The critical behavior of many physical systems involves two competing n1- and n2-component order-parameters, S1 and S2, respectively, with n = n1 +n2. Varying an external control parameter g, one encounters ordering of S1 below a critical (second-order) line for g < 0 and of S2 below another critical line for g > 0. These two ordered phases are separated by a first-order line, which meets the above critical lines at a bicritical point, or by an intermediate (mixed) phase, bounded by two critical lines, which meet the above critical lines at a tetracritical point. For n = 1 + 2 = 3, the critical behavior around the (bi- or tetra-) multicritical point either belongs to the universality class of a non-rotationally invariant (cubic or biconical) fixed point, or it has a fluctuation driven first-order transition. These asymptotic behaviors arise only very close to the transitions. We present accurate renormalization-group flow trajectories yielding the effective crossover exponents near multicriticality.

1.
L.
Néel
, “
Influence des fluctuations du champ moléculaire sur les propriétés magnétiques des corps
,”
Ann. Phys.
(Paris)
18
,
5
(
1932
);
Theory of constant paramagnetism. Application to manganese
,”
C. R. Acad. Sci.
203
,
304
(
1936
).
2.
C. J.
Gorter
and
T.
van Peski-Tinbergen
, “
Transitions and phase diagrams in an orthorhombic antiferromagnetic crystal
,”
Physica
22
,
273
(
1956
).
3.
A. R.
King
and
H.
Rohrer
, “
Spin-flop bicritical point in MnF2
,”
Phys. Rev. B
19
,
5864
(
1979
).
4.
For a review, see Y. Shapira et al., Experimental Studies of Bicritical Points in 3D Antiferromagnets, in Ref. 5 
(
1984
), p. 35.
5.
Multicritical Phenomena, Proc. NATO Advanced Studies Instutute Series B, Physics
, edited by
R.
Pynn
and
A.
Skjeltorp
(
Plenum Press
,
New York
,
1984
), Vol. 6.
6.
K. S.
Liu
and
M. E.
Fisher
, “
Quantum lattice gas and the existence of a supersolid
,”
J. Low Temp. Phys.
10
,
655
(
1973
).
7.
G. A.
Smolenski
,
Fiz. Tverd. Tela
4
,
1095
(
1962
) [Sov. Phys.-Solid State 4, 807 (1962)].
8.
H.
Weitzel
, “
Neutronenbeugung an mischkristallen (Mn,Fe)WO4, wolfrarait
,”
Z. Kristallogr.
131
,
289
(
1970
);
H. A.
Obermayer
,
H.
Dachs
, and
H.
Schrocke
, “
Investigations concerning the co-existence of two magnetic phases in mixed crystals (Fe,Mn)WO4
,”
Solid State Commun.
12
,
779
(
1973
);
Ch
.
Wissel
, “
A model for the phase diagram of Fe(PdxPt1–x)3 showing a quadruple point
,”
Phys. Status Solidi B
51
,
669
(
1972
).
9.
A.
Aharony
, “
Tetracritical points in mixed magnetic crystals
,”
Phys. Rev. Lett.
34
,
590
(
1975
).
10.
A.
Aharony
, “
Critical behavior of amorphous magnets
,”
Phys. Rev. B
12
,
1038
(
1975
);
A.
Aharony
and
S.
Fishman
, “
Decoupled tetracritical points in quenched random alloys with competing anisotropies
,”
Phys. Rev. Lett.
37
,
1587
(
1976
).
11.
K. A.
Müller
and
W.
Berlinger
, “
Static critical exponents at structural phase transitions
,”
Phys. Rev. Lett.
26
,
13
(
1971
).
12.
A.
Aharony
,
K. A.
Müller
, and
W.
Berlinger
, “
Trigonal-to-tetragonal transition in stressed SrTiO3 A realization of the 3-state Potts model
,”
Phys. Rev. Lett.
38
,
33
(
1977
).
13.
A.
Aharony
and
A. D.
Bruce
, “
Polycritical points and floplike displacive transitions in perovskites
,”
Phys. Rev. Lett.
33
,
427
(
1974
).
14.
E.
Demler
,
W.
Hanke
, and
S.-C.
Zhang
, “
SO(5) theory of anti-ferromagnetism and superconductivity
,”
Rev. Mod. Phys.
76
,
909
(
2004
).
15.
V. P.
Mineev
and
M. E.
Zhitomirsky
, “
Interplay between spin-density wave and induced local moments in URu2Si2
,”
Phys. Rev. B
72
,
014432
(
2005
).
16.
E.
Fradkin
,
S. A.
Kivelson
, and
J. M.
Tranquada
, “
Colloquium theory of intertwined orders in high temperature superconductors
,”
Rev. Mod. Phys.
87
,
457
(
2015
).
17.
S.
Chandrasekharan
,
V.
Chudnovsky
,
B.
Schlittgen
, and
U.-J.
Wiese
, “
Flop transitions in cuprate and color superconductors from SO(5) to SO(10) unification?
,”
Nucl. Phys. B-Proc. Suppl.
94
,
449
(
2001
);
S.
Chandrasekharan
and
U.-J.
Wiese
, “
SO(10) unification of color superconductivity and chiral symmetry breaking?
,” https://arxiv.org/abs/hep-ph/0003214hep-ph/0003214.
18.
M. E.
Fisher
, “
Scaling axes and the spin-flop bicritical phase boundaries
,”
Phys. Rev. Lett.
34
,
1634
(
1975
).
19.
A. D.
Bruce
and
A.
Aharony
, “
Coupled order parameters, symmetry-breaking irrelevant scaling fields, and tetracritical points
,”
Phys. Rev. B
11
,
478
(
1975
).
20.
D.
Mukamel
,
M. E.
Fisher
, and
E.
Domany
, “
Magnetization of cubic ferromagnets and the three-component potts model
,”
Phys. Rev. Lett.
37
,
565
(
1976
).
21.
M. E.
Fisher
and
D. R.
Nelson
, “
Spin flop, supersolids, and bi-critical and tetracritical points
,”
Phys. Rev. Lett.
32
,
1350
(
1974
).
22.
D. B.
Nelson
,
J. M.
Kosterlitz
, and
M. E.
Fisher
, “
Renormalization-group analysis of bicritical and tetracritical points
,”
Phys. Rev. Lett.
33
,
813
(
1974
);
J. M.
Kosterlitz
,
D. R.
Nelson
, and
M. E.
Fisher
, “
Bicritical and tetracritical points in anisotropic antiferromagnetic systems
,”
Phys. Rev. B
13
,
412
(
1976
).
23.
A.
Aharony
, “Dependence of universal critical behavior on symmetry and range of interaction,” in Phase Transitions and Critical Phenomnea edited by
C.
Domb
and
M. S.
Green
(
Academic Press
,
New York
,
1976
), Vol. 6, p. 357.
24.
A.
Aharony
, “
Critical behavior of anisotropic cubic systems
,”
Phys. Rev. B
8
,
4270
(
1973
).
25.
The name “biconical” should not be confused with the name “bicritical.” In fact, the biconical fixed point represent a tetra-critical point in the T-g phase diagram.
26.
M.
Hasenbusch
and
E.
Vicari
, “
Anisotropic perturbations in three-dimensional O(N)-symmetric vector models
,”
Phys. Rev. B
84
,
125136
(
2011
).
27.
J. M.
Carmona
,
A. Pelissato and E. Vicari, “N-component Ginzburg-Landau Hamiltonian with cubic anisotropy: A six-loop study
,”
Phys. Rev. B
61
,
15136
(
2000
).
28.
L. T.
Adzhemyan
,
E. V.
Ivanova
,
M. V.
Kompaniets
,
A.
Kudlis
, and
A. I.
Sokolov
, “
Six-loop ɛ expansion study of three-dimensional n-vector model with cubic anisotropy
,”
Nucl. Phys. B
940
,
332
(
2019
);
the detailed coefficients of the ɛ expansions appear in the Ancillary files of arXiv:1901.02754.
29.
S. M.
Chester
,
W.
Landry
,
J.
Liu
,
D.
Poland
,
D.
Simmons-Duffin
,
N.
Su
, and
A.
Vichi
, “
Bootstrapping heisenberg magnets and their cubic anisotropy
,”
Phys. Rev. D
104
,
105013
(
2021
).
30.
A.
Pelissetto
and
E.
Vicari
, “
Critical phenomena and renormalization-group theory
,”
Phys. Rep.
368
,
542
(
2002
).
31.
P.
Calabrese
,
A.
Pelissetto
, and
E.
Vicari
, “
Multicritical phenomena in O(n1)⊕O(n2)-symmetric theories
,”
Phys. Rev. B
67
,
054505
(
2003
).
32.
R.
Folk
,
Yu.
Holovatch
, and
G.
Moser
, “
Field theory of bicritical and tetracritical points. I. Statics
,”
Phys. Rev. E
78
,
041124
(
2008
).
33.
A.
Aharony
, “
Comment on ‘Bicritical and tetracritical phenomena and scaling properties of the SO(5) theory
,’”
Phys. Rev. Lett.
88
,
059703
(
2002
);
[PubMed]
Old and new results on multi-critical points
,”
Stat. Phys.
110
,
659
(
2003
).
34.
A.
Aharony
,
O.
Entin-Wohlman
, and
A.
Kuldis
, “
Different critical behaviors in cubic to trigonal and tetragonal perovskites
,”
Phys. Rev. B
105
, 104101 (2022); arXiv:2201.08252.
35.
K. G.
Wilson
, “
The renormalization group and critical phenomena (1982), Nobel Prize lecture
,”
Rev. Mod. Phys.
55
,
583
(
1983
).
36.
K. G.
Wilson
and
J.
Kogut
, “
The renormalization group and the ɛ expansion
,”
Phys. Rep.
12
,
75
(
1974
).
37.
M. E.
Fisher
, “
The renormalization group in the theory of critical behavior
,”
Rev. Mod. Phys.
46
,
597
(
1974
).
38.
M. E.
Fisher
, “
Renormalization group theory: Its basis and formulation in statistical physics
,”
Rev. Mod. Phys.
70
,
653
(
1998
).
39.
K. G.
Wilson
and
M. E.
Fisher
, “
Critical exponents in 3.99 dimensions
,”
Phys. Rev. Lett.
28
,
240
(
1972
);
K. G.
Wilson
, “
Feynman-Graph expansion for critical exponents
,”
ibid.
28
,
548
(
1972
).
40.
R. A.
Cowley
and
A. D.
Bruce
, “
Application of the wilson theory of critical phenomena to a structural phase transition
,”
J. Phys. C
6
,
L191
(
1973
).
41.
I. J.
Ketley
and
D. J.
Wallace
, “
A modified ɛ expansion for a Hamiltonian with cubic point-group symmetry,
J. Phys. A
6
,
1667
(
1973
).
42.
M. V.
Kompaniets
and
E.
Panzer
, “
Minimally subtracted six-loop renormalization of O(n)-symmetric ϕ4 theory and critical exponents
,”
Phys. Rev. D
96
,
036016
(
2017
).
43.
M. V.
Kompaniets
and
K. J.
Wiese
, “
Fractal dimension of critical curves in the O(n)-symmetric ϕ4 model and cross-over exponent at 6-loop order loop-erased random walks, self-avoiding walks, Ising, XY, and heisenberg models
,”
Phys. Rev. E
101
,
012104
(
2020
).
44.
J.
Rudnick
and
D. R.
Nelson
, “
Equations of state and renormalization-group recursion relations
,”
Phys. Rev. B
13
,
2208
(
1976
);
J.
Rudnick
, “
First-order transition induced by cubic anisotropy
,”
ibid.
18
,
1406
(
1978
).
45.
E.
Domany
,
D.
Mukamel
, and
M. E.
Fisher
, “
Destruction of first-order transitions by symmetry-breaking fields
,”
Phys. Rev. B
15
,
5432
(
1977
).
46.
D.
Blankschtein
and
A.
Aharony
, “
Crossover from fluctuation-driven continuous transitions to first-order transitions
,”
Phys. Rev. Lett.
47
,
439
(
1981
).
47.
A.
Aharony
, “
Axial and diagonal anisotropy crossover exponents for cubic systems
,”
Phys. Lett. A
59
,
163
(
1976
).
48.
A.
Bednyakov
, and
A.
Pikelner
, “
Six-loop beta functions in general scalar theory
,”
J. High Energ. Phys.
2021,
233
(
2021
).
49.
R.
Ben Ali Zinati
,
A.
Codello
, and
O.
Zanusso
, “
Multicritical hyper-cubic models
,” arXiv:2104.03118. In their notations, φdiag = θY / θS, φaxis = θX / θS.
50.
M. E.
Fisher
and
P.
Pfeuty
, “
Critical behavior of the an-isotropicn-vector model
,”
Phys. Rev. B
6
,
1889
(
1972
);
P.
Pfeuty
,
D.
Jasnow
, and
M. E.
Fisher
, “
Crossover scaling functions for exchange anisotropy
,”
ibid.
10
,
2088
(
1974
).
51.
F. J.
Wegner
, “
Critical exponents in isotropic spin systems
,”
Phys. Rev. B
6
,
1891
(
1972
).
52.
M.
Kerszberg
and
D.
Mukamel
, “
Fluctuation-induced first-order transitions and symmetry-breaking fields. I. Cubic model
,”
Phys. Rev. B
23
,
3943
(
1981
).
53.
A.
Codello
,
M.
Safari
,
G. P.
Vacca
, and
O.
Zanusso
, “
Critical models with N ≤ 4 scalars in d = 4−ɛ
,”
Phys. Rev. D
102
,
065017
(
2020
).
54.
E.
Brézin
,
J. C.
Le Guillou
, and
J.
Zinn-Justin
, “
Discussion of critical phenomena for general n-vector models
,”
Phys. Rev. B
10
,
892
(
1974
).
55.
J.-C.
Toledano
,
L.
Michel
,
P.
Toledano
, and
E.
Brézin
, “
Renormalization-group study of the fixed points and of their stability for phase transitions with four-component order parameters
,”
Phys. Rev. B
31
,
7171
(
1985
).
56.
Such a harmonic form was first written by F. J. Wegner, Ref. 51.
57.
S.
Stokka
and
K.
Fossheim
, “
Crossover exponent and structural phase diagram of SrTiO3
,”
Phys. Rev. B
25
,
4896
(
1982
).
58.
K. A.
Müller
,
W.
Berlinger
,
J. E.
Drumheller
, and
J. J.
Bednorz
, “
Bi- and tetra-critical behavior of uniaxially stressed LaAlO3
,” in Ref. 5, p. 143.
59.
G.
Bannasch
and
W.
Selke
, “
Heisenberg antiferromagnets with uniaxial exchange and cubic anisotropies in a field
,”
Eur. Phys. J. B
69
,
439
(
2009
).
60.
J.
Xu
,
S.-H.
Tsai
,
D. P.
Landau
, and
K.
Binder
, “
Finite-size scaling for a first-order transition where a continuous symmetry is broken: The spin-flop transition in the three-dimensional XXZ heisenberg antiferromagnet
,”
Phys. Rev. E
99
,
023309
(
2019
).
61.
B.
Lüthi
and
W.
Rehwald
, “
Ultrasonic studies near structural phase transitions
,” in Ref. 62, Vol. 1, p. 131, Sec. 4.3.2.
62.
K. A.
Müller
and
H.
Thomas
(eds.),
Structural Phase Transitions I, and Structural Phase Transitions II
(
Springer-Verlag
,
Berlin
,
1991
).
63.
U. T.
Höchli
and
A. D.
Bruce
, “
Elastic critical behaviour in SrTiO3
,”
J. Phys. C
13
,
1963
(
1980
).
64.
W.
Rehwald
, “
Critical behavior of strontiom titanate under stress
,”
Solid State Commun.
21
,
667
(
1977
).
65.
A.
Aharony
and
M. E.
Fisher
, “
Critical behavior of magnets with dipolar interactions. I. Renormalization group near four dimensions
,”
Phys. Rev. B
8
,
3323
(
1973
).
66.
K.
Fossheim
and
R. M.
Holt
, “
Critical dynamics of sound in KMNF3
,”
Phys. Rev. Lett.
45
,
730
(
1980
);
67.
V.
Privman
,
P. C.
Hohenberg
, and
A.
Aharony
, “Universal critical-point amplitude relations,” in Phase Transitions and Critical Phenomena, edited by
C.
Domb
and
J. L.
Lebowitz
(
Academic
,
New York
,
1991
), Vol. 14, pp. 1–134, 364–367.
You do not currently have access to this content.