Nikolaus Hartman -- University of British Columbia
https://nikhartman.github.io/stanford_03-18
Stanford University
March 22nd, 2018
Electrons and quasiparticles with strong interactions and non-trivial topology often have statistical properties that distinguish them from fermonic and bosonic particles.
Entropy measurements are common in bulk crystals (mm scale) through heat capacity or magnetization measurements
Some properties of MR quasiparticles:
Measure low temperature residual entropy of MR quasiparticles
$s_{MR} = \frac{1}{2} k_B n_{qp} \ln{2}$
where $n_{qp} = 4 \left| n - \frac{5}{2} \frac{eB}{h} \right|$
Maxwell relation: $\left(\frac{\partial \mu}{\partial T}\right)_{p,n} = -\left(\frac{\partial s}{\partial n}\right)_{p,T}$
$\left(\frac{\partial \mu}{\partial T}\right)_{p,n} = -\frac{\partial s}{\partial n_{qp}}\left(\frac{\partial n_{qp}}{\partial n}\right)_{p,T} = \mp 2 k_B \ln{2}$
Negative result: Asymmetry of measured lineshape suggests $\delta S \neq 0$. No change in signal across $\nu = 5/2$ state.
Experimental insight:
$\left(\frac{\partial \mu}{\partial T}\right)_{p,N} = -\left(\frac{\partial S}{\partial N}\right)_{p,T} \Rightarrow \delta \mu_N = -(\Delta S_{N-1 \rightarrow N}) \delta T$
“[Local quasiparticle trap] spectra reflect the QP statistics just as electronic dot spectra reflect the spin and fermionic statistics of electrons ”
$\delta \mu_N = -(\Delta S_{N-1 \rightarrow N}) \delta T$
$ \begin{align} \Gamma_{in} &= \Gamma_{N-1 \rightarrow N} \\ &= \Gamma d_{N} f(E_F - \mu_{N}) \\ \Gamma_{out} &= \Gamma_{N \rightarrow N-1} \\ &= \Gamma d_{N-1} [1-f(E_F - \mu_{N})] \end{align} $
When $\Gamma_{in}=\Gamma_{out}$:
$ \frac{d_{N-1}}{d_N} = \frac{f(E_F - \mu_N)}{[1-f(E_F - \mu_{N})]} $
$ \ln{\frac{d_N}{d_{N-1}}} = (\mu_N-E_F)/k_B T$
At $V_{mid}$:
$\Gamma_{in} = \Gamma_{out}$
$P(N-1) = P(N)$
Two things happen when $T$ changes:
$G_{sens}(V_p,T) \sim \tanh\left(\frac{\alpha (V_p - V_{mid}(T))}{2 k_B T}\right)$
$\delta G_{sens}(V_p, T) = \frac{dG_{sens}}{dT} \delta T$
$\delta G_{sens}(V_p, T) \sim -\delta T \left[ \frac{\alpha(V_p - V_{mid}(T))}{2 k_B T} - \frac{1}{2}\color{#13DAEC}{\frac{\Delta S}{k_B}} \right] \cosh^{-2}\left(\frac{\alpha(V_p - V_{mid}(T))}{2 k_B T}\right)$
$S_0 = 0$, $S_1 = k_B \ln{d_1}$
where $d_1=2$ is the degeneracy of the 1-electron state
Best fit: $\frac{\Delta S_{01}}{k_B} = (1.02 \pm 0.03) \ln{2}$
$S = k_B \sum_{-,+} p_{i}(B_\parallel, T) \ln{ p_{i}(B_\parallel,T) }$
with $p_{\pm}(B_\parallel, T) = (1+ e^{\mp \frac{g\mu_B B_{\parallel}}{k_B T}})^{-1}$
$S = k_B \sum_{-,+,\mathcal{S},\mathcal{T_+}} p_{i}(B_\parallel, T) \ln{ p_{i}(B_\parallel,T) }$
with $p_{\mathcal{S}/\mathcal{T_+}}(B_\parallel, T) = (1+ e^{\mp \frac{g\mu_B B_\parallel - \Delta_{ST}}{k_B T}})^{-1}$
$\delta G_{sens}(V_p, T) \sim -\delta T \left[ \frac{\alpha(V_p - V_{mid}(T))}{2 k_B T} - \frac{1}{2}\color{#13DAEC}{\frac{\Delta S}{k_B}} \right] \cosh^{-2}\left(\frac{\alpha(V_p - V_{mid}(T))}{2 k_B T}\right)$
$\delta G_{sens} \sim G_0 \frac{k_B \delta T}{h \Gamma} \color{#13DAEC}{\frac{\Delta S}{k_B}} \frac{1}{1+(\frac{\alpha e(V-V_0)}{h \Gamma})^2}$
C-X Liu, et al., arXiv:1803.05423
F. Nichele, et al., PRL, 119, 2017
Possible measurements:
This talk: https://nikhartman.github.io/stanford_03-18
Contact me: nik.hartman@gmail.com