In a recent Letter, Gopalakrishnan, Martin, and Demler [Phys. Rev. Lett. 111
(2013) 185304] show that quasi-two-dimensional dipolar Bose gases, subject to a
Rashba spin-orbit coupling, exhibit a variety of spatially ordered, or
crystalline, ground states, including a pentagonal quasicrystal. Indeed, as the
authors say, realizing quasicrystalline condensates would provide new ways to
explore the physics of quasicrystals, and in particular to study the quantum
dynamics of their unique collective phason modes. Yet, the authors conclude
that "there are typically additional phasons in quantum-mechanical
quasicrystals, when compared with their classical equivalents." In this Comment
I review the notion of phason modes in quasicrystals, and explain why their
number does not depend on whether they are classical or quantum

Lifshitz, Ron

English

arXiv

A Comment on the Letter by S. Gopalakrishnan, I. Martin, and E. A. Demler, Phys. Rev. Lett. 111, 185304 (2013).. The authors of the Letter offer a Reply

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Comment on “Quantum Quasicrystals of Spin-
Orbit-Coupled Dipolar Bosons”
In a recent Letter, Gopalakrishnan, Martin, and Dem-
ler [1, henceforth GMD] show that quasi-two-dimensional
dipolar Bose gases, subject to a Rashba spin-orbit cou-
pling, exhibit a variety of spatially ordered, or crystalline,
ground states, including a pentagonal quasicrystal. In-
deed, as the authors say, realizing quasicrystalline con-
densates would provide new ways to explore the physics
of quasicrystals, and in particular to study the quan-
tum dynamics of their unique collective phason modes.
Yet, the authors conclude that “there are typically ad-
ditional phasons in quantum-mechanical quasicrystals,
when compared with their classical equivalents.” In this
Comment I review the notion of phason modes in qua-
sicrystals, and explain why their number does not depend
on whether they are classical or quantum.
Phonons and phasons are Goldstone modes that result
from the fact that the crystalline phase breaks the con-
tinuous translational symmetry of the liquid phase, and
therefore also of the hamiltonian or the free energy of
the system. Any broken symmetry operation, by defini-
tion, takes a spontaneously chosen minimum free-energy
state into a different minimum free-energy state. Conse-
quently, the set of all broken symmetry operations pro-
vides a natural tool for exploring the space of all mini-
mum free-energy states and for counting the number of
independent phonon and phason modes [2]. See Ref. [3]
for a detailed discussion of this in the classical context.
To be concrete, let us describe the crystalline state by a
function ψα(r), possibly multicomponent—a real-valued
scalar density or tensor field in the classical case, or a
complex-valued wave function or spinor in the quantum
mechanical case—whose Fourier expansion is given by
ψα(r) =
∑
k∈L
ψα(k)e
ik·r, (1)
where the (reciprocal) lattice L is a finitely generated Z-
module. This means that L can be expressed as the set
of all integral linear combinations of a finite number D of
d-dimensional wave vectors [4]. In the special case where
the smallest possible D, called the rank of the crystal, is
equal to the physical dimension d, the crystal is periodic.
More generally, for quasiperiodic crystalsD ≥ d, and one
refers to quasiperiodic crystals that are aperiodic (with
D > d) as quasicrystals [5].
A generic free energy, expressed in Fourier space as
F=
∑
j≤n
∑
α1...αn
∑
k1...kn
Aα1...αnj (k1, . . . ,kn)
×ψα1(k1) . . . ψαj (kj)ψ
∗
αj+1
(−kj+1) . . . ψ
∗
αn
(−kn),(2)
is restricted only by the requirements imposed by the
symmetry of the disordered liquid phase. In the absence
of any external fields, the translation invariance of the
disordered phase imposes the constraint that the free en-
ergy coefficients Aα1...αnj (k1, . . . ,kn) must vanish unless
k1 + . . . + kn = 0 [6]. In the case of GMD the disor-
dered phase has, in addition, an overall U(1) symmetry
ψα → ψα exp[iθ], which further constrains the free energy
coefficients to vanish unless n = 2j.
A generic set of coefficients A can vary freely with ex-
ternal parameters such as temperature and pressure sub-
ject to these restrictions and no others, while the field
ψα adjusts itself accordingly to minimize F . Thus, for
two different fields ψα and ψ
′
α to both minimize the same
generic free-energy, and therefore be indistinguishable by
F , they must agree independently on all the products in
F with non-vanishing coefficients. In the absence of U(1)
symmetry, one can then show [7, 8] using the identity of
products of order 2 and 3 alone, that this condition for
indistinguishability reduces to the requirement that
ψ′α(k) = e
2πiχ(k)ψα(k), (3)
where χ(k) is a linear function to within an additive in-
teger on the lattice L. Rokhsar, Wright, and Mermin [9]
called χ(k) a gauge function, in analogy with gauge trans-
formations in electrodynamics, because it encodes all the
changes one can apply to the field ψα without affecting
any of its physical observables.
Any difference between classical crystals and quantum
crystals with broken U(1) symmetry can arrise only from
the additional constraint that n = 2j. Nevertheless,
one can still use the identity of products of the form
ψα1(k)ψ
∗
α2
(k) to establish Eq. (3); use products of order
4 to show that χ(−k) = −χ(k); and products of order
6 to obtain χ(k1 + k2) = χ(k1) + χ(k2); with the latter
equalities holding to within an overall constant coming
from the broken U(1) symmetry. Note that classical free
energies may be restricted as well, by an additional sym-
metry of ψα → −ψα, to contain products of only even
order, where one similarly requires products of order 6
to establish the linearity of χ(k).
The linearity of gauge functions implies that χ(k) is
uniquely determined by specifying its values on a chosen
basis for L, consisting of D linearly independent vectors
over the integers. Thus, there are exactly D independent
symmetry operations in F that are broken, and there-
fore exactly D Goldstone modes. One may choose the
basis so that d of the Goldstone modes are the familiar
phonon modes, which in the infinite wavelength limit re-
duce to rigid translations of the crystal. The remaining
D − d modes are phason modes, reducing in the infinite
wavelength limit to relative phase shifts of the Fourier
components of the crystal that leave some origin fixed.
I should note that suppression of 6th or higher-order
products at low density—whether in classical or in quan-
tum crystals— does not affect the conclusion here. Such
terms might be irrelevant at the critical fixed point, but
dangerously so [10] in the sense that they still allow to
distinguish between states in the ordered phase.
2
I am grateful to Michael Cross and Gil Refael for valu-
able discussions. This research is supported by the Israel
Science Foundation through Grant No. 556/10.
Ron Lifshitz
Condensed Matter Physics 149-33
California Institute of Technology
Pasadena, CA 91125, USA
On leave from:
School of Physics & Astronomy
Tel Aviv University, Tel Aviv 66978, Israel
Dated: December 4, 2013
PACS numbers: 61.44.Br, 64.70.Dv, 67.85.-d, 03.75.Mn
[1] S. Gopalakrishnan, I. Martin, and E. A. Demler,
Phys. Rev. Lett. 111, 185304 (2013).
[2] J. P. Sethna, Entropy, Order Parameters and Complexity
(Clarendon Press, Oxford, 2006) chapter 9.
[3] R. Lifshitz, Isr. J. Chem. 51, 1156 (2011).
[4] In GMD, the dimension d = 2, and the rank D of an M -
fold symmetric star of 2-dimensional wave vectors is given
by the Euler totient function φ(M) = M
∏
Pi
(1− 1/Pi),
where Pi are the distinct prime factors of M .
[5] R. Lifshitz, Found. Phys. 33, 1703 (2003);
Z. Kristallogr. 222, 313 (2007).
[6] The constraints imposed by rotational invariance are
immaterial here and treated elsewhere, for example in
Ref. [3].
[7] N. D. Mermin, Rev. Mod. Phys. 64, 3 (1992).
[8] R. Lifshitz, Rev. Mod. Phys. 69, 1181 (1997), see ap-
pendix.
[9] D. S. Rokhsar, D. C. Wright, and N. D. Mermin,
Acta Crystallogr. Sect. A 44, 197 (1988).
[10] D. J. Amit and L. Peliti,
Annals of Physics 140, 207 (1982).


Comment on "Quantum Quasicrystals of Spin-Orbit-Coupled Dipolar Bosons"

Comment on “Quantum Quasicrystals of
Spin-Orbit-Coupled Dipolar Bosons”
In a recent Letter [1], Gopalakrishnan et al. show that
quasi-two-dimensional dipolar Bose gases may exhibit a
variety of crystalline phases, including a pentagonal quasi-
crystal. Realizing quasicrystalline condensates would
provide new ways to study the quantum dynamics of
their collective phason modes, where according to the
authors “there are typically additional phasons in quantum-
mechanical quasicrystals, when compared with their
classical equivalents.” I explain here that, on the contrary,
the number of phason modes does not depend on whether
they are classical or quantum.
Let us describe the crystalline state by a function ψαðrÞ,
possibly multicomponent—a real-valued scalar density or
tensor field in the classical case or a complex-valued wave
function or spinor in the quantum mechanical case—whose
Fourier expansion is given by
ψαðrÞ ¼
X
k∈L
ψαðkÞeik·r; ð1Þ
where the (reciprocal) lattice L can be expressed as the set
of all integral linear combinations of a finite number D of
d-dimensional wave vectors [2]. If the smallest possible D,
called the “rank” of the crystal, is equal to the physical
dimension d, the crystal is periodic. More generally, for
quasiperiodic crystals, D ≥ d, and one refers to quasiperi-
odic crystals that are aperiodic (with D > d) as “quasi-
crystals” [3].
A generic free energy, expressed in Fourier space as
F ¼
X
j≤n
X
α1;…; αn
X
k1;…;kn
Aα1;…;αnj ðk1;…;knÞ
× ψα1ðk1Þ   ψαjðkjÞψαjþ1ð−kjþ1Þ   ψαnð−knÞ; ð2Þ
is restricted only by the requirements imposed by the
symmetry of the disordered liquid phase. Phonons and
phasons are Goldstone modes that result from breaking
the continuous symmetries of F that are imposed by
translation invariance, which requires the coefficients
Aα1;…;αnj ðk1;…;knÞ to vanish unless k1 þ    þ kn ¼ 0
[4]. We count the number of such broken symmetries by
characterizing the set of all symmetry-broken minimum
free-energy states since any broken symmetry operation, by
definition, takes a particular minimum free-energy state
into a different one. Surprisingly [5], as explained below,
the number of these Goldstone modes is not equal to the
number of independent translations in d dimensions, but
rather to the rank D of the crystal.
A generic set of coefficients A can vary freely with
external parameters subject to symmetry restrictions alone,
while the field ψα adjusts itself accordingly to minimize F .
Thus, for two different fields ψα and ψ 0α to be indistinguish-
able [6], in the sense that they both minimize the same
generic free energy, they must agree independently on all the
products in F with nonvanishing coefficients. In the absence
of additional symmetry, one can then show [6,7], using the
identity of products of order 2 and 3 alone, that the condition
for indistinguishability reduces to the requirement that
ψ 0αðkÞ ¼ e2πiχðkÞψαðkÞ; ð3Þ
where χðkÞ, called a “gauge function” [8], is linear to within
an additive integer on the lattice L.
In quantum crystals with additional U(1) symmetry
ψα → eiθψα or in classical crystals with an additional Z2
symmetry ψα → −ψα, the free energy (2) is restricted to
contain products of only even order, questioning the
linearity of gauge functions. Nevertheless, one can still
use the identity of products of order 2 to establish Eq. (3); in
the case of complex-valued fields, use products of order 4
to show that χð−kÞ ¼ −χðkÞ, which is trivially so for real-
valued fields; and finally, use products of order 6 to obtain
χðk1 þ k2Þ ¼ χðk1Þ þ χðk2Þ, where these equalities hold
to within an overall constant coming from the additional
symmetry. Suppression of sixth- or higher-order products at
low density—whether in classical or in quantum crystals—
does not affect this outcome. Such terms might be irrelevant
at the critical fixed point, but dangerously so [9] in the
sense that despite being small they still distinguish between
states in the ordered phase.
The linearity of gauge functions implies that χðkÞ is
uniquely determined by specifying its values on a chosen
basis for L, consisting of D linearly independent vectors
over the integers. Thus, there are exactly D independent
symmetry operations in F that are broken and therefore
exactly D Goldstone modes. One may choose the basis so
that d of the Goldstone modes are the familiar phonon
modes. The remaining D − d modes are the phason modes,
regardless of whether the crystal is quantum or classical.
I am grateful to Michael Cross and Gil Refael for
valuable discussions. This research is supported by the
Israel Science Foundation through Grant No. 556/10.
Ron Lifshitz*
Condensed Matter Physics 149-33
California Institute of Technology
Pasadena, California 91125, USA
Received 4 December 2013; published 13 August 2014
DOI: 10.1103/PhysRevLett.113.079602
PACS numbers: 67.85.−d, 03.75.Mn, 61.44.Br,
*Permanent address: School of Physics & Astronomy, Tel
Aviv University, Tel Aviv 69978, Israel.
ronlif@tau.ac.il
[1] S. Gopalakrishnan, I. Martin, and E. A. Demler, Phys. Rev.
Lett. 111, 185304 (2013).
PRL 113, 079602 (2014) P HY S I CA L R EV I EW LE T T ER S
week ending
15 AUGUST 2014
0031-9007=14=113(7)=079602(2) 079602-1 © 2014 American Physical Society
[2] In Ref. [1], the dimension d is 2, and the rank D of anM-fold
symmetric star of two-dimensional wave vectors is given by
the Euler totient function ϕðMÞ ¼ MΠPið1 − 1=PiÞ, where
Pi are the distinct prime factors of M.
[3] R. Lifshitz, Found. Phys. 33, 1703 (2003); Z. Kristallogr.
222, 313 (2007).
[4] The constraints imposed by rotational invariance are imma-
terial here and treated elsewhere, for example, in Ref. [5].
[5] R. Lifshitz, Isr. J. Chem. 51, 1156 (2011).
[6] N. D. Mermin, Rev. Mod. Phys. 64, 3 (1992).
[7] R. Lifshitz, Rev. Mod. Phys. 69, 1181 (1997), see the
Appendix.
[8] D. S. Rokhsar, D. C. Wright, and N. D. Mermin, Acta
Crystallogr. Sect. A 44, 197 (1988).
[9] D. J. Amit and L. Peliti, Ann. Phys. (N.Y.) 140, 207
(1982).
PRL 113, 079602 (2014) P HY S I CA L R EV I EW LE T T ER S
week ending
15 AUGUST 2014
079602-2


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