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CritPt / Challenge_38_main / The Hamiltonian for the Hatsugai-Kohmoto (HK) model is

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem description

# Problem setup: The Hamiltonian for the Hatsugai-Kohmoto (HK) model is H=∑k,σ(εk−μ)nkσ+U∑knk↑nk↓. H = \sum_{k,\sigma} (\varepsilon_k-\mu) n_{k \sigma} + U \sum_{k}n_{k \uparrow} n_{k \downarrow}. The first term represents the non-interacting part, which corresponds to a non-interacting band 0<εk<W0<\varepsilon_k<W, and the second term represents the interaction. We further add a perturbation term, which preserves momentum and respects fermion symmetries: $$ H' = \sum_{2,3,4}V(1,2,3,4) \delta_{k_1+k_2,k_3+k_4}c^{\dagger}_4 c^{\dagger}_3 c_2 c_1, $$ where the index i≡(ki,σi)i \equiv (k_i, \sigma_i) and the scattering process can be understood as 1→2¯+3+41 \rightarrow \bar{2} + 3+4 and 2¯+3+4→1\bar{2} + 3+4 \rightarrow 1. The energy phase space integral of the scattering rate of the propagating modes of the HK model is given by \begin{equation} I(T)=\langle n_2 (1-n_3)(1-n_4)+(1- n_2) n_3n_4 \rangle_{\epsilon_2,\epsilon_3,\epsilon_4}=\int d\epsilon_2 d\epsilon_3 d\epsilon_4 \langle n_2 (1-n_3)(1-n_4)+(1- n_2) n_3n_4 \rangle \delta(\epsilon_1+\epsilon_2-\epsilon_3-\epsilon_4). \end{equation} # Main problem: Compute the temperature dependence of the energy phase space integral in the limit U≫W≫kBT≫ϵ1U\gg W\gg k_BT\gg \epsilon_1, where the chemical potential μ\mu only crosses the lower Hubbard band, i.e., 0<μ<W0<\mu<W.
Plain-text mathematical notation (without MathML)
# Problem setup:
The Hamiltonian for the Hatsugai-Kohmoto (HK) model is
H=∑_(k,σ)(ε_(k)−μ)n_(kσ)+U∑_(k)n_(k↑)n_(k↓).
The first term represents the non-interacting part, which corresponds to a non-interacting band 0<ε_(k)<W, and the second term represents the interaction.
We further add a perturbation term, which preserves momentum and respects fermion symmetries:
$$
H' = \sum_{2,3,4}V(1,2,3,4) \delta_{k_1+k_2,k_3+k_4}c^{\dagger}_4 c^{\dagger}_3 c_2 c_1,
$$
where the index i≡(k_(i),σ_(i)) and the scattering process can be understood as 1→(2)¯+3+4 and (2)¯+3+4→1.
The energy phase space integral of the scattering rate of the propagating modes of the HK model is given by
\begin{equation}
    I(T)=\langle n_2 (1-n_3)(1-n_4)+(1- n_2) n_3n_4 \rangle_{\epsilon_2,\epsilon_3,\epsilon_4}=\int d\epsilon_2 d\epsilon_3 d\epsilon_4 \langle n_2 (1-n_3)(1-n_4)+(1- n_2) n_3n_4 \rangle \delta(\epsilon_1+\epsilon_2-\epsilon_3-\epsilon_4).
\end{equation}

# Main problem:

Compute the temperature dependence of the energy phase space integral in the limit U≫W≫k_(B)T≫ϵ₁, where the chemical potential μ only crosses the lower Hubbard band, i.e., 0<μ<W.
Original LaTeX notation
# Problem setup:
The Hamiltonian for the Hatsugai-Kohmoto (HK) model is
$$
H = \sum_{k,\sigma} (\varepsilon_k-\mu) n_{k \sigma} + U \sum_{k}n_{k \uparrow} n_{k \downarrow}.
$$
The first term represents the non-interacting part, which corresponds to a non-interacting band $0<\varepsilon_k<W$, and the second term represents the interaction.
We further add a perturbation term, which preserves momentum and respects fermion symmetries:
$$
H' = \sum_{2,3,4}V(1,2,3,4) \delta_{k_1+k_2,k_3+k_4}c^{\dagger}_4 c^{\dagger}_3 c_2 c_1,
$$
where the index $i \equiv (k_i, \sigma_i)$ and the scattering process can be understood as $1 \rightarrow \bar{2} + 3+4$ and $\bar{2} + 3+4 \rightarrow 1$.
The energy phase space integral of the scattering rate of the propagating modes of the HK model is given by
\begin{equation}
    I(T)=\langle n_2 (1-n_3)(1-n_4)+(1- n_2) n_3n_4 \rangle_{\epsilon_2,\epsilon_3,\epsilon_4}=\int d\epsilon_2 d\epsilon_3 d\epsilon_4 \langle n_2 (1-n_3)(1-n_4)+(1- n_2) n_3n_4 \rangle \delta(\epsilon_1+\epsilon_2-\epsilon_3-\epsilon_4).
\end{equation}

# Main problem:

Compute the temperature dependence of the energy phase space integral in the limit $U\gg W\gg k_BT\gg \epsilon_1$, where the chemical potential $\mu$ only crosses the lower Hubbard band, i.e., $0<\mu<W$.

code template

Code

import sympy as sp

T, U = sp.symbols('T U')

def answer(T):
    """
    Return the temperature dependence of the energy phase space integral in SymPy format.

    Inputs
    ----------
    T: sympy.Symbol, temperature $T$
    U: sympy.Symbol, on-site interaction strength $U$

    Outputs
    ----------
    I: sympy.Expr, the energy phase space integral, $I(T)$
    """

    # ------------------ FILL IN YOUR RESULTS BELOW ------------------
    I = ... # a SymPy expression of inputs
    # ---------------------------------------------------------------

    return I

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Source and history

upstream_id / Original problem ID
Challenge_38_main
task_id
55985a73-5bf0-5af2-9126-408dfad041b5
task_key
train--Challenge~5f38~5fmain
task_revision_id
4
config / split
/ train

Official source

initial import