite path in ${\mathbb{Z}}^{2}$.
Here a lifting $\hat{\gamma}$ means a connected component of the inverse image
of $\gamma$ w.r.t. the canonical map
${\mathbb{Z}}^{2}+(1/2,1/2)\to\Lambda^{*}$. Because the liftings
$\hat{\gamma}$ and $\hat{\gamma}^{\prime}$ of any two different connected
components $\gamma$ and $\gamma^{\prime}$ of $\partial C_{i}$ are disjoint,
their periods $p(\gamma),p(\gamma^{\prime})$ are linearly dependent. Assume
that we have at least three different infinite connected components
$\hat{\gamma},\hat{\gamma}^{\prime},\hat{\gamma}^{\prime\prime}$ in the
boundary $\partial\hat{C_{i}}$ of a lifting $\hat{C_{i}}$ of $C_{i}$. Since
their asymptotic directions are pairwise linearly dependent, two of them are
separated by the third, a contradiction.

Therefore there are two cases, illustrated in Figure 1:

  * •

Case 1, connected cycles: either precisely one of the contours
$\gamma_{1},\ldots,\gamma_{m}$, called $\gamma_{i,j}$, separates $i$ from $j$,
or

  * •

Case 2, winding pairs: $\gamma_{-}$ and $\gamma_{+}$ exist and their union
$\gamma_{i,j}:=\gamma_{-}\cup\gamma_{+}$ separates $i$ and $j$.

Let $\mathcal{L}_{i}(\gamma_{i,j})$ denote the set of all vertices
$k\in\Lambda$ such that $\gamma_{i,j}$ does not separate $i$ from $k$. Note
that $\mathcal{L}_{i}(\gamma_{i,j}(x))$ depends on the configuration $x$ only
through $\gamma_{i,j}(x)$. Consider the mapping

$\displaystyle F_{i,j}:\Omega_{i,j}\longrightarrow\Omega_{\Lambda},\qquad
x\longmapsto F_{i,j}(x),$ (28)

where $F_{i,j}(x)_{l}=x_{l}^{\prime}-x_{0}^{\prime}$, with
$x_{l}^{\prime}=x_{l}-1$ for $k\in\mathcal{L}_{i}(\gamma_{i,j}(x))$ and
$x_{l}^{\prime}=x_{l}$, otherwise. In other words, the map $F_{i,j}$ lowers
the configuration $x$ in the region bounded by the oriented contour(s)
$\gamma_{i,j}$. In compact notation,
$F_{i,j}(x)_{l}=x_{l}-\mathbf{1}_{\mathcal{L}_{i}(\gamma_{i,j}(x))}(l)-x_{0}+\textbf{1}_{\mathcal{L}_{i}(\gamma_{i,j}(x))}(0)$.
Note that for any edge $\\{l,m\\}\in\mathcal{E}_{\Lambda}$ one has

$\displaystyle|F_{i,j}(x)_{l}-F_{i,j}(x)_{m}|=|x_{l}-x_{m}|-\mathbf{1}_{\\{\\{l,m\\}^{*}\in\gamma_{i,j}(x)\\}}.$
(29)

Here, the event ${\\{\\{l,m\\}^{*}\in\gamma_{i,j}\\}}$ means that one dual
edge in $\gamma_{i,j}$ intersects the edge $\\{l,m\\}$.

Figure 1: Possible spin and contour configuration. Here, case 1 occurs for
$x_{i}=1$ and $x_{j}=0$ with the green contour $\gamma_{i,j}$, while case 2
occurs for $x_{i}=0$ and $j$ in the left lower corner with $\gamma_{i,j}$ the
union of the red contours.

Define $\Gamma_{i,j}:=\\{\gamma_{i,j}(x):x\in\Omega_{i,j}\\}$. The length
$|\gamma_{i,j}|$ of a contour/pair of contours $\gamma_{i,j}\in\Gamma_{i,j}$
is defined to be its total number of edges.

######  Lemma 5 (Peierls argument).

Let $\beta>0$ and $i,j\in\Lambda$ with $i\neq j$. Then for any
$x\in\Omega_{i,j}$ one has

$H_{\Lambda}(x)-H_{\Lambda}(F_{i,j}(x))\geq|\gamma_{i,j}(x)|.$ (30)

Further, for any $\gamma\in\Gamma_{i,j}$ and $k\in{\mathbb{N}}$, the following
estimate holds

$\displaystyle P_{\Lambda,\beta}(x_{i}-x_{j}\geq
k,\gamma_{i,j}(x)=\gamma)\leq{\mathrm{e}}^{-\beta|\gamma|}P_{\Lambda,\beta}(x_{i}-x_{j}\geq
k-1).$ (31)

In particular,

$P_{\Lambda,\beta}(x_{i}-x_{j}\geq k)\leq P_{\Lambda,\beta}(x_{i}-x_{j}\geq
k-1)\sum_{\gamma\in\Gamma_{i,j}}{\mathrm{e}}^{-\beta|\gamma|}.$ (32)

Hence, we have the inequality

$P_{\Lambda,\beta}(x_{i}-x_{j}\geq
k)\leq\bigg{(}\sum_{\gamma\in\Gamma_{i,j}}{\mathrm{e}}^{-\beta|\gamma|}\bigg{)}^{k}.$
(33)

###### Proof.

Given an edge $\\{l,m\\}\in\mathcal{E}_{\Lambda}$ and $x\in\Omega_{i,j}$,
formula (29) implies

$\displaystyle(x_{l}-x_{m})^{2}$
$\displaystyle=(|F_{i,j}(x)_{l}-F_{i,j}(x)_{m}|+\textbf{1}_{\\{\\{l,m\\}^{*}\in\gamma_{i,j}(x)\\}})^{2}$
$\displaystyle\geq(F_{i,j}(x)_{l}-F_{i,j}(x)_{m})^{2}+\textbf{1}_{\\{\\{l,m\\}^{*}\in\gamma_{i,j}(x)\\}}.$
(34)

Summing over $\\{l,m\\}\in\mathcal{E}_{\Lambda}$ in (34) yields (30).

Let $\gamma\in\Gamma_{i,j}$. Note that $F_{i,j}$ restricted to
$\\{x_{i}-x_{j}\geq k,\gamma_{i,j}(x)=\gamma\\}$ maps one-to-one into
$\\{x_{i}-x_{j}\geq k-1\\}$ with the inverse
$x\mapsto(x_{l}+\mathbf{1}_{\mathcal{L}_{i}(\gamma)}(l)-x_{0}-\textbf{1}_{\mathcal{L}_{i}(\gamma)}(0))_{l\in\Lambda}$.
By plugging formula (30) into the definition of $P_{\Lambda,\beta}$ we obtain
claim (31) as follows:

$\displaystyle P_{\Lambda,\beta}(x_{i}-x_{j}\geq
k,\gamma_{i,j}(x)=\gamma)=\frac{1}{Z_{\Lambda,\beta}}\sum_{\begin{subarray}{c}x:\;x_{i}-x_{j}\geq
k,\\\ \gamma_{i,j}(x)=\gamma\end{subarray}}{\mathrm{e}}^{-\beta
H_{\Lambda}(x)}\leq\frac{1}{Z_{\Lambda,\beta}}\sum_{\begin{subarray}{c}x:\;x_{i}-x_{j}\geq
k,\\\
\gamma_{i,j}(x)=\gamma\end{subarray}}{\mathrm{e}}^{-\beta|\gamma|}{\mathrm{e}}^{-\beta
H_{\Lambda}(F_{i,j}(x))}$
$\displaystyle={\mathrm{e}}^{-\beta|\gamma|}P_{\Lambda,\beta}(F_{i,j}(\\{x_{i}-x_{j}\geq
k,\gamma_{i,j}(x)=\gamma\\}))\leq{\mathrm{e}}^{-\beta|\gamma|}P_{\Lambda,\beta}(x_{i}-x_{j}\geq
k-1).$ (35)

Summing this over $\gamma\in\Gamma_{i,j}$ yields (32). The last claim (33)
follows by iterating (32). ∎

In the following, $\Lambda\Subset{\mathbb{Z}}^{2}$ means that $\Lambda$ is an
$N\times N$ box of length $N\geq 4$ with periodic boundary conditions
containing the origin.

######  Proposition 6 (Counting contours).

Given $\Lambda\Subset{\mathbb{Z}}^{2}$, let $i,j\in\Lambda$ with $i\neq j$ and
$\ell\in{\mathbb{N}}$. The number of contours/pairs of contours in
$\Gamma_{i,j}$ of length $\ell$ is bounded by

$\displaystyle|\\{\gamma\in\Gamma_{i,j}:|\gamma|=\ell\\}|\leq
3\ell^{2}3^{\ell}.$ (36)

For all $\beta\geq\log 6$ and $k\in{\mathbb{N}}$ we have

$\sup_{\Lambda\Subset{\mathbb{Z}}^{2}}\sup_{i,j\in\Lambda}P_{\Lambda,\beta}(|x_{i}-x_{j}|\geq
k)\leq 2\varphi(\beta)^{k},$ (37)

where $\varphi(\beta):=480(3{\mathrm{e}}^{-\beta})^{4}$.

###### Proof.

The number of connected cycles (Case 1 above) is, as in the Ising model case
(see [5]), bounded by $\frac{2}{3}\ell 3^{\ell}$, for all $\ell\geq 4$. We
note that the number of winding pairs $\gamma_{+}\cup\gamma_{-}$ (Case 2
above) is non-null only if their total length $\ell$ is larger or equal to
twice the box length $N$ (at least one $N$ for each contour $\gamma_{\pm}$).
Moreover, it is bounded by $(2N)^{2}\cdot 4^{2}\cdot 3^{\ell-2}$. One can see
this as follows: each contour $\gamma_{+}$ and $\gamma_{-}$ meets the $x-$axis
or $y-$axis on the torus in at least one point ($\leq(2N)^{2}$ choices).
Starting from this point, there are four choices for the next step ($4^{2}$
choices together). For the remaining $\ell-2$ steps, starting with
$\gamma_{+}$, there are at most $3$ choices per step, finishing $\gamma_{+}$
and starting with $\gamma_{-}$ as soon as the path becomes closed. Altogether,
this means

$|\\{\gamma\in\Gamma_{i,j}:|\gamma|=\ell\\}|\leq\frac{2}{3}\ell
3^{\ell}+\mathbf{1}_{\\{\ell\geq 2N\\}}64N^{2}3^{\ell-2}\leq
3\ell^{2}3^{\ell}.$ (38)

Take $\beta\geq\log 6$, which means $3{\mathrm{e}}^{-\beta}\leq\frac{1}{2}$.
Summing over $\ell\geq 4$, one obtains

$\sum_{\gamma\in\Gamma_{i,j}}{\mathrm{e}}^{-\beta|\gamma|}\leq\sum_{\ell\geq
4}{\mathrm{e}}^{-\beta\ell}|\\{\gamma\in\Gamma_{i,j}:|\gamma|=\ell\\}|\leq
3\sum_{\ell\geq 4}\ell^{2}(3{\mathrm{e}}^{-\beta})^{\ell}\leq
480(3{\mathrm{e}}^{-\beta})^{4},$ (39)

where we have used
$\sum_{\ell=4}^{\infty}3\ell^{2}z^{\ell}=3z^{4}(9z^{2}-23z+16)(1-z)^{-3}\leq
480z^{4}$ for $0\leq z\leq 1/2$. We haven’t tried to optimize the constant.

Using first that $P_{\Lambda,\beta}$ is symmetric with respect to the
reflection $x\mapsto-x$, and then inequality (33) we conclude for
$k\in{\mathbb{N}}$

$\displaystyle\sup_{\begin{subarray}{c}\Lambda\Subset{\mathbb{Z}}^{2}\\\ N\geq
4\end{subarray}}\sup_{i,j\in\Lambda}P_{\Lambda,\beta}(|x_{i}-x_{j}|\geq k)$
$\displaystyle\leq 2\sup_{\begin{subarray}{c}\Lambda\Subset{\mathbb{Z}}^{2}\\\
N\geq 4\end{subarray}}\sup_{i,j\in\Lambda}P_{\Lambda,\beta}(x_{i}-x_{j}\geq
k)$ $\displaystyle\leq
2\sup_{\begin{subarray}{c}\Lambda\Subset{\mathbb{Z}}^{2}\\\ N\geq
4\end{subarray}}\sup_{i,j\in\Lambda}\bigg{(}\sum_{\gamma\in\Gamma_{i,j}}{\mathrm{e}}^{-\beta|\gamma|}\bigg{)}^{k}\leq
2\varphi(\beta)^{k}.$ (40)

∎

For $i,j\in\Lambda$ we define the observable
$\mathcal{O}_{i,j}\colon\Omega_{\Lambda}^{\mathbb{R}}\to{\mathbb{R}}$,

$\displaystyle\mathcal{O}_{i,j}(x)=(x_{i}-x_{j})^{2}.$ (41)

######  Corollary 7 (Spontaneous breaking of the internal ${\mathbb{Z}}^{1}$
symmetry).

For all $\beta\geq 3$ one has

$\sup_{\Lambda\Subset{\mathbb{Z}}^{2}}\sup_{i,j\in\Lambda}E_{\Lambda,\beta}[\mathcal{O}_{i,j}(x)]\leq
M_{\beta}:=\frac{2\varphi(\beta)(1+\varphi(\beta))}{(1-\varphi(\beta))^{3}}.$
(42)

The bound $M_{\beta}$ is asymptotically equivalent to
$2\varphi(\beta)=960(3{\mathrm{e}}^{-\beta})^{4}$ as $\beta\to\infty$. In
particular,

$\lim_{\beta\to\infty}\beta M_{\beta}=0.$ (43)

###### Proof.

Let $\Lambda\Subset{\mathbb{Z}}^{2}$ and $i,j\in\Lambda$. For all $\beta\geq
3$ one has $\varphi(\beta)<1$ and we obtain

$\displaystyle 0\leq E_{\Lambda,\beta}[\mathcal{O}_{i,j}(x)]=$
$\displaystyle\sum_{n\in{\mathbb{N}}}n^{2}P_{\Lambda,\beta}(|x_{i}-x_{j}|=n)$
$\displaystyle\leq$
$\displaystyle\sum_{n\in{\mathbb{N}}}n^{2}P_{\Lambda,\beta}(|x_{i}-x_{j}|\geq
n)$ $\displaystyle\leq$ $\displaystyle
2\sum_{n\in{\mathbb{N}}}n^{2}\varphi(\beta)^{n}=\frac{2\varphi(\beta)(1+\varphi(\beta))}{(1-\varphi(\beta))^{3}}=M_{\beta}<\infty.$
(44)

Since $\varphi$ neither depends on $i,j$, nor on $\Lambda$, the above
estimates imply the desired conclusion (42). ∎

##  4 Main result

For $i,j\in\Lambda$ define the observable
$U_{ij}:{\mathbb{R}}^{\Lambda}\to{\mathbb{R}}$ by

$\displaystyle U_{ij}(k):=\sum_{\ell\in\Lambda}(G_{i\ell}-G_{j\ell})k_{\ell}.$
(45)

We interpret $\sum_{\ell\in\Lambda}G_{i\ell}k_{\ell}$ as the electric
potential at location $i$ of a charge distribution encoded by
$(k_{\ell})_{\ell\in\Lambda}$. In this interpretation, $U_{ij}$ encodes the
voltage between $i$ and $j$. We remark that the thermodynamic limit of the
potential kernel $G_{ii}-G_{ij}$ is well-defined cf. Theorem 1.6.1 in [12].
Since $G_{ii}-G_{ij}\sim C\log|i-j|$ in the double limit
$\Lambda\nearrow{\mathbb{Z}}^{2}$ and then $|i-j|\to\infty$, the following
result states that the variance of the energy cost of transporting a unit
charge from $i$ to $j$ asymptotically behaves like
$c(\beta^{*})^{-1}\log|i-j|$, for some positive constants $C,c$.

######  Theorem 8 (Variance of voltages in a lattice Coulomb gas).

For all $N\times N$ boxes $\Lambda\ni 0$ of length $N\geq 4$ with periodic
boundary conditions, all $i,j\in\Lambda$, and all $\beta^{*}\leq\frac{1}{12}$
one has

$\frac{4}{\beta^{*}}(G_{ii}-G_{ij})-\frac{4}{(\beta^{*})^{2}}M_{(4\beta^{*})^{-1}}\leq\operatorname{Var}^{*}_{\Lambda,\beta^{*}}(U_{ij})=E^{*}_{\Lambda,\beta^{*}}[U_{ij}^{2}]\leq\frac{4}{\beta^{*}}(G_{ii}-G_{ij}),$
(46)

where $E^{*}_{\Lambda,\beta^{*}}$ and
$\operatorname{Var}^{*}_{\Lambda,\beta^{*}}$ denote the expectation and the
variance with respect to $P_{\Lambda,\beta^{*}}^{*}$, respectively. As a
consequence, we obtain the following asymptotic equivalence

$E^{*}_{\Lambda,\beta^{*}}[U_{ij}^{2}]\sim\frac{4}{\beta^{*}}(G_{ii}-G_{ij})\;\text{
as }\;\beta^{*}\to 0.$ (47)

###### Proof.

Given $\Lambda$, let
${\mathrm{d}}x:=\delta_{0}({\mathrm{d}}x_{0})\prod_{\ell\in\Lambda\setminus\\{0\\}}{\mathrm{d}}x_{\ell}$
denote the Lebesgue measure on $\Omega^{\mathbb{R}}_{\Lambda}$. Set
$\beta:=(4\beta^{*})^{-1}\geq 3$. Given $i,j\in\Lambda$ let
$h(x)=\mathcal{O}_{i,j}(x){\mathrm{e}}^{-\beta H_{\Lambda}(x)}$. Using formula
(14) with $n=|\Lambda|-1$ and $A=-\beta\Delta_{0^{c}0^{c}}$ we obtain its
Fourier transform at $k\in{\mathbb{R}}^{\Lambda}$

$\displaystyle\hat{h}(k)=$
$\displaystyle\int_{\Omega^{\mathbb{R}}_{\Lambda}}{\mathrm{e}}^{-{\mathrm{i}}k\cdot
x}\mathcal{O}_{i,j}(x){\mathrm{e}}^{-\beta
H_{\Lambda}(x)}{\mathrm{d}}x=-(\partial_{k_{i}}-\partial_{k_{j}})^{2}\int_{\Omega^{\mathbb{R}}_{\Lambda}}{\mathrm{e}}^{-{\mathrm{i}}k\cdot
x}{\mathrm{e}}^{-\beta H_{\Lambda}(x)}{\mathrm{d}}x$ $\displaystyle=$
$\displaystyle-\left(\frac{\pi}{\beta}\right)^{\frac{|\Lambda|-1}{2}}\det(-\Delta_{0^{c}0^{c}})^{-\frac{1}{2}}(\partial_{k_{i}}-\partial_{k_{j}})^{2}{\mathrm{e}}^{\beta^{*}k^{t}_{0^{c}}(\Delta_{0^{c}0^{c}})^{-1}k_{0^{c}}}.$
(48)

Applying the Poisson summation formula (13) again, we obtain that

$\displaystyle\sum_{x\in\Omega_{\Lambda}}\mathcal{O}_{i,j}(x){\mathrm{e}}^{-\beta
H_{\Lambda}(x)}=-\left(\frac{\pi}{\beta}\right)^{\frac{|\Lambda|-1}{2}}\det(-\Delta_{0^{c}0^{c}})^{-\frac{1}{2}}\sum_{k\in\Omega_{\Lambda}^{*}}(\partial_{k_{i}}-\partial_{k_{j}})^{2}{\mathrm{e}}^{\beta^{*}k^{t}_{0^{c}}(\Delta_{0^{c}0^{c}})^{-1}k_{0^{c}}}.$
(49)

According to Corollary 4, the expression
${\mathrm{e}}^{\beta^{*}k^{t}_{0^{c}}(\Delta_{0^{c}0^{c}})^{-1}k_{0^{c}}}$
coincides with ${\mathrm{e}}^{-\frac{\beta^{*}}{4}k^{t}Gk}$ for
$k\in\mathcal{K}_{\Lambda}$. Consequently, all directional derivatives of
these two quantities in directions of the hyperplane $\mathcal{K}_{\Lambda}$
coincide as well. In particular, all $\partial_{k_{i}}-\partial_{k_{j}}$ are
such directional derivatives. Using this in (49) and plugging in the formula
for the partition sum (23), we calculate

$\displaystyle E_{\Lambda,\beta}[\mathcal{O}_{i,j}]=$
$\displaystyle\frac{1}{Z_{\Lambda,\beta}}\sum_{x\in\Omega_{\Lambda}}\mathcal{O}_{i,j}(x){\mathrm{e}}^{-\beta
H_{\Lambda}(x)}$ $\displaystyle=$
$\displaystyle-\frac{1}{Z_{\Lambda,\beta^{*}}^{*}}\sum_{k\in\Omega_{\Lambda}^{*}}(\partial_{k_{i}}-\partial_{k_{j}})^{2}{\mathrm{e}}^{\beta^{*}k^{t}_{0^{c}}(\Delta_{0^{c}0^{c}})^{-1}k_{0^{c}}}$
$\displaystyle=$
$\displaystyle-\frac{1}{Z_{\Lambda,\beta^{*}}^{*}}\sum_{k\in\Omega_{\Lambda}^{*}}(\partial_{k_{i}}-\partial_{k_{j}})^{2}{\mathrm{e}}^{-\frac{\beta^{*}}{4}k^{t}Gk}.$
(50)

Moreover, by translation invariance ($G_{ii}=G_{jj}$) and symmetry
($G_{ij}=G_{ji}$) of $G$ it follows

$\displaystyle(\partial_{k_{i}}-\partial_{k_{j}})^{2}{\mathrm{e}}^{-\frac{\beta^{*}}{4}k^{t}Gk}$
$\displaystyle=\frac{\beta^{*}}{4}{\mathrm{e}}^{-\frac{\beta^{*}}{4}k^{t}Gk}\bigg{[}\frac{\beta^{*}}{4}\Big{(}\sum_{\ell\in\Lambda}(G_{i\ell}+G_{\ell
i}-G_{j\ell}-G_{\ell
j})k_{\ell}\Big{)}^{2}-2\big{(}G_{ii}+G_{jj}-G_{ij}-G_{ji}\big{)}\bigg{]}$
$\displaystyle={\mathrm{e}}^{-\frac{\beta^{*}}{4}k^{t}Gk}\Big{(}\frac{(\beta^{*})^{2}}{4}U_{ij}^{2}(k)-\beta^{*}(G_{ii}-G_{ij})\Big{)}.$
(51)

Putting (42), (50) and (51) together yields

$\displaystyle
0\leq-\frac{1}{Z_{\Lambda,\beta^{*}}^{*}}\sum_{k\in\Omega_{\Lambda}^{*}}{\mathrm{e}}^{-\frac{\beta^{*}}{4}k^{t}Gk}\Big{(}\frac{(\beta^{*})^{2}}{4}U_{ij}^{2}(k)-\beta^{*}(G_{ii}-G_{ij})\Big{)}\leq
M_{\beta}=M_{(4\beta^{*})^{-1}},$ (52)

or equivalently,

$\displaystyle 0\geq
E^{*}_{\Lambda,\beta^{*}}[U_{ij}^{2}]-\frac{4}{\beta^{*}}(G_{ii}-G_{ij})\geq-\frac{4}{(\beta^{*})^{2}}M_{(4\beta^{*})^{-1}}.$
(53)

This implies the desired conclusion (46), taking into account that
$E^{*}_{\Lambda,\beta^{*}}[U_{ij}]=0$ by translation invariance. ∎

##  5 Discussion

In this section we compare the main result of this paper with results on Debye
screening in three dimensions by Brydges [2] and Brydges-Federbush [3] and
results on two-dimensional abelian spin systems and the Coulomb gas by
Fröhlich and Spencer [6].

As is mentioned in [6], their Theorem A in Section 1.3 also applies for the
discrete Gaussian model in the special case of weights equal to $1$ on the
integers. More precisely, it applies to the _large_ temperature regime in the
discrete Gaussian model, because the sine-Gordon transformation used there
inverts the temperature, so that their $\beta$ plays the role of the
temperature. The lower bound for the high-temperature regime in formula (1.20)
in [6] should be seen in contrast to our bound (42) in the low-temperature
regime. The former provides a lower bound which increases logarithmically with
distance, while the latter states an upper bound which is uniform in the
distance. This illustrates a difference between the high- and low-temperature
phases in the two-dimensional discrete Gaussian model. Fröhlich and Spencer
[6] use another way of dualizing the discrete Gaussian model, obtaining the
Villain model. In contrast to the discrete Coulomb gas, it deals with
continuous angle variables and thus their Theorems C-E cannot be directly
compared to our results.

As was mentioned in the introduction of [6], “the Coulomb gas has a high
temperature, low density plasma phase characterized by exponential Debye
screening”. This was examined in three space dimensions in several papers, in
particular, by Brydges in [2] and Brydges-Federbush in [3]. More precisely,
they prove a decay of correlations for observables depending only on the
charge particle configuration on compact regions, which is exponential in the
distance between these regions. The voltage $U_{ij}$ examined in this paper is
not an observable in this class, because it does not only depend on the charge
configuration close to $i$ and $j$, but on the whole charge configuration. In
this sense, it is not a local observable. Theorem 8 above shows that in the
high temperature regime of the two-dimensional lattice Coulomb gas the
variance of $U_{ij}$ increases logarithmically in the distance $|i-j|$. We
find this an interesting observation, which complements the previously
mentioned results on Debye screening. In view of phase transitions, it remains
an interesting open problem to study the voltage $U_{ij}$ also in the low-
temperature regime.

#### Acknowledgment.

We thank Aernout van Enter and anonymous referees for remarks on the
literature. We also thank the referees for their constructive comments, which
helped us to improve the paper.

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