Drag Force and Superfluidity in the Supersolid Stripe Phase of a Spin–Orbit-Coupled Bose–Einstein Condensate

G. I. Martone; G. V. Shlyapnikov

doi:10.1134/S1063776118110146

Journal of Experimental and Theoretical Physics

November 2018, Volume 127, Issue 5, pp 865–876 | Cite as

Drag Force and Superfluidity in the Supersolid Stripe Phase of a Spin–Orbit-Coupled Bose–Einstein Condensate

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G. I. Martone
G. V. Shlyapnikov

Article

First Online: 23 January 2019

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Abstract

The phase diagram of a spin–orbit-coupled two-component Bose gas includes a supersolid stripe phase, which is featuring density modulations along the direction of the spin–orbit coupling. This phase has been recently found experimentally [31]. In the present work, we characterize the superfluid behavior of the stripe phase by calculating the drag force acting on a moving impurity. Because of the gapless band structure of the excitation spectrum, the Landau critical velocity vanishes if the motion is not strictly parallel to the stripes, and energy dissipation takes place at any speed. Moreover, due to the spin–orbit coupling, the drag force can develop a component perpendicular to the velocity of the impurity. Finally, by estimating the time over which the energy dissipation occurs, we find that for slow impurities, the effects of friction are negligible on a time scale up to several seconds, which is comparable with the duration of a typical experiment.

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APPENDIX

Calculation of the Drag Force

The calculation of the drag force is as follows. We first make the replacement V^–1$\sum\nolimits_{\mathbf{q}}^{} \, $→ (2π)^–3$\int {{{d}^{3}}q} $ in Eq. (25). Then, we switch to cylindrical coordinates by setting q = (q_x, q_⊥cosφ_q, q_⊥sinφ_q). Equation (25) then becomes

$$\begin{gathered} {\mathbf{F}}\, = \, - {\kern 1pt} \frac{{{{\hbar }^{3}}{{b}^{2}}\bar {n}}}{{{{m}^{2}}}}\sum\limits_\ell ^{} {\int\limits_{ - \infty }^\infty {d{{q}_{x}}\int\limits_0^\infty {d{{q}_{ \bot }}} } } \int\limits_0^{2\pi } {d{{\varphi }_{q}}{{q}_{ \bot }}{\text{|}}{{f}_{\ell }}({{q}_{x}},{{q}_{ \bot }}){{{\text{|}}}^{2}}} \\ \, \times \delta ({{\omega }_{\ell }}({{q}_{x}},{{q}_{ \bot }}) - {{{v}}_{x}}{{q}_{x}} - {{{v}}_{y}}{{q}_{ \bot }}\cos {{\varphi }_{q}}){\mathbf{q}}, \\ \end{gathered} $$

(32)

where we have used the fact that ${{\omega }_{{\ell ,{\mathbf{q}}}}}$ and |${{f}_{{\ell ,{\mathbf{q}}}}}$|² do not depend on φ_q.

Let us first consider the ${{{v}}_{y}}$ ≠ 0 case. By using the properties of the δ function one can write

$$\begin{gathered} \delta ({{\omega }_{\ell }}({{q}_{x}},{{q}_{ \bot }}) - {{{v}}_{x}}{{q}_{x}} - {{{v}}_{y}}{{q}_{ \bot }}\cos {{\varphi }_{q}}) = \frac{1}{{{{{v}}_{y}}{{q}_{ \bot }}{\text{|}}\sin {{{\tilde {\varphi }}}_{{\ell ,q}}}{\text{|}}}} \\ \times \sum\limits_{\bar {m} \in \mathbb{Z}}^{} {[\delta ({{\varphi }_{q}} - ({{{\tilde {\varphi }}}_{{\ell ,q}}} + 2\bar {m}\pi ))} + \delta ({{\varphi }_{q}} - ( - {{{\tilde {\varphi }}}_{{\ell ,q}}} + 2\bar {m}\pi ))], \\ \end{gathered} $$

(33)

where

$${{\tilde {\varphi }}_{{\ell ,{\mathbf{q}}}}} = \arccos \frac{{{{\omega }_{\ell }}({{q}_{x}},{{q}_{ \bot }}) - {{{v}}_{x}}{{q}_{x}}}}{{{{{v}}_{y}}{{q}_{ \bot }}}}.$$

(34)

By plugging Eqs. (33) and (34) into (25) and performing the integration over φ_q, we find

$$\begin{gathered} {{F}_{x}} = - \frac{{{{\hbar }^{3}}{{b}^{2}}\bar {n}}}{{{{m}^{2}}}}\sum\limits_\ell ^{} {\int\limits_{ - \infty }^\infty {d{{q}_{x}}\int\limits_0^\infty {d{{q}_{ \bot }}{{q}_{ \bot }}{\text{|}}{{f}_{\ell }}({{q}_{x}},{{q}_{ \bot }}){{{\text{|}}}^{2}}} } } \\ \times \frac{{2\Theta ({{{v}}_{y}}{{q}_{ \bot }} - \,{\text{|}}{{\omega }_{\ell }}({{q}_{x}},{{q}_{ \bot }}) - {{{v}}_{x}}{{q}_{x}}{\text{|}})}}{{\sqrt {{{{({{{v}}_{y}}{{q}_{ \bot }})}}^{2}} - {{{[{{\omega }_{\ell }}({{q}_{x}},{{q}_{ \bot }}) - {{{v}}_{x}}{{q}_{x}}]}}^{2}}} }}{{q}_{x}}, \\ \end{gathered} $$

(35)

$$\begin{gathered} {{F}_{y}} = - \frac{{{{\hbar }^{3}}{{b}^{2}}\bar {n}}}{{{{m}^{2}}}}\sum\limits_\ell ^{} {\int\limits_{ - \infty }^\infty {d{{q}_{x}}\int\limits_0^\infty {d{{q}_{ \bot }}{{q}_{ \bot }}{\text{|}}{{f}_{\ell }}({{q}_{x}},{{q}_{ \bot }}){{{\text{|}}}^{2}}} } } \\ \times \frac{{2\Theta ({{{v}}_{y}}{{q}_{ \bot }} - \,{\text{|}}{{\omega }_{\ell }}({{q}_{x}},{{q}_{ \bot }}) - {{{v}}_{x}}{{q}_{x}}{\text{|}})}}{{\sqrt {{{{({{{v}}_{y}}{{q}_{ \bot }})}}^{2}} - {{{[{{\omega }_{\ell }}({{q}_{x}},{{q}_{ \bot }}) - {{{v}}_{x}}{{q}_{x}}]}}^{2}}} }}{{q}_{x}} \\ \times \frac{{{{\omega }_{\ell }}({{q}_{x}},{{q}_{ \bot }}) - {{{v}}_{x}}{{q}_{x}}}}{{{{{v}}_{y}}}}, \\ \end{gathered} $$

(36)

while F_z = 0, in agreement with the symmetry argument presented in Section 4. The symbol Θ in Eqs. (35) and (36) denotes the Heaviside function.

All the formulas derived up to now are exact. At very low impurity speed ${v}$, the dominant contribution to the integrals (35) and (36) comes from the modes of the lowest-lying band of the excitation spectrum ($\ell $ = 1) with q_x close to 2k₁ and small q_⊥ (q close to 2k₁). In this regime we can write ω₁(q_x, q_⊥) ≈ c₁$\sqrt {{{{({{q}_{x}} - 2{{k}_{1}})}}^{2}} + q_{ \bot }^{2}} $, where c₁ is the sound velocity, and we neglect its anisotropy. This assumption is reasonable when Ω_R is small and c_1,_x ≈ c_{1, ⊥}, as shown in [33]. In [33] it was also proven, by means of sum-rule techniques, that the static structure factor S(q) = N^–1$\sum\nolimits_{\ell ,{\mathbf{q}}}^{} {\text{|}} {{f}_{{\ell ,{\mathbf{q}}}}}{{{\text{|}}}^{2}}$ behaves as 1/$\sqrt {{{{({{q}_{x}} - 2{{k}_{1}})}}^{2}} + q_{ \bot }^{2}} $ when q approaches 2k₁. Since S(q) is dominated by the $\ell $ = 1 term close to the Brillouin point, we can write

$${\text{|}}{{f}_{1}}({{q}_{x}},{{q}_{ \bot }}){{{\text{|}}}^{2}} \approx \frac{{{\text{|}}{{{\tilde {f}}}_{1}}{{{\text{|}}}^{2}}}}{{\hbar \sqrt {{{{({{q}_{x}} - 2{{k}_{1}})}}^{2}} + q_{ \bot }^{2}} }},$$

where |${{\tilde {f}}_{1}}$|² is a numerical coefficient. We now put the above expressions for ω₁(q_x, q_⊥) and |f₁(q_x, q_⊥)|² into Eqs. (35) and (36). Then, using the polar representation q_x = 2k₁ + qcosθ_q, q_⊥ = qsinθ_q, and taking ${{{v}}_{x}}$q_x ≈ 2${{{v}}_{x}}$k₁, we arrive at the following expressions:

$$\begin{gathered} {{F}_{x}} \approx - \frac{{{{\hbar }^{2}}{{b}^{2}}\bar {n}}}{{{{m}^{2}}}}\int\limits_0^\pi {d{{\theta }_{q}}\int\limits_0^\infty {dqq\sin {{\theta }_{q}}} } \\ \times \frac{{2{\text{|}}{{{\tilde {f}}}_{1}}{{{\text{|}}}^{2}}\Theta ({{{v}}_{y}}q\sin {{\theta }_{q}} - \,{\text{|}}{{c}_{1}}q - 2{{{v}}_{x}}{{k}_{1}}{\text{|}})}}{{\sqrt {{{{({{{v}}_{y}}q\sin {{\theta }_{q}})}}^{2}} - {{{({{c}_{1}}q - 2{{{v}}_{x}}{{k}_{1}})}}^{2}}} }}(2{{k}_{1}} + q\cos {{\theta }_{q}}), \\ \end{gathered} $$

(37)

$$\begin{gathered} {{F}_{y}} \approx - \frac{{{{\hbar }^{2}}{{b}^{2}}\bar {n}}}{{{{m}^{2}}}}\int\limits_0^\pi {d{{\theta }_{q}}\int\limits_0^\infty {dqq\sin {{\theta }_{q}}} } \\ \times \frac{{2{\text{|}}{{{\tilde {f}}}_{1}}{{{\text{|}}}^{2}}\Theta ({{{v}}_{y}}q\sin {{\theta }_{q}} - \,{\text{|}}{{c}_{1}}q - 2{{{v}}_{x}}{{k}_{1}}{\text{|}})}}{{\sqrt {{{{({{{v}}_{y}}q\sin {{\theta }_{q}})}}^{2}} - {{{({{c}_{1}}q - 2{{{v}}_{x}}{{k}_{1}})}}^{2}}} }}\frac{{{{c}_{1}}q - 2{{{v}}_{x}}{{k}_{1}}}}{{{{{v}}_{y}}}}. \\ \end{gathered} $$

(38)

Integrals (37) and (38) can be easily evaluated. It is convenient to perform first the integration with respect to q. The Heaviside function reduces to unity in the range q_–(θ_q) < q < q₊(θ_q), with q_±(θ_q) = 2${{{v}}_{x}}$k₁/(c₁$ \mp $${{{v}}_{y}}$sinθ_q), and vanishes otherwise. One finally gets

$${{F}_{x}} \approx - \frac{{16\pi {{\hbar }^{2}}k_{1}^{2}{{b}^{2}}\bar {n}{\text{|}}{{{\tilde {f}}}_{1}}{{{\text{|}}}^{2}}}}{{{{m}^{2}}}}\frac{{{{{v}}_{x}}}}{{c_{1}^{2} - {v}_{y}^{2}}},$$

(39)

$${{F}_{y}} \approx - \frac{{16\pi {{\hbar }^{2}}k_{1}^{2}{{b}^{2}}\bar {n}{\text{|}}{{{\tilde {f}}}_{1}}{{{\text{|}}}^{2}}}}{{{{m}^{2}}}}\frac{{{v}_{x}^{2}{{{v}}_{y}}}}{{{{{(c_{1}^{2} - {v}_{y}^{2})}}^{2}}}},$$

(40)

which become Eqs. (28) and (29) of the main text if one expands the denominators in powers of ${{{v}}_{y}}$/c₁ and retains the terms up to cubic order in ${v}$.

If ${{{v}}_{y}}$ = 0, the integration over φ_q in Eq. (32) is trivial. It gives F_y = F_z = 0 and

$$\begin{gathered} {{F}_{x}} = - \frac{{{{\hbar }^{3}}{{b}^{2}}\bar {n}}}{{{{m}^{2}}}}\sum\limits_\ell ^{} {\int\limits_{ - \infty }^\infty {d{{q}_{x}}\int\limits_0^\infty {d{{q}_{ \bot }}{{q}_{ \bot }}{\text{|}}{{f}_{\ell }}({{q}_{x}},{{q}_{ \bot }}){{{\text{|}}}^{2}}} } } \\ \, \times \delta ({{\omega }_{\ell }}({{q}_{x}},{{q}_{ \bot }}) - {{{v}}_{x}}{{q}_{x}}){{q}_{x}}. \\ \end{gathered} $$

(41)

In the small-${v}$ limit, the integral (41) can be calculated using the same approximations as in the ${{{v}}_{y}}$ ≠ 0 case. The final result coincides with Eq. (39) with ${{{v}}_{y}}$ = 0.

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Authors and Affiliations

G. I. Martone
- 1
Email author
G. V. Shlyapnikov
- 1
- 2
- 3
- 4
- 5
- 6

1.LPTMS, CNRS, Univ. Paris-Sud, Université Paris-SaclayOrsayFrance
2.Russian Quantum CenterSkolkovoMoscowRussia
3.SPEC, CEA, CNRS, Université Paris-Saclay, CEA SaclayGif sur YvetteFrance
4.Van der Waals–Zeeman Institute, Institute of Physics, University of AmsterdamAmsterdamNetherlands
5.Wuhan Institute of Physics and Mathematics, Chinese Academy of SciencesWuhanChina
6.Russian Quantum Center, National University of Science and Technology MISISMoscowRussia

Drag Force and Superfluidity in the Supersolid Stripe Phase of a Spin–Orbit-Coupled Bose–Einstein Condensate

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