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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 ... | $U_0;U_1$ | $\boxed{U_1/2}$ | HF | 0 | |
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... | $k;N; (c_s^\dagger, NC); (c_s, NC)$ | $ \uparrow; \downarrow$ | $\boxed{\langle c_\uparrow^\dagger(k) c_\uparrow(k) \rangle; \langle c_\downarrow^\dagger(k) c_\downarrow(k) \rangle}$ | HF | 1 |
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 coor... | $\boxed{(2.09, 1.21); (0., 2.41); (-2.09, 1.21); (-2.09, -1.21); (0., -2.41); (2.09, -1.21)}$ | HF | 2 | ||
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}... | $N_q$ | $\boxed{16N_q^2}$ | HF | 3 | |
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}}+\Delt... | $a;b;c;d;e;f;g;h$ | $\boxed{c}$ | HF | 4 | |
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... | $a;b;c;d;e$ | $\boxed{a; b; d}$ | ED | 5 | |
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) $\sum_n \psi(R x)$ ... | $a;b;c;d;e$ | $\boxed{a; b; d}$ | VMC | 6 | |
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... | $a;b;c;d;e$ | $\boxed{d}$ | DMRG | 7 | |
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: ... | $a;b;c;d;e;f;g;h;i$ | $\boxed{b;f;i}$ | QMC | 8 | |
Consider the following Hamiltonian $\mathcal{H}= \sum\limits_{\langle \boldsymbol{r} \boldsymbol{r}' \rangle} \Big[ {J}_{z} S^{z}_{\boldsymbol{r}} S^{z}_{\boldsymbol{r}'} + { \frac{J_{\pm\pm}}{2}}( S^{+}_{\boldsymbol{r}} S^{+}_{\boldsymbol{r}'} + h.c. ) \Big] -h\sum\limits_{\boldsymbol{r} }S^{z}_{\boldsymbol{r}}$ defin... | $a;b;c;d$ | $\boxed{b;e}$ | QMC | 9 | |
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 calc... | $a;b;c;d;e;f;g;h$ | $\boxed{g}$ | QMC | 10 | |
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 ... | $N$ | $\boxed{2^{N/2-1}; c=1/2}$ | DMRG | 11 | |
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... | $a;b;c;d$ | $\boxed{c;e}$ | DMRG | 12 | |
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 th... | $a;b;c;d;e$ | $\boxed{d;e}$ | VMC | 13 | |
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 en... | $a; b; c; d; e$ | $\boxed{b; e}$ | ED | 14 | |
Consider the following Hamiltonian $\hat{H}=t \sum_{\sigma=\uparrow, \downarrow} ( \hat{c}^\dagger_{1,\sigma}\hat{c}_{2,\sigma}+ \hat{c}^\dagger_{2,\sigma}\hat{c}_{3,\sigma}+ \hat{c}^\dagger_{3,\sigma}\hat{c}_{4,\sigma}+ \hat{c}^\dagger_{4,\sigma}\hat{c}_{1,\sigma})+\mathrm{h.c.}+U\sum_{i=1}^4\hat{n}_{i,\uparrow}\hat{n... | $a;b;c;d;e$ | $\boxed{a; d}$ | ED | 15 | |
Consider a two-dimensional lattice system with 3x3 unit cells, where each unit cell consists of one site that can be empty, occupied by a spin-1/2 electron with either spin, or doubly occupied with two electrons of opposite spin. The system has periodic boundary conditions in both directions. The system has translation... | $\boxed{7}$ | ED | 16 | ||
Consider the Fermi-Hubbard Hamiltonian with nearest-neighbor hopping $t$ in its particle-hole symmetric form on a bipartite lattice with a chemical potential term. Express the Hamiltonian after the following transformation: $c^{\dagger}_{i,\uparrow}=p^{\dagger}_{i,\uparrow}$ and $c^{\dagger}_{i,\downarrow}=\pm p_{i,\do... | $t;U;\mu;m_{i,\uparrow};m_{i,\downarrow};(p^{\dagger}_{i,\sigma},NC);(p_{j,\sigma},NC);(p^{\dagger}_{j,\sigma},NC);(p_{i,\sigma},NC)$ | $\boxed{-t (p^{\dagger}_{i,\sigma}p_{j,\sigma}+p^{\dagger}_{j,\sigma}p_{i,\sigma})-U (m_{i,\uparrow}-\frac{1}{2})(m_{i,\downarrow}-\frac{1}{2})-\mu (m_{i,\uparrow}-m_{i,\downarrow}) -\mu}$ | Other | 17 | |
Consider the Hamiltonian for spinless fermions on a 2D lattice at half-filling $H=-t \sum_{\langle i j\rangle}\left[c_i^{\dagger} c_j+H . c .\right] + V \sum_{\langle i j\rangle}\left(n_i-\frac{1}{2}\right)\left(n_j-\frac{1}{2}\right)$ where $c_i^{\dagger}$ and $c_i$ are the creation and annihilation operators at site ... | $a;b;c;d$ | $\boxed{a;b}$ | QMC | 18 | |
Consider a noninteracting time-reversal symmetric system of spin-1/2 fermions with a unique gapped ground state that has also four-fold rotation symmetry. Which rotation eigenvalues could the ground state have? (a) $+1$, (b) $e^{i \pi/4}$, (c) $i$, (d) $e^{i 3\pi/4}$, (e) $-1$, (f) $e^{-i 3 \pi/4}$, (g) $-i$, (h) $e^{-... | $a;b;c;d;e;f;g;h$ | $\boxed{a}$ | ED | 19 | |
The following systems have degenerate ground states in the thermodynamic limit. Which of them have exactly degenerate ground states even in finite size systems? (a) Heisenberg ferromagnet, (b) Heisenberg antiferromagnet, (c) Laughlin states from Coulomb interaction in the lowest Landau level on the torus geometry, (d) ... | $a;b;c;d;e$ | $\boxed{a; c}$ | ED | 20 | |
Starting from the iPEPS with a tensor A which approximates the ground state of a given model, we modify the state by replacing one of the ground-state $A$ tensors by a different tensor $B$ at a location $x = (i,j)$ such that $\left|\Psi_0(A)\right\rangle \rightarrow\left|\Phi(A, B)_{{x}}\right\rangle$. The excited stat... | $N_k;B;B*$ | $\boxed{N_k B}$ | PEPS | 21 | |
Suppose that we want to perform exact diagonalization of a Hamiltonian for a kagome lattice model with only nearest-neighbor interactions and local terms. What is the maximum number of isomorphically distinct connected 5-site clusters that can be considered? Give the final answer in a $\boxed{}$ LaTeX environment. Your... | $\boxed{4}$ | ED | 22 | ||
Consider the following Hamiltonian: $H=-J\sum_{j}c_{2j,\sigma}^{\dagger}c_{2j+1,\sigma}-J'\sum_{j}c_{2j+1,\sigma}^{\dagger}c_{2j+2,\sigma}+h.c.+g\sum_{j}n_{j\uparrow}n_{j\downarrow}$. The electron filling is $n=1/5$ per site. In the limit $g\rightarrow\infty$, and setting $J'=2J/3$, compute the equal time density-densi... | $\boxed{(0.200,0.003,0.017,0.027,0.038)}$ | Other | 23 | ||
Consider the following two-band Hamiltonian: $H=\sum_{k,\sigma}\varepsilon_{c,k}c_{k\sigma}^{\dagger}c_{k\sigma}+\sum_{k,\sigma}\varepsilon_{d,k}d_{k\sigma}^{\dagger}d_{k\sigma}+\sum_{r}\left[U\left(n_{c,r}^{2}+n_{d,r}^{2}\right)+Vn_{c,r}n_{d,r}\right]+\left[\sum_{k,\sigma}ig_{1}c_{k\sigma}^{\dagger}d_{k\sigma}+\sum_{k... | $a;b;c;d;e$ | $\boxed{c; d; e}$ | QMC | 24 | |
Consider a tight binding model on a square lattice, whose sites are labeled by integers $x,y$. The Hamiltonian is given by $H=-t\sum_{x,y}c_{x,y}^{\dagger}c_{x,y+1}-t\sum_{x,y}e^{\frac{2i\pi y}{n}}c_{x,y}^{\dagger}c_{x+1,y}-t'\sum_{x,y}ie^{\frac{2i\pi y}{n}}\left(c_{x,y}^{\dagger}c_{x+1,y+1}+c_{x,y}^{\dagger}c_{x+1,y-1... | $a;b;c;d;e$ | $\boxed{b; c; e}$ | ED | 25 | |
Consider the following expansion for an extensive property of a lattice model (e.g., energy per site for a model with finite-range interactions) in the thermodynamic limit: $O_n = \sum_c M_c R_c$, where the sum is over all distinct clusters, with up to $n$ sites, that can be embedded on the lattice, $M_c$ is the number... | $O_n;E_n;E_{n-1}$ | $\boxed{O_n=E_n-E_{n-1}}$ | Other | 26 | |
We consider a model of electrons on a square lattice with nearest-neighbor hopping and long-range interactions. The Hamiltonian is given by: $H=-\sum_{i,j,\sigma}t_{ij}c_{i\sigma}^{\dagger}c_{j\sigma}+\sum_{i,j}V(\bm{r}_{i}-\bm{r}_{j})\left(n_{i}-\overline{n}\right)\left(n_{j}-\overline{n}\right)-h\sum_{j}(c_{j\uparrow... | $a;b;c;d;e;f;g$ | $\boxed{d; g}$ | QMC | 27 | |
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, $\si... | $a;b; c; d$ | $\boxed{a}$ | DMRG | 28 | |
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... | $a;b;c;d;e$ | $\boxed{b;c}$ | Other | 29 | |
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 $i$, $n_{i}=b^{\dagger}_{i}b_{i}$... | $a;b;c;d$ | $\boxed{a;b;c}$ | Other | 30 | |
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 ... | $a;b;c;d$ | $\boxed{a;c}$ | Other | 31 | |
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. ... | $q;\mu;t;B$ | $\boxed{\mu-tq^2B/4}$ | Other | 32 | |
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 (ann... | $a;b;c;d$ | $(c_s^\dagger, NC); (c_s, NC), (s,\uparrow,\downarrow)$ | $\boxed{c}$ | Other | 33 |
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^{\dagg... | $a;b;c;d$ | $\boxed{a;c}$ | Other | 34 | |
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 $\mathbf{U}^\gamma=e^{i \pi \mathbf{S}^{\gamma }}$... | $a;b;c;d;e;f$ | $\boxed{a;b}$ | Other | 35 | |
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 converg... | $a;b;c;d;e;f;g;h$ | $\boxed{g}$ | PEPS | 36 | |
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 $n_{AF}$... | $\boxed{1; 3}$ | Other | 37 | ||
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 impo... | $a;b;c;d;e;f;g;h;i;j$ | $\boxed{a;d;i}$ | Other | 38 | |
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 $t_2>t_1>0$?... | $t_1;t_2;n$ | $\boxed{2t_1(t_1/t_2)^n}$ | Other | 39 | |
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 topo... | $a;b;c;d;e;f$ | $\boxed{a;d;e}$ | Other | 40 | |
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)... | $\omega;\beta;\hbar;V$ | $\boxed{\frac{2\tanh(\frac{\beta \omega}{2})}{\omega}}$ | Other | 41 | |
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} ... | $D_o;\gamma;v_0;m;\rho;\tau;\Omega$ | $\boxed{D_o = \frac{\tau^2 \rho v_0^2 \Omega (2+\rho)}{[(1+\rho)^2 + (\Omega \tau)^2][1+(\Omega \tau)^2]}}$ | SM | 42 | |
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 ... | $a;b;c;d$ | $\boxed{c}$ | SM | 43 | |
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 fu... | $K_c;\gamma;\sigma$ | $\boxed{K_c = \sigma [1 - \frac{1}{\sqrt{3}}]\sqrt{\gamma + \frac{2\sigma^2}{3}(1 + \sqrt{1 + \frac{3\gamma}{\sigma^2}})}}$ | SM | 44 | |
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 t... | $a;b;c;d;e;f$ | $\boxed{d}$ | SM | 45 | |
Consider the overdamped Langevin dynamics $\dot x = -x^3 + x^2 \eta(t)$, where $\eta$ is a white Gaussian noise of mean $0$ and variance $2$. Using the Stratonovich discretization, write an expression for the Onsager-Machlup action, choosing an endpoint choosing an endpoint discretization for the normalization factor. ... | $p;x$ | $\boxed{\frac{(p + x^3)^2}{4 x^4} + 4 x^2}$ | SM | 46 | |
Consider a two-dimensional tight-binding model on the square lattice with a single orbital at each site. The lattice vectors are along $x$ and $y$ directions. Label each site by its coordinate $\vec{r}=(n_x,n_y)$ with two integers $n_x,n_y$. The model has nearest neighbor hopping $t_1$ along $x$ direction. Along $y$ di... | $\boxed{2,2,1}$ | Other | 47 | ||
We want to build infinite two-dimensional tensor network wavefunction for a systems of spin-1 moments on square lattice interacting with SU(2)-symmetric interactions. We choose infinite projected entangled-pair state as the TN. We want to guarantee, that our TN respects i) translational symmetry of the lattice, ii) SU(... | $\boxed{6}$ | PEPS | 48 | ||
We are interested in counting fully packed dimer configurations on honeycomb lattice. Dimer can be placed on any of the nearest-neighbor bonds of the lattice and each site has to be a member of exactly one dimer. This is one of the ways to count ground state configurations of classical antiferromagnetic Ising model on ... | $\boxed{0.33, 0.27}$ | SM | 49 |
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