--- title: Weak Conductance in Quantum Systems url: https://www.emergentmind.com/topics/weak-conductance type: topic --- # Weak Conductance in Quantum Systems Searching arXiv for the cited papers to ground the article in published sources. arXiv search query: 1201.3180 weak antilocalization conductance fluctuation Bi2Se3 arxiv_search({"query":"1201.3180"}) In the literature represented here, conductance is analyzed in regimes characterized by weak antilocalization and conductance fluctuations, a weak tunnel junction, weakly interacting Luttinger liquids, and a quantum dot weakly coupled to a conducting channel. These regimes are not unified by a single microscopic mechanism; rather, they are linked by the fact that conductance is controlled by small quantum corrections, weak tunneling, weak coupling, or weak perturbations superposed on a larger transport background. The resulting observables include low-field magnetoconductance corrections, zero-bias conductance suppression, temperature-dependent transmission through impurities, and scanning-gate conductance maps [1201.3180] [1607.01988] [1809.05484] [1405.3050]. ## 1. Regimes in which conductance is controlled by weak effects The sources describe four distinct settings. | Setting | Weak element | Conductance quantity | |---|---|---| | Epitaxial Bi\(_2\)Se\(_3\) wire | Weak antilocalization and universal conductance fluctuation | $\Delta \sigma(B)$, $B_c(T)$, $\delta G_{\rm rms}(T)$ | | Tunnel junction in resistor environment | Weak tunnel junction with Nyquist noise | $G(T)$, $G/G_t$ | | Inhomogeneous Luttinger liquid | Weakly interacting limit and weak barrier | $G(T)=G_0\,\tau(T)$ | | Side-coupled quantum dot | Weak coupling to channel and weak tip perturbation | $G(x_t,y_t)$, $\delta T$ | In the Bi\(_2\)Se\(_3\) system, weak conductance phenomena appear as “small, but experimentally resolvable, quantum corrections to the classical Drude conductance.” In the tunnel-junction problem, the defining assumption is a conductance much smaller than $e^2/\hbar$. In the Luttinger-liquid problem, the weakly interacting limit is the regime in which the Matveev–Yue–Glazman result is recovered, while the full non-chiral bosonization technique extends beyond that limit. In the conductance-microscopy problem, weak coupling means that the connection of the channel to the dot transmits a single transport mode only, and the tip is treated as a weak, delta-like perturbation [1201.3180] [1607.01988] [1809.05484] [1405.3050]. A plausible implication is that “weak conductance” is best understood as a transport descriptor tied to perturbative or quasi-perturbative control parameters, rather than as a synonym for simply small absolute conductance. ## 2. Phase-coherent weak conductance corrections in a Bi\(_2\)Se\(_3\) wire A sub-micrometer-sized Hall bar made of epitaxial Bi\(_2\)Se\(_3\) thin film was used to probe phase coherent transport below 22 K. The geometry was a wire of width $w=260$ nm, thickness $t=20$ nm, and length $L=2\,\mu{\rm m}$. Two signatures were examined: weak antilocalization (WAL) and universal conductance fluctuations (UCF) [1201.3180]. For WAL, the magnetoconductance correction is described by the Hikami–Larkin–Nagaoka formula $$ \Delta \sigma(B)\equiv \sigma(B)-\sigma(0) = \alpha \frac{e^2}{2\pi^2\hbar} \left[ \psi\left(\frac12+\frac{B_\phi}{B}\right)-\ln\left(\frac{B_\phi}{B}\right) \right], $$ with $$ B_\phi=\frac{\hbar}{4eL_\phi^2}. $$ Here, $\alpha$ is a dimensionless prefactor and $L_\phi$ is the phase coherence length. In experiment, $\Delta \sigma(B)$ is obtained by converting the measured resistance $R_{xx}(B)$ to conductance and subtracting its zero-field value, and the fit is performed over the low-field range $|B|L_T$ and $L>L_\phi$, theory gives $$ \delta G_{\rm rms}\propto T^{-2/3}, $$ with $L_T\equiv \sqrt{\hbar D/k_B T}$. Experimentally, the autocorrelation yields $B_c(T)$ and hence $L_\phi(T)$ consistent with WAL, while $\delta G_{\rm rms}(T)$ scales as $T^{-0.69}$, in excellent agreement with the $T^{-2/3}$ law for 1D UCF [1201.3180]. These results establish quasi-one-dimensional phase-coherent transport below approximately 22 K. The same work states that quasi-1D topological-insulator wires open routes to Aharonov–Bohm oscillations of Dirac surface modes, one-dimensional Majorana bound states when proximitized by a superconductor, and other interference-based spintronic functionalities relying on long $L_\phi$ [1201.3180]. ## 3. Zero-bias conductance of a weak tunnel junction with Nyquist noise A different weak-conductance problem concerns the Coulomb blockade of a tunnel junction with conductance much smaller than $e^2/\hbar$, capacitance $C$, and a series resistance $R$ producing Nyquist noise. The single-electron charging energy is $$ E_c=\frac{e^2}{2C}. $$ In the semiclassical regime $$ k_B T\gg \frac{\hbar}{RC}, \qquad R\gg R_K\equiv \frac{h}{e^2}, $$ the resistor’s phase fluctuations charge the junction with a random offset charge $Q$ whose steady-state distribution is Gaussian [1607.01988]. The zero-bias conductance is written in orthodox theory as $$ G=\left.\frac{dI}{dV}\right|_{V=0} = G_t\int_{-\infty}^{\infty} dE\; \frac{2E}{1-e^{-E/(k_B T)}}\, \bigl[-P'(-E)\bigr]. $$ In the semiclassical high-$T$ limit, $$ P(E)= \frac{1}{\sqrt{4\pi\,k_B T\,E_c}} \exp\!\left[-\frac{(E-E_c)^2}{4\,k_B T\,E_c}\right]. $$ With $$ \alpha\equiv \frac{E_c}{k_B T}, $$ the conductance becomes $$ \frac{G}{G_t} = \exp\!\left(-\frac{\alpha}{4}\right)\,I(\alpha), $$ where $$ I(\alpha)\equiv \left[ \sqrt{\pi}\int_{-\infty}^{\infty}\!dy\, e^{-y^2}\, \frac{\sqrt{\alpha}\,y}{\sinh(\sqrt{\alpha}\,y)} \right]^{-1}. $$ Since $I(\alpha)\to 1$ as $\alpha\to 0$ and remains $O(1)$ for all $\alpha$, the leading temperature dependence is $$ G(T)\propto \exp\!\left(-\frac{E_c}{4k_B T}\right). $$ The formula is stated to be valid for $$ k_B T \gtrsim \frac{R_K}{2\pi R}E_c,\qquad R\gtrsim R_K $$ and also reproduces earlier asymptotic forms [1607.01988]. The comparison with earlier results is explicit. For $k_B T\gg E_c$, Joyez–Esteve give $$ \frac{G}{G_t}\simeq 1-\frac{E_c}{3k_B T}+\frac{E_c^2}{15(k_B T)^2}+\cdots. $$ For $\alpha\gg 1$ but still $k_B T\gg (R_K/\pi R)E_c$, Averin and Odintsov give $$ \frac{G}{G_t}\simeq \sqrt{\frac{\pi^3}{4\alpha}}\, \exp\!\left(-\frac{\alpha}{4}\right). $$ The present result is stated to hold for all $\alpha$ subject to the Nyquist-noise condition above [1607.01988]. The factor of $1/4$ in the activation energy is a central physical point. If one ignored environmental fluctuations and took $P(E)=\delta(E-E_c)$, the activation would be $G\propto e^{-E_c/(k_B T)}$. Including Nyquist noise broadens $P(E)$ to a Gaussian and shifts the most probable energy cost to $E_c/4$; physically, the resistor’s charge fluctuations “help” the tunneling. Numerical data for $R/R_K=1$ and 10 are reported to agree with the analytic result down to $k_B T\approx (R_K/2\pi R)E_c$ within 30%, and experimental measurements on two samples with $R\approx R_K$ show an Arrhenius-like slope close to $E_c/4$ over a broad temperature range [1607.01988]. ## 4. Weak interactions, weak barriers, and conductance in inhomogeneous Luttinger liquids For a one-dimensional spinful fermion with forward-scattering interactions and an arbitrary impurity cluster $V(x)$ near the origin, the Hamiltonian is $$ H=\int_{-\infty}^{\infty}\!dx\;\psi^\dagger(x)\Bigl[-\frac{1}{2m}\partial_x^2+V(x)\Bigr]\psi(x) +\frac12\!\int\!dx\,dx'\;v(x-x')\,\rho(x)\,\rho(x'). $$ The interaction Fourier component satisfies $v_q=v_0$ for $|q|\le \Lambda\ll k_F$, zero otherwise. In the RPA limit one introduces the holon velocity $$ v_h=v_F\sqrt{1+\frac{2v_0}{\pi v_F}}, $$ and the dimensionless interaction parameter $$ g=\frac{v_F}{v_h}\qquad (\equiv K \text{ in many conventions}). $$ The finite-bandwidth conductance of this Luttinger liquid with a cluster of impurities is studied as a function of temperature [1809.05484]. In the weakly interacting limit, Matveev–Yue–Glazman obtain $$ G(T)=G_0\; \frac{\tau_0\,(T/D_0)^{2\alpha}} {1-\tau_0+\tau_0\,(T/D_0)^{2\alpha}}, $$ where $G_0=e^2/h$, $\tau_0=|t_0|^2$, and $$ 2\alpha=\frac{v_0}{\pi v_F} =\frac12\left(\frac{1}{g^2}-1\right)\ll 1. $$ For a very weak barrier with $\tau_0\approx 1$, $$ G(T)=G_0\Bigl[1-C\,T^{2(1/K-1)}+\cdots \Bigr],\qquad C\propto 1-\tau_0,\qquad K=g. $$ The non-chiral bosonization technique gives instead a transcendental equation for the effective transmission: $$ \tau=\tau_0\left(\frac{k_B T}{D_0}\right)^{\eta(\tau)}, $$ with $$ \eta(\tau)= \frac{(g-1)\bigl(\tau-g-3g^2(1-\tau)\bigr)} {4g\bigl(\tau+g(1-\tau)\bigr)}. $$ Since $G(T)=G_0\,\tau(T)$, this single equation covers all interaction strengths, and by expanding around $g=1+\delta$ it reduces exactly to the Matveev–Yue–Glazman formula when $|\delta|\ll 1$ [1809.05484]. Two nonstandard regimes are highlighted. First, if $\eta_0=\eta(\tau_0)=0$, then $$ \tau_0=\frac{v_h}{v_F}\qquad (\text{i.e. } |t_0|^2=v_h/v_F\le 1), $$ and the $T$-dependence in the ansatz disappears. In this special case, $$ G(T)\approx G_0\,\tau_0\,T^\alpha,\qquad \alpha=\eta_0\approx 0. $$ The conductance is therefore nearly temperature-independent. Second, if $g<1/2$ and $\tau_0\lesssim 1$, then one finds $\eta_0<0$ and approximately $$ \eta_0\simeq -\frac{(1-g)^2}{4g}<0, $$ so that $$ G(T)=G_0\,\tau_0\,(T/D_0)^{\eta_0}\longrightarrow \infty \quad (\text{formally})\quad \Rightarrow \quad G(T)\gtrsim G_0 $$ as $T\to 0$ [1809.05484]. The stated physical interpretation is that the standard “cutting” versus “healing” dichotomy is incomplete once finite bandwidth and arbitrary interaction strength are retained. In particular, a weak barrier under very strong repulsion need not simply suppress transport. The abstract additionally states that inclusion of backward scattering leads to non-monotonic temperature dependence of conductance when dealing with fermions with spin [1809.05484]. ## 5. Weak coupling and conductance microscopy of an open quantum dot A further meaning of weak conductance arises in scanning-gate conductance microscopy of an open quantum dot connected to a conducting channel. The system is modeled in two dimensions with an effective-mass Hamiltonian $$ H= -\frac{\hbar^2}{2m}\nabla^2 + V_c(x,y)+V_t(x,y;x_t,y_t), $$ where $m=0.067\,m_0$ is the GaAs effective mass, $V_c$ is zero inside the 70 nm-wide channel and the $300\times 300$ nm\(^2\) dot and tends to infinity elsewhere, and $V_t$ is the scanning-tip perturbation. The wave function is solved with a finite-difference approach and asymptotic subband boundary conditions in the leads [1405.3050]. At zero temperature, the two-terminal conductance is given by the Landauer–Büttiker formula $$ G=\frac{2e^2}{h}\,T,\qquad T=\sum_{p=1}^P\sum_{q=1}^P T_{pq}, $$ with $$ T_{pq}=\left| \frac{e_q}{c_p}\right|^2 \frac{k_q}{k_p}. $$ Weak coupling of the dot to the channel is realized when the narrow 130 nm-long opening of width $W$ supports exactly one transverse mode at the Fermi energy. In that case the $P-1$ bypass subbands in the main channel transmit with $T=1$ each, unaffected by the dot, while the single channel that enters the dot has $0\le T_{\rm dot}(E)\le 1$. Hence $$ \frac{G(E)}{2e^2/h}=(P-1)+T_{\rm dot}(E) \quad\Rightarrow\quad G\in[(P-1)(2e^2/h),\,P(2e^2/h)]. $$ The paper identifies this interval as the numerical signature of the weak-coupling one-mode regime [1405.3050]. The imaging result is formulated through Lippmann–Schwinger perturbation theory. The exact scattering states satisfy $$ |\Psi_p^+\rangle = |\Phi_p^+\rangle + G_0(E+i0)\,V_t\,|\Psi_p^+\rangle, $$ and the first-order change in transmission is $$ \delta T = -4\pi\,\mathrm{Im}\sum_{p,q} t_{pq}^{(0)\,*}\,\langle \Phi_q^-|V_t|\Phi_p^+\rangle. $$ For a delta-like tip, $$ V_t(r)=U\,\delta(r-r_t), $$ so that $$ \langle \Phi_q^-|V_t|\Phi_p^+\rangle = U\,\Phi_q^-(r_t)^*\,\Phi_p^+(r_t). $$ When the dot-channel coupling supports a single mode, the left and right solutions inside the dot differ only by a global phase, $\phi_r(r)=e^{i\theta}\phi_l(r)$, and therefore $$ \langle \phi_r|V_t|\phi_l\rangle \propto |\phi_{\rm dot}(r_t)|^2. $$ Because this quantity equals the local density of states, the conductance variation satisfies $\delta G(r_t)\propto {\rm LDOS}(r_t)$ to first order in $U$ [1405.3050]. The same source contrasts this with strong coupling. If the opening is wide enough that two or more transverse modes feed the dot simultaneously, $\phi_l$ and $\phi_r$ are different superpositions, the inter-mode interference terms no longer track $|\phi(r)|^2$, and the conductance map typically bears little resemblance to the LDOS except at isolated Fano-resonance energies where one quasi-bound state dominates the transport. In the weak-coupling perturbative limit, by contrast, the single-mode condition is stated to be both necessary and sufficient for the normalized conductance map to reproduce the unperturbed LDOS [1405.3050]. ## 6. Comparative interpretation and recurrent misconceptions Across these studies, weak conductance does not denote a single universal conductance law. The weak element is system-specific: a weak-localization-type correction in a topological-insulator wire, a weak tunnel junction in a noisy electromagnetic environment, a weakly interacting or weak-barrier sector of a Luttinger liquid, or a weakly coupled and weakly perturbed quantum dot geometry [1201.3180] [1607.01988] [1809.05484] [1405.3050]. The observables likewise differ. In the Bi\(_2\)Se\(_3\) wire, the relevant quantities are low-field $\Delta \sigma(B)$, the coherence length $L_\phi(T)$, the correlation field $B_c$, and $\delta G_{\rm rms}$. In the tunnel-junction problem, the central quantity is the zero-bias $G(T)$ normalized to the bare conductance $G_t$. In the Luttinger-liquid problem, the conductance is determined through an effective transmission $\tau(T)$. In the microscopy problem, conductance is spatially resolved as $G(x_t,y_t)$ or through the first-order $\delta T$ induced by the tip [1201.3180] [1607.01988] [1809.05484] [1405.3050]. Several misconceptions are explicitly corrected by the sources. Weak conductance effects are not necessarily synonymous with weak absolute transmission: in the Bi\(_2\)Se\(_3\) wire they are small corrections to the classical Drude conductance, while in the quantum-dot problem weak coupling still permits $G$ to lie between $(P-1)(2e^2/h)$ and $P(2e^2/h)$. Nor does Coulomb blockade in a noisy environment generically imply activation by $E_c$; in the semiclassical Nyquist-noise regime the leading activation is $E_c/4$. Likewise, a weak barrier in a Luttinger liquid does not always simply cut the chain: for $g<1/2$ and $\tau_0\lesssim 1$, the formal low-temperature trend is toward $G(T)\gtrsim G_0$ [1201.3180] [1607.01988] [1809.05484] [1405.3050]. A plausible synthesis is that the weak-conductance label is most useful when it identifies the controlling approximation. In the four cases summarized here, those approximations are respectively the low-field phase-coherent interference correction, the Gaussian Nyquist-noise form of $P(E)$, the effective-transmission equation of non-chiral bosonization, and first-order Lippmann–Schwinger perturbation theory for a delta-like tip. The conductance response is therefore weak not in a universal numerical sense, but in the sense that it is generated, renormalized, or imaged by a weak sector of the transport problem. Source: https://www.emergentmind.com/topics/weak-conductance