Appearance of odd-frequency superconductivity in a relativistic scenario

Abstract

Odd-frequency superconductivity is an exotic superconducting state in which the symmetry of the gap function is odd in frequency. Here, we show that an inherent odd-frequency mode emerges dynamically under application of a Lorentz transformation of the anomalous Green function with the general frequency-dependent gap function. To see this, we consider a Dirac model with quartic potential and perform a mean-field analysis to obtain a relativistic Bogoliubov–de Gennes system. Solving the resulting Gor'kov equations yields expressions for relativistic normal and anomalous Green functions. The form of the relativistically invariant pairing term is chosen such that it reduces to BCS form in the nonrelativistic limit. We choose an ansatz for the gap function in a particular frame which is even frequency and analyze the effects on the anomalous Green function under a boost into a relativistic frame. The odd-frequency pairing emerges dynamically as a result of the boost. In the boosted frame, the order parameter contains terms which are both even and odd in frequency. The relativistic correction to the anomalous Green function to first order in the boost parameter is completely odd in frequency. In this paper, we provide evidence that odd-frequency pairing may form intrinsically within relativistic superconductors.

Full text

PHYSICAL REVIEW B 108, 014510 (2023)
Appearance of odd-frequency superconductivity in a relativistic scenario
Patrick J. Wong *and Alexander V. Balatsky
Department of Physics, University of Connecticut, Storrs, Connecticut 06269, USA
and Nordita, Stockholm University and KTH Royal Institute of Technology, Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden
(Received 23 January 2023; revised 28 June 2023; accepted 28 June 2023; published 27 July 2023)
Odd-frequency superconductivity is an exotic superconducting state in which the symmetry of the gap function
is odd in frequency. Here, we show that an inherent odd-frequency mode emerges dynamically under application
of a Lorentz transformation of the anomalous Green function with the general frequency-dependent gap function.
To see this, we consider a Dirac model with quartic potential and perform a mean-field analysis to obtain
a relativistic Bogoliubov–de Gennes system. Solving the resulting Gor’kov equations yields expressions for
relativistic normal and anomalous Green functions. The form of the relativistically invariant pairing term is
chosen such that it reduces to BCS form in the nonrelativistic limit. We choose an ansatz for the gap function in
a particular frame which is even frequency and analyze the effects on the anomalous Green function under a boost
into a relativistic frame. The odd-frequency pairing emerges dynamically as a result of the boost. In the boosted
frame, the order parameter contains terms which are both even and odd in frequency. The relativistic correction to
the anomalous Green function to first order in the boost parameter is completely odd in frequency. In this paper,
we provide evidence that odd-frequency pairing may form intrinsically within relativistic superconductors.
DOI: 10.1103/PhysRevB.108.014510
I. INTRODUCTION
Berezinskii odd-frequency pairing [1–6] is a class of pair-
ing in Fermi systems in which the paired condensate is odd
in relative time between the paired particles. The parity of a
paired state of fermions may be classified according to the dis-
crete permutation symmetries of spin permutation S, relative
coordinate permutation P∗, orbital permutation O, and time
permutation T∗. To satisfy the Pauli principle, the symmetries
of a Fermi pair must obey the condition of SP∗OT ∗=−1
[3,4,7]. With respect to these permutation symmetries, odd-
frequency pairing possesses T∗=−1. The time permutation
and coordinate permutation symmetry operations T∗,P∗are
swap operations where relative indices are permuted, yet no
global inversion (for coordinates) nor the global time reversal
is invoked: These permutations are distinct from time reversal
and spatial inversion operations; hence, we use ∗to mark
the difference. Hereafter, we consider the single-band pairing
states and hence drop O. We thus focus on the SP∗T∗=−1
product. Table Ishows the allowed symmetry states. The
table implies that odd-frequency states are allowed by the
symmetry classification and appear on equal footing with
their even-frequency counterparts (BCS), so they therefore
should in principle appear as ubiquitously as even-frequency
*patrick.wong@uconn.edu
Published by the American Physical Society under the terms of the
Creative Commons Attribution 4.0 International license. Further
distribution of this work must maintain attribution to the author(s)
and the published article’s title, journal citation, and DOI. Funded
by Bibsam.
states. We emphasize that the T∗=−1 states appear with
equal weight as the T∗=+1 states in the table. However, the
majority of work on superconductivity, both theoretical and
experimental, has focused only on even-frequency pairing,
with odd-frequency pairing generally seen to require special
circumstances.
To date, much of the literature on odd-frequency supercon-
ductivity has focused on its manifestation in special select
circumstances, a prominent example being in the process
of scattering of conventional BCS pairs into odd-frequency
pairs at the interface between superconductors and other
media. Examples include superconductor-ferromagnet inter-
faces [8–11], Josephson junctions [12], topological insulators
[13,14], and other heterostructures. For a broader list, see
Ref. [3]. Odd-frequency pairing has additionally been de-
scribed as appearing in systems placed in the presence of
external electric fields, where the odd frequency is induced
by Zeeman splitting [15]. Odd-frequency superconductivity
has also been observed in multiband systems with interor-
bital pairing [16,17]. Odd-frequency pairing has been shown
to appear ubiquitously in multiband superconductors with
interband hybridization [3,16]. In this paper, we aim to
demonstrate that odd-frequency pairing is also ubiquitous to
the single-band situation with a minimal set of degrees of
freedom.
While odd-frequency superconducting order has been
observed or proposed in a variety of circumstances, the occur-
rence of odd-frequency superconductivity as an independent
bulk phase is disputed. It is not universally accepted that it
exists as its own independent bulk state without any external
influences [3,18,19]. It has been argued that the odd-frequency
pairing state is inherently thermodynamically unstable [18]. It
has also been claimed that odd-frequency pairing inherently
2469-9950/2023/108(1)/014510(10) 014510-1 Published by the American Physical Society

PATRICK J. WONG AND ALEXANDER V. BALATSKY PHYSICAL REVIEW B 108, 014510 (2023)
TABLE I. Permutation parities of spin-singlet and spin-triplet
pairing states are listed. The induction of the odd-frequency state is
possible once we allow the conversions of P∗=+1intoP∗=−1, as
shown by the arrow. Lorentz boost mixes up space and time indices
and induces finite amplitude of P∗=−1 states. Similar considera-
tions can be done for spin-triplet states, S=+1, which we list here
for completeness.
possesses an unphysical anti-Meissner effect [19]. However,
authors of other works have reached the opposite conclusion
[20–22]. There does not appear to be a clear consensus on this
matter [3].
To address the broader controversy regarding the intrinsic
existence of odd-frequency superconductivity, we argue that
a relativistic change of reference frame should not change
the underlying physics. We therefore aim to demonstrate on
physical grounds that odd-frequency pairing is on equal foot-
ing with even-frequency pairing and should be considered
equally as ubiquitous as implied by the symmetry table. By
ubiquity, we mean that the physical mechanism underlying
odd-frequency pairing is not categorically distinct to that of
conventional even-frequency pairing.
A. Dynamic and relativistic considerations
Odd-frequency superconductivity is a dynamic order with
the inherent time dependence associated with the state. At
equal times, the order parameter must vanish. The odd-
frequency symmetry of (t1,t2)=−(t2,t1)impliesthat
(t1,t2)=0fort1=t2and therefore only makes an appear-
ance over a finite time interval. We see now the growing list
of proposals where Berezinskii state can be realized in driven
systems, such as superconductors coupled to an external driv-
ing potential [23,24] and Floquet engineered states [25].
We now consider the effects of a relativistic boost. Boost
of electronic states in superconductors can be viewed as a
dynamic process. Hence, one can ask a similar question: Will
an odd-frequency state appear in the boosted superconducting
state?
We start with the Galilean boost. For a homogeneous order
parameter under a Galilean boost, there is no induction of
odd-frequency components. A Galilean boost of an electron
liquid when all electrons are in the superconducting state
can be realized as a steady DC supercurrent [14]. Assuming
a fully homogeneous system with a Galilean boosted elec-
tronic liquid, the system transforms according to p→p+q
and (p)→(p+q), and the gap function of the conden-
sate as →exp(iq·r). We therefore find no induction of
an odd-frequency state. In the inhomogeneous case, on the
other hand, an odd-frequency component can be induced by a
Galilean boost. Additionally, as will be shown below, in con-
trast to the Galilean boost, a Lorentz boost of a homogeneous
order parameter induces an odd-frequency component.
FIG. 1. A Cooper pair in frame S(red) as viewed from frame
S(black). The points denote the constituent fermions of the Cooper
pair. Planes of equal time in Sare those parallel to the xaxis. The
time coordinates of the Cooper pair are coincident in the Sframe,
t
1=t
2, but are displaced when viewed in frame S,t1= t2. The parity
of the Cooper pair under P∗in the Sframe is inherited by T∗in
the Sframe. This illustrates that temporal ordering can appear due
to a relativistic change of reference frame. Since such a temporal
displacement is necessary for odd-ωpairing, this illustration implies
that an odd-ωpairing may emerge from a relativistic boost.
The purpose of this paper is to demonstrate that odd-
frequency pairing can arise intrinsically in a dynamical
manner within superconducting systems from a relativistic
perspective [26].
As discussed previously, even- and odd-frequency pairings
appear with equal weight in the symmetry classification of
Table I. On the other hand, strong coupling, needed to pro-
duce odd-frequency states, often makes the odd-frequency
state a subleading pairing channel. Dynamic induction of an
odd-frequency channel as a result of Lorentz boost is another
opportunity to induce the odd-frequency correlations.
To heuristically illustrate the influence a Lorentz boost
can have on superconducting states, we can consider as a
particular example a p-wave equal spin superconductor with
a pairing of the form (k)=0sin(k). Under a Lorentz
transformation, sin(k) transforms as
sin(k)→ sin(γk+βγω)
=sin(γk) cos(βγω)
 
even-ω
+sin(βγ ω) cos(γk)
 
odd-ω
,(1)
which we see manifestly generates an odd-ωterm. This
transformation of order parameter may also be understood
graphically from the following thought experiment. Consider
a Cooper pair (we use the term Cooper pair as a general
term for pairing, not necessarily indicative of BCS-like state)
in a reference frame Swhich is moving at a relativistic
velocity with respect to frame S, as shown in Fig. 1.The
spatial displacement of the constituent electrons comprising
the Cooper pair in frame Sinduces a temporal displacement
between them in frame S. This concept of a dynamically in-
duced temporal parity in a Cooper pair by means of a boosted
reference frame described by this illustration will be explored
more quantitatively in Sec. III.
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APPEARANCE OF ODD-FREQUENCY … PHYSICAL REVIEW B 108, 014510 (2023)
FIG. 2. Interchange in temporal order of space-time events by
relativistic change of reference frame. Consider spacelike separated
events Aand Bwith respect to a reference frame S(black) and frame
S(red), which is boosted with respect to S. The lines parallel to
the x(x) axis are planes of equal time in reference frame S(S).
In frame S,eventAoccurs before event B,tA<tB.However,inthe
boosted frame S, the ordering is reversed, t
B<t
A. This shows that a
relativistic change of reference frame is a manifestation of the time
permutation operator T∗.
An additional interconnection between relativity and the
theory of odd-frequency superconductivity is the relation be-
tween boosts and the T∗permutation operator. The time
permutation operator T∗interchanges the two time coordi-
nates of the Cooper pair. The time permutation operation T∗
can manifest in the form of a relativistic frame transformation:
For spacelike separated spacetime events Aand Bwhich occur
at times tAand tB, respectively, and in order tA<tBin refer-
ence frame S, there exists a boosted reference frame Ssuch
that the events occur in the order t
A>t
B. This configuration
of events and frames is shown in Fig. 2. We see that, for such
a configuration of spacetime events Aand B, changing to a
boosted reference frame is a physical manifestation of the time
permutation operator T∗.
The preceding illustrations imply that there exists some re-
lation between relativistic transformations and odd-frequency
pairing. As described above, odd-frequency pairing can be
interpreted as dynamic state. In this paper, the relevant dy-
namics is that of a relativistic change in reference frame. An
example situation of the preceding illustrations is that of a
charged particle traveling at a relativistic velocity through a
particle accelerator with superconducting magnets, such as
SLAC or LHC. The relevant reference frames would be the
laboratory frame and the frame comoving with the accelerated
particles. In the frame of the particles, the magnets of the
accelerator would be observed as traveling at a relativistic
velocity. The construction in this paper then describes how
the superconducting magnets would appear to the accelerated
particles.
The remainder of this paper is organized as follows. In
Sec. II, we construct the theory of relativistic fermions with a
pairing condensate. In Sec. III, we perform a Lorentz boost of
an even-ωanisotropic anomalous Green function and observe
the generation of odd-ωmodes. We close in Sec. IV with some
concluding remarks.
II. RELATIVISTIC SUPERCONDUCTOR
To form a baseline for our discussion, we must first con-
struct a theory of superconductivity which is compatible with
special relativity. This construction inherently necessitates
the use of relativistic Dirac fermions. The need for Dirac
fermions in chemistry and condensed matter has previously
been reported in Ref. [27]. The requirement of relativistic
4-component Dirac spinors in the description of condensed
matter systems is relevant, for example, in heavy nucleon
compounds, where Z1. Although we are considering a
relativistic scenario in the mechanical sense, with a generic
superconducting object traveling at a real relativistic velocity
with respect to an observer, the framework describing super-
conducting pairing of relativistic fermions is also relevant to
heavy fermion compounds. In comparison with the Lorentz
transformation implemented in this paper, there would need
to be some analog of this transformation within the material
sample. Such an effect could potentially be caused by strain-
ing the material or applying external magnetic fields [28],
but the exact implementation of such a scheme is beyond the
scope of this paper, where we are simply interested in the me-
chanical Lorentz transformation. The general formalism of the
framework for discussing superconductivity in the relativistic
regime was developed in Refs. [29–32], and we follow their
construction in our presentation below.
The starting point of our analysis is an effective field theory
defined by the action:
S[A,ψ]
=d4xψi/
D−m+g0
2ψψψ−1
4Fμν Fμν ,(2)
consisting of a single flavor of relativistic Dirac fermions ψ
coupled to a U(1) gauge field Awith field strength F=dA
and D=d−ieA. The quartic term in the fermion field is
given an attractive coupling g0>0. Units are chosen such that
¯h=1=c. We take the standard notation ψ(x):=ψ†(x)γ0
and use the Dirac representation of the Clifford algebra with
the convention γ5:=iγ0γ1γ2γ3such that (γ5)2=+1. Fur-
ther details on conventions and explicit matrix forms of the
gamma matrices are found in Appendix A. Varying the action
in Eq. (2) yields the equation of motion for the fermion field
as
(iγμDμ−m)ψ(x)+g0[ψ(x)Tψ(x)T]ψ(x)=0,(3)
where we rewrite ψ(x)ψ(x)=ψ(x)Tψ(x)T. The equation of
motion for the adjoint field ψcan be obtained similarly;
however, for the following discussion, it is useful to use its
transpose:
−(iγμTD†
μ+m)ψ(x)T+g0[ψ(x)ψ(x)]ψ(x)T=0.(4)
We proceed by analyzing this system with a mean-field
approximation of the interaction term. The pairing potential
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PATRICK J. WONG AND ALEXANDER V. BALATSKY PHYSICAL REVIEW B 108, 014510 (2023)
obeys the self-consistency relation:
(x)=g0ψ(x)Tψ(x),(5)
(x)=g0ψ(x)ψ(x)T.(6)
The pairing considered in this model is that of a generic pair-
ing between Dirac spinors. The mechanism behind the pairing
is not specified. The initial starting action defined by Eq. (2)
should be considered as defining an effective field theory of
an electron-positron gas and not a complete theory specifying
the microscopic relativistic dynamics, e.g., we do not consider
relativistic electron/positron-phonon coupling.
A. Structure of the pairing term
For the action in Eq. (2) to be Lorentz invariant, the pairing
terms must be in terms of Lorentz invariant bilinears. The
general expression for the pairing potential is
=g0ψTMψ,(7)
where Mis a Lorentz invariant matrix which forms various
combinations of pairs of spinor components. There exist five
possible bilinear classes of pairing matrices Mwithato-
tal of 16 linearly independent components. These bilinears
are classified by their transformation properties under the
Lorentz group as scalar, vector, bivector, pseudovector, and
pseudoscalar [29,31,33].
In the ordinary nonrelativistic theory of superconductivity,
Cooper pairs are classified according to the symmetry of the
spin projections of the constituent fermions, either spin singlet
or spin triplet. In the relativistic regime, the spin projection is
generally not a good quantum number, so it is less straight-
forward how to classify paired states according to spin singlet
or triplet as done in the standard nonrelativistic theory of su-
perconductivity. The pairing between the fermions is instead
expressed through their global symmetries with respect to
Lorentz transformation.
To this end, we express the pairing states in a configuration
space in terms of time reversal ˆ
Tand parity ˆ
Poperations (not
to be confused with T∗,P∗). In the fully relativistic scenario,
we must also consider the charge conjugation symmetry ˆ
C.
The charge conjugation operator comes into play to describe
pairing terms involving antiparticles. In terms of the Clifford
algebra basis, these operators take the form [34]:
ˆ
T=iγ1γ3,(8a)
ˆ
P=γ0,(8b)
ˆ
C=−iγ2γ0.(8c)
For a state of spatial momentum kand spin σ,|k,σ,the
actions of the time reversal and parity operators are
|ˆ
Tk,σ=|−k,−σ,(9a)
|ˆ
Pk,σ=|−k,σ.(9b)
A conventional BCS s-wave spin-singlet pairing state is ex-
pressed as
|s=|k,↑| − k,↓ − |k,↓| − k,↑.(10)
In terms of the symmetry operators, this state may instead be
expressed as
|s=|k,σ| ˆ
Tk,σ−|ˆ
Tˆ
Pk,σ| ˆ
Pk,σ.(10)
For a general fermion state |ψ, the singlet pairing is then
expressed in configuration space as
|s=|ψ| ˆ
Tψ−|ˆ
Tˆ
Pψ| ˆ
Pψ.(11)
This is the basis in which the superconducting pairing bi-
linears are to be constructed. The pairing matrix Mis then
expressed in terms of gamma matrices corresponding to the
appropriate combination of symmetries ˆ
T,ˆ
P, and ˆ
C. Pairing
terms utilizing the charge conjugation operator are pairs in-
volving antimatter particles, either particle-antiparticle pairs
or antiparticle-antiparticle pairs. These pairs will not appear
in our discussion; however, we note that they are needed for a
complete set of relativistically invariant pairing terms.
The spin symmetry of the pairing can also be deduced
from the form of the bilinears. Bilinears satisfying MT=+M
correspond to triplet pairing in the nonrelativistic limit, and
MT=−Mcorrespond to singlet pairing. The parity of M
under transposition reflects the parity of the Cooper pair under
the Soperation in the SP∗T∗classification.
A reasonable prescription for choosing the form of a pair-
ing term in the relativistic theory is one which reduces to the
standard BCS pairing term in the nonrelativistic regime. With
respect to the nonrelativistic limit, it is useful to consider that
the 4-component Dirac bispinor may be decomposed into two
spinors φand χ:
ψ=φ
χ,(12)
which correspond to positive and negative energy eigenstates
of the Dirac equation.
In the Dirac basis, the negative energy spinors χare order
v/ccompared with the positive energy spinors χ∼v/cφ
[35]. Thus, in the nonrelativistic limit, the negative energy
state spinors are negligible. In the nonrelativistic limit, the
8×8 matrix Gor’kov system then reduces to the standard
4×4 matrix system.
In the nonrelativistic limit, the pairing term ψTψbecomes
φTφ. The spinor φhas components:
φ=φ1,↑
φ2,↓.(13)
Therefore, the terms of the pairing function which persist
in the nonrelativistic limit are those in the upper left block.
For the relativistic model to reduce to BCS s-wave singlet
pairing in the nonrelativistic limit, it is necessary to choose
a term which is off-diagonal and antisymmetric in the upper
left block of the matrix, i.e., a matrix whose upper left block
is of the form ( 0±1
∓10
).
The representation of a pairing between a state and its time-
reversed counterpart in the Clifford algebra basis is iγ5γ0γ2,
which is a Lorentz scalar. The explicit matrix form of this term
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APPEARANCE OF ODD-FREQUENCY … PHYSICAL REVIEW B 108, 014510 (2023)
in the Dirac basis is
−iγ5γ2γ0=⎛
⎜
⎜
⎜
⎝
0100
−1000
0001
00−10
⎞
⎟
⎟
⎟
⎠
.(14)
Not all possible Lorentz invariant pairing terms lead to com-
posites which can be identified as Cooper pairs. Some of the
pairing terms are pairs between positive and negative energy
states, which have the interpretation of excitons in condensed
matter. These pairing terms are block off-diagonal in their
matrix structure.
B. The Nambu-Gor’kov system
Analogously to the nonrelativistic formalism of supercon-
ductivity, we construct bispinor doublets in Nambu space [36].
Here, in the relativistic case, the Nambu-Dirac spinors are
eight component spinors which we write as
(x):=ψ(x)
ψT(x),(x):=
⎛
⎜
⎜
⎜
⎝
ψ(x)ψT(x).
⎞
⎟
⎟
⎟
⎠
(15)
From these 8-component Nambu-Dirac spinors, we can con-
struct a relativistic Gor’kov Green function [37]:
G(x,y)=−iT(x)(y)
=−iTψ(x)ψ(y)−iTψ(x)ψT(y)
−iTψT(x)ψ(y)−iTψT(x)ψT(y)
=:G(x,y)−iF(x,y)
−iF(x,y)−G(y,x)T,(16)
where xand yare position 4-vectors, and Tis the time-
ordering operator. G(x,y) and F(x,y) are the relativistic
normal and anomalous Green functions, respectively. The
Gor’kov Green function is an 8 ×8 matrix with the normal
and anomalous Green functions G(x,y) and F(x,y) each be-
ing 4 ×4 matrices. In the following analysis, we assume the
absence of external fields A=0 such that /
D=/
∂.
The Gor’kov Green function obeys the equation of motion:
i/
∂−m(x)
(x)i/
∂T+mG(x,y)=1δ(x−y),(17)
where /
∂T=(γμ)T∂μ, and it is understood that m↔m1.
The Green function equations of motion are obtained from
multiplying Eqs. (3) and (4)byψand taking the time-
ordered expectation value as well as utilizing the mean-field
self-consistency relations in Eqs. (5) and (6). Upon Fourier
transform to momentum space, the Gor’kov Eq. (17) becomes
/
k−m
/
kT+mG(k)=1,(18)
/
k−m
/
kT+mG(k)−iF(k)
−iF(k)−G(−k)T=1,(19)
where /
kT=(γμ)Tkμ. This expression yields the set of cou-
pled equations
(/
k−m)G(k)−iF(k)=1,(20a)
−(/
kT+m)G(−k)T−iF(k)=1,(20b)
−(/
k−m)iF(k)−G(−k)T=0,(20c)
−(/
kT+m)iF(k)+G(k)=0.(20d)
The second pair of equations, Eqs. (20c) and (20d), can
be rewritten to provide closed-form solutions for the first
pair:
−(/
kT+m)iF(k)+G(k)=0 (20d)
F(k)=i1
/
kT+mG(k),(20d)
(/
k−m)iF(k)+G(−k)T=0 (20c)
G(−k)T=−−1(/
k−m)iF(k).(20c)
Inserting these expressions into the first pair of the Gor’kov
Eqs. (20a) and (20b) yields solutions for the normal and
anomalous Green functions:
G(k)=/
k−m+1
/
kT+m−1
,(21)
iF(k)=[(/
kT+m)−1(/
k−m)−]−1.(22)
It is difficult to obtain explicit analytic forms for the Green
functions with general pairing due to their matrix structure.
However, solutions may be obtained in a straightforward
manner for a given form of the pairing function . A com-
prehensive more detailed analysis of the relativistic Gor’kov
system can be found in Refs. [31,32].
For a gap function which transforms as the Lorentz scalar
=iγ5γ0γ2, the anomalous Green function in Eq. (22)
takes the form:
F(k)=⎛
⎜
⎜
⎜
⎜
⎝
0f(k)0 0
−f∗(k)0 0 0
000f(k)
00−f∗(k)0
⎞
⎟
⎟
⎟
⎟
⎠
,(23)
where
if(k)=
−k2
0+k2+m2+||2.(24)
Here, f(k) is a scalar function, and F(k) is the full matrix
function. Here, we write k0for the zeroth component of
the 4-momentum kand write kfor the spatial components.
This form of the pairing potential has previously been shown
to yield a nontrivial solution to the self-consistency relation
[29–32]. For concreteness, this function can be expressed in
the explicit form of the components of ψ(k) given by the
Fourier transform of the definition Eq. (16). The general ex-
plicit form of F(k) from its definition in all its components
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PATRICK J. WONG AND ALEXANDER V. BALATSKY PHYSICAL REVIEW B 108, 014510 (2023)
is
F(k)=Tψ(k)ψT(−k)
=T"⎛
⎜
⎜
⎜
⎝
φ1φ1φ1φ2φ1χ1φ1χ2
φ2φ1φ2φ2φ2χ1φ2χ2
χ1φ1χ1φ2χ1χ1χ1χ2
φ1χ2φ2χ2χ1χ2χ2χ2
⎞
⎟
⎟
⎟
⎠#,(25)
where the elements φand χare the elements of ψnotated
in Eq. (12). Comparing this expression with the particular
solution Eq. (23) yields the expressions:
Tφ1(k)φ2(−k)=f(k),(26a)
Tφ2(k)φ1(−k)=−f∗(k),(26b)
Tχ1(k)χ2(−k)=f(k),(26c)
Tχ2(k)χ1(−k)=−f∗(k),(26d)
with all other components of Eq. (25) being 0.
In the nonrelativistic limit, the anomalous Green function
Eq. (23) reduces to the familiar 2 ×2form:
F(k)=0f(k)
−f∗(k)0
,(27)
which stems from the upper left block of the full relativistic
function Eq. (23).
III. DYNAMIC INDUCTION OF ODD-FREQUENCY
BY LORENTZ TRANSFORMATION
We now wish to consider the situation where the gap func-
tion is anisotropic, in space and/or time. Since a Lorentz boost
mixes spatial and temporal degrees of freedom, anisotropy
is needed for the Lorentz transformation to have an effect.
To demonstrate our concept of dynamically induced odd-
frequency pairing, we construct an ansatz for the gap function
which is valid in a particular reference frame. We choose a
frame in which the gap function takes the form:
(k)=0
1+αk2
0
,(28)
where 0is a constant, and αparameterizes the degree of
anisotropy. This ansatz for the pairing function is explicitly
even in the frequency k0. At high frequency, the gap func-
tion decays to zero. In general, the gap function is a
complex function which contains real and imaginary compo-
nents. We work in the regime where 0(0,k)≈const., and
0(0,k)≈0. The majority of the literature on the analysis
of pairing states stays within these assumptions. Relaxing
these assumptions will not change the general conclusion
about induction of odd-frequency states but will make analysis
of the main effect of the Lorentz boost more tedious.
For clarity, the following discussion considers only the real
part of the anomalous Green function. The imaginary part is
analyzed in Appendix Bfor completeness.
The anomalous Green function in the frame where is
given by Eq. (28) takes the form:
if(k)=
0
1+αk2
0
−k2
0+k2+m2+0
1+αk2
02.(29)
We now wish to see the transformation of this function under
a Lorentz boost. Applying a Lorentz transformation results in
a shift in the momentum 4-vector by
⎛
⎜
⎝
k0
k
k⊥
⎞
⎟
⎠→ ⎛
⎜
⎝
γ−βγ 0
−βγ γ 0
001
⎞
⎟
⎠⎛
⎜
⎝
k0
k
k⊥
⎞
⎟
⎠,(30)
where kis comprised of the components of the momentum
which are collinear in the direction of β, and k⊥consists of
the remaining orthogonal components. The boost is parame-
terized by the Lorentz factor γ=1/1−β2with normalized
velocity β=v/c.
Under this transformation, the gap function Eq. (28) trans-
forms as
(k)=0γ2k2
0+β2γ2k2
+2βγ2kk0
1+αγ2k2
0−β2γ2k2
2.(31)
The denominator is even in both T∗and P∗. The first two
terms of (k)haveT∗=+1 and P∗=+1. The last term
has T∗=−1 and P∗=−1 on account of being linear in both
k0and k, meaning that it is odd in frequency as well as
parity. Overall, the SP∗T∗condition holds as S=−1fora
spin singlet is retained after the boost.
Applying this transformation to Eq. (29) results in the
anomalous Green function taking the form:
if(k)=0(1 −β2)
1−1−αk2
β2−2αkk0β+αk2
0k2
0−k2−m2−(1−β2)22
0
[1−(1−αk2
)β2−2αkk0β+αk2
0]2.(32)
This procedure undertaken here for our analysis is analogous to a standard method of finding solutions to the Dirac equa-
tion where the solutions are first obtained in the frame where k=0, and then general finite-ksolutions are obtained by boosting
the rest-frame solution to an arbitrary frame where the momentum is finite [33].
The characteristic properties of this transformed Green function in the boosted frame can be understood from examining
the weak (low velocity) and ultra-relativistic limits. For low boost velocity, 0 <β1, we expand the transformed anomalous
Green function in orders of β. To first order in the boost velocity β, the anomalous Green function is
if(k)=
0
1+αk2
0
−k2
0+k2+m2+0
1+αk2
02+k0kβ2α0−k2
0+k2+m21+αk2
02−2
0
−k2
0+k2+m21+αk2
0+2
02+O(β2).(33)
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APPEARANCE OF ODD-FREQUENCY … PHYSICAL REVIEW B 108, 014510 (2023)
In the boosted frame, we now analyze the system in imaginary time τ=−it in terms of the Matsubara frequency k0→iωn=
πkBT(2n+1). The anomalous Green function in Matsubara frequency is defined as F(τ, x)=Tτψ(τ, x)ψT(0,0), where Tτ
is the time-ordering operator in imaginary time τ. In Matsubara frequency, the boosted anomalous Green function is
f(iωn,k)=
0
1−αω2
n
ω2
n+k2+m2+0
1−αω2
n2+iωnkβ2α0ω2
n+k2+m21−αω2
n2−2
0
ω2
n+k2+m21−αω2
n+2
02+O(β2).(34)
The transition to Matsubara frequency is introduced only at this stage after the relativistic change of reference frame to avoid
ambiguities associated with defining the transformation properties of temperature, and hence Matsubara frequency, under
Lorentz transformations [38–42].
In the transformed function Eq. (34), we see that the first-order relativistic correction under a Lorentz boost is exclusively odd
in Matsubara frequency: The first-order correction term contains terms of order ωn,ω3
n, and ω5
n. This shows that, for any finite
boost β>0, an odd-frequency component is induced dynamically in the anomalous Green function. This is the primary result
of this paper. The appearance of odd frequency already at O(β1) reinforces the notion that odd-frequency pairing is a ubiquitous
state. For comparison, spin-orbit coupling, which is known to occur in a wide variety of materials, is an effect which occurs at
O(β2)[35,43].
The magnitude of the odd-frequency contribution to the anomalous propagator is proportional to the magnitude of the boost,
i.e., v/c.
Next, we wish to examine the behavior of the anomalous Green function under a boost in the ultra-relativistic limit where
β∼1−. Performing an expansion of f(k) around (1 −β), we find that
f(iωn,k)=20(1 −β)
α(iωn−k)2−ω2
n−k2−m2−04−3αk2
+2αkiωn−αω2
n(1 −β)2
α2(iωn−k)4−ω2
n−k2−m2+O[(1 −β)3],(35)
where the first-order term can be rewritten as
20(1 −β)
α(iωn−k)2−ω2
n−k2−m2
=20−ω2
n+k2
+2kiωn(1 −β)
α−ω2
n+k2
2(−ω2
n−k2−m2)
.(36)
Here, we see that this term contains terms which are both even
and odd powers of Matsubara frequency. Therefore, in the
ultra-relativistic limit, the superconducting order parameter is
neither exclusively even or odd frequency for a given order
of (1 −β). Terms of both symmetries must be present. The
second-order term in the ultra-relativistic expansion shows
the same characteristic, indicating that this feature is generic
in the ultra-relativistic limit. In the extreme limit of β=
1, the anomalous Green function of the spin-singlet sector
vanishes.
Note that terms which are odd under T∗are also always
odd under P∗with the combination T∗P∗=+1. This reflects
that a Lorentz transformation does not violate the SP∗T∗=
−1 restriction, as the spin degree of freedom remains in the
singlet sector S=−1.
As shown from the above, the anomalous Green function
contains both even-ωand odd-ωcomponents for any finite
boost. The odd-ωcomponents appear on equal footing with
the even-ωterms.
IV. CONCLUSIONS
In this paper, we analyzed the consequences of viewing
a conventional even-frequency superconductor from a rela-
tivistically boosted observer. We therefore conclude that the
pairing between electrons in odd-frequency Cooper pairs is
physically equivalent to the pairing between conventional
even-frequency Cooper pairs since relativistic transformations
should not change the underlying physics. The concept of
odd-frequency pairing should then be thought of as ubiquitous
as even-frequency pairing. An analogy here is with electric
and magnetic fields. ubiquity of magnetic fields in compar-
ison with electric fields. Macroscopic electric and magnetic
fields are physically distinct with nonidentical properties;
however, their interpretation as components of the electro-
magnetic field imply that one is no more ubiquitous than the
other. Similarly, we can say that even-frequency pairing in
superconductivity is no more ubiquitous than odd-frequency
pairing.
We note that the notion of a relativistically boosted Cooper
pair has previously been studied with respect to the en-
tanglement structure of the constituent fermions [44,45].
The authors of these studies concluded that an isotropic
s-wave spin-singlet Cooper pair would acquire spin-triplet
components under a Lorentz boost due to Wigner rota-
tion [46]. This result appears to differ from the relativistic
theory of superconductivity constructed above based on
Refs. [29–32], as for example, the spin-singlet pairing cor-
relation function in Eq. (23) is invariant under Lorentz
transformations and does not acquire triplet components (re-
call that spin-triplet pairing manifests as even parity under
matrix transposition). This difference in the result is pre-
sumably related to the fact that, in Refs. [44,45], states are
labeled with the nonrelativistic scheme of different quantum
numbers for orbital- and spin-angular momentum separately,
despite these not being good quantum numbers relativis-
tically. With respect to the conclusions of Refs. [44,45],
where a boosted spin singlet transforms into a pair with
singlet and triplet components, our work presented above
can be said to occur in the spin-singlet projection of the
transformed pair, and thus, our conclusions regarding the
014510-7

PATRICK J. WONG AND ALEXANDER V. BALATSKY PHYSICAL REVIEW B 108, 014510 (2023)
emergence of an odd-frequency channel are valid regard-
less.
The role of relativity with respect to odd-frequency pair-
ing has ramifications to other systems beyond the simple
condensed matter model here. It is suspected that there
exists a region of the quantum chromodynamic phase dia-
gram in which quarks form pairs in a condensate, which is
known as color superconductivity [47]. As we showed in this
paper, there exists a relationship between relativity and odd-
frequency pairing, which implies that odd-frequency color
Cooper pairs in principle should appear in the color super-
conducting state.
In summary, we presented a calculation demonstrating the
dynamical emergence of an odd-ωsuperconductivity channel
as a result of a Lorentzian change in reference frame. We first
constructed a theory of superconductivity which is relativisti-
cally invariant for constant gap function . We then supposed
a temporally anisotropic form of the gap function even in
frequency in a particular reference frame. The anomalous
Green function for this temporally anisotropic gap function
was then boosted into a new frame, where we found that
an odd-ωchannel appears. The result of this calculation is
that it provides an existence proof for the notion that odd-ω
superconductivity is a valid channel inherent to superconduc-
tors which appears ubiquitously. Furthermore, in this paper,
we imply that for superconductivity to be compatible with
the fundamental symmetry of special relativity, odd frequency
should generally be considered.
ACKNOWLEDGMENTS
We are grateful to A. Black-Schaffer and M. Geilhufe for
useful discussions. We acknowledge support from the Knut
and Alice Wallenberg Foundation KAW 2019.0068, European
Research Council under the European Union Seventh Frame-
work ERS-2018-SYG 810451 HERO, and the University of
Connecticut.
APPENDIX A: CONVENTIONS
In this Appendix are some conventions of the gamma ma-
trices used in the main text:
γ0=10
0−1,γ
j=0σj
−σj0,(A1)
γ5:=iγ0γ1γ2γ3=01
10
,(A2)
{γμ,γν}=2ημν ,η=diag[+,−,−,−],(A3)
(γ0)†=γ0,(γj)†=−γj.(A4)
APPENDIX B: IMAGINARY PART OF f(k)
In this Appendix, we analyze the imaginary part of the
anomalous Green function in Eq. (29) for the form of the
gap function in Eq. (28) under application of the boost in
Eq. (30). For an even-ωgap function, it holds that (ω)
is even in ω, and (ω) is odd in ω. Likewise, for an odd-ω
gap function, (ω) is odd in ω, and (ω)iseveninω.
For completeness, it is necessary to check that the Lorentz
transformation of the anomalous Green function as performed
in the main text satisfies these conditions.
Taking the anomalous Green function in Eq. (29), we apply
the prescription of k0→ k0+iδfor 0 <δ1 to obtain
f(k)=
0
1+α(k0+iδ)2
−(k0+iδ)2+k2+m2+0
1+α(k0+iδ)22.(B1)
We obtain the imaginary part by expanding the expression to
first order in δ. To first order in δ,O(δ), the imaginary part of
the anomalous Green function is
f(k)|O(δ)=−2k001−α−2k2
0+k2+m21+αk2
02+α2
0
−k2
0+k2+m21+αk2
02+2
02.(B2)
To this function, we apply the Lorentz boost from Eq. (30). The weak asymptotic limit of 0 <β 1 takes the form:
f(k)|O(δ)=2k0β−k2
0+k2+m21+αk2
03{1−α(k2+m2)+α[1 +3α(k2+m2)]}k2
0−4α2k4
0
−k2
0+k2+m21+αk2
02+2
03
−1+αk2
01+αk2
08−12α(k2+m2)+19αk2
02
0−α4
0
−k2
0+k2+m21+αk2
02+2
03+O(β2).(B3)
By inspection, this function is even in k0and odd in k, which is the opposite parity of the first-order term in the real part in
Eq. (33) as expected.
In the ultra-relativistic asymptotic limit β∼1−, the imaginary part to first order in (1 −β) reads as
f(k)|O(δ)=2√20(k0+k)√1−β
α−k2
0+k2
−k2
0+k2+m22+O[(1 −β)3/2].(B4)
014510-8

APPEARANCE OF ODD-FREQUENCY … PHYSICAL REVIEW B 108, 014510 (2023)
FIG. 3. The anomalous spectrum B(ω) in arbitrary units with the
parameters 0=1.0, m=0.5, and α=1.0, without boost (β=0,
black dashed) and with finite boost (β=0.1, red solid). Odd-ωpair-
ing corresponds to even-ωsymmetry in the imaginary part of f(k)
plotted here. Under a finite boost, B(ω) is a function which contains
both even- and odd-ωcontributions. The even-ωcontribution is small
in magnitude compared with the odd-ωcontribution, cf. Fig. 4.
As in the real part of Eq. (35), this function contains pieces
which are both even and odd in k0. This matches the result that
the even- and odd-frequency components get mixed together
to the same order in the boost parameter in the ultra-relativistic
limit.
For odd-frequency pairing, the anomalous spectrum, which
is given by the imaginary part of the anomalous Green func-
tion, is even frequency. The anomalous spectral function is
obtained from the imaginary part of the retarded anomalous
Green function as [48]
B(ω)=−1
πd3kf(ω+i0+,k).(B5)
The symmetry of the spectral function can be observed from
its decomposition into even and odd parts as Beven(ω)=
1
2[B(ω)+B(−ω)] and Bodd(ω)=1
2[B(ω)−B(−ω)].
In the absence of a relativistic boost, there is no even-ω
contribution to B(ω). For a finite boost, B(ω) acquires an
FIG. 4. The even-ωpart Beven(ω)=1
2[B(ω)+B(−ω)] of the
anomalous spectrum for finite boost, β=0.1, plotted in arbitrary
units with the parameters 0=1.0, m=0.5, and α=1.0. The
even-ωin the imaginary part of the spectrum plotted here corre-
sponds to odd-frequency pairing. In the unboosted case Beven (ω)=
0∀ω. Note that the even contribution is an order of magnitude
smaller than the total anomalous spectral function plotted in Fig. 3.
even-frequency contribution. The anomalous spectral func-
tion for a small finite boost, β=0.1, in comparison with
the unboosted spectrum is plotted in Fig. 3with the pa-
rameters 0=1.0, m=0.5, and α=1.0. The spectrum is
computed numerically with δ=0.005. We compute these
integrals numerically with finite bandwidths. The integrals
were computed for a variety of bandwidths and were found
to produce the same solution. We therefore conclude that the
integrals are convergent.
The even contribution Beven(ω) is plotted in Fig. 4.
While the overall spectrum appears odd, there is a slight
even component which emerges. As shown by Fig. 4,the
even contribution is an order of magnitude smaller than
the total function. Graphically, we see that Beven(ω)= 0for
β>0, indicating that a finite boost induces an even-frequency
component in the imaginary part of the anomalous Green
function, as would be expected in the case of odd-frequency
pairing.
These computations show that the imaginary part of the
anomalous Green function behaves as expected for the gener-
ation of odd-frequency components in the real part.
[1] V. L. Berezinskii, New model of the anisotropic phase of super-
fluid He3, ZhETF Pis. Red. 20, 628 (1974) [JETP Lett. 20, 287
(1974)].
[2] A. Balatsky and E. Abrahams, New class of singlet supercon-
ductors which break time reversal and parity, Phys. Rev. B 45,
13125(R) (1992).
[3] J. Linder and A. V. Balatsky, Odd-frequency superconductivity,
Rev. Mod. Phys. 91, 045005 (2019).
[4] Y. Tanaka, M. Sato, and N. Nagaosa, Symmetry and topology
in superconductors—Odd-frequency pairing and edge states,
J. Phys. Soc. Jpn. 81, 011013 (2012).
[5] Y. Tanaka and S. Tamura, Theory of surface Andreev
bound states and odd-frequency pairing in supercon-
ductor junctions, J. Supercond. Nov. Magn. 34, 1677
(2021).
[6] J. Cayao, C. Triola, and A. M. Black-Schaffer, Odd-frequency
superconductivity pairing in one-dimensional systems, Eur.
Phys. J. Spec. Top. 229, 545 (2020).
[7] H. Ebisu, B. Lu, J. Klinovaja, and Y. Tanaka, Theory of time-
reversal topological superconductivity in double Rashba wires:
Symmetries of Cooper pairs and Andreev bound states, Prog.
Theor. Exp. Phys. 2016, 083I01 (2016).
[8] A. F. Volkov, F. S. Bergeret, and K. B. Efetov, Odd Triplet Su-
perconductivity in Superconductor-Ferromagnet Multilayered
Structures, Phys. Rev. Lett. 90, 117006 (2003).
[9] F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Odd triplet
superconductivity and related phenomena in superconductor-
ferromagnet structures, Rev. Mod. Phys. 77, 1321 (2005).
[10] J. Linder and J. W. A. Robinson, Superconducting spintronics,
Nat. Phys. 11, 307 (2015).
014510-9

PATRICK J. WONG AND ALEXANDER V. BALATSKY PHYSICAL REVIEW B 108, 014510 (2023)
[11] A. S. Di Bernando, S. Diesch, Y. Gu, J. Linder, G. Divitini, C.
Ducati, E. Scheer, M. G. Blamire, and J. W. A. Robinson, Sig-
nature of magnetic-dependent gapless odd frequency states at
superconductor/ferromagnet interfaces, Nat. Commun. 6, 8053
(2015).
[12] A. V. Balatsky, S. S. Pershoguba, and C. Triola, Odd-frequency
pairing in conventional Josephson junctions, arXiv:1804.07244.
[13] A. M. Black-Schaffer and A. V. Balatsky, Proximity-induced
unconventional superconductivity in topological insulators,
Phys.Rev.B87, 220506(R) (2013).
[14] A. M. Black-Schaffer and A. V. Balatsky, Odd-frequency su-
perconductivity pairing in topological insulators, Phys.Rev.B
86, 144506 (2012).
[15] J. Linder and J. W. A. Robinson, Strong odd-frequency corre-
lations in fully gapped Zeeman-split superconductors, Sci. Rep.
5, 15483 (2015).
[16] A. M. Black-Schaffer and A. V. Balatsky, Odd-frequency super-
conducting pairing in multiband superconductors, Phys.Rev.B
88, 104514 (2013).
[17] L. Komendová, A. V. Balatsky, and A. M. Black-Schaffer, Ex-
perimentally observable signatures of odd-frequency pairing in
multiband superconductors, Phys. Rev. B 92, 094517 (2015).
[18] R. Heid, On the thermodynamic stability of odd-frequency su-
perconductors, Z. Phys. B 99, 15 (1995).
[19] Y. V. Fominov, Y. Tanaka, Y. Asano, and M. Eschrig,
Odd-frequency superconducting states with different types of
Meissner response: Problem of coexistence, Phys. Rev. B 91,
144514 (2015).
[20] D. Belitz and T. R. Kirkpatrick, Properties of spin-triplet, even-
parity superconductors, Phys.Rev.B60, 3485 (1999).
[21] D. Solenov, I. Martin, and D. Mozyrsky, Thermodynamical
stability of odd-frequency superconducting state, Phys.Rev.B
79, 132502 (2009).
[22] H. Kusunose, Y. Fuseya, and K. Miyake, On the puzzle of
odd-frequency superconductivity, J. Phys. Soc. Jpn. 80, 054702
(2011).
[23] C. Triola and A. V. Balatsky, Odd-frequency superconductivity
in driven systems, Phys.Rev.B94, 094518 (2016).
[24] C. Triola and A. V. Balatsky, Pair symmetry conversion in
driven multiband superconductors, Phys. Rev. B 95, 224518
(2017).
[25] J. Cayao, C. Triola, and A. M. Black-Schaffer, Floquet engi-
neering bulk odd-frequency superconducting pairs, Phys. Rev.
B103, 104505 (2021).
[26] Throughout the remainder of this paper, we use the term rel-
ativistic to refer to Einsteinian relativity and nonrelativistic to
refer to its vclimit as well as to Galilean relativity (which
forms the basis of conventional condensed matter theory).
[27] J. Swain, A. Widom, and Y. Srivastasva, The Dirac equation in
low energy condensed matter physics, PoS (ICHEP2016), 346
(2017).
[28] J. Wyzula, X. Lu, D. Santos-Cottin, D. K. Mukherjee, I.
Mohelský, F. L. Mardelé, J. Novák, M. Novak, R. Sankar, Y.
Krupko et al., Lorentz-boost-driven magneto-optics in a Dirac
nodal-line semimetal, Adv. Sci. 9, 2105720 (2022).
[29] K. Capelle and E. K. U. Gross, Relativistic framework for mi-
croscopic theories of superconductivity. I. The Dirac equation
for superconductors, Phys.Rev.B59, 7140 (1999).
[30] K. Capelle and E. K. U. Gross, Relativistic framework for mi-
croscopic theories of superconductivity. II. The Pauli equation
for superconductors, Phys.Rev.B59, 7155 (1999).
[31] T. Ohsaku, BCS and generalized BCS superconductivity in
relativistic quantum field theory: Formulation, Phys.Rev.B65,
024512 (2001).
[32] T. Ohsaku, BCS and generalized BCS superconductivity in rel-
ativistic quantum field theory. II. Numerical calculations, Phys.
Rev. B 66, 054518 (2002).
[33] M. E. Peskin and D. V. Schroeder, Introduction to Quantum
Field Theory (CRC Press, Boca Raton, 1995).
[34] C. Itzykson and Jean-Bernard Zuber, Quantum Field Theory
(McGraw-Hill, New York, 1980).
[35] R. P. Feynman, Quantum Electrodynamics (Westview Press,
Boulder, 1998).
[36] Y. Nambu, Quasi-particles and gauge invariance in the theory
of superconductivity, Phys. Rev. 117, 648 (1960).
[37] L. P. Gor’kov, Microscopic derivation of the Ginzburg-Landau
equations in the theory of superconductivity, J. Exptl. Theoret.
Phys. 36, 1918 (1959) [Sov. Phys. JETP 36, 1364 (1959)].
[38] C. Farías, V. A. Pinto, and P. S. Moya, What is the temperature
of a moving body? Sci. Rep. 7, 17657 (2017).
[39] J. Bros and D. Buchholz, Towards a relativistic KMS-condition,
Nucl. Phys. B 429, 291 (1994).
[40] G. L. Sewell, Statistical thermodynamics of moving bodies,
Rep. Math. Phys. 64, 285 (2009).
[41] G. L. Sewell, On the question of temperature transformations
under Lorentz and Galilei boosts, J. Phys. A: Math. Theor. 41,
382003 (2008).
[42] J. A. Heras and M. G. Osorno, A note on the relativistic tem-
perature, Eur.Phys.J.Plus137, 423 (2022).
[43] A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A. Duine,
New perspectives for Rashba spin-orbit coupling, Nat. Mater.
14, 871 (2015).
[44] J. Dunningham, V. Palge, and V. Vedral, Entanglement and
nonlocality of a single relativistic particle, Phys. Rev. A 80,
044302 (2009).
[45] V. Palge, V. Vedral, and J. A. Dunningham, Behavior and entan-
glement of Cooper pairs under relativistic boosts, Phys. Rev. A
84, 044303 (2011).
[46] E. Wigner, On unitary representations of the inhomogeneous
Lorentz group, Ann. Math. 40, 149 (1939).
[47] M. G. Alford, A. Schmitt, K. Rajagopal, and T. Schäfer, Color
superconductivity in dense quark matter, Rev. Mod. Phys. 80,
1455 (2008).
[48] As the boosted anomalous propagator is odd in k||, the inte-
gration domain is only over the half line on this component,
k|| ∈[0,∞), otherwise Beven (ω) would trivially be zero.
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