Thermodynamic properties of non-Hermitian NambuJona-Lasinio models Alexander Felski1Alireza Beygi2yand S. P. Klevansky1z 1Institute for Theoretical Physics Heidelberg University

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Thermodynamic properties of non-Hermitian Nambu–Jona-Lasinio models

Alexander Felski1,∗Alireza Beygi2,†and S. P. Klevansky1‡

1Institute for Theoretical Physics, Heidelberg University,

Philosophenweg 12, 69120 Heidelberg, Germany and

2Department of Molecular Bioinformatics, Institute of Computer Science,

Goethe University Frankfurt, Robert-Mayer-Strasse 11-15, 60325 Frankfurt a. M., Germany

We investigate the impact of non-Hermiticity on the thermodynamic properties of interacting

fermions by examining bilinear extensions to the 3+1 dimensional SU(2)-symmetric Nambu–Jona-

Lasinio (NJL) model of quantum chromodynamics at ﬁnite temperature and chemical potential.

The system is modiﬁed through the anti-PT-symmetric pseudoscalar bilinear ¯

ψγ5ψand the PT-

symmetric pseudovector bilinear iBν¯

ψγ5γνψ, introduced with a coupling g. Beyond the possibility

of dynamical fermion mass generation at ﬁnite temperature and chemical potential, our ﬁndings

establish model-dependent changes in the position of the chiral phase transition and the critical

end-point. These are tunable with respect to gin the former case, and both gand |B|/B0in the

latter case, for both lightlike and spacelike ﬁelds. Moreover, the behavior of the quark number,

entropy, pressure, and energy densities signal a potential fermion or antifermion excess compared

to the standard NJL model, due to the pseudoscalar and pseudovector extension respectively. In

both cases regions with negative interaction measure I=−3pare found. Future indications of

such behaviors in strongly interacting fermion systems, for example in the context of neutron star

physics, may point toward the presence of non-Hermitian contributions. These trends provide a

ﬁrst indication of curious potential mechanisms for producing non-Hermitian baryon asymmetry.

In addition, the formalism described in this study is expected to apply more generally to other

Hamiltonians with four-fermion interactions and thus the eﬀects of the non-Hermitian bilinears are

likely to be generic.

I. INTRODUCTION

The concept of PT (parity-time reﬂection) symme-

try has, since its inception by Bender and Boettcher

in 1998 [1], become a highly active ﬁeld of research in

both theoretical and experimental physics. In general, it

has overthrown the prevailing principle that physical sys-

tems must be governed by a Hermitian Hamiltonian and

has demonstrated that rich and unexpected features are

found in non-Hermitian systems with PT symmetry. In

particular, the possible occurrence of exceptional points

has illustrated consequences beyond those observed in

Hermitian models. Various experimental realizations,

displaying these particular properties of PT-symmetric

systems have ﬁrmly established PT symmetry as an im-

portant feature of classical and quantum-mechanical sys-

tems [2–15].

On a fundamental level, however, the development of

a quantum-ﬁeld-theoretical approach is essential. In the

context of 3+1 dimensional fermionic ﬁeld theories, the

oddness of the time-reversal operator T, i.e., T2=−1,

becomes a core feature when discussing non-Hermitian

models, centered around their behavior under combined

parity reﬂection and time reversal [16, 17]. In a re-

cent study [18] we have shown that modifying free Dirac

∗felski@thphys.uni-heidelberg.de

†alireza.beygi@kgu.de

‡spk@physik.uni-heidelberg.de

fermions through the inclusion of non-Hermitian bilin-

ears, PT-symmetric or otherwise, results in a breakdown

of the existence of a real physical fermion mass. In hind-

sight, this is due to the odd nature of the fermionic time-

reversal operator that also underlies Kramer’s degener-

acy. It is not ensured in the relativistic context, that both

necessary conditions for a real spectrum, [H, PT ] = 0

and the simultaneity of eigenfunctions to both Hand

PT, are met. However, in further studies [19, 20] we

demonstrated that such real-mass solutions can in fact

exist, when higher-order interactions are also present in

the Hamiltonian. Then mass can be generated dynami-

cally through the inclusion of the non-Hermitian but PT-

symmetric pseudovector extension igBν¯

ψγ5γνψ.

However, the existence of a real mass solution and

even of dynamical mass generation is not restricted

to non-Hermitian PT-symmetric modiﬁcations: a pseu-

doscalar bilinear term g¯

ψγ5ψ, for example, while being

neither Hermitian, nor PT-symmetric, still generates real

fermion mass dynamically when taken in combination

with higher-order interactions [19, 20]. For this reason,

we choose here particularly to study these two model

interactions, g¯

ψγ5ψand igBν¯

ψγ5γνψ, placed in such a

context, as a function of ﬁnite temperature and density.

This is most conveniently done within the Nambu–Jona-

Lasinio (NJL) model, which provides a fermionic system

with a two-body contact interaction [21, 22], whose re-

sults may easily be taken over for other similar systems

[23, 24]. As a commonly used eﬀective ﬁeld theory of

quantum chromodynamics (QCD), that models in par-

arXiv:2210.15503v2 [hep-ph] 8 Dec 2022

ticular the spontaneous chiral-symmetry-breaking phase

transition at ﬁnite temperature and density, the use of

the NJL model furthermore ties this study to the develop-

ment of a general framework of PT-symmetric ﬁeld the-

ories containing four-point contact interactions and the

possibility of non-Hermitian physics beyond the Standard

Model, see for example [23–27]. Here, analyzing the ef-

fects of ﬁnite temperature and density on non-Hermitian

and in particular PT-symmetric theories marks a cru-

cial step towards developing feasible non-Hermitian ap-

proaches and PT quantum ﬁeld theories applicable to

experimental realizations, such as heavy-ion collisions

and astrophysical models of compact stars. The crucial

task of course is to identify characteristics beyond the

existence and generation of real eﬀective fermion mass,

that may both diﬀerentiate between Hermitian and non-

Hermitian ﬁeld theories, and examine whether new fea-

tures of quantum ﬁeld theories may arise, when the un-

derlying system is generally non-Hermitian, and speciﬁ-

cally when it is PT symmetric.

This paper is structured as follows. In Sec. II the

standard SU (2) NJL model is reviewed. This discus-

sion serves as the baseline for the examination and

the modiﬁed approach used in the study of the non-

Hermitian extensions of the NJL model. The gap

equation for the eﬀective fermion mass is presented

in a self-consistent Hartree approximation and within

the Matsubara-formalism for ﬁnite temperature Tand

at ﬁnite chemical potential µ, introducing a three-

dimensional regulator Λ. The thermodynamic (grand)

potential Ωis obtained following the coupling-constant

integration method. Based on this, the phase diagram of

the physical fermion mass is determined and the behav-

ior of the thermodynamic observables – quark number,

pressure, entropy, and energy density, as well as the in-

teraction measure – is established.

Section III adapts the NJL formalism for the study

of the model which is modiﬁed through the inclusion

of a non-Hermitian non-PT-symmetric bilinear extension

based on the pseudoscalar term γ5, introduced with the

coupling constant g. The qualitative results obtained at

zero temperature and chemical potential are seen to ver-

ify the behavior found previously with an Euclidean four-

momentum cutoﬀ under similar constraints [20]. The

behavior of the eﬀective fermion mass, in particular the

spontaneous chiral-symmetry-breaking phase transition

and its critical end-point (CEP), as well as the eﬀect

on the thermodynamic observables is analyzed in depen-

dence of the temperature Tand chemical potential µ, as

well as Tand the quark number density nfor illustrative

values of the coupling g. Despite a dynamical gener-

ation of fermion mass within the spontaneously broken

approximate chiral symmetry regime, the behavior of the

thermodynamic observables is demonstrated to coincide

with the standard NJL model behavior at low temper-

ature and small chemical potential. In the vicinity of

the phase transition and throughout the restored sym-

metry region, however, the pseudoscalar extension drives

a fermion excess compared to the standard NJL model

and exhibits interaction measures I=ε−3p < 0.

In Sec. IV the NJL model is extended through the in-

clusion of the non-Hermitian but PT-symmetric pseu-

dovector bilinear igBν¯

ψγ5γνψ. The inﬂuence of this

modiﬁcation on the eﬀective fermion mass at ﬁnite T

and µis analyzed for a spacelike, a lightlike and a time-

like background ﬁeld Bν. We conﬁrm that the results

for the spacelike background ﬁeld obtained in the zero-

temperature and vanishing chemical potential limit coin-

cide qualitatively with those previously found using an

Euclidean cutoﬀ, demonstrating the robustness of the

regularization procedure in this limit. The eﬀect on the

position of the chiral phase transition, the CEP, and on

the behavior of the thermodynamic observables within

the T-µ–plane is investigated, ﬁnding a notable deviation

from the standard NJL model behavior and an empha-

sis on the antifermionic component of the system. This

contrasts with the ﬁndings within the pseudoscalar ex-

tension.

We conclude and summarize our results in Sec. V.

II. THE NJL MODEL

In the grand canonical ensemble, the two-ﬂavor version

of the standard NJL model [21] is characterized by the

Hamiltonian density

HNJL−µN =¯

ψ(−iγk∂k+m0−µ)ψ−G[( ¯

ψψ)2

+( ¯

ψiγ5~τψ)2],

(1)

where Nis the quark number density operator, µis the

baryon chemical potential, Gis the two-body coupling

strength, and m0is a bare fermion mass term. ~τ denotes

the isospin SU (2) matrices. The Dirac matrices γin 3+1

dimensional spacetime have the form

γ0=10

0−1, γk=0σk

−σk0, γ5=iγ0γ1γ2γ3,

(2)

where σkwith k∈[1,3] are the Pauli matrices. In

the limit of vanishing bare mass m0, this model can be

used to study the spontaneous chiral symmetry breaking,

which occurs through fermion-antifermion pair produc-

tion, parallelling the Bardeen-Cooper-Schrieﬀer mecha-

nism of superconductivity [28]. It is therefore a com-

monly used eﬀective model for the study of QCD in the

low-energy regime. Due to the non-renormalizability in-

troduced by the contact interaction, a cutoﬀ length of

Λ = 653 MeV and the coupling strength GΛ2= 2.14,

determined traditionally through the quark condensate

density per ﬂavor and the pion-decay constant, are ﬁxed

within a three-momentum cutoﬀ regularization scheme

in this context, cf. [22].

Following the Feynman-Dyson perturbation theory,

the gap equation for the eﬀective fermion mass mis de-

termined in a self-consistent Hartree approximation to

50 100 150 190

100

200

313

Figure 1: Behavior of the eﬀective fermion mass mwithin the NJL

model in MeV at the chemical potential µ= 0 and µ= 0.2Λ as a

function of the temperature Tin MeV.

take the well-established form

mNJL =m0−2GNcNfZΛd3p

(2π)3TX

eiωnηtr[S(pn)],(3)

where Nc= 3,Nf= 2, and tr denotes the spinor trace

over the fermion propagator S(pn)=(/

pn+µγ0−mNJL)−1

with pn= (iωn,p)and ωn= (2n+ 1)πT . The eﬀects of

the ﬁnite temperature Tare included here through the

imaginary-time (or Matsubara) formalism, cf. [29, 30];

the parameter ηdenotes an inﬁnitesimally small posi-

tive imaginary-time diﬀerence, kept for deﬁniteness and

ultimately taken to vanish. Upon evaluation of the

Matsubara-frequency summation and in the chiral limit

m0→0, the gap equation (3) becomes

mNJL = 2GNcNfmNJL

×ZΛd3p

(2π)3

EhtanhE+µ

2T+ tanhE−µ

2Ti,

(4)

where E2=p2+m2

NJL, cf. [22].

When evaluating the self-consistent gap equation at

vanishing chemical potential µ, one obtains a ﬁnite ef-

fective fermion mass solution of mNJL(0,0) ≈313 MeV

at T=µ= 0, which decreases monotonically as a func-

tion of increasing temperature T, until a second-order

phase transition is reached at a critical value Tc(µ= 0)

≈190 MeV. At higher temperatures the initial sponta-

neously broken chiral symmetry of the system is restored,

and the eﬀective fermion mass mNJL vanishes. This be-

havior is shown in Fig. 1. A qualitatively similar second-

order phase transition is found for small ﬁnite chemical

potential µ, diﬀering in a decrease of the critical temper-

ature Tc(µ)and of the mass mNJL(T, µ)in the sponta-

neously broken chiral-symmetry phase.

When evaluating the gap equation (4) at vanishing

(or small) temperature Tas a function of the chemical

potential µ, however, a parametric region is reached in

which the gap equation admits multiple real mass solu-

tions. The stable physical mass solution in this region

can then be determined as the global minimum of the

thermodynamic potential

ΩNJL(T, µ)=−Tln[ Z]=−Tlntre−(HNJL −µN )/T  (5)

under variation of m, where Zdenotes the (grand canon-

ical) partition function. Using the coupling-constant in-

tegration method, see, e.g., [22, 30], ΩNJL can be deter-

mined as follows: By considering the Hamiltonian den-

sity Hλ=H0+λHint, with Hint denoting the two-body

contact interaction term in (1), eq. (5) implies that

dΩλ

dλ=1

λZλ

tr(λHint Zλ) = 1

λhHint i.(6)

Accordingly, the thermodynamic potential ΩNJL of the

system (1), associated with λ= 1, can be determined

from the thermodynamic average of the interaction en-

ergy to be

ΩNJL −Ω0=1

4Gh(mNJL −m0)2−2Z1

dλ

λ(mλ−m0)dmλ

dλi,

(7)

cf. [22]. By substituting the λ-dependent equivalent of

the gap equation (3) for mλ−m0, the coupling-constant

integration thus results in the expression

ΩNJL −Ω0=(mNJL −m0)2

−2T NcNfZΛd3p

(2π)3ln "cosh(E+µ

2T) cosh(E−µ

2T)

cosh(E0+µ

2T) cosh(E0−µ

2T)#,

(8)

with E2

0=p2+m2

0. Subtracting the contribution of

the denominator in the logarithm, which is associated

with the thermodynamic potential Ω0of the free theory

obtained at λ= 0, one thus ﬁnds

ΩNJL(T, µ) = (mNJL −m0)2

4G−2NcNfZΛd3p

(2π)3E

−2T NcNfZΛd3p

(2π)3ln1+e−(E+µ)/T 1+e−(E−µ)/T .

(9)

The gap equation (4) can be regained from the extremal

condition dΩ/dm= 0 in the chiral limit.

In Fig. 2 the behavior of the thermodynamic poten-

tial is visualized for vanishing temperature Tand various

chemical potentials µas a function of the eﬀective mass

m. For small chemical potentials µ < µ−≈314 MeV

(dotted black line), the only minimum lies at a ﬁnite

value of the fermion mass, which identiﬁes the physi-

cal solution in this region of spontaneously broken chiral

symmetry. For µ−<µ<µ+≈333 MeV the thermo-

dynamic potential admits a second minimum at vanish-

ing mass, which for µ−< µ < µc≈326 MeV (dashed

black line) is only a local, not a global minimum of

ΩNJL(T= 0, µ). Therefore, the vanishing mass solution

Figure 2: Qualitative behavior of the thermodynamic poten-

tial ΩNJL at vanishing temperature Tas a function of the ef-

fective mass mfor various chemical potentials µ. The diﬀerent

cases illustrate the possible existence of extrema at vanishing

and ﬁnite mass.

260 280 300 340

100

200

313

Figure 3: Behavior of the eﬀective fermion mass mwithin the

NJL model in MeV at vanishing temperature Tas a function

of the chemical potential µin MeV. The stable physical mass

solution associated with the global minimum of ΩNJL is shown

as a solid black line, undergoing a ﬁrst-order phase transition

at µc.

is a metastable state in this region. At µc(solid black

line) both minima of ΩNJL lie at the same height. The

abrupt transition from the ﬁnite fermion mass to the van-

ishing mass solution at µcthus marks a chiral-symmetry-

breaking ﬁrst-order phase transition. For µc< µ < µ+

(dashed red line), the global minimum characterizing the

physical solution lies at vanishing mass, while the ﬁnite-

mass solution describes a local minimum associated with

a metastable solution only. When µ > µ+(solid red

line), the only minimum lies at vanishing mass. No ad-

ditional solutions exist. Moreover, a local maximum of

ΩNJL(T= 0, µ)can be found for µ<µ−at m= 0 (see

dotted black curve) and for µ−< µ < µ+in between the

two existing minima (see dashed curves and solid black

curve). While this maximum denotes a possible mass so-

lution of the gap equation, such a solution is an unstable

state.

The behavior of the stable physical fermion mass, as

well as the metastable and unstable mass solutions of the

Figure 4: Eﬀective fermion mass mNJL as a function of the tem-

perature Tand the chemical potential µin MeV. The chiral phase

transition is denoted in red, with a red dot indicating the CEP. At

low temperatures the mass undergoes a discontinuous ﬁrst-order

chiral phase transition, while the transition is of second order at

small chemical potentials.

gap equation, are visualized as a function of the chemical

potential µin Fig. 3 as solid, dashed, and dotted lines re-

spectively. A qualitatively similar ﬁrst-order phase tran-

sition behavior is found at small ﬁnite temperatures T,

causing a small decrease in the critical chemical poten-

tial µc(T)and the mass mNJL(T, µ)in the spontaneously

broken chiral-symmetry phase.

In Fig. 4 the behavior of the physical mass mNJL is

shown as a function of both the temperature Tand the

chemical potential µ, with the chiral phase transition be-

ing denoted in red. The red dot marks the critical end-

point (CEP) at which the ﬁrst-order transition behavior,

found at small temperatures, changes to a second-order

transition, found at small chemical potentials. It lies at

TCEP ≈79 MeV and µCEP ≈281 MeV (with a ratio of

(µ/T )CEP ≈3.56).

Another instructive representation of the chiral phase

transition that is commonly used in the context of heavy-

ion collisions and astrophysical models of compact stars,

considers the behavior of the eﬀective fermion mass as

a function of the quark number density n(T, µ)instead

of the chemical potential µ, which is determined through

the thermodynamic potential as

nNJL(T, µ) = −∂ΩNJL(T, µ)

∂µ T

=NcNfZΛd3p

(2π)3htanhE+µ

2T−tanhE−µ

2Ti.

(10)

It is the physical quantity that enters into the equa-

tions of state and which is accessible experimentally.

Such a representation also has the advantage that the

stable, metastable and unstable solutions of the fermion

mass found in the region with a ﬁrst-order phase transi-

tion arise as separate regions of a single-valued function

in the quark number density, while these solutions form

overlapping branches in the chemical potential between

µ−and µ+. This is illustrated for the case of vanishing

temperature in Fig. 5(a), where nNJL(T= 0, µ)is shown

as a function of the chemical potential. The solutions

for stable, metastable, and unstable fermion masses are

indicated as solid, dashed, and dotted lines respectively

and the corresponding regions of the quark number den-

sity nNJL are indicated. The chiral phase diagram in the

T-n–plane is shown in Fig. 5(b). The phase transition

is denoted as a solid black line. Red lines denote the

spinoidals associated with µ±, that bind the regions of

metastable solutions (shaded in red) and mark the tran-

sition to the region of only unstable results.

Beyond the identiﬁcation of the physical eﬀective

fermion mass, the thermodynamic potential in (9) allows

for the study of various thermodynamic observables. In

addition to the quark number density nNJL, the entropy

density sNJL is determined through ΩNJL(T, µ)to be

sNJL(T, µ) = −∂ΩNJL(T, µ)

∂T µ= 2NcNfZΛd3p

(2π)3nln1+e−(E+µ)/T 1+e−(E−µ)/T +E

−E+µ

2TtanhE+µ

2T−E−µ

2TtanhE−µ

2To.(11)

The pressure density pNJL(T, µ)corresponds to the

thermodynamic potential (9) up to an overall sign and

relative to the physical vacuum at vanishing temperature

and chemical potential,

pNJL(T, µ) = −ΩNJL(T, µ)−ΩNJL(0,0) .(12)

The energy density NJL(T, µ)and the interaction mea-

sure (or trace anomaly of the energy-momentum tensor)

INJL(T, µ), quantifying the deviation from an ideal-gas

behavior, have the respective forms

NJL(T, µ) = −pNJL +T sNJL +µ nNJL,(13)

INJL(T, µ) = NJL −3pNJL.(14)

To study the behavior of these observables (10) – (14),

their analysis as a function of the temperature Talong

lines of constant µ/T in the T-µ–plane is instructive. Fur-

thermore, the inﬂuence of the three-momentum cutoﬀ

scale Λcan be examined by considering the behavior of

the thermodynamic observables along these lines in the

limit Λ→ ∞. Note that, while the quark number den-

sity (10) and entropy density (11) remain convergent in

this limit, the thermodynamic potential (9), and conse-

quently the pressure density (12), energy density (13),

and interaction measure (14), show an ultra-violet diver-

gence. In the form (9), however, one ﬁnds this divergence

contained in the last term contributing to ΩNJL(T, µ),

while the three-momentum integration of the logarith-

mic term remains ﬁnite in the large cutoﬀ limit. Since

the divergent term is, in particular, independent of the

temperature Tand the chemical potential µ, it is suﬃ-

cient to remove the cutoﬀ of the ﬁnite logarithmic term

in (9) to study the behavior at high temperatures.

Notably the algebraic behavior of ΩNJL in the large

temperature limit for ﬁxed µ/T and when removing the

cutoﬀ scale Λ→ ∞ is found to be

ΩNJL(T, µ)∼ −T4NcNfh7π2

180 +1

6µ

T2+1

12π2µ

T4i,

(15)

and coincides with the Stefan-Boltzmann (SB) behavior

of an ideal massless fermion gas, which is expected to

dominate the behavior of the NJL model in the chirally

symmetric region. Together with the corresponding ex-

pansions of (10) and (11),

nNJL(T, µ)∼T3NcNf1

3µ

T+1

3π2µ

T3,(16)

sNJL(T, µ)∼T3NcNf7π2

45 +1

3µ

T2,(17)

it is thus sensible to present the scaled quantities n/T 3,

s/T 3,p/T 4,/T 4, and I/T 4when considering the be-

havior of these observables as functions of Tfor ﬁxed

values µ/T , because they approach ﬁnite limits at large

temperatures. To compare the behavior of the thermo-

dynamic observables along diﬀerent lines of ﬁxed values

µ/T , they are here furthermore normalized to their re-

spective SB limit values, as determined by (15) – (17).

An exception is the asymptotically vanishing interaction

measure, which is scaled to its value at the phase tran-

sition in the Λ→ ∞ case instead. Their behaviors are

shown in Figs. 6(a) – 6(e). In each case, the solid lines

denote the behavior with ﬁxed cutoﬀ length Λas a func-

tion of the scaled temperature T/T cfor µ/T = 2 (red),

i.e., in the second-order phase-transition region, and for

µ/T = 5 (blue), i.e., in the ﬁrst-order transition region.

For comparison with the frequently presented case along

the temperature axis, the behavior for close-to vanish-

ing chemical potential, µ/T = 10−4, is included in black.

This value of µ/T is not taken to vanish exactly in order

to present a nontrivial reference for the quark number

density nas well, whose SB limit otherwise vanishes as

µ→0, see (16). The respective normalizations, as well

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Thermodynamicpropertiesofnon-HermitianNambuJona-LasiniomodelsAlexanderFelski1,AlirezaBeygi2,yandS.P.Klevansky1z1InstituteforTheoreticalPhysics,HeidelbergUniversity,Philosophenweg12,69120Heidelberg,Germanyand2DepartmentofMolecularBioinformatics,InstituteofComputerScience,GoetheUniversityFrankfurt,R...

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Thermodynamic properties of non-Hermitian NambuJona-Lasinio models Alexander Felski1Alireza Beygi2yand S. P. Klevansky1z 1Institute for Theoretical Physics Heidelberg University

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