Operator evolution via the similarity renormalization group: The deuteron

Similarity renormalization group (SRG) flow equations can be used to unitarily soften nuclear Hamiltonians by decoupling high-energy intermediate-state contributions to low-energy observables while maintaining the natural hierarchy of many-body forces. Analogous flow equations can be used to consistently evolve operators so that observables are unchanged if no approximations are made. The question in practice is whether the advantages of a softer Hamiltonian and less-correlated wave functions might be offset by complications in approximating and applying other operators. Here we examine the properties of SRG-evolved operators, focusing in this article on applications to the deuteron but leading toward methods for few-body systems. We find the advantageous features generally carry over to other operators with additional simplifications in some cases from factorization of the unitary transformation operator.

Figure 1

SRG evolution of the momentum-space ${}^3{S}_1$ potential starting with (a) Argonne $v_{18}$ (AV18) from $\lambda =15\hspace{0.5em}{\mathrm{fm}}^{-1}$ to $\lambda =1.5\hspace{0.5em}{\mathrm{fm}}^{-1}$ [16] and (b) ${\mathrm{N}}^3$LO (500 MeV) from $\lambda =6\hspace{0.5em}{\mathrm{fm}}^{-1}$ to $\lambda =1.5\hspace{0.5em}{\mathrm{fm}}^{-1}$ [17].

Figure 2

The momentum distribution in the deuteron as given by the expectation value of the bare number operator $a_q^{†}{a}_q$ for the (a) Argonne $v_{18}$ (AV18) [16] and (b) ${\mathrm{N}}^3$LO (500MeV) [17] potentials, both evolved and unevolved.

Figure 3

(a) SRG evolution of the operator $\langle k\left|a_q^{†}{a}_q\right|k^{\prime }\rangle$ for $q=0.34\hspace{0.5em}{\mathrm{fm}}^{-1}$ in the ${}^3{S}_1$ partial wave from $\lambda =6\hspace{0.5em}{\mathrm{fm}}^{-1}$ to $\lambda =1.5\hspace{0.5em}{\mathrm{fm}}^{-1}$, with the ${\mathrm{N}}^3$LO (500 MeV) [17] initial potential. (b) Integrand of $\langle {\psi }_d(s)|(a_q^{†}{a}_q){}_s|{\psi }_d(s)\rangle$ with linear (top) and logarithmic magnitude (bottom) scales.

Figure 4

Same as Fig. 3 but for $q=3.02\hspace{0.5em}{\mathrm{fm}}^{-1}$.

Figure 5

Decoupling in operator matrix elements is tested by calculating the momentum distribution in the deuteron after evolving the AV18 potential to several different $\lambda$ and then truncating the Hamiltonian and evolved occupation operators (i.e., set them to zero above $\Lambda =2.5\hspace{0.5em}{\mathrm{fm}}^{-1}$).

Figure 6

Decoupling in operator matrix elements is tested by calculating the momentum distribution in the deuteron after evolving the ${\mathrm{N}}^3$LO potential to several different $\lambda$ and then truncating the Hamiltonian and evolved occupation operators (i.e., set them to zero above $\Lambda =2.0\hspace{0.5em}{\mathrm{fm}}^{-1}$).

Figure 7

Operator evolution of $\langle k\left|r^2\right|k^{\prime }\rangle$ in the ${}^3{S}_1$ partial wave from $\lambda =6\hspace{0.5em}{\mathrm{fm}}^{-1}$ to $\lambda =1.5\hspace{0.5em}{\mathrm{fm}}^{-1}$ using the ${\mathrm{N}}^3$LO (500-MeV) potential.

Figure 8

Integrand given by $\langle {\psi }_d\left|r^2\right|{\psi }_d\rangle$ and evolved from $\lambda =6\hspace{0.5em}{\mathrm{fm}}^{-1}$ to $\lambda =1.5\hspace{0.5em}{\mathrm{fm}}^{-1}$ using the N3LO 500-MeV potential with a linear color scale (top) and a logarithmic scale of the magnitude (bottom). Notice the difference in momentum scales.

Figure 9

Integrand of the quadrupole moment expectation value in the deuteron and evolved from $\lambda =6\hspace{0.5em}{\mathrm{fm}}^{-1}$ to $\lambda =1.5\hspace{0.5em}{\mathrm{fm}}^{-1}$ using the N3LO 500-MeV potential with a linear color scale (top) and a logarithmic scale of the magnitude (bottom). Notice the difference in momentum scales.

Figure 10

Integrand given by $\langle {\psi }_d\left|r^{-1}\right|{\psi }_d\rangle$ and evolved from $\lambda =6\hspace{0.5em}{\mathrm{fm}}^{-1}$ to $\lambda =1.5\hspace{0.5em}{\mathrm{fm}}^{-1}$ using the N3LO 500-MeV potential with a linear color scale (top) and a logarithmic scale of the magnitude (bottom). Notice the difference in momentum scales.

Figure 11

Deuteron form factors $G_{\mathrm{C}}$, $G_{\mathrm{Q}}$, and $G_{\mathrm{M}}$ using the isoscalar electric form factor parametrization from Ref. [24]. $M$ is the nucleon mass and $M_d$ is the mass of the deuteron. The wave function is derived from the NNLO 550/600-MeV potential and the evolution is run to $\lambda =1.5\hspace{0.5em}{\mathrm{fm}}^{-1}$ [25].

Figure 12

Integrand of $G_{\mathrm{M}}$ at $q=6.90\hspace{0.5em}{\mathrm{fm}}^{-1}$ evolved from $\lambda =6\hspace{0.5em}{\mathrm{fm}}^{-1}$ to $\lambda =1.5\hspace{0.5em}{\mathrm{fm}}^{-1}$ using the NNLO 550/600-MeV potential with a linear scale (top) and a logarithmic scale of the magnitude (bottom).

Figure 13

Integrand of $G_{\mathrm{M}}$ at $q=0.34\hspace{0.5em}{\mathrm{fm}}^{-1}$ evolved from $\lambda =6\hspace{0.5em}{\mathrm{fm}}^{-1}$ to $\lambda =1.5\hspace{0.5em}{\mathrm{fm}}^{-1}$ using the NNLO 550/600-MeV potential with a linear scale (top) and a logarithmic scale of the magnitude (bottom).

Figure 14

Deviation from $E_d$ of the best variational energy as a function of SRG decoupling parameter for the wave function ansatz of Eq. (31). $E_d$ is the deuteron-binding energy for each interaction derived via a full eigenvalue calculation of the Hamiltonian.

Figure 15

The momentum distribution in the deuteron as given by the expectation value of the evolved occupation operator ${\mathit{Ua}}_q^{†}{a}_q{U}^{†}$ using the variational wave functions derived from the Salpeter ansatz and the AV18 potential evolved to $\lambda =6.0$, $4.0$, and $1.5\hspace{0.5em}{\mathrm{fm}}^{-1}$. The direct calculation is from a full eigenvector solution of the Hamiltonian.

Figure 16

Deuteron form factor $G_{\mathrm{M}}$ using the isoscalar electric form factor parametrization from Ref. [24]. $M$ is the nucleon mass and $M_d$ is the mass of the deuteron. The variational wave functions are derived from the Salpeter ansatz and the NNLO 550/600-MeV potential evolved to $\lambda =5.0\hspace{0.5em}{\mathrm{fm}}^{-1}$ and $\lambda =1.2\hspace{0.5em}{\mathrm{fm}}^{-1}$. The operators are evolved consistently. The direct calculation is from a full eigenvector solution of the Hamiltonian.

Figure 17

Numerical tests of factorization of the unitary transformation $U_{\lambda }(k,q)$ by plotting the ratio in Eqs. (32) and (33) as a function of $q$ for fixed $k_0$ and several values of $k_i$. Plateaus in $q$ indicate factorization. The unitary transformations are generated from the Argonne $v_{18}$ (AV18) [16] potential evolved to $\lambda =2\hspace{0.5em}{\mathrm{fm}}^{-1}$ in the (a) ${}^1{S}_0$ and (b) ${}^1{P}_1$ partial waves. The shaded region marks $q<\lambda$.

Figure 18

Same as Fig. 17 but for the ${}^{3}S$${}_1$ partial wave and $\lambda$ of (a) $1.5\hspace{0.5em}{\mathrm{fm}}^{-1}$ and (b) $3\hspace{0.5em}{\mathrm{fm}}^{-1}$.

Figure 19

The scaling of momentum distributions at high momenta in a 1D model is tested by using the leading-order factorized approximation to the momentum occupation number operator to predict high-momentum scaling in $A=2$, 3, 4 (symbols). The full momentum distributions for $A=2$, 3, and 4 are shown with solid, dot-dashed, and dashed lines, respectively.