Instability of Holographic Superfluids in Optical Lattice

As an artificial lattice structure, optical lattice, has a close physical resemblance to the periodic coulomb potential that felt by electrons in solid crystal. While it is not like electrons moving between positive ions in a nanometer dimension, optical lattice can be manipulated with a typical dimension much larger than the conventional crystal—micrometer dimension. Due to this virtue, optical lattice has provided a broad platform for experimental and theoretical physicists to explore much richer and more fundamental properties in condensed matter physics, such as instability PhysRevA.64.061603 ; Wu_2003 , Landau-Zener tunneling 1996PhRvL..76.4504N ; 2000PhRvA..61b3402W ; 2002PhRvA..65f3612C ; 2003PhRvL..91w0406J ; Wu_2003 ; 2020PhRvL.125u3401G , superfluid-Mott insulator quantum phase transition 1998PhRvL..81.3108J ; 2002PhRvA..66c1601K ; 2002PhRvL..89q0402S ; 2004PhRvA..70d3612C ; 2005PhRvA..71d3601Z ; 2006PhRvA..73b3610C and so on. Additionally, by controlling the frequency of the lasers in the course of time, the time-dependent optical lattice can be used for driving the system to explore the floquet dynamics 2015AdPhy..64..139B ; PhysRevLett.115.205301 ; PhysRevLett.121.233603 ; PhysRevX.9.011047 ; 2019NatPh..15.1168S , which are getting more interesting recently.

The instability (or stability) of superfluids is always one of the most fundamental and important properties and usually many physical phenomena are related with it. Since the optical lattice starts to be used for studying cold atom physics, many kinds of research works about the instability have been carried out in various respects. There are Landau or dynamical instability 2001PhRvE..64e6615B ; 2001PhRvE..63c6612B ; PhysRevA.64.061603 ; 2003NJPh….5…71C ; Wu_2003 ; 2004PhRvL..93n0406F ; 2004PhRvA..70d3625M ; PhysRevLett.95.180403 ; PhysRevLett.95.110403 ; 2006PhRvA..74e3611I ; 2006JPhB…39S.101K ; 2007PhRvA..75c3621H ; ZHANG20081147 ; 2009ChPhL..26a6701L ; 2014JPSJ…83l4001A ; PhysRevA.93.013628 ; PhysRevLett.86.4447 ; Cataliotti_2003 ; PhysRevLett.93.140406 ; PhysRevA.72.013603 ; Konabe_2006 , parametric instabilities 2004JPhB…37S.257R ; 2005PhRvA..71f1602K ; 2005PhRvL..95q0404G ; 2017PhRvX…7b1015L ; PhysRevX.10.011030 , modulational instability 2002PhRvA..65b1602K ; 2004PhRvE..69a7601R ; 2006PhRvL..96o0402B ; PhysRevA.87.013633 ; PhysRevE.89.052917 and low-acceleration instability PhysRevLett.93.230401 , etc. In this paper we will focus on the Landau and dynamical instability only and simultaneously.

Since the first theoretical study about the Landau and dynamical instability for superfluids in optical lattice PhysRevA.64.061603 using the Gross-Pitaevskii (GP) equation, this topic has been studied extensively. 2001PhRvE..63c6612B ; 2001PhRvE..64e6615B explore the cases both for repulsive and attractive atomic interactions, ZHANG20081147 adds the influence of three-body interactions. 2009ChPhL..26a6701L shows that the states in excited bands always have Landau instability and 2004PhRvA..70d3625M investigates the effect of transverse excitations for higher dimensional systems. PhysRevLett.95.180403 reveals that the dynamical instability is the origin of the spatial domain formation in spin-1 atomic condensates. All these research works relied on the mean-field theory, i.e., GP equation, while in addition to this equation the Bose-Hubbard model is also used in 2014JPSJ…83l4001A ; PhysRevA.93.013628 that is beyond the mean-field theory.

In the above research works, all the superfluid systems have no dissipation. From the instability diagram (Fig.5 in PhysRevA.64.061603 ) we can see that there are two distinct regimes about the instability of superflow. The first is the small optical lattice height regime and in this regime the dynamical instability regime is so narrow that the Landau instability can be studied solely; the second is the large optical lattice height regime where the dynamical instability regime spreads out and it can be detected in experiments, obviously. Experiment PhysRevLett.86.4447 shows that the regime where the dissipative process occurs to break down the superfluid phase agrees with the theoretical prediction about the regime of onset of Landau instability for a superfluid in shallow optical lattice; Experiment Cataliotti_2003 finds that for deep optical lattice there will be transition from a superfluid into an insulator when the superfluid is under dynamical instability. After that, experiments PhysRevLett.93.140406 ; PhysRevA.72.013603 both show that in the presence of thermal component the dissipative process can be related with the occurrence of Landau instability. Since the original GP equation cannot apply to dissipative processes, until Konabe_2006 based on the modified GP equation with dissipation, the instability of superfluid in optical lattice with thermal component has not been studied theoretically. Its result dramatically shows that when the thermal component is added the dynamical instability will occur throughout the Landau instability regime predicted from the original GP equation, which means the Landau instability and dynamical instability will always appear at the same time when there exists dissipation. So it is no longer suitable to investigate Landau instability or dynamical instability separately.

In this paper, we will use the simplest holographic superfluid model PhysRevLett.101.031601 ; Hartnoll_2008 to re-explore the Landau and dynamical instability of superfluids in optical lattice simultaneously based on the advantage that such a model contains superfluid and normal fluid (as thermal component with a finite temperature) intrinsically, which introduces the dissipation naturally and consistently Adams:2012pj . In contrast, the dissipation in the modified GP equation is purely put by hand, without any relation to thermodynamics. In holography, the unstable region can be calculated by quasi-normal mode (QNM) analysis and the results show that there is a critical Bloch wave vector $k_{c}$ above which the Bloch states are unstable due to the Landau instability and dynamical instability. With the help of the calculation of sound speed we confirm that the Landau instability and dynamical instability also appear at the same time in the presence of optical lattice just like the homogenous case in 2014JHEP…02..063A ; 2020arXiv201006232L .

Beyond the linear perturbation analysis, the unstable superflow are also studied by the numerical simulation. In this part we use the unstable superflow state as the initial state for the evolution equation and evolve it for a long time with some small perturbation given at early time. The unstable modes of perturbation will grow exponentially at first and it will lead the whole system into a chaotic state, in which the solitons will appear and disappear (for higher dimension there will be vortices and the system can be considered as under a transient turbulence state 2020arXiv201006232L ). Finally, along with some dissipative processes the system will reduce its current to become stable with the final wave vector $k<k_{c}$. These results can be tested in experiments.

This paper is organized as follows. In Sec. II we introduce the holographic superfluid model that we use and show how the optical lattice is added. Then in Sec. III by solving the corresponding equations of motion we get the static and steady-flow suferfluid states that expressed as Bloch waves with different wave vector $k$. The QNM analysis and the calculation of sound speed are in Sec. IV. And in Sec. V we present the dynamic processes of the evolution of an unstable superflow with numerical simulation. Finally, in Sec. VI we give a summary.

The simplest holographic model to describe superfluids is given in PhysRevLett.101.031601 , where a complex scalar field $\Psi$ is coupled to a $U(1)$ gauge field $A_{M}$ in the $(3+1)$D gravity with a cosmological constant related to the AdS radius as $\Lambda=-3/{L^{2}}$. The action is

where $G_{4}$ is the gravitational constant in four dimensional spacetime. The first two terms in the parenthesis are the gravitational part of the Lagrangian with the Ricci scalar $R$ and the AdS radius $L$, and the third consists of all matter fields:

where $D_{\mu}=\partial_{\mu}\Psi-iA_{\mu}\Psi$.

Since the backreaction of matter fields onto the spacetime geometry is not necessary for our problem, taking the probe limit ${e}\to\infty$ is a suitable and convenient choice, which means that we fix the background spacetime as the standard Schwarzschild-AdS black brane with metric

Here $z_{H}$ is the radius of the black brane horizon. The Hawking temperature of this black brane is $T=\frac{3}{4\pi z_{H}}$, which is also the temperature of the boundary system by the holographic dictionary. Furthermore, due to the existence of the bulk black hole the boundary field theory will intrinsically have dissipation, since there will be energy flow absorbed into the horizon Adams:2012pj ; Li:2019swh ; Tian:2019fax during dynamic processes. For numerical simplicity, we set $L=1=z_{H}$.

The equations of motion for $A_{\mu}$ and $\Psi$ are

One can easily see that there is a trivial solution with $\Psi=0$. As we increase the chemical potential $\mu$, there will be a critical chemical potential $\mu_{c}$ above which a solution with nonzero $\Psi$ appears, indicating the break of $U(1)$ symmetry and the formation of superfluid condensate. Hereafter, we will focus on the case with $\mu=4.5>\mu_{c}$, i.e., the superfluid phase of the boundary field theory.

From the Bloch theorem and with the source $\Psi_{+}$ turning off, we can use Bloch waves to describe the static response

Here, $\psi_{B}\left(z,x\right)$ is the Bloch wave along the $x$ direction, which has the same period as the optical lattice, i.e. $\psi_{B}\left(z,x+l\right)=\psi_{B}\left(z,x\right)$, $k$ is the Bloch wave vector and $l$ the lattice constant (hereafter we choose $l=\pi$). Due to the fact that Bloch waves are complex functions, we can separate them into real and imaginary parts:

Substituting the above equation into the static equations of motion (10)-(13), we get the following five differential equations:

Among them, Eq.(21) will be satisfied automatically if the other four equations are solved. To obtain the static or steady-flow states of the superfluid, we impose the source free boundary conditions for $\Psi_{+}$ and $A_{x}$ at the conformal boundary, regular conditions for $\Psi_{-}$ and $A_{t}=0$ at the horizon and periodic boundary conditions for $A_{t,x}$ and $\Psi$ in the $x$ direction. And the optical lattice structure is imposed by choosing $V(x)=v\cos(2x)$, i.e. $\left.A_{t}\right|_{z=0}=4.5+v\cos(2x)$, with $v$ the height (or strength) of the optical lattice. We solve these four static equations of motion (20), (22), (23) and (24) numerically with the Newton-Raphson method.

Fig.1 shows the numerical solutions of the bulk fields $\psi$ and $A_{t}$, respectively, with two different Bloch wave vectors. Since $\psi$ is complex, we plot its amplitude and phase angle separately. The inexistence of node of order parameter means that these Bloch waves sit in the lowest Bloch band. In Fig. 2(a), we plot the Bloch band in our holographic model. To obtain it, we fix the total particle number, defined as

and solve the static equations (20) and (22)-(24) with $k$ given different values. The loop structure of the blue line ($v=0.5$), at which comes from particle interaction energy is larger than the optical lattice strength PhysRevA.86.063636 , is witnessed around the edge $k=1$ of the Brillouin zone. And for the red line ($v=4$) there is no loop structure. Here, the Bloch band also confirms that the static solution in Fig. 1 are Bloch wave solutions. Actually, the higher value of chemical potential $\mu$, the larger density of total particle $\rho$ in one lattice and hence the larger value of particle interaction energy, which means that when we increase the chemical potential the critical value of the optical lattice strength $v_{c}$ that exceeds the particle interaction energy to prevent the loop structure from appearing will also increase. And we confirm this in Fig. 2(b).

Now we study the linear instability of the Bloch wave solutions obtained in the previous section via QNM analyses. To calculate the QNM, a better choice is to rewrite the static equations of motions (10)-(13) with the Bloch wave form for $\psi$ in the infalling Eddington-Finkelstein coordinates Li:2020ayr :

where the metric is

Here and in the following sections we will not separate $\psi$ into real and imaginary parts. Similarly, there is one constraint equation, which is chosen to be Eq.(27). Next, we give all the bulk field perturbations

Substitute these perturbed fields into the equations of motion, we will get the linear perturbation equations

In accordance with the background steady solutions, we impose Dirichlet and periodic boundary conditions for $a$, $b$, $u$ and $v$ at $z=0$ and in the $x$ direction, respectively, while regular boundary conditions are chosen at the horizon $z=1$. Then, we can obtain $\omega$ from the perturbation equations by solving generalized eigenvalue problems if $k$ and $q$ are given. (See Appendix A for more details about this procedure.)

The perturbations in Eq.(30) will grow exponentially in the linear regime if there exists an $\omega$ whose imaginary part is positive. We use $\omega_{M}$ to denote the $\omega$ with the maximal imaginary part when $q$ and $\mu$ are given, so a system is (linearly) unstable if $\operatorname{Im}(\omega_{M})>0$. Fig. 3 shows the result for the distributions of $\omega$ with two different values for Bloch wave vector $k$ and $q$. In Fig. 3(a) and Fig. 3(b), $\operatorname{Im}(\omega_{M})\leq 0$, so the $k=0$ solution is stable under perturbations $q=0$ and $q=0.15$. By comparison, the solution at $k=0.75$ is unstable because $\operatorname{Im}(\omega_{M})>0$ in Fig. 3(d), despite the fact that in Fig. 3(c) $\operatorname{Im}(\omega_{M})\leq 0$. A system is stable only when $\operatorname{Im}(\omega_{M})\leq 0$ for all values of $q$.

Actually we find that there are two special values $k_{c}$ and $q_{M}$ for the Bloch waves. $k_{c}$ means the critical wave vector exceeding which the Bloch wave vector $k$ will lead to an unstable state; $q_{M}$ means the Bloch wave vector of the most unstable perturbation mode, which has a maximal imaginary value of $\omega_{M}$, since it will increase with time $t$ with the form $e^{\omega_{I}t}$ controlled by the linear perturbation equations in early times (the late time behavior will be more interesting and it will be investigated in the next section.). Fig. 4 shows the value of $\omega_{M}$ with respect to $q\in[0,4]$, the periodicity and $q_{M}$ can be seen directly.

As we have explained, when $q$ and $k$ are given, $\omega_{M}$ can be obtained to determine the stability under the corresponding perturbation. In this spirit, we plot stable and unstable regions as a function of $q$ and $k$, i.e. the instability diagram, in the half first Brillouin zone in Fig. 5. In this plot, the left region in the figure, which is colored darker grey, is stable ($\operatorname{Im}(\omega_{M})\leq 0$), and the right region in the figure, which is colored lighter grey, is unstable ($\operatorname{Im}(\omega_{M})>0$).¹¹1When $q=0$, $\operatorname{Im}(\omega_{M})\leq 0$ for all $k$, which cannot be shown clearly in the figure. The figure also tells us that in the unstable region the cutoff $q$ increases with $k$, and when $k$ is large enough (but still in the first Brillouin zone), the system is unstable for all values of $q$ except $q=0$.

As a signal of onset of the instability mentioned in 2014JHEP…02..063A ; 2020arXiv201006232L , we also plot the dispersion relation of the sound mode in Fig.6. When there is a non-zero $k$ the sound speed becomes direction-dependent. There are two directions corresponding to the maximal and minimal values of the sound speed, which are parallel ($q>0$) and anti-parallel ($q<0$) to the velocity of the superflow, respectively. Hereafter we denote the sound speed as $c_{+}$ for $q>0$ and $c_{-}$ for $q<0$.

The onset of instability is accompanied by the sign-change of $c_{-}$. Along the negative axis of $q$, $\operatorname{Re}(\omega)$ becomes negative as $k$ is larger than 0.655. As $\operatorname{Re}(\omega)$ corresponds to the energy of the perturbation, $c_{-}<0$ (so $\operatorname{Re}(\omega)<0$) indicates the existence of a perturbed state with lower energy and thus the instability of the system (Landau instability). At the same time, we find that $\operatorname{Im}(\omega)>0$, which implies the dynamical instability of the system. So, our result confirms that the Landau instability and dynamical instability occur at the same time, as we have mentioned.

For the purpose of opening the next section we give a brief discussion about this section. Since it is not always true for a superflow to flow in the optical lattice stably, when $k\geq k_{c}$ excitations will always appear to slow the superflow down 2020arXiv201006232L , but what kind of intermediate states will this unstable superflow go through and what kind of final states will it end up with? In the next section we will use full nonlinear evolution to find the final state out for the Bloch wave-type superflow and the intermediate states will also be discussed.

To simulate the evolution of the holographic superfluid, we adopt the same scheme as in Li:2020ayr , i.e. taking Eqs.(26), (28) and (29) as evolution equations while choosing Eq.(27) as the constraint. Since Eq.(28) does not have time derivative terms but have space derivatives up to the second order, it can be solved with one boundary condition from $A_{t}(z_{b})=4.5+0.5\cos(2x)$ and another from the restriction of Eq.(27) on the conformal boundary. In the spatial directions the pseudo-spectral method is chosen for discretization, while in the time direction we adopt the fourth order Runge-Kutta method. When doing the time evolution, we choose static solutions, including $\psi$, $A_{t}$ and $A_{x}$ which are solved from the static equations of motion, as the initial states (at $t=0$), and at an early time $t_{1}$ we add perturbations, which are some random functions with small amplitudes, to $\psi$. Here, we require that the perturbations should not break the source free boundary condition of $\psi$. The perturbation is not necessarily periodic and there are many lattice cells in experiments, so it is necessary to include more lattice cells into the time evolution.²²2Actually a larger number of lattices for the simulation is more realistic and reliable, but in practice we choose $11$ cells, which is enough to study some universal properties in our case.

The dynamical process of superfluids that simultaneously experiences Landau and dynamical instability in optical lattice is investigated using the simplest holographic superfluid model. Due to the existence of dissipation the sound mode of an unstable superflow will always have negative energy and grows up exponentially at early times. We give a universal picture of the instability process for the unstable superflow. From real time evolution the unstable superflow is shown to reduce its particle current density to settle down to a steady state with a current density $\langle j^{x}\rangle_{f}$ and a Bloch wave vector $k_{f}$ below the critical value $k_{c}$, where this $k_{f}$ can be solved from the equations of motion by fixing $\langle j^{x}\rangle=\langle j^{x}\rangle_{f}$. In the course of the evolution, a chaotic process involving soliton generation is observed, in which the solitons are supposed to play the role of reducing the wave vector $k$ (similar to the case without an optical lattice 2020arXiv201006232L ).

As we have mentioned, when the number of dimensions of the boundary system is greater than one, the process for an unstable state to evolve into a stable one will become much more complicated, which to some extent is described as transient turbulence 2020arXiv201006232L . So beside the one dimensional boundary superfluid system in optical lattice, it is worthwhile to consider higher dimensional cases. Another direction for future investigation is to include backreaction in this holographic superfluid model, which will enable us to explore the complete interplay between the normal fluid and superfluid components of the superflow at a finite temperature in optical lattice.