Interactions in low-dimensional spinor bosonic gases
Renormalization group approach for low-dimensional dilute Bose gases is generalized for the case of multicomponent bosons. As an example, systems of two- and three-component spinor bosons are considered. It is shown that interatomic interactions are strongly renormalized with a tendency to enhance t...
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nasplib_isofts_kiev_ua-123456789-1174382025-02-10T00:51:55Z Interactions in low-dimensional spinor bosonic gases Kolezhuk, A.K. К 80-летию со дня рождения В.Г. Барьяхтара Renormalization group approach for low-dimensional dilute Bose gases is generalized for the case of multicomponent bosons. As an example, systems of two- and three-component spinor bosons are considered. It is shown that interatomic interactions are strongly renormalized with a tendency to enhance the interaction symmetry. Support by Deutsche Forschungsgemeinschaft (the Heisenberg Program, KO 2335/1-2) is gratefully acknowledged. 2010 Article Interactions in low-dimensional spinor bosonic gases / A.K. Kolezhuk // Физика низких температур. — 2010. — Т. 36, № 8-9. — С. 947–951. — Бібліогр.: 34 назв. — англ. 0132-6414 PACS: 67.85.Fg, 67.60.Bc, 37.10.Jk https://nasplib.isofts.kiev.ua/handle/123456789/117438 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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К 80-летию со дня рождения В.Г. Барьяхтара К 80-летию со дня рождения В.Г. Барьяхтара Kolezhuk, A.K. Interactions in low-dimensional spinor bosonic gases Физика низких температур |
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Renormalization group approach for low-dimensional dilute Bose gases is generalized for the case of multicomponent bosons. As an example, systems of two- and three-component spinor bosons are considered. It is shown that interatomic interactions are strongly renormalized with a tendency to enhance the interaction symmetry. |
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Kolezhuk, A.K. |
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Interactions in low-dimensional spinor bosonic gases |
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Interactions in low-dimensional spinor bosonic gases |
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Interactions in low-dimensional spinor bosonic gases |
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Interactions in low-dimensional spinor bosonic gases |
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Interactions in low-dimensional spinor bosonic gases |
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interactions in low-dimensional spinor bosonic gases |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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К 80-летию со дня рождения В.Г. Барьяхтара |
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Interactions in low-dimensional spinor bosonic gases / A.K. Kolezhuk // Физика низких температур. — 2010. — Т. 36, № 8-9. — С. 947–951. — Бібліогр.: 34 назв. — англ. |
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Физика низких температур |
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AT kolezhukak interactionsinlowdimensionalspinorbosonicgases |
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2025-12-02T07:30:23Z |
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© Alexei K. Kolezhuk, 2010
Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9, p. 947–951
Interactions in low-dimensional spinor bosonic gases
Alexei K. Kolezhuk
Institute of Magnetism, National Academy of Sciences
and Ministry of Education of Ukraine, Kiev 03142, Ukraine
Institute of High Technologies, T. Shevchenko Kiev National University Kiev 01601, Ukraine
E-mail: kolezhuk@imag.kiev.ua
Received December 28, 2009
Renormalization group approach for low-dimensional dilute Bose gases is generalized for the case of multicom-
ponent bosons. As an example, systems of two- and three-component spinor bosons are considered. It is shown that
interatomic interactions are strongly renormalized with a tendency to enhance the interaction symmetry.
PACS: 67.85.Fg Multicomponent condensates; spinor condensates;
67.60.Bc Boson mixtures;
37.10.Jk Atoms in optical lattices.
Keywords: ultracold atoms, spinor bosons, low dimension.
1. Introduction
Recently, there has been a continuing interest in multi-
component degenerate quantum gases. Multicomponent Bose
condensates were initially realized in 87 Rb [1,2] and 23 Na
[3] by using two different hyperfine states as different spe-
cies, and later also in heteronuclear condensates of 41K and
87 Rb [4,5]. Such systems exhibit a very rich behavior rang-
ing from phase separation [6,7] and transitions between dif-
ferent magnetic states in spinor condensates [8,9] to a variety
of novel transitions predicted to appear in the presence of an
optical lattice [10–14]. Both in traps and optical lattices, the
realization of highly anisotropic geometries («cigars» and
«pancakes») is possible [15], which provides an opportunity
to study the above phenomena in low-dimensional systems.
Usually, the analysis of the ground state and dynamics
of multicomponent Bose systems is performed at the
mean-field level, by studying coupled Gross–Pitaevskii
equations [6–9]. However, it is well known that in low
dimensions mean-field arguments might become inapplic-
able in the low density regime [16]. For instance, in a one-
dimensional Bose gas, with the decrease of the density, the
healing length ξ becomes smaller than the average inter-
particle distance d , thus invalidating the mean-field ap-
proach. In a dilute one-dimensional Bose gas, the effective
interaction constant flows to strong coupling, leading to the
effectively hardcore behavior of bosons [17,18]. In the
two-dimensional Bose gas, although the ratio / dξ does
not depend on the density and falls below unity only in the
case of very tight two-dimensional confinement, in the
dilute limit the effective coupling constant becomes strong-
ly energy- and density-dependent [19,20].
The goal of the present study is to provide a general
framework for the analysis of dilute multicomponent Bose
systems in the case of quasi-low-dimensional geometries.
To that end, we will generalize the well-known renormali-
zation group (RG) approach to the one-component dilute
Bose gas [17,18,21,22]. The idea is to derive the RG equa-
tions at the critical point of the zero chemical potential
(vanishing particle density), and to stop the RG flow at a
certain density-dependent cutoff scale. It will be shown
that interactions in a multicomponent Bose gas are in gen-
eral strongly renormalized in the low density limit. For
spinor bosons, the renormalization tends to enhance the
interaction symmetry, effectively diminishing the spin-
dependent part of the coupling.
2. Renormalization group for the multicomponent Bose
gas at the zero density critical point
The starting point of our analysis will be the theory of a
N-species bosonic field in d-spatial dimensions, with a
general two-body (quartic) contact interaction, that effec-
tively describes a system of atoms with typical momenta
much smaller than the inverse characteristic potential range
(atom size). The theory is described by the following con-
tinuum action
2
* | |
= { ( ) },
2
d
N d d x U
m
α
α τ α α
α
∇ψ
τ ψ ∂ −μ ψ + +∫ ∫A
* *,
1= .
2
U g ′ ′ ′ ′αβ α β α β α β
′ ′αβα β
ψ ψ ψ ψ∑ (1)
Here αψ are the fields describing bosonic particles with
masses mα , = 1, Nα … , and for the sake of clarity we have
Alexei K. Kolezhuk
948 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9
set = 1 . The interaction matrix for Bose fields satisfies the
obvious symmetry conditions , ,= =g g′ ′ ′ ′αβ α β βα α β
, =g ′ ′αβ β α= * ,g ′ ′α β αβ . The action (1) can describe a mixture
in a continuum as well as in an optical lattice (in the latter case
the particle density should be incommensurate with the lattice,
i.e., the average occupation per lattice site should not be an
integer number). The action (1) also arises in quantum mag-
netism problems dealing with the so-called Bose–Einstein
condensation of magnons, induced by a strong external mag-
netic field H in the vicinity of the saturation field sH [23].
In a frustrated magnet the magnon dispersion may have dege-
nerate minima at inequivalent wave vectors, which gives rise
to multiple «species» in an effective model [24–26], while
sH H− plays the role of the chemical potential.
If all chemical potentials αμ are zero, the system is at
the special critical point (the particle density vanishes).
There is no self-energy correction so the full propagator
just coincides with the free one ( , ) =G kαβ ω
( )/ ( )ki α
αβ= δ ω−ε , where ( ) 2= / 2k k mα
αε . Interaction
between the particles is, however, renormalized due to
multiple scatterings. The peculiar character of the critical
point leads to the fact that the contribution of any diagram
containing a closed loop vanishes [27,21], so only ladder
diagrams have to be included. As a result, the renormalized
vertex Γ , defined at the fixed sum of external momenta
,Q satisfies the following Bethe-Salpeter equation shown
schematically in Fig. 1.
, , , ,( ) = ( ) ( ),Q g g f Q Q′ ′ ′ ′ ′ ′αβ γδ αβ γδ αβ α β α β α β γδΓ − Γ
( ) ( ) 1( ) = [ ] .
(2 )
d
k Q kd
d kf Q α β −
αβ −ε + ε
π∫ (2)
Above the upper critical dimension = 2d , ( )f Qαβ con-
verges at 0Q → and (2) simply yields the dressed (obser-
vable) interaction matrix. For 2d ≤ there is a singularity
at 0Q → : ( ) 1/f Q Qαβ ∝ for = 1d and ( )f Qαβ ∝
0ln( / )Q∝ Λ for = 2d , where 0Λ is the ultraviolet cutoff
(for a system in an optical lattice, the parameter 0Λ has
the physical meaning of a lattice cutoff, and in a continuum
it has the sense of an inverse characteristic potential
range). Thus, for 2d ≤ one should look at the RG flow of
the running coupling matrix ( ) = ( ) exp [(2 ) ]l Q d lΓ Γ −
with the change of the scale 0 exp( )Q lΛ − . Defining
the matrices
1/2
, ,= 2 ( ) ,m m′ ′αβ α β ′ ′ ′ ′αβ α β αβ α βΓ Γ
, ,( ) = ( ) ,dF Q R Q′ ′ ′ ′αβ α β αβ α β αβδ (3)
where / ( )m m m m mαβ α β α β≡ + are the reduced masses,
1/2=2 ( ) / ( ),Q Q m m m mαβ α β α β+ and
2 2 0
1 , = 1
21( ) = ,
1(2 ) ln , = 2
2
d
d d
d
qd kR q
k q d
q
⎧
⎪⎪→ ⎨ Λπ + ⎪
⎪ π⎩
∫
one can write down the RG equation as
{ }1(1 ) = 0.d F
dl
−Γ − Γ (4)
Further, using the identity = = 1AX XA X − with
1(1 )X A −≡ − and its derivative / = ( / ) ,dX dl X dA dl X
those equations can be rewritten in the following form:
/ ( / ) = 0.d dl dF dlΓ +Γ Γ (5)
For further simplification, it is convenient to redefine the
interaction matrix once again. In one dimension ( = 1d ),
one can introduce 1/2 1/2= F FΓ Γ ; then, since F is a di-
agonal matrix that for = 1d is proportional to 1/ Q , one
obtains / =dF dl F and Eq. (5) can be rewritten as
2/ = ,d dlΓ Γ −Γ (6)
which is familiar from the one-component case [28]; the
only difference is that the interaction is now a matrix.
In two dimensions ( = 2d ), it is convenient to define
= / 2Γ Γ π ; then, using the fact that / = 1/ (2 )dF dl π for
= 2d , one can reduce (5) to
2/ = .d dlΓ −Γ (7)
Summarizing the above derivation, we see that the RG
equations for the interaction matrix of the low-dimensional
multicomponent Bose gas can be cast into the common
form
2
/ = (2 ) ,d dl dΓ − Γ −Γ (8)
where
1/4
, , 0( ) = ( )( ) / , = 1,l l m m m m dαβ γδ αβ γδ α β γ δΓ Γ Λ
, ,( ) = ( ) / , = 2.l l m m dαβ γδ αβ γδ αβ γδΓ Γ π (9)
Since the Γ matrices are symmetric, they can always be
diagonalized by an appropriate orthogonal transformation.
It is worth noting that at the zero-density critical point the
Eqs (8) are exact to all orders in the interaction Γ , similar
to the one-component case [28].
The above derivation is nothing but a direct generaliza-
tion of the well-known RG approach to the one-component
Fig. 1. The renormalized interaction vertex for the multicompo-
nent Bose gas model (1) in the zero density limit, see Eq. (2).
Γ(Q)
��
��
��
��
��
��
��
��
Q−k,−ω
k,ω
+
Q−p
=
Q−p’
p’p
g
Interactions in low-dimensional spinor bosonic gases
Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9 949
dilute Bose gas [17,18,21,22]. Similar to the one-com-
ponent case, we will use the RG Eqs (8), derived at the
critical point (zero particle density), to describe the gas of a
finite but small density, and to stop the RG flow at some
scale *=l l where the system is effectively no more dilute,
i.e., where the «running» total density tot( ) = edllρ ρ be-
comes comparable with 0
dΛ . This condition determines the
stopping scale *l as
*
0 tote = / ,dl d
dC Λ ρ (10)
where 2
=1 = 2 /dC π and =2 = 1/ 2dC π can be identified
by comparing to the known results for the one-component
case [22].
3. Applications to spinor bosons
Let us illustrate the RG approach outlined in the pre-
vious section by applying it to simple spinor bosonic gases
with two and three components.
3.1. Spin-l bosons
Consider a gas of bosons with the hyperfine spin = 1F .
The contact interaction between two paticles depends on
their total spin S , so there are two characteristic scattering
lengths Sa with = 0S , 2 , and the interaction can be writ-
ten as [8,9]
† †† † 2
0 2
1= { ( ) },
2
z
a a a a b baba a a bU c c F F h Sμ μ
′ ′′ ′ ′ ′ψ ψ ψ ψ + ψ ψ ψ ψ −
(11)
where , {0, 1}a b∈ ± denote the three components of the
bosonic field, Fμ are the spin-l matrices, and =Sμ
†
a babFμ= ψ ψ are the spin operators. The couplings are given
by 0 0 2= ( 2 ) / 3c g g+ , 2 2 0= ( ) / 3c g g− , with S Sg a∝ ,
and the mass m is the same for all three components. The
last term in (11) describes the quadratic Zeeman effect
caused by an external magnetic field. The bilinear part of the
quadratic Zeeman term leads to a relative shift in chemical
potentials of different species, suppressing the | 0〉 states for
> 0h and | 1± 〉 states for < 0h , respectively. Thus, this
type of interaction favors ferromagnetic spin correlations for
2 < 0c , and polar (nematic) ones for 2 > 0c [8,9]. In addi-
tion to that, the quartic part of the Zeeman term affects inte-
ractions as well.
It is worth noting that on an optical lattice, at integer
odd filling, spin-1 bosons can be effectively described by
the generalized Heisenberg model with biquadratic interac-
tions [11,12], which also supports exotic magnetic states
with nematic-type spin ordering [29,30].
The interaction (11) includes just three parameters, but
the 9 9× interaction matrix ,ab a bg ′ ′ contains generally five
different couplings:
, ,= = ,g g r++ ++ −− −−
0, 0 0, 02 = 2 = ,g g f+ + − −
,2 = ,g v+− +− (12)
00,00 = ,g u
,00 = ,g w+−
whose bare values are given by
(0) (0)
0 2 0 2= , =r c c h f c c+ − + ,
(0) (0) (0)
0 2 0 2= , = , = 2 .v c c h u c w c− + (13)
The RG equations (8) take the form
2 2/ = (2 ) , / = (2 ) ,dr dl d r r df dl d f f− − − −
2 21/ = (2 ) ,
2
du dl d u u w− − −
2 21/ = (2 ) ,
2
dv dl d v v w− − − (14)
/ = (2 ) ( ) ,dw dl d w u v w− − +
where the «tilded» variables are defined as 0= /x xm Λ for
= 1d and = / 2x xm π for = 2d .
At zero external field ( = 0h ) the (2)SU symmetry
dictates that the effective potential can contain only two
constants 0c and 2c , and the RG equations can be cast
into the simpler form
2/ = (2 ) ,dx dl d x x− − (15)
with 0 2 0 2{ , 2 }x c c c c∈ + − . It is easy to see that those
combinations are proportional to the scattering lengths 2a
and 0a in the = 2S and = 0S channels, respectively. In
one dimension ( = 1d ), there is a nontrivial stable fixed
point 0 2( = 1, = 0)c c , which exhibits the enhanced
(3)SU symmetry. The other two nontrivial fixed points,
0 2( = 2 / 3, = 1/ 3)c c and 0 2( = 1/ 3, = 1/ 3)c c − are un-
stable. The corresponding RG flow for = 1d is shown in
Fig. 2. Both for = 1d and = 2d , there are two runaway
flows: the one at 2 0> / 2c c marks the onset of the pairing
state characterized by the formation of bound singlet pairs
[31–33], while the other runaway flow at 2 0<c c−
0( > 0)c corresponds to the collapse instability in the fer-
romagnetic channel. We see that renormalization tends to
enhance the interaction symmetry: the spin-dependent
coupling 2c is diminished during the RG flow.
In the more general case of a nonzero external field h ,
the RG equations can be rewritten in the same form (15)
with { , , }x r f ±∈ χ , where
2= ( 2) / 2, ( ) / = ,u v w u v w±χ + ± λ + − λ (16)
and 2 2= ( ) / 2c h cλ − is a constant (an invariant of the
flow) that can be controlled by the external field. The cor-
responding RG flow for = 1d is shown in Fig. 3. One can
see that in the presence of external field the pairing insta-
Alexei K. Kolezhuk
950 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9
bility point is shifted: instead of 2 0> / 2c c , the instability
occurs at
2 1/2
2 0 0 0> [(3 / 4) / 2] / 4.c c c h c+ − (17)
Similarly, the ferromagnetic collapse point shifts to
2 1/2
2 0 0 0< [(3 / 4) / 2] / 4c c c h c− + − .
3.2. Two-component gas with convertible species
Consider now the double-species mixture with convert-
ible species, described by the quartic interaction in (1) of
the form
4 4
1 2 11 1 22 2( , ) = ( | | | | ) / 2U u uψ ψ ψ + ψ +
2 2 2 2 2 2**
12 1 2 11 2 2| | | | {( ) ( ) } / 2.u w+ ψ ψ + ψ ψ + ψ ψ (18)
Here, only the total number of particles is conserved. Terms
describing the conversion of a pair of type-1 particles into a
pair of type-2 particles arise, for example, in models of bo-
sons in optical lattices with considerably populated higher
bands [34]. In that case, the effective masses of the two spe-
cies, as well as the intraspecies interactions, might differ. In
frustrated magnets pair conversion terms may arise if the
magnon dispersion has two degenerate minima at such in-
commensurate wave vectors 1q , 2q that 1 22( ) =−q q g ,
where g is the reciprocal lattice vector.
In the notation of (1), the bare couplings are 11,11 11=g u ,
22,22 22=g u , 12,12 12= / 2g u , and 11,22 =g w . Denoting
the entries of the Γ matrix by 11u , etc., we can write the RG
Eqs (8) in the form (15) with 12{ , , }x u u u+ −∈ , where
2 2 2= , = ,u u W W w s± ± +
11 22 11 22= ( ) / 2, = ( ) / 2 .u u u s u u+ − (19)
The tilded variables for = 1d are defined as
1 2 0= /w w m m Λ and 0= /u u m mαβ αβ α β Λ ; for = 2d
one has 1 2= / 2w w m m π and = /u u mαβ αβ αβ π . The
ratio /s w remains constant during the RG flow. For
= 1d , the flow diagram has the same form as shown in
Fig. 3; in this case =x u , =y W , and the unstable fixed
point is at 0 0( , ) = (1/ 2,1/ 2)x y . Again, the RG flow tends
to enhance the symmetry: it is easy to see that at the stable
fixed point the potential (18) becomes (2)SU -symmetric,
22 2
0 1 2
1 2
1 2
| | | |
( , ) .
2
U
m m
⎛ ⎞Λ ψ ψ
ψ ψ → +⎜ ⎟⎜ ⎟
⎝ ⎠
(20)
Both for = 1d and = 2d , there are «runaway» flows at
| | >W u , which translates into the condition
2
11 22> .w u u (21)
By the analogy with spin-1 bosons, one may assume that
those runaways correspond to the pairing instability.
4. Summary
Renormalization group approach for low-dimensional
dilute Bose gases is generalized for the case of multicom-
ponent bosons. The approach is applied to systems of two-
and three-component spinor bosons. It is shown that intera-
tomic interactions are strongly renormalized with a tenden-
cy to enhance the interaction symmetry.
Support by Deutsche Forschungsgemeinschaft (the
Heisenberg Program, KO 2335/1-2) is gratefully acknow-
ledged.
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Interactions in low-dimensional spinor bosonic gases
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