PHYSICAL REVIEW LETTERS
PRL 113, 079601 (2014)
week ending 15 AUGUST 2014
Comment on “Quantum Quasicrystals of Spin-Orbit-Coupled Dipolar Bosons” In a recent Letter, Gopalakrishnan, Martin, and Demler propose a quantum-mechanical system with several stable crystalline and quasicrystalline ground states and conclude that “there are typically additional phasons in quantummechanical (quasi)crystals, when compared with their classical equivalents” [1]. However, as we show in this Comment, all modes can also be realized in classical systems, except for the quantum-mechanical Goldstone mode due to the overall U(1) symmetry. Furthermore, Gopalakrishnan, Martin, and Demler claim that in their system a M component quantum state exhibits M − 2 phasons [1]. We explain in this Comment that there are only φðMÞ − 2 phasons, where φðxÞ is Euler’s totient function, i.e., the number of integers smaller or equal to M that are coprime to M. Phasons are modes whose free energy cost is zero [2], which is the case if and only if the local isomorphism (LI) class is preserved [3,4]. As we show in the following, the additional modes described in [1] change the LI class and thus are not phasons. In case of even M, the number of all modes whose free energy cost is approximately zero is smaller than the number of phonons and phasons given in [1]. All M modes of Eq. (10) in [1] also occur in density fields ρðrÞ that are composed of M plane waves ΦGj ;ϕj ¼ Aj exp ðiGj · r þ iϕj Þ with phases ϕj and symmetrically arranged wave vectors Gj [cf. Fig. 2(b) in [1] ]. There are φðMÞ basis vectors [5,6], P which we index with 0 to φðMÞ−1 φðMÞ − 1. Therefore, Gj ¼ k¼0 ajk Gk with integers ajk for j ¼ φðMÞ; …; M − 1. To determine which modes preserve the LI class, we expand ρðrÞ with respect to plane waves along the basis vectors: ρðrÞ ¼
φðMÞ−1 X j¼0
¼
φðMÞ−1 X j¼0
ΦGj ;ϕj þ ΦGj ;ϕj þ
M −1 X
Aj ei
PφðMÞ−1 k¼0
A~j
φðMÞ−1 Y
j¼φðMÞ
k¼0
We acknowledge support by the Deutsche Forschungsgemeinschaft through the Emmy-Noether program (Schm 2657/2). M. Sandbrink,1 J. Roth2 and M. Schmiedeberg1,*
a
ΦGjkk ;ϕk :
ð1Þ
QφðMÞ−1 a The prefactor A~j ¼ Aj exp ðiΔϕj Þ= k¼0 Ak jk with PφðMÞ−1 Δϕj ¼ ϕj − k¼0 ajk ϕk is invariant under a change of phases if Δϕjk is an integer multiple of 2π (cf. [7]). Therefore, there are only φðMÞ independent continuous modes—two phonons and φðMÞ − 2 phasons [2]—that preserve the structure up to a displacement and thus stay within a given LI class [3,4]. Only these modes do not change the free energy [4]. In an expansion with respect to quantum-mechanical Bose fields, terms of even order vanish, and terms of order 2n correspond to classical terms of order n. Therefore, the number of phasons in quantummechanical systems also is φðMÞ − 2. A mode similar to the one shown in Figs. 3(d)–3(f) in [1] can be realized classically (see Fig. 1). This mode 0031-9007=14=113(7)=079601(2)
corresponds to a phonon in combination with a global phase shift. The latter does not preserve the LI class and thus is associated with a possibly small change of the free energy. A gradient along a similar phase change already has been employed in [8]. Furthermore, concerning laserinduced colloidal quasicrystals [6,9–11], all modes can in principle be realized [10]. Note that the global phase shift can be achieved by changing the polarization [11]. In [1], the third-order and higher-order terms of the free energy expansion with respect to the density and thus also the free energy cost of some of the additional modes are small. Similar approximations might be possible in special classical systems. However, in case of even M, the energy cost of half of the additional modes proposed in [1] still cannot be neglected due to the quadratic term in the expansion. An open question is whether some of the additional modes that change the LI class can be thermally excited, e.g., in an experimental realization of the system proposed in [1].
ajk Gk ·rþiϕj
j¼φðMÞ M −1 X
FIG. 1 (color online). Classical counterpart of the mode shown in Figs. 3(d)–3(f) in [1]. Here, the real part of the sum of three plane waves with three phases is used. Two phases are always 0, while the third phase is (a) −π, (b) −3π=2, and (c) −2π. The LI class is not preserved, and thus, this is not a phason.
1
Institut für Theoretische Physik 2: Weiche Materie Heinrich-Heine-Universität Düsseldorf 40204 Düsseldorf, Germany 2 Institut für Funktionelle Materialien und Quantentechnologien, Universität Stuttgart 70550 Stuttgart, Germany Received 3 December 2013; published 13 August 2014 DOI: 10.1103/PhysRevLett.113.079601 PACS numbers: 67.85.−d, 03.75.Mn, 71.70.Ej, *
[email protected]‑duesseldorf.de [1] S. Gopalakrishnan, I. Martin, and E. A. Demler, Phys. Rev. Lett. 111, 185304 (2013). [2] J. E. S. Socolar, T. C. Lubensky, and P. J. Steinhardt, Phys. Rev. B 34, 3345 (1986). [3] D. Levine and P. J. Steinhardt, Phys. Rev. B 34, 596 (1986).
079601-1
© 2014 American Physical Society
PRL 113, 079601 (2014)
PHYSICAL REVIEW LETTERS
[4] J. E. S. Socolar and P. J. Steinhardt, Phys. Rev. B 34, 617 (1986). [5] M. Baake and U. Grimm, Aperiodic Order: A Mathematical Invitation (Cambridge University Press, Cambridge, 2013), Vol. 1, pp. 36, 49–53. [6] M. Schmiedeberg and H. Stark, J. Phys. Condens. Matter 24, 284101 (2012). [7] D. S. Rokhsar, D. C. Wright, and N. D. Mermin, Acta Crystallogr. Sect. A 44, 197 (1988).
week ending 15 AUGUST 2014
[8] B. Freedman, R. Lifshitz, J. W. Fleischer, and M. Segev, Nat. Mater. 6, 776 (2007). [9] J. Mikhael, J. Roth, L. Helden, and C. Bechinger, Nature (London) 454, 501 (2008). [10] J. A. Kromer, M. Schmiedeberg, J. Roth, and H. Stark, Phys. Rev. Lett. 108, 218301 (2012). [11] M. Schmiedeberg, Ph.D. thesis, Technische Universität Berlin, 2008.
079601-2
PRL 113, 079601 (2014)
week ending 15 AUGUST 2014
Comment on “Quantum Quasicrystals of Spin-Orbit-Coupled Dipolar Bosons” In a recent Letter, Gopalakrishnan, Martin, and Demler propose a quantum-mechanical system with several stable crystalline and quasicrystalline ground states and conclude that “there are typically additional phasons in quantummechanical (quasi)crystals, when compared with their classical equivalents” [1]. However, as we show in this Comment, all modes can also be realized in classical systems, except for the quantum-mechanical Goldstone mode due to the overall U(1) symmetry. Furthermore, Gopalakrishnan, Martin, and Demler claim that in their system a M component quantum state exhibits M − 2 phasons [1]. We explain in this Comment that there are only φðMÞ − 2 phasons, where φðxÞ is Euler’s totient function, i.e., the number of integers smaller or equal to M that are coprime to M. Phasons are modes whose free energy cost is zero [2], which is the case if and only if the local isomorphism (LI) class is preserved [3,4]. As we show in the following, the additional modes described in [1] change the LI class and thus are not phasons. In case of even M, the number of all modes whose free energy cost is approximately zero is smaller than the number of phonons and phasons given in [1]. All M modes of Eq. (10) in [1] also occur in density fields ρðrÞ that are composed of M plane waves ΦGj ;ϕj ¼ Aj exp ðiGj · r þ iϕj Þ with phases ϕj and symmetrically arranged wave vectors Gj [cf. Fig. 2(b) in [1] ]. There are φðMÞ basis vectors [5,6], P which we index with 0 to φðMÞ−1 φðMÞ − 1. Therefore, Gj ¼ k¼0 ajk Gk with integers ajk for j ¼ φðMÞ; …; M − 1. To determine which modes preserve the LI class, we expand ρðrÞ with respect to plane waves along the basis vectors: ρðrÞ ¼
φðMÞ−1 X j¼0
¼
φðMÞ−1 X j¼0
ΦGj ;ϕj þ ΦGj ;ϕj þ
M −1 X
Aj ei
PφðMÞ−1 k¼0
A~j
φðMÞ−1 Y
j¼φðMÞ
k¼0
We acknowledge support by the Deutsche Forschungsgemeinschaft through the Emmy-Noether program (Schm 2657/2). M. Sandbrink,1 J. Roth2 and M. Schmiedeberg1,*
a
ΦGjkk ;ϕk :
ð1Þ
QφðMÞ−1 a The prefactor A~j ¼ Aj exp ðiΔϕj Þ= k¼0 Ak jk with PφðMÞ−1 Δϕj ¼ ϕj − k¼0 ajk ϕk is invariant under a change of phases if Δϕjk is an integer multiple of 2π (cf. [7]). Therefore, there are only φðMÞ independent continuous modes—two phonons and φðMÞ − 2 phasons [2]—that preserve the structure up to a displacement and thus stay within a given LI class [3,4]. Only these modes do not change the free energy [4]. In an expansion with respect to quantum-mechanical Bose fields, terms of even order vanish, and terms of order 2n correspond to classical terms of order n. Therefore, the number of phasons in quantummechanical systems also is φðMÞ − 2. A mode similar to the one shown in Figs. 3(d)–3(f) in [1] can be realized classically (see Fig. 1). This mode 0031-9007=14=113(7)=079601(2)
corresponds to a phonon in combination with a global phase shift. The latter does not preserve the LI class and thus is associated with a possibly small change of the free energy. A gradient along a similar phase change already has been employed in [8]. Furthermore, concerning laserinduced colloidal quasicrystals [6,9–11], all modes can in principle be realized [10]. Note that the global phase shift can be achieved by changing the polarization [11]. In [1], the third-order and higher-order terms of the free energy expansion with respect to the density and thus also the free energy cost of some of the additional modes are small. Similar approximations might be possible in special classical systems. However, in case of even M, the energy cost of half of the additional modes proposed in [1] still cannot be neglected due to the quadratic term in the expansion. An open question is whether some of the additional modes that change the LI class can be thermally excited, e.g., in an experimental realization of the system proposed in [1].
ajk Gk ·rþiϕj
j¼φðMÞ M −1 X
FIG. 1 (color online). Classical counterpart of the mode shown in Figs. 3(d)–3(f) in [1]. Here, the real part of the sum of three plane waves with three phases is used. Two phases are always 0, while the third phase is (a) −π, (b) −3π=2, and (c) −2π. The LI class is not preserved, and thus, this is not a phason.
1
Institut für Theoretische Physik 2: Weiche Materie Heinrich-Heine-Universität Düsseldorf 40204 Düsseldorf, Germany 2 Institut für Funktionelle Materialien und Quantentechnologien, Universität Stuttgart 70550 Stuttgart, Germany Received 3 December 2013; published 13 August 2014 DOI: 10.1103/PhysRevLett.113.079601 PACS numbers: 67.85.−d, 03.75.Mn, 71.70.Ej, *
[email protected]‑duesseldorf.de [1] S. Gopalakrishnan, I. Martin, and E. A. Demler, Phys. Rev. Lett. 111, 185304 (2013). [2] J. E. S. Socolar, T. C. Lubensky, and P. J. Steinhardt, Phys. Rev. B 34, 3345 (1986). [3] D. Levine and P. J. Steinhardt, Phys. Rev. B 34, 596 (1986).
079601-1
© 2014 American Physical Society
PRL 113, 079601 (2014)
PHYSICAL REVIEW LETTERS
[4] J. E. S. Socolar and P. J. Steinhardt, Phys. Rev. B 34, 617 (1986). [5] M. Baake and U. Grimm, Aperiodic Order: A Mathematical Invitation (Cambridge University Press, Cambridge, 2013), Vol. 1, pp. 36, 49–53. [6] M. Schmiedeberg and H. Stark, J. Phys. Condens. Matter 24, 284101 (2012). [7] D. S. Rokhsar, D. C. Wright, and N. D. Mermin, Acta Crystallogr. Sect. A 44, 197 (1988).
week ending 15 AUGUST 2014
[8] B. Freedman, R. Lifshitz, J. W. Fleischer, and M. Segev, Nat. Mater. 6, 776 (2007). [9] J. Mikhael, J. Roth, L. Helden, and C. Bechinger, Nature (London) 454, 501 (2008). [10] J. A. Kromer, M. Schmiedeberg, J. Roth, and H. Stark, Phys. Rev. Lett. 108, 218301 (2012). [11] M. Schmiedeberg, Ph.D. thesis, Technische Universität Berlin, 2008.
079601-2
