RIKEN Accel. Prog. Rep. 50 (2017)

Ⅱ-2. Nuclear Physics (Theory)

Liquid-Gas Phase Transitions in Quantum Field Theories†
∗2 ∗2
∗1∗1
Hiromichi
MichaelOgilvie,Kamal ∗2
Pangeni ∗2
H.Nishimura,Nishimura,
M. Ogilvie,
and andK. Pangeni

We address the generic problem of liquid-gas phasetransitions from a ﬁeld theory perspective. The general class of ﬁeld theories we will study is the classof CK-symmetric models obtained from dimensional
reduction of a four-dimensional ﬁeld theory at ﬁnite
temperature and density. The fermions with mass mof the underlying theory are integrated out, and after dimensional reduction and redeﬁnition of ﬁelds, weobtain a Lagrangian of the general form
111122(∇φ1 ) + m21 φ21 + (∇φ2 ) + m22 φ222222−F (φ1 , φ2 )T

where β is the inverse temperature and v is a parameter with dimensions of volume, which we set it to beone. The coupling constants g and e determine thestrength of the attractive and repulsive forces respectively. The thermodynamic behavior is extracted fromCK-symmetric complex saddle points of the eﬀective
ﬁeld theory at tree level. The relativistic and static
fermions show a liquid-gas transition, manifesting as
a ﬁrst-order line at low temperature and high density,
terminated by a critical end point as shown in Fig. 1and Fig. 2. In the cases of nonrelativistic fermionsand classical particles, we ﬁnd no ﬁrst-order liquid-gas
transitions at tree level.
The mass matrix controlling the behavior of correlation functions is obtained from ﬂuctuations around thesaddle points. Due to the CK symmetry of the models, the eigenvalues of the mass matrix can be complex2) . This leads to the existence of disorder lines,
which mark the boundaries where the eigenvalues gofrom purely real to complex. The regions where themass matrix eigenvalues are complex are associatedwith the critical line. In the case of static fermions,a powerful duality between particles and holes allows
for the analytic determination of both the critical line
∗1
∗2

50.4

4

where φ1 is associated with the attractive force, and φ2
with the repulsive force. Although charge conjugation
C is broken at ﬁnite density, there remains a symmetry
under CK, where K is complex conjugation1) . We consider four models: relativistic fermions, nonrelativistic
fermions, static fermions and classical particles, corresponding to four diﬀerent choices for the function
F (φ1 , φ2 ). For example, F for the static fermions isgiven as


1/2
1/2
1
F = log 1 + e−βm+βμ+β gφ1 +iβ eφ2v

†m1 1, m2 0.75

6Condensed from the article in arXiv:1612.09575.
RIKEN Nishina Center
Department of Physics, Washington University- 102 -0.8321051015Μ202530Fig. 1. Phase diagram for relativistic fermions for m = 20,m1 = 1, m2 = 0.75 with g1 = 1 and e = 0.3. The
shaded region indicates where the mass matrix eigenvalues form complex conjugate pairs, and the contour
lines refer to the imaginary parts of the mass matrix
eigenvalues. The thick line shows a ﬁrst-order line.

m1 1, m2 0.75
0.8

0.05

0.6

0.1

T

L3d =and the disorder lines as shown in Fig. 2. Depending on the values of the parameters, either zero, one
or two disorder lines are found. Numerical results forrelativistic fermions give a similar picture as shown inFig. 1.

0.4
0.2
19.0

19.5

20.0

20.5

μFig. 2. Phase diagram for static fermions with the same
parameter set as the relativistic fermions. The dashedvertical line emerging from the critical end point is theline of particle-hole duality.

References
1) H. Nishimura, M. C. Ogilvie and K. Pangeni, Phys. Rev.
D 90, no. 4, 045039 (2014).
2) H. Nishimura, M. C. Ogilvie and K. Pangeni, Phys. Rev.
D 91, no. 5, 054004 (2015).