id int64 | question string | answer string | final_answer list | answer_type string | topic string | symbol string |
|---|---|---|---|---|---|---|
501 | Please calculate the interaction between two electrons in the weak coupling case. | [] | Expression | Superconductivity | $E$: Energy of the two additional electrons
$\hbar$: Reduced Planck's constant
$\omega_D$: Debye frequency
$g(0)$: Density of states at the Fermi surface for one spin orientation
$V$: Interaction strength (attractive interaction potential between electrons) | |
502 | Calculate the superconducting gap $\Delta$ at zero temperature; | [] | Expression | Superconductivity | $\Delta$: Superconducting energy gap
$\hbar$: Reduced Planck's constant
$\omega_D$: Debye frequency
$g(0)$: Density of states for one spin orientation at the Fermi surface
$V$: Interaction strength (attractive interaction between electrons) | |
503 | Please calculate the superconducting critical temperature $T_c$ based on the gap equation. | [] | Expression | Superconductivity | $k_B$: Boltzmann constant
$T_c$: Superconducting critical temperature
$\mathrm{e}$: Base of natural logarithm
$\gamma$: Euler's constant, $\gamma \approx 0.5772$
$\pi$: Pi constant
$\hbar$: Reduced Planck's constant
$\omega_D$: Debye frequency
$g(0)$: Density of states at the Fermi level
$V$: Interaction strength in the gap equation | |
504 | At $T \ll T_c$, calculate the approximate expression for $\Delta(T)$ according to the previous results | [] | Expression | Superconductivity | $\Delta(T)$: Temperature-dependent energy gap
$\Delta(0)$: Energy gap at zero temperature
$\pi$: Mathematical constant pi
$k_B$: Boltzmann constant
$T$: Temperature | |
505 | Calculate the approximate expression for $\Delta(T)$ as $T \rightarrow T_c$ | [] | Expression | Superconductivity | $\Delta(T)$: Temperature-dependent energy gap
$k_B$: Boltzmann constant
$T_c$: Critical temperature for superconductivity
$T$: Temperature
$\pi$: Mathematical constant pi
$\zeta(3)$: Value of the Riemann zeta function at 3 | |
506 | In the case $T \ll T_c$, the low-temperature approximation of the electronic specific heat | [] | Expression | Superconductivity | $c_{es}$: Electronic specific heat of superconductors
$g(0)$: Density of states at the Fermi level (or at $\epsilon=0$)
$\Delta(0)$: Quasiparticle energy gap at zero temperature
$T$: Temperature
$k_B$: Boltzmann constant | |
507 | Solve for the ground state energy of the Heisenberg model. | [] | Expression | Magnetism | $E_0$: Ground state energy eigenvalue of the Hamiltonian $H$.
$J$: Exchange coupling constant.
$N$: Total number of lattice sites.
$Z$: Coordination number of the lattice.
$S$: Total spin quantum number (ion spin) for a lattice site. | |
508 | First-order moment $\langle S_2 \rangle$ | [] | Numeric | Strongly Correlated Systems | ||
509 | The average ground state energy per electron under the Hartree-Fock approximation (expressed in terms of Fermi wave vector); | [] | Expression | Others | $\hbar$: Reduced Planck's constant
$k_F$: Fermi wave vector
$m$: Mass of the electron
$e$: Elementary charge of an electron
$\pi$: Mathematical constant pi | |
510 | Calculate the Hamiltonian density | [] | Expression | Others | $\dot{\mathbf{W}}$: Time derivative of the reduced displacement vector $\mathbf{W}$
$\gamma_{11}$: Coefficient for the elastic energy density, and appearing in the total potential energy density
$\mathbf{W}$: Vector describing the optical branch vibration, also called the reduced displacement, defined as $\mathbf{W} \equiv \rho^{1/2} (u_+ - u_-)$
$\gamma_{12}$: Coefficient relating polarization intensity to reduced displacement and electric field, and appearing in the potential energy density
$\mathbf{E}$: Macroscopic electric field
$\gamma_{22}$: Coefficient relating polarization intensity to the electric field, and appearing in the potential energy density | |
511 | Calculate the squared transverse vibration frequency $\omega^2_L$ | [] | Expression | Others | $\gamma_{11}$: Coefficient for the elastic energy, elastic energy is $\frac{1}{2}\gamma_{11}\mathbf{W}\cdot \mathbf{W}$.
$\gamma_{12}$: Coefficient relating polarization intensity to reduced displacement and macroscopic field, $P = \gamma_{12} W + \gamma_{22}E$.
$\gamma_{22}$: Coefficient relating polarization intensity to macroscopic field, $P = \gamma_{12} W + \gamma_{22}E$. | |
512 | Write the line element in parabolic coordinates | [] | Expression | Theoretical Foundations | $\xi$: Parabolic coordinate, ranging from 0 to $\infty$
$\eta$: Parabolic coordinate, ranging from 0 to $\infty$
$\varphi$: Parabolic coordinate, azimuthal angle, ranging from 0 to $2\pi$
$d\xi$: Differential change in the parabolic coordinate $\xi$
$d\eta$: Differential change in the parabolic coordinate $\eta$
$d\varphi$: Differential change in the parabolic coordinate $\varphi$ | |
513 | Write out the Laplacian in parabolic coordinates | [] | Expression | Theoretical Foundations | $\xi$: Parabolic coordinate, defined as $\xi = r + z$
$\eta$: Parabolic coordinate, defined as $\eta = r - z$
$\varphi$: Parabolic coordinate, azimuthal angle, defined as $\varphi = \arctan\left(\frac{y}{x}\right)$ | |
514 | Write down the single-particle Schrödinger equation You should return your answer as an equation. | [] | Equation | Theoretical Foundations | $\xi$: One of the parabolic coordinates, defined as $\xi = r + z$.
$\eta$: One of the parabolic coordinates, defined as $\eta = r - z$.
$\psi$: Wave function of the single particle.
$\varphi$: Azimuthal angle, one of the parabolic coordinates, defined as \varphi = \arctan(y/x).
$E$: Total energy of the single particle. | |
515 | Use the method of separation of variables to solve the Schrödinger equation corresponding to the discrete spectrum; | [] | Expression | Theoretical Foundations | $n$: Principal quantum number, $n = \frac{1}{\sqrt{-2E}}$
$\xi$: Parabolic coordinate, $\xi = r + z$
$\eta$: Parabolic coordinate, $\eta = r - z$
$m$: Magnetic quantum number
$L_{n_1}^{|m|}$: Associated Laguerre polynomial
$L_{n_2}^{|m|}$: Associated Laguerre polynomial
$n_1$: Parabolic quantum number, $n_1 = \frac{|m|+1}{2} + n\beta_1$
$n_2$: Parabolic quantum number, $n_2 = \frac{|m|+1}{2} + n\beta_2$
$\varphi$: Azimuthal angle in parabolic coordinates, $\varphi = \arctan\left(\frac{y}{x}\right)$ | |
516 | Assuming a hydrogen atom in a uniform electric field, solve for the energy level correction to second order approximation (Stark effect). | [] | Expression | Theoretical Foundations | $E$: Energy of the system.
$n$: Principal quantum number, related to $n_1, n_2, m$ by $n = n_1 + n_2 + |m| + 1$ for the hydrogen atom.
$\mathscr{E}$: Strength of the uniform electric field.
$n_1$: Parabolic quantum number associated with the $\xi$ coordinate.
$n_2$: Parabolic quantum number associated with the $\eta$ coordinate.
$m$: Magnetic quantum number. | |
517 | Charge density within a metal | [] | Expression | Theoretical Foundations | $\rho_0$: Initial charge density at time $t=0$.
$t$: Time.
$\varepsilon$: Dielectric constant of the uniform isotropic medium.
$\sigma$: Conductivity of the uniform isotropic medium. | |
518 | Introduce the complex dielectric constant $\hat{\epsilon} = \epsilon + i \frac{4\pi \sigma}{\omega}$, the complex phase velocity $\hat{v} = \frac{c}{\sqrt{\mu\hat{\epsilon}}}$, and the complex refractive index $\hat{n} = \frac{c}{\hat{v}} = \frac{c}{\omega}k$, let $\hat{n}=n(1+i\kappa)$, where $\kappa$ is the attenuation coefficient. Please express the refractive index $n$ using the material constants $\epsilon,\mu,\sigma$. You should return your answer as an equation. | [] | Equation | Theoretical Foundations | $n$: Refractive index (real part of complex refractive index).
$\mu$: Magnetic permeability of the medium.
$\varepsilon$: Dielectric constant of the medium.
$\sigma$: Conductivity of the medium.
$v$: Frequency, related to angular frequency by $v = \omega / (2\pi)$. | |
519 | If the electric field is a plane wave, calculate the energy density of the wave. | [] | Expression | Theoretical Foundations | $w_0$: Initial energy density of the wave (amplitude of the energy density at $\mathbf{r}=0$).
$\chi$: Attenuation coefficient, defined as $\chi = \frac{2\omega}{c} n\kappa$.
$|\mathbf{r}|$: Magnitude of the position vector. | |
520 | Consider the action of the Hubbard model:
\begin{align}
S_{\text{loc}}[\Phi^\dagger, \Phi] &= \int_0^\beta d\tau \sum_{\mathbf{r}} \Phi_{\mathbf{r}}^\dagger(\partial_\tau - \mu - i\Delta_c - \Delta\sigma^z)\Phi_{\mathbf{r}}, \nonumber \\
S_1[\Phi^\dagger, \Phi, \Omega] &= \int_0^\beta d\tau \sum_{\mathbf{r}} \Phi_{\mathbf{r}}^\dagger R_{\mathbf{r}}^\dagger \dot{R}_{\mathbf{r}} \Phi_{\mathbf{r}}, \nonumber \\
S_2[\Phi^\dagger, \Phi, \Omega] &= - \int_0^\beta d\tau \sum_{\mathbf{r},\mathbf{r}'} t_{\mathbf{r}\mathbf{r}'} \Phi_{\mathbf{r}}^\dagger R_{\mathbf{r}}^\dagger R_{\mathbf{r}'} \Phi_{\mathbf{r}'}.
\label{eq:6.141}
\end{align}
where:
\begin{equation}
R_{\mathbf{r}} = e^{-\frac{i}{2}\varphi_{\mathbf{r}}\sigma^z} e^{-\frac{i}{2}\theta_{\mathbf{r}}\sigma^y} e^{-\frac{i}{2}\psi_{\mathbf{r}}\sigma^z} = \begin{pmatrix} \cos\left(\frac{\theta_{\mathbf{r}}}{2}\right)e^{-\frac{i}{2}\varphi_{\mathbf{r}}} & -\sin\left(\frac{\theta_{\mathbf{r}}}{2}\right)e^{-\frac{i}{2}\varphi_{\mathbf{r}}} \\ \sin\left(\frac{\theta_{\mathbf{r}}}{2}\right)e^{\frac{i}{2}\varphi_{\mathbf{r}}} & \cos\left(\frac{\theta_{\mathbf{r}}}{2}\right)e^{\frac{i}{2}\varphi_{\mathbf{r}}} \end{pmatrix},
\end{equation}
We consider the fields $\Delta_c$ and $\Delta$ at the saddle-point approximation level: $\Delta_c = i(U/2)\langle\Phi_{\mathbf{r}}^\dagger\Phi_{\mathbf{r}}\rangle$ and $\Delta = (U/2)\langle\Phi_{\mathbf{r}}^\dagger\sigma^z\Phi_{\mathbf{r}}\rangle$. At half-filling, we have $\mu + i\Delta_c = 0$.
Please answer the problem
\begin{enumerate}
\item the effective action $S[\Omega]$ accurate to the first order in $\partial_\tau$ and the second order in $t$:
\begin{equation}
S[\Omega] = \langle S_1 + S_2 \rangle - \frac{1}{2}\langle S_2^2 \rangle_c,
\label{eq:6.142}
\end{equation}
where the expectation value $\langle \cdots \rangle$ is taken with respect to the local action $S_{\text{loc}}$.
This coincides with the action of the spin- $1/2$ Heisenberg model, what is its exchange coupling $J$?
\end{enumerate} | [] | Expression | Strongly Correlated Systems | $J$: Exchange coupling
$t$: Hopping parameter (simplified notation for $t_{\mathbf{r}\mathbf{r}'}$ for nearest neighbors)
$U$: On-site Coulomb interaction strength |
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