<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> symmetry broken supersolid in spin–orbit-coupled Bose–Einstein condensate

Abstract Supersolid is an exotic state of matter characterized by both superfluid properties and periodic particle density modulation, due to spontaneous breaking of U (1) gauge symmetry and spatial translation symmetry, respectively. For conventional supersolids, continuous translation symmetry breaking is accompanied by one gapless Goldstone mode in the excitation spectra. An interesting question naturally arises: what is the consequence of breaking discrete translation symmetry for supersolids? In this work, we study the consequence of discrete symmetry breaking in a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> supersolid resulting from spontaneous breaking of a discrete <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> symmetry, or equivalently, a discrete translation symmetry. This <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> supersolid is realized in the stripe phase of spin–orbit-coupled Bose–Einstein condensate under an external periodic potential with period <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> of intrinsic stripe period. For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mtext>⩾</mml:mtext> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , there are n degenerate ground states with spontaneously broken lattice translation symmetry. The low-energy excitations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> supersolid include a pseudo-Goldstone mode, whose excitation gap at long wavelength limit is found to decrease rapidly with n . We further numerically show that, when confined in a harmonic trap, a spin-dependent perturbation can result in the transition between degenerate ground states of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> supersolid. With the integer n tunable using the experimental technique of generating subwavelength optical lattice, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> supersolid proposed here offers a cold atom platform to simulate physics related with generic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> symmetry breaking in a highly controllable setting.