#
Canonical Density Matrices from Eigenstates of Mixed Systems^{ §}

^{*}

^{§}

## Abstract

**:**

## 1. Introduction

## 2. The Fermi–Hubbard Model with Impurity

## 3. Measures of Quantum Chaos

#### 3.1. Spectral Measures

#### 3.2. Measures for Wavefunctions

## 4. The Reduced Density Matrix of the Impurity

## 5. Eigenstate Canonicity and Degree of Quantum Chaoticity

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DOS | Density of states |

ETH | Eigenstate thermalization hypothesis |

GOE | Gaussian orthogonal ensemble |

MF | Mean field |

NNLS | Nearest-neighbor level spacing |

RDM | Reduced density matrix |

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**Figure 1.**(

**a**) The spectral staircase function $N\left(E\right)$ for the Fermi–Hubbard model with impurity with ${W}_{\mathrm{BB}}=1$ and a total number of states ${d}_{\mathrm{H}}=\mathrm{96,525}$ (${M}_{s}=15$, ${N}_{\mathrm{B}}=7$). The inset shows a magnification of $N\left(E\right)$ with a fit for the smoothed “average” staircase function $\overline{N}\left(E\right)$ entering the spectral unfolding. (

**b**) The normalized density of states (DOS), $\mathsf{\Omega}\left(E\right)$, using a bin size of $\Delta E=0.4$ for different interaction strengths of the bath particles. The impurity–bath interaction in (

**a**,

**b**) is ${W}_{\mathrm{IB}}=1$.

**Figure 2.**The numerically determined nearest-neighbor level statistics $P\left(s\right)$ for the Fermi–Hubbard model with impurity (Equations (1)–(4)) (

**a**) ${W}_{\mathrm{BB}}=0$, (

**b**) ${W}_{\mathrm{BB}}=0.1$ and (

**c**) ${W}_{\mathrm{BB}}=1$ compared to the Poisson (exponential) distribution ${P}_{\mathrm{P}}\left(s\right)$, the Wigner–Dyson distribution ${P}_{\mathrm{WD}}\left(s\right)$, as well as the fit to the Brody distribution ${P}_{\mathrm{B}}\left(s\right)$. The bin size used is $\Delta s=0.01$. Other parameters are ${M}_{s}=15$, ${N}_{\mathrm{B}}=7$, and ${W}_{\mathrm{IB}}=1$.

**Figure 3.**Statistical distribution function of restricted gap ratios for the Fermi–Hubbard model with impurity (${M}_{s}=15$, ${N}_{\mathrm{B}}=7$) for different ${W}_{\mathrm{BB}}$ and ${W}_{\mathrm{IB}}=1$ compared to the analytical predictions for random matrices within the GOE ensemble (Equation (10)) and for integrable spectra (Equation (11)).

**Figure 4.**Distribution of Shannon entropies (Equation (15)) as a measure of the complexity of eigenstates of the system for different ${W}_{\mathrm{BB}}$, (

**a**) ${W}_{\mathrm{BB}}=1$, (

**b**) ${W}_{\mathrm{BB}}=0.3$, (

**c**) ${W}_{\mathrm{BB}}=0.1$, and (

**d**) ${W}_{\mathrm{BB}}=0.05$. The horizontal lines mark the value ${S}_{\mathrm{GOE}}\approx ln0.48{d}_{\mathrm{H}}$ expected for the GOE ensemble. All other parameters as in Figure 2.

**Figure 5.**The Brody parameter $\gamma $ (Equation (12)) or scaled Shannon entropy $\overline{S}$ (Equation (16), left y-axis) as a function of ${W}_{\mathrm{BB}}$. The error bars for $\gamma $ correspond to the standard deviation by comparison between the fits to $P\left(s\right)$ with fits to ${\int}_{0}^{s}d{s}^{\prime}P\left({s}^{\prime}\right)$, and the black line corresponds to a fit to a tanh function $\gamma \left({W}_{\mathrm{BB}}\right)\approx {\gamma}_{0}tanh({W}_{\mathrm{BB}}/{W}_{\mathrm{BB}}^{0})$ with the parameters ${\gamma}_{0}=0.88$ and ${W}_{\mathrm{BB}}^{0}=0.15$. The error bars in $\overline{S}$ reflect the width of S in Figure 4 and correspond to the scaled standard deviation around ${S}_{\mathrm{max}}$. Error of the fit to the Brody distribution as measured by the square root of the ${\chi}^{2}$ function (Equation (14)) (gray line and right y-axis).

**Figure 6.**The inverse temperature as a function of the energy of the entire system predicted by the microcanonical Boltzmann entropy (Equation (18), solid black) and the Gibbs entropy (Equation (20), blue) as well as the canonical expectation value (Equation (23), dashed black). The energy is restricted to the interval $[{E}_{\mathrm{min}},{E}_{\mathrm{peak}}+{E}_{\mathrm{FWHM}}/2,]$ with ${E}_{\mathrm{min}}$ the lower bound where the DOS of the entire system is $\ge $15% of its peak value at ${E}_{\mathrm{peak}}$, and ${E}_{\mathrm{FWHM}}$ the full-width-at-half-maximum of the DOS. Bath–bath interaction strength ${W}_{\mathrm{BB}}=1$ and impurity–bath interaction ${W}_{\mathrm{IB}}=1$ (see Figure 1b).

**Figure 7.**The occupation numbers ${n}_{m,\alpha}$ of the natural orbitals as a function of their energies ${\overline{\u03f5}}_{m,\alpha}$ (Equation (26)) for the eigenstate number $\alpha =4364$ of the total system with energy ${E}_{\alpha}\approx -2.396$ and ${W}_{\mathrm{BB}}=1$ (${M}_{s}=15$, ${N}_{\mathrm{B}}=7$). The impurity–bath coupling strength is ${W}_{\mathrm{IB}}=1$. The horizontal error bars indicate the fluctuations $\Delta {\overline{\u03f5}}_{m,\alpha}$ (Equation (29)). The blue solid line corresponds to the best exponential fit yielding the exponent $\beta \approx 0.58$ in agreement with ${\beta}_{\mathrm{Boltzmann}}$ deduced for this state from Equation (19). The inset shows the same plot on a logarithmic scale.

**Figure 8.**The inverse Boltzmann temperature ${\beta}_{\alpha}$ of the impurity as a function of energy ${E}_{\alpha}$ for the eigenstates of the entire system as obtained from fits to the RDM of the impurity for varying interaction strengths ${W}_{\mathrm{BB}}$, (

**a**) ${W}_{\mathrm{BB}}=1$ with $\gamma \approx 0.9$, (

**b**) ${W}_{\mathrm{BB}}=0.3$ with $\gamma \approx 0.8$, (

**c**) ${W}_{\mathrm{BB}}=0.1$ with $\gamma \approx 0.5$ and (

**d**) ${W}_{\mathrm{BB}}=0.05$ with $\gamma \approx 0.3$. The color bar on the right-hand side represents the variance of ${\beta}_{\alpha}$, $\Delta {\beta}_{\alpha}$, obtained from the fit. Variances above $\Delta \beta >0.05$ are shown in red. The lines correspond to $\beta \left(E\right)$ obtained from the microcanonical ensemble Equation (19) (solid) and the canonical ensemble Equation (23) (dashed), respectively. Other parameters are ${M}_{s}=15$, ${N}_{\mathrm{B}}=7$, ${W}_{\mathrm{IB}}=1$.

**Figure 9.**The fraction of canonical density matrices G obtained for the RDM of the impurity as a function of the Brody parameter $\gamma $ (lower horizontal axis, dots) or as a function of the scaled Shannon entropy $\overline{S}$ (upper horizontal axis, triangles). The dots are color-coded by the interaction strength ${W}_{\mathrm{BB}}$ between the bath particles. Horizontal error bars for $\gamma $ indicate the uncertainty in the extraction of the Brody parameter, horizontal error bars in $\overline{S}$ indicate the standard deviation of the Shannon entropy (see Figure 5). The vertical error bars give the variation of G under variation of the threshold $\Delta \beta $. Other parameters are ${M}_{s}=15$, ${N}_{\mathrm{B}}=7$, and ${W}_{\mathrm{IB}}=1$.

**Figure 10.**Site representation of the impurity RDM, $\langle {j}_{2}\left|{D}_{\alpha}^{\left(\mathrm{I}\right)}\right|{j}_{1}\rangle $, resulting from the reduction of two adjacent states (

**a**) $\alpha =\mathrm{13,638}$ and (

**b**) $\alpha =\mathrm{13,637}$ of the entire system. In (

**c**), we compare the site correlation function ${C}_{\alpha}(\Delta j)$ (Equation (31)) for these two states with the prediction for the ideal Gibbs state (Equation (30)) (black open circles). The system is in the transition regime between quantum integrable and quantum chaotic (${W}_{\mathrm{BB}}=0.1$). Other parameters are ${M}_{s}=15$, ${N}_{\mathrm{B}}=7$, ${W}_{\mathrm{IB}}=1$.

**Figure 11.**Distribution of distances $\Delta {D}_{\alpha}$ (Equation (32)) from the (ideal) Gibbs state of the impurity density matrices reduced from the eigenstates $|{\psi}_{\alpha}\rangle $ of the large system with energy ${E}_{\alpha}$. Shown are only those states with variance $\Delta {\beta}_{\alpha}<0.01$. The points are color-coded by the Shannon entropy of their parent state $|{\psi}_{\alpha}\rangle $. (

**a**) Near the quantum chaotic limit (${W}_{\mathrm{BB}}=1$); (

**b**) in the transition regime between quantum integrability and quantum chaos (${W}_{\mathrm{BB}}=0.1$). The dashed horizontal line in (

**a**) marks the minimal distance ${(\Delta {D}_{\alpha})}_{\mathrm{min}}$ plotted in the inset of (

**a**) as a function of the dimension of the Hilbert space ${d}_{\mathrm{H}}$ for ${W}_{\mathrm{BB}}=1$. Other parameters are ${M}_{s}=15$, ${N}_{\mathrm{B}}=7$, ${W}_{\mathrm{IB}}=1$.

**Figure 12.**Variation of the Brody parameter $\gamma $ characterizing the transition from quantum integrability to quantum chaos as a function of the bath–bath interaction ${W}_{\mathrm{BB}}$ breaking quantum integrability shown for different system sizes. The impurity–bath interaction is ${W}_{\mathrm{IB}}=1$.

**Figure 13.**Universal relation between the fraction G of RDMs of the impurity converging to a Gibbs state and the Brody parameter $\gamma $ for different combinations of system sizes (${M}_{s}$ and ${N}_{\mathrm{B}}$) and bath interaction strengths ${W}_{\mathrm{BB}}$. Impurity–bath interaction in all systems considered is ${W}_{\mathrm{IB}}=1$. Dashed line to guide the eye.

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**MDPI and ACS Style**

Kourehpaz, M.; Donsa, S.; Lackner, F.; Burgdörfer, J.; Březinová, I.
Canonical Density Matrices from Eigenstates of Mixed Systems. *Entropy* **2022**, *24*, 1740.
https://doi.org/10.3390/e24121740

**AMA Style**

Kourehpaz M, Donsa S, Lackner F, Burgdörfer J, Březinová I.
Canonical Density Matrices from Eigenstates of Mixed Systems. *Entropy*. 2022; 24(12):1740.
https://doi.org/10.3390/e24121740

**Chicago/Turabian Style**

Kourehpaz, Mahdi, Stefan Donsa, Fabian Lackner, Joachim Burgdörfer, and Iva Březinová.
2022. "Canonical Density Matrices from Eigenstates of Mixed Systems" *Entropy* 24, no. 12: 1740.
https://doi.org/10.3390/e24121740