Effective Lagrangian for Nambu-Goldstone modes in nonequilibrium open systems

TL;DR

This paper develops an effective field theory for diffusive Nambu-Goldstone modes in nonequilibrium open systems, revealing their unique diffusive gapless behavior and illustrating two typical physical scenarios.

Contribution

It introduces a novel effective Lagrangian framework for NG modes in open systems, highlighting diffusive behaviors absent in closed systems.

Findings

NG modes exhibit diffusive gapless behavior in open systems

Derived matching conditions for low-energy coefficients

Identified scenarios with diffusive type-A and type-B NG modes

Abstract

We develop the effective field theory of diffusive Nambu-Goldstone (NG) modes associated with spontaneous internal symmetry breaking taking place in nonequilibrium open systems. The effective Lagrangian describing semi-classical dynamics of the NG modes is derived and matching conditions for low-energy coefficients are also investigated. Due to new terms peculiar to open systems, the associated NG modes show diffusive gapless behaviors in contrast to the propagating NG mode in closed systems. We demonstrate two typical situations relevant to the condensed matter physics and high-energy physics, where diffusive type-A or type-B NG modes appear.

Equations

$$\begin{array}{rl}Z[j_1,j_2]& \equiv \mathrm{Tr}\left({\hat{\rho }}_0{\hat{U}}_{j_2}^{†}(\infty ,-\infty ){\hat{U}}_{j_1}(\infty ,-\infty )\right)\\ & =\int \mathcal{D}{\phi }_1\mathcal{D}{\phi }_2{e}^{\mathrm{i}(S[{\phi }_1;j_1]-S[{\phi }_2;j_2])}{\rho }_0[\phi ],\end{array}$$

$$Z[j_1,j_2]=\int \mathcal{D}{\varphi }_1\mathcal{D}{\varphi }_2{e}^{\mathrm{i}{S}_{\mathrm{open}}[{\varphi }_1,{\varphi }_2;j_1,j_2]}.$$

$$Z[j_1,j_2]=\int \mathcal{D}{\pi }_R\mathcal{D}{\pi }_A{e}^{\mathrm{i}{S}_{\mathrm{eff}}[{\pi }_R,{\pi }_A;j_1,j_2]}.$$

$$S_{\mathrm{eff}}[{\pi }_R,{\pi }_A=0]=0,$$

$$S_{\mathrm{eff}}[{\pi }_R,{\pi }_A{]}^{\ast }=-S_{\mathrm{eff}}[{\pi }_R,-{\pi }_A],$$

$$\mathrm{Im}{S}_{\mathrm{eff}}[{\pi }_R,{\pi }_A]\ge 0,$$

$${\xi }_i(\mathbf{\pi })\stackrel{G_i}{\to }{g}_i{\xi }_i(\mathbf{\pi })h_i^{-1}(\mathbf{\pi },g_i)\mathrm{with}{g}_i\in G_i,h_i\in H_i.$$

$$\begin{array}{rl}\xi ({\pi }_R)& \stackrel{G_A}{\to }g\xi ({\pi }_R)h^{-1}({\pi }_R,g),\\ {\pi }_A& \stackrel{G_A}{\to }h({\pi }_R,g){\pi }_A{h}^{-1}({\pi }_R,g).\end{array}$$

$$\begin{array}{rl}{\alpha }_{\perp }({\pi }_R)& \stackrel{G_A}{\to }h{\alpha }_{\perp }({\pi }_R)h^{-1},\\ {\alpha }_{\parallel }({\pi }_R)& \stackrel{G_A}{\to }h{\alpha }_{\perp }({\pi }_R)h^{-1}+{\mathrm{i}}^{-1}h\mathrm{d}{h}^{-1},\end{array}$$

$$\begin{array}{rl}& [X_I,X_J]=\mathrm{i}{X}_K{f}_{IJ}^K,\mathrm{tr}(X_I{X}_J)\equiv g_{IJ},\end{array}$$

$$\{\begin{array}{l}{\pi }_R\stackrel{G_R}{\to }{\pi }_R+O({\hslash }^2),\\ {\pi }_A\stackrel{G_R}{\to }{\pi }_A+2[e^{-\mathrm{i}{\pi }_R}\theta e^{i{\pi }_R}{]}_{\perp }+O({\hslash }^3),\end{array}$$

$$\begin{split}\mathcal{L}_{\mathrm{eff}}=-F&\Bigg{[}\frac{1}{f^{2}}\mathop{\mathrm{tr}}(\pi_{A}D_{0}\alpha_{0\perp})+\mathrm{i}\mathop{\mathrm{tr}}\big{(}m^{\alpha^{\prime}}X_{\alpha^{\prime}}[\pi_{A},\alpha_{0\perp}]\big{)}-\delta^{ij}\mathop{\mathrm{tr}}(\pi_{A}D_{i}\alpha_{j\perp})\\ &+\gamma\mathop{\mathrm{tr}}(\pi_{A}\alpha_{0\perp})+\mathrm{i}\mathop{\mathrm{tr}}\big{(}\zeta_{t}^{\alpha^{\prime}}X_{\alpha^{\prime}}[\pi_{A},D_{0}\alpha_{0\perp}]\big{)}+\mathrm{i}\delta^{ij}\mathop{\mathrm{tr}}\big{(}\zeta_{s}^{\alpha^{\prime}}X_{\alpha^{\prime}}[\pi_{A},D_{i}\alpha_{j\perp}]\big{)}-\mathrm{i}\frac{A}{2F}\mathop{\mathrm{tr}}(\pi_{A}\pi_{A})\Bigg{]},\end{split}$$

$$\begin{array}{rl}{\mathcal{L}}_{\mathrm{eff}}& =-{\rho }_{ab}{\pi }_A^a{\partial }_0{\pi }_R^b+g_{ab}^t{\partial }_0{\pi }_A^a{\partial }_0{\pi }_R^b-g_{ab}^s\nabla {\pi }_A^a\cdot \nabla {\pi }_R^b+\frac{\mathrm{i}}{2}{g}_{ab}^A{\pi }_A^a{\pi }_A^b\\ & =\frac{\mathrm{i}}{2}\left(\begin{array}{cc}{\pi }_R^a& {\pi }_A^a\end{array}\right)\left(\begin{array}{cc}0& \mathrm{i}(g_{ab}^t{\partial }_0^2-{\rho }_{ab}{\partial }_0-g_{ab}^s{\nabla }^2)\\ \mathrm{i}(g_{ab}^t{\partial }_0^2+{\rho }_{ab}{\partial }_0-g_{ab}^s{\nabla }^2)& g_{ab}^A\end{array}\right)\left(\begin{array}{c}{\pi }_R^b\\ {\pi }_A^b\end{array}\right),\end{array}$$

$${\rho }_{ab}$$

$$g_{ab}^t$$

$$(G_{R,A}^{ab}{)}^{-1}(\omega ,\mathbf{k})=-g_{ab}^t{\omega }^2\mp \mathrm{i}{\rho }_{ab}\omega +g_{ab}^s{\mathbf{k}}^2.$$

$$\{\begin{array}{l}{\delta }_{A_a}{\pi }_R^b=F{\delta }_a^b,\\ {\delta }_{A_a}{\pi }_A^b=0,\end{array}\mathrm{and}\{\begin{array}{l}{\delta }_{R_a}{\pi }_R^b=0,\\ {\delta }_{R_a}{\pi }_A^b=F{\delta }_b^a,\end{array}$$

$$\begin{array}{rl}{J}_{a,A}^{\mu }& =\left(\begin{array}{c}F(g_{ba}^t{\partial }_0{\pi }_A^b-{\rho }_{ba}{\pi }_A^b)\\ -F{g}_{ba}^s\nabla {\pi }_A^b\end{array}\right)+\cdots ,\\ J_{a,R}^{\mu }& =\left(\begin{array}{c}F(f^{-2}{g}_{ab}{\partial }_0{\pi }_R^b-m^{{\alpha }^{\prime }}{f}_{{\alpha }^{\prime }ab}{\pi }_R^b)\\ -F{g}_{ab}\nabla {\pi }_R^b\end{array}\right)+\cdots ,\end{array}$$

$$\begin{split}&\langle\delta_{A_{a}}\pi_{R}^{b}\rangle\big{|}_{\pi=0}=F\delta_{a}^{b},\quad\langle\delta_{R_{a}}J^{0}_{b,A}\rangle\big{|}_{\pi=0}=-F^{2}\rho_{ab},\\ &G_{J_{a,R}^{0}J_{b,A}^{0}}(\omega,\bm{k}=0)\big{|}_{\omega=0}-\frac{F^{2}}{\omega}m^{\alpha^{\prime}}f_{\alpha^{\prime}ab}=\mathrm{i}\frac{F^{2}}{f^{2}}g_{ab},\end{split}$$

$$\begin{split}&C^{0}_{ab}(\omega,\bm{k}=0)\big{|}_{\omega=0}=-\mathrm{i}F^{2}\gamma g_{ab},\\ &\partial_{\omega}C^{0}_{ab}(\omega,\bm{k}=0)\big{|}_{\omega=0}=-F^{2}\zeta_{t}^{\alpha^{\prime}}f_{\alpha^{\prime}ab},\\ &\partial_{k^{i}}C^{i}_{ab}(\omega=0,\bm{k})\big{|}_{\bm{k}=0}=-F^{2}\zeta_{s}^{\alpha^{\prime}}f_{\alpha^{\prime}ab},\end{split}$$

$$\begin{array}{rl}& G_{J_{a,R}^{\mu }{J}_{b,A}^{\nu }}(\omega ,\mathbf{k})\equiv \int {\mathrm{d}}^{d}x{e}^{\mathrm{i}\omega t-\mathrm{i}\mathbf{k}\cdot \mathbf{x}}⟨J_{a,R}^{\mu }(x)J_{b,A}^{\nu }(0)⟩,\\ & C_{ab}^{\nu }(\omega ,\mathbf{k})\equiv \int {\mathrm{d}}^{d}x{e}^{\mathrm{i}\omega t-\mathrm{i}\mathbf{k}\cdot \mathbf{x}}⟨{\partial }_{\mu }{J}_{a,R}^{\mu }(x)J_{b,A}^{\nu }(0)⟩.\end{array}$$

$${\mathcal{L}}_{\mathrm{open}}$$

$$+(\text{mbox}h.c.)+\frac{\mathrm{i}(A+2{\gamma }_0|{\varphi }_R{|}^2)}{2}|{\varphi }_A{|}^2+O({\varphi }_A^3),$$

$$\omega =-\mathrm{i}\frac{g}{m{\gamma }_0}{\mathbf{k}}^2+O({\mathbf{k}}^4).$$

$${\mathcal{L}}_{\mathrm{open}}$$

$$+(h.c.)+\mathrm{i}A{\mathbf{\phi }}_A^{†}{\mathbf{\phi }}_A,$$

$$\{\begin{array}{l}\omega =\frac{\pm m-\mathrm{i}\gamma }{m^2+{\gamma }^2}{\mathbf{k}}^2+O({\mathbf{k}}^4),\\ \omega =f^2(\pm m-\mathrm{i}\gamma )+\frac{\pm m+\mathrm{i}\gamma }{m^2+{\gamma }^2}{\mathbf{k}}^2+O({\mathbf{k}}^4).\end{array}$$

$${\xi }_{1,2}(\mathbf{\pi })\to {\xi }_{1,2}({\mathbf{\pi }}^{\prime })=e^{\pm \mathrm{i}\theta }{e}^{\mathrm{i}{\pi }_R}{e}^{\pm \mathrm{i}{\pi }_A/2}{e}^{-\mathrm{i}{\beta }_{1,2}(\mathbf{\pi },\theta )},$$

$$\begin{array}{rl}{\xi }_{1,2}(\mathbf{\pi })& =e^{\mathrm{i}{\pi }_R}\pm \frac{\mathrm{i}}{2}{e}^{\mathrm{i}{\pi }_R}{\pi }_A+O({\hslash }^2)\\ {\xi }_{1,2}({\mathbf{\pi }}^{\prime })& =\left[e^{\mathrm{i}{\pi }_R}+O({\hslash }^2)\right]\pm \frac{\mathrm{i}}{2}{e}^{\mathrm{i}{\pi }_R}\left[{\pi }_A+2e^{-\mathrm{i}{\pi }_R}\theta e^{\mathrm{i}{\pi }_R}\mp 2{\beta }_{1,2}+O({\hslash }^3)\right].\end{array}$$

$$\{\begin{array}{l}{\pi }_R^{\prime }={\pi }_R+O({\hslash }^2),\\ {\pi }_A^{\prime }={\pi }_A+2e^{-\mathrm{i}{\pi }_R}\theta e^{\mathrm{i}{\pi }_R}-2{\beta }_1+O({\hslash }^3)={\pi }_A+2e^{-\mathrm{i}{\pi }_R}\theta e^{\mathrm{i}{\pi }_R}+2{\beta }_2+O({\hslash }^3).\end{array}$$

$${\beta }_1=-{\beta }_2={\left[e^{-\mathrm{i}{\pi }_R}\theta e^{\mathrm{i}{\pi }_R}\right]}_{\parallel },$$

$$\begin{split}G_{J_{a,R}^{0}J_{b,A}^{0}}(\omega,\bm{k})&=\int\mathrm{d}^{d}xe^{\mathrm{i}\omega(t-t^{\prime})-\mathrm{i}\bm{k}\cdot(\bm{x}-\bm{x}^{\prime})}\Big{\langle}F\big{(}-\mathrm{i}\omega f^{-2}g_{ac}-m^{\alpha^{\prime}}f_{\alpha^{\prime}ac}\big{)}\pi_{R}^{c}(x)F\big{(}\mathrm{i}\omega g_{db}^{t}-\rho_{db}\big{)}\pi_{A}^{d}(x^{\prime})\Big{\rangle}\\ &=F^{2}\big{(}-\mathrm{i}\omega f^{-2}g_{ac}-m^{\alpha^{\prime}}f_{\alpha^{\prime}ac}\big{)}\big{(}\mathrm{i}\omega g_{db}^{t}-\rho_{db}\big{)}\frac{-\mathrm{i}}{-g_{cd}^{t}\omega^{2}-\mathrm{i}\rho_{cd}\omega+g_{cd}^{s}\bm{k}^{2}}\\ &\xrightarrow{\bm{k}\to 0}F^{2}\big{(}\mathrm{i}\omega f^{-2}g_{ac}+m^{\alpha^{\prime}}f_{\alpha^{\prime}ac}\big{)}\big{(}\mathrm{i}\omega g_{db}^{t}-\rho_{db}\big{)}\frac{1}{\mathrm{i}\omega g_{cd}^{t}-\rho_{cd}}\frac{1}{\omega}\\ &=F^{2}\big{(}\mathrm{i}f^{-2}g_{ab}+\omega^{-1}m^{\alpha^{\prime}}f_{\alpha^{\prime}ab}\big{)},\end{split}$$

$${\partial }_{\mu }{J}_{a,R}^{\mu }=F\left({\zeta }_t^{{\alpha }^{\prime }}{f}_{{\alpha }^{\prime }ab}{\partial }_0^2-\gamma g_{ab}{\partial }_0+{\zeta }_s^{{\alpha }^{\prime }}{f}_{{\alpha }^{\prime }ab}{\nabla }^2\right){\pi }_R^b+\mathrm{i}F{g}_{ab}{\pi }_A^b.$$

Full text

††preprint: KOBE-COSMO-19-10

Introduction.— Symmetry and its realization give fundamental descriptions of physical systems from condensed matter physics to high-energy physics. Global symmetry of the system, if it exists, remains unbroken, or spontaneously broken in a given situation—e. g. in the ground state—and the resulting symmetry realization restricts a possible low-energy spectrum contained in the system: if spontaneous symmetry breaking (SSB) of continuous global symmetry takes place, it leads to the inevitable appearance of gapless excitations known as the Nambu-Goldstone (NG) modes according to the NG theorem Nambu and Jona-Lasinio (1961); Goldstone (1961); Goldstone et al. (1962) with a low-dimensional exception protected by the Mermin-Wagner theorem Mermin and Wagner (1966); Hohenberg (1967); Coleman (1973).

Although the original NG theorem is applicable to systems respecting the Lorentz symmetry in the ground state, the scope of its application has been just recently extended to several interesting directions. One example is a generalization of the NG theorem to a nonrelativistic system. In the absence of the Lorentz symmetry, there is generally a mismatch between the number of NG modes and the number of the broken symmetries, and the associated (so-called type-B) NG mode often shows a quadratic dispersion relation $\omega=a\bm{k}^{2}\,(a\in\mathbb{R})$ Nielsen and Chadha (1976); Leutwyler (1994); Miransky and Shovkovy (2002); Schafer et al. (2001); Nambu (2004); Brauner (2010); Watanabe and Brauner (2011); Hidaka (2013); Watanabe and Murayama (2012, 2014); Hayata and Hidaka (2015). Furthermore, the notion of the symmetry and its spontaneous breaking are also extended to nonequilibrium closed systems Endlich et al. (2013); Grozdanov and Polonyi (2015); Haehl et al. (2015); Crossley et al. (2017); Haehl et al. (2016); Jensen et al. (2018a); Glorioso et al. (2017); Haehl et al. (2018); Jensen et al. (2018b) and open systems Minami and Hidaka (2018); Sieberer et al. (2016); Hongo et al. (2019); Hayata and Hidaka (2018), where diffusive gapless modes appear. To achieve these developments, one most powerful tool, the effective field theory (EFT) Coleman et al. (1969); Callan et al. (1969); Weinberg (1979), has been actively used. One can apply EFT to show a nonrelativistic generalization of the NG theorem Leutwyler (1994); Watanabe and Murayama (2012, 2014) and to describe nonequilibrium closed systems respecting conservation laws Endlich et al. (2013); Grozdanov and Polonyi (2015); Haehl et al. (2015); Crossley et al. (2017); Haehl et al. (2016); Jensen et al. (2018a); Glorioso et al. (2017); Haehl et al. (2018); Jensen et al. (2018b). Dissipative effects in the low-energy spectrum are captured with the help of the Schwinger-Keldysh (real-time) formalism Schwinger (1961); Keldysh (1964); Chou et al. (1985), and the doubled symmetry structure inherent in it plays a central role.

In this Letter, taking one step further, we develop EFT for the NG modes resulting from spontaneous symmetry breaking in nonequilibrium open systems (or non-Hermitian systems) where the conservation law is violated by considering e. g. the diffusive coupling between the system and environment Minami and Hidaka (2018); Sieberer et al. (2016); Hongo et al. (2019); Hayata and Hidaka (2018). Generalizing the Callan-Coleman-Wess-Zumino’s (CCWZ) coset construction Coleman et al. (1969); Callan et al. (1969) to the Schwinger-Keldysh formalism, we lay out a solid basis to construct the general effective Lagrangian for the open system NG modes, and apply it to two typical situations where the type-A and type-B NG modes appear.

Symmetry structure in open systems.— Real-time dynamics of quantum systems can be systematically described by the use of the Schwinger-Keldysh formalism Schwinger (1961); Keldysh (1964); Chou et al. (1985). One most basic quantity is the closed-time-path generating functional (CTPGF) defined by

[TABLE]

where $\hat{U}_{j}(t_{2},t_{1})$ denotes the time-evolution operator from $t_{1}$ to $t_{2}$ in the presence of the external field $j(t)$, $\hat{\rho}_{0}$ the initial density operator at $t=-\infty$. In the second line, we used the path-integral expression for a system composed of $\varphi$ with its action $S[\varphi;j]$. One crucial point for the Schwinger-Keldysh formalism is the doubled degrees of freedom $\varphi\to\{\varphi_{1},\varphi_{2}\}$ on the CTP. As a result, if the system originally enjoys $G$-symmetry, it is also doubled; the phase weight $e^{\mathrm{i}(S[\varphi_{1};j_{1}]-S[\varphi_{2};j_{2}])}$ is invariant under $(G_{1}\times G_{2})$-transformation acting on $\varphi_{1}$ and $\varphi_{2}$, respectively111 However, note that non-diagonal part of $(G_{1}\times G_{2})$ is explicitly broken due to the existence of the boundary, e. g. $\rho_{0}[\varphi]$. This broken symmetry is recently shown to be nonlinearly realized in the effective field theory for conserved systems such as dissipative hydrodynamics (See e. g. Crossley et al. (2017); Glorioso et al. (2017)). .

The charges attached to that symmetry can be diffused if the system is put under the influence of environments, and, as a result, the above symmetry structure is modified. For example, let us consider the total system composed of two kinds of dynamical variables $\varphi=\{\phi,\,\sigma\}$ with system variables $\phi$ and environment variables $\sigma$. After integrating out the environment variables, we obtain

[TABLE]

Then, the open system action $S_{\mathrm{open}}[\phi_{1},\phi_{2};j_{1},j_{2}]$ is, in general, not invariant under the nondiagonal part of $(G_{1}\times G_{2})$-transformation, which is a manifestation of violating conservation laws. Even in that cases, the diagonal subgroup of doubled symmetry, which we call $G_{A}$-symmetry, still survives, and we can consider its spontaneous breaking in open quantum systems Minami and Hidaka (2018); Sieberer et al. (2016); Hongo et al. (2019); Hayata and Hidaka (2018).

Suppose that the spontaneous $G_{A}$-symmetry breaking of the quantum open system takes place. Integrating out gapped degrees of freedom, we would like to develop the low-energy EFT for the associated NG fields:

[TABLE]

Here we introduced the effective action $S_{\mathrm{eff}}=\int\mathrm{d}^{d}x\mathcal{L}_{\mathrm{eff}}$ for $\{\pi_{R},\,\pi_{A}\}$—a combination of doubled NG fields in the so-called Keldysh basis whose properties will be elucidate in the next section. The vital point here is that we need to pay attention to several basic restrictions to the CTPGF (e. g. the probablity conservation), which can be manifestly respected by demanding the following conditions for the effective action (See e. g. Refs. Crossley et al. (2017); Glorioso et al. (2017) in detail)

[TABLE]

where we switched off the external field. In the following, we will construct the effective action with vanishing external field $j$ based on the coset construction.

Coset and Maurer-Cartan 1-form.— Let us first specify building blocks of the effective Lagrangian attached to spontaneous symmetry breaking of open systems $G_{A}\to H_{A}$ with unbroken symmetry $H_{A}$222We employ the standard assumption in the coset construction that the broken generators are closed under the action of broken symmetry generators.. Naïvely speaking, it may be natural to identify the associated NG fields as a coordinate of the coset $G_{A}/H_{A}$. Nevertheless, due to the basic structure of the Schwinger-Keldysh EFT, we need to include the doubled NG fields into the effective Lagrangian. With the help of a set of NG fields $\bm{\pi}=(\pi_{R},\pi_{A})$, we then introduce doubled cosets $\xi_{i}(\bm{\pi})\in G_{i}/H_{i}~{}(i=1,2)$, where the $G_{i}$-transformation acts as

[TABLE]

Since we are considering the situation where all non-diagonal parts of $G_{1}\times G_{2}$ are explicitly broken in the open system, we only need to respect the $G_{A}$-symmetry. By choosing a special representative $\xi_{1,2}(\bm{\pi})\equiv\xi(\pi_{R})e^{\pm\mathrm{i}\pi_{A}/2}$ with $\xi(\pi_{R})\equiv e^{\mathrm{i}\pi_{R}}$, which gives our definition of $\pi_{R}$ and $\pi_{A}$, we find the following simple transformation rule under $(g,g)\in G_{A}$-transfomation:

[TABLE]

In other words, we introduced the nonlinearly transforming NG field $\pi_{R}$ and the associated linearly transforming adjoint field $\pi_{A}$ under $G_{A}$-transformation. Note that $\pi_{R}$ and $\pi_{A}$ indeed parametrize the average and the difference parts of the doubled NG fields, which demand the basic constraints (4a)-(4c). Then, projections of the Maurer-Cartan 1-form $\alpha(\pi_{R})\equiv\mathrm{i}^{-1}\xi^{-1}(\pi_{R})\mathrm{d}\xi(\pi_{R})$, transform in the usual manner:

[TABLE]

where $\alpha_{\perp}\equiv\mathop{\mathrm{tr}}(X_{a}\alpha)X^{a}$ and $\alpha_{\parallel}\equiv\mathop{\mathrm{tr}}(X_{\alpha}\alpha)X^{\alpha}$. Among the generic generators $X_{I}$ of $G_{A}$ satisfying

[TABLE]

we introduced $X_{a}$ and $X_{\alpha}$ as generators which belong to broken and unbroken parts, respectively333 We will use the notation $I,J,\cdots$ to label generic generators of $G_{A}$, and $a,b,\cdots(\alpha,\beta,\cdots)$ for broken and (unbroken) ones. . In the following, assuming $g_{a\alpha}=0$, we lower (raise) the broken and unbroken indices by using the block diagonal part of the Cartan (inverse) metric $g_{ab}$ and $g_{\alpha\beta}$ ($g^{ab}$ and $g^{\alpha\beta}$).

On the other hand, it seems unnecessary to examine the explicitly broken $(g,g^{-1})\in G_{R}$-transformation property. Nevertheless, as will be shown later, we need them to clarify matching conditions for low-energy coefficients. Restricting ourselves to the semi-classical description of the NG fields, we obtain the following $G_{R}$-transformation rule SM (1):

[TABLE]

where we introduced $A_{\perp}\equiv\mathop{\mathrm{tr}}(X_{a}A)X^{a}$ with a transformation parameter $\theta\equiv\theta^{I}X_{I}$. Note that we count $\pi_{A}$ and $G_{R}$-transformation parameter as $O(\hbar)$ as is usual for the Schwinger-Keldysh EFT Crossley et al. (2017); Glorioso et al. (2017).

Effective Lagrangian and diffusive NG modes.— Taking account of basic constraints of the Schwinger-Keldysh EFT (4a)-(4c) and $G_{A}$-transformation properties developed in the previous section, we now construct the general $G_{A}$-invariant effective Lagrangian by the use of the $\alpha_{\perp,\parallel}$ and $\pi_{A}$ as basic building blocks. Within the semi-classical level—including up to $O(\pi_{A}^{2})$ terms, the leading-order $G_{A}$-invariant Schwinger-Keldysh effective Lagrangian is given as444For notational simplicity, we assume that the broken symmetry generators are irreducible under the unbroken symmetry transformation $H$. If not, the EFT parameters $\{f,m^{\alpha},\gamma,\zeta_{t}^{\alpha},\zeta_{s}^{\alpha},A\}$ may have different values among irreducible sectors

[TABLE]

where we defined a covariant derivative of $\alpha_{\perp}$ by $D_{\mu}\alpha_{\nu\perp}\equiv\partial_{\mu}\alpha_{\nu\perp}+\mathrm{i}[\alpha_{\mu\parallel},\alpha_{\nu\perp}]$ which transforms covariantly: $D_{\mu}\alpha_{\nu\perp}\xrightarrow{G_{A}}hD_{\mu}\alpha_{\nu\perp}h^{-1}$. We here rescaled the NG fields as $\pi_{R,A}\to\pi_{R,A}/F$, and indices $\alpha^{\prime}$ denotes the possible unbroken generators belonging to the center of $H_{A}$. A set of real parameters $\{f,m^{\alpha},\gamma,\zeta_{t}^{\alpha},\zeta_{s}^{\alpha},A\}$ gives low-energy coefficients, whose matching will be discussed shortly. The condition (4c) leads to $A\geq 0$, and when the charge is assumed to diffuse into the environment, we may also have $\gamma>0$. We note that it corresponds to (or defines) type-A or type-B NG mode whether terms proportional to $m^{\alpha^{\prime}}$ vanish or not Leutwyler (1994); Miransky and Shovkovy (2002); Schafer et al. (2001); Nambu (2004); Brauner (2010); Watanabe and Brauner (2011); Hidaka (2013); Watanabe and Murayama (2012, 2014); Hayata and Hidaka (2015). Terms appearing in the second line of Eq. (10) are not invariant under $G_{R}$-transformation. Hence, they are peculiar to the Schwinger-Keldysh EFT for nonequilibrium open systems, where $G_{R}$-symmetry is explicitly broken. To see the quadratic part of the effective Lagrangian, expanding the Maurer-Cartan 1-form as $\alpha_{\mu\perp}=\partial_{\mu}\pi_{R}/F+O(\pi_{R}^{2})$ and $\alpha_{\mu\parallel}=O(\pi_{R}^{2})$, we obtain

[TABLE]

where we defined the following quantities

[TABLE]

and neglected surface terms resulting from the integration by parts. We thus obtain the inverse retarded/advanced Green’s functions for the NG modes as

[TABLE]

Solving $\det(G_{R}^{ab})^{-1}(\omega,\bm{k})=0$ enables us to get the dispersion relation for the several types of NG modes depending on which parameters are present. Fig. 1 shows two typical dispersion relations of the type-A and type-B NG modes with non-vanishing $\gamma$ and vanishing $\zeta$-terms. We see the diffusive nature of the resulting NG modes due to the negative imaginary part $\mathrm{Im}\,\omega(\bm{k})<0$.

Let us then specify the matching condition for low-energy coefficients in $\rho_{ab}$ and $g_{ab}$’s based on the linearized effective Lagrangian (11). In contrast to SSB in the ground state, we generally have new couplings peculiar to nonequilibrium open systems: $\rho_{ab}$ can have symmetric components whereas $g_{ab}^{t}$ and $g_{ab}^{s}$ can have anti-symmetric components. To clarify the matching condition, noting that the leading infinitesimal transformation of the NG fields are given by $\pi^{b}_{R,A}\to\pi^{b}_{R,A}+\epsilon_{A,R}^{a}(\delta_{A_{a},R_{a}}\pi_{R,A}^{b}+O(\epsilon,\pi))$ with

[TABLE]

we turn our attention to the leading part of Noether currents attached to the $G_{A,R}$-transformations:

[TABLE]

where the ellipsis stands for higher-order terms with respect to $\pi_{R,A}$. Here $J^{\mu}_{a,R}$ is defined by the $G_{R}$-invariant part of the effective Lagrangian [corresponding to terms in the first line of Eq. (10)], and hence, not conserved in open systems. Using these, we can identify the matching conditions, which are sorted into the first category SM (1)555 It should be noticed that the currents $J_{A,R}^{\mu}$ are defined in the IR EFT (10). While $J_{A}^{\mu}$ is equivalent to the UV theory one up to equations of motion for gapped UV modes that are integrated out, it is nontrivial to identify the UV operator equivalent to the IR current $J_{R}^{\mu}$ essentially because the $G_{R}$ symmetry is explicitly broken. Also, the second condition in Eq. (17) involves a $G_{R}$ transformation, which is again an obstruction to the UV-IR matching. Therefore, further studies are required for establishing the UV-IR matching conditions, leaving it for future works.

[TABLE]

and the second one

[TABLE]

where we defined

[TABLE]

Here the angle bracket (with $\pi=0$) represents the path integral without dynamics of the NG fields. Note that $\gamma$ and $\zeta$-terms are matched by $C^{\nu}_{ab}(\omega,\bm{k})$ in the second category containing nonvanishing $\partial_{\mu}J_{a,R}^{\mu}(x)$ peculiar to open systems. While Eqs. (17)-(18) give matching conditions including $\zeta$-terms, we will consider two models with vanishing $\zeta$-terms in the following.

Examples.— As an example for a diffusive type-A NG mode, we consider a driven-dissipative BEC system with $U(1)$ symmetry Sieberer et al. (2016) whose Lagrangian reads

[TABLE]

where we introduced $\phi_{R}\equiv(\phi_{1}+\phi_{2})/2$ and $\phi_{A}\equiv\phi_{1}-\phi_{2}$ for the doubled Bosonic Schrödinger field. When $\kappa<0$ and $\gamma_{0}>0$, a driven-dissipative condensate $\overline{\phi}_{R}=ve^{-\mathrm{i}\omega_{0}t}$ could arise. Note that this solution is regarded as an example of time crystal in nonequilibrium open systems Minami and Hidaka (2018); Sieberer et al. (2016); Hongo et al. (2019); Hayata and Hidaka (2018) since it spontaneously breaks a mixed part of time-translation and $U(1)$ symmetry SM (2). In this case, terms proportional to $\zeta^{\alpha^{\prime}}_{t},\,\zeta^{\alpha^{\prime}}_{s}$, and $m^{\alpha^{\prime}}$ all vanish, and the dispersion relation for the resulting NG modes is obtained as

[TABLE]

We thus see that there exists a diffusive gapless mode, whose dispersion relation corresponds to small $|k|(<k_{c})$ part of Fig. 1 (a) with $k_{c}=f\gamma/2$ (the exceptional point). Note that the number of the gapless modes agrees with that of broken symmetries; ${\mathrm{rank}(g_{ab})}=\dim G_{A}/H_{A}$.

On the other hand, the dissipative $SU(2)\times U(1)$ model with a chemical potential defined by

[TABLE]

gives an example for the type-B NG mode Minami and Hidaka (2018). This model can be regarded as an effective model describing the Kaon condensation in the dense QCD matter Miransky and Shovkovy (2002); Schafer et al. (2001) coupled to the environment. Here $\bm{\varphi}_{R,A}$ denote doubled two-component complex scalar fields. One can find a stationary solution for $\bm{\varphi}_{R}$’s equation of motion parametrized by e. g. $\overline{\bm{\varphi}}_{R}=(0,v)$ with $v=\mu/\sqrt{2\lambda}$, which spontaneously breaks $G_{A}=SU(2)\times U(1)$ symmetry down to $H_{A}=U(1)$. Due to a nonvanishing anti-symmetric part of $\rho_{ab}$, this system contains the type-B NG mode and its gapped partner, whose dispersion relation is shown to be

[TABLE]

This gapless diffusive-propagating behavior of the type-B NG mode is shown in Fig. 1 (b). Taking account of one diffusive type-A NG mode, we see that the total number of the gapless NG modes is smaller than that of broken symmetry, which is peculiar to type-B NG modes.

Summary and discussion.— We have developed the EFT for the NG modes associated with SSB taking place in nonequilibrium open systems based on the coset construction. The derived effective Lagrangian enables us to describe the semi-classical dynamics of the diffusive NG modes, which can be applied to general open systems from open quantum system including non-Hermitian quantum systems to classical stochastic systems. As an application, we discussed the diffusive dispersion relation for type-A and type-B NG modes in two possible examples in condensed matter and high-energy physics.

Let us comment on some future problems. While we only investigate the dispersion relation of the diffusive NG modes with vanishing $\zeta_{t/s}$-terms, the constructed effective Lagrangian (10) contains much more information on many-body processes of the diffusive NG modes. It is interesting to investigate such information, e.g., low-energy theorems and possible instability caused by $\zeta_{t/s}$-terms based on our formalism. It would also be important to understand how the loop correction from NG fields affects the low-energy behavior of systems like the Mermin-Wagner theorem Mermin and Wagner (1966); Hohenberg (1967); Coleman (1973). Also, it is worthwhile clarifying an additional constraint, known as dynamical KMS/thermal symmetry Sieberer et al. (2015, 2016); Crossley et al. (2017); Glorioso et al. (2017); Aron et al. (2018) in thermal systems. Another interesting direction is to consider the SSB of spacetime symmetry in open systems. While the driven-dissipative BEC discussed in this Letter provides such a simplest example of time crystal, more general models may cause an instability towards the pattern formation Kuramoto (2012); Mori et al. (2013). We left these problems and wide applications of the developed EFT in many physical systems from cold atomic, condensed matter, high-energy, and active matter systems, as future works.

Appendix A Derivation of $G_{R}$-transformation rules (9)

We here give the derivation of $G_{R}$-transformation rules of the NG fields given in Eq. (9). For that purpose, let us explicitly write the $G_{A}$-transformation given in Eq. (5) as

[TABLE]

where $\theta=\theta^{I}X_{I}$ denotes the transformation parameter with the pullback $\beta(\bm{\pi},\theta)=\beta^{\alpha}(\bm{\pi},\theta)X_{\alpha}$. To determine the pullback $\beta(\bm{\pi},\theta)$, based on the power counting scheme in which $\pi_{A},~{}\theta$ and $\beta_{1,2}$ are all counted as $O(\hbar)$, we expand $\xi_{1,2}(\bm{\pi})$ and their transformations (S1) as follows:

[TABLE]

Comparing these, we can express the transformation rules for $\pi_{R}$ and $\pi_{A}$ as

[TABLE]

We therefore identify that the pullback $\beta_{1,2}$ takes the following form in our parametrization:

[TABLE]

where $A_{\parallel}\equiv\mathop{\mathrm{tr}}(X_{\alpha}A)X^{\alpha}$ is the unbroken component of $A$. This completes the semi-classical transformation rule of the NG fields given in Eq. (9).

Appendix B Derivation of the matching condition (17) and (18)

Since the first two matching conditions in Eq. (17) immediately follows from the transformation rule (15), we here provide the derivation of the others in Eqs. (17)-(18). By using the retarded Green’s function for NG modes given in Eq. (14), we can directly evaluate $G_{J_{a,R}^{0}J_{b,A}^{0}}(\omega,\bm{k})$ as follows:

[TABLE]

which gives the matching condition for $f$, or the third equation in the matching condition (17).

We can show the remaining ones in Eq. (18) in the similar manner if we notice that the equation of motion for $\pi_{R}^{a}$ together with the definition of $J_{a,R}^{\mu}$ in Eq. (16), brings about

[TABLE]

This equation is a manifestation of the non-conserving feature of open systems. Then, we can evaluate the low-frequency/wave number behavior of $C^{\nu}_{ab}(\omega,\bm{k})$ in the similar manner with $G_{J_{a,R}^{0}J_{b,A}^{0}}(\omega,\bm{k})$, using the retarded Green’s function of NG modes. Noting that $\langle\pi_{A}^{a}(x)\pi_{A}^{b}(x^{\prime})\rangle=0$ thanks to the unitarity condition (4a), we can evaluate the time-component $C^{0}_{ab}(\omega,\bm{k})$ as

[TABLE]

whose $\omega\to 0$ limit and the derivative with respect to $\omega$ give the matching conditions for $\gamma$ and $\zeta_{t}$—the first two equations in Eq. (18). Also, the spatial component $C^{i}_{ab}(\omega,\bm{k})$ is similarly evaluated as

[TABLE]

which gives the matching condition for $\zeta_{s}$—the third equation in Eq. (18).

Appendix C Diffusive type-A NG mode in driven-dissipative Bose-Einstein condensate

We here provide a detailed analysis of the type-A NG mode followed by the presence of the driven-dissipative Bose-Einstein condensate. Our starting point is the Schwinger-Keldysh Lagrangian

[TABLE]

where we defined the Keldysh-basis fields $\phi_{R}=(\phi_{1}+\phi_{2})/2$ and $\phi_{A}\equiv\phi_{1}-\phi_{2}$. We truncated the action at the quadratic order with respect to $A$-type field, which is regarded as the semiclassical approximation to open quantum systems described by the Lindblad equation (See, e. g., Ref. Sieberer et al. (2016) for a review). Here $(\mathrm{h.c.})$ represents the Hermitian conjugate, and $\kappa<0$ and $\gamma_{0}>0$ denote a driven particle-injection term and a dissipative term describing a nonlinear particle loss.

Using infinitesimal parameters $\epsilon$ for time-translational symmetry $\mathbb{R}_{t}$, and $\theta$ for internal $U(1)$ symmetry, we define infinitesimal $G_{A}$ and $G_{R}$ transformations in the semiclassical regime as

[TABLE]

where only $G_{A}$-symmetry is respected in Eq. (S9).

Let us then investigate a homogeneous symmetry breaking solution $\overline{\phi}_{R}$ and $\overline{\phi}_{A}$, on the top of which the type-A diffusive NG mode appears. Noting $\overline{\phi}_{A}=0$ resulting from the unitarity condition, we obtain the mean-field equation of motion for $\overline{\phi}_{R}$:

[TABLE]

The driven particle injection $\kappa<0$ make the trivial solution $\overline{\phi}_{R}=0$ unstable, and as a result, there appears a nontrivial solution with a time-oscillating homogeneous condensate given by

[TABLE]

Due to the time-oscillating condensate, this solution breaks $G_{A}=\mathbb{R}_{t}\times U(1)_{M}$ symmetry down to $H_{A}=U(1)_{M+t}$. Here $U(1)_{M+t}$ denotes the combination of the time-translation and $U(1)$ transformation satisfying $\omega_{0}\epsilon_{A}-\theta_{A}=0$, which let the driven-dissipative condensate (S12) invariant.

Next, we consider the fluctuation on the top of the above symmetry breaking solution and derive the effective Lagrangian for the diffusive NG mode. The embedding of Nambu-Goldstone mode reads

[TABLE]

where $\pi_{R,A}$ and $\sigma_{R,A}$ represent a phase (Nambu-Goldstone) and gapped amplitude field, respectively. To see the semiclassical dynamics of those fields, we dropped the higher-order terms including more than two $A$-type fields. Substituting the embedding (S13) into Eq. (S9) and focusing on the leading-order quadratic terms, we obtain the following result:

[TABLE]

After integrating out the gapped mode $\sigma$ and taking the low-energy limit $\bm{\nabla}^{2}/m\ll 4gv^{2}$, we obtain the effective Lagrangian of the NG fields as

[TABLE]

where we introduced $\bar{A}\equiv A+4\gamma_{0}v^{2}$. The derived effective Lagrangian provides the following inverse retarded Green’s function in the Fourier space:

[TABLE]

which results in the expected diffusive dispersion relation for the type-A NG mode

[TABLE]

We can also specify the $G_{A,R}$ transformation rules of NG modes as

[TABLE]

from which one see that $\pi_{R/A}$ nonlinearly transforms under the $G_{A/R}$ transformation while linearly transforms under the $G_{R/A}$ transformation.