Luttinger Liquid with Attraction and One Impurity: Exact Solution

Abstract

The exact solution to the problem of scattering of one-dimensional attracted fermions on a point impurity, which exists in the Kane–Fisher model is analyzed for a Luttinger parameter equal to 2. The expression obtained for the frequency dependence of the coefficient of fermion reflection from an impurity demonstrates impossibility of extrapolation of results for a relatively weak fermion–fermion interaction to the range of coupling constants on the order of unity.

Appendices

APPENDIX A

1.1 Transition from the Feynman Response to Retarded Response

Using the result obtained for polarization operator (9), we can write the Feynman part of the response, which depends on impurity scattering and has been calculated for real-valued ω, in form

$$\delta {{j}_{F}}(\omega ,k) = \frac{{\omega V_{{{\text{imp}}}}^{2}{{\mathcal{S}}_{F}}({\text{|}}\omega {\text{|}})}}{{{v}_{c}^{2}{{k}^{2}} - {{\omega }^{2}} - i\delta }}\int {(dq)\frac{{{{E}_{{{\text{ext}}}}}(q,\omega )}}{{{v}_{c}^{2}{{q}^{2}} - {{\omega }^{2}} - i\delta }}} .$$

(44)

This expression effectively decreases for |ω| → ∞ and is exact in the entire 1D domain. (All limitations in ω are determined by the approximations made in the calculation of function ${{\mathcal{S}}_{F}}$(|ω|).) To obtain the retarded response (δj_ret(ω, k)), we must continue Feynman response (44) calculated for real frequencies to the analytic function in the upper half-plane of ω. For this, it is convenient to pass to the coordinate representation:

$$\begin{gathered} \delta {{{\tilde {j}}}_{F}}(\omega ,x) = \frac{{iV_{{{\text{imp}}}}^{2}{{\mathcal{S}}_{F}}({\text{|}}\omega {\text{|}})}}{{8{v}_{c}^{3}{\text{|}}\omega {\text{|}}}} \\ \times \,(\theta (x)\exp (i{\text{|}}\omega {\text{|}}x{\text{/}}{{{v}}_{c}}) + \theta ( - x)\exp ( - i{\text{|}}\omega {\text{|}}x{\text{/}}{{{v}}_{c}}) \\ \times \int {(dq} ){{E}_{{{\text{ext}}}}}(q,\omega )\left( {\frac{1}{{q - (\omega {\text{/}}{{{v}}_{c}} + i\delta {\text{sgn}}\omega )}}} \right. \\ \left. { - \frac{1}{{q + (\omega {\text{/}}{{{v}}_{c}} + i\delta {\text{sgn}}\omega )}}} \right). \\ \end{gathered} $$

(45)

In addition, we are interested in the conductance, i.e., the real part of the retarded response in the limit ω → 0 for real ω. For a transition to the retarded response, we must perform substitution ω² + iδsgn(ω) → (ω + iδ)² in expression (45) at the poles of the integrand. It remains for us to find out how to interpret factor ${{\mathcal{S}}_{F}}$(|ω|)/|ω|. It should be noted that the expressions for the real part of the Feynman, retarded, and advanced responses on the real axis must be identical and can differ only in an infinitely small shift of the singularity existing in the loop diagram for ω = 0 to the complex plane. This means that factor ${{\mathcal{S}}_{F}}$(|ω|)/|ω| in the retarded Green’s function, which has been calculated for real-valued frequencies, we must assume that |ω| is +$\sqrt {{{{(\omega + i\delta )}}^{2}}}$ (the root is defined as a function of complex variable ω with the phase equal to zero at the upper bank of the cut directed to the right). As a result, the entire resulting expression has no singularities for |ω|/$\mathcal{M}$ ≪ 1 in the upper half-plane in frequency (in this domain, its analytic expression is known). In region |ω|/$\mathcal{M}$ ≥ 1, factor ${{\mathcal{S}}_{F}}$(|ω|)/|ω| stops increasing, and the entire response remains a decreasing function in frequently (we are interested in the response to an external field slowly varying in the coordinate, |q|${{{v}}_{c}}$ ≪ |ω| in the high frequency region). Naturally, it is impossible to obtain an analytic result for ${{\mathcal{S}}_{F}}$(|ω|)/|ω| in the region of ω ~ $\mathcal{M}$ because we must know for this the Hamiltonian in the UV range. However, for the analytic continuation of the low-frequency response, the fact of decreasing conductance σ_F(ω, k, q) (Eq. (44)) at high frequencies and at low pulses of k and q (see the second half of Section 2).

For obtaining the expression for conductance, we must take into account the fact that the frequency-dependent part of Re(${{\mathcal{S}}_{F}}$(|ω|)/|ω|) tends to zero for |ω| → 0. This decrease in the real part of the response can be compensated by the pole contribution of the integrand; therefore, the most slowly varying asymptotic form of the linear response is defined as

$$\begin{gathered} \operatorname{Re} \delta {{j}_{{{\text{ret}}}}}(\omega ,x) = - \frac{{V_{{{\text{imp}}}}^{2}\operatorname{Re} (\mathcal{S}({\text{|}}\omega {\text{|}})/{\text{|}}\omega {\text{|}})}}{{8{v}_{c}^{2}}} \\ \times \,(\theta (x)\exp (i{\text{|}}\omega {\text{|}}x{\text{/}}{{{v}}_{c}}) + \theta ( - x)\exp ( - i{\text{|}}\omega {\text{|}}x{\text{/}}{{{v}}_{c}})) \\ \times \,\int {dq{{E}_{{{\text{ext}}}}}(q,\omega )(\delta (q - \omega {\text{/}}{{{v}}_{c}}) + \delta (q + \omega {\text{/}}{{{v}}_{c}})).} \\ \end{gathered} $$

In this expression, factor Re($\mathcal{S}$(|ω|)/|ω|) must be calculated in accordance with the Feynman rules for real ω (i.e., $\mathcal{S}$(ω) is the Fourier transform of the T‑ordered anticommutator $\hat {s}$ of Bose fields) and then continued from the real axis to the upper half-plane with replacement |ω| → +$\sqrt {{{{(\omega + i\delta )}}^{2}}}$.

Passing to the limit ω → 0 (i.e., ω ≪ ${{{v}}_{c}}$/L ~ T_c), we write the result in form

$$\operatorname{Re} \delta j(\omega ,x) = \frac{{V_{{{\text{imp}}}}^{2}\operatorname{Re} (\mathcal{S}({\text{|}}\omega {\text{|}})/{\text{|}}\omega {\text{|}})}}{{8{v}_{c}^{2}}}\int\limits_{ - L/2}^{L/2} {dx\partial {{U}_{{{\text{ext}}}}}(x,\omega ).} $$

Assuming that the potential difference slowly depends on time (or, which is the same, it is the “smeared” δ function of frequency: U_ext(±L/2, ω) = 2πΔ(ω)U_ext(±L/2); $\int d$ωΔ(ω) = 1), we obtain the conventional expression for conductance:

$$\mathcal{G}(\omega ) = \frac{{e_{0}^{2}}}{{2\pi {{{v}}_{c}}}}(1 - \,{\text{|}}{{R}_{\omega }}{{{\text{|}}}^{2}}),$$

(46)

in which the coefficient of reflection of carriers from an impurity, which slowly depends on frequency, has been introduced:¹

$${\text{|}}{{\mathcal{R}}_{\omega }}{{{\text{|}}}^{2}} = - \frac{\pi }{{2{{{v}}_{c}}{\text{|}}\omega {\text{|}}}}V_{{{\text{imp}}}}^{2}\operatorname{Re} \mathcal{S}({\text{|}}\omega {\text{|}}).$$

(47)

APPENDIX B

1.1 Derivation of the Equation for Rotation Matrix

Majorana fields λ and β introduced by us are real-valued, normalized to the channel length, and commute with each other:

$$\{ \hat {\lambda }({{\epsilon }_{n}}),\hat {\lambda }({{\epsilon }_{m}})\} = L{{\delta }_{{n, - m}}},$$

and the rotation matrix has been defined as

$$\hat {\lambda }({{\epsilon }_{n}}) = \frac{1}{L}\sum\limits_{m \ne 0}^{} {S({{\epsilon }_{n}},{{p}_{m}})\hat {\Phi }(m).} $$

(48)

The equation for the rotation matrix follows from the “time-independent Schrödinger equation” [$\hat {\lambda }$(n), $\mathcal{H}$]_– = ${{\epsilon }_{n}}\hat {\lambda }$(n), the substitution of fields λ(n) into it in accordance of definition (48), and the variation of the resulting expression with respect to field Φ_m.

In this way, for obtaining the impurity part of the equation, we must calculate commutators

$$\begin{gathered} \left[ {\frac{1}{L}\sum\limits_{m \ne 0}^{} {S({{\epsilon }_{n}},{{p}_{m}})\hat {\Phi }(m),\frac{{{{{v}}_{c}}}}{{2L}}\sum\limits_n^{} {\hat {\Phi }( - {{p}_{n}}){{p}_{n}}\hat {\Phi }({{p}_{n}})} } } \right. \\ \left. {\, + \gamma \frac{1}{L}\sum\limits_{{{m}_{1}} \ne 0}^{} {\hat {\Phi }({{p}_{{{{m}_{1}}}}})\frac{1}{L}\sum\limits_{{{m}_{2}} \ne 0}^{} {{{p}_{{{{m}_{2}}}}}\hat {\Phi }({{p}_{{{{p}_{{{{m}_{2}}}}}}}})} } } \right]. \\ \end{gathered} $$

(i) Commuting the first term with the kinetic energy and considering that

$$\begin{gathered} LS({{\epsilon }_{n}},{{p}_{m}})({{\delta }_{{m,{{m}_{1}}}}}{{{\hat {\Phi }}}_{{{{m}_{1}}}}} - {{\delta }_{{m, - {{m}_{1}}}}}{{{\hat {\Phi }}}_{{ - {{m}_{1}}}}}){{p}_{{{{m}_{1}}}}} \\ = L({{p}_{m}}S({{\epsilon }_{n}},{{p}_{m}}) + {{p}_{m}}S({{\epsilon }_{n}},{{p}_{m}})){{{\hat {\Phi }}}_{m}}, \\ \end{gathered} $$

we obtain the contribution to the commutator equal to

$$\frac{{{{{v}}_{c}}}}{L}\sum\limits_{{{m}_{1}} \ne 0}^{} {{{p}_{{{{m}_{1}}}}}S({{\epsilon }_{n}},{{p}_{{{{m}_{1}}}}})\hat {\Phi }({{p}_{{{{m}_{1}}}}}).} $$

(ii) The commutator of the first term with impurity scattering is given by

$$\begin{gathered} \gamma \frac{1}{{{{L}^{2}}}}\sum\limits_{m,{{m}_{1}} \ne 0}^{} {S({{\epsilon }_{n}},{{p}_{m}})} ({{\delta }_{{m, - {{m}_{1}}}}}{{p}_{{{{m}_{2}}}}}{{{\hat {\Phi }}}_{{{{m}_{2}}}}} - {{\delta }_{{m, - {{m}_{2}}}}}{{p}_{{{{m}_{2}}}}}{{{\hat {\Phi }}}_{{ - {{m}_{1}}}}}) \\ = \gamma \frac{1}{L}\sum\limits_{m \ne 0}^{} {S({{\epsilon }_{n}},{{p}_{m}})\frac{1}{L}\sum\limits_{{{m}_{1}} \ne 0}^{} {{{p}_{{{{m}_{1}}}}}{{{\hat {\Phi }}}_{{{{m}_{1}}}}}} } \\ + \,\gamma \frac{1}{L}\sum\limits_{m \ne 0}^{} {S({{\epsilon }_{n}},{{p}_{m}})} {{p}_{m}}\frac{1}{L}\sum\limits_{{{m}_{1}} \ne 0}^{} {{{{\hat {\Phi }}}_{{{{m}_{1}}}}}.} \\ \end{gathered} $$

(iii) The right-hand side of the equation is given by

$${{\epsilon }_{n}}\frac{1}{L}\sum\limits_{{{m}_{1}} \ne 0}^{} {S({{\epsilon }_{n}},{{p}_{{{{m}_{1}}}}})\hat {\Phi }({{p}_{{{{m}_{1}}}}}).} $$

By varying the resulting “time-independent” Schrödinger equation in $\hat {\Phi }$(k_m) (m is any number including n), we obtain the following equation for the rotation matrix:

$$\begin{gathered} ({{\epsilon }_{n}} - {{{v}}_{c}}{{k}_{m}})S({{\epsilon }_{n}},{{k}_{m}}) \\ = \gamma {{k}_{m}}\frac{1}{L}\sum\limits_{{{m}_{1}}}^{} {S({{\epsilon }_{n}},{{p}_{{{{m}_{1}}}}}) + \gamma \frac{1}{L}\sum\limits_{{{m}_{1}}}^{} {{{p}_{{{{m}_{1}}}}}S({{\epsilon }_{n}},{{p}_{{{{m}_{1}}}}}).} } \\ \end{gathered} $$

(49)

It remains to us to obtain the expression for the reciprocal scattering matrix (S^–1):

$$\hat {\Phi }({{p}_{{{{m}_{1}}}}}) = \frac{1}{L}\sum\limits_{{{n}_{1}} \ne 0}^{} {{{S}^{{ - 1}}}({{\epsilon }_{{{{n}_{1}}}}},{{p}_{{{{m}_{1}}}}})\hat {\lambda }({{\epsilon }_{{{{n}_{1}}}}}).} $$

(50)

Substituting this expression into (48), we obtain

$$\hat {\lambda }({{\epsilon }_{n}}) = \frac{1}{{{{L}^{2}}}}\sum\limits_{m,{{n}_{1}} \ne 0}^{} {S({{\epsilon }_{n}},{{p}_{m}}){{S}^{{ - 1}}}({{\epsilon }_{{{{n}_{1}}}}},{{p}_{m}})} \hat {\lambda }({{\epsilon }_{{{{n}_{1}}}}}).$$

(51)

We must now take into account the fact that Majorana operators are real and must anticommute. For this, we write the anticommutator of fields λ in terms of “unrotated” fields:

$$\frac{1}{{{{L}^{2}}}}\sum\limits_{{{m}_{1}};{{m}_{2}} \ne 0}^{} {S({{\epsilon }_{n}},{{p}_{{{{m}_{1}}}}})S({{\epsilon }_{m}},{{p}_{{{{m}_{2}}}}})\{ \hat {\Phi }({{m}_{1}}),\hat {\Phi }({{m}_{2}})\} } = L{{\delta }_{{n, - m}}},$$

i.e.,

$${{\delta }_{{n,m}}} = \frac{1}{{{{L}^{2}}}}\sum\limits_{{{m}_{1}} \ne 0}^{} {S({{\epsilon }_{n}},{{p}_{{{{m}_{1}}}}})S({{\epsilon }_{{ - m}}},{{p}_{{ - {{m}_{1}}}}}).} $$

This immediately implies that if the elements of the reciprocal rotation matrix satisfy condition

$${{S}^{{ - 1}}}({{\epsilon }_{n}},{{p}_{m}}) = S({{\epsilon }_{{ - n}}},{{p}_{{ - m}}}),$$

(52)

equality (51) becomes an identity. (However, this relation only shows that the rotation matrix is unitary.)

1.2 B.1. Spectrum

System of equations (29), (30), which defines the rotation matrix, depends on sums diverging in the thermodynamic limit (N₁ → ∞). For symmetric regularization, we have

$$\mathcal{A}(y) = \sum\limits_{ - {{N}_{1}},m \ne 0}^{{{N}_{1}}} {\frac{1}{{y - m}} = \pi \cot (\pi y) - \frac{1}{y} + \mathcal{O}(1{\text{/}}{{N}_{1}})} ,$$

$$B(y,{{N}_{1}}) = \sum\limits_{ - {{N}_{1}}}^{{{N}_{1}}} {\frac{m}{{y - m}} = y\mathcal{A}(y,{{N}_{1}}) - 2{{N}_{1}}} ,$$

(53)

$$C(y,{{N}_{1}}) = \sum\limits_{ - {{N}_{1}}}^{{{N}_{1}}} {\frac{{{{m}^{2}}}}{{y - m}} = yB(y).} $$

In this notation, the equations for the rotation matrix can be written in form

$$\begin{gathered} \flat {{N}_{1}}\mathcal{Z}({{y}_{n}}) = B(y,{{N}_{1}})\mathcal{Z}({{y}_{n}}) + \frac{{2\pi }}{L}C({{y}_{n}})Y({{y}_{n}}) = 0, \\ \flat {{N}_{1}}Y({{y}_{n}}) = \frac{{2\pi }}{L}\mathcal{A}({{y}_{n}})\mathcal{Z}({{y}_{n}}) + B({{y}_{n}})Y({{y}_{n}}) = 0. \\ \end{gathered} $$

(54)

The condition for the existence of the solution to these equations gives the exact equation for the spectrum:

$$B({{y}_{n}}) = \frac{{{{\flat }^{2}}{{N}_{1}}}}{{2(\flat + 1)}},$$

(55)

while the last equation gives the relation between Y(y_n) and $\mathcal{Z}$(y_n):

$$Y({{y}_{n}}) = \frac{{\flat + 2}}{\flat }\frac{{{{{v}}_{c}}}}{{{{\epsilon }_{n}}}}\mathcal{Z}({{y}_{n}}).$$

Therefore, the whole rotation matrix has the form

$$S({{\epsilon }_{n}},{{k}_{m}}) = \frac{{\mathcal{Z}({{y}_{n}})L}}{{2\pi {{{v}}_{c}}(n - m)}}\left[ {1 + \frac{{\flat + 2}}{\flat }\frac{m}{n}} \right].$$

(56)

From unitarity condition of rotation,

$$\frac{1}{L}\sum\limits_{ - {{N}_{1}};m \ne 0}^{{{N}_{1}}} {S({{\epsilon }_{{ - n}}};{{p}_{{ - m}}})S({{\epsilon }_{n}};{{p}_{m}}) = L} $$

(57)

we can obtain the following expression for $\mathcal{Z}$(y_n):

$$\begin{gathered} - {{\left( {\frac{1}{{2\pi {{{v}}_{c}}}}} \right)}^{2}}\mathcal{Z}({{y}_{n}})\mathcal{Z}({{y}_{{ - n}}}) \\ \times \sum\limits_{ - {{N}_{1}};m \ne 0}^{{{N}_{1}}} {{{{\left[ {1 + \frac{{\flat + 2}}{\flat }\frac{m}{n}} \right]}}^{2}}{\text{/}}{{{(n - m)}}^{2}} = 1.} \\ \end{gathered} $$

(58)

To calculate the last term, we introduce

$$\begin{gathered} {{S}_{0}}(y,{{N}_{1}}) = \sum\limits_{ - {{N}_{1}},m \ne 0}^{{{N}_{1}}} {\frac{1}{{{{{(y - m)}}^{2}}}} = - {{\partial }_{y}}\mathcal{A}(y)} \\ = - {{\partial }_{y}}[(B(y) + 2{{N}_{1}}){\text{/}}y], \\ {{S}_{1}}(y,{{N}_{1}}) = \sum\limits_{ - {{N}_{1}}}^{{{N}_{1}}} {\frac{m}{{{{{(y - m)}}^{2}}}} = - {{\partial }_{y}}[yA(y,{{N}_{1}})],} \\ {{S}_{2}}(y,{{N}_{1}}) = \sum\limits_{ - {{N}_{1}}}^{{{N}_{1}}} {\frac{{{{m}^{2}}}}{{{{{(y - m)}}^{2}}}} = - {{\partial }_{y}}[yB(y)]} \\ = - {{\partial }_{y}}[ - 2y{{N}_{1}} + {{y}^{2}}\mathcal{A}(y)]. \\ \end{gathered} $$

(59)

In terms of these sums, Eq. (58) can be written in form

$$\begin{gathered} - {{\left( {\frac{1}{{2\pi {{{v}}_{c}}}}} \right)}^{2}}\mathcal{Z}({{y}_{n}})\mathcal{Z}({{y}_{{ - n}}})\left[ {{{S}_{0}}(y,{{N}_{1}})_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}^{{^{{^{{^{{}}}}}}}}} \right. \\ + \left. {2\frac{{\flat + 2}}{{y\flat }}{{S}_{1}}\left( {y,{{N}_{1}}} \right) + {{{\left( {\frac{{\flat + 2}}{{y\flat }}} \right)}}^{2}}{{S}_{2}}\left( {y,{{N}_{1}}} \right)} \right] = 1, \\ \end{gathered} $$

The last factor in this expression is given by

$$\begin{gathered} \text{[}...] = - {{\partial }_{y}}\mathcal{A}(y){{\left( {1 + \frac{{\flat + 2}}{\flat }} \right)}^{2}} \\ - 2\mathcal{A}(y)\frac{{\flat + 2}}{{y\flat }}\left( {1 + \frac{{\flat + 2}}{\flat }} \right) + 2{{N}_{1}}{{\left( {\frac{{\flat + 2}}{{y\flat }}} \right)}^{2}}, \\ \end{gathered} $$

where

$$\mathcal{A}(y) = \pi \cot (\pi y) - \frac{1}{y},\quad \partial \mathcal{A}(y) = - \frac{{{{\pi }^{2}}}}{{{{{\sin }}^{2}}\pi y}} + \frac{1}{{{{y}^{2}}}}.$$

It is more convenient for us to write sin²πy in the form following from the exact equation for the spectrum:

$$\begin{gathered} \left[ {1 + {{{\left( {\frac{\mathcal{P}}{{\pi y}}} \right)}}^{2}}} \right]{{\sin }^{2}}\pi y = 1, \\ {{\mathcal{P}}^{2}} = {{\left[ {{{N}_{1}}\frac{{{{{(\flat + 2)}}^{2}}}}{{2(\flat + 1)}} + 1} \right]}^{2}}. \\ \end{gathered} $$

(60)

Ultimately, we obtain

$$\begin{gathered} \text{[}...] = \frac{1}{{{{y}^{2}}}}\left[ {\frac{{4{{{(\flat + 1)}}^{2}}}}{{{{\flat }^{2}}}}({{\mathcal{P}}^{2}} - 1) - 2{{N}_{1}}\frac{{(\flat + 2)}}{{{{\flat }^{2}}}}(\flat + 1)} \right. \\ \left. { + \,2{{N}_{1}}\frac{{{{{(\flat + 2)}}^{2}}}}{{{{\flat }^{2}}}}} \right] + 4{{\pi }^{2}}\frac{{{{{(\flat + 1)}}^{2}}}}{{{{\flat }^{2}}}}. \\ \end{gathered} $$

We are interested in the thermodynamic limit of the expression for the rotation matrix. We assume that N₁|$\flat$ + 2| ≫ 1. In this case, $\mathcal{P}$ ~ N₁; therefore, the second and third terms in this expression acquire post-thermodynamic corrections of order O(1/N₁). The last term leads to the regularization of the expression for $\mathcal{Z}$ for small y_n and should be retained. Therefore, we have

$$\begin{gathered} \text{[}...] = {{\left( {\frac{{2\pi (\flat + 1)}}{{y\flat }}} \right)}^{2}}[\tilde {N}_{1}^{2} + {{y}^{2}}], \\ {{{\tilde {N}}}_{1}} = {{N}_{1}}{{(\flat + 2)}^{2}}{\text{/}}2\pi (\flat + 1), \\ \mathcal{Z}({{\epsilon }_{n}}) = \frac{{\flat {{{v}}_{c}}}}{{\flat + 1}}\frac{{{{y}_{n}}}}{{{{{[\tilde {N}_{1}^{2} + y_{n}^{2}]}}^{{1/2}}}}}. \\ \end{gathered} $$

(61)

Substituting this expression into Eq. (56), we obtain the final expression for the rotation matrix:

$$S({{\epsilon }_{n}},{{k}_{m}}) = \frac{{L{{z}_{0}}}}{{{{{[\tilde {N}_{1}^{2} + y_{n}^{2}]}}^{{1/2}}}}}\frac{{{{y}_{n}}}}{{{{y}_{n}} - m}}\left[ {1 + \frac{{\flat + 2}}{2}\frac{m}{{{{y}_{n}}}}} \right],$$

(62)

where z₀ = $\flat$/2π($\flat$ + 1). It can be shown by direct calculation that this expression satisfies unitarity condition (57).

For n, N₁ ≫ 1, the exact expression for spectrum (60) is transformed to

$${{y}_{n}} = n\left( {1 + \frac{1}{{\pi n}}\arcsin \frac{{\pi n}}{{\sqrt {{{\mathcal{P}}^{2}} + {{\pi }^{2}}{{n}^{2}}} }}} \right)\sim n + \frac{n}{\mathcal{P}}$$

(63)

(in this notation, arcsin is taken in the sense of the principal value).

APPENDIX C

1.1 Calculation of Correlator $\mathcal{S}$

Substituting expressions (36)–(38) for one-particle Green’s functions into (39), we obtain the following expression for the regularized correlator of pseudoscalar density:

$$\begin{gathered} \mathcal{S}(t) = - {{(2{{\gamma }_{s}})}^{2}}\Phi _{0}^{2}(\flat )\Phi _{0}^{2}( - \flat ) \\ \, \times \int {(dkd{{k}_{1}})\exp (it{{{v}}_{c}}(k + {{k}_{1}})){{{(k{{\Phi }_{0}}(\flat ) - {{k}_{1}}{{\Phi }_{0}}( - \flat ))}}^{2}}} \\ \, \times [\theta (t)\theta (k) - \theta ( - t)\theta ( - k)][\theta (t)\theta ({{k}_{1}}) \\ \, - \theta ( - t)\theta ( - {{k}_{1}})]\mathcal{F}(k,{{M}_{{PV}}})\mathcal{F}({{k}_{1}},{{M}_{{PV}}}). \\ \end{gathered} $$

(64)

In further calculations, it will be convenient to take advantage of the fact that except for factor (kΦ₀($\flat$) – k₁Φ₀(–$\flat$))², the remaining integrand is symmetric relative to replacement k ↔ k₁. Then we write this factor in form

$$\begin{gathered} {{({{\Phi }_{0}}(\flat )k - {{\Phi }_{0}}( - \flat ){{k}_{1}})}^{2}} \\ \to \frac{1}{2}({{\Phi }_{0}}{{(\flat )}^{2}}\, + \,{{\Phi }_{0}}{{( - \flat )}^{2}}){{(k\, + \,{{k}_{1}})}^{2}}\, - \,k{{k}_{1}}{{({{\Phi }_{0}}(\flat )\, + \,{{\Phi }_{0}}( - \flat ))}^{2}}. \\ \end{gathered} $$

Therefore, the entire expression can be written as

$$\begin{gathered} \mathcal{S}(t) = - {{(2{{\gamma }_{s}})}^{2}}\Phi _{0}^{2}(\flat )\Phi _{0}^{2}( - \flat )\int {(dkd{{k}_{1}})\exp (it{{{v}}_{c}}(k + {{k}_{1}}))} \\ \times \left[ {\frac{1}{2}(\Phi _{0}^{2}(\flat ) + \Phi _{0}^{2}( - \flat )){{{(k + {{k}_{1}})}}^{2}} - k{{k}_{1}}{{{({{\Phi }_{0}}(\flat ) + {{\Phi }_{0}}( - \flat ))}}^{2}}} \right] \\ \, \times [\theta (t)\theta (k)\theta ({{k}_{1}}) + \theta ( - t)\theta ( - k)\theta ( - {{k}_{1}})] \\ \, \times \mathcal{F}(k,{{M}_{{PV}}})F({{k}_{1}},{{M}_{{PV}}}). \\ \end{gathered} $$

(65)

For calculating the renormalized coefficient of reflection of fermions from an impurity, we will need the real part of the Fourier transform of this expression. Performing substitution t; k_1,2 → –t: –k_1,2 in the last term of the resulting expression, we obtain

$$\begin{gathered} \operatorname{Re} \mathcal{S}(\omega ) = - 2{{(2{{\gamma }_{s}})}^{2}}\Phi _{0}^{2}(\flat )\Phi _{0}^{2}( - \flat ) \\ \times \operatorname{Re} \int\limits_0^\infty {dt\cos (\omega t)\int\limits_0^\infty {(dkd{{k}_{1}})\exp (it{{{v}}_{c}}(k + {{k}_{1}}))} } \\ \times \left[ {\frac{1}{2}(\Phi _{0}^{2}(\flat ) + \Phi _{0}^{2}( - \flat )){{{(k + {{k}_{1}})}}^{2}} - k{{k}_{1}}{{{({{\Phi }_{0}}(\flat ) + {{\Phi }_{0}}( - \flat ))}}^{2}}} \right] \\ \times \,\mathcal{F}(k,{{M}_{{PV}}})\mathcal{F}({{k}_{1}},{{M}_{{PV}}}). \\ \end{gathered} $$

(66)

It can be seen from this expression that in the integrand with respect to k, two structures (with (k + k₁)² and kk₁) have been separated.

(i) The term with (k + k₁)². We can lower the power of the denominator for function $\mathcal{F}$(k, M_PV) by writing the square of the expression appearing in $\mathcal{F}$(k, M_PV) as a derivative with respect to M_PV, and (k + k₁)² as $\partial _{t}^{2}$ of the exponential. Therefore, the entire integral with respect to dkdk₁ can be written as

$$\begin{gathered} - \frac{1}{{{{{(4\pi {{{v}}_{c}})}}^{2}}}}\frac{{{{\partial }^{2}}}}{{\partial {{t}^{2}}}} \\ \times {{\left[ {\int {dk\exp (it{{{v}}_{{ck}}})\left( {M_{{PV}}^{3}\frac{\partial }{{\partial {{M}_{{PV}}}}}\frac{1}{{{{k}^{2}}{v}_{c}^{2} + M_{{PV}}^{2}}}} \right)} } \right]}^{2}}, \\ \end{gathered} $$

and the entire first term is given by

$$\begin{gathered} \operatorname{Re} {{\mathcal{S}}_{1}}(w) = {{(2{{\gamma }_{s}})}^{2}}\Phi _{0}^{2}(\flat )\Phi _{0}^{2}( - \flat )(\Phi _{0}^{2}(\flat ) + \Phi _{0}^{2}( - \flat )) \\ \times \frac{{M_{{PV}}^{6}}}{{{{{(4\pi {{{v}}_{c}})}}^{2}}}}{\text{Re}}\int\limits_0^\infty {{\text{cos}}\omega tdt} \frac{{{{\partial }^{2}}}}{{\partial {{t}^{2}}}}{{\left[ {\frac{\partial }{{\partial {{M}_{{PV}}}}}\int\limits_0^\infty {dk\frac{{\exp (it{{{v}}_{c}}k)}}{{{{k}^{2}}{v}_{c}^{2} + M_{{PV}}^{2}}}} } \right]}^{2}}. \\ \end{gathered} $$

It is convenient to express this equality in terms of Laplace integral (${{\mathcal{L}}_{a}}$) and Raabe integral (${{\mathcal{R}}_{a}}$) [22]:

$$\begin{gathered} \int\limits_0^\infty {dk\frac{{\exp (it{{{v}}_{c}}k)}}{{{{k}^{2}}{v}_{c}^{2} + M_{{PV}}^{2}}}} \\ = \frac{{{{{v}}_{c}}}}{{{{M}_{{PV}}}}}({{\mathcal{L}}_{a}}({{M}_{{PV}}}t) + i{{\mathcal{R}}_{a}}({{M}_{{PV}}}t)). \\ \end{gathered} $$

The Laplace integral exponentially decreases for M_PVt ≫ 1 and tends to a constant value in the opposite limiting case. The Raabe integral for M_PVt ≫ 1 decreases as 1/M_PVt, while for small value of the argument, we have

$${{\mathcal{R}}_{a}}({{M}_{{PV}}}t) \simeq - ({{\gamma }_{E}} + \ln ({{M}_{{PV}}}t)){{M}_{{PV}}}t.$$

This allows us to write the previous expression in form

$$\begin{gathered} \operatorname{Re} {{\mathcal{S}}_{1}}(w) = \frac{{\gamma _{s}^{2}}}{{{{{(2\pi )}}^{2}}}}\Phi _{0}^{2}(\flat )\Phi _{0}^{2}( - \flat )(\Phi _{0}^{2}(\flat ) + \Phi _{0}^{2}( - \flat ))M_{{PV}}^{6} \\ \times \operatorname{Re} \int\limits_0^\infty {{\text{cos}}\omega tdt} \partial _{t}^{2}{{\left[ {{{\partial }_{{{{M}_{{PV}}}}}}\frac{1}{{{{M}_{{PV}}}}}{\kern 1pt} ({{\mathcal{L}}_{a}}({{M}_{{PV}}}t)\, + \,i{{\mathcal{R}}_{a}}({{M}_{{PV}}}t){\kern 1pt} )} \right]}^{2}}. \\ \end{gathered} $$

Further, we must take into account the fact that although we are interested in the limit M_PV → ∞, dimensionless time τ = M_PVt can take any value in this case. After differentiation with respect to M_PV, we can pass to dimensionless time

$$\begin{gathered} {{\partial }_{{{{M}_{{PV}}}}}}\frac{1}{{{{M}_{{PV}}}}}({{\mathcal{L}}_{a}}({{M}_{{PV}}}t) + i{{\mathcal{R}}_{a}}({{M}_{{PV}}}t)) \\ = \frac{1}{{M_{{PV}}^{2}}}\left[ { - 1 + \tau \frac{\partial }{{\partial \tau }}} \right]({{\mathcal{L}}_{a}}(\tau ) + i{{\mathcal{R}}_{a}}(\tau )). \\ \end{gathered} $$

It should be noted that the entire expression for the correlator has no integrable singularities for τ → 0 and converges well for large τ:

$$ - {{\mathcal{R}}_{a}}(\tau ) + \tau \frac{\partial }{{\partial \tau }}{{\mathcal{R}}_{a}}(\tau ) = - \frac{2}{\tau },\quad \tau \gg 1,$$

$$ - {{\mathcal{R}}_{a}}(\tau ) + \tau \frac{\partial }{{\partial \tau }}{{\mathcal{R}}_{a}}(\tau ) = - \tau ,\quad \tau \ll 1.$$

For this reason, in calculating the real part of the frequency-dependent contribution, which appears from this term in the real part of expression (66), we can integrate twice by parts. As a result, the entire expression turns out to be proportional to

$$\begin{gathered} {{M}_{{PV}}}{{\omega }^{2}}\int\limits_0^\infty {d\tau \cos (\omega \tau {\text{/}}{{M}_{{PV}}})} \\ \times \left\{ {{{{\left[ {\left( {1 - \tau \frac{\partial }{{\partial \tau }}} \right){{\mathcal{L}}_{a}}(\tau )} \right]}}^{2}} - {{{\left[ {\left( {1 - \tau \frac{\partial }{{\partial \tau }}} \right){{\mathcal{R}}_{a}}(\tau )} \right]}}^{2}}} \right\}. \\ \end{gathered} $$

Considering that the integrand rapidly decreases for τ ≫ 1 (i.e., cos(ωτ/M_PV) can be replaced by 1), we see that the entire integral with respect to τ gives a coefficient of M_PVτ² on the order of 1. As a result, the frequency-dependent part of this term is

$$\operatorname{Re} \delta {{\mathcal{S}}_{1}}(\omega ) = - \gamma _{s}^{2}{{F}_{1}}({{V}_{{{\text{imp}}}}}){{M}_{{PV}}}{{\omega }^{2}},$$

(67)

$$\begin{gathered} {{F}_{1}}({{V}_{{{\text{imp}}}}}) = \frac{1}{{{{{(2\pi )}}^{2}}}}\Phi _{0}^{2}(\flat )\Phi _{0}^{2}( - \flat )(\Phi _{0}^{2}(\flat ) + \Phi _{0}^{2}( - \flat )) \\ \times \int\limits_0^\infty {d\tau \left\{ {{{{\left[ {\left( {1 - \tau \frac{\partial }{{\partial \tau }}} \right){{\mathcal{L}}_{a}}(\tau )} \right]}}^{2}} - {{{\left[ {\left( {1 - \tau \frac{\partial }{{\partial \tau }}} \right){{\mathcal{R}}_{a}}(\tau )} \right]}}^{2}}} \right\}} . \\ \end{gathered} $$

(68)

(ii) The term with kk₁. In this case, the integrals with respect to k are proportional to

$$\begin{gathered} - \frac{1}{{{{{(4\pi {{{v}}_{c}})}}^{2}}}}\left[ {\frac{\partial }{{\partial t}}\int {dk\exp (it{{{v}}_{c}}k)} } \right. \\ {{\left. { \times \left( {M_{{PV}}^{3}\frac{\partial }{{\partial {{M}_{{PV}}}}}\frac{1}{{{{k}^{2}}{v}_{c}^{2} + M_{{PV}}^{2}}}} \right)} \right]}^{2}}. \\ \end{gathered} $$

Noting that

$$\begin{gathered} {{\partial }_{t}}{{\partial }_{{{{M}_{{PV}}}}}}\frac{1}{{{{M}_{{PV}}}}}({{\mathcal{L}}_{a}}({{M}_{{PV}}}t) + i{{\mathcal{R}}_{a}}({{M}_{{PV}}}t)) \\ = \frac{\tau }{{{{M}_{{PV}}}}}\frac{{{{\partial }^{2}}}}{{\partial {{\tau }^{2}}}}({{\mathcal{L}}_{a}}(\tau ) + i{{\mathcal{R}}_{a}}(\tau )), \\ \end{gathered} $$

we can conveniently write this expression in form

$$\operatorname{Re} \delta {{\mathcal{S}}_{2}}(\omega ) = \gamma _{s}^{2}{{F}_{2}}({{V}_{{{\text{imp}}}}}){{M}_{{PV}}}{{\omega }^{2}},$$

(69)

$$\begin{gathered} {{F}_{2}}({{V}_{{{\text{imp}}}}}) = \frac{1}{{{{{(2\pi )}}^{2}}}}{{\Phi }_{0}}{{(\flat )}^{2}}{{\Phi }_{0}}{{( - \flat )}^{2}}{{({{\Phi }_{0}}(\flat ) + {{\Phi }_{0}}( - \flat ))}^{2}} \\ \times \int\limits_0^\infty {{{\tau }^{4}}d\tau \left\{ {{{{\left[ {\frac{{{{\partial }^{2}}}}{{\partial {{\tau }^{2}}}}{{\mathcal{L}}_{a}}(\tau )} \right]}}^{2}} - {{{\left[ {\frac{{{{\partial }^{2}}}}{{\partial {{\tau }^{2}}}}{{\mathcal{R}}_{a}}(\tau )} \right]}}^{2}}} \right\}} . \\ \end{gathered} $$

(70)

In this transition, we have again taken into account the fact that the integrand rapidly decrease for τ ≫ 1. This enabled us to confine analysis to the first term of expansion sin²(ωτ/2M_PV) for the frequency-dependent part of the correlator.

Ultimately, we have verified that the real part of the loop diagram is proportional to –$\gamma _{s}^{2}$M_PVω² with coefficient

$$F({{V}_{{{\text{imp}}}}}) = {{F}_{1}}({{V}_{{{\text{imp}}}}}) - {{F}_{2}}({{V}_{{{\text{imp}}}}}).$$

In all these calculations, it is important that all our integrands do not diverge for τ → 0, and the integrals themselves are determined by domain τ ~ 1.