# Dynamical Symmetries and Causality in Non-Equilibrium Phase Transitions

## Abstract

**:**

## 1. Introduction

#### 1.1. Conformal Algebra

#### 1.2. Schrödinger Algebra

- Why should it be obvious that the time difference ${t}_{1}-{t}_{2}>0$, to make the power-law prefactor real-valued ?
- Given the convention that ${\mathcal{M}}_{1}\ge 0$, the condition ${t}_{1}-{t}_{2}>0$ is also required in order to have a decay of the two-point function with increasing distance $\left|\mathit{r}\right|=|{\mathit{r}}_{1}-{\mathit{r}}_{2}|\to \infty $.
- In applications to non-equilibrium statistical physics, one studies indeed two-point functions of the above type, which are then interpreted as the linear response function of the scaling operator ϕ with respect to an external conjugate field $h(t,\mathit{r})$$$R({t}_{1},{t}_{2};{\mathit{r}}_{1},{\mathit{r}}_{2})={\left.\frac{\delta \langle \varphi ({t}_{1},{\mathit{r}}_{1})\rangle}{\delta h({t}_{2},{\mathit{r}}_{2})}\right|}_{h=0}=\u2329\varphi ({t}_{1},{\mathit{r}}_{1})\tilde{\varphi}({t}_{2},{\mathit{r}}_{2})\u232a$$Then, the formal condition ${t}_{1}-{t}_{2}>0$ simply becomes the causality condition, namely that a response will only arise at a later time ${t}_{1}>{t}_{2}$ after the stimulation at time ${t}_{2}\ge 0$.

#### 1.3. Conformal Galilean Algebra

**Lemma 1.**[33] One has the isomorphism, where ⋉ denotes the semi-direct sum

- Why should one have ${t}_{1}-{t}_{2}>0$ for the time difference, as required to make the power-law prefactor real-valued ?
- Even for a fixed vector ${\mathit{\gamma}}_{1}$ of rapidities, and even if ${t}_{1}-{t}_{2}>0$ could be taken for granted, how does one guarantee that the scalar product ${\mathit{\gamma}}_{1}\xb7({\mathit{r}}_{1}-{\mathit{r}}_{2})>0$, such that the two-point function decreases as $\left|\mathit{r}\right|=|{\mathit{r}}_{1}-{\mathit{r}}_{2}|\to \infty $ ?

#### 1.4. Ageing Algebra

#### 1.5. Langevin Equation and Reduction formulæ

**Theorem 1.**[44] If in the functional $\mathcal{J}[\varphi ,\tilde{\varphi}]={\mathcal{J}}_{0}[\varphi ,\tilde{\varphi}]+{\mathcal{J}}_{b}\left[\tilde{\varphi}\right]$, the part ${\mathcal{J}}_{0}$ is Galilei-invariant with non-vanishing masses and ${\mathcal{J}}_{b}\left[\tilde{\varphi}\right]$ does not contain the field ϕ, then the computation of all responses and correlators can be reduced to averages which only involve the Galilei-invariant part ${\mathcal{J}}_{0}$.

**Proof.**We illustrate the main idea for the calculation of the two-time response. Define the average ${\langle X\rangle}_{0}=\int \phantom{\rule{-0.166667em}{0ex}}\mathcal{D}\varphi \mathcal{D}\tilde{\varphi}\phantom{\rule{0.222222em}{0ex}}X\left[\varphi \right]{e}^{-{\mathcal{J}}_{0}[\varphi ,\tilde{\varphi}]}$ with respect to the functional ${\mathcal{J}}_{0}[\varphi ,\tilde{\varphi}]$. Then, from Equation (23)

## 2. Representations

**Proposition 1.**Let γ be a constant and $g\left(z\right)$ a non-constant function. Then the generators

**Proposition 2.**If $\varphi \left(z\right)$ is a quasi-primary scaling operator under the representation Equation (24) of the conformal algebra $\langle {\ell}_{\pm 1,0}\rangle $, its co-variant two-point function is, where ${\phi}_{0}$ is a normalisation constant

**Proof.**For brevity, denote $F({z}_{1},{z}_{2})=\u2329{\varphi}_{1}\left({z}_{1}\right){\varphi}_{2}\left({z}_{2}\right)\u232a$. Then the co-variance of F is expressed by the three Ward identities, with ${\partial}_{i}:=\partial /\partial {z}_{i}$

**Proposition 3.**If one replaces in the representation Equation (5) the generator ${X}_{n}$ as follows

**Proposition 4.**Consider the representation Equation (5), but with the generators ${X}_{n}$ replaced by Equation (26), of the ageing algebra $\mathfrak{age}\left(d\right)$ and the Schrödinger algebra $\mathfrak{sch}\left(d\right)$. The invariant Schrödinger operator has the form

**Proof.**To shorten the calculations, we restrict here to $d=1$. It is enough to restrict attention to the generators ${X}_{\pm 1,0}$, and we must reproduce Equation (8) in this more general setting. We first look at $\mathfrak{age}\left(1\right)$. Consideration of ${X}_{0}$ gives $t\dot{v}\left(t\right)+v-\dot{\Xi}\left(t\right)=0$ and considering ${X}_{1}$ gives $x+\xi -\frac{1}{2}+\Xi \left(t\right)+t\dot{\Xi}\left(t\right)-2tv\left(t\right)-{t}^{2}\dot{v}\left(t\right)=0$, where the dot denotes the derivative with respect to t. The second relation can be simplified to $x+\xi -\frac{1}{2}+\Xi \left(t\right)-tv\left(t\right)=0$ which gives the assertion. Going over to $\mathfrak{sch}\left(1\right)$, the condition $[\mathcal{S},{X}_{-1}]=0$ leads to $\xi /{t}^{2}+\dot{\Xi}\left(t\right)/t-\Xi \left(t\right)/{t}^{2}-\dot{v}\left(t\right)=0$. This is only compatible with the result found before for $\mathfrak{age}\left(1\right)$, if $\xi =-x-\xi +\frac{1}{2}$, hence $x=\frac{1}{2}-2\xi $, as claimed. ☐

**Example 1.**For a physical illustration of the meaning of the explicitly time-dependent terms in the Schrödinger operator Equation (27), we consider the growth of an interface [46]. One may imagine that an interface can be created by randomly depositing particle onto a substrate. The height of this interface will be described by a function $h(t,\mathit{r})$. One usually works in a co-moving coordinate system such that the average height $\langle h(t,\mathit{r})\rangle =0$ which we shall assume from now on. Then physically interesting quantities are either the interface width $w\left(t\right)=\langle h{(t,\mathit{r})}^{2}\rangle \sim {t}^{\beta}$, which for sufficiently long times t defines thegrowth exponentβ, or else two-time height-height correlators $C(t,s;\mathit{r})=\langle h(t,\mathit{r})h(s,\mathbf{0})\rangle $ or two-time response functions $R(t,s;\mathit{r})={\left.\frac{\delta \langle h(t,\mathit{r})\rangle}{\delta j(s,\mathbf{0})}\right|}_{j=0}$, with respect to an external deposition rate $j(t,\mathit{r})$. Their scaling behaviour is described by several non-equilibrium exponents [1,2]. Herein, spatial translation-invariance was assumed for the sake of simplicity of the notation.

**Example 2.**We give a different illustration of the new representations of $\mathfrak{age}\left(d\right)$ with $\xi \ne 0$ (and $\Xi \left(t\right)=0$). Although we shall not be able to write down explicitly the invariant Schrödinger operator of the form specified in Equation (27), this example makes it clear that the domain of application of these representations extends beyond the context of that single differential equation.

**Ψ**. Furthermore, the autocorrelator scaling function should be non-singular as $t\to s$. This implies $\mathbf{\Psi}\left(w\right)\sim {w}^{{\tilde{x}}_{2}-x-4\xi -d/2+\mu}$ for $w\gg 1$. The most simple case arises when this form remains valid for all w. Using the values of the scaling exponents identified from the autoresponse $R(t,s)$ before, the exact $1D$ Glauber–Ising autocorrelator Equation (32) is recovered from Equation (36), with the choice $\mu =-\frac{1}{4}$ and ${C}_{0}=2/\sqrt{\pi}$ [38].

**Table 1.**The “lattice” representations of the Schrödinger algebra $\mathfrak{sch}\left(1\right)$, and its “continuum” representation, to which it reduces in the limit $a\to 0$ [61].

Generator | Continuum | Lattice |
---|---|---|

${X}_{-1}$ | $-{\partial}_{t}$ | $-{\partial}_{t}$ |

${X}_{0}$ | $-t{\partial}_{t}-\frac{1}{2}r{\partial}_{r}$ | $-t{\partial}_{t}-\frac{1}{acosh\left(\frac{a}{2}\phantom{\rule{0.166667em}{0ex}}{\partial}_{r}\right)}rsinh\left(\frac{a}{2}\phantom{\rule{0.166667em}{0ex}}{\partial}_{r}\right)$ |

${X}_{1}$ | $-{t}^{2}{\partial}_{t}-tr{\partial}_{r}-\frac{1}{2}\mathcal{M}{r}^{2}$ | $-{t}^{2}{\partial}_{t}-\frac{2t}{acosh\left(\frac{a}{2}\phantom{\rule{0.166667em}{0ex}}{\partial}_{r}\right)}rsinh\left(\frac{a}{2}\phantom{\rule{0.166667em}{0ex}}{\partial}_{r}\right)-\frac{\mathcal{M}}{2}{\left(\frac{1}{cosh\left(\frac{a}{2}\phantom{\rule{0.166667em}{0ex}}{\partial}_{r}\right)}r\right)}^{2}$ |

${Y}_{-1/2}$ | $-{\partial}_{r}$ | $-\frac{2}{a}sinh\left(\frac{a}{2}\phantom{\rule{0.166667em}{0ex}}{\partial}_{r}\right)$ |

${Y}_{1/2}$ | $-t{\partial}_{r}-\mathcal{M}r$ | $-\frac{2t}{a}sinh\left(\frac{a}{2}\phantom{\rule{0.166667em}{0ex}}{\partial}_{r}\right)-\frac{\mathcal{M}}{cosh\left(\frac{a}{2}\phantom{\rule{0.166667em}{0ex}}{\partial}_{r}\right)}r$ |

${M}_{0}$ | $-\mathcal{M}$ | $-\mathcal{M}$ |

**Proposition 5.**[63] For any $\nu \in \mathbb{N}$, the generators Equation (41) of the algebra $\mathfrak{age}\left(d\right)$ satisfy the commutators Equation (4) in $d=1$ spatial dimensions, but with the only exception

**Proposition 6.**[63] For $\nu \in \mathbb{N}$, a two-point function F, covariant under the non-local representation Equation (41) of the Lie algebra $\mathfrak{age}\left(1\right)$, defined on the solution space of $\mathcal{S}\varphi =0$, where $\mathcal{S}$ is the Schrödinger operator Equation (43), has the form $F=\delta ({\mathcal{M}}_{1}-{\mathcal{M}}_{2}^{*})\phantom{\rule{0.166667em}{0ex}}{t}_{2}^{-({x}_{1}+{x}_{2})/\nu}F(u,v,r)$, where

## 3. Dual Representations

#### 3.1. Schrödinger Algebra

**Lemma 2.**[80] For the Schrödinger algebra in $d=1$ space dimension, the holographic principle takes the form

**Proof.**We merely outline the main ideas. First, construct the Green’s function in the bulk, by solving

#### 3.2. Conformal Galilean Algebra I

#### 3.3. Conformal Galilean Algebra II

#### 3.4. Parabolic Sub-Algebras

**Figure 1.**Root diagrammes of the Lie algebras (

**a**) $\mathfrak{age}\left(1\right)$; (

**b**) $\mathfrak{sch}\left(1\right)$ and (

**c**) $cga\left(1\right)$. The generators are represented by the black filled dots. The red circles indicate the extra generator N which extends these algebras to maximal parabolic sub-algebras of the complex Lie algebra ${B}_{2}$. The thick green line indicates the separation between positive and non-positive roots.

**Proposition 7.**Consider the dual representations Equation (48) of the Schrödinger–Virasoro algebra, the $z=2$ dual representation Equation (56) of the conformal Galilean algebra $\text{CGA}\left(1\right)$, the $z=1$ dual representation Equation (57) of $\text{CGA}\left(d\right)$ and the dualisation of the non-local representation Equation (41) of $\mathfrak{age}\left(1\right)$, dualised with respect to the mass $\mathcal{M}$. There is a generator N which extends these representations to representations of the associated maximal parabolic sub-algebra. The explicit form of the generator N is as follows:

## 4. Causality

**Proposition 8.**[11,77] Consider the co-variant dual two-point functions. For the dual representation Equation (48) of $\tilde{\mathfrak{sch}}\left(d\right)$, it has the form, up to a normalisation constant

**Theorem 2.**[11] With the convention that masses $\mathcal{M}\ge 0$ of scaling operators ϕ should be non-negative, and if $\frac{1}{2}({x}_{1}+{x}_{2})+{\xi}_{1}^{\prime}+{\xi}_{2}^{\prime}>0$, the full two-point function, co-variant under the representation Equation (5) of the parabolically extended Schrödinger algebra $\tilde{\mathfrak{sch}}\left(d\right)$, has the form

**Proof.**This follows directly from Equation (59). Carrying out the inverse Fourier transform and using the translation-invariance in the dual coordinate ζ, one recovers the habitual two-point function multiplied by an integral representation of the Θ-function. ☐

**Definition 1.**Let ${\mathbb{H}}_{+}$ be the upper complex half-plane $w=u+\mathrm{i}v$ with $v>0$. A function $g:{\mathbb{H}}_{+}\to \mathbb{C}$ is said to be in theHardy class ${H}_{2}^{+}$, written as $g\in {H}_{2}^{+}$, if (i) $g\left(w\right)$ is holomorphic in ${\mathbb{H}}_{+}$ and (ii) if it satisfies the bound

**Lemma 3.**[82] If $g\in {H}_{2}^{\pm}$, then there are square-integrable functions ${\mathcal{G}}_{\pm}\in {L}^{2}(0,\infty )$ such that for $v>0$ one has the integral representation

**Proposition 9.**[77] Let $\xi :=\frac{1}{2}({\xi}_{1}+{\xi}_{2})>\frac{1}{4}$. If $\lambda >0$, then ${f}_{\lambda}\in {H}_{2}^{+}$ and if $\lambda <0$, then ${f}_{\lambda}\in {H}_{2}^{-}$.

**Proof.**The holomorphy of ${f}_{\lambda}$ being obvious, we merely must verify the bound Equation (62). Let $\lambda >0$. Clearly, $\left|{f}_{\lambda}(u+\mathrm{i}v)\right|=\left|{(u+\mathrm{i}(v+\lambda ))}^{-2\xi}\right|={\left({u}^{2}+{(v+\lambda )}^{2}\right)}^{-\xi}$. Hence, computing explicitly the integral,

**Theorem 3.**[77] The full two-point function, co-variant under the representation Equation (13) of the parabolically extended conformal Galilean algebra $\tilde{\text{CGA}}\left(d\right)$, has the form

**Proof.**Since the final result is rotation-invariant, because of the representation Equation (13), it is enough to consider the case $d=1$. Let $\lambda >0$. From Equation (63) of Lemma 3 we have

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

## References and Notes

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Henkel, M.
Dynamical Symmetries and Causality in Non-Equilibrium Phase Transitions. *Symmetry* **2015**, *7*, 2108-2133.
https://doi.org/10.3390/sym7042108

**AMA Style**

Henkel M.
Dynamical Symmetries and Causality in Non-Equilibrium Phase Transitions. *Symmetry*. 2015; 7(4):2108-2133.
https://doi.org/10.3390/sym7042108

**Chicago/Turabian Style**

Henkel, Malte.
2015. "Dynamical Symmetries and Causality in Non-Equilibrium Phase Transitions" *Symmetry* 7, no. 4: 2108-2133.
https://doi.org/10.3390/sym7042108