Competition of chiral soliton lattice and Abrikosov vortex lattice in QCD with isospin chemical potential
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Eur. Phys. J. C manuscript No.
(will be inserted by the editor)
Competition of chiral soliton lattice and Abrikosov vortex lattice
in QCD with isospin chemical potential
Martin Spillum Grønlia,1 , Tomáš Braunerb,2
1 Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
2 Department of Mathematics and Physics, University of Stavanger, N-4036 Stavanger, Norway
arXiv:2201.07065v1 [hep-ph] 18 Jan 2022
Abstract We investigate the thermodynamics of two-flavor lattice Monte Carlo simulations of QCD suffer from at non-
quark matter in presence of nonzero isospin chemical po- zero baryon chemical potential, µB . This conundrum has
tential and external magnetic field. It is known that at suf- triggered interest in alternative theories of dense quark mat-
ficiently high isospin chemical potential, charged pions un- ter where the sign problem is absent.
dergo Bose–Einstein condensation (BEC). The condensate One of such alternative theories is QCD with two light
behaves as a superconductor, exhibiting Meissner effect in quark flavors and nonzero isospin chemical potential, µI [1].
weak external magnetic fields. Stronger fields stress the su- We will refer to this theory as isospin QCD (iQCD). In iQCD,
perconducting state, turning it first into an Abrikosov lat- low-density matter is realized by bosons rather than fermio-
tice of vortices, and eventually destroying the condensate nic nucleons. When µI exceeds the vacuum pion mass, mπ ,
altogether. On the other hand, for sufficiently strong mag- charged pions undergo Bose–Einstein condensation (BEC).
netic fields and low-to-moderate isospin chemical potential, Since the energy scale at which the phase transition to the
the ground state of quantum chromodynamics (QCD) is ex- BEC phase occurs is controlled by the pion mass, it is possi-
pected to be a spatially modulated condensate of neutral pi- ble to analyze the properties of such dense matter using the
ons, induced by the chiral anomaly: the chiral soliton lattice low-energy effective field theory of QCD: the chiral pertur-
(CSL). We map the phase diagram of QCD as a function bation theory (ChPT) [2]. Together with the absence of sign
of isospin chemical potential and magnetic field in the part problem in iQCD, this opens the possibility to confront pre-
of the parameter space accessible to a low-energy effective cision analytic computations within ChPT [3–5] with state-
field theory description of QCD. Our main result is an ex- of-the-art lattice simulations of iQCD [6].
plicit account of the competition between the CSL and the
Abrikosov vortex lattice. This is accomplished by adopting The effect of strong magnetic fields on quark matter [7,
a fast numerical algorithm for finding the vortex lattice solu- 8] is of great phenomenological interest due to its relevance
tion of the equation of motion and the corresponding Gibbs for heavy-ion collisions and the astrophysics of neutron stars.
energy. We find that the Abrikosov vortex lattice phase per- In particular, the magnetism of quark matter exhibits intrigu-
sists in the phase diagram, separating the uniform charged ing properties arising from the chiral anomaly of QCD [9].
pion BEC phase from the CSL phase. The precise layout of In Ref. [10] it was shown that at nonzero µB and in a suffi-
the phase diagram depends sensitively on the choice of the ciently strong magnetic field, the ground state of QCD car-
vacuum pion mass. ries a spatially modulated condensate of neutral pions, dub-
bed chiral soliton lattice (CSL). This novel state of matter
arises from the anomalous coupling of neutral pions to the
1 Introduction electromagnetic field, and carries a topologically generated
baryon number. Similar topological crystalline phases were
Understanding the phase diagram of quantum chromody- subsequently predicted to occur in dense QCD under rota-
namics (QCD) remains one of the major unresolved prob- tion [11, 12] and in QCD-like theories with quarks in a real
lems in particle physics. The slow pace of progress towards or pseudoreal representation of the color gauge group [13].
this goal is largely due to the infamous sign problem, which The aim of the present paper is to investigate the ther-
a E-mail: martin.gronli96@gmail.com modynamics of iQCD in an external magnetic field (and at
b E-mail: tomas.brauner@uis.no zero temperature). While the presence of the magnetic field2
reinstates the sign problem, the physical properties of iQCD review the physics of uniform charged pion BEC and calcu-
in external magnetic fields make this goal worth pursuing late its Gibbs energy in the external field. This serves as a
on its own. Namely, the pion BEC carries electric charge starting point for finding the thermodynamic equilibrium by
and as such behaves as an electromagnetic superconductor. comparing several candidate ground states. In Sec. 4 we do
Weak magnetic fields should be entirely expelled from such the same for the neutral pion CSL. While trading µB for µI
a medium by the Meissner effect. On the other hand, suf- as compared to Ref. [10] is next-to-trivial, a novel element
ficiently strong magnetic fields should destroy the conden- here is that we likewise treat the magnetic field as dynamical
sate. In so-called type-II superconductors, the transition be- and pin down the CSL state by minimizing the Gibbs energy.
tween the uniform BEC phase and the normal state without This stands in contrast to all previous accounts of the CSL
BEC proceeds in two steps as the magnetic field is cranked in quark matter, where the magnetic field was treated as a
up. In the intermediate stage, the magnetic flux penetrates fixed background. Section 5 is the core of the paper. Here we
the superconducting medium through a periodic array of describe in detail the algorithm for finding a self-consistent
vortices, called the Abrikosov lattice. A first sketch of the vortex lattice solution to the equations of motion. Finally, in
phase diagram of the pion superfluid as a function of µI and Sec. 6 we put all the bits together and map the phase dia-
magnetic field appeared nearly a half-century ago [14], in- gram of iQCD as a function of isospin chemical potential
cluding an explicit solution for a single isolated vortex [15]. and magnetic field. The concluding Sec. 7 is reserved for
These early analyses were recently revisited and improved additional comments. In Appendix A, we return to the ques-
using the model independent approach of ChPT [16]. The tion whether iQCD in an external magnetic field actually is
Abrikosov vortex lattice in iQCD in external magnetic fields a type-I or a type-II superconductor. We derive a sufficient
was considered in Ref. [17]. condition for the existence of a phase with an inhomoge-
The present paper improves upon the existing literature neous pion condensate in the phase diagram, and show that
in two main aspects. First, the chiral anomaly affects the this condition is satisfied for the physical pion mass.
magnetism of iQCD in the same way as that of QCD with
nonzero µB . Hence, sufficiently strong magnetic fields will
inevitably lead to the formation of CSL. It is therefore man- 2 Effective field theory for QCD at nonzero isospin
datory to consider the competition of charged pion BEC and chemical potential and magnetic field
neutral pion CSL to have a physically adequate picture of
the phase diagram of iQCD. Yet, the possible existence of The analysis presented in this paper is based on the follow-
CSL in iQCD has not been investigated before.1 Without ing schematic Lagrangian,
a detailed analysis, it is not even clear whether the phase L = LQED + LChPT + LWZW. (1)
carrying the Abrikosov lattice of charged pion vortices will
survive in the phase diagram. The first contribution is the usual free Maxwell Lagrangian
Second, the analysis of the vortex lattice in Ref. [17] with a gauge-fixing term if needed,
was carried out using a semi-analytic approximation, valid 1
near the upper critical field where the back-reaction of the LQED = − Fµν F µν + Lg.f. , (2)
4
pion condensate to the external magnetic field is very weak.
In this paper, we provide a numerical solution for the vor- with Fµν ≡ ∂µ Aν − ∂ν Aµ . Moreover,
tex lattice that is complete within the tree level of ChPT
fπ2
Tr(Dµ Σ † Dµ Σ ) + m2π Tr(Σ + Σ † )
without further approximations. We do so by adapting a fast LChPT = (3)
4
momentum-space numerical algorithm for finding vortex lat-
tice solutions in the Ginzburg–Landau theory, known in con- is the leading-order Lagrangian of ChPT for two light quark
densed-matter physics [19, 20]. An alternative numerical ap- flavors. This depends on two empirically determined param-
proach, based on coordinate space discretization, was put eters, the pion decay constant fπ and the vacuum pion mass
forward recently in Ref. [21]. mπ . The matrix Σ ∈ SU(2) encodes the three pion degrees
The plan of the paper is as follows. Section 2 reviews the of freedom. Its covariant derivative Dµ Σ couples it to gauge
basic setup of ChPT for two light quark flavors including the fields associated with the chiral symmetry of QCD. In pres-
effects of nonzero isospin chemical potential and magnetic ence of the electromagnetic gauge potential Aµ , the covari-
field. We take this opportunity to summarize at one place the ant derivative reads
technical assumptions that the following sections base upon. i
Dµ Σ ≡ ∂µ Σ − eAµ τ3 , Σ , (4)
A brief reminder of the thermodynamics of physical systems 2
in external magnetic fields is also included. In Sec. 3, we
where τ3 is the third Pauli matrix and e is the electromag-
1 Theonly exception we are aware of is Ref. [18] which, however, only netic coupling. A nonzero isospin chemical potential µI is
considered a limiting case of CSL, the neutral pion domain wall. implemented as a constant temporal background, A0 = µI /e.3
Finally, the last term in Eq. (1) is the anomalous Wess– In practice, it does not seem feasible to find the absolute
Zumino–Witten (WZW) term. The full WZW term depends minimum of the Gibbs energy directly. What we will rather
on all three pion fields in a rather complicated manner which do is to find several specific solutions to the classical equa-
we shall not reproduce here; see e.g. Ref. [9]. We will only tions of motion and evaluate their respective Gibbs energies.
need a reduced version of the WZW term obtained by dis- The phase diagram is then mapped by comparing the Gibbs
carding the charged pion degrees of freedom and only keep- energies of these candidate equilibrium states. The follow-
ing the neutral pion field, π 0 . This reduced WZW term reads ing candidate states will be included in the analysis:
e µI – The vacuum state, corresponding to Σ = 1.
LWZW = B · ∇φ , (5)
8π 2 – The uniform charged pion BEC state, analyzed in Sec. 3.
where φ ≡ π 0 / fπ is a dimensionless neutral pion field. This – The neutral pion CSL state, investigated in Sec. 4.
differs from the WZW term at nonzero µB used in Ref. [10] – The Abrikosov vortex lattice state, explored in Sec. 5.
to study the CSL in QCD only by a factor of two [22].
2.2 Power counting
2.1 Thermodynamics of iQCD in external magnetic fields
An effective field theory such as ChPT relies on a power-
In this paper, we will address the equilibrium properties of counting scheme to be predictive. Within ChPT (see for in-
iQCD in an external magnetic field. This means that we will stance Ref. [24]) it is usual to count each derivative as well
assume the magnetic field, and hence the potential Aµ , to be as the pion mass as O(p1 ). Including the kinetic term for
static. We will also assume the absence of any electric field. the electromagnetic field in the Lagrangian (1) can be made
The latter is ensured if the iQCD matter is locally electri- consistent with this power-counting if we count Aµ = O(p0 )
cally neutral in equilibrium. This is a nontrivial assumption and e = O(p1 ). Likewise, µI = O(p1 ). Then, both LQED and
since both the charged pion BEC and the CSL carry elec- LChPT in Eq. (1) enter at the same counting order, O(p2 ).
tric charge. Other degrees of freedom are therefore needed This makes it among others possible to consistently analyze
to maintain local charge neutrality. The assumptions under vortices in the pion BEC phase of iQCD within ChPT.2
which such a neutralizing background can be treated as non- As to the CSL phase, an explicit power-counting scheme
dynamical were discussed, respectively for the charged pion that applies to nonzero µB was proposed recently in Ref. [25].
BEC and the CSL phase, in Refs. [16] and [23]. However, this power-counting scheme treats the chemical
With the above assumptions, the equilibrium free energy potential as a large quantity rather than a small perturba-
density of iQCD in the region of the parameter space acces- tion. Unfortunately, it appears impossible to define a power-
sible by ChPT can be written as counting in which all three parts of the Lagrangian (1) would
appear at the same order.
B2 This serves as a warning to the reader that the results
F= + HChPT + HWZW , (6)
2 we report in this paper should strictly speaking not be un-
where H indicates the canonical Hamiltonian density and derstood as predictions of an effective field theory of iQCD
the various contributions match those to the Lagrangian (1). with a well-defined derivative expansion. We can however
The free energy is a local functional of the pion fields and the still treat Eq. (1) as a model within which we can map the
magnetic field, B = ∇ × A. This field is however affected by phase diagram of iQCD. Our model is defined as the min-
the back-reaction of the iQCD matter. In order to correctly imal chiral Lagrangian that captures all the physics needed
account for the boundary conditions imposed by the external to address the competition of charged pion condensation and
magnetic field, we have to Legendre-transform to the Gibbs anomaly-induced neutral pion condensation.
energy density,
G ≡ F − B · H, (7) 3 Uniform charged pion condensate
where H = B − M and M is the magnetization of the iQCD
In the vacuum state, Σ = 1, the HChPT term in Eq. (8) re-
matter in the external field. In order to determine the ther-
duces to −mπ2 fπ2 , whereas HWZW vanishes. Minimizing the
modynamic equilibrium, we therefore have to minimize the
Gibbs energy (8) with respect to the dynamical vector poten-
spatial integral of
tial A then gives (not surprisingly) B = H, and subsequently
B2
G= − B · H + HChPT + HWZW (8) H2
2 Gvac = − − mπ2 fπ2 . (9)
2
as a functional of A and the pion fields. The external field H
is fixed, and assumed to be uniform throughout this paper. 2 We are grateful to Naoki Yamamoto for pointing this out to us.4
For µI > mπ , another uniform stationary state, carrying a 4 Chiral soliton lattice
BEC of charged pions, exists. This state is a superconductor
exhibiting Meissner effect, that is B = 0. This corresponds The CSL state carries a spatially inhomogeneous condensate
to the electromagnetic potential Aµ = (µI /e, 0). The pion of neutral pions. We can therefore restrict the SU(2)-valued
condensate is parameterized by [1] matrix field Σ to the ansatz Σ = eiτ3 φ , where φ is the di-
mensionless neutral pion field, introduced in Eq. (5). Upon
m2π 1 inserting this ansatz in the Lagrangian (1) and using the re-
Σ = 1 cos α + iτ2 sin α , cos α = 2
= 2, (10) sult (9) for the Gibbs energy of the vacuum, we can bring the
µI x
relative Gibbs energy density of neutral pions to the form
where x ≡ µI /mπ is a dimensionless isospin chemical po-
1 f2
tential. This result is by now so well-known that we skip the ∆ GCSL = (B − H)2 + π (∇φ )2 + 2mπ2 (1 − cos φ )
details of its derivation; see Ref. [26] for a pedagogical ac- 2 2 (14)
e µI
count. A simple algebra then leads to an expression for the − 2 B · ∇φ .
8π
Gibbs energy density of the uniform pion BEC state,
Keeping in mind that this should be treated as a local func-
1 2 2 2 1 tional of φ and A, the stationarity condition for the magnetic
GBEC = − mπ fπ x + 2 . (11)
2 x field becomes ∇ × B = 0. In order to simplify the analysis,
we will assume that the magnetic field B only varies in the
In the following, we will use the vacuum Gibbs energy (9) direction of H, which we take by choice to be oriented along
as a reference and relate the Gibbs energies of the other can- the z-axis. Together with the zero-divergence constraint on
didate states to it. Thus, the relative Gibbs energy density of B, this automatically implies that in equilibrium, B must be
the uniform charged pion BEC is a constant vector. Moreover, the −B · H term in the Gibbs
energy forces B to be parallel to H.
H2 1 2 2
1 2
The partial derivatives of φ in the transverse directions
∆ GBEC ≡ GBEC − Gvac = − mπ f π x − . (12) x, y only enter Eq. (14) squared with a positive sign. It fol-
2 2 x
lows that in the equilibrium, φ can only be modulated in the
z-direction. The corresponding one-dimensional stationarity
Should we assume that the charged pion BEC is a type-I condition for φ reads
superconductor, we could extract from Eq. (12) the critical
magnetic field that destroys the superconducting order, ∂z2 φ = m2π sin φ . (15)
m2π
1 This is the equation of motion of a simple pendulum, whose
Hc = mπ fπ x − = f π µI − , (13)
x µI closed-form solutions are, up to an overall translation [10],
in agreement with Ref. [16]. Above this field, the uniform φ (zmπ /k)
cos = sn(zmπ /k, k), (16)
charged pion BEC is thermodynamically disfavored by the 2
energy cost associated with expelling the magnetic field from
where sn is one of the Jacobi elliptic functions and k, con-
its bulk. It however turns out that the charged pion BEC is, in
strained to the range 0 ≤ k ≤ 1, the so-called elliptic modu-
fact, a type-II superconductor in a large part of the parameter
lus. The condensate Σ defined by this solution exhibits peri-
space of iQCD. Cranking up the external magnetic field does
odic modulation with the period
not destroy the order at once, but leads first to an interme-
diate phase where the condensate is penetrated by a lattice 2kK(k)
of vortices carrying magnetic flux. This phase is limited to ℓ= , (17)
mπ
the magnetic field range Hc1 < H < Hc2 , where Hc1 and Hc2
are known respectively as the lower and upper critical field. where K(k) is the complete elliptic integral of the first kind.
Below Hc1 the uniform BEC state prevails, whereas above Plugging the solution (16) back into Eq. (14), we get the
Hc2 the vacuum state is favored over any state containing a spatially averaged Gibbs energy density of the CSL state as
condensate of charged pions. a function of B and k,
It follows by definition of the various critical fields that
Hc1 < Hc < Hc2 . Until we actually determine the lower and ¯ 1 2 2 2 1 2 E(k)
∆ GCSL = (B − H) + 2mπ fπ 1 − 2 + 2
upper critical fields, Eq. (13) will therefore still serve as a 2 k k K(k)
(18)
useful reference, indicating the scale of magnetic field that e µ I mπ B
severely affects the uniform BEC state. − ,
8π kK(k)5
where E(k) is the complete elliptic integral of the second setting mπ = 0 in Eq. (14), upon which the Gibbs energy
kind. This is a quadratic polynomial in B, which gives at density only depends on B and the gradient of φ and its di-
once the solution for the magnetic field B in terms of H, rect minimization is trivial. For the record, we remark that
e µ I mπ the corresponding expression for the Gibbs energy density
B=H+ . (19) of the uniform charged pion BEC in the chiral limit follows
8π kK(k)
from Eq. (12),
Inserting this back in Eq. (18) leads finally to an expression
for the Gibbs energy density of the CSL state in terms of a 1
∆ GBEC = (H2 − fπ2 µI2 ). (25)
sole free parameter k, 2
H2 1 e µ I mπ 2
∆ G¯CSL = − H+
2 2 8π kK(k) 4.2 Comparison to CSL at fixed magnetic field
(20)
1 2 E(k)
+ 2m2π fπ2 1 − 2 + 2 . As already briefly mentioned in the introduction, previous
k k K(k)
treatments of the CSL in QCD were based on the setup where
In order to find the actual equilibrium CSL state, this is to the external magnetic field is a fixed background, which
be minimized with respect to k, which leads to the condition amounts to setting B = H. This raises a natural question to
what extent the approach used here, where B is a dynamical
e µI e µ I mπ
E(k)
= H + . (21) field that is self-consistently solved for, is more accurate.
k 32π mπ fπ2 8π kK(k) Upon replacing µI with 2µB , previously published re-
This has to be solved numerically. sults for CSL in QCD [10] are recovered by neglecting the
second term on the right-hand side of Eq. (19) as well as the
terms in Eqs. (20) and (21), proportional to e2 . The latter
4.1 Chiral limit capture the back-reaction of the neutral pions to the mag-
netic field, which is absent if B is treated as fixed.
Since the analytic expressions for the CSL state found above The validity of this approximation relies on a single con-
are rather involved, it is instructive to inspect their behavior dition that should be satisfied in equilibrium,
in the limiting case of vanishing vacuum pion mass, i.e. the
chiral limit, mπ → 0. This can be done either by making e µ I mπ
≪ H. (26)
use of the asymptotic properties of the elliptic integrals, or 8π kK(k)
by setting mπ = 0 and minimizing the Gibbs energy (14)
directly. While the latter approach is much easier, we choose Given that E(k)/k is monotonously decreasing on (0, 1] and
the former, which also covers small but nonzero mπ . bounded from below by one, it follows from Eq. (21) that
It follows from Eq. (21) that for very small mπ , the opti- within the same approximation, the CSL state can only exist
mal value of k is very small as well. By approximating both in equilibrium above the critical field
complete elliptic integrals with the leading term of their Tay-
lor series, E(k) ≈ K(k) = π /2 + O(k2), Eq. (21) reduces to 32π mπ fπ2
HCSL = . (27)
e µI
k 2 fπ e µI
≈ (1 − Ω 2), Ω≡ 2 , (22)
mπ ΩH 8π f π For k that is not too small, this automatically guarantees that
where Ω is a dimensionless measure of the magnitude of the the condition (26) holds, as long as the parameter Ω de-
WZW term. Inserting this in Eqs. (17) and (19), we get the fined by Eq. (22) is much smaller than one. For very small
corresponding values for the period of the CSL solution and k, we can use the approximate expression (22) to arrive at
the dynamical magnetic field, the same condition, Ω ≪ 1. This is in turn satisfied if µI is
much smaller than the cutoff scale of ChPT, 4π fπ , which is
2π f π H
ℓ≈ (1 − Ω 2), B≈ . (23) something we expect from a quantity of order O(p1 ).
ΩH 1 − Ω2
We conclude that in practice, the corrections to the equi-
Finally, the relative Gibbs energy density of the CSL state librium properties of the CSL, caused by treating B as a dy-
near the chiral limit follows from Eq. (20), namical field, are numerically negligible. This can be seen
as an a posteriori justification of the assumption made in pre-
Ω 2H 2
∆ G¯CSL ≈ − . (24) vious works that the external magnetic field is a fixed, non-
2(1 − Ω 2)
dynamical background. That is no longer true for the super-
Except for the asymptotic expression (22) for k/mπ , all conducting charged pion BEC phase, where the dynamical
the above results for the chiral limit can also be derived by nature of B is essential for getting the basic physics right.6
5 Abrikosov vortex lattice of the condensate to the magnetic field is negligible and we
can therefore treat B as fixed, B = H.
In this section, we will address in detail the possible appear- The linearized equation of motion for π , obtained by ex-
ance of a phase with an inhomogeneous charged pion con- panding the free energy density (30) to second order in π ,
densate supporting a lattice of magnetic vortices. Our ap- reads
proach follows closely that of Refs. [19, 20]. The basic idea
is to rewrite the free energy of the magnetic and pion fields −(∇ − ieA)2 π = (µI2 − mπ2 )π . (31)
in terms of manifestly gauge-invariant variables. Assuming Solving this equation is equivalent to the usual Landau level
spatial periodicity of the equilibrium state in the plane trans- eigenvalue problem. We define the gauge-invariant operator
verse to the external field H, the various fields are then ex- Π ≡ −i∇ − eA, and in terms of its components the annihi-
panded in a Fourier series; the specific ansatz we use is de- lation and creation operators,
scribed in Sec. 5.2. Next, the classical equations of motion
are cast in a form that streamlines iterative solution for the 1 1
a≡ √ (Πx + iΠy ), a† ≡ √ (Πx − iΠy ). (32)
Fourier coefficients; this is detailed in Sec. 5.3. This proce- 2eH 2eH
dure makes it possible to solve numerically the equations of Our new operators satisfy the simple commutation relation
motion at fixed average magnetic flux density, B̄. The latter [Πx , Πy ] = ieH, and hence [a, a† ] = 1. The linearized equa-
is related to the external field H using a version of the virial tion of motion (31) now acquires the form
theorem, derived in Ref. [27]; see Sec. 5.4. Having at hand
both the free energy and the external field finally allows us eH(2a†a + 1)π = (µI2 − mπ2 )π . (33)
to calculate the Gibbs energy of the vortex lattice.
Our analysis is based on a few simplifying assumptions. Since the eigenvalues of a† a are quantized in terms of non-
These do not constitute any approximation to our model (1) negative integers, it follows immediately that a nontrivial
per se, but rather help to select a class of relevant solutions solution exists only if µI2 − mπ2 ≥ eH. This gives the well-
to the equations of motion. First, we assume the absence of known result for the upper critical field,
any neutral pion condensate. Hence, the SU(2)-valued ma- µI2 − mπ2
trix field Σ takes the generic form Hc2 = . (34)
e
1 With a little extra effort, we can also extract some details
Σ= (σ 1 + iτ1 π1 + iτ2 π2 ), (28)
fπ about the profile of the condensate that minimizes the free
energy. This must be annihilated by a. Following Refs. [19,
where π1,2 are the real scalar fields representing charged pi-
20], we will now switch to the exponential parameterization
ons and σ is fixed implicitly by the normalization condition
for the charged pion field π ,
σ 2 + π12 + π22 = fπ2 . It will be convenient to group π1,2 into
√
a single complex field, π ≡ ω eiθ . (35)
π ≡ π1 + iπ2 . (29) We will also trade the vector potential A for the gauge-
invariant condensate supervelocity,
Second, we will be looking for static solutions to the equa-
tions of motion, and thus assume the electromagnetic gauge ∇θ
Q ≡ A− . (36)
potential Aµ to take the form Aµ = (µI /e, A) with a time- e
independent A. With these assumptions, the free energy den- Considering separately the real and imaginary parts of the
sity implied by the Lagrangian (1) becomes √
condition (Πx + iΠy ) ω eiθ = 0 then leads to a relation be-
B2 1 1 tween Q and the gradient of ω , which can be put in a neat
2
F= + (∇σ )2 + (∇ − ieA)π vector form,
2 2 2 (30)
1 2 ∗ 1 ∇ω × ẑ
− µI π π − fπ m2π σ . Q= . (37)
2 2e ω
This relates the supervelocity of the condensate to the spatial
5.1 Abrikosov solution near Hc2 variation of its magnitude in a way that is independent of the
choice of gauge for A.
Let us start with the extreme case of external field close to Equation (37) will be used as a benchmark below when
the upper critical field Hc2 . Taking this limit leads to two we search for the vortex lattice solution away from the upper
simplifications. First, the condensate of charged pions be- critical field. Further details about Abrikosov’s analytic so-
comes very small and the free energy density (30) can there- lution for the vortex lattice near Hc2 , tailored to iQCD, can
fore be expanded in powers of π . Second, the back-reaction be found in Ref. [17].7
5.2 Vortex lattice ansatz away from Hc2 the polar angle and n the winding number of the vortex. This
implies that ∇θ (r) ≃ nϕ̂ /r. Using the Stokes theorem, we
With the exponential parameterization (35) and the defini- then find that
tion (36) at hand, the free energy density (30) can be written
solely in terms of the gauge-invariant variables ω and Q, ∇ × ∇θ (r) = 2π nẑδ (r), (42)
B2 (∇ω )2
1 1
where the two-dimensional δ -function is supported on the
F= + + vortex core, corresponding to the line r = 0. In the follow-
2 8 fπ2 − ω ω (38)
1 2
q ing, we only consider vortices with a unit winding number,
2 2 2 2
− (µI − e Q )ω − fπ mπ fπ − ω , which are usually energetically favored. Taking into account
2
the whole lattice of vortices, the definition (36) then leads to
where we eliminated σ in terms of ω by using the constraint
σ 2 + ω = fπ2 . ∇ × Q(r) = B(r) − Φ0 ẑ ∑ δ (r − R), (43)
R
We will search for stationary states of the free energy,
obtained by integrating Eq. (38) over space, that carry a pe- where Φ0 ≡ 2π /e is the quantum of magnetic flux. Requir-
riodic lattice of magnetic vortices, aligned with the external ing accordingly that each unit cell of the vortex lattice car-
field H. We will assume that such states are strictly two- ries a single quantum of flux, that is B̄ = Φ0 /S, fixes the
dimensional in that both ω and B only depend on the trans- lattice spacing of the hexagonal lattice (41) in terms of B̄,
verse coordinates x, y. Moreover, we will assume that B is s
parallel to H everywhere, that is, B = (0, 0, B(x, y)). In terms 4π
x1 = √ . (44)
of the two-dimensional position vector r = (x, y), the mag- 3 eB̄
nitude of the condensate and the magnetic field can then be
Fourier-expanded as Working with a singular field such as Q is detrimental
to the prospect of solving for the vortex lattice numerically.
ω (r) = ∑ aK (1 − cosK · r), This problem can be bypassed by separating out the singular
K
(39) part of Q. We do so by writing Q as Q = QA + QB , where
B(r) = B̄ + ∑ bK cos K · r, QA is the supervelocity (37) of the Abrikosov solution. The
K
latter corresponds to a nearly uniform magnetic field, hence
where K are vectors from the reciprocal lattice; only K 6= 0
are included in the sum. Also, B̄ is the spatial average of the ∇ × QA(r) = B̄ẑ − Φ0 ẑ ∑ δ (r − R). (45)
R
magnetic flux density.
The direct lattice in real space, corresponding to points It follows that ∇ × QB (r) = [B(r) − B̄]ẑ. The two compo-
R such that K · R is an integer multiple of 2π for all K, car- nents of Q thus play very different roles: QA carries infor-
ries the cores of vortices at which the charged pion conden- mation about the average magnetic flux density and the lo-
sate vanishes. For a generic lattice, the direct and reciprocal cation of vortices, whereas QB encodes smooth variations of
lattice vector can be parameterized by two integers m, n as the magnetic field around the average. Such a smooth field
can be Fourier-expanded, and we start with a generic ansatz,
Rmn ≡ (mx1 + nx2, ny2 ),
2π (40) QB (r) = ∑ cK sin K · r. (46)
Kmn ≡ (my2 , −mx2 + nx1), K
S
Comparing this to Eq. (39) results in the condition K × cK =
where the lattice basis in real space has been chosen with-
bK ẑ. This determines cK up to addition of a multiple of K.
out loss of generality as (x1 , 0) and (x2 , y2 ), and S ≡ x1 y2 is
In order to remove the ambiguity, we make one last assump-
the area of the unit cell of the direct lattice. We will restrict
tion on our stationary state ansatz, that is, ∇ · Q = 0. This
our analysis to a hexagonal lattice ansatz, for which we can
is satisfied exactly by the solution (37) near Hc2 . It is also
choose the parameters of the lattice basis as
true thanks to continuous rotational invariance for a single
√
x1 x1 3 isolated vortex, that is near the lower critical field Hc1 , and
x2 = , y2 = . (41) for any H near the vortex cores. A detailed justification of
2 2
why ∇ · Q can be generally expected to be negligibly small
This lattice geometry is known to be energetically favored
is given in the appendix of Ref. [19]. We now have a unique
on a fairly general ground [28].
solution for the Fourier coefficients of QB , cK = bK (ẑ ×
To be able to evaluate the free energy for the ansatz (39), K)/K2. The full supervelocity then assumes the form
we need to relate the supervelocity Q to the magnetic field
B. To that end, note that near the vortex core at R = 0, the 1 ∇ωA (r) × ẑ ẑ × K
Q(r) = + ∑ bK 2 sin K · r, (47)
phase of the condensate behaves as θ (r) ≃ nϕ , where ϕ is 2e ωA (r) K K8
which is completely fixed by B̄ and the Fourier coefficients Taking a curl of this equation and using the fact that B has a
bK . We have not given a detailed solution for the conden- vanishing divergence, we can bring this to the form3
sate magnitude ωA of the Abrikosov solution (37). For our
(∇2 − e2 ζ 2 )B = e2 [∇ω × Q + (ω − ζ 2 )B]. (52)
purposes, it is however sufficient to know its Fourier com-
ponents as defined by Eq. (39), see Ref. [29], We have again added an extra term with an arbitrary co-
efficient ζ to both sides of the equation to ensure conver-
K2mn S
aA = −(−1)m+mn+n
exp − . (48) gence of the iterative procedure. In the next step, we use the
Kmn
8π Fourier expansion (39) on the left-hand side, multiply with
cosK′ · r and take a spatial average. This leads to an iterative
Equations (39)–(41), (44), (47) and (48) determine our equation for bK ,
vortex lattice ansatz in terms of the average magnetic flux
density B̄ and the Fourier coefficients aK , bK . It remains to 2e2 D
bK = − 2 Qy ∂x ω − Qx ∂y ω
be seen for what values of the Fourier coefficients this field K + e2 ζ 2 (53)
configuration is a solution to the classical equations of mo-
E
+ (ω − ζ 2 )B cos K · r .
tion. This is addressed in Sec. 5.3. Finally, in Sec. 5.4, we
will show how B̄ can be related to the external field H. Equations (50) and (53) constitute the basis for the iter-
ative solution to the equations of motion. The convergence
of the iteration can however be further accelerated by a uni-
5.3 Ginzburg–Landau equations for iQCD form rescaling of ω after each application of Eq. (50). This
is done by the replacement
Here we will derive a set of equations that constitute a basis
ω → (1 + c)ω , (54)
for numerical iterative solution for the Fourier coefficients
aK , bK , whose results are presented in Sec. 6. We start with or equivalently aK → (1 + c)aK. The value of the constant
the equation of motion for the magnitude of the condensate, c is optimized by requiring that upon the rescaling, the spa-
ω . A direct variation of the free energy (38) gives tial average of the free energy density (38) has a vanishing
derivative with respect to c. Since we expect c to be small,
ω2
2 1
2 we can simplify this condition by expanding to first order in
(−∇ + ξ )ω = ω− 2 4(µI2 − e2 Q2 )
2 fπ c. The resulting linear equation for c is solved by
2 1 1
+ (∇ω ) − (49)
2 2
( fπ2 − ω )2 ω 2 (∇ω )2 fπ 4 f π mπ2 ω 2 2 2
ω f π2 −ω
+ √ 2 − 4 ω ( µI − e Q )
f π −ω
4 fπ m2π
c=− .
+ ξ 2ω ,
D 4
f π mπ2 ω 2
E
−p 2 (fπf 2(∇ ω) 2
+
fπ2 − ω π −ω ) 3
π
2
( f −ω ) 3/2
(55)
where the term ξ 2 ω with an arbitrary coefficient ξ 2 was
added on both sides to ensure stability and convergence of
the iterative procedure for solving for the Fourier coeffi- 5.4 Virial theorem and the Gibbs energy
cients. In the next step, we insert the Fourier expansion (39)
on the left-hand side, multiply by cos K′ · r and take a spatial So far we have kept the average magnetic flux density B̄ as a
average, which we will indicate with angular brackets, h· · · i. fixed input parameter. In order to evaluate the Gibbs energy
Using the property that hcos K · r cosK′ · ri = (1/2)δKK′ , we of the Abrikosov vortex lattice, we must however relate it to
arrive at an iterative equation for aK , the imposed external field H. This can be done elegantly us-
ing a version of the virial theorem, put forward in Ref. [27].
ω2 Let us start by a uniform coordinate dilatation, r′ = r/λ ,
2
aK = − 2 ω − (µI2 − e2 Q2 ) under which the charged pion field transforms as a scalar,
K2 + ξ 2 fπ2
(∇ω )2 ω fπ2 − ω π ′ (r′ ) = π (r), or π ′ (r) = π (λ r).
(56)
+ − (50)
2 fπ2 fπ2 − ω ω
2
The vector potential A is a covariant vector (1-form), and
2m ω
q
− π fπ2 − ω + ξ 2 ω cos K · r . hence transforms differently,
fπ
A′ (r′ ) = λ A(r), or A′ (r) = λ A(λ r). (57)
Next we consider the equation of motion for A. Here 3 Taking the curl of ω Q also produces a term proportional to ω ∇ × ∇θ .
Eq. (38) gives at once This can however be dropped in spite of the singular behavior of θ near
the vortex cores. This is thanks to the presence of the factor ω which
∇ × B = −e2 ω Q. (51) vanishes at the cores.9
Finally, the magnetic field B is an antisymmetric rank-two
0.0
covariant tensor (2-form), and thus transforms as
0.1
B′ (r′ ) = λ 2 B(r), or B′ (r) = λ 2 B(λ r). (58)
0.2
This last rule agrees with the observation and upon the co-
[106 MeV4]
ordinate dilatation, the magnetic flux through the unit cell 0.3
of the vortex lattice must remain unchanged (because it is
0.4
quantized in units of Φ0 ), whereas the area of the unit cell is
rescaled by the factor 1/λ 2 . 0.5
In thermodynamic equilibrium, the variation of the free
energy under the above coordinate rescaling must vanish. 0.6
This can be ensured by inserting our scaling relations in
0.2 0.4 0.6 0.8 1.0
Eq. (38), taking the spatial average and setting the deriva- H/Hc2
tive thereof at λ = 1 to zero. This leads to the condition
Fig. 1 Average Gibbs energy density of the Abrikosov vortex lattice,
(∇ω )2
1 1 1 2 2 2 relative to the vacuum state, as a function of H at µI = 147 MeV.
H · B̄ = + + e Q ω + B , (59)
8 fπ2 − ω ω 2
where we have used the thermodynamic relation ∂ F¯ /∂ B̄ =
H. By combining this with the definition (7) of Gibbs energy the number of points of the reciprocal lattice included in the
and the previously calculated Gibbs energy density (9) of the calculation was of the order of a thousand.
vacuum state, we obtain our final result for the spatially av- Before we tackle the problem of finding the phase di-
eraged Gibbs energy density of the Abrikosov vortex lattice, agram of iQCD, let us first do some basic checks that the
relative to the vacuum, purely numerically obtained vortex lattice solution has the
expected physical properties. The upper critical field Hc2 ,
H2 B2 1 2
∆ G¯vortex = + − − µI ω given analytically by Eq. (34), represents a field at which
2 2 2 the vortex lattice solution ceases to be energetically favored
(60)
q over the vacuum state. Hence the relative averaged Gibbs
− fπ m2π fπ2 − ω − fπ .
energy density of the vortex lattice (60) should go to zero as
H → Hc2 . This behavior is displayed in Fig. 1. For numeri-
cal illustration, we have chosen a value of µI just above the
6 Numerical results onset of charged pion BEC, µI = 147 MeV, for which the
upper critical field is small compared to the characteristic
All the numerical results presented below were obtained with scale of QCD, Hc2 ≈ 6600 MeV2 .
a fixed value of the pion decay constant, fπ = 92 MeV. Like-
The case of the lower critical field Hc1 is more subtle.
wise, we mostly considered the physical pion mass, mπ =
In principle, this is defined as the external field at which the
140 MeV. Sometimes, it is however advantageous to treat
Gibbs energy of a single isolated vortex becomes smaller
the pion mass as a tunable parameter. We use this as a tool than that of the uniform BEC state [17]. In order to avoid
to access features of the phase diagram that might other-
having to deal separately with the isolated vortex solution,
wise remain hidden. Finally, in the high-energy units that
we have adopted a different approach. We picture the iso-
we use throughout the paper, the fine structure constant is
lated vortex as an extreme case of a vortex lattice with an in-
α = e2 /(4π ). Hence√ the electromagnetic coupling takes the finite lattice spacing. According to Eq. (44), this corresponds
numerical value e = 4πα ≈ 0.303. to B̄ → 0. We take vanishing of B̄ as an alternative definition
of the lower critical field.
6.1 Abrikosov vortex lattice The downside of this approach is that finding the lower
critical field through a limit is numerically challenging. The
In order to find the Abrikosov vortex lattice solution in equi- convergence of the search for the critical field turns out to be
librium, we used the following numerical procedure. Ini- particularly sensitive to the value of the pion mass. We were
tially, we set bK = 0 and iterated Eqs. (50) and (55) for aK a not able to obtain sufficient accuracy for the physical pion
few times. Subsequently, we iterated Eqs. (50), (55) and (53) mass. We shall instead present some numerical results for a
until convergence was reached. The spatial averages indi- lower pion mass, mπ = e fπ ≈ 27.9 MeV. This value is not
cated with angular brackets were found by numerical inte- chosen accidentally; it is a threshold value above which the
gration over a single unit cell of the direct lattice. In practice, existence of a phase with an inhomogeneous pion conden-10
1.0 3000
2500
0.8
2000
0.6
B)/ Hc2
H [MeV2]
1500
(H
0.4
1000
0.2 500
0
0.0
0.5 0.6 0.7 0.8 0.9 1.0 28 30 32 34 36 38 40
H/Hc2 μI [MeV]
Fig. 2 Reverse magnetization of the Abrikosov vortex lattice as a Fig. 3 Various critical fields for the charged pion condensate. The three
function of H. To generate these data, the pion mass was tuned to curves displayed correspond, from the top to the bottom, to Hc2 , Hc and
mπ = e f π ≈ 27.9 MeV. The different curves with data points dis- Hc1 , respectively. To generate these data, the pion mass was tuned to
played correspond to different values of µI with 1 MeV spacing. The mπ = e f π ≈ 27.9 MeV.
right-most curve corresponds to µI = 29 MeV and the left-most one to
µI = 40 MeV.
6.2 Phase diagram of iQCD
Let us at last address the phase diagram of iQCD, which is
sate can be guaranteed. See Appendix A for a detailed proof the primary target of the present paper. We now know the
of this claim. Gibbs energies of all four candidate states, listed at the end
of Sec. 2.1. We take the vacuum state as the reference. The
It is practically advantageous to focus on the magneti- relative Gibbs energies of the uniform charged pion BEC,
zation, M = B − H. In Fig. 2, we show the values of mag- the neutral pion CSL, and the Abrikosov vortex lattice of
netization as a function of H for various choices of µI ; for charged pions are then given respectively by Eqs. (12), (20)
convenience, the magnetization is displayed with an oppo- and (60). Our main result is the phase diagram in Fig. 4,
site overall sign. The straight solid line in Fig. 2 corresponds obtained using the physical pion mass.
to B̄ = 0, hence M̄ = −H. The point of contact of the vari- Let us comment on the main qualitative features of the
ous data curves with this line defines the lower critical field. phase diagram. The curve labeled as HCSL is defined by set-
Although the numerical approach does not allow us to set ting ∆ G¯CSL = 0. In the region of the phase diagram where
B̄ = 0, we can make a small extrapolation of the data curves charged pion condensation is disfavored by the strong mag-
in order to identify the points of contact with the straight netic field, this describes a phase transition between the vac-
solid line. The convergence of all the data curves to a single uum state and the CSL phase. The critical field Hc is shown
point with M = 0 as H/Hc2 → 1 is an independent verifi- in Fig. 4 just for orientation. Since for the physical pion
cation of the correct physical behavior of our numerically mass, we were not able to calculate the lower critical field
found solution near the upper critical field. Hc1 for the formation of vortex lattice accurately, the Hc
curve delimits the part of the phase diagram to which the
The values of Hc1 extracted using this approach are dis- uniform charged pion BEC phase is localized. The Abrikosov
played in Fig. 3 as a function of µI . We crosschecked these vortex lattice phase occupies a significant part of the phase
results independently by looking for the value of H where diagram. The first-order transition between the vortex lat-
the Gibbs energies of the uniform charged pion BEC and the tice phase and the CSL phase was pinned down by a nu-
vortex lattice coincide. For comparison, we also show in the merical comparison of ∆ G¯CSL and ∆ G¯vortex . This requires
figure the upper critical field Hc2 (34) and the critical field the full numerical machinery as laid out in Sec. 5. The po-
Hc (13) above which the uniform charged pion BEC state sition of the triple point at which the vacuum, vortex lattice
is no longer favored over the vacuum state. As explained in and CSL phases meet can however be determined simply
Sec. 3, these three fields are expected to satisfy the hierarchy by solving the condition HCSL = Hc2 . We find the coordi-
Hc1 < Hc < Hc2 , which the numerical results for Hc1 con- nates of the triple point to be µI,tr ≈ 505 MeV and Htr ≈
firm. All the three fields vanish simultaneously as µI → mπ . 0.78 GeV2 . Note that the latter value simultaneously repre-
Below this threshold, no charged pion condensation can oc- sents the threshold magnetic field for the appearance of the
cur, uniform or not. CSL phase in the phase diagram of iQCD.11
1.6 3456
HC2
1.4 ./12
1.2 *+,-
1.0 &'()
]
]
:9
2
2
0.8
[ VA 87 "#$%
H
H
0.6 !
H
0.4 V
0.2
0.0 0
0 100 200 300 400 500 600 700 300 320 340 360 380 400
μI [MeV] μI [MeV]
Fig. 4 Phase diagram of iQCD as a function of external magnetic Fig. 5 Competition of the CSL phase and the Abrikosov vortex lattice
field and isospin chemical potential for the physical pion mass, mπ = phase for a low pion mass, mπ = e f π ≈ 27.9 MeV. The top curve indi-
140 MeV. The thick solid lines indicate phase transitions between the cates the upper critical field Hc2 . The bottom curve corresponds to the
vacuum (VAC), Abrikosov vortex lattice (AVL) and CSL phases. For field HCSL , at which the CSL state becomes favored over the vacuum.
orientation, we also show the upper critical field Hc2 which defines The actual phase transition between the vortex lattice and CSL phases
the transition between the VAC and AVL phases, and the curve HCSL , is indicated by the middle curve with data points.
defining the transition between the VAC and CSL phases. The lower
critical field Hc1 separating the AVL phase from the uniform BEC
phase was not possible to determine accurately due to convergence is-
7 Summary and discussion
sues. The phase transition between the BEC and AVL phases however
lies below Hc , indicated by the thin dashed line near the horizontal axis.
In this paper, we have used ChPT to investigate the phase
diagram of QCD in presence of a nonzero isospin chemi-
It is instructive to see how the details of the phase di- cal potential (iQCD) and external magnetic field. Experience
agram depend on the pion mass, which we to some extent based on available literature on the subject leads to the ex-
treated as a tunable parameter. First, note that both Hc and pectation that the phase diagram includes a phase with a uni-
Hc2 only contain mπ through the combination µI2 − m2π . Tun- form condensate of charged pions, and the anomaly-induced
ing the pion mass to smaller values and ultimately towards CSL phase with a spatially modulated condensate of neutral
the chiral limit will therefore leave a large part of the phase pions. Our main goal was to study the interplay of these two
diagram, for which µI2 ≫ mπ2 , unaffected. On the other hand, orders, focusing specifically on the possibility that another
the critical curve HCSL will scale down roughly linearly as inhomogeneous phase, carrying a lattice of magnetic vor-
mπ is lowered; cf. Eq. (27). At the same time, the competi- tices, appears in an intermediate range of magnetic fields.
tion of the vortex lattice and CSL phases to the right of the While the uniform charged pion BEC state and the CSL
triple point is affected by changing the pion mass. A detailed state can be treated largely analytically, the Abrikosov vor-
numerical comparison shows that for small pion masses, the tex lattice requires a fully numerical solution of the classical
boundary separating the vortex lattice and CSL phases drifts equations of motion. Here we have improved on previously
toward the upper critical field; this trend is clearly visible in published results on iQCD by implementing a fast numer-
Fig. 5. For mπ = e fπ , used in this figure, we find the triple ical algorithm for finding the vortex lattice solution with-
point located at µI,tr ≈ 288 MeV and Htr ≈ 0.27 GeV2 . out any approximations to the underlying ChPT Lagrangian.
In the chiral limit, the critical fields Hc (13) and Hc2 (34) Our main qualitative result is that for the physical pion mass
become equal to each other at µI = e fπ . The above obser- of mπ = 140 MeV, the Abrikosov vortex lattice phase in-
vations then allow us to draw the following rough picture deed appears in the phase diagram, occupying a large range
of the phase diagram of iQCD close to the chiral limit. The of magnetic fields. Hence iQCD is a type-II superconduc-
vacuum state is only realized in equilibrium for µI = 0 and tor. Type-I superconductivity appears to be possible only for
H = 0. For 0 < µI < e fπ , the equilibrium state is either small, unphysical pion masses, mπ < e fπ ≈ 28 MeV.
the uniform charged pion BEC or the neutral pion CSL. Let us briefly mention two technical aspects of our setup
These are separated by a first-order phase transition at H = that might be worth pointing out. First, in writing down the
Hc = fπ µI . For e fπ < µI , an intermediate phase supporting anomaly-induced WZW term (5), we discarded charged pion
the Abrikosov vortex lattice appears. The first-order transi- degrees of freedom and only kept the neutral pion field, rel-
tion from the vortex lattice phase to the CSL phase appears evant for the CSL state. In hindsight, we have not commit-
roughly at H = Hc2 = µI2 /e. ted any approximation by doing so. Upon a closer look, it12
is possible to show that the full WZW term, spelled out prevails. On the other hand, in type-II superconductors, an
for instance in Ref. [9], does not contribute to the classi- intermediate phase featuring the Abrikosov lattice of mag-
cal Gibbs energy of either the uniform charged pion BEC, netic vortices appears in the range of magnetic fields Hc1 <
or the Abrikosov vortex lattice within the two-dimensional H < Hc2 . Throughout the present paper, we mostly implic-
ansatz used here. itly assumed that iQCD is a type-II superconductor. In this
Second, it has been proposed that augmenting QCD with appendix, we provide some supportive evidence for when
dynamical electromagnetic fields leads to a modification of type-I or type-II behavior of the charged pion condensate
the ChPT Lagrangian already at the leading order by a term may be expected.
that explicitly breaks the isospin SU(2) symmetry [30]. This The key observation is that at Hc2 derived in Sec. 5.1, the
is responsible for the electromagnetic splitting of pion masses vacuum state becomes unstable with respect to formation of
already at tree level. Such a correction to the effective La- a charged pion condensate. But if Hc < Hc2 , then the sta-
grangian is however numerically very small and we neglect ble equilibrium just below Hc2 cannot be the uniform BEC
it here for simplicity. state. It follows immediately that if, for a given fixed value
Finally, let us comment on the fact that the threshold of µI , Hc2 > Hc , then the phase diagram must feature a phase
magnetic field for the appearance of the CSL phase in the carrying some kind of inhomogeneous pion condensate. It is
phase diagram of iQCD was found to be Htr ≈ 0.78 GeV2 for not a priori given that this condensate is an Abrikosov lattice
the physical pion mass. This is a consequence of the compe- of vortices. To show that a state carrying vortices is stable
tition between the CSL state and the Abrikosov vortex lat- and favored over the uniform BEC state, one should rather
tice, as shown in Fig. 4. A natural question arises whether impose the condition Hc1 < Hc [17].
the ChPT setup used in this paper can still be trusted for The advantage of our criterion is that it takes a very sim-
such strong magnetic fields. On the one hand, it was argued ple analytic form. Namely, from Eqs. (13) and (34) we ob-
in Ref. [13] that magnetic fields well above the QCD scale tain immediately that
do not necessarily preclude the application of a low-energy
µI2 − mπ2
effective field theory based on the spontaneous breakdown e fπ
Hc2 − Hc = 1− . (A.1)
of chiral symmetry. On the other hand, it might then be nec- e µI
essary to take explicitly into account the anisotropy of the
QCD vacuum induced by the magnetic field, and in the very Since any kind of charged pion condensate can only be sta-
least the magnetic field dependence of the input parameters ble if µI > mπ , we see immediately that a sufficient con-
fπ , mπ of ChPT induced by loop corrections. dition for the existence of a phase with an inhomogeneous
We conclude from the above that the existence of the pion condensate in the phase diagram is
CSL phase in the phase diagram of iQCD for the physical
µI > e f π . (A.2)
pion mass of mπ = 140 MeV currently still has the status of
a conjecture that requires further investigation. However, the
If in addition mπ > e fπ , then some kind of inhomogeneous
CSL phase certainly is relevant for the thermodynamics of
equilibrium will appear for any value of µI that supports
iQCD, since we can treat the pion mass as a tunable param-
charged pion condensation. On the other hand, if mπ < e fπ ,
eter and lower it towards the chiral limit. This will reduce
we expect two qualitatively different regimes, depending on
the threshold magnetic field for the appearance of the CSL
whether µI is smaller or larger than e fπ . For mπ < µI < e fπ ,
phase to in principle arbitrarily weak fields.
it is in principle possible that as the external magnetic field
is cranked up, the uniform charged pion BEC is destroyed in
Acknowledgements Some of the results presented here appeared pre- favor of the vacuum. As we explain in Sec. 6.2, we expect
viously in the master thesis of one of us (M.S.G.) [31]. The numerical
solution of the Abrikosov vortex lattice was done using a P YTHON
in this case a direct first-order transition from the uniform
code, adapted from a M ATLAB code, available from the authors of BEC phase to the neutral pion CSL phase.
Ref. [32]; see the URL therein. We are indebted to Prabal Adhikari,
Jens Oluf Andersen, Andreas Schmitt, Asle Sudbø and Naoki Yama-
moto for useful discussions and correspondence. This work has been
supported in part by the grant no. PR-10614 within the ToppForsk-UiS References
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