上个学期期末做的有限温场论大作业。学习并讲了一下一直有兴趣的Keldysh场论。
主要就是学习了这篇文章的一部分:Sieberer, L. M., Buchhold, M., & Diehl, S. (2016). Keldysh field theory for driven open quantum systems. Reports on Progress in Physics, 79(9), 096001.
还有之前入门学的一本书:Kamenev, A. (2011). Field Theory of Non-Equilibrium Systems. Cambridge: Cambridge University Press.
有驱动开放系统的Keldysh场论
Basics of Keldysh Field Theory
Motivation: Why do we need Keldysh field theory?
- In some open quantum systems, besides coherence, driven–dissipative dynamics also occur on equal footing.
- Dynamical non-equilibrium quantum many-body phases emerge in such systems.
- Studying such systems calls for a merger of many-body field theory in equilibrium and quantum mater equation.
Closed Time Contour
- Equilibrium field theory use only one-way time path, while for non-equilibrium case we need to consider a closed time contour.
- Let
be an evolution from to , then the density at time is . - The expected value of an observable
at time is:
- In equilibrium field theory, only the forward evolution is considered, because the backward path can be eliminated by
which means the adiabatic evolution takes the GS at
- With the condition in
, the value of can be written in only forward evolution:
- The normalization denominator offsets
, and it is amounting to subtracting all disconnected vacuum diagrams. In non-equilibrium cases,
is not satisfied, so a closed time contour is needed. The expectation value of
in closed time contour:
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{pic/ClosedContour.png}
\label{fig:contour}
\end{figure}
The closed time contour is the central idea of non-equilibrium field theory.
Keldysh Partition Function
- With a closed time contour
, the partition function is defined as:
- With
, the partition function is normalized to . - Adding a source
, the partition function becomes:
- And the expected value of
is the derivative of :
- In contrast, in finite temperature field theory, the exp value is the derivative of
, which subtracts all disconnected terms.
Markovian Keldysh Action from the Master Equation
- Now we proceed from the master equation to Keyldysh action.
- The open quantum system is described by quantum master equation:
- Using the coherent states
as basis, the Keldysh (closed contour) version of is:
- The subscript
represents the forward/backward part of the contour.
\begin{frame}{Keldysh Action}
\small
\begin{itemize}
\item The partition function
\begin{equation}
\begin{aligned}
Z&=\int \mathcal{D}[\psi_+,\psi_+^,\psi_-,\psi_-^] e^{iS}=1,\
S&=\int_{-\infty}^{+\infty}dt (\psi_+^i\partial_t \psi_+-\psi_-^i\partial_t \psi_- -i\mathcal{L})
\end{aligned}
\end{equation}
Note that all the cross terms of $\psi_\sigma^ \psi_{\sigma’}, \sigma \neq \sigma’
\end{itemize}
\end{frame}
\begin{frame}{Keldysh Rotation and Green’s Functions}
\small
\begin{itemize}
\item After the Keldysh rotation, we can calculate the correlation, or Green’s function, between the classical and quantum field, and we’ll get a matrix like:
\begin{equation}
\begin{aligned}
&\left(\begin{array}{cc}
\langle\phi_{c}(t, \mathbf{x}) \phi_{c}^{} (t^{\prime}, \mathbf{x}^{\prime} ) \rangle_{d} & \langle\phi_{c}(t, \mathbf{x}) \phi_{q}^{} (t^{\prime}, \mathbf{x}^{\prime} ) \rangle_{d} \
\langle\phi_{q}(t, \mathbf{x}) \phi_{c}^{} (t^{\prime}, \mathbf{x}^{\prime} ) \rangle_{d} & \langle\phi_{q}(t, \mathbf{x}) \phi_{q}^{} (t^{\prime}, \mathbf{x}^{\prime} ) \rangle_{d}
\end{array} \right)\
&=-\left(\begin{array}{cc}
\frac{\delta^2Z}{\delta j_q^(t,\mathbf{x})\delta j_q(t’,\mathbf{x’})} & \frac{\delta^2Z}{\delta j_q^(t,\mathbf{x})\delta j_c(t’,\mathbf{x’})} \
\frac{\delta^2Z}{\delta j_c^(t,\mathbf{x})\delta j_q(t’,\mathbf{x’})} & \frac{\delta^2Z}{\delta j_c^(t,\mathbf{x})\delta j_c(t’,\mathbf{x’})}
\end{array}\right)
=\mathrm{i} \left(
\end{aligned}
\end{equation}
\item Here the block of 0 shows directly the redundancy in
\item The superscript
\item And the subscript
\end{itemize}
\end{frame}
\section{Keldysh Field Theory by Example}
\begin{frame}{Model of a Single Mode Cavity}
\small
\begin{itemize}
\item Now we show the Keldysh field theory on a simple model.
\item The Hamiltonian of single mode photons in the cavity is
with
\item For the master equation (
\item The Keldysh action is:
\begin{equation}
\begin{aligned}
S=\int_t&\{a_+^(i\partial_t-\omega_0)a_+-a_-^(i\partial_t-\omega_0)a_-\
&-i\kappa[2a_+a_-^-(a_+^a_++a_-^*a_-)]\}
\end{aligned}
\end{equation}
\end{itemize}
\end{frame}
\begin{frame}{Model of a Single Mode Cavity}
\small
\begin{itemize}
\item Performing Keldysh rotation and Fourier trans into the frequency space, the action is:
\begin{equation}
S=\int_\omega\left(a_c^(\omega),a_q^(\omega)\right)\left(
\left(
\end{equation}
\item Here
\item From the
\end{itemize}
\end{frame}
\begin{frame}{Properties of Keldysh Green’s Function}
\small
\begin{itemize}
\item \textbf{Conservation of probability.} The partition func
\item \textbf{Analytic structure.} The poles of
\item The inverse Fourier transformation of the retarded Green’s function is
\item With the Heaviside func
\end{itemize}
\end{frame}
\begin{frame}{External Source}
\small
\begin{itemize}
\item The partition function with source
\begin{equation}
\begin{aligned}
Z[j_c,j_q]&=\int \mathcal{D}[a_c^,a_q^,a_c,a_q] \exp(iS-iS_j),\
S_j&=\int_t j_c^a_q+j_q^a_c+c.c.
\end{aligned}
\end{equation}
\item And from the derivation of
\item Note that
\item The ‘classical’ source
\end{itemize}
\end{frame}
\begin{frame}{Response and Correlation Functions}
\small
\begin{itemize}
\item \textbf{The retarded Green’s function
\item For the case of single mode cavity, if the external photons interact with the photons inside,
\item Then the
\begin{equation}
S-j = S-\int_t (j^(t)a_q(t)+j(t)a_q^(t))
\end{equation}
\item Assume the coupling is weak. To the first order of the current, the photon field in the cavity is then
\begin{equation}
\begin{aligned}
\langle a_c(t)\rangle_j &= \langle a_c(t)\rangle_{j=0}-i\sqrt \kappa\int_{t’}\langle a_c(t)a_q^(t’)\rangle_{j=0}j(t’)\
&=\langle a_c(t)\rangle_{j=0}+\sqrt \kappa\int_{t’}G^R(t-t’)j(t’)
\end{aligned}
\end{equation}
\end{itemize}
\end{frame}
\begin{frame}{Driven-dissipative Condensate}
\small
\begin{equation}\label{eq:realAction}
\begin{aligned}
S=&\int_{t,x}\{\phi_q^*(i\partial_t+K_c\nabla^2-r_c+ir_d)\phi_c+c.c\\
&-[(u_c-iu_d)(\phi_q^*\phi_c^*\phi_c^2+\phi_q^*\phi_c^*\phi_q^2)+c.c]\\
&+i2(\gamma+2u_d\phi_c^*\phi_c)\phi_q^*\phi_q\}
\end{aligned}
\end{equation}
\begin{itemize}
\item Here $K_c=1/(2m_{LP})$, $r_c=\omega_{LP}^0$. And the noise parameter is defined as $\gamma=(\gamma_l+\gamma_p)/2$ and $r_d=(\gamma_l-\gamma_p)/2$. l-loss, p-pumping.
\end{itemize}
\end{frame}
\begin{frame}{Driven-dissipative Condensate}
\small
\begin{itemize}
\item Solving for this action is hard. Now we consider a simple mean-field analysis that can lead us to the condensation. Mean-field means fluctuations of the field is neglected:
\begin{equation}\label{eq:meanField}
\frac{\delta S}{\delta \phi_c^}=0,
\frac{\delta S}{\delta \phi_q^}=0
\end{equation}
\item Inserting mean-field condition into the action (
\item For the Im part of (
\item This result shows a driven-dissipative condensate.
\end{itemize}
\end{frame}
\section{Semi-classical Limit}
\begin{frame}{Canonical Scaling Analysis}
\small
\begin{itemize}
\item To study the long-wavelength phenomenon, we need to consider the \textbf{semiclassical limit}. And this could be done by RG that ignores small-
\item For the Keldysh action talked earlier in (
\item After the scaling analysis, any quartic term that includes more than one quantum field is irrelevant. The action becomes:
\begin{equation}
\begin{aligned}
S=&\int_{t,x}\{\phi_q^(i\partial_t+(K_c-iK_d)\nabla^2-r_c+ir_d)\phi_c+c.c\
&-[(u_c-iu_d)\phi_q^\phi_c^\phi_c^2+c.c]
+i2\gamma\phi_q^\phi_q\}
\end{aligned}
\end{equation}
\end{itemize}
\end{frame}
\begin{frame}{Canonical Scaling Analysis}
\small
\begin{itemize}
\item Using the Hubbard–Stratonovich transformation (detailed in class), the Gauss integral is changed to basis $[\phi_c^,\phi_c,\xi^,\xi]$.
\begin{equation}
\begin{aligned}
Z &=\int \mathscr{D}\left[\phi_{c}, \phi_{c}^{}, \xi, \xi^{}\right] \mathrm{e}^{-\frac{1}{2 \gamma} \int_{t, \mathrm{x}} \xi \xi} \
& \times \delta\left(\left[\mathrm{i} \partial_{t}+\left(K_{c}-\mathrm{i} K_{d}\right) \nabla^{2}-r_{c}+\mathrm{i} r_{d}-\left(u_{c}-\mathrm{i} u_{d}\right)\left|\phi_{c}\right|^{2}\right] \phi_{c}-\xi\right) \
& \times \delta\left(\left[-\mathrm{i} \partial_{t}+\left(K_{c}+\mathrm{i} K_{d}\right) \nabla^{2}-r_{c}-\mathrm{i} r_{d}-\left(u_{c}+\mathrm{i} u_{d}\right)\left|\phi_{c}\right|^{2}\right] \phi_{c}^{}-\xi^{}\right) .
\end{aligned}
\end{equation}
Here
\item The
\end{itemize}
\end{frame}
\section{Symmetries of the Keldysh Action}
\begin{frame}{Thermodynamic Equilibrium}
\small
\begin{itemize}
\item The FDRs holds for correlation functions of arbitrary order, not just the 2-point Green’s functions in
\item And the
\item Careful analysis gives a precise form of the symmetry transformation:
\begin{equation}\label{eq:thermalSym}
\begin{aligned}
&\mathcal{T}_{\beta} \psi_{\sigma}(t, \mathbf{x})=\psi_{\sigma}^{}(-t+\mathrm{i} \sigma \beta / 2, \mathbf{x}) \
&\mathcal{T}_{\beta} \psi_{\sigma}^{}(t, \mathbf{x})=\psi_{\sigma}(-t+\mathrm{i} \sigma \beta / 2, \mathbf{x})
\end{aligned}
\end{equation}
\item So thermal equilibrium can be diagnosed by a simple symmetry test on the Keldysh action.
\end{itemize}
\end{frame}
\begin{frame}{Semi-classical Limit of the Thermal Symmetry}
\small
\begin{itemize}
\item Semi-classical limit: typical frequency scale is much smaller than
\item Firstly, the symmetry operation
\item Secondly, the equivalent Langevin equation in (
\begin{equation}
i\partial_t \psi = \frac{\delta H_c}{\delta \psi^}-i\frac{\delta H_d}{\delta \psi^} +\xi
\end{equation}
\item The effective
\end{itemize}
\end{frame}
\begin{frame}{Noether’s Theorem in Keldysh Formalism}
\small
\begin{itemize}
\item Assume a transformation
\end{itemize}
\end{frame}
\begin{frame}{Extended Continuity Equation in Open Systems}
\small
\begin{itemize}
\item In open system, the particle number is not conserved.
\item Keldysh action is symmetry under classical phase rotation
\item Now consider the action (
\begin{equation}
\left\langle\partial_{\mu} j_{c}^{\mu}\right\rangle-\gamma_{p}\left\langle\psi_{+}^{} \psi_{-}\right\rangle+\gamma_{l}\left\langle\psi_{-}^{} \psi_{+}\right\rangle+4 u_{d}\left\langle\left(\psi_{-}^{} \psi_{+}\right)^{2}\right\rangle=0
\end{equation}
\item Here we have terms proportional to
\item In open systems, however, we can still add a dissipative current and measure the local mass flow by
\end{itemize}
\end{frame}