Dimensional Reduction and its Breakdown in the Driven Random Field O(N) Model

Abstract

We investigate the critical behavior of the driven random field O(N) model. First, from an intuitive argument, we derive a new dimensional reduction property, which predicts that the critical behavior of the driven disordered system at zero temperature is the same as that of the lower dimensional pure system in equilibrium. However, this dimensional reduction breaks down in low enough dimensions due to a nonperturbative effect associated with meta-stable states. By employing the nonperturbative formalism of the functional renormalization group, we derive the flow equation of the renormalized disorder correlator and clarify the condition that the dimensional reduction fails. We also calculate the critical exponents near three dimensions.We investigate the critical behavior of the driven random field O(N) model. First, from an intuitive argument, we derive a new dimensional reduction property, which predicts that the critical behavior of the driven disordered system at zero temperature is the same as that of the lower dimensional pure system in equilibrium. However, this dimensional reduction breaks down in low enough dimensions due to a nonperturbative effect associated with meta-stable states. By employing the nonperturbative formalism of the functional renormalization group, we derive the flow equation of the renormalized disorder correlator and clarify the condition that the dimensional reduction fails. We also calculate the critical exponents near three dimensions.

4.6 Appendix

4.1.1 4.6.1 Propagators

In this section, we show the expression of the one-replica propagator Eq. (4.42). The functional derivative $\Gamma _{1,k}^{(2)}[\Psi ]$ is evaluated for a uniform field ${\varvec{\psi }}_{rt} \equiv {}^t(\psi ^1,\ldots ,\psi ^N)$ and ${\varvec{\hat{\psi }}}_{rt} \equiv {\varvec{0}}$. $\Gamma _{1,k}[\Psi ]$ is given by Eq. (4.45). For simplicity, we omit the subscript k in the following. We introduce $P(\mathbf {q},\omega ;{\varvec{\psi }})$ as

$$\begin{aligned} \mathrm {P}[\Psi ]_{q_1,q_2} = P(\mathbf {q}_1,\omega _1;{\varvec{\psi }}) (2\pi )^{D+1} \delta (\mathbf {q}_1+\mathbf {q}_2) \delta (\omega _1+\omega _2), \end{aligned}$$

(4.151)

where $q=(\mathbf {q},\omega )$. $P(\mathbf {q},\omega ;{\varvec{\psi }})$ is a $2N \times 2N$ matrix, thus we write its element as $P_{ij}^{\mu \nu }$, where $i,j=1,2$ represent the two conjugate fields $\psi$ and $\hat{\psi }$, and $\mu , \nu =1,\ldots ,N$ are the field component indices.

$P_{ij}^{\mu \nu }(\mathbf {q},\omega ;{\varvec{\psi }})$ can be written as

$$\begin{aligned} P_{ij}^{\mu \nu }(\mathbf {q},\omega ;{\varvec{\psi }}) = P^{(T)}_{ij}(\mathbf {q},\omega ;\rho ) \biggl ( \delta ^{\mu \nu } - \frac{ \psi ^{\mu } \psi ^{\nu }}{2\rho } \biggr ) + P^{(L)}_{ij}(\mathbf {q},\omega ;\rho ) \frac{ \psi ^{\mu } \psi ^{\nu }}{2\rho }, \end{aligned}$$

(4.152)

where the transverse and longitudinal parts are given by

$$\begin{aligned} P^{(T)}_{11}(\mathbf {q},\omega ;\rho )= & {} \frac{2 X T}{D_0(\mathbf {q},\omega ;\rho )}, \nonumber \\ P^{(T)}_{12}(\mathbf {q},\omega ;\rho )= & {} \frac{M_0(\mathbf {q};\rho )-i(\omega -q_x v)X}{D_0(\mathbf {q},\omega ;\rho )}, \nonumber \\ P^{(T)}_{21}(\mathbf {q},\omega ;\rho )= & {} \frac{M_0(\mathbf {q};\rho )+i(\omega -q_x v)X}{D_0(\mathbf {q},\omega ;\rho )}, \nonumber \\ P^{(T)}_{22}(\mathbf {q},\omega ;\rho )= & {} 0, \end{aligned}$$

(4.153)

$$\begin{aligned} P^{(L)}_{11}(\mathbf {q},\omega ;\rho )= & {} \frac{2 X T}{D_1(\mathbf {q},\omega ;\rho )}, \nonumber \\ P^{(L)}_{12}(\mathbf {q},\omega ;\rho )= & {} \frac{M_1(\mathbf {q};\rho )-i(\omega -q_x v)X}{D_1(\mathbf {q},\omega ;\rho )}, \nonumber \\ P^{(L)}_{21}(\mathbf {q},\omega ;\rho )= & {} \frac{M_1(\mathbf {q};\rho )+i(\omega -q_x v)X}{D_1(\mathbf {q},\omega ;\rho )}, \nonumber \\ P^{(L)}_{22}(\mathbf {q},\omega ;\rho )= & {} 0. \end{aligned}$$

(4.154)

We have defined the following notations:

$$\begin{aligned} M_0(\mathbf {q};\rho )= & {} Z_{\parallel } q_x^2 + Z_{\perp } q_{\perp }^2 + R_k(\mathbf {q}) - F(\rho ), \nonumber \\ M_1(\mathbf {q};\rho )= & {} Z_{\parallel } q_x^2 + Z_{\perp } q_{\perp }^2 + R_k(\mathbf {q}) - F(\rho ) - 2\rho F'(\rho ), \nonumber \\ D_0(\mathbf {q},\omega ;\rho )= & {} M_0(\mathbf {q};\rho )^2 + ( \omega - q_x v )^2 X^2, \nonumber \\ D_1(\mathbf {q},\omega ;\rho )= & {} M_1(\mathbf {q};\rho )^2 + ( \omega - q_x v )^2 X^2. \end{aligned}$$

(4.155)

In Sect. 4.3.5, we also use the simplified notation $D(\mathbf {q};\rho )=D(\mathbf {q},\omega =0;\rho )$.

To express the RG equations in a compact form, we introduce the following integrals:

$$\begin{aligned} L_{n}^{(T)}(\rho ) = - \int \limits _{\mathbf {q}} \partial _l R_k(\mathbf {q}) M_0(\mathbf {q};\rho )^{-n}, \end{aligned}$$

(4.156)

$$\begin{aligned} I_{n n'}^{(T)}(\rho _1,\rho _2) = - \int \limits _{\mathbf {q}} \partial _l R_k(\mathbf {q}) P^{(T)}_{21}(\mathbf {q};\rho _1)^{n} P^{(T)}_{12}(\mathbf {q};\rho _2)^{n'}, \end{aligned}$$

(4.157)

$$\begin{aligned} J_{n n'}^{(T)}(\rho _1,\rho _2) = - \int \limits _{\mathbf {q}} \partial _l R_k(\mathbf {q}) P^{(T)}_{21}(\mathbf {q};\rho _1)^{n} P^{(T)}_{21}(\mathbf {q};\rho _2)^{n'}, \end{aligned}$$

(4.158)

where $\partial _l=-k\partial _k$ and the frequency $\omega$ in Eqs. (4.157) and (4.158) are set to zero. The integrals for the longitudinal mode are also defined by replacing $M_0$ and $D_0$ in Eqs. (4.156)–(4.158) with $M_1$ and $D_1$, respectively.

4.1.2 4.6.2 Flow Equation for $F_k(\rho )$

In this section, the flow equation for $F_k(\rho )$ is derived, which is given by Eq. (4.55). To do this, the exact flow equation for $\Gamma _1^{(1)}$ is required. It is convenient to introduce a graphical representation. The flow equation for $\Gamma _1$ is rewritten as

$$\begin{aligned} \partial _l \Gamma _{1} = \frac{1}{2} [ \gamma _{1,a} + \gamma _{1,b} ], \end{aligned}$$

(4.159)

where $\gamma _{1,a}$ and $\gamma _{1,b}$ are given by Eqs. (3.154) and (3.155), respectively. A scale parameter $l=-\ln (k/\Lambda )$ has been introduced. The flow equation for $\Gamma _{1;\hat{\psi }^1}^{(1)}=\delta \Gamma _1/\delta \hat{\psi }^1$ is then written as

$$\begin{aligned} \partial _l \Gamma _{1;\hat{\psi }^1}^{(1)} = \frac{1}{2} [ -\gamma _{1,a-1}^{(1)} - 2 \gamma _{1,b-1}^{(1)} + 2 \gamma _{1,b-2}^{(1)} ]. \end{aligned}$$

(4.160)

where $\gamma _{1,a-1}^{(1)}$, $\gamma _{1,b-1}^{(1)}$, and $\gamma _{1,b-2}^{(1)}$ are given in Fig. 4.4. For example, $\gamma _{1,b-1}^{(1)}$ is written as

$$\begin{aligned} \gamma _{1,b-1}^{(1)} = \mathrm {Tr} \partial _l {\varvec{\mathrm {R}}}_k(\mathbf {q}) \mathrm {P}[\Psi ] \Gamma _2^{(11)}[\Psi ,\Psi ] \mathrm {P}[\Psi ] \Gamma _{1;\hat{\psi }^1}^{(3)}[\Psi ] \mathrm {P}[\Psi ], \end{aligned}$$

(4.161)

where $\Gamma _{1;\hat{\psi }^1}^{(3)}=\delta \Gamma _{1}^{(2)}/\delta \hat{\psi }^1$ and $\mathrm {Tr}$ represents an integration over momentum and frequency as well as a sum over the field component and the two conjugate fields. All functional derivatives are evaluated for a uniform field ${\varvec{\psi }}_{rt} \equiv {}^t(\sqrt{2\rho },0,\ldots ,0)$, ${\varvec{\hat{\psi }}}_{rt} \equiv {}^t(0,\ldots ,0)$.

Fig. 4.4

Reprinted figure with permission from Ref. [1]. Copyright (2017) by the American Physical Society. https://doi.org/10.1103/PhysRevB.96.184202

Graphical representations for the flow equations of $\Gamma _{1}^{(1)}$.

The following notation is introduced:

$$\begin{aligned} \Gamma _{1; \psi ^2 \psi ^2 \hat{\psi }^1}^{(3)}(q_1,q_2,q_3) = \frac{\delta ^2 \Gamma _1[\Psi ]}{\delta \psi ^2(q_1) \delta \psi ^2(q_2) \delta \hat{\psi }^1(q_3)}, \nonumber \\ \Gamma _{2; \psi _1^2 \hat{\psi }_1^2 \hat{\psi }_2^1}^{(21)}(q_1,q_2,q_3) = \frac{\delta ^2 \Gamma _2[\Psi _1,\Psi _2]}{\delta \psi _1^2(q_1) \delta \hat{\psi }_1^2(q_2) \delta \hat{\psi }_2^1(q_3)}. \nonumber \end{aligned}$$

From Eqs. (4.45) and (4.46), they are calculated as

$$\begin{aligned} \Gamma _{1;\psi ^1 \psi ^1 \hat{\psi }^1}^{(3)}(q_1,q_2,q_3) =&- \sqrt{2\rho } [3F'(\rho )+2\rho F''(\rho )] (2\pi )^{D+1}\delta (\mathbf {q}_1+\mathbf {q}_2+\mathbf {q}_2) \nonumber \\&\times \delta (\omega _1+\omega _2+\omega _2), \nonumber \end{aligned}$$

$$\begin{aligned} \Gamma _{1;\psi ^{\nu } \psi ^{\nu } \hat{\psi }^1}^{(3)}(q_1,q_2,q_3) =&- \sqrt{2\rho } F'(\rho ) (2\pi )^{D+1}\delta (\mathbf {q}_1+\mathbf {q}_2+\mathbf {q}_2) \nonumber \\&\times \delta (\omega _1+\omega _2+\omega _2), \,\,\,(\nu =2,\ldots ,N), \nonumber \end{aligned}$$

$$\begin{aligned} \Gamma _{2; \hat{\psi }_1^1 \hat{\psi }_2^1}^{(11)}(q_1,q_2)\bigr |_{\Psi _1=\Psi _2=\Psi } = \Delta _L(\rho ) (2\pi )^{D+2}\delta (\mathbf {q}_1+\mathbf {q}_2) \delta (\omega _1) \delta (\omega _2), \nonumber \end{aligned}$$

$$\begin{aligned} \Gamma _{2; \hat{\psi }_1^{\nu } \hat{\psi }_2^{\nu }}^{(11)}(q_1,q_2)\bigr |_{\Psi _1=\Psi _2=\Psi } = \Delta _T(\rho ) (2\pi )^{D+2}\delta (\mathbf {q}_1+\mathbf {q}_2) \delta (\omega _1) \delta (\omega _2), \nonumber \\ (\nu =2,\ldots ,N), \nonumber \end{aligned}$$

$$\begin{aligned} \Gamma _{2; \psi _1^1 \hat{\psi }_1^1 \hat{\psi }_2^1}^{(21)}(q_1,q_2,q_3)\bigr |_{\Psi _1=\Psi _2=\Psi } =&\frac{1}{2} \sqrt{2 \rho } \Delta _L'(\rho ) (2\pi )^{D+2}\delta (\mathbf {q}_1+\mathbf {q}_2+\mathbf {q}_3) \nonumber \\&\times \delta (\omega _1+\omega _2) \delta (\omega _3), \nonumber \end{aligned}$$

$$\begin{aligned} \Gamma _{2; \psi _1^{\nu } \hat{\psi }_1^1 \hat{\psi }_2^{\nu }}^{(21)}(q_1,q_2,q_3)\bigr |_{\Psi _1=\Psi _2=\Psi } =&\frac{1}{\sqrt{2 \rho }} [ \Delta _{21}(\rho )+\Delta _{11}(\rho ) ] (2\pi )^{D+2}\delta (\mathbf {q}_1+\mathbf {q}_2+\mathbf {q}_3) \nonumber \\&\times \delta (\omega _1+\omega _2) \delta (\omega _3), \,\,\,(\nu =2,\ldots ,N), \nonumber \end{aligned}$$

where we have used the notations $\Delta _{\ldots }(\rho )=\Delta _{\ldots }(\rho ,\rho ,z=1)$, $\Delta _T(\rho ) = \Delta ^{22}({\varvec{\psi }},{\varvec{\psi }}) = \Delta _{00}(\rho )$, $\Delta _L(\rho ) = \Delta ^{11}({\varvec{\psi }},{\varvec{\psi }})=\Delta _{00}(\rho )+\Delta _{12}(\rho )+\Delta _{21}(\rho )+\Delta _{11}(\rho )+\Delta _{22}(\rho )$. From these expressions, we obtain

$$\begin{aligned} \gamma _{1,a-1}^{(1)} = \sqrt{2\rho } T \{ [3F'(\rho )+2\rho F''(\rho )] L_2^{(L)}(\rho ) + (N-1) F_k'(\rho )L_2^{(T)}(\rho ) \}, \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{1,b-1}^{(1)} = \sqrt{2\rho } \{ [3F'(\rho )+2\rho F''(\rho )] \Delta _L(\rho ) I_{12}^{(L)}(\rho ) + (N-1) F'(\rho ) \Delta _T(\rho ) I_{12}^{(T)}(\rho ) \}, \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{1,b-2}^{(1)} = - \sqrt{2\rho } \biggl \{ \frac{1}{2} \Delta _L'(\rho ) J_{11}^{(L)}(\rho ) + (N-1) \frac{1}{2 \rho } [\Delta _{21}(\rho )+\Delta _{11}(\rho )] J_{11}^{(T)}(\rho ) \biggr \}, \nonumber \end{aligned}$$

where we have already calculated the $\omega$-integral. The integrals L, I, and J are defined by Eqs. (4.156), (4.157), and (4.158), respectively, and simplified notations such as $I_{nn'}^{(T)}(\rho )=I_{nn'}^{(T)}(\rho ,\rho )$ are used. From Eqs. (4.55) and (4.160), we have the flow equation for $F(\rho )$,

$$\begin{aligned} \partial _l F(\rho ) = \partial _l F^{(1)}(\rho ) + \partial _l F^{(2)}(\rho ), \end{aligned}$$

(4.162)

where $\partial _l F^{(1)}(\rho )$ and $\partial _l F^{(2)}(\rho )$ are the contributions from the one and two-replica parts, respectively,

$$\begin{aligned} \partial _l F^{(1)}(\rho ) = \frac{1}{2} T \{ [3 F'(\rho ) + 2 \rho F''(\rho )] L_2^{(L)}(\rho ) + (N-1) F'(\rho ) L_2^{(T)}(\rho ) \}, \end{aligned}$$

(4.163)

$$\begin{aligned} \partial _l F^{(2)}(\rho ) =&[3 F'(\rho ) + 2 \rho F''(\rho )] \Delta _L(\rho ) I_{12}^{(L)}(\rho ) + (N-1) F'(\rho ) \Delta _T(\rho ) I_{12}^{(T)}(\rho ) \nonumber \\&- \frac{1}{2} \Delta _L'(\rho ) J_{11}^{(L)}(\rho ) - (N-1) \frac{1}{2 \rho } [ \Delta _{21}(\rho )+\Delta _{11}(\rho ) ] J_{11}^{(T)}(\rho ). \end{aligned}$$

(4.164)

It can be easily checked that, in the equilibrium case ($v=0$), the equation can be reduced to that of the RFO(N)M, which is given in Ref. [9].

4.1.3 4.6.3 Flow Equation for $\Delta _k({\varvec{\psi }}_1,{\varvec{\psi }}_2)$

In this section, we derive the RG equation for $\Delta _k({\varvec{\psi }}_1,{\varvec{\psi }}_2)$, which is given by Eq. (4.57). To do this, the exact flow equation for $\Gamma _2^{(11)}$ is required. The flow equation for $\Gamma _2$ is rewritten as

$$\begin{aligned} \partial _l \Gamma _{2}[ \Psi _1, \Psi _2 ] = - \frac{1}{2} [ \gamma _{2,a} + \gamma _{2,b} + 2 \gamma _{2,c} + \mathrm {perm} ], \end{aligned}$$

(4.165)

where $\gamma _{2,a}$, $\gamma _{2,b}$, and $\gamma _{2,c}$ are given by Eqs. (3.157), (3.158), and (3.159), respectively. The flow equation for $\Gamma _{2;\hat{\psi }_1^{\mu } \hat{\psi }_2^{\nu }}^{(11)}=\delta ^2 \Gamma _2/\delta \hat{\psi }_1^{\mu } \delta \hat{\psi }_2^{\nu }$ is then written as

$$\begin{aligned} \partial _l \Gamma _{2;\hat{\psi }_1^{\mu } \hat{\psi }_2^{\nu }}^{(11)}[ \Psi _1, \Psi _2 ] =&- \frac{1}{2} [ -2 \gamma _{2,a-1}^{(11)} + \gamma _{2,a-2}^{(11)} - 2 \gamma _{2,b-1}^{(11)} -2 \gamma _{2,b-2}^{(11)} + 2 \gamma _{2,b-3}^{(11)} \nonumber \\&+ 2 \gamma _{2,b-4}^{(11)} -2 \gamma _{2,b-5}^{(11)} + 2 \gamma _{2,b-6}^{(11)} - 2 \gamma _{2,c-1}^{(11)} -2 \gamma _{2,c-2}^{(11)} \nonumber \\&- 2 \gamma _{2,c-3}^{(11)} + 2 \gamma _{2,c-4}^{(11)} + 2 \gamma _{2,c-5}^{(11)} + 2 \gamma _{2,c-6}^{(11)} +\mathrm {perm} ], \end{aligned}$$

(4.166)

where $\gamma _{2,a-1}^{(11)},\ldots ,\gamma _{2,c-6}^{(11)}$ are given in Fig. 4.5 and “perm” denotes the permutation between the indices 1 and 2, $\mu$ and $\nu$. For example, $\gamma _{2,b-1}^{(11)}$ is written as

$$\begin{aligned} \gamma _{2,b-1}^{(11)} =&\mathrm {Tr} \partial _l {\varvec{\mathrm {R}}}_k(\mathbf {q}) \mathrm {P}_k[\Psi _1] \Gamma _{1;\hat{\psi }_1^{\mu }}^{(3)}[\Psi _1] \mathrm {P}_k[\Psi _1] \Gamma _{2;\hat{\psi }_2^{\nu }}^{(12)}[\Psi _1,\Psi _2] \nonumber \\&\times \mathrm {P}_k[\Psi _2] \Gamma _{2}^{(11)}[\Psi _2,\Psi _1] \mathrm {P}_k[\Psi _1]. \end{aligned}$$

(4.167)

All functional derivatives are evaluated for a uniform field ${\varvec{\psi }}_{1,rt} \equiv {\varvec{\psi }}_1$, ${\varvec{\hat{\psi }}}_{1,rt} \equiv {\varvec{0}}$, ${\varvec{\psi }}_{2,rt} \equiv {\varvec{\psi }}_2$, and ${\varvec{\hat{\psi }}}_{2,rt} \equiv {\varvec{0}}$.

Fig. 4.5

Reprinted figure with permission from Ref. [1]. Copyright (2017) by the American Physical Society. https://doi.org/10.1103/PhysRevB.96.184202

Graphical representation for the flow equation of $\Gamma _{2}^{(2)}$.

The functional derivatives of $\Gamma _1[\Psi ]$ and $\Gamma _2[\Psi _1,\Psi _2]$ are calculated as

$$\begin{aligned} \Gamma _{1; \psi ^{\alpha } \psi ^{\beta } \hat{\psi }^{\mu }}^{(3)}(q_1,q_2,q_3) =&[ -F''(\rho ) \psi ^{\alpha } \psi ^{\beta } \psi ^{\mu } \nonumber \\&-F'(\rho ) (\delta ^{\alpha \beta } \psi ^{\mu } + \delta ^{\alpha \mu } \psi ^{\beta } + \delta ^{\beta \mu } \psi ^{\alpha }) ] \nonumber \\&\times (2\pi )^{D+1}\delta (\mathbf {q}_1+\mathbf {q}_2+\mathbf {q}_2)\delta (\omega _1+\omega _2+\omega _2), \nonumber \end{aligned}$$

$$\begin{aligned} \Gamma _{2; \hat{\psi }_1^{\mu } \hat{\psi }_2^{\nu }}^{(11)}(q_1,q_2) = \Delta ^{\mu \nu }({\varvec{\psi }}_1,{\varvec{\psi }}_2) (2\pi )^{D+2} \delta (\mathbf {q}_1+\mathbf {q}_2) \delta (\omega _1) \delta (\omega _2), \nonumber \end{aligned}$$

$$\begin{aligned} \Gamma _{2; \psi _1^{\alpha } \hat{\psi }_1^{\beta } \hat{\psi }_2^{\mu }}^{(21)}(q_1,q_2,q_3) = \partial _{\psi _1^{\alpha }} \Delta ^{\beta \mu }({\varvec{\psi }}_1,{\varvec{\psi }}_2) (2\pi )^{D+2}\delta (\mathbf {q}_1+\mathbf {q}_2+\mathbf {q}_3) \delta (\omega _1+\omega _2) \delta (\omega _3), \nonumber \end{aligned}$$

$$\begin{aligned} \Gamma _{2; \psi _1^{\alpha } \psi _1^{\beta } \hat{\psi }_1^{\mu } \hat{\psi }_2^{\nu }}^{(31)}(q_1,q_2,q_3,q_4) =&\partial _{\psi _1^{\alpha }} \partial _{\psi _1^{\beta }} \Delta ^{\mu \nu }({\varvec{\psi }}_1,{\varvec{\psi }}_2) (2\pi )^{D+2}\delta (\mathbf {q}_1+\mathbf {q}_2+\mathbf {q}_3+\mathbf {q}_4) \nonumber \\&\times \delta (\omega _1+\omega _2+\omega _3) \delta (\omega _4), \nonumber \end{aligned}$$

$$\begin{aligned} \Gamma _{2; \psi _1^{\alpha } \hat{\psi }_1^{\mu } \psi _2^{\beta } \hat{\psi }_2^{\nu }}^{(22)}(q_1,q_2,q_3,q_4) =&\partial _{\psi _1^{\alpha }} \partial _{\psi _2^{\beta }} \Delta ^{\mu \nu }({\varvec{\psi }}_1,{\varvec{\psi }}_2) (2\pi )^{D+2}\delta (\mathbf {q}_1+\mathbf {q}_2+\mathbf {q}_3+\mathbf {q}_4) \nonumber \\&\times \delta (\omega _1+\omega _2) \delta (\omega _3+\omega _4), \nonumber \end{aligned}$$

where $\Delta ^{\mu \nu }({\varvec{\psi }}_1,{\varvec{\psi }}_2)$ is expressed as Eq. (4.47).

As mentioned in Sect. 4.3.5, near the lower critical dimension $D=D_{\mathrm {lc}}+\epsilon$, $\gamma _{2}^{(11)}$ is expanded in terms of $\rho ^{-1} \sim \epsilon$. In addition, $\gamma _{2}^{(11)}$ is rewritten as

$$\begin{aligned} \gamma _{2}^{(11)} =&\gamma _{2,00}^{(11)} \delta ^{\mu \nu } + (4 \rho _1 \rho _2)^{-1/2} [ \gamma _{2,12}^{(11)} \psi _1^{\mu } \psi _2^{\nu } + \gamma _{2,21}^{(11)} \psi _2^{\mu } \psi _1^{\nu } \nonumber \\&+ \gamma _{2,11}^{(11)} \psi _1^{\mu } \psi _1^{\nu } + \gamma _{2,22}^{(11)} \psi _2^{\mu } \psi _2^{\nu } ]. \end{aligned}$$

(4.168)

By noting

$$\begin{aligned} \psi ^{\alpha } P_{12}^{\alpha \beta }=P_{12}^{(L)} \psi ^{\beta } \simeq -\frac{1}{2\rho F'(\rho )} \psi ^{\beta }, \end{aligned}$$

(4.169)

and

$$\begin{aligned} \psi ^{\alpha } P_{11}^{\alpha \beta }=P_{11}^{(L)} \psi ^{\beta } \simeq \frac{2XT}{ 4\rho ^2 F'(\rho )^2} \psi ^{\beta }, \end{aligned}$$

(4.170)

each coefficients of Eq. (4.168) can be calculated in the leading order of $\rho ^{-1}$ as follows:

$$\begin{aligned} \gamma _{2,a-1,00}^{(11)} = T (2\rho _1)^{-1} (\sqrt{\rho _1/\rho _2} \Delta _{11} + z \Delta _{21} ) L_2^{(T)}(\rho _1), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,a-2,00}^{(11)} =&T (2 \rho _1)^{-1} [ (N-1) ( 2\rho _1 \partial _{\rho _1} - z \partial _z ) \Delta _{00} + (1-z^2) \partial _z^2 \Delta _{00} \nonumber \\&+ 2 \sqrt{\rho _1/\rho _2} \Delta _{11} ] L_2^{(T)}(\rho _1), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,b-1,00}^{(11)} = (4 \rho _1 \rho _2)^{-1/2} ( \Delta _{12} + \sqrt{\rho _2/\rho _1} z \Delta _{22} ) \Delta _{00} I_{21}^{(T)}(\rho _1,\rho _2), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,b-2,00}^{(11)} = (4 \rho _1 \rho _2)^{-1/2} (1-z^2) ( \Delta _{00} + z \Delta _{21} + \sqrt{\rho _1/\rho _2} \Delta _{11} ) \partial _z \Delta _{00} J_{21}^{(T)}(\rho _1,\rho _2), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,b-3,00}^{(11)} =&(4 \rho _1 \rho _2)^{-1/2} [ z \Delta _{00} + z^2 \Delta _{21}+ \Delta _{12} + \sqrt{\rho _1/\rho _2} z \Delta _{11} \nonumber \\&+ \sqrt{\rho _2/\rho _1} z \Delta _{22} ] \Delta _{00} I_{21}^{(T)}(\rho _1,\rho _2), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,b-4,00}^{(11)} =&(4 \rho _1 \rho _2)^{-1/2} [ (N-2+z^2) \Delta _{00} \partial _z \Delta _{00} \nonumber \\&- z (1-z^2) ( \Delta _{21} \partial _z \Delta _{00} + \Delta _{00} \partial _z^2 \Delta _{00}) \nonumber \\&+ (1-z^2)^2 \Delta _{21} \partial _z^2 \Delta _{00} + (\Delta _{21}+\Delta _{12}) \Delta _{00} ] I_{21}^{(T)}(\rho _1,\rho _2), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,b-5,00}^{(11)} =&(4 \rho _1 \rho _2)^{-1/2} ( \Delta _{12} + \sqrt{\rho _1/\rho _2}z \Delta _{11} ) \Delta _{00} I_{21}^{(T)}(\rho _1,\rho _2) \nonumber \\&+ (4 \rho _1 \rho _2)^{-1/2} (1-z^2) ( \Delta _{00} + z \Delta _{21} + \sqrt{\rho _2/\rho _1} \Delta _{22} ) \partial _z \Delta _{00} J_{21}^{(T)}(\rho _1,\rho _2), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,b-6,00}^{(11)} = (4 \rho _1 \rho _2)^{-1/2} (1-z^2) ( \sqrt{\rho _1/\rho _2} \Delta _{11} + \sqrt{\rho _2/\rho _1} \Delta _{22} ) \partial _z \Delta _{00} J_{21}^{(T)}(\rho _1,\rho _2), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,c-1,00}^{(11)} = (2 \rho _1)^{-1} ( z \Delta _{21} + \sqrt{\rho _1/\rho _2} \Delta _{11} ) \Delta _{00}(\rho _1,\rho _1,z=1) I_{21}^{(T)}(\rho _1,\rho _1), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,c-2,00}^{(11)} = 0, \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,c-3,00}^{(11)} = (2 \rho _1)^{-1} ( z \Delta _{21} + \sqrt{\rho _1/\rho _2} \Delta _{11} ) \Delta _{00}(\rho _1,\rho _1,z=1) I_{21}^{(T)}(\rho _1,\rho _1), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,c-4,00}^{(11)} =&(2 \rho _1)^{-1} [ (N-1) ( 2\rho _1 \partial _{\rho _1} - z \partial _z ) \Delta _{00} + (1-z^2) \partial _z^2 \Delta _{00} + 2 \sqrt{\rho _1/\rho _2} \Delta _{11} ] \nonumber \\&\times \Delta _{00}(\rho _1,\rho _1,z=1) I_{21}^{(T)}(\rho _1,\rho _1), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,c-5,00}^{(11)} = \gamma _{2,c-6,00}^{(11)} = 0, \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,a-1,21}^{(11)} = T (2\rho _1)^{-1} ( \partial _z \Delta _{00} + z \partial _z \Delta _{21} +\sqrt{\rho _1/\rho _2} \partial _z \Delta _{11} ) L_2^{(T)}(\rho _1), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,a-2,21}^{(11)} =&T (2\rho _1)^{-1} [ (1-z^2) \partial _{z}^2 \Delta _{21} - 2 z \partial _z \Delta _{21} \nonumber \\&+ (N-1) (2\rho _1 \partial _{\rho _1} \Delta _{21} - \Delta _{21} - z \partial _z \Delta _{21}) \nonumber \\&+2 \sqrt{\rho _1/\rho _2} \partial _z \Delta _{11} ] L_2^{(T)}(\rho _1), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,b-1,21}^{(11)} =&-(4 \rho _1 \rho _2)^{-1/2} \{ \Delta _{21} ( \Delta _{12} + \sqrt{\rho _2/\rho _1} z \Delta _{22} ) + [ - z \Delta _{00} + (1-z^2) \Delta _{21} ] \nonumber \\&\times ( \partial _z \Delta _{00} + \Delta _{21} + z \partial _z \Delta _{21} + \sqrt{\rho _1/\rho _2} \partial _z \Delta _{11} ) \} I_{21}^{(T)}(\rho _1,\rho _2), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,b-2,21}^{(11)} =&(4 \rho _1 \rho _2)^{-1/2} (\Delta _{00} + z \Delta _{21} + \sqrt{\rho _1/\rho _2} \Delta _{11}) [ (1-z^2) \partial _z \Delta _{21} - z \Delta _{21} \nonumber \\&+ \sqrt{\rho _2/\rho _1} \Delta _{22} ] J_{21}^{(T)}(\rho _1,\rho _2), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,b-3,21}^{(11)} =&(4 \rho _1 \rho _2)^{-1/2} ( z \Delta _{00} + \Delta _{12} + z^2 \Delta _{21} + \sqrt{\rho _1/\rho _2} z \Delta _{11} \nonumber \\&+ \sqrt{\rho _2/\rho _1} z \Delta _{22} ) \Delta _{21} I_{21}^{(T)}(\rho _1,\rho _2) \nonumber \\&+ (4 \rho _1 \rho _2)^{-1/2} (\Delta _{00} + z \Delta _{21} + \sqrt{\rho _1/\rho _2} \Delta _{11} ) \nonumber \\&\times (\Delta _{00} + z \Delta _{21} + \sqrt{\rho _2/\rho _1} \Delta _{22} ) J_{21}^{(T)}(\rho _1,\rho _2), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,b-4,21}^{(11)} =&(4 \rho _1 \rho _2)^{-1/2} \{ (N-2+z^2) \Delta _{00} \partial _z \Delta _{21} \nonumber \\&- z (1-z^2) (\Delta _{00} \partial _z^2 \Delta _{21} + \Delta _{21} \partial _z \Delta _{21} ) \nonumber \\&+ 2 \partial _z \Delta _{21} [ z^2 \Delta _{00} - z (1-z^2) \Delta _{21} ] + \Delta _{21} ( z \Delta _{00} + \Delta _{12} + z^2 \Delta _{21} ) \nonumber \\&+ \sqrt{\rho _1/\rho _2} \partial _z \Delta _{11} [ - \Delta _{00} + (1-z^2) \Delta _{21} ] \nonumber \\&+ \sqrt{\rho _2/\rho _1} \partial _z \Delta _{22} [ - \Delta _{00} + (1-z^2) \Delta _{21} ] \nonumber \\&+ (1-z^2)^2 \Delta _{21} \partial _z^2 \Delta _{21} \} I_{21}^{(T)}(\rho _1,\rho _2), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,b-5,21}^{(11)} =&(4 \rho _1 \rho _2)^{-1/2} \{ [ - z \Delta _{00} + (1-z^2) \Delta _{21} ] ( \partial _z \Delta _{00} + \Delta _{21} + z \partial _z \Delta _{21} + \partial _z \Delta _{22} ) \nonumber \\&+ \Delta _{21} ( \Delta _{12} + \sqrt{\rho _1/\rho _2} z \Delta _{11} ) \} I_{21}^{(T)}(\rho _1,\rho _2) \nonumber \\&+ (4 \rho _1 \rho _2)^{-1/2} [ (1-z^2) \partial _z \Delta _{21} - z \Delta _{21} + \sqrt{\rho _1/\rho _2} \Delta _{11} ] \nonumber \\&\times (\Delta _{00} + z \Delta _{21} + \sqrt{\rho _2/\rho _1} \Delta _{22}) J_{21}^{(T)}(\rho _1,\rho _2), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,b-6,21}^{(11)} =&(4 \rho _1 \rho _2)^{-1/2} \{ (N-2+z^2) (\Delta _{21})^2 + \Delta _{11} \Delta _{22} - 2 z (1-z^2) \Delta _{21} \partial _z \Delta _{21} \nonumber \\&+ (1-z^2)^2 (\partial _z \Delta _{21})^2 + z^2 (\partial _z \Delta _{00})^2 + [ z \Delta _{21} - (1-z^2) \partial _z \Delta _{21} ] \nonumber \\&\times ( - \sqrt{\rho _1/\rho _2} \Delta _{11} - \sqrt{\rho _2/\rho _1} \Delta _{22} + 2 z \partial _z \Delta _{00} ) \} J_{21}^{(T)}(\rho _1,\rho _2), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,c-1,21}^{(11)} =&(2\rho _1)^{-1} ( \partial _z \Delta _{00} + z \partial _z \Delta _{21} + \sqrt{\rho _1/\rho _2} \partial _z \Delta _{11} ) \nonumber \\&\times \Delta _{00}(\rho _1,\rho _1,z=1) I_{21}^{(T)}(\rho _1,\rho _1), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,c-2,21}^{(11)} = 0, \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,c-3,21}^{(11)} =&(2\rho _1)^{-1} ( \partial _z \Delta _{00} + z \partial _z \Delta _{21} + \sqrt{\rho _1/\rho _2} \partial _z \Delta _{11} ) \nonumber \\&\times \Delta _{00}(\rho _1,\rho _1,z=1) I_{21}^{(T)}(\rho _1,\rho _1), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,c-4,21}^{(11)} =&(2\rho _1)^{-1} [ (1-z^2) \partial _{z}^2 \Delta _{21} - 2 z \partial _z \Delta _{21} \nonumber \\&+ (N-1) (2\rho _1 \partial _{\rho _1} \Delta _{21} - \Delta _{21} - z \partial _z \Delta _{21}) \nonumber \\&+\sqrt{\rho _1/\rho _2} \partial _z \Delta _{11} ] \Delta _{00}(\rho _1,\rho _1,z=1) I_{21}^{(T)}(\rho _1,\rho _1), \nonumber \end{aligned}$$

$$\begin{aligned} \gamma _{2,c-5,21}^{(11)} = \gamma _{2,c-6,21}^{(11)} = 0, \nonumber \end{aligned}$$

$\gamma _{2,00}^{(11)}$ and $\gamma _{2,21}^{(11)}$ yield the flow equations for $\Delta _{00}$ and $\Delta _{21}$, respectively. In the leading order of $\rho ^{-1}$, $\Delta _{12}$, $\Delta _{11}$, and $\Delta _{22}$ do not appear in $\gamma _{2,00}^{(11)}$ and $\gamma _{2,21}^{(11)}$. Thus, the flow equations for $\Delta _{00}$ and $\Delta _{21}$ compose a closed set of equations.

4.1.4 4.6.4 Flow Equations for $X_k$, $v_k$, $Z_k$, and $T_k$

In this section, we derive the flow equations for $X_k$, $v_k$, $Z_k$, and $T_k$, which are given by Eqs. (4.51), (4.52), (4.54) and (4.53), respectively. From Eq. (4.159), we have the exact flow equation for $\Gamma _{1;\Psi (p),\Psi '(p')}^{(2)} = \delta ^2 \Gamma _1/\delta \Psi (p) \delta \Psi '(p')$, where $\Psi$ represents $\psi ^{\mu }$ or $\hat{\psi }^{\mu }$ and $p=(\mathbf {p},\omega _p)$, as follows:

$$\begin{aligned} \partial _l \Gamma _{1;\Psi (p),\Psi '(p')}^{(2)} =&\frac{1}{2} [ 2 \gamma _{1,a-1}^{(2)} - \gamma _{1,a-2}^{(2)} + 2 \gamma _{1,b-1(+)}^{(2)} + 2 \gamma _{1,b-1(-)}^{(2)} \nonumber \\&+ 2 \gamma _{1,b-2}^{(2)} - 2 \gamma _{1,b-3(+)}^{(2)} - 2 \gamma _{1,b-3(-)}^{(2)} -2 \gamma _{1,b-4(+)}^{(2)} \nonumber \\&-2 \gamma _{1,b-4(-)}^{(2)} - 2 \gamma _{1,b-5}^{(2)} + 2 \gamma _{1,b-6}^{(2)} + 2 \gamma _{1,b-7}^{(2)} ], \end{aligned}$$

(4.171)

where the terms on the right-hand side are given in Fig. 4.6. All functional derivatives are evaluated for a uniform field ${\varvec{\psi }}_{rt} \equiv {}^t(\sqrt{2\rho _m},0,\ldots ,0)$ and ${\varvec{\hat{\psi }}}_{rt} \equiv {}^t(0,\ldots ,0)$.

Fig. 4.6

Reprinted figure with permission from Ref. [1]. Copyright (2017) by the American Physical Society https://doi.org/10.1103/PhysRevB.96.184202

Graphical representation for the flow equation of $\Gamma _{1}^{(2)}$.

Calculation of $\partial _l (X_k v_k)$ and $\partial _l Z_k$

We set $\Psi =\hat{\psi }^2$, $\Psi '=\psi ^2$, and $p'=-p$ for the calculation of $\partial _l (X_k v_k)$, $\partial _l Z_k$, and $\partial _l X_k$. We denote each term in Eq. (4.171) as $\gamma _{1,a-1,\hat{\psi } \psi }^{(2)}$, and so on. Note that $\gamma _{1,a-2,\hat{\psi } \psi }^{(2)}$, $\gamma _{1,b-5,\hat{\psi } \psi }^{(2)}$, and $\gamma _{1,b-7,\hat{\psi } \psi }^{(2)}$ do not depend on the external momentum $\mathbf {p}$ and frequency $\omega _p$, thus they do not contribute to the flow equations for $X_k v_k$, $Z_k$, and $X_k$. The other terms are given by

$$\begin{aligned} \gamma _{1,a-1,\hat{\psi } \psi }^{(2)} =&4 \rho _m F'(\rho _m)^2 X T \int \limits _{\mathbf {q},\omega _q} \partial _l R_k(\mathbf {q}) \Bigl [ P_{21}^{(L)}(\mathbf {q},\omega _q)^2 D_0(\mathbf {p+q},\omega _p+\omega _q)^{-1} \nonumber \\&+ 2 M_1(\mathbf {q}) D_1(\mathbf {q},\omega _q)^{-2} P_{12}^{(T)}(\mathbf {p+q},\omega _p+\omega _q) \nonumber \\&+ P_{21}^{(T)}(\mathbf {q},\omega _q)^2 D_1(\mathbf {p+q},\omega _p+\omega _q)^{-1} \nonumber \\&+ 2 M_0(\mathbf {q}) D_0(\mathbf {q},\omega _q)^{-2} P_{12}^{(L)}(\mathbf {p+q},\omega _p+\omega _q) \Bigr ], \end{aligned}$$

(4.172)

$$\begin{aligned} \gamma _{1,b-1(\pm ),\hat{\psi } \psi }^{(2)} =&2 \rho _m F'(\rho _m)^2 \int \limits _{\mathbf {q}} \partial _l R_k(\mathbf {q}) \Bigl [ \nonumber \\&\Delta _{T}(\rho _m) P_{21}^{(T)}(\mathbf {q},0)^2 P_{12}^{(T)}(\mathbf {q},0) P_{12}^{(L)}(\mathbf {p \pm q},\omega _p) \nonumber \\&+ \Delta _{L}(\rho _m) P_{21}^{(L)}(\mathbf {q},0)^2 P_{12}^{(L)}(\mathbf {q},0) P_{12}^{(T)}(\mathbf {p \pm q},\omega _p) \Bigr ], \end{aligned}$$

(4.173)

$$\begin{aligned} \gamma _{1,b-2,\hat{\psi } \psi }^{(2)} =&2 \rho _m F'(\rho _m)^2 \int \limits _{\mathbf {q}} \partial _l R_k(\mathbf {q}) \Bigl [ \nonumber \\&\Delta _{T}(\rho _m) P_{12}^{(T)}(\mathbf {-p+q},0) P_{21}^{(T)}(\mathbf {-p+q},0) P_{12}^{(L)}(\mathbf {q},\omega _p)^2 \nonumber \\&+ \Delta _{L}(\rho _m) P_{12}^{(L)}(\mathbf {-p+q},0) P_{21}^{(L)}(\mathbf {-p+q},0) P_{12}^{(T)}(\mathbf {q},\omega _p)^2 \Bigr ], \end{aligned}$$

(4.174)

$$\begin{aligned} \gamma _{1,b-3(+),\hat{\psi } \psi }^{(2)} =&F'(\rho _m) \int \limits _{\mathbf {q}} \partial _l R_k(\mathbf {q}) \Bigl [ \nonumber \\&(\Delta _{12}(\rho _m)+\Delta _{11}(\rho _m)) P_{12}^{(L)}(\mathbf {q},0)^2 P_{12}^{(T)}(\mathbf {p+q},\omega _p) \nonumber \\&+ \rho _m \Delta _{T}'(\rho _m) P_{12}^{(T)}(\mathbf {q},0)^2 P_{12}^{(L)}(\mathbf {p+q},\omega _p) \Bigr ], \end{aligned}$$

(4.175)

$$\begin{aligned} \gamma _{1,b-3(-),\hat{\psi } \psi }^{(2)} =&F'(\rho _m) \int \limits _{\mathbf {q}} \partial _l R_k(\mathbf {q}) \Bigl [ \nonumber \\&(\Delta _{12}(\rho _m)+\Delta _{11}(\rho _m)) P_{12}^{(L)}(\mathbf {q},0)^2 P_{12}^{(T)}(\mathbf {p-q},\omega _p) \nonumber \\&+ (\Delta _{21}(\rho _m)+\Delta _{11}(\rho _m)) P_{12}^{(T)}(\mathbf {q},0)^2 P_{12}^{(L)}(\mathbf {p-q},\omega _p) \Bigr ], \end{aligned}$$

(4.176)

$$\begin{aligned} \gamma _{1,b-4(+),\hat{\psi } \psi }^{(2)} =&F'(\rho _m) \int \limits _{\mathbf {q}} \partial _l R_k(\mathbf {q}) \Bigl [ \nonumber \\&(\Delta _{12}(\rho _m)+\Delta _{11}(\rho _m)) P_{12}^{(T)}(\mathbf {q},\omega _p)^2 P_{12}^{(L)}(\mathbf {-p+q},0) \nonumber \\&+ \rho _m \Delta _{T}'(\rho _m) P_{12}^{(L)}(\mathbf {q},\omega _p)^2 P_{12}^{(T)}(\mathbf {-p+q},0) \Bigr ], \end{aligned}$$

(4.177)

$$\begin{aligned} \gamma _{1,b-4(-),\hat{\psi } \psi }^{(2)} =&F'(\rho _m) \int \limits _{\mathbf {q}} \partial _l R_k(\mathbf {q}) \Bigl [ \nonumber \\&(\Delta _{12}(\rho _m)+\Delta _{11}(\rho _m)) P_{12}^{(T)}(\mathbf {q},\omega _p)^2 P_{21}^{(L)}(\mathbf {-p+q},0) \nonumber \\&+ (\Delta _{21}(\rho _m)+\Delta _{11}(\rho _m)) P_{12}^{(L)}(\mathbf {q},\omega _p)^2 P_{21}^{(T)}(\mathbf {-p+q},0) \Bigr ], \end{aligned}$$

(4.178)

$$\begin{aligned} \gamma _{1,b-6,\hat{\psi } \psi }^{(2)} =&(2\rho _m)^{-1} [ \partial _z \Delta _{00}(\rho _m) + \Delta _{12}(\rho _m) + (N-1) \Delta _{21}(\rho _m) ] \nonumber \\&\times \int \limits _{\mathbf {q}} \partial _l R_k(\mathbf {q}) P_{12}^{(T)}(\mathbf {q},\omega _p)^2, \end{aligned}$$

(4.179)

where we have used the notations $\Delta _{\ldots }(\rho )=\Delta _{\ldots }(\rho ,\rho ,z=1)$, $\Delta _T(\rho ) = \Delta ^{22}({\varvec{\psi }},{\varvec{\psi }}) = \Delta _{00}(\rho )$, and $\Delta _L(\rho ) = \Delta ^{11}({\varvec{\psi }},{\varvec{\psi }})=\Delta _{00}(\rho )+\Delta _{12}(\rho )+\Delta _{21}(\rho )+\Delta _{11}(\rho )+\Delta _{22}(\rho )$.

The flow equations of $X_k v_k$ and $Z_k$ can be obtained from the momentum derivative of $\gamma _{1,\hat{\psi } \psi }^{(2)}$. We next expand $\partial _{(-i p_x)} \gamma _{1,\hat{\psi } \psi }^{(2)}$ and $\partial _{p_{\perp }^2} \gamma _{1,\hat{\psi } \psi }^{(2)}$ in terms of $\rho ^{-1}$ and retain the leading order. By noting that $P_{12}^{(T)}(q)=\mathcal {O}(1)$ and $P_{12}^{(L)}(q)=\mathcal {O}(\rho ^{-1})$, one finds that the leading order contributions to $\partial _l(X_k v_k)$ and $\partial _l Z_k$ come from $\gamma _{1,a-1,\hat{\psi } \psi }^{(2)}$, $\gamma _{1,b-1,\hat{\psi } \psi }^{(2)}$, and $\gamma _{1,b-2,\hat{\psi } \psi }^{(2)}$. The corresponding flow equations are given by Eqs. (4.90), (4.91), and (4.92).

Calculation of $\partial _l T_k$

We set $\Psi =\hat{\psi }^2$, $\Psi '=\hat{\psi }^2$ and $p=p'=0$ for the calculation of $\partial _l(X_k T_k)$. We denote each term in Eq. (4.171) as $\gamma _{1,a-1,\hat{\psi } \hat{\psi }}^{(2)}$, and so on. They are given as follows:

$$\begin{aligned} \gamma _{1,a-2,\hat{\psi } \hat{\psi }}^{(2)} = \gamma _{1,b-5,\hat{\psi } \hat{\psi }}^{(2)} = \gamma _{1,b-7,\hat{\psi } \hat{\psi }}^{(2)} = 0, \end{aligned}$$

(4.180)

$$\begin{aligned} \gamma _{1,a-1,\hat{\psi } \hat{\psi }}^{(2)} =&2 \rho _m F'(\rho _m)^2 \int \limits _{\mathbf {q},\omega _q} \partial _l R_k(\mathbf {q}) \Bigl [ P_{11}^{(T)}(\mathbf {q},\omega _q) \bigl \{ P_{12}^{(L)}(\mathbf {q},\omega _q) P_{11}^{(L)}(\mathbf {q},\omega _q) \nonumber \\&+ P_{11}^{(L)}(\mathbf {q},\omega _q) P_{21}^{(L)}(\mathbf {q},\omega _q) \bigr \} +P_{11}^{(L)}(\mathbf {q},\omega _q) \nonumber \\&\times \bigl \{ P_{12}^{(T)}(\mathbf {q},\omega _q) P_{11}^{(T)}(\mathbf {q},\omega _q) + P_{11}^{(T)}(\mathbf {q},\omega _q) P_{21}^{(T)}(\mathbf {q},\omega _q) \bigr \} \Bigr ], \end{aligned}$$

(4.181)

$$\begin{aligned} \gamma _{1,b-1(\pm ),\hat{\psi } \hat{\psi }}^{(2)} =&2 \rho _m F'(\rho _m)^2 \int \limits _{\mathbf {q}} \partial _l R_k(\mathbf {q}) \Bigl [ \Delta _{T}(\rho _m) P_{21}^{(T)}(\mathbf {q})^2 P_{12}^{(T)}(\mathbf {q}) P_{11}^{(L)}(\mathbf {q}) \nonumber \\&+ \Delta _{L}(\rho _m) P_{21}^{(L)}(\mathbf {q})^2 P_{12}^{(L)}(\mathbf {q}) P_{11}^{(T)}(\mathbf {q}) \Bigr ], \end{aligned}$$

(4.182)

$$\begin{aligned} \gamma _{1,b-2,\hat{\psi } \hat{\psi }}^{(2)} =&2 \rho _m F'(\rho _m)^2 \int \limits _{\mathbf {q}} \partial _l R_k(\mathbf {q}) \Bigl [ \Delta _{T}(\rho _m) P_{12}^{(T)}(\mathbf {q}) P_{21}^{(T)}(\mathbf {q}) \nonumber \\&\times \bigl \{ P_{12}^{(L)}(\mathbf {q}) P_{11}^{(L)}(\mathbf {q}) + P_{11}^{(L)}(\mathbf {q}) P_{21}^{(L)}(\mathbf {q}) \bigr \} \nonumber \\&+ \Delta _{L}(\rho _m) P_{12}^{(L)}(\mathbf {q}) P_{21}^{(L)}(\mathbf {q}) \nonumber \\&\times \bigl \{ P_{12}^{(T)}(\mathbf {q}) P_{11}^{(T)}(\mathbf {q}) + P_{11}^{(T)}(\mathbf {q}) P_{21}^{(T)}(\mathbf {q}) \bigr \} \Bigr ], \end{aligned}$$

(4.183)

$$\begin{aligned} \gamma _{1,b-3(\pm ),\hat{\psi } \hat{\psi }}^{(2)} =&F'(\rho _m) \int \limits _{\mathbf {q}} \partial _l R_k(\mathbf {q}) \Bigl [ (\Delta _{12}(\rho _m)+\Delta _{11}(\rho _m)) P_{12}^{(L)}(\mathbf {q})^2 P_{11}^{(T)}(\mathbf {q}) \nonumber \\&+ \rho _m \Delta _{T}'(\rho _m) P_{12}^{(T)}(\mathbf {q})^2 P_{11}^{(L)}(\mathbf {q}) \Bigr ], \end{aligned}$$

(4.184)

$$\begin{aligned} \gamma _{1,b-4(\pm ),\hat{\psi } \hat{\psi }}^{(2)} =&F'(\rho _m) \int \limits _{\mathbf {q}} \partial _l R_k(\mathbf {q}) \Bigl [ (\Delta _{12}(\rho _m)+\Delta _{11}(\rho _m)) P_{12}^{(L)}(\mathbf {q}) \nonumber \\&\times \bigl \{ P_{12}^{(T)}(\mathbf {q}) P_{11}^{(T)}(\mathbf {q}) + P_{11}^{(T)}(\mathbf {q}) P_{21}^{(T)}(\mathbf {q}) \bigr \} \nonumber \\&+ \rho _m \Delta _{T}'(\rho _m) P_{12}^{(T)}(\mathbf {q}) \nonumber \\&\times \bigl \{ P_{12}^{(L)}(\mathbf {q}) P_{11}^{(L)}(\mathbf {q}) + P_{11}^{(L)}(\mathbf {q}) P_{21}^{(L)}(\mathbf {q}) \bigr \} \Bigr ], \end{aligned}$$

(4.185)

$$\begin{aligned} \gamma _{1,b-6,\hat{\psi } \hat{\psi }}^{(2)} =&(2\rho _m)^{-1} [ (N-1) \partial _z \Delta _{00}(\rho _m) + \Delta _{12}(\rho _m) + \Delta _{21}(\rho _m) ] \nonumber \\&\times \int \limits _{\mathbf {q}} \partial _l R_k(\mathbf {q}) \bigl [ P_{12}^{(T)}(\mathbf {q}) P_{11}^{(T)}(\mathbf {q}) + P_{11}^{(T)}(\mathbf {q}) P_{21}^{(T)}(\mathbf {q}) \bigr ], \end{aligned}$$

(4.186)

where the frequency $\omega _q$ in Eqs. (4.182)–(4.186) is set to zero.

By noting that $P_{12}^{(T)}(q)=\mathcal {O}(1)$, $P_{12}^{(L)}(q)=\mathcal {O}(\rho ^{-1})$, and $P_{11}^{(L)}(q)=\mathcal {O}(\rho ^{-2})$, one finds that the leading order contributions to $\partial _l(X_k T_k)$ come from $\gamma _{1,b-1,\hat{\psi } \hat{\psi }}^{(2)}$, $\gamma _{1,b-2,\hat{\psi } \hat{\psi }}^{(2)}$, $\gamma _{1,b-4,\hat{\psi } \hat{\psi }}^{(2)}$, and $\gamma _{1,b-6,\hat{\psi } \hat{\psi }}^{(2)}$. To obtain the flow equation for $T_k$, we also need the flow equation for $X_k$. The leading order contributions to $\partial _l X_k$ come from $\gamma _{1,b-1,\hat{\psi } \psi }^{(2)}$, $\gamma _{1,b-2,\hat{\psi } \psi }^{(2)}$, $\gamma _{1,b-4,\hat{\psi } \psi }^{(2)}$, and $\gamma _{1,b-6,\hat{\psi } \psi }^{(2)}$. Finally, the flow equation for $T_k$ is calculated from

$$\begin{aligned} \partial _l T_k= & {} X_k^{-1} \bigl \{ \partial _l (X_k T_k) - T_k \partial _l X_k \bigr \} \nonumber \\= & {} X_k^{-1} \biggl ( -\frac{1}{2} \partial _l \Gamma _{1;\hat{\psi }(p),\hat{\psi }(p)}^{(2)} \Bigl |_{p=0} - T_k \partial _{i\omega _p} \Bigl [\partial _l \Gamma _{1;\hat{\psi }(p),\psi (-p)}^{(2)}\Bigr ] \Bigl |_{p=0} \biggr ), \end{aligned}$$

(4.187)

which yields Eq. (4.93).

4.1.5 4.6.5 Numerical Scheme to Calculate the Fixed Point

In this section, the numerical method used to obtain the fixed functions $\delta _{00}^*(z)$ and $\delta _{21}^*(z)$ is presented. Since the solution exhibits a nonanalytic behavior near $z=1$, standard numerical techniques are not applicable. We define $\tilde{\delta }_{00}(z)=(N-2)\delta _{00}(z)$ and $\tilde{\delta }_{21}(z)=(N-2)\delta _{21}(z)$, and the trial functions $\tilde{\delta }_{00}^{(\mathrm {t})}(z)$ and $\tilde{\delta }_{21}^{(\mathrm {t})}(z)$ as follows,

$$\begin{aligned} \tilde{\delta }_{00}^{(\mathrm {t})}(z) = a_0 + \sum _{n=1}^{n_{\mathrm {max}}} a_n (1-z)^{n/2} \nonumber \\ \tilde{\delta }_{21}^{(\mathrm {t})}(z) = b_0 + \sum _{n=1}^{n_{\mathrm {max}}} b_n (1-z)^{n/2}. \end{aligned}$$

(4.188)

We rewrite Eqs. (4.108) and (4.109) as

$$\begin{aligned} \partial _l \delta _{00}(z)= & {} \beta _{00} [ \delta _{00}, \delta _{21}; \epsilon ](z), \nonumber \\ \partial _l \delta _{21}(z)= & {} \beta _{21} [ \delta _{00}, \delta _{21}; \epsilon ](z). \end{aligned}$$

(4.189)

Then, the fixed functions $\tilde{\delta }_{00}^*(z)$ and $\tilde{\delta }_{21}^*(z)$ satisfy

$$\begin{aligned} \beta _{00} [\tilde{\delta }_{00}^*, \tilde{\delta }_{21}^*;\epsilon =N-2](z)= & {} 0, \nonumber \\ \beta _{21} [\tilde{\delta }_{00}^*, \tilde{\delta }_{21}^*;\epsilon =N-2](z)= & {} 0. \end{aligned}$$

(4.190)

The integral $S(\{ a_n \},\{ b_n \})$ is introduced,

$$\begin{aligned} S(\{ a_n \},\{ b_n \}) =&\int \limits _{-1}^{1} \{ \beta _{00}[ \tilde{\delta }_{00}^{(\mathrm {t})}, \tilde{\delta }_{21}^{(\mathrm {t})}; N-2](z)^2 \nonumber \\&+ \beta _{21}[ \tilde{\delta }_{00}^{(\mathrm {t})}, \tilde{\delta }_{21}^{(\mathrm {t})}; N-2 ](z)^2 \} \mathrm {d}z, \end{aligned}$$

(4.191)

which vanishes if the true fixed functions are attained. The set of optimal parameters $\{ a_n \}$ and $\{ b_n \}$ can be obtained by minimizing $S(\{ a_n \},\{ b_n \})$. From Eq. (4.118), $a_0$ and $a_1$ satisfy $-a_0 + a_0^2 - (1/2) a_1^2 = 0$. Since $a_0=1$ when $a_1=0$, we obtain the following constraint,

$$\begin{aligned} a_0 = \frac{1}{2} \Bigl ( 1+\sqrt{1+2a_1^2} \, \Bigr ), \end{aligned}$$

(4.192)

which enables us to avoid the trivial solution $\{ a_n \} = \{ b_n \} = 0$. Note that the integral $S(\{ a_n \},\{ b_n \})$ has several local minima. One is chosen such that it recovers the fixed function Eq. (4.129) at $N=2$. The truncation number is fixed at $n_{\mathrm {max}}=4$. The inclusion of the higher order terms only changes $\eta _{\perp }=\delta _{00}^*(1)$ by less than one percent.

For the case of the random anisotropy, the disorder correlators satisfy

$$\begin{aligned} \delta _{00}^*(-z) = -\delta _{00}^*(z), \,\,\,\,\, \delta _{21}^*(-z) = \delta _{21}^*(z). \end{aligned}$$

(4.193)

Therefore, we introduce the following trial functions:

$$\begin{aligned} \tilde{\delta }_{00}^{(\mathrm {t})}(z) = a_0 + \sum _{n=1}^{n_{\mathrm {max}}} a_n z (1-z^2)^{n/2} \nonumber \\ \tilde{\delta }_{21}^{(\mathrm {t})}(z) = b_0 + \sum _{n=1}^{n_{\mathrm {max}}} b_n (1-z^2)^{n/2}. \end{aligned}$$

(4.194)

With these trial functions, Eq. (4.192) becomes

$$\begin{aligned} a_0 = \frac{1}{2} \Bigl ( 1+\sqrt{1+4a_1^2} \, \Bigr ). \end{aligned}$$

(4.195)

By employing a similar method, the anomalous dimensions $\eta$ for the random field and random anisotropy O(N) models can be also calculated from Eq. (4.101). We have checked that they agree with the known values given in Refs. [12] and [13].