Exciton crystal melting and destruction by disorder in bilayer quantum hall system with total filling factor one

When $\Delta\nu$ is fixed, the low temperature superfluid behavior can be described by an effective XY model. In this section we calculate the phase stiffness from two different models: spin $1/2$ easy-plane ferromagnet [7] and dilute exciton [10]. Once the phase stiffness $\rho_{s}$ is obtained, critical temperature of SF is bounded by $T_{c}=\frac{\pi\rho_{s}}{2}$. It turns out in the vicinity of $\Delta\nu=1$, two models lead to the same result. Let $Q^{2}=e^{2}/(4\pi\epsilon)$ for simplicity.

In this subsection we compare melting temperature of exciton Wigner crystal, $T_{m}$, with the KT temperature estimated above, and determine the finite temperature phase diagram of the system.

The melting temperature of classical exciton Wigner crystal was reported to be $T_{m}\approx 0.0907\frac{d^{2}Q^{2}}{a^{3}}$ [46, 47]. Relation $a=[\frac{\sqrt{3}}{8\pi}(1-\Delta\nu)]^{-1/2}$ can obtained from $\frac{1-\Delta\nu}{2}=\frac{n_{e}}{1/2\pi}$ where $n_{e}=2/(\sqrt{3}a^{2})$. We then have dimensionless temperatures

$$t_{m}=0.0907d^{2}[\frac{\sqrt{3}}{8\pi}(1-\Delta\nu)]^{3/2}$$

(9)

where $t_{m}=T_{m}/Q^{2}$. Compare Equation 9 with Equation 8, we are able to determine the finite temperature phase diagrams Figure 1 for two different situations, both of which are included in the zero temperature phase diagram Figure 2. Two situations are separated by $d_{c}\simeq 2$. When $d>d_{c}$ the Wigner crystal could melt into either superfluid or normal liquid, otherwise it only melts into superfluid. In the dilute limit $1-\Delta\nu\ll 1$, the exciton Wigner crystal always melts into a superfluid phase since $t_{m}<t_{\mathrm{KT}}$ is always true.

Treating WC as classical leads to an overestimation of $T_{m}$, because quantum fluctuation tends to lower $T_{m}$ as well. Since our goal is to demonstrate the possibility of $T_{m}<T_{\mathrm{KT}}$, they are justified, and does not change the phase diagram qualitatively. A more serious issue is neglecting the effects of disorder, which are very important when $\Delta\nu\rightarrow 1$, where the excitons are destroyed. This is the focus of the next section. The resultant phase there is a single-layer integer quantum Hall state, which dominates the experimental phase diagram there. One should keep this in mind when comparing with the theoretical phase diagrams in this section obtained without taking these into account.

We introduce the Gaussian white noise random potential $V_{i}(\mathbf{r})$ that is uncorrelated between the two layers:

$$\langle V_{i}(\mathbf{r})V_{j}(\mathbf{r^{\prime}})\rangle=\Delta^{2}\delta(\mathbf{r-r^{\prime}})\delta_{ij},$$

(10)

where $i,j=1,2$ are layer indices. Pinning length $R$ of 2d Wigner crystal, defined as $\langle[u(0)-u(R)]^{2}\rangle\simeq a^{2}$ where $a$ is the lattice constant and $u(R)$ is the field of lattice distortion, is given by balancing the energy gain of random potential and energy cost of lattice distortion [48, 49]

$$Rn_{e}\Delta=ca^{2},$$

(11)

where $n_{e}=1/(A_{c}a^{2}),A_{c}=\sqrt{3}/2$ is the density of electron, $c$ is the shear modulus. Left and right hand sides of this equation stand respectively for random potential energy gain and elastic energy cost due to lattice distortion. Since this amount of energy is for a region of linear size $R$, dividing by $R^{2}$ we obtain the density of random potential energy (for convenience in density comparison we keep one factor of $n_{e}$ here)

$$\varepsilon_{r}=\frac{Rn_{e}\Delta}{R^{2}}=\frac{n_{e}\Delta^{2}}{cA_{c}a^{4}}.$$

(12)

For single layer Wigner crystal of electron-type interaction and dipole-type interaction we simply take the shear modulus from [50]

$$\begin{array}[]{ll}c_{1}(d\lesssim a)\approx 2.5\frac{D^{2}}{a^{5}}&\operatorname{dipole}\\ c_{2}=0.3\frac{Q^{2}}{a^{3}}&\operatorname{charge}\end{array}$$

(13)

where $Q^{2}=\frac{e^{2}}{4\pi\epsilon},D^{2}=\frac{e^{2}d^{2}}{4\pi\epsilon}$. Transition from coupled to decoupled picture lowers the disorder potential energy (density) by

$$\Delta\varepsilon_{r}=\frac{2n_{e}\Delta^{2}}{c_{1}A_{c}a^{4}}-\frac{n_{e}\left(\sqrt{2}\Delta\right)^{2}}{c_{2}A_{c}a^{4}}=\frac{2n_{e}\Delta^{2}}{A_{c}a^{4}}\left(\frac{1}{c_{1}}-\frac{1}{c_{2}}\right),$$

(14)

where $\sqrt{2}\Delta$ is the effective random potential strength seen by the bilayer (since $V(\mathbf{r})=V_{1}(\mathbf{r})+V_{2}(\mathbf{r})$ has $\langle V(\mathbf{r})V(\mathbf{r^{\prime}})\rangle=2\Delta^{2}\delta(\mathbf{r}-\mathbf{r^{\prime}})$). On the other hand, in $d\rightarrow\infty$ the interlayer Coulomb energy is diminished and what we have is merely two copies of single layer Wigner crystal. Therefore $c_{1}(\infty)=2c_{2}$. In this limit $\Delta E_{r}$ is exactly half of that for individual pinning. In practice for a specific $d$ in experiments, the effective spacing $d/a$ has an upper bound $d/\sqrt{2}$, which is generally smaller than 1 (see below). For such considerations, we will simply take the dipole approximation $c_{1}=2.5D^{2}/a^{5}$.

$$\Delta\varepsilon_{r}=\frac{2n_{e}\Delta^{2}}{A_{c}Q^{2}a}\left(\frac{1}{0.3}-\frac{1}{2.5d^{2}/a^{2}}\right)=q\frac{n_{e}Q^{2}}{a}\left(\frac{1}{0.3}-\frac{1}{2.5d^{2}/a^{2}}\right),$$

(15)

where

$$q=\frac{2\Delta^{2}}{A_{c}Q^{4}}=\frac{4\Delta^{2}}{\sqrt{3}Q^{4}}$$

(16)

is the dimensionless random potential strength.

As we demonstrated above, the system can lower the disorder potential energy by distorting the electron and hole WCs in the two layers independently, compared to that of the exciton WC. Doing that, however, decouples the two layers and destroy the excitons, resulting in an increase in the interlayer Coulomb interaction energy. In this subsection we calculate this energy cost.

In this subsection we let $\frac{Q^{2}}{a}$ be energy scale and $a$, the lattice constant of 2d triangular lattice, be length scale. We are evaluating the interlayer correlation energy difference of Wigner crystal vs. homogeneous electron gas (since random relative distribution of charges in one layer is seen on average as homogeneous gas of charge by the other layer), i.e.

$$\Delta E_{e}=\int\mathrm{d}\mathbf{r}[g_{1}(\mathbf{r})-g_{2}(\mathbf{r})]\frac{-1}{\sqrt{r^{2}+d^{2}}}=\int\mathrm{d}\mathbf{r}\left[\sum_{i}\delta(\mathbf{r}-\mathbf{R}_{i})-1/A_{c}\right]\frac{1}{\sqrt{r^{2}+d^{2}}},$$

(17)

where $g_{1}(\mathbf{r})=1/A_{c},g_{2}(\mathbf{r})=\sum_{i}\delta(\mathbf{r}-\mathbf{R}_{i})$, $A_{c}=\sqrt{3}/2$ is the area of unit cell. Compared with Equation 15, a transition between locked/decoupled phase will be determined.

In the small $d$ limit, apart from a divergent $1/d$ term, this energy difference is the classic problem of static energy of 2d Wigner crystal. That is (see e.g. [51, 52])

$$\lim_{d\rightarrow 0}[\Delta E_{e}(d)-1/d]=-4.213423,$$

(18)

We now calculate this energy difference for general $d$. Let $\Delta E_{e}=E_{0}+E_{1}+E_{2}$, where $E_{0}=1/d$ and

$$\begin{array}[]{ccl}E_{1}&=&\frac{1}{\sqrt{\pi}}\left(\int_{0}^{\pi}+\int_{\pi}^{\infty}\right)\mathrm{d}tt^{-1/2}\mathrm{e}^{-td}\sum^{\prime}\mathrm{e}^{-tR_{i}^{2}}\equiv E_{11}+E_{12}\\ E_{2}&=&-\frac{1}{A_{c}}\int\mathrm{d}\mathbf{r}\frac{e^{2}}{\sqrt{r^{2}+d^{2}}}=-\frac{1}{\sqrt{\pi}A_{c}}\int_{0}^{\infty}\mathrm{d}t\int\mathrm{d}\mathbf{r}\mathrm{e}^{-tr^{2}}\mathrm{e}^{-td^{2}}t^{-1/2}=-\frac{\sqrt{\pi}}{A_{c}}\int_{0}^{\infty}\mathrm{d}tt^{-3/2}\mathrm{e}^{-td^{2}}\\ &=&-\frac{\sqrt{\pi}}{A_{c}}\int_{0}^{\pi}\mathrm{d}tt^{-3/2}\mathrm{e}^{-td^{2}}-\frac{2}{A_{c}}\left(e^{-\pi d^{2}}-\pi d\text{erfc}\left(\sqrt{\pi}d\right)\right)\end{array}$$

(19)

where $\Gamma(n)z^{-n}=\int_{0}^{\infty}t^{n-1}\mathrm{e}^{-zt}\mathrm{d}t$ is used in rewriting $1/\sqrt{d^{2}+r^{2}}=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}t^{-1/2}\mathrm{e}^{-t(d^{2}+r^{2})}\mathrm{d}t$, and $\sqrt{\pi}\int_{\pi}^{\infty}\mathrm{d}tt^{-3/2}\mathrm{e}^{-td^{2}}=\frac{2}{\sqrt{\pi}}\left(e^{-\pi d^{2}}-\pi d\text{erfc}\left(\sqrt{\pi}d\right)\right)$. $\sum^{\prime}$ stands for the summation excluding $R_{i}=0$.

Let $t=\pi x$, we have

$$E_{12}=\int_{1}^{\infty}\mathrm{d}xx^{-1/2}\sum\nolimits^{\prime}\mathrm{e}^{-\pi x(d^{2}+R_{i}^{2})}=\sum\nolimits^{\prime}\operatorname{erfc}\left[\sqrt{\pi(d^{2}+R_{i}^{2})}\right]/\sqrt{(d^{2}+R_{i}^{2})},$$

(20)

where $\int_{1}^{\infty}x^{-1/2}\mathrm{e}^{-\pi xa^{2}}\mathrm{d}x=\operatorname{erfc}\left(\sqrt{\pi}a\right)/a$ is utilized. To calculate $E_{11}$ we first complete it with a $R_{i}=0$ term

$$\begin{array}[]{ccl}E_{11}&=&\frac{1}{\sqrt{\pi}}\int_{0}^{\pi}\mathrm{d}tt^{-1/2}\mathrm{e}^{-td^{2}}\Theta_{\Gamma}(t/\pi)-\frac{1}{\sqrt{\pi}}\int_{0}^{\pi}t^{-1/2}\mathrm{e}^{-td^{2}}\mathrm{d}t\\ &=&\frac{\sqrt{\pi}}{A_{c}}\int_{0}^{\pi}\mathrm{d}tt^{-3/2}\mathrm{e}^{-td^{2}}\Theta_{\Gamma^{\prime}}(\pi/t)-\operatorname{erf}\left(\sqrt{\pi}d\right)/d\\ &=&\frac{1}{A_{c}}\int_{1}^{\infty}\mathrm{d}xx^{-1/2}\mathrm{e}^{-\pi d^{2}/x}\sum^{\prime}\mathrm{e}^{-\pi xK_{i}^{2}}-\operatorname{erf}\left(\sqrt{\pi}d\right)/d+\frac{\sqrt{\pi}}{A_{c}}\int_{0}^{\pi}\mathrm{d}tt^{-3/2}\mathrm{e}^{-td^{2}}\end{array}$$

(21)

with $\Theta_{\Gamma}(t)\equiv\sum_{\mathbf{R}_{i}\in\Gamma}\mathrm{e}^{-\pi tR_{i}^{2}},\Gamma$ being a lattice. From first line to second line we used $\int_{0}^{1}t^{-1/2}\mathrm{e}^{-\pi td^{2}}\mathrm{d}t=\operatorname{erf}\left(a\sqrt{\pi}\right)/a$ and $\Theta_{\Gamma}(t)=t^{-n/2}v(\Gamma)^{-1}\Theta_{\Gamma^{\prime}}(1/t)$, where $\Gamma^{\prime}$ is the dual of lattice $\Gamma$, $v(\Gamma)$ is the measure of unit cell of $\Gamma$ and $n$ is dimension of the lattice $\Gamma$ (see e.g. pg. 115 of [53]); from second line to third line, points of dual lattice are denoted as $\mathbf{K}_{i}$ and we let $t=\pi/x$ for all $K_{i}\equiv|\mathbf{K}_{i}|\neq 0$ terms. Note that the very last divergent term in $E_{11}$ cancel the divergent part of $E_{2}$.

Since

$$\begin{array}[]{ccl}\int_{1}^{\infty}\mathrm{d}xx^{-1/2}\mathrm{e}^{-\pi(d^{2}/x+K_{i}^{2}x)}&=&\frac{\mathrm{e}^{-2\pi dK_{i}}\left(1+\operatorname{erf}\left[\sqrt{\pi}(d-K_{i})\right]\right)+\mathrm{e}^{2\pi dK_{i}}\left(1-\operatorname{erf}\left[\sqrt{\pi}(d+K_{i})\right]\right)}{2K_{i}}\\ &\equiv&\phi_{-1/2}(d,K_{i})\end{array}$$

(22)

we have

$$\begin{array}[]{ccl}E_{1}+E_{2}&=&-\frac{\operatorname{erf}\left(\sqrt{\pi}d\right)}{d}-\frac{2}{A_{c}}\left(e^{-\pi d^{2}}-\pi d\text{erfc}\left(\sqrt{\pi}d\right)\right)\\ &&+\sum^{\prime}\frac{\operatorname{erfc}\left[\sqrt{\pi(d^{2}+R_{i}^{2})}\right]}{\sqrt{d^{2}+R_{i}^{2}}}+\frac{1}{A_{c}}\sum^{\prime}\phi_{-1/2}(d,K_{i})\end{array}$$

(23)

For a sanity check, let $d\rightarrow 0$ we have

$$\begin{array}[]{ccl}E_{1}+E_{2}&=&-2\left(1+\frac{1}{A_{c}}\right)+\sum^{\prime}\operatorname{erfc}\left(\sqrt{\pi}R_{i}\right)/R_{i}+\frac{1}{A_{c}}\sum^{\prime}\operatorname{erfc}\left(\sqrt{\pi}K_{i}\right)/K_{i}\\ &\cong&-2\left(1+\frac{1}{A_{c}}\right)+6\operatorname{erfc}\left(\sqrt{\pi}\right)+6\operatorname{erfc}\left(\sqrt{\pi}/A_{c}\right)\\ &=&-4.213475\end{array}$$

(24)

where we took nearest lattice point approximation, i.e. only six terms with smallest $R_{i},K_{i}$ in the those lattice summations are kept. Nevertheless the result match the known static energy for 2d Wigner crystal up to fourth digit.

For general $d$, let $\delta E(d)$ be the nearest lattice point approximation of $E_{1}+E_{2}$ in Equation 23

$$\begin{array}[]{ccl}\delta E(d)&=&-\frac{\operatorname{erf}\left(\sqrt{\pi}d\right)}{d}-\frac{2\left(e^{-\pi d^{2}}-\pi d\text{erfc}\left(\sqrt{\pi}d\right)\right)}{A_{c}}+\frac{6\operatorname{erfc}\left[\sqrt{\pi(d^{2}+1)}\right]}{\sqrt{d^{2}+1}}\\ &&+3\left\{\mathrm{e}^{-2\pi d/A_{c}}\left(1+\operatorname{erf}\left[\sqrt{\pi}\left(d-\frac{1}{A_{c}}\right)\right]\right)+\mathrm{e}^{2\pi d/A_{c}}\operatorname{erfc}\left[\sqrt{\pi}\left(d+\frac{1}{A_{c}}\right)\right]\right\}\end{array}$$

(25)

$1/A_{c}=2/\sqrt{3}$ comes from lattice constant of the dual lattice. It behaves asymptotically in the $d\rightarrow\infty$ limit as $\delta E(d)+1/d\sim 6\mathrm{e}^{-4\pi d/\sqrt{3}}$. Also for $d\rightarrow\infty$, $\operatorname{erfc}\left[\sqrt{\pi(d^{2}+R_{i}^{2})}\right]/\sqrt{d^{2}+R_{i}^{2}}\sim\mathrm{e}^{-\pi(d^{2}+R_{i}^{2})}/(\pi(d^{2}+R_{i}^{2}))$ and $\operatorname{erfc}\left(\sqrt{\pi}x\right)\sim\mathrm{e}^{-\pi x^{2}}/(\pi x)$ results in

$$\phi_{-1/2}(d,K_{i})\leqslant[2\mathrm{e}^{-2\pi dK_{i}}+\mathrm{e}^{-\pi(d^{2}+K_{i}^{2})}/(\pi(d+K_{i}))]/(2K_{i}),$$

(26)

All terms generated from farther lattice points are dominated by $6\mathrm{e}^{-4\pi d/\sqrt{3}}$. In the sense that $\delta E(d)$ is a good approximation to $E_{1}+E_{2}$ for both $d\rightarrow 0$ and $d\rightarrow\infty$, we could safely take

$$\Delta E_{e}\cong 1/d+\delta E(d)$$

(27)

Putting back dimensions, the Coulomb energy density difference is, with $\delta E$ defined in Equation 25,

$$\Delta\varepsilon_{e}\cong n_{e}\frac{Q^{2}}{a}(a/d+\delta E(d/a))$$

(28)

Comparing (15) with (28) we can immediately see that the transition between coupled/decoupled phases is determined by the root of the dimensionless equation

$$\frac{q}{0.3}x^{2}-x-q/2.5-x^{2}\delta E(x)=0,\quad x=d/a=d\sqrt{\frac{\sqrt{3}}{8\pi}(1-\Delta\nu)}$$

(29)

where $q$, defined in Equation 16, is, up to a constant, the energy scale of random potential comparing with Coulomb energy. Putting together, we can draw a phase diagram Figure 3 for the decoupled electron-hole Wigner crystal and exciton Wigner crystal.