Fix problems with index 2, 6, 8, 9, 11, 31, 36, 39, 40, 42

cmt_data.jsonl

@@ -1,15 +1,15 @@
 {"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 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 $k_x$ 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": "$\\boxed{(2.09, 1.21); (0., 2.42); (-2.09, 1.21); (-2.09, -1.21); (0., -2.42); (2.09, -1.21)}$", "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}
-{"prompt": "Given a wavefunction $\\psi(x)$ ansatz on 2d heisenberg J1-J2 model which breaks rotation symmetry, please select among the following ways that construct a wavefunction ansatz that restores the C4 rotation symmetry with the rotation operator $R$: (a) $\\sum_n \\psi(R^n x)$ (b) $\\prod_n \\psi(R^n x)$ (c) $\\sum_n \\psi(R x)$ (d) $\\sum_n (-1)^n \\psi(R^n x)$ (e) $\\sum_n e^{i\\pi*n^2} \\psi(R^n x)$, 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": "VMC", "index": 6}
+{"prompt": "Given a wavefunction $\\psi(x)$ ansatz on 2d heisenberg J1-J2 model which breaks rotation symmetry, please select among the following ways that construct a wavefunction ansatz that restores the C4 rotation symmetry with the rotation operator $R$: (a) $\\sum_n \\psi(R^n x)$ (b) $\\prod_n \\psi(R^n x)$ (c) $\\sum_n \\psi(R x)$ (d) $\\sum_n (-1)^n \\psi(R^n x)$ (e) $\\sum_n e^{i\\pi*n^2} \\psi(R^n x)$, 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": "$\\boxed{a; b; d; e}$", "type": "VMC", "index": 6}
 {"prompt": "Consider a two dimensional gappless quantum system with system size $L x L$, if one wants to solve the ground state of such system using DMRG and PEPS, how should the bond dimension scale with the system size respectively? The answer is expressed in terms of $L$ and nonzero constant c, $\\alpha$: (a) $\\chi_\\text{DMRG}(L) \\sim \\exp(\\alpha L), \\; \\chi_\\text{PEPS}(L) \\sim L^{c}$ (b) $\\chi_\\text{DMRG}(L) \\sim L^{c}, \\; \\chi_\\text{PEPS}(L) \\sim \\exp(c L)$ (c) $\\chi_\\text{DMRG}(L) \\sim \\exp(cL \\log L), \\; \\chi_\\text{PEPS}(L) \\sim L^{c} \\log L$ (d) $\\chi_\\text{DMRG}(L) \\sim (L^{L})^c, \\; \\chi_\\text{PEPS}(L) \\sim L^{c}$ (e) $\\chi_\\text{DMRG}(L) \\sim \\exp(\\alpha \\log^2 L), \\; \\chi_\\text{PEPS}(L) \\sim \\log^{c} L$ . 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": "$\\boxed{d}$", "type": "DMRG", "index": 7}
-{"prompt": "Consider the transverse field ising model with antiferromagnetic interactions on the triangular lattice. Consider also the same model on the 4-8 lattice with both antiferromagnetic bonds as well as strong ferromagnetic bonds arrange such that every plaquette has a $\\pi$ flux. Which of the following statements is true: a. The triangular lattice model has only one Kosterlitz-Thouless transition with emergent U(1) symmetry and the 4-8 lattice has an Ising transition at finite temperature. b. The 4-8 model has two Kosterlitz-Thouless transitions with emergent U(1) symmetry and the triangular lattice has two Kosterlitz-Thouless transitions at finite temperature with emergent U(1) symmetry. c. The 4-8 model has two Kosterlitz-Thouless transitions with emergent U(1) symmetry and the triangular lattice has an Ising transition at finite temperature. d. The 4-8 model has an Ising transition at finite temperature and the triangular lattice has an Ising transition at finite temperature. e. The 4-8 model has only one Kosterlitz-Thouless transition with emergent U(1) symmetry and the triangular lattice has an Ising transition at finite temperature. f. Both models exhibit order-by-disorder at zero temperature and a 3D Ising transition at high transverse field. g. Only the triangular model exhibits a transition to a spin liquid at high transverse field at zero temperature. h. No magnetically ordered phase arises in both models at zero temperature due to frustration. i. Both models have ordered phases at zero temperature. Return your answer in a $\\boxed{}$ latex environment consisting of a list of the appropriate choices (a,b,c,d,e,f,g,h, or i) 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": "$a;b;c;d;e;f;g;h;i$", "functions": "", "solution": "$\\boxed{b;f;i}$", "type": "QMC", "index": 8}
-{"prompt": "Consider the following Hamiltonian $\\mathcal{H}= \\sum\\limits_{\\langle \\boldsymbol{r} \\boldsymbol{r}' \\rangle} \\Big[ {J}_{z} S^{z}_{\\boldsymbol{r}} S^{z}_{\\boldsymbol{r}'} + { \\frac{J_{\\pm\\pm}}{2}}( S^{+}_{\\boldsymbol{r}} S^{+}_{\\boldsymbol{r}'} + h.c. ) \\Big] -h\\sum\\limits_{\\boldsymbol{r} }S^{z}_{\\boldsymbol{r}}$ defined on on the square and kagome lattices. Now consider their simulation via the stochastic series expansion (SSE) quantum Monte Carlo at some finite inverse temperature $\\beta$. Which of the following statements is true: a. On the square lattice, the Hamiltonian has a sign problem. b. On the square lattice, the problem does not have a sign problem. c. On the kagome lattice, the model has a sign problem. d. Both on the square and kagome lattices, the model has a sign problem. e. Both on the square and kagome lattices, the model does not have as a sign problem. 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 ;. You must comply with the formatting guidelines or be marked incorrect. 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": "$a;b;c;d$", "functions": "", "solution": "$\\boxed{b;e}$", "type": "QMC", "index": 9}
+{"prompt": "Consider the transverse field ising model with antiferromagnetic interactions on the triangular lattice. Consider also the same model on the 4-8 lattice with both antiferromagnetic bonds as well as strong ferromagnetic bonds arrange such that every plaquette has a $\\pi$ flux. Which of the following statements is true: a. The triangular lattice model has only one Kosterlitz-Thouless transition with emergent U(1) symmetry and the 4-8 lattice has an Ising transition at finite temperature. b. The 4-8 model has two Kosterlitz-Thouless transitions with emergent U(1) symmetry and the triangular lattice has two Kosterlitz-Thouless transitions at finite temperature with emergent U(1) symmetry. c. The 4-8 model has two Kosterlitz-Thouless transitions with emergent U(1) symmetry and the triangular lattice has an Ising transition at finite temperature. d. The 4-8 model has an Ising transition at finite temperature and the triangular lattice has an Ising transition at finite temperature. e. The 4-8 model has only one Kosterlitz-Thouless transition with emergent U(1) symmetry and the triangular lattice has an Ising transition at finite temperature. f. Both models exhibit order-by-disorder at zero temperature and a 3D Ising transition at high transverse field. g. Only the triangular model exhibits a transition to a spin liquid at high transverse field at zero temperature. h. No magnetically ordered phase arises in both models at zero temperature due to frustration. i. Both models have ordered phases at zero temperature. Return your answer in a $\\boxed{}$ latex environment consisting of a list of the appropriate choices (a,b,c,d,e,f,g,h, or i) 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": "$a;b;c;d;e;f;g;h;i$", "functions": "", "solution": "$\\boxed{b;i}$", "type": "QMC", "index": 8}
+{"prompt": "Consider the following Hamiltonian $\\mathcal{H}= \\sum\\limits_{\\langle \\boldsymbol{r} \\boldsymbol{r}' \\rangle} \\Big[ {J}_{z} S^{z}_{\\boldsymbol{r}} S^{z}_{\\boldsymbol{r}'} + { \\frac{J_{\\pm\\pm}}{2}}( S^{+}_{\\boldsymbol{r}} S^{+}_{\\boldsymbol{r}'} + h.c. ) \\Big] -h\\sum\\limits_{\\boldsymbol{r} }S^{z}_{\\boldsymbol{r}}$ defined on on the square and kagome lattices. Now consider their simulation via the stochastic series expansion (SSE) quantum Monte Carlo at some finite inverse temperature $\\beta$. Which of the following statements is true: a. On the square lattice, the Hamiltonian has a sign problem. b. On the square lattice, the problem does not have a sign problem. c. On the kagome lattice, the model has a sign problem. d. Both on the square and kagome lattices, the model has a sign problem. e. Both on the square and kagome lattices, the model does not have as a sign problem. 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 ;. You must comply with the formatting guidelines or be marked incorrect. 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": "$\\boxed{b;e}$", "type": "QMC", "index": 9}
 {"prompt": "Consider the transverse field Ising model $d=1$ lattice and on the 4-8 lattice with both antiferromagnetic bonds as well as strong ferromagnetic bonds arranged such that every plaquette has a $\\pi$ flux. For a large number of spins (i.e., more than $200$ spins) which of options have a provably efficient method for calculating the ground state properties of the models? a. DMRG for $d=1$ and QMC $d=2$ because QMC has no sign problem. b. Jordan-Wigner transformation for $d=1$ and QMC for $d=2$. c. No provably efficient method known in because QMC has sign problem $d=2$ and Jordan-Wigner transformation in $d=1$ d. QMC for $d=2$ because absence of sign problem and Jordan-Wigner transformation for $d=1$ e. Jastrow wavefunction for $d=1$ and no provably efficient method known in $d=2$. f. DMRG for $d=1$ and $d=2$. g. No provably efficient method known because lack of convergence guarantee of QMC even though there is no sign problem in $d=2$ and transfer matrix approach in $d=1$. h. No provably efficient method known QMC sign problem at low temperatures in $d=2$ and transfer matrix approach in $d=1$. Return your answer in a $\\boxed{}$ latex environment consisting of a list of the appropriate choices (a,b,c,d,e,f,g, or h) 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": "$a;b;c;d;e;f;g;h$", "functions": "", "solution": "$\\boxed{g}$", "type": "QMC", "index": 10}
-{"prompt": "Consider a peculiar example of Kitaev alternating chain, whose Hamiltonian is given by $H=-\\sum_{i}^{N/2}(\\sigma^x_{2i-1}\\sigma^x_{2i}+\\sigma^y_{2i}\\sigma^y_{2i+1})$, where $\\sigma^x_i$ and $\\sigma^y_i$ are Pauli matrices on site $i$, and $N$ is the number of sites. Calculate its ground state degeneracy for open chain in terms of $N$ and the value of central charge $c$. Denote these degenercies as a function of $N$ and the value of central charge $c$, and return your answer in LaTeX as $\\boxed{}$. 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": "$N$", "functions": "", "solution": "$\\boxed{2^{N/2-1}; c=1/2}$", "type": "DMRG", "index": 11}
+{"prompt": "Consider a peculiar example of Kitaev alternating chain, whose Hamiltonian is given by $H=-\\sum_{i}^{N/2}(\\sigma^x_{2i-1}\\sigma^x_{2i}+\\sigma^y_{2i}\\sigma^y_{2i+1})$, where $\\sigma^x_i$ and $\\sigma^y_i$ are Pauli matrices on site $i$, and $N$ is the number of sites. Calculate its ground state degeneracy for open chain in terms of $N$ and the value of central charge $c$. Denote these degenercies as a function of $N$ and the value of central charge $c$, and return your answer in LaTeX as $\\boxed{}$. 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": "$N$", "functions": "", "solution": "$\\boxed{2^{N/2}; c=1/2}$", "type": "DMRG", "index": 11}
 {"prompt": "Consider an antiferromagnetic Heisenberg model with nearest-neighbor coupling $J$ on a finite width $W$ and length $L$ cylinder on the square lattice with model Hamiltonian $H=J\\sum_{\\langle i,j\\rangle } \\vec{S}_i.\\vec{S}_j$, where $i$ and $j$ are site indices, $\\vec{S}_i$ is the spin operator on site $i$. The cylinder has open boundary condition along the length $L$ direction and periodic boundary condition along the width $W$ direction. Now let us compute the spin-spin correlation length in two different ways. First way is to directly calculate the spin-spin correlation function in the bulk of the system and extract the correlation length $\\xi_1$ by using an exponential decaying function to fit the spin-spin correlation. Second way is to apply a small but finite staggered Zeeman pinning field (that is consistent with the antiferromagnetic pattern of the spin correlations) at the open boundary, and calculate how the absolute value of induced local staggered spin moment decays from the boundary to bulk using an exponential decaying function and extract the corresponding correlation length $\\xi_2$. Now the question is when both width $W$ and length $L$ are finite and even numbers, what is the relation between $\\xi_1$ or $\\xi_2$? Choose from the following and return your answer in a $\\boxed{}$ latex environment consisting of a list of the appropriate choices (a,b,c, or d) separated by a ;. For example $\\boxed{a;b}$ a. $\\xi_1>\\xi_2$ b. $\\xi_1\\approx\\xi_2$ c. $\\xi_1<\\xi_2$ d. It depends on how quantum mechanical $\\vec{S}$ is 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": "$a;b;c;d$", "functions": "", "solution": "$\\boxed{c}$", "type": "DMRG", "index": 12}
 {"prompt": "Suppose you model a wavefunction of a quantum system with a recurrent neural network (RNN) out of which unbiased samples can be obtained. Within the framework of variational Monte Carlo, it is possible to estimate both the variational energy and the gradients of the energy with respect to the parameters $\\lambda$ in the recurrent neural network wavefunction. The gradients of the energy are given by $\\partial_{\\lambda} E = \\sum_{{\\sigma},{\\sigma'}} \\partial_{\\lambda} \\left[ \\psi^{*}_\\lambda({\\sigma}) H_{\\sigma,\\sigma'} \\psi^{}_\\lambda({\\sigma'}) + \\psi^{*}_\\lambda({\\sigma}) H_{\\sigma,\\sigma'} \\partial_{\\lambda} \\psi^{}_\\lambda({\\sigma'})\\right]$ A Monte Carlo estimator of the energy gradient is given by: $\\partial_{\\lambda} E \\approx \\frac{2}{N_S} \\mathfrak{Re} \\left ( \\sum_{i=1}^{N_S} \\frac{\\partial_{\\lambda}\\psi^{*}_\\lambda({\\sigma^{(i)}})}{\\psi^{*}_\\lambda({\\sigma^{(i)}})} E_{loc}({\\sigma^{(i)}}) \\right)$ where $N_s$ is the number of samples $\\sigma^{(i)}$ drawn from the amplitude square of the RNN wavefunction. Which of the following statements is true: a. The gradient estimator is unbiased. b. The gradient estimator is biased because it misses a baseline involving the average energy. c. The gradient estimator is unbiased but has high variance. d. The gradient estimator is biased because in general the RNN and its gradients may have non-identical support in the computational basis. e. The gradient estimator is biased and has high variance. 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 ;. For example $\\boxed{a;b}$ 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": "$\\boxed{d;e}$", "type": "VMC", "index": 13}
 {"prompt": "Consider spin-polarized electrons in a lowest Landau level of a finite system with square aspect ratio and periodic boundary conditions that is penetrated by $4N$ quanta of magnetic flux at quarter filling, that interact via the Coulomb interaction. We are interested in the scaling of the spacing between consecutive energy levels with $N$ as $N$ becomes large. Which of the following scaling relations can be found among the 20 lowest energy eingenvalues? (a) $O(1)$, (b) $O(1/N)$, (c) $O(1/N^2)$, (d) $O(e^-N)$, (e) exact degeneracy? Return your answer in a $\\boxed{}$ LaTeX environment consisting of a list of the appropriate choices (a,b,c, or d) 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": "$\\boxed{b; e}$", "type": "ED", "index": 14}
@@ -29,18 +29,18 @@
 {"prompt": "Consider a single-band fermionic spinful Hubbard model on a two-leg ladder on the square lattice, with model Hamiltonian $H=-\\sum_{\\langle ij\\rangle,\\sigma} t_{ij} (c_{i\\sigma}^+.c_{j\\sigma}+h.c.) + U\\sum_i n_{i,\\uparrow}n_{i,\\downarrow}$, where $U>0$ is the on-site coulomb repulsion, $i$ and $j$ are site indices, $\\sigma$ is the spin of an electron. $t_{ij}$ is the nearest-neighbor electron hopping integral, and takes the value $t_{ij}=t$ along the chain direction, and $t_{ij}=2t$ perpendicular to the chain direction. Now consider a case with electron filling $n=7/8$ per site, and $U=8t$, choose which one is its ground state from the options below (a) Luttinger liquid; (b) Luther-Emery liquid; (c) spin liquid; (d) non-Luttinger liquid. Return your choice in a $\\boxed{}$ LaTeX environment, such as $\\boxed{a}$. 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": "$a;b; c; d$", "functions": "", "solution": "$\\boxed{a}$", "type": "DMRG", "index": 28}
 {"prompt": "Consider a classical Ising model with the following Hamiltonian: $H= - \\frac{1}{2}\\sum_{i\\ne j}J(|i-j|)\\sigma_{i}\\sigma_{j}$, where $J(n)=|J_{0}|/(1+n^2)^{\\alpha}$. Which of the following statements are correct? Indicate all that apply. a) For $\\alpha=2$, there is a non-zero critical temperature $T_{c}$. b) For $\\alpha=1$, at sufficiently low temperature, $\\langle\\sigma_{j}\\sigma_{j+n}\\rangle\\rightarrow m^{2}>0$ as $n\\rightarrow\\infty$. c) For $\\alpha=1$, there is a temperature for which $\\langle\\sigma_{j}\\sigma_{j+n}\\rangle-\\langle\\sigma_{j}\\rangle\\langle\\sigma_{j+n}\\rangle$ decays as the inverse of the logarithm of distance. d) For $\\alpha=4$, $\\langle\\sigma_{j}\\sigma_{j+n}\\rangle$ decays exponentially with distance. e) For $\\alpha=2$, $\\langle\\sigma_{j}\\sigma_{j+n}\\rangle$ decays exponentially with distance. 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": "$\\boxed{b;c}$", "type": "Other", "index": 29}
 {"prompt": "Consider a triangular lattice model for hardcore bosons with charge $e$ and the following nearest-neighbor Hamiltonian: $H=-\\sum_{ij}t_{ij}(b^{\\dagger}_{i}b_{j}+\\textrm{H.c.})+V\\sum_{ij} n_{i}n_{j}-\\mu \\sum_i n_i$, where $b^{\\dagger}_{i}$ ($b_{i}$) creates (annihilates) a boson on site $i$, $n_{i}=b^{\\dagger}_{i}b_{i}$ is the number operator, $t_{ij}=t$ is the amplitude of the hopping between nearest-neighbor sites $i$ and $j$, and $V$ is finite for nearest-neighbor sites. Which of the following statements are correct? Indicate all that apply. a) The above model has a resistivity much larger than $h/e^2$ at high temperatures when $V\\gg t$ at any filling b) The charge compressibilty at $1/2$ filling for $V\\ggt$ is proportional to $1/T$ at low temperatures c) For temperatures $T\\gg t, V$, the derivative of the boson resistivity with respect to temperature is a constant at any filling d) The $T=0$ ground state at incommensurate boson fillings is a superfluid for any value of $t/V\\geq0$ Return your answer in a $\\boxed{}$ latex environment consisting of a list of the appropriate choices (a,b, c, or d) 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": "$a;b;c;d$", "functions": "", "solution": "$\\boxed{a;b;c}$", "type": "Other", "index": 30}
-{"prompt": "Consider a quantum Ising model at zero temperature with the following nearest-neighbor Hamiltonian: $H=-\\sum_{i,j}\\sigma^z_i\\sigma^z_j + h\\sum_i \\sigma^x_i + g\\sum_i \\sigma^z_i$. Which of the following statements are correct? Indicate all that apply. a) At $h=1$ and $g=0$, the excitation energy vanishes. b) There is a symmetry breaking transition at a finite value of $h$. c) At $h=10$ and $g=1$, $\\langle\\sigma^z_{i}\\sigma^z_{j}\\rangle$ vanishes exponentially as a function of $|i-j|$. d) At $h=0.1$ and $g=1$, $\\langle\\sigma^x_{i}\\sigma^x_{j}\\rangle$ vanishes as a power-law for a large $|i-j|$. Return your answer in a $\\boxed{}$ latex environment consisting of a list of the appropriate choices (a,b, c, or d) separated by a ;. Your solution must comply with this formatting guideline or it will be marked incorrect. 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": "$a;b;c;d$", "functions": "", "solution": "$\\boxed{a;c}$", "type": "Other", "index": 31}
+{"prompt": "Consider a quantum Ising model at zero temperature with the following nearest-neighbor Hamiltonian: $H=-\\sum_{i,j}\\sigma^z_i\\sigma^z_j + h\\sum_i \\sigma^x_i + g\\sum_i \\sigma^z_i$. Which of the following statements are correct? Indicate all that apply. a) At $h=1$ and $g=0$, the excitation energy vanishes. b) There is a symmetry breaking transition at a finite value of $h$. c) At $h=10$ and $g=1$, $\\langle\\sigma^z_{i}\\sigma^z_{j}\\rangle$ vanishes exponentially as a function of $|i-j|$. d) At $h=0.1$ and $g=1$, $\\langle\\sigma^x_{i}\\sigma^x_{j}\\rangle$ vanishes as a power-law for a large $|i-j|$. Return your answer in a $\\boxed{}$ latex environment consisting of a list of the appropriate choices (a,b, c, or d) separated by a ;. Your solution must comply with this formatting guideline or it will be marked incorrect. 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": "$a;b;c;d$", "functions": "", "solution": "$\\boxed{a}$", "type": "Other", "index": 31}
 {"prompt": "For the fermion Hamiltonian \\[ H = \\sum_{i=1}^4\\left[\\left(-tc_{i}^\\dagger e^{iqa_{i,i+1}}c_{i+1}+h.c.\\right)-\\mu BS^z_i\\right] \\] where we have suppressed the spin index, $S^z_i$ is the usual spin operator for fermions, and $B = a_{12}+a_{23}+a_{34}+a_{41}$ is the magnetic field with $a_{i,i+1}$ classical parameters. We consider the case where $\\mu > qt$ and $B>0$. What is the expected measured total magnetization $M^z$ in an experiment at zero temperature and half filling in linear response to the applied magnetic field $B$? Using the variables $B$, $q$, $t$ and $\\mu$, present your 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": "$q;\\mu;t;B$", "functions": "", "solution": "$\\boxed{\\mu-tq^2B/4}$", "type": "Other", "index": 32}
 {"prompt": "Consider a two-dimensional model with attractive on-site interactions with the following Hamiltonian: $H=-\\sum_{ij\\sigma}t_{ij}(c^{\\dagger}_{i\\sigma}c_{j\\sigma}+\\textrm{H.c.}) - |U|\\sum_i (n_{i\\uparrow}-\\frac{1}{2})(n_{i\\downarrow}-\\frac{1}{2})-\\mu \\sum_i n_i$, where $c^{\\dagger}_{i\\sigma}$ ($c_{i\\sigma}$) creates (annihilates) a fermion with spin $\\sigma$ on site $i$, $n_{i\\sigma}=c^{\\dagger}_{i\\sigma}c_{i\\sigma}$ is the number operator, $t_{ij}$ is the amplitude of the hopping between sites $i$ and $j$ ($i\\neq j$), $|U|>0$ is the strength of attraction between fermions on the same site, $\\mu$ is the chemical potential, and $n_i=(n_{i\\uparrow}+n_{i\\downarrow})$ is the total particle number operator on site $i$. Assume that the electron filling is $\\langle n\\rangle=0.12$. Which of the following statements are correct? Indicate all that apply. a) The ground-state at $T=0$ is always a superconductor regardless of ratio of $t/U$. b) For $|U|\\gg t$, the superconducting transition temperature behaves as $t$. c) For $|U|\\gg t$, the superconducting transition temperature behaves as $t^2/|U|$. d) For $|U|\\ll t$, the superconducting transition temperature behaves as $|U|$. Return your answer in a $\\boxed{}$ latex environment consisting of a list of the appropriate choices (a,b, c, or d) 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": "$a;b;c;d$", "functions": "$(c_s^\\dagger, NC); (c_s, NC), (s,\\uparrow,\\downarrow)$", "solution": "$\\boxed{c}$", "type": "Other", "index": 33}
 {"prompt": "Consider a two-dimensional model on a square lattice at zero temperature with the following nearest-neighbor Hamiltonian: $H=-\\sum_{ij}t_{ij}(b^{\\dagger}_{i}b_{j}+\\textrm{H.c.})+\\frac{U}{2}\\sum_i n_{i}(n_{i}-1)-\\mu \\sum_i n_i$, where $b^{\\dagger}_{i}$ ($b_{i}$) creates (annihilates) a boson on site $i$, $n_{i}=b^{\\dagger}_{i}b_{i}$ is the number operator, $t_{ij}=t$ is the amplitude of the hopping between nearest-neighbor sites $i$ and $j$ , $U$ is the strength of the repulsion between bosons on the same site, and $\\mu$ is the chemical potential. Assume that we are working at an average filling of two bosons per site. Which of the following statements are correct? Indicate all that apply. a) There exists a value of $t/U$ across which $\\langle \\sum_i b_i\\rangle$ becomes non-zero with emergent Lorentz invariance. b) For $U=30t$, the boson conductivity vanishes at all frequency. c) For $U=0.001t$, the boson conductivity diverges at zero frequency. d) By tuning either $t/U$ or the filling, we can realize a theory with emergent Lorentz invariance. Return your answer in a $\\boxed{}$ latex environment consisting of a list of the appropriate choices (a,b, c, or d) 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": "$a;b;c;d$", "functions": "", "solution": "$\\boxed{a;c}$", "type": "Other", "index": 34}
 {"prompt": "Consider a $S=1$ Hamiltonian on a honeycomb lattice, $\\mathbf{H}=-\\sum_{\\langle i, j\\rangle_x} \\mathbf{S}_i^x \\mathbf{S}_j^x- \\sum_{\\langle i, j\\rangle_y} \\mathbf{S}_i^y \\mathbf{S}_j^y- \\sum_{\\langle i, j\\rangle_z} \\mathbf{S}_i^z \\mathbf{S}_j^z$ and an unitary operator $\\mathbf{U}^\\gamma=e^{i \\pi \\mathbf{S}^{\\gamma }}$. Consider an operator defined on a plaquette $p$, $W_p=(-1)^{2 S} \\mathbf{U}_1^z \\mathbf{U}_2^x \\mathbf{U}_3^y \\mathbf{U}_4^z \\mathbf{U}_5^x \\mathbf{U}_6^y$ where the subscripts indicate the sites on the hexagonal plaquette $p$. $W_p$ commutes the Hamiltonian. Also consider an operator defined on a loop $LG$, $Q_{\\mathrm{LG}}=\\operatorname{tTr} \\prod_\\alpha Q_{k_\\alpha i_\\alpha j_\\alpha}^{s s^{\\prime}}|s\\rangle\\left\\langle s^{\\prime}\\right|$, where $s$ and $s^\\prime$ are spin indices and $k_\\alpha, i_\\alpha, j_\\alpha ={0,1}$ are virtual bond indices. $Q_{\\mathrm{LG}}$ is a projector satisfying $W_p Q_{\\mathrm{LG}}=Q_{\\mathrm{LG}} W_p=Q_{\\mathrm{LG}}$. Which of the following are correct ? (a) $\\mathbf{Q}_{000}=\\mathbf{I}$ (b) $\\mathbf{Q}_{011}=\\mathbf{U}^z$ (c) $\\mathbf{Q}_{101}=i\\mathbf{U}^x$, (d) $ \\mathbf{Q}_{110}=i\\mathbf{U}^y$, (e) $\\mathbf{Q}_{111}=\\mathbf{U}^z$, (f) $\\mathbf{Q}_{002}= -i\\mathbf{U}^x$. Return your answer in a $\\boxed{}$ latex environment consisting of a list of the appropriate choices (a,b, c, or d) separated by a ;. You must comply with the formatting guidelines or be marked incorrect. 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": "$a;b;c;d;e;f$", "functions": "", "solution": "$\\boxed{a;b}$", "type": "Other", "index": 35}
-{"prompt": "Suppose that we are considering a classical spin model in two dimension. We want to find the scaling dimension of the primary field of the corresponding CFT. The calculation may involve some of the following operations: (a) analyze the singular value spectrum; (b) iteratively coarse-grain a tensor network until convergence; (c) build a transfer matrix; (d) analyze the eigenvalue spectrum; (e) perform a real time evolution; (f) compute the specific heat; (g) compute the magnetic susceptibility; (h) compute the order parameter; (g) do not make further operations. Choose the operations in the correct sequence. Return your answer in a $\\boxed{}$ latex environment consisting of a list of the appropriate choices (a, b, c, ..., h) separated by a `;`, e.g., $\\boxed{a;b}$ 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": "$\\boxed{g}$", "type": "PEPS", "index": 36}
+{"prompt": "Suppose that we are considering a classical spin model in two dimension. We want to find the scaling dimension of the primary field of the corresponding CFT. The calculation may involve some of the following operations: (a) analyze the singular value spectrum; (b) iteratively coarse-grain a tensor network until convergence; (c) build a transfer matrix; (d) analyze the eigenvalue spectrum; (e) perform a real time evolution; (f) compute the specific heat; (g) compute the magnetic susceptibility; (h) compute the order parameter; do not make further operations. Choose the operations in the correct sequence. Return your answer in a $\\boxed{}$ latex environment consisting of a list of the appropriate choices (a, b, c, ..., h) separated by a `;`, e.g., $\\boxed{a;b}$ 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": "$\\boxed{b,c,d}$", "type": "PEPS", "index": 36}
 {"prompt": "Consider a classical O(3) spin Hamiltonian in two spatial dimensions on a triangular lattice: $H = - \\sum_{i,j} J_{ij} S_i \\cdot S_j$, where $J_{ij}=J$ for x-directed bonds and $J_{ij}=J'$ otherwise. At $T=0$, find the number of gapless Goldstone modes, $n_{FM}$, for ferromagnetic couplings ($J>0,J>J'>0$), and $n_{AF}$, for antiferromagnetic couplings ($J<0,J<J'<0$). Return $n_{FM}$ and $n_{AF}$ in a $\\boxed{}$ latex environment 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": "", "functions": "", "solution": "$\\boxed{1; 3}$", "type": "Other", "index": 37}
 {"prompt": "For the fermion Hamiltonian \\[ H = \\sum_{i=1}^n\\left[\\left(-tc_{i}^\\dagger e^{iqa_{i,i+1}}c_{i+1}+h.c.\\right)-\\mu BS^z_i\\right] \\] where we have suppressed the spin index, $S^z_i$ is the usual spin operator for fermions, $B = \\sum_ia_{i,i+1}$ is the magnetic field with $a_{i,i+1}$ classical parameters, and we have imposed periodic boundary conditions with the index arraned in a clockwise manner. Choose from the following options. Make sure to only present the final answer in a single $\\boxed{}$ LaTeX environment. E.g. $\\boxed{a;c;f}$: a) The magnetization is not proportional to $B$. b) The linear-in-$B$ contribution to the magnetization is independent of $q$. c) The linear-in-$B$ contribution to the magnetization is proportional to $q$ for large $q$. d) The linear-in-$B$ contribution to the magnetization is proportional to $q^2$ for large $q$. e) The linear-in-$B$ contribution to the magnetization is independent of $t$. f) The linear-in-$B$ contribution to the magnetization is proportional to $\\mu$ for large $\\mu$. g) The linear-in-$B$ contribution to the magnetization is proportional to $\\mu^2$ for large $\\mu$. h) There are no currents in the system. i) There are currents in the system and they circulate clockwise if $B>0$ and $q>0$. j) There are currents in the system and they circulate counterclockwise if $B>0$ and $q>0$. 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": "$a;b;c;d;e;f;g;h;i;j$", "functions": "", "solution": "$\\boxed{a;d;i}$", "type": "Other", "index": 38}
-{"prompt": "Consider a half filled fermionic system with Hamiltonian \\[ H = -t_1\\sum_{i=1}^n (c^\\dagger_{2i-1}c_{2i} + h.c.) -t_2\\sum_{i=1}^{n-1} ( c^\\dagger_{2i}c_{2i+1} + h.c.). \\] What is the leading contribution to the energy gap between the lowest excited state and the ground state of a chain as $n\\to\\infty$ when $t_2>t_1>0$? Using only the variables $t_1$, $t_2$, and $n$, 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": "$t_1;t_2;n$", "functions": "", "solution": "$\\boxed{2t_1(t_1/t_2)^n}$", "type": "Other", "index": 39}
-{"prompt": "Consider the fermionic Hamiltonian \\[ H = -t_1\\sum_{i=1}^{N_u} (c^\\dagger_{2i-1}c_{2i} + h.c.) -t_2\\sum_{i=1}^{N_u-1} ( c^\\dagger_{2i}c_{2i+1} + h.c.) \\] For $t_2>t_1>0$, and at half filling, which of the following are true: a) This model has a topologically protected mode in the large-$N_u$ limit b) This model is topologically trivial in the large-$N_u$ limit c) The energy gap is of order $t_2(t_1/t_2)^2$ for small $t_1/t_2$ d) The energy gap is of order $t_1(t_2/t_1)^{N_u}$ for small $t_1/t_2$ e) The leading excitation is an \"\"end state\"\" localized on the end of the system f) The leading excitation is a \"\"bulk state\"\" that evenly occupies most sites of the system. Provide your answer in a $\\boxed{}$ latex environment with a list of a subset of the letters a, b, c, d, e, and f, separated by a semicolon ;. For example $\\boxed{a}$ 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": "$a;b;c;d;e;f$", "functions": "", "solution": "$\\boxed{a;d;e}$", "type": "Other", "index": 40}
+{"prompt": "Consider a half filled fermionic system with Hamiltonian \\[ H = -t_1\\sum_{i=1}^n (c^\\dagger_{2i-1}c_{2i} + h.c.) -t_2\\sum_{i=1}^{n-1} ( c^\\dagger_{2i}c_{2i+1} + h.c.). \\] What is the leading contribution to the energy gap between the lowest excited state and the ground state of a chain as $n\\to\\infty$ when $t_2>t_1>0$? Using only the variables $t_1$, $t_2$, and $n$, 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": "$t_1;t_2;n$", "functions": "", "solution": "$\\boxed{2\\frac{t_2^2-t_1^2}{t_2}(t_1/t_2)^n}$", "type": "Other", "index": 39}
+{"prompt": "Consider the fermionic Hamiltonian \\[ H = -t_1\\sum_{i=1}^{N_u} (c^\\dagger_{2i-1}c_{2i} + h.c.) -t_2\\sum_{i=1}^{N_u-1} ( c^\\dagger_{2i}c_{2i+1} + h.c.) \\] For $t_2>t_1>0$, and at half filling, which of the following are true: a) This model has a topologically protected mode in the large-$N_u$ limit b) This model is topologically trivial in the large-$N_u$ limit c) The energy gap is of order $t_2(t_1/t_2)^2$ for small $t_1/t_2$ d) The energy gap is of order $t_1(t_1/t_2)^{N_u}$ for small $t_1/t_2$ e) The leading excitation is an \"\"end state\"\" localized on the end of the system f) The leading excitation is a \"\"bulk state\"\" that evenly occupies most sites of the system. Provide your answer in a $\\boxed{}$ latex environment with a list of a subset of the letters a, b, c, d, e, and f, separated by a semicolon ;. For example $\\boxed{a}$ 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": "$a;b;c;d;e;f$", "functions": "", "solution": "$\\boxed{a;d;e}$", "type": "Other", "index": 40}
 {"prompt": "Consider the AC spin conductivity of the Fermi-Hubbard model $\\sigma({\\bf q},\\omega)$ in terms of the spin current operator $j_x$. Find $\\alpha$ in terms of $\\omega$ and the inverse temperature $\\beta$ such that $\\textrm{Re } \\sigma({\\bf q},\\omega)=\\alpha \\int_0^\\infty dte^{i\\omega t}\\textrm{Re } \\left< j_x({\\bf -q}|t)j_x({\\bf q}|0) \\right>$. Make sure to 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": "$\\omega;\\beta;\\hbar;V$", "functions": "", "solution": "$\\boxed{\\frac{2\\tanh(\\frac{\\beta \\omega}{2})}{\\omega}}$", "type": "Other", "index": 41}
-{"prompt": "Consider the following two-dimensional chiral active Ornstein-Uhlenbeck process with inertia: $m \\mathbf{\\dot v} = -\\gamma \\mathbf{v} + \\gamma v_0 \\mathbf{f} + \\sqrt{2T}\\boldsymbol{\\xi}$, where the self propulsion speed $\\mathbf{f}$ evolves according to $\\mathbf{\\dot f} = -\\mathbf{f}/\\tau + \\Omega \\mathbf{A}\\mathbf{f} +\\sqrt{2/\\tau}\\boldsymbol{\\chi}$, where $\\mathbf{A}=\\begin{bmatrix} 0&-1\\\\1&0\\end{bmatrix}$ is an antisymmetric matrix. Compute the odd diffusion constant of this process. Do not use other symbols than $v_0$, $\\tau$, $\\Omega$, $m$ and $\\gamma$ and, if needed, the ratio $\\rho \\equiv \\frac{\\gamma}{m \\tau}$. Note that not all symbols might be needed. Make srue to only return the expression of the odd diffusivity $D_o$ enclosed in a \"$\\boxed{}$\" 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": "$D_o;\\gamma;v_0;m;\\rho;\\tau;\\Omega$", "functions": "", "solution": "$\\boxed{D_o = \\frac{\\tau^2 \\rho v_0^2 \\Omega (2+\\rho)}{[(1+\\rho)^2 + (\\Omega \\tau)^2][1+(\\Omega \\tau)^2]}}$", "type": "SM", "index": 42}
+{"prompt": "Consider the following two-dimensional chiral active Ornstein-Uhlenbeck process with inertia: $m \\mathbf{\\dot v} = -\\gamma \\mathbf{v} + \\gamma v_0 \\mathbf{f} + \\sqrt{2T}\\boldsymbol{\\xi}$, where the self propulsion speed $\\mathbf{f}$ evolves according to $\\mathbf{\\dot f} = -\\mathbf{f}/\\tau + \\Omega \\mathbf{A}\\mathbf{f} +\\sqrt{2/\\tau}\\boldsymbol{\\chi}$, where $\\mathbf{A}=\\begin{bmatrix} 0&-1\\\\1&0\\end{bmatrix}$ is an antisymmetric matrix. Compute the odd diffusion constant of this process. Do not use other symbols than $v_0$, $\\tau$, $\\Omega$, $m$ and $\\gamma$ and, if needed, the ratio $\\rho \\equiv \\frac{\\gamma\\tau}{m }$. Note that not all symbols might be needed. Make srue to only return the expression of the odd diffusivity $D_o$ enclosed in a \"$\\boxed{}$\" 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": "$D_o;\\gamma;v_0;m;\\rho;\\tau;\\Omega$", "functions": "", "solution": "$\\boxed{D_o = \\frac{\\tau^2 \\rho v_0^2 \\Omega (2+\\rho)}{[(1+\\rho)^2 + (\\Omega \\tau)^2][1+(\\Omega \\tau)^2]}}$", "type": "SM", "index": 42}
 {"prompt": "Consider the dynamics in two dimensions of the following modified active Brownian particle: $\\mathbf{\\dot x} = v_0 \\mathbf{u}$, where $v_0$ is a postive constant, and $\\mathbf{u}$ is a vector of unit norm, whose orientation $\\theta$ with respect to the $x$ axis evolves according to the overdamped dynamics: $\\dot\\theta = -\\frac{v_0}{T} \\nabla V \\cdot \\mathbf{A}\\mathbf{u} + \\chi(t)$, where $V(\\mathbf{x})$ is an external potential that depends only on $\\mathbf{x}$, and $T$ is a positive constant. The matrix $\\mathbf{A}=\\begin{bmatrix} 0&-1\\\\ 1 & 0\\end{bmatrix}$ is a fully antisymmetric two-dimensional matrix. Consider a perturbation of the potential $V\\to V + h(t)\\phi(\\mathbf{x})$. Cconsider the steady state linear response function $R(s) =\\langle \\frac{\\delta \\phi(\\mathbf{x}(t+s))}{\\delta h(t)}\\rangle$ and the steady state autocorrelation function $C(s) = \\langle \\phi(\\mathbf{x}(t))\\phi(\\mathbf{x}(t+s))\\rangle$. Is the fluctuation dissipation theorem between these correlation and response functions violated? Choose one of the following options: (a) Yes, because the dynamics has a positive entropy production rate. (b) No, because the dynamics is time-reversible. (c) No, because the Boltzmann distribution is the stationary distribution. (d) Yes, because there is a nonzero self-propulsion speed. Return your choice among the options \"a\",\"b\",\"c\" and \"d\" enclosed in a $\\boxed{}$ 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": "$a;b;c;d$", "functions": "", "solution": "$\\boxed{c}$", "type": "SM", "index": 43}
 {"prompt": "Consider two coupled systems, made respectively by $N$ soft spins $x_i$ and $N$ soft spins $y_i$. The dynamics of the systems is $\\dot x_i = -\\lambda(\\mathbf{x})x_i + N^{-1} \\sum_{j,k} J_{i}^{jk} x_j x_k + K(y_i -x_i)$ and $\\dot y_i = -\\lambda(\\mathbf{y})y_i + N^{-1} \\sum_{j,k} J_{i}^{jk} y_j y_k + K(x_i -y_i)$. The function $\\lambda$ is defined as $\\lambda(\\mathbf{x}) = N^{-1}|\\mathbf{x}^2| - \\gamma$, with $\\gamma$ a positive parameter. The tensor $J_i^{jk}$ is a random tensor, and it is symmetric with respect to permutations of the upper indexes. Each of its entries has mean $0$ and variance $\\sigma^2$. The coupling constant $K$ is positive. Consider the dynamics of the system in the limit $N\\to \\infty$. We say that the system synchronizes if the quantity $N^{-1} \\sum_{i=1}^N (x_i(t)-y_i(t))^2$ is zero at all times in the steady state. What is the critical value of $K$, $K_c$ above which the synchronous state is stable? Return the critical value of K as a function of $\\gamma$ and $\\sigma^2$. Only present the expression of $K_c$ inside a $\\boxed{}$ 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": "$K_c;\\gamma;\\sigma$", "functions": "", "solution": "$\\boxed{K_c = \\sigma [1 - \\frac{1}{\\sqrt{3}}]\\sqrt{\\gamma + \\frac{2\\sigma^2}{3}(1 + \\sqrt{1 + \\frac{3\\gamma}{\\sigma^2}})}}$", "type": "SM", "index": 44}
 {"prompt": "Consider a liquid composed by diatomic classical molecules in $d$ dimensions. Consider the problem of deriving an effective equation for the motion of one molecule In the limit $d\\to\\infty$. What is the appropriate cavity variable to be considered? Choose among the following: a - One tagged atom of a molecule b - One tagged molcule c - One tagged degree of freedom of the atom of a molecule. d - One tagged degree of freedom of the atom of a molecule, and one tagged degree of freedom of the partner atom in the same molecule. e - A cluster of molecules. f - None of the above. Return your answer among a,b,c,d and f in a $\\boxed{}$ 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": "$a;b;c;d;e;f$", "functions": "", "solution": "$\\boxed{d}$", "type": "SM", "index": 45}