A spin-1 ferromagnetic Bose-Einstein condensate subject to a certain magnetic field exhibits a broken-axisymmetry phase in which the magnetization tilts against the applied magnetic field due to the competition between ferromagnetism and linear and quadratic Zeeman effects. The Bogoliubov analysis shows that in this phase two Goldstone modes associated with U(1) and SO(2) symmetry breakings exist, in which phonons and magnons are coupled to restore the two broken symmetries.

arXiv:cond-mat/0607184Physical Review AJan 5, 200799 citations

Ground state of spin-1 Bose-Einstein condensates with spin-orbit coupling in a Zeeman field

Lin Wen, Qing‐Feng Sun, H. Q. Wang +2 more

arXiv:1208.2514Physical Review AOct 3, 201251 citations

arXiv:2002.06088 · Physical Review Research

Phase diagram of solitons in the polar phase of a spin-1 Bose-Einstein condensate

I-Kang Liu, Shih-Chuan Gou, Hiromitsu Takeuchi

Published Sep 28, 2020• 13 citations

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Abstract

DOI: 10.1103/physrevresearch.2.033506

Related papers

Phase Diagram of Solitons in the Polar Phase of a Spin-1 Bose-Einstein Condensate

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We theoretically study the structure of a stationary soliton in the polar phase of spin-1 Bose-Einstein condensate in the presence of quadratic Zeeman effect at zero temperature. The phase diagram of such solitons is mapped out by finding the states of minimal soliton energy in the defining range of polar phase. The states are assorted into normal, anti-ferromagnetic, broken-axisymmetry, and ferromagnetic phases according to the number and spin densities in the core. The order of phase transitions between different solitons and the critical behaviour of relevant continuous transitions are proved within the mean-field theory.We theoretically study the structure of a stationary soliton in the polar phase of spin-1 Bose-Einstein condensate in the presence of quadratic Zeeman effect at zero temperature. The phase diagram of such solitons is mapped out by finding the states of minimal soliton energy in the defining range of polar phase. The states are assorted into normal, anti-ferromagnetic, broken-axisymmetry, and ferromagnetic phases according to the number and spin densities in the core. The order of phase transitions between different solitons and the critical behaviour of relevant continuous transitions are proved within the mean-field theory.

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We systematically investigate the weakly trapped spin-1 Bose-Einstein condensates with spin-orbit coupling in an external Zeeman field. We find that the mean-field ground state favors either a magnetized standing-wave phase or plane-wave phase when the strength of the Zeeman field is below a critical value related to the strength of spin-orbit coupling. The Zeeman field can induce a phase transition between standing-wave and plane-wave phases, and we determine the phase boundary analytically and numerically. The magnetization of these two phases responds to the external magnetic field in a very unique manner: the linear Zeeman effect magnetizes the standing-wave phase along the direction of the magnetic field, but the quadratic Zeeman effect demagnetizes the plane-wave phase. When the strength of Zeeman field surpasses the critical value, the system is completely polarized to a ferromagnetic state or polar state with zero momentum.We systematically investigate the weakly trapped spin-1 Bose-Einstein condensates with spin-orbit coupling in an external Zeeman field. We find that the mean-field ground state favors either a magnetized standing-wave phase or plane-wave phase when the strength of the Zeeman field is below a critical value related to the strength of spin-orbit coupling. The Zeeman field can induce a phase transition between standing-wave and plane-wave phases, and we determine the phase boundary analytically and numerically. The magnetization of these two phases responds to the external magnetic field in a very unique manner: the linear Zeeman effect magnetizes the standing-wave phase along the direction of the magnetic field, but the quadratic Zeeman effect demagnetizes the plane-wave phase. When the strength of Zeeman field surpasses the critical value, the system is completely polarized to a ferromagnetic state or polar state with zero momentum.

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{m}_{z}

, we observe for low magnetic fields an ``antiferromagnetic'' phase where atoms condense in the $m=\ifmmode\pm\else\textpm\fi{}1$ Zeeman states, and occupation of the

m=0

state is suppressed. Conversely, for large enough magnetic fields, a phase transition to a ``broken-axisymmetry'' phase takes place: The

m=0

component becomes populated and rises sharply above a critical field

{B}_{c}({m}_{z})

. This behavior results from the competition between antiferromagnetic spin-dependent interactions (dominant at low fields) and the quadratic Zeeman energy (dominant at large fields). We compare the measured

{B}_{c}

as well as the global shape of the phase diagram with mean-field theory, and find good quantitative agreement.