Colloquium: The spin-incoherent Luttinger liquid

In contrast to the well-known Fermi-liquid theory of three dimensions, interacting one-dimensional and quasi-one-dimensional systems of fermions are described at low energy by an effective theory known as Luttinger liquid theory. This theory is expressed in terms of collective many-body excitations that show exotic behavior such as spin-charge separation. Luttinger liquid theory is commonly applied on the premise that “low energy” describes both the spin and charge sectors. However, when the interactions in the system are very strong, as they typically are at low particle densities, the ratio of spin to charge energy may become exponentially small. It is then possible at very low temperatures for the single-spin excitation energy to be low compared to the characteristic single excitation charge energy, but still high compared to the characteristic spin energy. This energy window of near ground-state charge degrees of freedom but highly thermally excited spin degrees of freedom is called a spin-incoherent Luttinger liquid. The spin-incoherent Luttinger liquid exhibits a higher degree of universality than the Luttinger liquid and its properties are qualitatively distinct. In this Colloquium some recent theoretical developments in the field are detailed and experimental indications of such a regime in gated semiconductor quantum wires are described.

Figure 1

(Color online) Schematic of a fluctuating Wigner solid model. Shown is a snapshot in time of the density modulations $\delta \rho (x,t)$ of a fluctuating Wigner solid in one dimension of mean particle spacing $a$. A cartoon of the antiferromagnetic spin sector is shown with the light arrows representing spin orientation on the Wigner solid sites given by $la$. There is an antiferromagnetic exchange $J_l$ between the $l\text{th}$ and $(l+1)\text{th}$ sites. The lattice has a stiffness characterized by the frequency ${\omega }_0⪢J_l∕\hslash$.

Figure 2

(Color online) World lines for strongly interacting one-dimensional spin-$S$ fermions for $E_{\text{spin}}⪡k_{B}T⪡E_{\text{charge}}$ in which the noncrossing approximation is made. Particle trajectories in space and imaginary time are shown as curved lines. Dashed lines represent the world line paths for creating a particle and removing it for large $x=x_f-x_i$, $\tau ={\tau }_f-{\tau }_i$. The solid lines represent trajectories of other particles. Because of the large action cost associated with the trajectories in (b), at low energies (relative to $E_{\text{charge}}$) a process like that shown in (a) where world lines wrap around from $\tau =\beta$ to $\tau =0$ will dominate. Such a process, however, requires that all dashed world lines have the same spin. For $E_{\text{spin}}⪡k_{B}T$ in zero magnetic field, this occurs with probability ${(2S+1)}^{-\mid N\mid }$. Physically, the process shown in (a) is equivalent to adding an electron at $(x_i,{\tau }_i)$ then pushing all electrons to the right and removing the electron at $(x_f,{\tau }_f)$ as shown in (c). In order for the final spin configuration to look the same as the original one, all spins must be aligned between the initial and final points.

Figure 3

Schematic of the tunneling density of states $\nu \left(\epsilon \right)\equiv A\left(\omega \right)$. The solid (dashed) line is the high- (low-) temperature regime $k_{B}T⪢J$ $(k_{B}T⪡J)$. In the high-temperature regime when $\epsilon ⪢k_{B}T$, $\nu \left(\epsilon \right)=2\nu (-\epsilon )$. When the temperature is lowered below the spin energy $J$, the tunneling density of states at $\epsilon <0$ grows by roughly a factor of 3, while for $\epsilon >0$ it decreases dramatically. For $k_{B}T⪡\epsilon$ and $\epsilon ⪡J$ the standard Luttinger liquid behavior of the power-law suppression is observed at small $\epsilon$. From 47.

Figure 4

Schematic of threshold photoexcitation for (a) finite hole mass and (b) infinite hole mass. (a) An electron is excited from the valence band to the conduction band leaving behind a hole in valence band. (b) An electron is excited from a deep (infinite mass) core level. We assume the conduction band is occupied by a spin-incoherent Luttinger liquid.

Figure 5

(Color online) The interference contrast (31) obeys a power law with a temperature- and magnetic-field-dependent exponent $2K_c(1-{\ln }^2{p}_{\uparrow }∕{\pi }^2)$, Eq. (32). (Interference schematic in the inset.) Plotted is $c=C\left(V\right)∕C\left(V_0\right)$, the interference contrast normalized with respect to an arbitrary reference voltage $V_0$. It is compared to its value at zero magnetic field $c_{B=0}$ for various polarizations $p_{\uparrow }$ with $K_c=1∕2$. From 38.

Figure 6

Schematic of a quantum wire attached to Fermi-liquid leads. The electron density is assumed to be small near the center of the wire so that $r_s$ is large and strong Wigner solid correlations are present. In the leads to the left and right of the wire the electrons are assumed to be noninteracting. From 46.

Figure 7

Schematic of the temperature dependence of a quantum wire of length $L$ attached to Fermi-liquid leads with a strong impurity in the center. For a range of temperatures $\hslash v_s∕L⪡k_{B}T⪡\hslash v_s∕L$ the charge modes have propagated into the leads, but the spin modes have not, leading to a universal temperature dependence of the conductance in this energy window.

Figure 8

Coulomb drag schematic. Two quantum wires are arranged parallel to one another and separated by a distance $d$. A dc current $I_1$ flows in the active wire 1 and a voltage bias $V_2$ is measured in the passive wire 2 when $I_2=0$.

Figure 9

Schematic of the possible temperature dependence of the Coulomb drag in identical wires of sufficiently low electron density that $J⪡E_{\text{charge}}=\hslash {\omega }_0$. The nonmonotonic temperature dependence shown can be obtained for $K_c>1∕2$ and $\tilde{U}\left(4k_F\right)⪡\tilde{U}\left(2k_F\right)$ which may be realized when $k_{F}d>1$. The temperature $T^{*}$ is the locking temperature of two identical wires, below which the drag exhibits activated behavior and $E_s$ is an energy gap associated with the locking. When $J⪡E_{\text{charge}}$ a sharp drop in the drag resistance should be observable for $k_{B}T\sim J$.

Figure 10

Schematic temperature dependence of the dephasing time of a qubit. The constant $c$ in the exponential is $\mathcal{O}\left(1\right)$. The plot assumes $1∕4<K_c<1$ and $K_s=1$.

Figure 11

(Color online) Schematic of the measurement setup with cleave plane front and perpendicular to the magnetic field $B$. The top gates $(G_1,G_2,G_3)$ are $2\mu \mathrm{m}$ wide. The upper wire at the edge of the two-dimensional electron gas is $20\mathrm{nm}$ thick, the lower wire is $30\mathrm{nm}$ thick, and the barrier between them is $6\text{−}\mathrm{nm}$ insulating AlGaAs. Here $U_S\left(x\right)∕U_M\left(x\right)$ is the gate-induced potential for the single-mode/multimode wires, $E_F^U$ is the Fermi energy of the upper wire, $O_1$ is an Ohmic contact that serves as a source, and the Ohmic contacts $O_{2,3}$ serve as drains. The electron density in the quantum wires is modulated by a gate voltage $V_G$. The tunneling current is $I_T$ and the two-terminal current is $I$. From 58.

Figure 12

Experimental data indicating a localization transition to a strongly correlated regime at low particle density. (a) Plot of $d{G}_T∕d{V}_G$ vs $V_G$ and $B$ for a single-mode quantum wire. The applied bias $V_{SD}=100\mu \mathrm{V}$ is selected to avoid the zero-bias anomaly, but is still small enough to allow tunneling predominantly between the Fermi points of both wires. The upper and lower curves are momentum-conserving tunneling features $B^{\pm }\left(V_G\right)$ and each curve is accompanied by finite-size features marked by FS. At low densities localized features (LFs) appear instead of these curves. The localization transition occurs at $V_G^{*}$. (b) indicates the upper wire (UW) and lower wire (LW) densities extracted from the data. From 58.

Figure 13

Detail of the localization features in the strongly correlated regime at low particle density. (a) A high-resolution measurement of $d{G}_T∕d{V}_G$ of localization features for a single-mode wire, $V_{SD}=50\mu \mathrm{V}$, $d{V}_G=300\mu \mathrm{V}$. (e) $T\left(N\right)$ of (a) where $N$ is the number of added electrons for each LF. (c) $\Gamma \left(B\right)\propto {\mid \Psi \left(k\right)\mid }^2$. $\Gamma$ and $T$ are defined in the text. From 58.

Figure 14

Schematic geometry of electron tunneling between two parallel quantum wires, as in Fig. 11. Electrons are assumed to tunnel between an infinitely long lower wire and a short upper wire of length $L$. The wires are separated by a center-to-center distance $d$. The upper (lower) wire has a width $W_u$ $\left(W_l\right)$ and an average electron density ${\overline{n}}_u$ $\left({\overline{n}}_l\right)$.

Figure 15

Ground-state density of a four-electron system with spin. Scaled densities are measured in units of $1∕L$. In the label of the individual plot, density means the average physical density $N∕L$ of the system, in units of $\mu {\mathrm{m}}^{-1}$.

Figure 16

Estimate of the spin-exchange energy $J$ from a two- and four-electron system as a function of density. Typical experimental temperatures $\left(T_{\exp }\right)$ are in the range $250\mathrm{mK}–2\mathrm{K}\approx 25–200\mu \mathrm{eV}$.

Figure 17

Momentum dependence of tunneling conductance between one-electron and two-electron states for a wire of length $L=0.4\mu \mathrm{m}$. The dashed curve shows the line shape ${\mid M_s\left(k\right)\mid }^2$ for tunneling into the singlet ground state of the two-particle system. The short-dashed curve shows ${\mid M_t\left(k\right)\mid }^2$ for tunneling into the triplet ground state, applicable at $T=0$ when $E_Z>J$. The solid curve is a weighted average, applicable if $k_{B}T$ is large compared to both $E_Z$ and $J$ but small compared to the energy of the lowest charge excitation. The $\text{singlet}+\text{triplet}$ line shape should be compared with the $\Gamma \left(B\right)$ line shape in Fig. 13c.

Figure 18

Spectral function $A(k,\overline{u})$, which determines the momentum dependence of tunneling in the spin-incoherent regime. The quantity $\overline{u}$ is the root-mean-square electron displacement, due to quantum fluctuations, from the sites of a classical Wigner crystal, and $a$ is the lattice spacing. When $\overline{u}\gtrsim a$, the momentum distribution is single lobed and peaked about zero momentum. In the opposite limit, when $\overline{u}⪡a$, the momentum distribution exhibits a doubled-lobed structure with peaks near $k=\pm 2k_F$.