Update problems with index 0 & 2

cmt_data.jsonl

@@ -1,6 +1,6 @@
-{"prompt": "Consider a spinful electron system in a two-dimensional triangular lattice. We want to study the commensurate charge density wave in real space at half-filling per site, induced by Coulomb repulsion denoted by $U_0$, $U_1$ for the onsite and nearest-neighbor interaction. In the strong coupling limit, where the Coulomb energy scale is much larger than the hopping parameter between sites. What is the ground state energy per site? Only present the final answer in a single $\\boxed{}$ LaTeX environment, Your final solution must be placed in a $\\boxed{}$ LaTeX environment. Use only the variables and constants given in the problem; do not define additional constants.", "parameters": "$U_0;U_1$", "functions": "", "solution": "$\\boxed{U_1/2}$", "type": "HF", "index": 0}
+{"prompt": "Consider a spinful electron system in a two-dimensional triangular lattice. We want to study the commensurate charge density wave in real space at half-particle per site, induced by Coulomb repulsion denoted by $U_0$, $U_1$ for the onsite and nearest-neighbor interaction. In the strong coupling limit, where the Coulomb energy scale is much larger than the hopping parameter between sites. What is the ground state energy per site? Only present the final answer in a single $\\boxed{}$ LaTeX environment, Your final solution must be placed in a $\\boxed{}$ LaTeX environment. Use only the variables and constants given in the problem; do not define additional constants.", "parameters": "$U_0;U_1$", "functions": "", "solution": "$\\boxed{U_1/2}$", "type": "HF", "index": 0}
 {"prompt": "We are interested in solving the self-consistency equation for Hartree-Fock mean-field theory on a 2D triangular lattice associated with the following mean-field Hamiltonian $H = H_{\\text{Kinetic}} + H_{\\text{Hartree}} +H_{\\text{Fock}}$, with $H_{\\text{Kinetic}} = \\sum_{s, k} E_s(k) c^\\dagger_s(k) c_s(k)$, where $E_s(k)=\\sum_{n} t_s(n) e^{-i k \\cdot n}$ with the spin index $s = \\{\\uparrow, \\downarrow\\}$ and momentum $k$. The mean-field terms are $H_{\\text{Hartree}} = \\frac{1}{N} \\sum_{s, s'} \\sum_{k_1, k_2} U(0) \\langle c_s^\\dagger(k_1) c_s(k_1) \\rangle c_{s'}^\\dagger(k_2) c_{s'}(k_2)$ $H_{\\text{Fock}} = -\\frac{1}{N} \\sum_{s, s'} \\sum_{k_1, k_2} U(k_1 - k_2) \\langle c_s^\\dagger(k_1) c_{s'}(k_1) \\rangle c_{s'}^\\dagger(k_2) c_s(k_2)$, where $U(k) = \\sum_{n} U_n e^{-i k \\cdot n}$ is the repulsive interaction strength ($U_n>0$) in the momentum basis. What are the possible order parameters that preserve translational symmetry for a Hartree-Fock mean-field Hamiltonian on a two-dimensional triangular lattice? Give all valid order parameters separate by a ; within a single $\\boxed{}$ environment. Print your answer using only the operators provided. Your final solution must be placed in a $\\boxed{}$ LaTeX environment. Use only the variables and constants given in the problem; do not define additional constants.", "parameters": "$k;N; (c_s^\\dagger, NC); (c_s, NC)$", "functions": "$ \\uparrow; \\downarrow$", "solution": "$\\boxed{\\langle c_\\uparrow^\\dagger(k) c_\\uparrow(k) \\rangle; \\langle c_\\downarrow^\\dagger(k) c_\\downarrow(k) \\rangle}$", "type": "HF", "index": 1}
-{"prompt": "Consider a two-dimensional triangular lattice with lattice constant $a = 1$. The first Brillouin zone is a regular hexagon, oriented so that two of its corners lie on the $k_y$ axis. Suppose a charge density wave forms with $\\sqrt{3} \\times \\sqrt{3}$ periodicity, resulting in a reduced Brillouin zone. What are the coordinates of the six original Brillouin zone corners ($K$ and $K'$ points) in the reduced Brillouin zone? Express each coordinate as $(x, y)$, rounded to two decimal places, and present all six in a single $\\boxed{}$ LaTeX environment, separated by semicolons. Your final solution must be placed in a $\\boxed{}$ LaTeX environment. Use only the variables and constants given in the problem; do not define additional constants.", "parameters": "", "functions": "", "solution": "$\\boxed{(2.09, 1.21); (0., 2.41); (-2.09, 1.21); (-2.09, -1.21); (0., -2.41); (2.09, -1.21)}$", "type": "HF", "index": 2}
+{"prompt": "Consider a two-dimensional triangular lattice with lattice constant $a = 1$. The first Brillouin zone is a regular hexagon, oriented so that two of its corners lie on the $k_y$ axis. Suppose a charge density wave forms with $\\sqrt{3} \\times \\sqrt{3}$ periodicity, resulting in a reduced Brillouin zone. What are the coordinates of the six Brillouin zone corners ($K$ and $K'$ points) in the reduced Brillouin zone? Express each coordinate as $(x, y)$, rounded to two decimal places, and present all six in a single $\\boxed{}$ LaTeX environment, separated by semicolons. Your final solution must be placed in a $\\boxed{}$ LaTeX environment. Use only the variables and constants given in the problem; do not define additional constants.", "parameters": "", "functions": "", "solution": "$\\boxed{(2.09, 1.21); (0., 2.41); (-2.09, 1.21); (-2.09, -1.21); (0., -2.41); (2.09, -1.21)}$", "type": "HF", "index": 2}
 {"prompt": "We are interested in solving the self-consistency equation for Hartree-Fock mean-field theory on a 2D triangular lattice associated with the following continuum mean-field Hamiltonian using plane-wave basis covering $N_q$ Brillouin zones. The non-interacting term is \\begin{equation} \\hat{\\mathcal{H}}_0=\\sum_{\\tau={\\pm}} \\int d^2 \\bm{r} \\Psi_{\\tau}^\\dagger(\\bm{r}) H_{\\tau} \\Psi_{\\tau}(\\bm{r}), \\end{equation} \\begin{equation} H_{\\tau}=\\begin{pmatrix} -\\frac{\\hbar^2\\bm{k}^2}{2m_\\mathfrak{b}}+\\Delta_{\\mathfrak{b},\\tau}(\\bm{r}) & \\Delta_{\\text{T},\\tau}(\\bm{r})\\\\ \\Delta_{\\text{T},\\tau}^\\dag(\\bm{r}) & -\\frac{\\hbar^2\\left(\\bm{k}-\\tau \\bm{\\kappa}\\right)^2}{2m_\\mathfrak{t}}+ \\Delta_\\mathfrak{t,\\tau}(\\bm{r}) \\end{pmatrix}, \\end{equation} \\begin{equation} \\Psi_{+}(\\bm{r})=\\begin{pmatrix} \\psi_{\\mathfrak{b},\\uparrow,+}(\\bm{r}) \\\\ \\psi_{\\mathfrak{t},\\downarrow,+}(\\bm{r}) \\end{pmatrix}, \\quad \\Psi_{-}(\\bm{r})=\\begin{pmatrix} \\psi_{\\mathfrak{b},\\downarrow,-}(\\bm{r}) \\\\ \\psi_{\\mathfrak{t},\\uparrow,-}(\\bm{r}) \\end{pmatrix}, \\end{equation} where $\\tau=\\pm $ represents $\\pm K$ valleys, $\\hbar \\bm{k} = -i \\hbar \\partial_{\\bm{r}}$ is the momentum operator, $\\bm{\\kappa}=\\frac{4\\pi}{3a_M}\\left(1,0\\right)$ is at a corner of the moir\\'e Brillouin zone, and $a_M$ is the moir\\'e lattice constant. The spin index of the fermion operators $\\Psi_{\\tau}$ is both layer and valley dependent. $(m_{\\mathfrak{b}},m_{\\mathfrak{t}})=(0.65,0.35)m_e$ ($m_e$ is the rest electron mass) Here, we are only interested in the topmost valence bands from both layers with spin-valley locking. The interacting term is Coulomb repulsion. To solve the self-consistency equation numerically, what's the spatial complexity for each $k$ to represent the Hamiltonian. Only present the final answer in a single $\\boxed{}$ LaTeX environment Your final solution must be placed in a $\\boxed{}$ LaTeX environment. Use only the variables and constants given in the problem; do not define additional constants.", "parameters": "$N_q$", "functions": "", "solution": "$\\boxed{16N_q^2}$", "type": "HF", "index": 3}
 {"prompt": "Given the following single-particle Hamiltonian in the second quantization formalism as \\begin{equation} \\hat{\\mathcal{H}}_0=\\sum_{\\tau={\\pm}} \\int d^2 \\bm{r} \\Psi_{\\tau}^\\dagger(\\bm{r}) H_{\\tau} \\Psi_{\\tau}(\\bm{r}), \\end{equation} \\begin{equation} H_{\\tau}=\\begin{pmatrix} -\\frac{\\hbar^2\\bm{k}^2}{2m_\\mathfrak{b}}+\\Delta_{\\mathfrak{b},\\tau}(\\bm{r}) & \\Delta_{\\text{T},\\tau}(\\bm{r})\\\\ \\Delta_{\\text{T},\\tau}^\\dag(\\bm{r}) & -\\frac{\\hbar^2\\left(\\bm{k}-\\tau \\bm{\\kappa}\\right)^2}{2m_\\mathfrak{t}}+ \\Delta_\\mathfrak{t,\\tau}(\\bm{r}) \\end{pmatrix}, \\end{equation} \\begin{equation} \\Psi_{+}(\\bm{r})=\\begin{pmatrix} \\psi_{\\mathfrak{b},\\uparrow,+}(\\bm{r}) \\\\ \\psi_{\\mathfrak{t},\\downarrow,+}(\\bm{r}) \\end{pmatrix}, \\quad \\Psi_{-}(\\bm{r})=\\begin{pmatrix} \\psi_{\\mathfrak{b},\\downarrow,-}(\\bm{r}) \\\\ \\psi_{\\mathfrak{t},\\uparrow,-}(\\bm{r}) \\end{pmatrix}, \\end{equation} where $\\tau=\\pm $ represents $\\pm K$ valleys, $\\hbar \\bm{k} = -i \\hbar \\partial_{\\bm{r}}$ is the momentum operator, $\\bm{\\kappa}=\\frac{4\\pi}{3a_M}\\left(1,0\\right)$ is at a corner of the moir\\'e Brillouin zone, and $a_M$ is the moir\\'e lattice constant. The spin index of the fermion operators $\\Psi_{\\tau}$ is both layer and valley dependent. $(m_{\\mathfrak{b}},m_{\\mathfrak{t}})=(0.65,0.35)m_e$ ($m_e$ is the rest electron mass) The periodic interlayer potential $\\Delta_{\\mathfrak{b}}(\\bm{r})$ is parametrized as \\begin{equation}\\label{eq:Delta_b} \\Delta_{\\mathfrak{b}}(\\bm{r})=2V_{\\mathfrak{b}}\\sum_{j=1,3,5} \\cos(\\bm{g}_j \\cdot \\bm{r}+\\psi_{\\mathfrak{b}}), \\end{equation} For the interlayer tunneling at $+K$ valley $\\Delta_{\\text{T},+}(\\bm{r})=w\\left(1+\\omega e^{i\\bm{g}_2\\cdot\\bm{r}}+\\omega^{2} e^{i\\bm{g}_3\\cdot\\bm{r}} \\right)$, we obtain the tunneling at $-K$ valley as $\\Delta_{\\text{T},-}(\\bm{r})=-w\\left(1+\\omega^{-1} e^{-i\\bm{g}_2\\cdot\\bm{r}}+\\omega^{-2} e^{-i\\bm{g}_3\\cdot\\bm{r}} \\right)$, where $\\omega=e^{i\\frac{2\\pi}{3}}$, and we assume $w$ takes a real value. Now consider the following symmetries:\\\\ 1. Time-reversal \\begin{equation}\\label{eq:Ham} \\hat{\\mathcal{T}} \\psi_{l,s,\\tau}(\\bm{r}) \\hat{\\mathcal{T}}^{-1} = - \\sum_{s'}\\epsilon_{ss'} \\psi_{l,s',-\\tau}(\\bm{r}),\\quad \\hat{\\mathcal{T}} \\psi_{l,s,\\tau}^{\\dagger}(\\bm{r}) \\hat{\\mathcal{T}}^{-1} = - \\sum_{s'}\\epsilon_{ss'} \\psi_{l,s',-\\tau}^{\\dagger}(\\bm{r}) ,\\quad \\hat{\\mathcal{T}}i\\hat{\\mathcal{T}}^{-1}=-i, \\end{equation} \\\\ 2. Layer-inversion $C_{2y}$ symmetry.\\\\ 3. Rotational $C_{3z}$ symmetry.\\\\ What are the correct symmetry manifested in the Hamiltonian $\\hat{\\mathcal{H}}_0$? Choose from the following options. Make sure to only present the final answer in a single $\\boxed{}$ LaTeX environment. E.g. $\\boxed{a}$ (a) Time-reversal $\\mathcal{T}$ symmetry, $C_{2y}$ symmetry, $C_{3z}$ symmetry.\\\\ (b) Time-reversal $\\mathcal{T}$ symmetry, $C_{2y}$ symmetry, no $C_{3z}$ symmetry.\\\\ (c) Time-reversal $\\mathcal{T}$ symmetry, no $C_{2y}$ symmetry, $C_{3z}$ symmetry.\\\\ (d) Time-reversal $\\mathcal{T}$ symmetry, no $C_{2y}$ symmetry, no $C_{3z}$ symmetry.\\\\ (e) No time-reversal $\\mathcal{T}$ symmetry, $C_{2y}$ symmetry, $C_{3z}$ symmetry.\\\\ (f) No time-reversal $\\mathcal{T}$ symmetry, $C_{2y}$ symmetry, no $C_{3z}$ symmetry.\\\\ (g) No time-reversal $\\mathcal{T}$ symmetry, no $C_{2y}$ symmetry, $C_{3z}$ symmetry.\\\\ (h) No time-reversal $\\mathcal{T}$ symmetry, no $C_{2y}$ symmetry, no $C_{3z}$ symmetry. Your final solution must be placed in a $\\boxed{}$ LaTeX environment. Use only the variables and constants given in the problem; do not define additional constants.", "parameters": "$a;b;c;d;e;f;g;h$", "functions": "", "solution": "$\\boxed{c}$", "type": "HF", "index": 4}
 {"prompt": "Consider a hamiltonian of $N$ fermions $H = -t \\sum_{i,\\sigma} \\left( c^{\\dagger}_{i\\sigma} c_{i+1\\,\\sigma} + \\text{H.c.} \\right) + \\sum_i U\\, n_{i\\uparrow} n_{i\\downarrow} + \\sum_i h_i S^z_i$, which of the followings are good quantum numbers (multiple): (a) $N$ (b) $S^z$ (c) $\\sum_{i,\\sigma} c^{\\dagger}_{i\\sigma} c_{i+1\\,\\sigma} $ (d) $\\eta^2 = \\frac{1}{2} \\left( \\eta^+ \\eta^- + \\eta^- \\eta^+ \\right) + (\\eta^z)^2, \\eta_{-} = \\sum_i (-1)^i c_{i\\uparrow} c_{i\\downarrow}, \\quad \\eta_{+} = \\eta_{-}^\\dagger, \\quad \\eta_0 = \\frac{1}{2} (\\hat{N} - L)$ (e) $\\sum_i n_{i\\uparrow} n_{i\\downarrow} $ Return your answer in a $\\boxed{}$ latex environment consisting of a list of the appropriate choices (a,b,c, d, or e) separated by a ;. Your final solution must be placed in a $\\boxed{}$ LaTeX environment. Use only the variables and constants given in the problem; do not define additional constants.", "parameters": "$a;b;c;d;e$", "functions": "", "solution": "$\\boxed{a; b; d}$", "type": "ED", "index": 5}