Lattice calculation of thermal properties of low-density neutron matter with pionless $\mathit{NN}$ effective field theory

Thermal properties of low-density neutron matter are investigated by determinantal quantum Monte Carlo lattice calculations on 3+1 dimensional cubic lattices. Nuclear effective field theory (EFT) is applied using the pionless single- and two-parameter neutron-neutron interactions, determined from the ${}^1{S}_0$ scattering length and effective range. The determination of the interactions and the calculations of neutron matter are carried out consistently by applying EFT power counting rules. The thermodynamic limit is taken by the method of finite-size scaling, and the continuum limit is examined in the vanishing lattice filling limit. The ${}^1{S}_0$ pairing gap at $T\approx 0$ is computed directly from the off-diagonal long-range order of the spin pair-pair correlation function and is found to be approximately 30% smaller than BCS calculations with the conventional nucleon-nucleon potentials. The critical temperature $T_c$ of the normal-to-superfluid phase transition and the pairing temperature scale $T^{*}$ are determined, and the temperature-density phase diagram is constructed. The physics of low-density neutron matter is clearly identified as being a BCS-Bose-Einstein condensation crossover.

Figure 1

Spin pair-pair correlation function $P_s$ as a function of the lattice separation $R$ for the lattice size $N_s=8^3$ for the site-occupation fraction $n=0.25$ at $k_F=30$ MeV. The DQMC results are shown with statistical uncertainties for $T/t=2.0,\hspace{0.5em}0.444,\hspace{0.5em}0.25$, and $0.125$ in the unit of hopping amplitude $t=0.126$ MeV. The dashed line is the asymptotic value of $P_s=0.0244(6)$ extracted from the values for $R=4$ and $4\sqrt{3}$ at $T/t=0.25,\hspace{0.5em}0.125$, and $0.0625$ (not shown).

Figure 2

Spin pair-pair correlation sum $C_{\Delta }$ as a function of temperature $T$ in the unit of the hopping parameter $t$. The Monte Carlo data with the statistical uncertainties are shown for the case of $N_s=8^3$ and $k_F=30$ MeV. The vertical dashed line signifies $T_c(N_s=8^3)=0.335(1)t$, corresponding to the inflexion point of the interpolated curve of $C_{\Delta }(T/t)$, Eq. (30) with $C_1=13.6\pm 1.0,C_2=15.1\pm 2.9,$ and $C_3=16.2\pm 0.6$. The interpolated curve is shown as the solid curve.

Figure 3

Pauli spin susceptibility ${\chi }_P$ as a function of temperature $T$ in the unit of hopping parameter $t$ for $N_s=4^3$. The solid curve is the free fermion gas limit ($\left|c_0\right|/(a^{3}t)\to 0$) of ${\chi }_P(T),\approx n(1-n/2)/T$ (with the filling fraction $n$) [48]. In comparison to this, the cases for $\left|c_0\right|/(a^{3}t)=2,\hspace{0.5em}4,\hspace{0.5em}6,\hspace{0.5em}8,\hspace{0.5em}10$, and $12$ are shown in the increasing order of the interaction strength, from top to bottom.

Figure 4

Pauli spin susceptibility ${\chi }_P$ as a function of temperature $T$ in the unit of hopping parameter $t$ for $N_s=8^3$ and $k_F=30$ MeV. The vertical dashed line signifies $T^{*}(N_s=8^3)=0.5253(3)t$, which is determined as the maximum point of ${\chi }_P(T)$, using the fitting function Eq. (32) with $C_1=0.258\pm 0.008$ (shown as the solid curve).

Figure 5

Lattice-size ($N_s$) dependence of Δ at LO for $k_F=15,\hspace{0.5em}30$, and $60$ MeV with $n=1/4$. The Monte Carlo data shown with the statistical uncertainties are obtained for $\mathit{Ns}=4^3,\hspace{0.5em}{6}^3,\hspace{0.5em}{8}^3$, and ${10}^3$. The dashed lines are the best fits by the use of linear functions of $N_s^{-1/2}$.

Figure 6

Same as Fig. 5, but at NLO for $k_F=60,\hspace{0.5em}90$, and $120$ MeV.

Figure 7

Pairing gap Δ in the unit of the Fermi energy ${\epsilon }_F$ as a function of the filling fraction $n$ for $k_F=60$ MeV. The solid circles show Monte Carlo data for $N_s=6^3$ at LO, with statistical uncertainties. The dashed line is the best fit by the use of a linear $n^{1/3}$ dependence. The interception of the dashed line with the vertical axis corresponds to Δ at the continuum limit ($n\to 0$) for $N_s=6^3$ at $k_F=60$ MeV.

Figure 8

Finite-size scaling of the critical temperature $T_c$ and the pairing temperature scale $T^{*}$. The Monte Carlo data for $T_c$ and $T^{*}$ with $n=1/4$ at LO are shown for $k_F=15,\hspace{0.5em}30$, and $60$ MeV from bottom to top. The dotted lines are the best fits of Eqs. (38) and (40).

Figure 9

Same as Fig. 8, but for $k_F=60,\hspace{0.5em}90$, and $120$ MeV, from bottom to top. The dotted lines are the best fits of Eqs. (39) and (41).

Figure 10

$n$ dependence of the critical temperature $T_c$ and pairing temperature scale $T^{*}$ at LO in the unit of the Fermi energy ${\epsilon }_F$ for $N_s=6^3$ at $k_F=60$ MeV. The dashed lines are the best fits to the $T_c$ and $T^{*}$ data, Eqs. (42) and (43), respectively.

Figure 11

${}^1{S}_0$ pairing gap, Δ, in the thermodynamic and continuum limits, resulting from the LO (solid circles) and NLO (open circles) calculations. The neutron density is denoted in terms of the Fermi momentum $k_F$. The BCS calculation of Ref. [54] ( solid curve) and a higher order calculation including polarization effects of Ref. [55] (dashed curve) are also shown for comparison. For a more detailed comparison, see Fig. 17 in Sec. 7b.

Figure 12

${}^1{S}_0$ pairing gap, Δ, for $N_s=4^3$ and $n=1/4$, resulting from the LO (solid circles) and NLO (open circles) calculations. The neutron density is denoted in terms of the Fermi momentum $k_F$.

Figure 13

Critical temperature $T_c$ (circles) and the pairing temperature scale $T^{*}$ (squares) by the LO (solid symbols) and NLO (open symbols) calculations for $N_s=4^3$ and $n=1/4$, shown as a function of the neutron matter density (represented by the Fermi momentum $k_F$). The error bars are statistical uncertainties only.

Figure 14

${}^1{S}_0$ phase diagram of low-density neutron matter. The solid and open symbols with statistical uncertainties show the LO and NLO results, respectively. The dotted curves for $T_c$ and $T^{*}$ are drawn by extrapolation. Neutron matter is in the superfluid phase below the critical temperature $T_c$ of the second-order phase transition. Above $T_c$, neutron matter is in the pseudogap phase [49], in which pairing remains locally without forming long-range order, and undergoes a smooth transition from the pseudogap phase to the normal phase around $T^{*}$, as pairing gets much less.

Figure 15

EFT parameter ($c_0$) dependence of the critical temperature $T_c$. For easier comparison, $T_c$ and $c_0$ are expressed as dimensionless by use of the spatial lattice spacing $a$ and the hopping parameter $t$. The open circles are shown for $N_s=6^3$ at the quarter-filling ($n=0.5$). The dashed curves are $T_c/t$ at the BCS and BEC limits of Eq. (44).

Figure 16

Chemical potential μ as a function of the interaction strength $c_0$ in a dimensionless unit, with the spatial lattice spacing $a$ and the hopping amplitude $t$. The calculation is of the LO for $N_s=6^3$ and $n=0.5$.

Figure 17

Comparison of our Monte Carlo Δ to other calculations as a function of the neutron matter density (represented by the Fermi momentum $k_F$). The solid diamonds show our results, with statistical uncertainties. The other calculations consist of three types: quantum Monte Carlo (symbols with statistical uncertainties), BCS (solid curve), and BCS with higher-order effects (R's, C's, and RG; shown by dotted and dashed curves). See text for the description of each calculation shown.

Figure 18

Pair correlation function $C_{\Delta }$ as a function of temporal lattice spacing $\Delta \beta$, with $N_s=4^3$ at $k_F=30$ MeV in the unit of hopping amplitude $t$ in the DQMC calculation.

Figure 19

Pair correlation function $C_{\Delta }$ as a function of the sample number at $N_s=4^3$ and $N_t=12$ in the unit of hopping amplitude $t$ in our DQMC calculation.

Figure 20

Autocorrelation as a function of thermalization steps between samples taken with the number of spatial lattice sites $N_s=4^3$ and of temporal lattice sites $N_t=12$ at the Fermi momentum $k_F=30$ MeV in our DQMC calculation.

Figure 21

$\Delta \beta$ dependence of the ratio of energy per particle $E/A$, pair correlation function $C_{\Delta }$, Pauli spin susceptibility ${\chi }_P$, and chemical potential μ to those at $\Delta \beta \to 0$ at $T/t=0.4$ with the interaction strength $c_0/(a^{3}t)=-6.0$ at the one-eighth filling ($n=1/4$) in the dimensionless unit.

Figure 22

Pair correlation function $C_{\Delta }$ as a function of temperature $T$ in the unit of hopping amplitude $t$ at different $\Delta \beta$ at $k_F=30$ MeV, $N_s=4^3$, and $n=1/4$. The open and solid circles with statistical errors are the results at LO and NLO, respectively.

Figure 23

Pauli spin susceptibility ${\chi }_P$ as a function of temperature $T$ in the unit of hopping amplitude $t$ at different $\Delta \beta$ at $k_F=30$ MeV, $N_s=4^3$, and $n=1/4$. The open and solid circles with statistical errors are the results at LO and NLO, respectively.