Dissipation and friction of a quantum spin system

Abstract

We investigated the dissipation dynamics of a magnetic tip scanning the surface of a 3D ferromagnetic substrate at a velocity ${{\varvec{v}}}$ via a functional integral approach based on the Holstein–Primakoff boson representation of spin operators. The magnetic surface was parameterized by the XXZ model. And the degrees of freedom in the tip were taken as a single spin operator (M). The magnetic surface was coupled to the tip via a local exchange potential $\frac{w}{2}\delta _{il}$. Moreover, the tip induced a surface potential $f_{il}$ acting on the magnetic surface spin operators. After tracing over total internal degrees of freedom, we obtained the in–out quantum action. We calculated the imaginary part of the in–out quantum action and the frictional force as functions of speed $v=|{{\varvec{v}}}|$. We found that the imaginary part of the in–out quantum action is suppressed as $v\rightarrow 0$. The imaginary part of the in–out quantum action is proportional to the probability of ground state decay. Therefore, it implies dissipation of the system. The frictional force linearly depends on the velocity as $v<0.04$ . While $v>0.04$ the dependence of the frictional force on v becomes nonlinear.

GraphicAbstract

Appendix A: The calculation of deriving the effective EOM for the tip trajectory

Via Eqs. (19) and (20), the Feynman amplitude ${\mathcal {Z}}_{\texttt {in-in}}$ on CTP time interval reads

$$\begin{aligned} {\mathcal {Z}}_{\texttt {in-in}}&=K({{\varvec{q}}}_1,\phi ,x^0;{{\varvec{q}}}'_1,\phi '_1,0)K({{\varvec{q}}}'_2,\phi '_2,0;{{\varvec{q}}}_2,\phi ,x^0)\nonumber \\&=\int _{{{\varvec{q}}}_+(0)={{\varvec{q}}}'_1}^{{{\varvec{q}}}_+(x^0)={{\varvec{q}}}_1} \texttt {d}{{\varvec{q}}}_+ \int _{\phi _+(0)=\phi '_1}^{\phi _+(x^0)=\phi } \texttt {d}\phi _+\nonumber \\&\qquad \int _{{{\varvec{q}}}_-(0)={{\varvec{q}}}'_2}^{{{\varvec{q}}}_-(x^0)={{\varvec{q}}}_2} \texttt {d}{{\varvec{q}}}_- \int _{\phi _-(0)=\phi '_2}^{\phi _-(x^0)=\phi } \texttt {d}\phi _-\nonumber \\&\quad \times \exp [\texttt {i}S_\texttt {T}({{\varvec{q}}}_+,\phi _+)-\texttt {i}S_\texttt {T}({{\varvec{q}}}_-,\phi _-)]. \end{aligned}$$

(A1)

We always consider the case that the velocity of the tip is small enough. For this purpose, firstly we introduce new variables ${{\varvec{Q}}}=\frac{1}{2}({{\varvec{q}}}_+ + {{\varvec{q}}}_-)$ and ${{\varvec{r}}}={{\varvec{q}}}_+ - {{\varvec{q}}}_-$. ${{\varvec{Q}}}$ is the average location of the tip, and ${{\varvec{r}}}$ represents the fluctuation around ${{\varvec{Q}}}$. Secondly we expand $\phi$ by the orthonormalized eigen-functions $\psi _{mu}[{{\varvec{q}}}(x^0)]$ of the internal DOFs one-particle Hamiltonian $E+V$.

$$\begin{aligned} \phi [x^0,{{\varvec{q}}}(x^0)]= \sum _{\mu } a_{\mu }(x^0) \psi _{\mu }[{{\varvec{q}}}(x^0)]. \end{aligned}$$

(A2)

The time dependence of $\psi _{\mu }[{{\varvec{q}}}(x^0)]$ comes from $V({{\varvec{q}}})$. The total action can be written as

$$\begin{aligned} S_{\texttt {CTP}}&=S_\texttt {T}({{\varvec{q}}}_+,\phi _+)-S_\texttt {T}({{\varvec{q}}}_-,\phi _-)\nonumber \\&=\int _{0}^{x^0} \texttt {d}z^0 \left[ m\frac{\texttt {d}{{\varvec{Q}}}}{\texttt {d}z^0}\frac{\texttt {d}{{\varvec{r}}}}{\texttt {d}z^0}-U({{\varvec{Q}}}+\frac{{{\varvec{r}}}}{2})+U\left( {{\varvec{Q}}}-\frac{{{\varvec{r}}}}{2}\right) \right] \nonumber \\&\quad +\int _{0}^{x^0} \texttt {d}z^0 \sum _{\mu \nu } a_{+,\mu }^{\dagger }\left[ \delta _{\mu \nu }{} \texttt {i}\frac{\partial }{\partial z^0}-\psi _{\mu }^{\dagger }(E+V)\psi _{\nu }\right. \nonumber \\&\quad \left. +\psi _{\mu }^{\dagger }{} \texttt {i}\frac{\partial }{\partial z^0}\psi _{\nu }\right] a_{+,\nu }\nonumber \\&\quad -\int _{0}^{x^0} \texttt {d}z^0 \sum _{\mu \nu } a_{-,\mu }^{\dagger }\left[ \delta _{\mu \nu }{} \texttt {i}\frac{\partial }{\partial z^0}-\psi _{\mu }^{\dagger }(E+V)\psi _{\nu }\right. \nonumber \\&\quad \left. +\psi _{\mu }^{\dagger }{} \texttt {i}\frac{\partial }{\partial z^0}\psi _{\nu }\right] a_{-,\nu }. \end{aligned}$$

(A3)

After performing the integrals over the internal DOFs $(a,a^{\dagger })$, ${\mathcal {Z}}_{\texttt {in-in}}$ can be written as

$$\begin{aligned} {\mathcal {Z}}_{\texttt {in-in}}&=\int \texttt {d}{{\varvec{Q}}} \int \texttt {d}{{\varvec{r}}}\times \exp \{\texttt {i}\int _{0}^{x^0} \texttt {d}z^0 \left[ m\frac{d{{\varvec{Q}}}}{\texttt {d}z^0}\frac{d{{\varvec{r}}}}{dz^0}\right. \nonumber \\&\quad \left. -U\left( {{\varvec{Q}}}+\frac{{{\varvec{r}}}}{2}\right) +U\left( {{\varvec{Q}}}-\frac{{{\varvec{r}}}}{2}\right) \right] \}\nonumber \\ {}&\quad \times \left[ {\det }^+\left( \delta _{\mu \nu }{} \texttt {i}\frac{\partial }{\partial z^0}-\psi _{\mu }^{\dagger }(E+V)\psi _{\nu }+\psi _{\mu }^{\dagger }{} \texttt {i}\frac{\partial }{\partial z^0}\psi _{\nu }\right) \right] ^{f}\nonumber \\ {}&\quad \times \left[ {\det }^-\left( \delta _{\mu \nu }{} \texttt {i}\frac{\partial }{\partial z^0}-\psi _{\mu }^{\dagger }(E+V)\psi _{\nu }+\psi _{\mu }^{\dagger }{} \texttt {i}\frac{\partial }{\partial z^0}\psi _{\nu }\right) \right] ^{f}, \end{aligned}$$

(A4)

f is a numerical factor decided by that the internal DOFs are Bosonic or Fermionic. $\det ^+$ and $\det ^-$ correspond the functional determinant over the two time path. Therefore the effective action of the tip reads

$$\begin{aligned}&S_\texttt {eff}({{\varvec{Q}}},{{\varvec{r}}})\nonumber \\&=\int _{0}^{x^0} \texttt {d}z^0 \texttt {i}\left[ m\frac{\texttt {d}{{\varvec{Q}}}}{\texttt {d}z^0}\frac{\texttt {d}{{\varvec{r}}}}{\texttt {d}z^0}-U\left( {{\varvec{Q}}}+\frac{{{\varvec{r}}}}{2}\right) +U\left( {{\varvec{Q}}}-\frac{{{\varvec{r}}}}{2}\right) \right] \nonumber \\ {}&\quad + f \ln {\det }^+\left( \delta _{\mu \nu }{} \texttt {i}\frac{\partial }{\partial z^0}-\psi _{\mu }^{\dagger }(E+V)\psi _{\nu }+\psi _{\mu }^{\dagger }{} \texttt {i}\frac{\partial }{\partial z^0}\psi _{\nu }\right) \nonumber \\ {}&\quad + f \ln {\det }^-\left( \delta _{\mu \nu }{} \texttt {i}\frac{\partial }{\partial z^0}-\psi _{\mu }^{\dagger }(E+V)\psi _{\nu }+\psi _{\mu }^{\dagger }{} \texttt {i}\frac{\partial }{\partial z^0}\psi _{\nu }\right) . \end{aligned}$$

(A5)

At the semiclassical level ${{\varvec{r}}}$ is small, thus we can expand the total action around ${{\varvec{Q}}}$. To the first order of ${{\varvec{r}}}$, the effective action can be written as

$$\begin{aligned}&\texttt {i}S_\texttt {eff}({{\varvec{Q}}},{{\varvec{r}}})\nonumber \\&=-\int \texttt {d}z^0\texttt {i}\left[ m\left( \frac{\texttt {d}}{\texttt {d}z^0}\right) ^2{{\varvec{Q}}}+\frac{\partial U({{\varvec{Q}}})}{\partial {{\varvec{Q}}}}\right] {{\varvec{r}}}\nonumber \\ {}&\quad + f \ln {\det }^+\left( \delta _{\mu \nu }{} \texttt {i}\frac{\partial }{\partial z^0}-\psi _{\mu }^{\dagger }(E+V)\psi _{\nu }+\psi _{\mu }^{\dagger }{} \texttt {i}\frac{\partial }{\partial z^0}\psi _{\nu }\right) \nonumber \\ {}&\quad + f \ln {\det }^-\left( \delta _{\mu \nu }{} \texttt {i}\frac{\partial }{\partial z^0}-\psi _{\mu }^{\dagger }(E+V)\psi _{\nu }+\psi _{\mu }^{\dagger }{} \texttt {i}\frac{\partial }{\partial z^0}\psi _{\nu }\right) . \end{aligned}$$

(A6)

Moreover we consider the case that the velocity of the tip is small enough, that is to say $\dot{{{\varvec{q}}}}=\frac{\texttt {d}{{\varvec{q}}}}{\texttt {d}z^0}$ is small enough. And because the following equation

$$\begin{aligned} \texttt {i}\frac{\partial }{\partial z^0}\psi _{\nu }=\texttt {i}\dot{{{\varvec{q}}}}\frac{\partial }{\partial {{\varvec{q}}}}\psi _{\nu }. \end{aligned}$$

(A7)

We have

$$\begin{aligned}&\ln \det \left( \delta _{\mu \nu }{} \texttt {i}\frac{\partial }{\partial z^0}-\psi _{\mu }^{\dagger }(E+V)\psi _{\nu }+\psi _{\mu }^{\dagger }{} \texttt {i}\frac{\partial }{\partial z^0}\psi _{\nu }\right) \nonumber \\ {}&=\texttt {tr}\ln \left( D^{-1}+\psi _{\mu }^{\dagger }{} \texttt {i}\dot{{{\varvec{q}}}}\frac{\partial }{\partial {{\varvec{q}}}}\psi _{\nu }\right) \nonumber \\ {}&=\texttt {tr}\ln [D^{-1}\Big (1+D\psi _{\mu }^{\dagger }{} \texttt {i}\dot{{{\varvec{q}}}}\frac{\partial }{\partial {{\varvec{q}}}}\psi _{\nu }\Big ]\nonumber \\ {}&=\texttt {tr}\ln (D^{-1})+tr\ln \left( 1+D\psi _{\mu }^{\dagger }{} \texttt {i}\dot{{{\varvec{q}}}}\frac{\partial }{\partial {{\varvec{q}}}}\psi _{\nu }\right) \nonumber \\ {}&=\texttt {tr}\ln (D^{-1})+tr\left( D\psi _{\mu }^{\dagger }{} \texttt {i}\dot{{{\varvec{q}}}}\frac{\partial }{\partial {{\varvec{q}}}}\psi _{\nu }\right) \nonumber \\&\quad -\frac{1}{2} \left( D\psi _{\mu }^{\dagger }{} \texttt {i}\dot{{{\varvec{q}}}}\frac{\partial }{\partial {{\varvec{q}}}}\psi _{\nu }D\psi _{\mu }^{\dagger }{} \texttt {i}\dot{{{\varvec{q}}}}\frac{\partial }{\partial {{\varvec{q}}}}\psi _{\nu }\right) +.... \end{aligned}$$

(A8)

Here we introduce the operator $D=(\delta _{\mu \nu }{} \texttt {i}\frac{\partial }{\partial z^0}-\psi _{\mu }^{\dagger }(E+V)\psi _{\nu })^{-1}$ which is the Green’s function of the internal DOFs in the energy representation. Considering ${{\varvec{q}}}_+={{\varvec{Q}}}+\frac{{{\varvec{r}}}}{2}$ and ${{\varvec{q}}}_-={{\varvec{Q}}}-\frac{{{\varvec{r}}}}{2}$, and defining $C({{\varvec{q}}}):=\psi _{\mu }^{\dagger }\frac{\partial }{\partial {{\varvec{q}}}}\psi _{\nu }$, to the first order of ${{\varvec{r}}}$ we have

$$\begin{aligned} C({{\varvec{q}}}_+)=C({{\varvec{Q}}})+\frac{{{\varvec{r}}}}{2}\frac{\partial C({{\varvec{Q}}})}{\partial {{\varvec{Q}}}}, \end{aligned}$$

(A9)

and

$$\begin{aligned} C({{\varvec{q}}}_-)=C({{\varvec{Q}}})-\frac{{{\varvec{r}}}}{2}\frac{\partial C({{\varvec{Q}}})}{\partial {{\varvec{Q}}}}. \end{aligned}$$

(A10)

Therefore the term $\texttt {tr} D\psi _{\mu }^{\dagger }{} \texttt {i}\dot{{{\varvec{q}}}}_+\frac{\partial }{\partial {{\varvec{q}}}_+}\psi _{\nu })-\texttt {tr}(D\psi _{\mu }^{\dagger }{} \texttt {i}\dot{{{\varvec{q}}}}_-\frac{\partial }{\partial {{\varvec{q}}}_-}\psi _{\nu })$ in the effective action becomes

$$\begin{aligned} \texttt {tr}[\texttt {i}D\dot{{{\varvec{Q}}}} {{\varvec{r}}} \frac{\partial C({{\varvec{Q}}})}{\partial {{\varvec{Q}}}}], \end{aligned}$$

(A11)

the term $\texttt {tr}\ln D^{-1}$ is contributed from the ground state of the free internal DOFs, and it can be represented as $-\texttt {i}\int \texttt {d}z^0\bar{{{\varvec{F}}}}\cdot {{\varvec{r}}}$. The effective action becomes

$$\begin{aligned} \texttt {i}S_{\texttt {eff}}&=-\int \texttt {d}z^0\texttt {i}\left[ m\left( \frac{\texttt {d}}{\texttt {d}z^0}\right) ^2{{\varvec{Q}}}+\frac{\partial U({{\varvec{Q}}})}{\partial {{\varvec{Q}}}}-\bar{{{\varvec{F}}}}\right] {{\varvec{r}}}\nonumber \\ {}&\quad +\texttt {i}f\int \texttt {d}y^0 \int \texttt {d}z^0 D(y^0-z^0)\frac{\partial C({{\varvec{Q}}})}{\partial {{\varvec{Q}}}} \dot{{{\varvec{Q}}}} {{\varvec{r}}}\nonumber \\ {}&\quad +\frac{\texttt {i}}{2}\int \texttt {d}y^0 \int \texttt {d}z^0 {{\varvec{r}}}\Pi (y^0,z^0){{\varvec{r}}}, \end{aligned}$$

(A12)

$\Pi (y^0,z^0)$ is a integral kernel which can be determined by some complicated but not difficult calculations. This kernel can be obtained by Hubbard–Stratonovich transformation [19].

We set

$$\begin{aligned} \Theta \equiv m\left( \frac{\texttt {d}}{\texttt {d}z^0}\right) ^2{{\varvec{Q}}}+\frac{\partial U({{\varvec{Q}}})}{\partial {{\varvec{Q}}}}-\bar{{{\varvec{F}}}}-\int \texttt {d}y^0 D(y^0-z^0)\frac{\partial C({{\varvec{Q}}})}{\partial {{\varvec{Q}}}} \dot{{{\varvec{Q}}}}, \end{aligned}$$

(A13)

after integrating out the fluctuation ${{\varvec{r}}}$ in the Feynman amplitude, we obtain

$$\begin{aligned} {\mathcal {Z}}_{\texttt {in-in}}&=\int \texttt {d}{{\varvec{Q}}} \exp [-\texttt {i}\int \texttt {d}y^0 \int \texttt {d}z^0(\Theta \Pi ^{-1} \Theta )]\nonumber \\ {}&=\int \texttt {d}{{\varvec{Q}}} \int \texttt {d}\xi \delta (\Theta -\xi ) \exp [-\texttt {i}\int \texttt {d}y^0\nonumber \\&\quad \int \texttt {d}z^0(\xi \Pi ^{-1} \xi )]\equiv \int \texttt {d}{{\varvec{Q}}} \exp (\texttt {i}{\mathcal {A}}). \end{aligned}$$

(A14)

Here we introduce an auxiliary field $\xi$. It describes the stochastic force acting on the tip. The Gaussian expectation of $\xi$ and the second moment of $\xi$ are given as

$$\begin{aligned} <\xi >&=\int \texttt {d}\xi \exp [-\int \texttt {d}z^0 \int \texttt {d}w^0 \xi \Pi ^{-1} \xi ] \xi =0, \end{aligned}$$

(A15)

$$\begin{aligned} <\xi (x^0)\xi (y^0)>&=\int \texttt {d}\xi \exp [-\int \texttt {d}z^0 \int \texttt {d}w^0 \xi \Pi ^{-1} \xi ] \xi (x^0)\xi (y^0)\nonumber \\&=\Pi (x^0,y^0). \end{aligned}$$

(A16)

The angle brackets $<>$ mean the Gaussian expectation because the factor $\texttt {e}^{-\texttt {i}\int \texttt {d}y^0 \int \texttt {d}z^0(\xi \Pi ^{-1} \xi )}$ in Eq. (34) is a Gaussian distribution function. The EOM of the tip is given by $\exp (\texttt {i}{\mathcal {A}})=<\delta (\Theta -\xi )> \ne 0$, and ${\mathcal {A}}$ is the effective action of the DOF ${{\varvec{Q}}}$. The result is

$$\begin{aligned} m\left( \frac{\texttt {d}}{\texttt {d}z^0}\right) ^2{{\varvec{Q}}}+\frac{\partial U({{\varvec{Q}}})}{\partial {{\varvec{Q}}}}-\bar{{{\varvec{F}}}}-\int \texttt {d}y^0 D(y^0-z^0)\frac{\partial C({{\varvec{Q}}})}{\partial {{\varvec{Q}}}} \dot{{{\varvec{Q}}}}=\xi . \end{aligned}$$

(A17)

If we define the linear friction coefficient

$$\begin{aligned} \eta (y^0-z^0):=-D(y^0-z^0)\frac{\partial C({{\varvec{Q}}})}{\partial {{\varvec{Q}}}}, \end{aligned}$$

(A18)

the EOM becomes

$$\begin{aligned} m\left( \frac{\texttt {d}}{\texttt {d}z^0}\right) ^2{{\varvec{Q}}} =-\frac{\partial U({{\varvec{Q}}})}{\partial {{\varvec{Q}}}}+\bar{{{\varvec{F}}}}-\int dy^0 \eta (y^0-z^0) \dot{{{\varvec{Q}}}}+\xi . \end{aligned}$$

(A19)