# Toward an Automated-Algebra Framework for High Orders in the Virial Expansion of Quantum Matter

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Formalism

## 3. Calculation Methods for the Virial Expansion

#### 3.1. Second Order

#### 3.2. Third Order and Beyond

## 4. Homogeneous Fermi Gases with a Zero-Range Interaction

#### 4.1. Factorizing the Transfer Matrix

#### 4.2. From Transfer Matrices to Canonical Partition Functions

#### 4.3. Computational Details of Automated Algebra

**Term generation**: Expand the product ${M}_{mj}^{{N}_{\tau}}$ symbolically, which will yield a large number of terms as ${N}_{\tau}$ is increased.**Delta crunch**: Contract indices to saturate all Kronecker deltas, thus simplifying each term into a product of Gaussian functions, namely the propagator $K\left(\mathbf{p}\right)$, by integrating out a subset of variables. This is the most computationally expensive step.**Gaussian integration**: For each term, take the summation over the rest of the variables and take the continuum limit, ultimately turning each term into a multidimensional Gaussian integral whose results are analytically available as the well-known formula$$\int \mathcal{D}\overrightarrow{x}exp(-\frac{1}{2}{\overrightarrow{x}}^{T}A\overrightarrow{x})=\sqrt{\frac{{\left(2\pi \right)}^{n}}{detA}},$$

**Step 1**

**Step 2**

**Step 3**

**Parallelization**

#### 4.4. Selected Results

## 5. Generalization to Other Systems

#### 5.1. Harmonic Traps

#### 5.2. Neutron Matter

#### 5.3. Unitary Bose Gas

## 6. Pressure, Density, and Generalization to Other Observables

#### 6.1. From Pressure to Density and Tan Contact

**Figure 4.**(

**Left**): Density $n/{n}_{0}$ as a function of fugacity z. The colorful solid lines are the results using truncated VE at third (blue), fourth (orange) and fifth (green) order. The black dashed line is the result using second-order VE. The dashed lines are the results using diagonal or off-diagonal Padé resummation. The black dotted line is the experimental measurement by Reference [52]. (

**Right**): Dimensionless contact $\mathcal{I}/\left(N{k}_{F}\right)$ as a function of reduced temperature $T/{T}_{F}$. The blue, orange and green solid lines are the results using truncated VE for both the contact and the density, which is connected to the reduced temperature $T/{T}_{F}$. The same color code is used. Colorful dotted lines are the results using the experimental density measurement and VE contact. The red curves are the results using resummation: the dotted is with experimental density and the solid with Padé resummed density. The light gray and brown points are experimental determinations from Reference [65] and Reference [69], respectively.

#### 6.2. One-Body Operator Example: The Momentum Distribution

#### 6.3. Two-Body Operators and Real-Time Evolution

## 7. Summary and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**Left**): $\Delta {b}_{n}$ for $n=3,4,5$, from weak coupling (scattering length ${a}_{0}\to 0$) to unitarity (${a}_{0}\to \infty $) as parameterized by $\Delta {b}_{2}/\Delta {b}_{2}^{\mathrm{UFG}}$, where $\Delta {b}_{2}^{\mathrm{UFG}}=1/\sqrt{2}$ is the value of $\Delta {b}_{2}$ at unitarity. The inset shows a zoom into the region around unitarity, where several experimental and theoretical estimates are shown for comparison (see main text and Reference [18] for details). (

**Right**): Subspace contributions $\Delta {b}_{ij}$.

**Figure 2.**(

**a**–

**c**): $\Delta {b}_{3}$, $\Delta {b}_{4}$ and $\Delta {b}_{5}$ as a function of $\beta \omega $, respectively. Our calculations are presented as blue crosses with error bars. The dotted and dashed black lines are the leading and next-leading-order results. The black stars at $\beta \omega \to 0$ are the result of the homogeneous system. The red solid circles in panel (

**a**,

**b**) show the results by Yan and Blume [47]. In panel (

**b**), the dash-dotted black line is the high-temperature fitting from the same work. (

**d**,

**e**): Subspace contribution $\Delta {b}_{ij}$ for (

**d**) $n=4$ and (

**e**) $n=5$. The open green squares with dotted lines represent the $\Delta {b}_{x1}$ contribution, and the open purple diamond for the $\Delta {b}_{x2}$ contribution. The black bar shows the results of the homogeneous system. In panel (

**d**), the dotted red lines are the theoretical conjecture from Reference [36], and the open circle is to emphasize the results in the $\beta \omega \to 0$ limit. The PIMC results from Reference [47] are shown as red circles.

**Figure 3.**Scaled virial coefficients $exp(-\beta {E}_{T}){b}_{n}$ as a function of ${\left(\beta {E}_{T}\right)}^{-1}$ where ${E}_{T}$ is the ground state energy of the trimer. The black dashed line corresponds to the constant second-order value ${b}_{2}=\left(9\sqrt{2}\right)/8$. The blue line is the scaled ${b}_{3}$ from Reference [59], which we used in Equation (61) to obtain the orange line showing ${b}_{4}$.

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**MDPI and ACS Style**

Czejdo, A.J.; Drut, J.E.; Hou, Y.; Morrell, K.J. Toward an Automated-Algebra Framework for High Orders in the Virial Expansion of Quantum Matter. *Condens. Matter* **2022**, *7*, 13.
https://doi.org/10.3390/condmat7010013

**AMA Style**

Czejdo AJ, Drut JE, Hou Y, Morrell KJ. Toward an Automated-Algebra Framework for High Orders in the Virial Expansion of Quantum Matter. *Condensed Matter*. 2022; 7(1):13.
https://doi.org/10.3390/condmat7010013

**Chicago/Turabian Style**

Czejdo, Aleks J., Joaquin E. Drut, Yaqi Hou, and Kaitlyn J. Morrell. 2022. "Toward an Automated-Algebra Framework for High Orders in the Virial Expansion of Quantum Matter" *Condensed Matter* 7, no. 1: 13.
https://doi.org/10.3390/condmat7010013