Broken Spacetime Symmetry in Classical Matter

Abstract

The most common type of a low-energy collective mode occurring in nature is a classical matter wave, usually manifested as sound. This features many of the attributes of Nambu–Goldstone bosons, notably a vanishing energy in the long-wavelength limit. Yet, it is not obvious at first sight how thermodynamic sound fits into the general framework worked out and applied in this book so far. This chapter develops a description of classical matter that makes application of the techniques of nonlinear realization of symmetry possible. Starting from a simple toy model of a solid, it is shown that a classical medium possesses a set of emergent symmetries, reflecting its internal structure. Different choices of the emergent symmetry naturally give rise to systems that exhibit respectively solid and fluid behavior. Once all the symmetries are properly identified, the construction of their nonlinear realization proceeds as usual. By working out specific examples, the text demonstrates how well-known theories of solids (elasticity) and fluids (hydrodynamics) can be naturally recovered from the effective field theory perspective. The last section of the chapter shows how Nambu–Goldstone bosons of a spontaneously broken internal symmetry can be coupled to the collective modes of a background classical medium.

The first example of a Nambu–Goldstone (NG) boson I mentioned in the introduction (Chap. 1) was hydrodynamic sound. However, our subsequent exploration of spontaneous symmetry breaking (SSB) took us very far away from this initial target. Indeed, I focused almost exclusively on the quantum world. This had a good reason: many conceptual subtleties of SSB are rooted in the description of quantum symmetries in terms of operators on the Hilbert space of states. We then did much work to establish the methods of nonlinear realization of symmetry and effective field theory (EFT) for NG bosons. Eventually, we discovered that the leading order (LO) of the derivative expansion of the EFT for quantum systems with SSB is a classical field theory. In this chapter, we shall close the circle and see how the techniques we have developed can be applied to purely classical systems.

To build the necessary intuition, I will start in Sect. 14.1 with a primitive toy model for elastic solids. While physically inadequate in the details, this is good enough to show that a classical medium may possess an emergent symmetry, reflecting its internal structure. A proper identification of such emergent symmetries is key for distinguishing thermodynamic phases of matter with otherwise identical microscopic dynamics, such as solids and fluids. The subsequent analysis is straightforward and follows the standard workflow from nonlinear realization to effective actions. The nonlinear realization of a purely spacetime symmetry augmented with the emergent symmetry of classical matter is detailed in Sect. 14.2. In the following Sect. 14.3, I then work out explicit examples of EFTs for different phases of matter.

1 Emergent Symmetry of Classical Matter

Classical matter is characterized by the possibility to uniquely label its elements and individually track their evolution. While doing so, one has to carefully distinguish two different types of coordinates. The first type are the genuine spacetime coordinates, independent of whatever matter is present in the spacetime. The second type are the coordinates that label the individual elements of the medium. These are known in classical mechanics as body coordinates (see e.g. Sect. 8.1 of [1]) or material coordinates (Chap. 4 of [2]). They capture internal variations of the material properties of the medium. The difference between the two types of coordinates is best elucidated by an example.

1.1 Introduction: Spring Model of Elasticity

One of the most basic mechanical models, illustrating a transition between mechanics of particles and continuous field theory, is the linear spring chain, shown in Fig. 14.1. For the sake of discussion, I will assume that the particles connected by the springs have generally unequal masses. Likewise, the springs themselves may have different spring constants, although I will for simplicity assume equal rest (unloaded) length a. Denoting the position of the i-th particle as $x_i$, the Lagrangian of the system is

$$\displaystyle \begin{aligned} L=\sum_i\left[\frac 12m_i\dot x_i^2-\frac 12k_i(x_{i+1}-x_i-a)^2\right]\;. {} \end{aligned} $$

(14.1)

The ground state of this system corresponds to a chain of point masses at rest, placed equidistantly at distance a between the nearest neighbors. One can choose the origin of coordinates so that in the ground state, $\langle {x_i}\rangle =ia$. We would now like to perform the continuum limit, assuming that a is very small. This requires that the displacement $x_i-ia$ from the equilibrium position only varies appreciably over distances much longer than a. I will replace the discrete index i with a continuous body coordinate X through $i\to X/a$. The individual masses $m_i$ are replaced with the linear mass density $\varrho (X)$ via $m_i\to a\varrho (X)$, and the spring constants $k_i$ with the local elastic (Young) modulus $E(X)$ via $k_i\to E(X)/a$. Treating the Lagrangian (14.1) as a Riemann sum then leads to the continuous approximation

$$\displaystyle \begin{aligned} L\to\int\mathrm{d} X\,\left\{\frac 12\varrho(X)[\partial_0 x(X,t)]^2-\frac 12E(X)[\partial_Xx(X,t)-1]^2\right\}\;. {} \end{aligned} $$

(14.2)

This is a field theory in one spatial dimension with the dynamical degree of freedom $x(X,t)$. Provided both $\varrho (X)$ and $E(X)$ are positive for any $X\in \mathbb {R}$, which follows from their origin in the parameters $m_i$ and $k_i$, the ground state is $\langle {x(X,t)}\rangle =X$ up to an additive constant. This residual freedom stems from spontaneous breakdown of spatial translations. Parameterizing the displacement of the medium from equilibrium by the field $\phi (X,t)\equiv x(X,t)-X$, the fluctuations around the ground state are governed by the equation of motion (EoM),

$$\displaystyle \begin{aligned} \varrho(X)\partial_0^2\phi(X,t)=\partial_X[E(X)\partial_X\phi(X,t)]\;. {} \end{aligned} $$

(14.3)

Fig. 14.1

One-dimensional chain of point masses, connected by linear springs. The springs are only allowed to vibrate in the horizontal direction. The masses and spring constants may vary along the chain as indicated

In the limit of equal masses $m_i$ and spring constants $k_i$, the functions $\varrho (X)$ and $E(X)$ become constant. The EoM (14.3) then describes compression waves, propagating along the chain with phase velocity $v=\sqrt {E/\varrho }$. In this limit, the action for $\phi (X,t)$ features a set of emergent continuous symmetries that were not present in the original discrete model (14.1). The most obvious of these is the invariance under the “translation” . A straightforward application of Noether’s theorem shows that the corresponding integral charge is, up to overall normalization,

$$\displaystyle \begin{aligned} P_{\mathrm{pseudo}}=-\int\mathrm{d} X\,\varrho\partial_0\phi(X,t)\partial_X\phi(X,t)\;. {} \end{aligned} $$

(14.4)

Despite the suggestive analogy with translation invariance, this is not the momentum carried by the oscillating masses of the original discrete chain. The latter rather equals

$$\displaystyle \begin{aligned} P=\sum_im_i\dot x_i\to\int\mathrm{d} X\,\varrho(X)\partial_0\phi(X,t)\;. {} \end{aligned} $$

(14.5)

The resolution of this puzzle is that in presence of a uniform classical medium, we have two different coordinates and consequently two different translation symmetries. The position of the mass $m_i$ with respect to the laboratory inertial reference frame is given by the coordinate $x_i$. Accordingly, a genuine spatial translation amounts to the shift . In the formulation (14.2) of our field theory, this acts like an “internal symmetry.” That is, it transforms the dependent variable $x(X,t)$ while leaving the independent variable X intact. Applying Noether’s theorem recovers the integral momentum (14.5). This remains conserved for any choice of functions $\varrho (X)$ and $E(X)$. That is because momentum conservation reflects the uniformity of the underlying space itself; it does not care about the properties of the medium. The corresponding local conservation law is equivalent to (14.3), which is the continuous limit of the Newtonian EoM for the point masses $m_i$.

The other integral charge, (14.4), is usually called pseudomomentum. This arises, as already stressed, from invariance under . Given that X was introduced as a material coordinate, conservation of $P_{\mathrm {pseudo}}$ should reflect the uniformity of the medium. Sure enough, it was essential for derivation of (14.4) to assume that $\varrho (X)$ and $E(X)$ are constant. The distinction between momentum and pseudomomentum has historically led to much confusion in the research on the continuum mechanics of fluids. See [3] for a pedagogical account of some of the related subtleties.

The comparison of momentum and pseudomomentum illustrates best the striking contrast between the fundamental spacetime symmetries and the emergent symmetries of matter. However, the field theory (14.2) also features other emergent symmetries than translations. Namely, for constant $\varrho (X)$ and $E(X)$ its action is invariant under the whole Poincaré group $\mathrm {ISO}(1,1)$ of transformations acting on $X,t$. These include the “translations” of X, giving rise to conservation of pseudomomentum, and the usual time translations, leading to conservation of energy. In addition, the theory has an emergent $\mathrm {SO}(1,1)$ Lorentz invariance under “boosts” that mix X and t. It is straightforward to work out the corresponding conservation law but I will not do so since we will not need it. Finally, the dynamics of our nonrelativistic spring chain should be invariant under Galilei boosts. Under the corresponding transformations, with v being the velocity of the boost, the Lagrangian (14.2) shifts by a total time derivative. Similarly to spatial translations, this is a genuine spacetime symmetry that is present for any choice of $\varrho (X)$ and $E(X)$.

Before closing the discussion of our toy model, let us briefly look at its generalization to a higher number of spatial dimensions d. For illustration, it is sufficient to take $d=2$ and consider a rectangular network of springs as in Fig. 14.2. The position of the particle $m_{i,j}$ is now given by a two-component vector $\boldsymbol x_{i,j}$. Assuming otherwise the same dynamics as before, whereby the potential energy of each spring is a quadratic function of its extension, the Lagrangian becomes

(14.6)

The ground state of this model corresponds to a rectangular network of equidistantly placed particles at rest. One can choose Cartesian coordinates in the plane so that $\langle {\boldsymbol x_{i,j}}\rangle =(ia,ja)$. To perform the continuum limit, we make the replacement $(i,j)\to \boldsymbol X/a$ and $m_{i,j}\to a^2\varrho (\boldsymbol X)$, where $\boldsymbol X$ is a two-component vector of body coordinates and $\varrho (\boldsymbol X)$ the local mass density. We also need two elastic moduli, and . In terms of the displacement vector $\phi ^r(\boldsymbol X,t)\equiv x^r(\boldsymbol X,t)-X^r$,¹ the Lagrangian of the resulting continuous two-dimensional theory reads

$$\displaystyle \begin{aligned} \begin{aligned} L\to\int\mathrm{d}^2\!\boldsymbol X\,\bigg\{\frac 12\varrho(\boldsymbol X)[\partial_0\boldsymbol\phi(\boldsymbol X,t)]^2&-\frac 12E_1(\boldsymbol X)[\partial_1\phi^1(\boldsymbol X,t)]^2\\ &-\frac 12E_2(\boldsymbol X)[\partial_2\phi^2(\boldsymbol X,t)]^2\bigg\}+\dotsb\;. \end{aligned} {} \end{aligned} $$

(14.7)

The ellipsis stands for terms of higher order in $\phi ^r$; in $d\geq 2$ dimensions, the dynamics of the spring network does not resolve into a superposition of one-dimensional harmonic motions. Moreover, the model (14.7) does not correctly capture the physics of elastic solids even in the harmonic approximation. Namely, for purely transverse oscillations such that $\partial {\phi ^r}/\partial {X^r}=0$ for each fixed r, there is no linear restoring force and the motion is strongly anharmonic. In spite of these flaws, the model (14.7) is sufficient to shed light on the fate of the emergent symmetries.

Fig. 14.2

Rectangular network of springs as a toy model of a two-dimensional elastic solid. All the point masses $m_{i,j}$ are allowed to differ, as are the spring constants and . The rest (unloaded) length of all the springs is assumed to be the same and equal to a

We always find exact invariance under genuine spacetime translations, spatial rotations and Galilei boosts regardless of the choice of the functions $\varrho (\boldsymbol X)$ and $E_{1,2}(\boldsymbol X)$. This much is obvious from the discrete version (14.6) of the model. These symmetries reflect the properties of spacetime itself and are insensitive to the material structure of the medium. Should the medium be uniform, with constant $\varrho (\boldsymbol X)$ and $E_{1,2}(\boldsymbol X)$, we will in addition have invariance under continuous “internal translations” of $\boldsymbol X$. This symmetry ensures conservation of pseudomomentum, and we can expect it to arise in the long-distance description of real crystalline materials. However, we cannot in general expect invariance under continuous “internal rotations” of $\boldsymbol X$, unless we demand $E_1=E_2$ and restrict the Lagrangian (14.7) to the lowest order in the power expansion in $\phi ^r$. This suggests that the macroscopic properties of real crystals tend to be homogeneous but anisotropic even in the continuum limit $a\to 0$, which agrees with empirical observations. The dependence of the Lagrangian on $X^r$ is then constrained by the continuous internal translation invariance and the discrete group of point symmetries of the crystal lattice.

1.2 Emergent Symmetries of Solids and Fluids

We shall now synthesize the above observations into a general setup for describing classical matter. Much of the discussion in this subsection is inspired by Chap. 4 of [2]. The basic assumption is that we are dealing with a thermodynamic state where quantum correlations are limited to short distances, typically due to thermal fluctuations. It is then possible to identify elements of the medium that are mutually distinguishable and can be assigned unique labels. This requirement limits the validity of the EFTs developed below to distances much longer than the scales characterizing quantum correlations and the discrete structure of matter.

The labels $X^i$ on the medium elements take values from a target space $\mathcal {M}$, which is typically some domain in the real space, $\mathbb {R}^d$. The domain would be finite for an isolated material object of a finite size. However, here we will mostly be concerned with properties of matter filling the entire space, where it is natural to take $\mathcal {M}\simeq \mathbb {R}^d$. In both cases, the labels $X^i$ represent comoving body coordinates, attached to a fixed medium element. The trajectory of the element is defined by giving its position as a function of time, $\boldsymbol x(X,t)$. This is the standard Lagrangian picture of continuum mechanics, which I used to formulate the toy models (14.2) and (14.7). In order to highlight the genuine spacetime symmetries, it is however more convenient to invert the relation between $x^r$ and $X^i$. The time evolution of the medium is then specified by a map from the spacetime M to $\mathcal {M}$ that assigns to every point $x^\mu \in M$ the material coordinates $X^i(\boldsymbol x,t)\equiv X^i(x)$ of the element occupying this point. The uniqueness of the labeling of the medium elements is ensured by requiring that for any fixed time t, the map $x^r\to X^i(\boldsymbol x,t)$ is a diffeomorphism between space and $\mathcal {M}$. The advantage of this picture is that the labels $X^i$ take the same values in any reference frame. In other words, the functions $X^i(x)$ are scalar fields with respect to any spacetime symmetry. On the other hand, emergent symmetries acting solely on $X^i$ can then be treated as internal symmetries, that is point transformations on $\mathcal {M}$.

The explicit realization of the fields $X^i(x)$ depends on a choice of coordinates. Importantly, the coordinates $x^\mu$ in the spacetime and the material coordinates $X^i$ can be fixed independently of each other. See Fig. 14.3 for a visualization of this freedom. In the following, I will deal exclusively with matter in flat spacetimes. Accordingly, I will use standard Minkowski coordinates $x^\mu$ for relativistic systems, and Cartesian coordinates $x^r$ augmented with Newtonian time t for nonrelativistic systems. This still leaves us with the freedom to choose the body coordinates $X^i$ at will, which should be distinguished from any internal symmetry acting on $\mathcal {M}$. In Sect. 14.1.1, I fixed this freedom tacitly by the definition of $X^r$ in terms of the discrete labels $i,j$. In order to avoid misunderstanding and to highlight the scalar nature of the body coordinates, I have now changed the notation from $X^r$ to $X^i$.

Fig. 14.3

The time evolution of the classical medium as a map from the spacetime M to the target space $\mathcal {M}$ of body coordinates. The spacetime coordinates $x^\mu$ in M and the body coordinates $X^i$ in $\mathcal {M}$ can be chosen arbitrarily and independently of each other

The scalar fields $X^i(\boldsymbol x,t)$ are the generalized coordinates of our continuum field-theoretic description of classical matter. The corresponding generalized velocity is $\partial _0X^i(\boldsymbol x,t)$. How is this related to the actual local (Eulerian) velocity of the medium with respect to the laboratory frame? Recall that $X^i$ are comoving labels that cannot change along the trajectory of the medium element. Parameterizing the trajectory as $\boldsymbol x(t)$, the condition $\mathrm{d} {X^i(\boldsymbol x(t),t)}/\mathrm{d} {t}=0$ leads to $\partial _0 X^i=-\dot x^r\partial _rX^i$, where $\dot {\boldsymbol x}\equiv \mathrm{d} {\boldsymbol x}/\mathrm{d} {t}$ is the desired local velocity. The matrix is invertible thanks to the assumption that the map $x^r\to X^i(\boldsymbol x,t)$ is a diffeomorphism. It follows that

(14.8)

There is also another, more elegant albeit slightly more mathematically advanced, way to express the local medium velocity in terms of derivatives of $X^i(x)$. First pick any function $f(X)$ on $\mathcal {M}$ and define the current

$$\displaystyle \begin{aligned} J^\mu_f\equiv\frac{f(X)}{d!}\varepsilon^{\mu\nu_1\dotsb\nu_d}\varepsilon_{i_1\dotsb i_d}\partial_{\nu_1}X^{i_1}\dotsb\partial_{\nu_d}X^{i_d}\;. {} \end{aligned} $$

(14.9)

By the antisymmetry of the Levi-Civita (LC) symbol, for any i. Expanding this as gives . Using (14.8), we can thus rewrite the current as

$$\displaystyle \begin{aligned} J^\mu_f=(J^0_f,J^0_f\dot{\boldsymbol x})\;. {} \end{aligned} $$

(14.10)

It also follows from the antisymmetry of the LC symbol that the current (14.9) is conserved, , for any choice of the function $f(X)$. It is therefore natural to identify with the density of a conserved charge as observed in the laboratory frame. The spatial part of the current, , then represents the flow of this charge. Finally, $f(X)$ itself is the density of the same charge in the comoving material coordinates, since $\det M$ is just the Jacobian of the transformation between the spacetime and material coordinate systems.

The origin of the currents can be traced back to the fact that the individual elements of our classical medium are distinguishable. Indeed, we could in principle attach to each element a dedicated “charge.” All such charges would be conserved by construction. The freedom to choose the function $f(X)$ in (14.9) at will is just a continuous version of this observation. Operationally, the existence of infinitely many conservation laws, one for each $f(X)$, is not a problem. Namely, are not Noether currents in that their conservation does not require the EoM. The presence of such identically conserved currents does not constrain the local classical dynamics of the medium.

The above said does not mean that the currents are inevitably artifacts of the Lagrangian picture of continuum mechanics that are not macroscopically observable. Suppose our medium carries a conserved charge. This could count for instance the number of particles or, in the nonrelativistic limit, their mass. By choosing $f(X)$ as the density of the charge in the material coordinates, we get a current that describes macroscopic flow of this charge.

Mathematically, the currents (14.9) descend from the d-forms $f(X)\mathrm{d} X^1\wedge \dotsb \wedge \mathrm{d} X^d$ on $\mathcal {M}$. Being top-dimensional, any such d-form is automatically closed. The closedness is preserved when the d-form is pulled back to the spacetime by the map $x^\mu \to X^i(x)$. Taking the Hodge dual of the ensuing closed spacetime d-form then gives the current $J^\mu _f$ with vanishing divergence.

Having established the basic setup, we are now finally in a position to discuss the emergent symmetries of various types of classical matter. I will focus on the two most common classical phases of matter: crystalline solids and fluids. Since this will to a certain extent merely collect and organize some observations I have already made previously, I can afford to be brief.

Solids

We will be interested in the quasiequilibrium dynamics of crystalline solids at distances much longer than the lattice spacing of the microscopic crystal structure. In this regime, one can expect the solid to be internally uniform, and thus be symmetric under the group $\mathbb {R}^d$ of continuous internal translations. These translations take a particularly simple form in “Cartesian coordinates” on $\mathcal {M}$, namely . Barring possible quasi-invariant contributions, the effective Lagrangian can then only depend on $X^i(x)$ through their derivatives. In addition, the dependence on $X^i(x)$ is constrained by the point symmetry of the crystal lattice. This can be imposed order by order in powers of $X^i(x)$ using tensor methods. I will not go into detail here, and rather refer the reader to Sect. 10 of [4] for further discussion. For the sake of illustration, I will only deal with the special case of isotropic solids, which feature a full $\mathrm {SO}(d)$ symmetry under continuous rotations of $X^i(x)$. Exact, full isotropy cannot really be achieved for any crystalline material. However, some polycrystalline materials are effectively isotropic at distances much longer than the typical size of a grain, thanks to the random orientation of the grains.

Fluids

The key difference between solids and fluids is that in the latter, there are no restoring elastic forces that would counteract shear strain. This enables macroscopic flow, whose physics is largely affected by dissipation. Unfortunately, dissipative effects are notoriously difficult to include in Lagrangian field theory. In order that the EFT for fluids we construct be meaningful, we thus need to make some simplifying assumptions. Namely, we shall restrict ourselves to physical processes that do not involve entropy production. Also, we will only consider states of the fluid that are perturbations of a uniform equilibrium where the entropy density in the material coordinates is constant. Such fluids are called barotropic; their thermodynamic state can be described by a single variable such as pressure or density. Thus, our EFT will be able to capture processes such as sound propagation, but not, for instance, convective heat transfer due to an initial temperature gradient.

By definition, the local density of energy and entropy, and thus also of any other extensive property, of a barotropic fluid corresponds to a constant function f on $\mathcal {M}$. At the same time, the macrostate of the fluid is completely specified by its local Eulerian velocity and density observed in the laboratory frame. The same should be true for the Lagrangian, which is directly connected to observables such as energy density and pressure. The field-theoretic description of fluids in terms of the scalar fields $X^i(x)$ should therefore be invariant under any point transformation on $\mathcal {M}$ that preserves the currents (14.9). This is not mere freedom of choice of coordinates on $\mathcal {M}$. It corresponds to an actual physical reshuffling of the elements of the fluid that does not affect its macroscopic properties. Now requires that the point transformation on $\mathcal {M}$ preserves $\det M$, and so must have a unit Jacobian. We conclude that the dynamics of a barotropic fluid possesses an emergent symmetry under the group of volume-preserving diffeomorphisms (VPDs) on $\mathcal {M}$, $\mathrm {SDiff}(\mathcal {M})$. This is an infinite-dimensional symmetry group that includes the internal translations and rotations of solids as a subgroup. See [5] for a detailed account of the role of the diffeomorphism group in hydrodynamics, and [6, 7] for two somewhat complementary overviews of the variational approach to fluids.

2 Nonlinear Realization of Emergent Symmetry

We have done a serious amount of work to carefully identify the emergent symmetries of classical matter. Our effort will now pay dividends in that the next steps will be fairly straightforward. In the absence of degrees of freedom sensitive to other symmetries, the symmetry group of a classical medium is $G\simeq G_{\mathrm {s.t.}}\times G_{\mathrm {int}}$. Here $G_{\mathrm {s.t.}}$ is whatever spacetime symmetry is appropriate for the system at hand, such as Aristotelian, Galilei or Poincaré. The group $G_{\mathrm {int}}$ collects all the emergent symmetries that act on the scalar fields $X^i(x)$ as internal. For the reader’s convenience, the choices of $G_{\mathrm {int}}$ in case of uniform solids and barotropic fluids are summarized in the first two columns of Table 14.1. The product structure, $G\simeq G_{\mathrm {s.t.}}\times G_{\mathrm {int}}$, of the symmetry group already appeared in Sect. 13.2, and we will be able to largely follow the path paved therein.

Table 14.1 Overview of the emergent symmetries and related subgroups in different phases of classical matter. The $G_{\mathrm {cryst}}$ symbol denotes the crystallographic point group of a crystal lattice. Also, is the group of VPDs in $\mathbb {R}^d$ that fix the origin. The coset space $G_{\mathrm {int}}/H_{\mathrm {int}}$ is equivalent to $\mathbb {R}^d$ in all cases. The unbroken subgroup $H_\varphi$ indicates the symmetry of a uniform, static equilibrium state. The subscript “diag” denotes diagonal symmetries whose actions on the spatial and body coordinates are locked to each other. The values shown in the last column assume that the subgroup of time-independent transformations in $G_{\mathrm {s.t.}}$, acting only on the spatial coordinates, is the Euclidean group $\mathrm {ISO}(d)$. The factor of $\mathbb {R}$ in $H_\varphi$ corresponds to time translations

2.1 Field Variables and Unbroken Symmetry

To start with, the isotropy group of the spacetime origin is $H_0\simeq H_{\mathrm {s.t.}}\times G_{\mathrm {int}}$, where $H_{\mathrm {s.t.}}$ collects all transformations from $G_{\mathrm {s.t.}}$ that fix the origin (typically rotations and boosts). I will always assume that $G_{\mathrm {int}}$ includes a set of mutually commuting translations that act transitively on $\mathcal {M}\simeq \mathbb {R}^d$. The isotropy subgroup $H_{(X_0,0)}$ is then the same, up to conjugation by an element of $G_{\mathrm {int}}$, for any choice of the reference point $X^i_0$. It is convenient to set $X^i_0=0$ so that $H_{(X_0,0)}\simeq H_{\mathrm {s.t.}}\times H_{\mathrm {int}}$ where $H_{\mathrm {int}}$ consists of all emergent symmetries that fix the origin in $\mathcal {M}$. The concrete isotropy subgroups for solids and fluids are listed in the third column of Table 14.1. Note that in case of fluids, both $G_{\mathrm {int}}\simeq \mathrm {SDiff}(\mathbb {R}^d)$ and $H_{\mathrm {int}}\simeq \mathrm {SDiff}_0(\mathbb {R}^d)$ are infinite-dimensional. However, the coset space $G_{\mathrm {int}}/H_{\mathrm {int}}$ is equivalent to $\mathbb {R}^d$ for any choice of $G_{\mathrm {int}}$ thanks to the assumed transitive action of $G_{\mathrm {int}}$ on $\mathcal {M}$. This reconfirms that the dynamical degrees of freedom of the EFT for classical matter will always be the d NG fields $X^i(x)$. That is in contrast to the agnostic nonlinear realization (cf. Sects. 12.3.2 and 13.3.4), whose application to fluids would suggest an EFT with an infinite number of would-be NG fields; see [8] for details.

Before we proceed to the construction of effective actions, let us check which of the symmetries in G actually are spontaneously broken. Suppose we are interested in the physics of fluctuations around a static equilibrium state, characterized by the time-independent vacuum expectation values (VEVs)²

$$\displaystyle \begin{aligned} \langle{X^i(\boldsymbol x,t)}\rangle \equiv\varphi^i(\boldsymbol x)\;. {} \end{aligned} $$

(14.11)

Due to the product structure of G, an element $(g_{\mathrm {s.t.}},g_{\mathrm {int}})\in G$ transforms the spacetime and body coordinates as $T_g:(X^i,x^\mu )\to (X'^i,x'^\mu )\equiv (\mathbb {F}^i(X,g_{\mathrm {int}}),\mathbb {X}^\mu (x,g_{\mathrm {s.t.}}))$. Here the $x^\mu$-independent functions $\mathbb {F}^i$ realize the action of $G_{\mathrm {int}}$ on $\mathcal {M}$, whereas the $X^i$-independent functions $\mathbb {X}^\mu$ realize the action of $G_{\mathrm {s.t.}}$ on the spacetime. The unbroken subgroup $H_\varphi$ consists of transformations preserving (14.11),

$$\displaystyle \begin{aligned} \mathbb{F}^i(\varphi(x),h_{\mathrm{int}})=\varphi^i(\mathbb{X}(x,h_{\mathrm{s.t.}}))\;,\qquad (h_{\mathrm{s.t.}},h_{\mathrm{int}})\in H_\varphi\;. {} \end{aligned} $$

(14.12)

This condition has the geometric meaning of equivariance of $\varphi ^i$ as a map $M\to \mathcal {M}$ under the action of $H_\varphi$, that is $\mathbb {F}\circ \varphi =\varphi \circ \mathbb {X}$. Since $\varphi ^i(\boldsymbol x)$ is a diffeomorphism between the coordinate space and $\mathcal {M}$, (14.12) defines a one-to-one correspondence between the maps $\mathbb {F}^i$ and $\mathbb {X}^r$. In this sense, $H_\varphi$ is the “diagonal subgroup” of $G_{\mathrm {s.t.}}\times G_{\mathrm {int}}$. It also includes those transformations from $G_{\mathrm {s.t.}}$ that act on $x^r$ trivially, that is time translations. An overview of the unbroken subgroups in solids and fluids is given in the last column of Table 14.1.

2.2 Building Blocks for Construction of Effective Actions

In the absence of other degrees of freedom, we are dealing with the d scalar fields $X^i(x)$.³ All of these are of the NG type, realizing nonlinearly the emergent internal translations on $\mathcal {M}\simeq \mathbb {R}^d$. Let us denote the generators of the translations as $\Pi _i$. I will now finally fix the freedom to choose coordinates on $\mathcal {M}\simeq G_{\mathrm {int}}/H_{\mathrm {int}}\simeq \mathbb {R}^d$ at will by parameterizing it as $U(X)\equiv \exp (\mathrm{i} X^i\Pi _i)$. This gives a precise definition of the previously mentioned “Cartesian coordinates” on $\mathcal {M}$, in which the translations from $G_{\mathrm {int}}$ act on $X^i$ by trivial shifts, . According to (13.24), the Maurer–Cartan (MC) form is then given by

$$\displaystyle \begin{aligned} \omega (X,x)=\Pi_i\mathrm{d} X^i+P\cdot\mathrm{d} x\;, {} \end{aligned} $$

(14.13)

where generates spacetime translations. The spacetime coframe is trivial, that is, . Accordingly, it is not necessary to use different notations for frame and coordinate-basis indices. The covariant derivatives of the NG fields are simply $\nabla _\mu X^i=\partial _\mu X^i$. This makes it possible to use the same power counting as for superfluids (Sect. 13.2.2), whereby an n-th partial derivative of $X^i$ is assigned the counting degree $n-1$. The LO of the EFT is thus defined by a Lagrangian density where every field $X^i$ carries exactly one derivative.

Invariance of the effective action is ensured as follows. First, invariance under the whole internal symmetry $G_{\mathrm {int}}$ is guaranteed by using the MC form as a building block and imposing solely the linearly realized isotropy group $H_{\mathrm {int}}$. Similarly, one has to impose explicitly invariance under the linearly realized spacetime isotropy group, $H_{\mathrm {s.t.}}$. Finally, invariance under spacetime translations requires that the Lagrangian density does not depend explicitly on the spacetime coordinates.

In case of fluids, the coset space $G_{\mathrm {int}}/H_{\mathrm {int}}\simeq \mathrm {SDiff}(\mathbb {R}^d)/\mathrm {SDiff}_0(\mathbb {R}^d)$ is not reductive, hence the line of reasoning using the transformation properties (13.10) of the MC form does not necessarily apply. In such a situation, we may have to impose by hand invariance under the entire emergent symmetry group $G_{\mathrm {int}}$. Luckily, we know a priori how VPDs act on the coordinates $X^i$ and thus also on the 1-forms $\mathrm{d} X^i$ on $\mathcal {M}$. This will make it possible for us to construct an EFT for fluids in Sect. 14.3.4.

3 Effective Field Theory of Classical Matter

I will now carry out the above-outlined program for several types of physical systems of interest. I will start with what in many ways is the simplest case: an isotropic relativistic solid. Having warmed up, we shall then have a look at the phenomenologically more relevant nonrelativistic solids. At the very end, we will return to the most nontrivial case of fluids with their infinite-dimensional symmetry group.

3.1 Relativistic Solids

Consider a system with relativistic microscopic dynamics that settles to a thermodynamic equilibrium state with the symmetries of an isotropic solid. The full symmetry is $G_{\mathrm {s.t.}}\times G_{\mathrm {int}}\simeq \mathrm {ISO}(d,1)\times \mathrm {ISO}(d)$, where the first factor is the spacetime Poincaré group and the second one is the emergent internal symmetry. With one derivative per field, the only way to obey the linearly realized Lorentz symmetry, $H_{\mathrm {s.t.}}\simeq \mathrm {SO}(d,1)$, is to pairwise contract indices on $\partial _\mu X^i$. It follows that the LO effective Lagrangian is some function of the Lorentz-invariant matrix operator

$$\displaystyle \begin{aligned} \Xi^{ij}\equiv\partial_\mu X^i\partial^\mu X^j\;. {} \end{aligned} $$

(14.14)

It remains to impose invariance under $H_{\mathrm {int}}\simeq \mathrm {SO}(d)$. Since $\Xi ^{ij}$ is a symmetric tensor, any $H_{\mathrm {int}}$-invariant function of $\Xi ^{ij}$ can only depend on its d real eigenvalues. These can be encoded in any set of d algebraically independent invariants constructed out of $\Xi ^{ij}$, for instance the traces of the first d powers of $\Xi ^{ij}$. The LO effective action for relativistic isotropic solids then reads

$$\displaystyle \begin{aligned} S_{\text{eff}}\{X\}=\int\mathrm{d}^D\!x\,F\bigl(\operatorname{\mathrm{tr}}\Xi(x),\operatorname{\mathrm{tr}}\Xi(x)^2,\dotsc,\operatorname{\mathrm{tr}}\Xi(x)^d\bigr)+\dotsb\;, {} \end{aligned} $$

(14.15)

where F is a function of the displayed arguments and the ellipsis indicates corrections of higher order in the derivative expansion. These include operators with on average more than one derivative per $X^i$, and Lorentz and internal indices contracted in a way respecting the $H_{\mathrm {s.t.}}\times H_{\mathrm {int}}\simeq \mathrm {SO}(d,1)\times \mathrm {SO}(d)$ symmetry.

With all the background we had built up, the path to (14.15) was very short. There are however a couple of potential loopholes to close. First, I tacitly assumed the effective Lagrangian to be strictly invariant under all the symmetries. Can there be any quasi-invariant contributions to the Lagrangian? Here I refer the reader to [10], which showed that the symmetries of a relativistic isotropic solid do not allow any genuinely quasi-invariant Lagrangians in $d=2$ or 3 spatial dimensions. It is plausible to assume that the same conclusion holds for any $d\geq 2$.

Second, we still need to check whether the tentative equilibrium (14.11) is a stable state of the EFT (14.15). To start with, note that the LO EoM of the EFT is

$$\displaystyle \begin{aligned} \partial_\mu\left(\frac{\partial{F}}{\partial{\Xi^{ij}}}\partial^\mu X^j\right)=0\;. \end{aligned} $$

(14.16)

In a uniform equilibrium, the VEV of the invariants (14.14), $\langle {\Xi ^{ij}}\rangle =-\boldsymbol \nabla \varphi ^i\cdot \boldsymbol \nabla \varphi ^j$, should be coordinate-independent. The EoM then implies that the functions $\varphi ^i(\boldsymbol x)$ must be harmonic, $\boldsymbol \nabla ^2\varphi ^i=0$. Acting with the Laplace operator on $\langle {\Xi ^{ij}}\rangle$, we get in turn that $(\partial _r\boldsymbol \nabla \varphi ^i)\cdot (\partial _r\boldsymbol \nabla \varphi ^j)=0$. Setting $i=j$ herein shows that all partial derivatives of $\boldsymbol \nabla \varphi ^i$ vanish. Hence, $\varphi ^i(\boldsymbol x)$ is a linear function of coordinates, with a constant invertible matrix and constant $c^i$. The set of constants $c^i$ can be removed by shifting the origin of coordinates. Finally, suppose that the generators $\Pi _i$ of internal translations were chosen as a basis of the standard vector representation of $H_{\mathrm {int}}\simeq \mathrm {SO}(d)$. By using the singular value decomposition augmented with appropriate orthogonal rotations of the fields $X^i$ and coordinates $x^r$, the matrix can be made diagonal and positive-semidefinite. Invariance under the unbroken diagonal $\mathrm {SO}(d)$ rotations of spatial and body coordinates then requires that . Finally, the proportionality factor can be absorbed into a simultaneous rescaling of the body coordinates $X^i$ and the generators $\Pi _i$. We therefore conclude that, without loss of generality, the uniform equilibrium of the solid can be described by

$$\displaystyle \begin{aligned} \langle{X^i(\boldsymbol x,t)}\rangle =\delta^i_rx^r\;. {} \end{aligned} $$

(14.17)

Below, I will implicitly assume the same ground state for both nonrelativistic solids and (relativistic or nonrelativistic) fluids.

Further constraints on the parameters of the effective Lagrangian arise from requiring that the state (14.17) is (meta)stable with respect to small fluctuations. To that end, we parameterize the fluctuations of $X^i$ by a set of NG fields, $\pi ^i(x)\equiv X^i(x)-\delta ^i_rx^r$. In terms of these fields, we have

$$\displaystyle \begin{aligned} \Xi^{ij}=-\delta^{ij}-(\delta^{ir}\partial_r\pi^j+\delta^{jr}\partial_r\pi^i)+\partial_\mu\pi^i\partial^\mu\pi^j\;. {} \end{aligned} $$

(14.18)

It is now a matter of straightforward algebra to expand the action (14.15) to second order in $\pi ^i$ and compute the excitation spectrum. The reader is most welcome to do this exercise, or check [11] where the special case of $d=3$ spatial dimensions is analyzed. I will not work out the details here, since I will address the same problem in the arguably more realistic setting of nonrelativistic solids below.

3.2 Nonrelativistic Supersolids

Most naturally occurring solid materials lie safely within the nonrelativistic domain. It therefore appears more appropriate to demonstrate the utility of our EFT formalism in a setting that connects more directly to classical theory of elasticity [4]. However, in order that the discussion below is not a mere rehash of Sect. 14.3.1 using nonrelativistic notation, I will add a new physical ingredient. Namely, some materials are known to enter at low temperatures a quantum phase called suggestively supersolid. In this phase, matter exhibits a combination of crystalline solid order and superflow. The latter stems from the presence of an internal, spontaneously broken $\mathrm {U}(1)$ symmetry. Once we have found an effective action for supersolids, we will be able to recover an EFT for ordinary solids by decoupling the superfluid NG boson. The content of this subsection is heavily influenced by [12].

The mathematical setup closely follows the discussion of nonrelativistic superfluids in Sect. 13.3.3, to which I refer the reader for details. In order to avoid excessive cross-references, I will however spell out the main features of the setup explicitly. The full symmetry of the nonrelativistic supersolid is given by the Bargmann group augmented with the emergent $\mathrm {ISO}(d)$ symmetry of a classical isotropic solid, that is

$$\displaystyle \begin{aligned} G\simeq\mathrm{SO}(d)\ltimes\{\mathbb{R}_K^d\ltimes[\mathbb{R}^D\times\mathrm{U}(1)_Q]\}\times\mathrm{ISO}(d)\;. \end{aligned} $$

(14.19)

Here the $\mathrm {SO}(d)$ factor represents spatial rotations, $\mathbb {R}_K^d$ Galilei boosts, $\mathbb {R}^D$ spacetime translations, and $\mathrm {U}(1)_Q$ the internal symmetry counting the number of particles. The symmetry group does not have the simple direct product structure $G_{\mathrm {s.t.}}\times G_{\mathrm {int}}$. This is due to the dual nature of $\mathrm {U}(1)_Q$, which is an internal symmetry but simultaneously centrally extends the spacetime Galilei group. We will however still be able to follow the philosophy of Sect. 14.2 with minor modifications. In particular the spacetime isotropy group $H_0$ is in this case .

In order to be able to apply our basic EFT framework for spacetime symmetry, we need to realize the Galilei boosts nonlinearly. To that end, we need a Galilei vector order parameter $A^\mu =(A^0,\boldsymbol A)$, choosing a timelike reference point, with $a\neq 0$. This is accompanied by two other order parameters, a complex scalar $\psi$ charged under $\mathrm {U}(1)_Q$, and the body coordinates $X^i$. Taking any $\psi _0\neq 0$ and leads to , where the two factors act respectively on the spatial and body coordinates. The coset space relevant for the nonlinear realization of the symmetry is . The last factor of $\mathbb {R}^d$ is new compared to Sect. 13.3.3 and carries the solid degrees of freedom. The whole coset space is parameterized by the NG variables $\pi$, $\xi ^r$ and $X^i$ through

$$\displaystyle \begin{aligned} U(\pi,\boldsymbol\xi,X)\equiv\mathrm{e}^{\mathrm{i} \pi Q}\mathrm{e}^{\mathrm{i}\boldsymbol{\xi}\cdot\boldsymbol{K}}\exp(\mathrm{i} X^i\Pi_i)\;, \end{aligned} $$

(14.20)

where Q is the generator of $\mathrm {U}(1)_Q$ and $K^r$ that of Galilei boosts. The MC form is calculated using the commutation relations of the Bargmann group,

$$\displaystyle \begin{aligned} \begin{aligned} \omega (\pi,\boldsymbol\xi,X,\boldsymbol x,t)={}&Q[\mathrm{d}\pi-\boldsymbol\xi\cdot\mathrm{d}\boldsymbol x+(1/2)\boldsymbol\xi^2t]+\boldsymbol K\cdot\mathrm{d}\boldsymbol\xi+\Pi_i\mathrm{d} X^i\\ &+H\mathrm{d} t+\boldsymbol P\cdot(\mathrm{d}\boldsymbol x-\boldsymbol\xi\mathrm{d} t)\;. \end{aligned} \end{aligned} $$

(14.21)

The second line herein defines the spacetime coframe, and . This allows one to extract the covariant derivatives of all the NG fields from the $\omega _\perp$ part of the MC form,

$$\displaystyle \begin{aligned} \begin{aligned} \nabla_{0}\pi&=\partial_0\pi+\boldsymbol{\xi}\cdot\boldsymbol{\nabla}\pi-\boldsymbol\xi^2/2\;,\qquad & \nabla_{r}\pi&=\partial_r\pi-\xi_r\;,\\ \nabla_{0}\xi^r&=(\partial_0+\boldsymbol{\xi}\cdot\boldsymbol{\nabla})\xi^r\;,\qquad & \nabla_{s}\xi^r&=\partial_s\xi^r\;,\\ \nabla_{0}X^i&=(\partial_0+\boldsymbol{\xi}\cdot\boldsymbol{\nabla})X^i\;,\qquad & \nabla_{r}X^i&=\partial_rX^i\;. \end{aligned} \end{aligned} $$

(14.22)

For simplicity of notation, I have already dropped the underscores distinguishing spacetime and frame indices.

The would-be NG field $\xi ^r(x)$ is unphysical and can be eliminated by imposing the inverse Higgs constraint (IHC) $\nabla _r\pi =0$, or $\boldsymbol \xi (x)=\boldsymbol \nabla \pi (x)$. This assigns $\xi ^r$ the counting degree zero. As a consequence, $\nabla _\mu \xi ^r$ has degree one, whereas both $\nabla _\mu \pi$ and $\nabla _\mu X^i$ are of degree zero. The building blocks we have to construct the LO effective Lagrangian are therefore

$$\displaystyle \begin{aligned} \nabla_0\pi\to\partial_0\pi+(\boldsymbol\nabla\pi)^2/2\;,\quad \nabla_0X^i\to(\partial_0+\boldsymbol\nabla\pi\cdot\boldsymbol\nabla)X^i\;,\quad \nabla_rX^i\to\partial_rX^i\;. {} \end{aligned} $$

(14.23)

These take automatically care of the nonlinearly realized symmetries under Galilei boosts, $\mathrm {U}(1)_Q$ transformations and internal translations of $X^i$. To ensure invariance under spatial rotations, note that both $\nabla _0\pi$ and $\nabla _0X^i$ are scalars, whereas $\nabla _rX^i$ is a vector under spatial $\mathrm {SO}(d)$. All spatial indices thus have to be contracted into

$$\displaystyle \begin{aligned} \tilde\Xi^{ij}\equiv\delta^{rs}\nabla_rX^i\nabla_sX^j=\boldsymbol\nabla X^i\cdot\boldsymbol\nabla X^j\;. {} \end{aligned} $$

(14.24)

This symmetric tensor is related to the previously defined matrix by $\tilde \Xi =MM^T$. Note that there is no additional, algebraically independent rotationally invariant operator where the spatial indices on $\nabla _rX^i$ would be contracted with the LC symbol, since . Altogether, the LO effective action for nonrelativistic supersolids is given by [12]

$$\displaystyle \begin{aligned} S_{\mathrm{eff}}\{\pi,X\}=\int\mathrm{d}^D\!x\,F\bigl(\nabla_0\pi(x),\nabla_0X(x),\tilde\Xi(x)\bigr)+\dotsb\;. {} \end{aligned} $$

(14.25)

The indices on $\nabla _0X^i$ and $\tilde \Xi ^{ij}$ are to be contracted in a way that respects invariance under the internal $\mathrm {SO}(d)$ rotations.

Again, I have not explicitly considered the possibility of quasi-invariant contributions to the Lagrangian. There are $d+1$ such terms, constructed solely out of the superfluid mode $\pi$; cf. Sect. 13.3.3. However, these are only relevant beyond the LO of the derivative expansion. An additional quasi-invariant Lagrangian existing in $d=2$ dimensions and including the solid variable $X^i$ was found in [10]. This likewise contributes only at higher orders of the derivative expansion.

3.3 Nonrelativistic Solids

Let us now see whether we can recover the physics of classical (isotropic) solids from the supersolid EFT (14.25). It appears we should be able to simply erase the NG field $\pi (x)$. This certainly does not interfere with the nonlinear realization of the symmetries of solids on $X^i$, since that is entirely independent of $\pi$. The building blocks (14.23) then boil down to and . Invariance under spatial rotations again requires that only enters the LO EFT through the combination $\tilde \Xi ^{ij}$. However, the time derivative, , is potentially problematic since unlike , it is not invariant under Galilei boosts. This actually makes sense: recall that the nonrelativistic kinetic term in (14.2) or (14.7) is only quasi-invariant under Galilei boosts. Guided by the analogy, the kinetic term for the solid should be . Here $\boldsymbol x(X,t)$ is the Lagrangian variable we worked with in Sect. 14.1 and $\varrho (\boldsymbol x,t)$ is the local mass density of the solid in the laboratory frame. Now ${\varrho (x)=\varrho _0\det M(x)=\varrho _0\sqrt {\det \tilde \Xi (x)}}$, where $\varrho _0$ is the mass density of the solid in the body coordinates, which is constant thanks to the assumed uniformity of the solid. Using finally (14.8), the LO effective action for the classical nonrelativistic isotropic solid can be written as

(14.26)

The function defines the free energy of static elastic deformations of the solid. Our derivation of the kinetic term in (14.26) was based on an educated guess. It can however also be recovered by starting from an exactly Galilei-invariant operator in the EFT (14.25) for supersolids and adding a surface term, proportional to the gradient of $\pi (x)$. This explains why setting $\pi \to 0$ does not destroy Galilei invariance altogether but renders (14.26) quasi-invariant. See [12] for details.

I will now review the basic elastic properties of isotropic solids, following closely [4]. As in Sect. 14.1.1, the displacement of the solid element carrying label $X^i$ from its equilibrium position is given by $\phi ^i(X,t)\equiv \delta ^i_rx^r(X,t)-X^i$. This parameterization is however not suitable for an EFT where the scalar fields $X^i(x)$ are the dynamical variables. We need to invert the relation between $x^r$ and $\phi ^i$, which is not possible in a closed form. Luckily, we will only need the deviation of $X^i$ from its VEV to first order in $\phi ^i$,

$$\displaystyle \begin{aligned} X^i(x)=\delta^i_rx^r-\phi^i(x)+\mathcal{O}(\phi^2)\;. {} \end{aligned} $$

(14.27)

To the same order, the matrix variable $\tilde \Xi ^{ij}$ reads

$$\displaystyle \begin{aligned} \tilde\Xi^{ij}(x)=\delta^{ij}-[\delta^{ir}\partial_r\phi^j(x)+\delta^{jr}\partial_r\phi^i(x)]+\mathcal{O}(\phi^2)\;. \end{aligned} $$

(14.28)

With the shorthand notation $\phi _r(x)\equiv \delta _{ri}\phi ^i(x)$, small deformations of the solid are therefore encoded in the strain tensor

$$\displaystyle \begin{aligned} e_{rs}(x)\equiv\frac 12[\partial_r\phi_s(x)+\partial_s\phi_r(x)]\;. {} \end{aligned} $$

(14.29)

By its definition, $\phi ^i(x)$ transforms as a vector under the unbroken diagonal $\mathrm {SO}(d)$ subgroup. The strain tensor therefore behaves as a rank-2 symmetric tensor.

The energy cost of a small static elastic deformation of the solid is governed by the expansion of $\mathcal {F}(\tilde \Xi )$ in powers of $\tilde \Xi ^{ij}-\delta ^{ij}$, or directly the strain tensor. The expansion cannot contain a linear term, as that would immediately imply an instability of the equilibrium state where $\langle {\phi ^i(x)}\rangle =0$. The physics of small elastic deformations is therefore dominated by the part of $\mathcal {F}(\tilde \Xi )$ quadratic in the strain tensor. Invariance under unbroken diagonal rotations admits two independent quadratic operators at the lowest order in derivatives,

$$\displaystyle \begin{aligned} \mathcal{F}=\mu\operatorname{\mathrm{tr}} e^2+\frac\lambda2(\operatorname{\mathrm{tr}} e)^2+\dotsb\;. {} \end{aligned} $$

(14.30)

Here $\lambda ,\mu$ are known as the Lamé coefficients, and the ellipsis denotes contributions of higher order in $\phi ^i$ or derivatives. It is convenient to rewrite (14.30) as

$$\displaystyle \begin{aligned} \mathcal{F}=\mu\operatorname{\mathrm{tr}}\left(e-\frac{\operatorname{\mathrm{tr}} e}d{\mathbb{1}}\right)^2+\frac K2(\operatorname{\mathrm{tr}} e)^2+\dotsb\;, {} \end{aligned} $$

(14.31)

where $K\equiv \lambda +(2\mu /d)$. Importantly, there are configurations of the solid for which either of the two terms in (14.31) vanishes whereas the other is nonzero. The $\mu$-term is zero if and only if $e_{rs}$ is proportional to $\delta _{rs}$, that is for uniform dilatations or contractions of the solid. Such overall volume change is detected by $\operatorname {\mathrm {tr}} e=\boldsymbol {\nabla }\cdot \boldsymbol {\phi }$, which measures the deviation of the Jacobian $\det M$ from unity. For this reason, the parameter K is usually called bulk modulus. On the other hand, for deformations of shape that do not affect the volume of the solid, the K-term in (14.31) vanishes. The energy of such deformations is measured by the shear modulus$\mu$. Since the two terms in (14.31) can be set to zero independently of each other, bulk stability of the solid requires that

$$\displaystyle \begin{aligned} K>0\;,\qquad \mu>0\;. {} \end{aligned} $$

(14.32)

Example 14.1

The constraints (14.32) may look trivial, but they have very observable consequences for small oscillations of the solid. To see this, it is sufficient to consider the part of (14.26), bilinear in the fields $\phi ^i(x)$,

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{eff}}=\frac{\varrho_0}2\delta^{rs}\partial_0\phi_r\partial_0\phi_s-\left[\mu\operatorname{\mathrm{tr}} e^2+\frac\lambda2(\operatorname{\mathrm{tr}} e)^2\right]+\dotsb\;. \end{aligned} $$

(14.33)

The corresponding EoM reads

$$\displaystyle \begin{aligned} \varrho_0\partial_0^2\phi_r\approx\mu\boldsymbol\nabla^2\phi_r+(\lambda+\mu)\partial_r\boldsymbol{\nabla}\cdot\boldsymbol{\phi}\;, \end{aligned} $$

(14.34)

where the symbol $\approx$ indicates a linear approximation. This has plane-wave solutions of the type

$$\displaystyle \begin{aligned} \phi_r(\boldsymbol x,t)=\hat\phi_r\mathrm{e}^{-\mathrm{i} Et}\mathrm{e}^{\mathrm{i}\boldsymbol{p}\cdot\boldsymbol{x}}\;, \end{aligned} $$

(14.35)

where $\hat \phi _r$ is an amplitude. Solutions for which $\boldsymbol {\hat \phi }\parallel \boldsymbol p$ and $\boldsymbol {\hat \phi }\perp \boldsymbol p$ represent respectively longitudinal and transverse sound. The corresponding phase velocities are

$$\displaystyle \begin{aligned} v_{\mathrm{L}}=\sqrt{\frac{\lambda+2\mu}{\varrho_0}}=\sqrt{\frac{K+2\mu(1-1/d)}{\varrho_0}}\;,\qquad v_{\mathrm{T}}=\sqrt{\frac{\mu}{\varrho_0}}\;. \end{aligned} $$

(14.36)

The stability condition $K>0$ implies a specific hierarchy between the velocities of longitudinal and transverse sound,

$$\displaystyle \begin{aligned} \frac{v_{\mathrm{L}}}{v_{\mathrm{T}}}>\sqrt{2\left(1-\frac 1d\right)}\;. \end{aligned} $$

(14.37)

This relation would have been completely invisible, had we focused solely on the spectrum of sound waves itself. When looking for the constraints on parameters of an EFT imposed by the stability requirement, it is therefore all-important to consider the whole configuration space of the EFT.

3.4 Perfect Fluids

As the last stop of our exploration of classical matter, we now return to fluids. I will restore full Poincaré invariance, partially because relativistic fluids are common in astrophysics and particle physics, and partially because it makes the analysis simpler.

As argued in Sect. 14.3.1, the NG fields $X^i(x)$ necessarily enter the effective Lagrangian through the combination . However, not all actions of the solid type (14.15) are consistent with the required invariance of the fluid action under $G_{\mathrm {int}}\simeq \mathrm {SDiff}(\mathbb {R}^d)$. The transition from solids to fluids can be viewed as a “fine-tuning” of the effective couplings of the former. There is only one algebraically independent function of $\Xi ^{ij}$ that fits the bill, namely $\det \Xi \equiv \left \lvert {\Xi }\right \rvert$. Thus, the LO effective Lagrangian of a relativistic perfect (barotropic) fluid reads

$$\displaystyle \begin{aligned} S_{\mathrm{eff}}\{X\}=\int\mathrm{d}^D\!x\,F(\left\lvert{\Xi(x)}\right\rvert )+\dotsb\;. {} \end{aligned} $$

(14.38)

With the help of the identity $\partial {\left \lvert {\Xi }\right \rvert }/\partial {\Xi ^{ij}}=\left \lvert {\Xi }\right \rvert (\Xi ^{-1})_{ji}$, we readily extract the corresponding LO EoM,

$$\displaystyle \begin{aligned} \partial_\mu\bigl[F'(\left\lvert{\Xi}\right\rvert )\left\lvert{\Xi}\right\rvert (\Xi^{-1})_{ij}\partial^\mu X^j\bigr]=0\;, {} \end{aligned} $$

(14.39)

where the prime indicates a derivative of $F(\left \lvert {\Xi }\right \rvert )$ with respect to its argument $\left \lvert {\Xi }\right \rvert$.

We could in principle stop here, for we have accomplished our goal to construct an EFT for perfect fluids. However, to shed light on the physical content of (14.38) and (14.39) requires additional work. The first step is to derive the energy–momentum (EM) tensor of the EFT. This is a standard problem (see for instance Example 4.3),⁴ the result being

$$\displaystyle \begin{aligned} T^{\mu\nu}=2F'(\left\lvert{\Xi}\right\rvert )\left\lvert{\Xi}\right\rvert (\Xi^{-1})_{ij}\partial^\mu X^i\partial^\nu X^j-g^{\mu\nu}F(\left\lvert{\Xi}\right\rvert )\;. {} \end{aligned} $$

(14.40)

This already tells us, upon some manipulation, that the EoM (14.39) is equivalent to the local conservation laws for energy and momentum. The next step is to trade the explicit dependence on the derivatives $\partial _\mu X^i$ for the local Eulerian velocity of the fluid. To that end, consider the current (14.9) with $f(X)=1$,

$$\displaystyle \begin{aligned} J^\mu\equiv J^\mu_1=\frac 1{d!}\varepsilon^{\mu\nu_1\dotsb\nu_d}\varepsilon_{i_1\dotsb i_d}\partial_{\nu_1}X^{i_1}\dotsb\partial_{\nu_d}X^{i_d}\;. {} \end{aligned} $$

(14.41)

It is a matter of straightforward algebra to verify that $J^\mu J_\mu =(-1)^d\left \lvert {\Xi }\right \rvert$. The extra sign compensates for our set of conventions, in which $\langle {\Xi ^{ij}}\rangle =-\delta ^{ij}$ but $\langle {J^\mu }\rangle =\delta ^{\mu 0}$. The spacetime vector

$$\displaystyle \begin{aligned} u^\mu\equiv\frac{J^\mu}{\sqrt{(-1)^d\left\lvert{\Xi}\right\rvert }} \end{aligned} $$

(14.42)

is then normalized to unity, $u^2=1$. Moreover, according to (14.10), its spatial part is proportional to the local Eulerian velocity $\dot {\boldsymbol x}$ of the fluid. It follows that $u^\mu$ is the velocity spacetime vector as observed in the laboratory frame.

Consider now the symmetric tensor $G_{\mu \nu }\equiv (\Xi ^{-1})_{ij}\partial _\mu X^i\partial _\nu X^j+u_\mu u_\nu$. Since $\partial _\mu X^i$ as a spacetime vector is orthogonal to $J^\mu$ and thus $u^\mu$, it follows that $G_{\mu \nu }u^\nu =u_\mu$. At the same time, it is easy to see that $G_{\mu \nu }\partial ^\nu X^i=\partial _\mu X^i$. Together, $\partial ^\mu X^i$ for $i=1,\dotsc ,d$ and $u^\mu$ constitute a basis of D linearly independent vectors on the spacetime, hence $G_{\mu \nu }A^\nu =A_\mu =g_{\mu \nu }A^\nu$ for any spacetime vector $A^\mu$. This implies the useful identity

$$\displaystyle \begin{aligned} (\Xi^{-1})_{ij}\partial_\mu X^i\partial_\nu X^j=g_{\mu\nu}-u_\mu u_\nu\;, \end{aligned} $$

(14.43)

which allows us to rewrite the EM tensor (14.40) as

$$\displaystyle \begin{aligned} T^{\mu\nu}=2F'(\left\lvert{\Xi}\right\rvert )\left\lvert{\Xi}\right\rvert (g^{\mu\nu}-u^\mu u^\nu)-g^{\mu\nu}F(\left\lvert{\Xi}\right\rvert )\;. \end{aligned} $$

(14.44)

In an inertial reference frame locally comoving with the fluid (local rest frame), the EM tensor should be diagonal and isotropic, $T^{\mu \nu }= \operatorname {\mathrm {diag}}(U,P,\dotsc ,P)$. Here U is the local energy density and P the pressure of the fluid. This can be written in a covariant form as $T^{\mu \nu }=(U+P)u^\mu u^\nu -Pg^{\mu \nu }$. The scalar functions U and P can be projected out of the EM tensor using $T^{\mu \nu }u_\mu u_\nu =U$ and . It follows that the as yet unknown function $F(\left \lvert {\Xi }\right \rvert )$ is related to the energy and pressure by

$$\displaystyle \begin{aligned} U(\left\lvert{\Xi}\right\rvert )=-F(\left\lvert{\Xi}\right\rvert )\;,\qquad P(\left\lvert{\Xi}\right\rvert )=F(\left\lvert{\Xi}\right\rvert )-2F'(\left\lvert{\Xi}\right\rvert )\left\lvert{\Xi}\right\rvert \;. {} \end{aligned} $$

(14.45)

Example 14.2

To check that we have got the basic physics right, let us use our EFT to calculate the speed of sound in the fluid. For that, we need to find the bilinear part of the effective Lagrangian in the fluctuations $\pi ^i(x)\equiv X^i(x)-\delta ^i_rx^r$. I will outline the main steps but skip straightforward details. First, the fluctuation of $\Xi ^{ij}$ itself is given by (14.18),

$$\displaystyle \begin{aligned} \Xi^{ij}=-\delta^{ij}+\updelta\Xi^{ij}\;,\qquad \updelta\Xi^{ij}\equiv-(\delta^{ir}\partial_r\pi^j+\delta^{jr}\delta_r\pi^i)+\partial_\mu\pi^i\partial^\mu\pi^j\;. \end{aligned} $$

(14.46)

Using the identity $\det \Xi =\exp \operatorname {\mathrm {tr}}\log \Xi$, we next obtain the expansion

$$\displaystyle \begin{aligned} (-1)^d\left\lvert{\Xi}\right\rvert =1-\operatorname{\mathrm{tr}}\updelta\Xi+\frac 12\bigl[(\operatorname{\mathrm{tr}}\updelta\Xi)^2-\operatorname{\mathrm{tr}}(\updelta\Xi^2)\bigr]+\mathcal{O}(\updelta\Xi^3)\;. \end{aligned} $$

(14.47)

What remains to be done is just a Taylor expansion of the Lagrangian, $\mathcal {L}_{\mathrm {eff}}=F(\left \lvert {\Xi }\right \rvert )$, in $\pi ^i$. Dropping a constant and a surface term, we find

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{eff}}\simeq-(-1)^dF^{\prime}_0(\partial_0\boldsymbol\pi)^2+[(-1)^dF^{\prime}_0+2F^{\prime\prime}_0](\boldsymbol{\nabla}\cdot\boldsymbol{\pi})^2+\dotsb\;, {} \end{aligned} $$

(14.48)

where $F^{\prime }_0$ and $F^{\prime \prime }_0$ are derivatives of $F(\left \lvert {\Xi }\right \rvert )$ taken in the equilibrium, $\langle {\left \lvert {\Xi }\right \rvert }\rangle =(-1)^d$. Note that (14.48) does not contain any transverse gradients of $\pi ^i(x)$. This is expected, saying merely that transverse fluctuations do not propagate via harmonic oscillations due to the absence of shear elastic forces in a fluid. Fluids support only longitudinal sound waves, whose phase velocity follows as

$$\displaystyle \begin{aligned} v_{\mathrm{L}}=\sqrt{1+2(-1)^d\frac{F^{\prime\prime}_0}{F^{\prime}_0}}\;. \end{aligned} $$

(14.49)

It is easy to see with the help of (14.45) that this equals , as expected for thermodynamic sound.

Example 14.3

Another informative check of our EFT is to take the nonrelativistic limit. To that end, we use the relation (14.45) between the function $F(\left \lvert {\Xi }\right \rvert )$ and the energy density $U(\left \lvert {\Xi }\right \rvert )$ to decompose the former as

$$\displaystyle \begin{aligned} F(\left\lvert{\Xi}\right\rvert )=-\varrho_0\sqrt{(-1)^d\left\lvert{\Xi}\right\rvert }-\mathcal{F}\bigl((-1)^d\left\lvert{\Xi}\right\rvert \bigr)\;. {} \end{aligned} $$

(14.50)

The constant $\varrho _0$ is the mass density of the fluid in the body coordinates. In the local rest frame, where is defined by (14.24), is the density of the rest mass of the fluid. This is expected to dominate the energy in the nonrelativistic limit. On the other hand, becomes the elastic free energy of the fluid. Its contribution to $F(\left \lvert {\Xi }\right \rvert )$ is assumed to be suppressed relatively to the leading term in (14.50) by two inverse powers of the speed of light.

In the laboratory frame, we write $\Xi ^{ij}=\partial _0X^i\partial _0X^j-\tilde \Xi ^{ij}$ and again use the identity $\det \Xi =\exp \operatorname {\mathrm {tr}}\log \Xi$. This gives

(14.51)

where the ellipsis stands for higher-order corrections, negligible in the nonrelativistic limit. In turn, (14.45) becomes

(14.52)

This has a simple interpretation. The three contributions to U correspond, up to a sign, to the relativistic rest energy, the nonrelativistic kinetic energy and the thermodynamic (internal) energy, while P is the nonrelativistic pressure. The same expansion converts the action (14.38) to

(14.53)

This copies the EFT (14.26) for nonrelativistic solids that we previously obtained by an educated guess. The only difference is that the gradient free energy $\mathcal {F}$ is now allowed to depend only on . (The ${\varrho _0\sqrt {\left \lvert {{\tilde \Xi (x)}}\right \rvert }}$ term gives upon integration a constant and can be dropped.) The EFT (14.53) for nonrelativistic fluids inherits the symmetry under VPDs of the target space $\mathcal {M}$ from its relativistic ancestor (14.38). The spacetime part of its symmetry is however different, and includes spacetime translations, spatial rotations, and Galilei boosts.

The symmetry of fluids under VPDs, $G_{\mathrm {int}}\simeq \mathrm {SDiff}(\mathbb {R}^d)$, implies the existence of infinitely many conserved currents. Indeed, the action of an infinitesimal VPD can be expressed as $\updelta X^i=\epsilon V^i(X)$, where $\epsilon$ is a parameter and $V^i(X)$ any smooth vector field on $\mathcal {M}\simeq \mathbb {R}^d$ with vanishing divergence, $\partial {V^i(X)}/\partial {X^i}=0$. A straightforward application of Noether’s theorem gives the current

$$\displaystyle \begin{aligned} J^\mu_V=F^{\prime}(\left\lvert{\Xi}\right\rvert )\left\lvert{\Xi}\right\rvert (\Xi^{-1})_{ij}\partial^\mu X^iV^j(X)\;. \end{aligned} $$

(14.54)

Its on-shell conservation is also seen as a direct consequence of the EoM (14.39). The conservation of the corresponding integral charges generalizes the so-called Kelvin circulation theorem to relativistic fluids; see [13] for further details.

4 Coupling Nambu–Goldstone Bosons to Classical Matter

Before we wrap up the discussion of EFTs for classical matter, let us step back to see how it connects to the rest of the book. We started the analysis of spontaneously broken spacetime symmetries with a fairly general construction of nonlinear realization thereof in Chap. 12. However, in its subsequent applications, we gradually increased the assumptions on the symmetry and the order parameter. In the present chapter, we ended up discarding altogether the possible presence of NG modes associated with long-range order in the quantum ground state. Here it is possible to tie up some loose ends with little effort. An attentive reader might have noticed two related observations I made in quite different contexts. In Sect. 13.3.3, I pointed out that it might be possible to make an EFT with Aristotelian symmetry invariant under boosts, whether Galilei or Lorentz. All one needs is an auxiliary variable representing the velocity of the medium in which the EFT lives. On the other hand, we learned in Sect. 14.1.2 that this velocity is naturally encoded in the identically conserved current (14.9). Following this link, I will now sketch concretely how an Aristotelian EFT for broken internal symmetry can be made boost-invariant by coupling the EFT to a classical medium. A general EFT framework for such hybrid systems does not seem to exist as yet. For the sake of illustration, I will restrict the discussion to systems where the classical medium is a fluid. The construction is loosely inspired by Pavaskar et al. [14].

Instead of lengthy reminders, I refer the reader to Chap. 8 for the details of construction of EFTs for broken internal symmetries. (See Sect. 8.1.4 for an executive summary.) All these EFTs were built assuming Aristotelian symmetry, that is symmetry under spacetime translations and spatial rotations. There are two details we have to attend to if we want to augment this spacetime symmetry with boosts. First, the temporal and spatial derivatives, which appear independently in the general Aristotelian effective Lagrangian (8.33), have to be combined in a way that respects boost invariance. Second, any operator invariant under all the spacetime and internal symmetries can be multiplied with an arbitrary function of $\left \lvert {\Xi }\right \rvert$ (for relativistic fluids) or (for nonrelativistic fluids). From this point on, it is convenient to split the discussion of the two cases.

Suppose we want to make the Aristotelian EFT invariant under Galilei boosts. We need not do anything about spatial derivatives, which are already Galilei-invariant on their own. Temporal derivatives can be fixed by contraction with the current (14.41). In terms of the body coordinates $X^i$ and the velocity $\dot {\boldsymbol x}$ of the medium, this amounts to replacing the time derivative of the NG field $\pi ^a$ of the broken internal symmetry with $J^\mu \partial _\mu \pi ^a=\det M(\partial _0\pi ^a+\dot {\boldsymbol x}\cdot \boldsymbol \nabla \pi ^a)\equiv \det M(\nabla _0\pi ^a)$. The minimal modification of the two-derivative part of the effective Lagrangian (8.33) then reads

(14.55)

The two factors of $\det M$ coming from the replacement $\partial _0\pi ^a\to J^\mu \partial _\mu \pi ^a$ are without loss of generality absorbed into . To preserve the internal symmetry acting on $\pi ^a$, the coupling functions , and must satisfy the constraints (8.34) and (8.35) for all values of $X^i(x)$. The precise dependence of the coupling functions on can be fixed by measuring the effective couplings $\kappa _{ab}$, $\bar \kappa _{ab}$ and $\lambda _{ab}$ as a function of the density of the underlying medium.

The only nontrivial bit of the “Galileanization” of the EFT (8.33) lies in the part of the Lagrangian with a single time derivative, . In case this is also strictly invariant under the internal symmetry, it can be promoted to

(14.56)

where is subject to the condition ; cf. (8.39). However, in case is merely quasi-invariant, we cannot multiply it by an arbitrary function of . The only way out then seems to be to set

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{eff}}^{(0,1)}=c_a(\pi)J^\mu\partial_\mu\pi^a\;, {} \end{aligned} $$

(14.57)

where the 1-form $c(\pi )\equiv c_a(\pi )\mathrm{d} \pi ^a$ is constrained by the required strict invariance of its exterior derivative, $\mathrm{d} c(\pi )$. This makes invariance under Galilei boosts manifest, while quasi-invariance under the internal symmetry acting on $\pi ^a$ is preserved thanks to the identical conservation of $J^\mu$. Note that in this special case, imposing Galilei invariance produces a unique coupling between the NG fields $\pi ^a$ and the matter variables $X^i$, without any new a priori unknown parameters. Mathematically, the Lagrangian (14.57) descends from the $(d+1)$-form $c(\pi )\wedge \mathrm{d} X^1\wedge \dotsb \wedge \mathrm{d} X^d$ on the coset space of broken internal symmetry.

Example 14.4

Our main example of a system with a Lagrangian have been ferromagnets (see Sect. 9.2). While most natural ferromagnets are metals, there are also ferromagnetic materials that are crystalline insulators. The low-energy EFT of such materials brings together ferromagnetic magnons and the phonons of the crystal lattice. In this case, the dependence of the two-derivative Lagrangian on the tensor is more complicated than in (14.55). The reason for this is the reduced emergent symmetry of crystals as compared to fluids; see [14] for further details. However, the one-derivative part of the Lagrangian is still (14.57), generating a unique coupling between magnons and phonons.

Let us finally see how the above argument needs to be modified, should we replace the requirement of Galilei invariance with the relativistic Poincaré invariance. In this case, the two-derivative Lagrangian may only feature properly contracted Lorentz indices. Embedding the antisymmetric $\lambda _{ab}$-term into a Lorentz-invariant operator is possible by replacing . Analogously, the spatial gradient operator $\boldsymbol \nabla \pi ^a\cdot \boldsymbol \nabla \pi ^b$ should be embedded in . In order to maintain the form of the two-derivative Lagrangian in the local rest frame of the underlying medium, it is then convenient to write it as

$$\displaystyle \begin{aligned} \notag \mathcal{L}_{\mathrm{eff}}^{(2,0)}={}&\frac 12\kappa_{cd}(\left\lvert{\Xi}\right\rvert )\omega ^c_a(\pi)\omega ^d_b(\pi)\partial_\mu\pi^a\partial^\mu\pi^b\\ {} &-\frac 12\lambda_{cd}(\left\lvert{\Xi}\right\rvert )\omega ^c_a(\pi)\omega ^d_b(\pi)\varepsilon^{\lambda\mu\nu}J_\lambda\partial_\mu\pi^a\partial_\nu\pi^b\;,\\ \notag \mathcal{L}_{\mathrm{eff}}^{(0,2)}={}&\frac 12[\bar\kappa_{cd}(\left\lvert{\Xi}\right\rvert )-\kappa_{cd}(\left\lvert{\Xi}\right\rvert )]\omega ^c_a(\pi)\omega ^d_b(\pi)(J^\mu\partial_\mu\pi^a)(J^\nu\partial_\nu\pi^b)\;. \end{aligned} $$

(14.58)

This naturally incorporates the possibility for type-A NG bosons to propagate with a velocity unrelated to the speed of light while maintaining full Poincaré invariance. The coupling functions $\kappa _{ab}(\left \lvert {\Xi }\right \rvert )$, $\bar \kappa _{ab}(\left \lvert {\Xi }\right \rvert )$ and $\lambda _{ab}(\left \lvert {\Xi }\right \rvert )$ are still subject to the constraints (8.34) and (8.35) for all values of $X^i(x)$. The one-derivative Lagrangian , should it be strictly invariant, can be coupled to the medium by a slight modification of (14.56),

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{eff}}^{(0,1)}=-\sigma_b(\left\lvert{\Xi}\right\rvert )\omega ^b_a(\pi)J^\mu\partial_\mu\pi^a\;. \end{aligned} $$

(14.59)

When is merely quasi-invariant, it can still be coupled to the classical medium via (14.57). Namely, (14.57) in fact respects the infinite-dimensional group of transformations preserving the spacetime volume form thanks to its origin in the $(d+1)$-form $c(\pi )\wedge \mathrm{d} X^1\wedge \dotsb \wedge \mathrm{d} X^d$. This group includes both Galilei and Lorentz boosts.