CCounting Rules of
Nambu-Goldstone Modes
Haruki Watanabe
Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan;email: [email protected]. Rev. Condens. Matter Phys. 2020.11:1–15https://doi.org/10.1146/annurev-conmatphys-031119-050644Copyright c (cid:13)
Keywords spontaneous symmetry breaking, Nambu–Goldstone modes,nonrelativsitic systems, internal and space-time symmetries,low-energy effective theory
Abstract
When global continuous symmetries are spontaneously broken, thereappear gapless collective excitations called Nambu–Goldstone modes(NGMs) that govern the low-energy property of the system. The appli-cation of this famous theorem ranges from high-energy, particle physicsto condensed matter and atomic physics. When a symmetry breakingoccurs in systems that lack the Lorentz invariance to start with, as isusually the case in condensed matter systems, the number of result-ing NGMs can be fewer than that of broken symmetry generators, andthe dispersion of NGMs is not necessarily linear. In this article, wereview recently established formulae for NGMs associated with brokeninternal symmetries that work equally for relativistic and nonrelativisticsystems. We also discuss complexities of NGMs originating from space-time symmetry breaking. Along the way we cover many illuminatingexamples from various context. We also present a complementary pointof view from the Lieb-Schultz-Mattis theorem. a r X i v : . [ c ond - m a t . o t h e r] N ov ontents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Internal symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1. Spontaneous symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2. Number of Nambu–Goldstone modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3. Historical review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5. Effective Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6. Mermin-Wagner-Coleman theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83. Space-time symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1. Translation symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2. Other spatial symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104. When does symmetry breaking occur? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.1. Time translation symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2. Lieb-Schultz-Mattis theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1. Introduction
The Nambu–Goldstone theorem is a powerful theorem predicting the appearance of massless particles uponspontaneous breaking of global continuous symmetries (1, 2, 3, 4, 5). It explains, for instance, why the pionmass is so small in terms of the chiral symmetry breaking (6). This is one of a few rare examples of general,non-perturbative theorems applicable to a variety of systems regardless of the microscopic details.The examples of spontaneous broken symmetries and accompanying gapless excitations are not limited toLorentz-invariant systems (7). For instance, the very existence of crystals around us is a result of spontaneousbreaking of spatial translation symmetry, and their universal low-energy properties, such as the Debye T law of the specific heat, can be attributed to acoustic phonons, i.e., the Nambu–Goldstone modes (NGMs)associated with the broken translation (8, 9, 10). The idea of classifying phases and exploring transitionsbetween them based on the symmetry breaking pattern encoded in order parameters is nowadays referredto as Landau’s paradigms and is understood as a prerequisite for the further classification of phases from atopological perspective (11, 12).Condensed matter systems usually lack the Lorentz symmetry due to their coupling to the surroundingenvironment that fixes a specific choice of the reference frame. In the absence of the restrictive constraintimposed by the Lorentz symmetry, even the basics properties of NGMs such as the number of modes andthe behavior of their dispersion relations get variety even for the same symmetry breaking pattern. Thespinwave excitation, or magnon, in ferromagnets is the classic example that deviates from the relativisticbehavior. In recent years, many new examples of “abnormal number” of NGMs were reported in differentcontext (13, 14, 15, 16, 17) ranging form high-density quantum chromodynamics to cold atom systems.This review article discusses recently established formulae that provide us with a coherent understandingof all of these examples based solely on the symmetry breaking pattern and an additional information ondensities of globally conserved charges in ground states. We also address spontaneous breaking of space-timesymmetries and their consequences, resolving, for example, why crystals only have phonons but do not havegapless excitations for equally broken rotations.Throughout this review we will set c = (cid:126) = 1 to simplify notations.
2. Internal symmetries2.1. Spontaneous symmetry breaking
Let us first specify the class of symmetries we consider in this section. Let G be the symmetry group of thesystem of our interest. In the most general case, an element g ∈ G would transform a local operator ˆ φ α ( x , t )in the following way . ˆ g ˆ φ α ( x , t )ˆ g † = (cid:88) β ˆ φ β ( x (cid:48) , t (cid:48) )[ U g ( x , t )] βα . (1)Here we assume that the symmetries are global , meaning that the unitary matrix U g ( x , t ) in Eq. 1 is independentof x or t . We also restrict ourselves to internal symmetries for which the coordinates ( x , t ) and ( x (cid:48) , t (cid:48) ) in Eq. 1 Throughout this review, quantities with ‘hat’ represent quantum mechanical operators re identical for every g ∈ G . When ( x (cid:48) , t (cid:48) ) differs from ( x , t ) for some elements of g ∈ G , the symmetry group G is referred to as space-time and it requires a more careful treatment. We will discuss such symmetries laterin Sec. 3.The symmetry of a physical ground state | GS (cid:105) can be lower than G , and when this is the case we say G isspontaneously broken. An element h ∈ G is unbroken if ˆ h | GS (cid:105) and | GS (cid:105) represent the same state, i.e.,ˆ h | GS (cid:105) = e iθ h | GS (cid:105) . (2)The set of unbroken symmetries forms a subgroup H of G . Other elements of G , G \ H as a set, is said to be broken . Spontaneous breaking of the symmetry G down to H ⊂ G results in a ground state degeneracy, andthe set of degenerate ground states constitutes the coset space G/H .In practice, the definition of (un)broken symmetries written in the form of Eq. 2 is not very useful, becauseket vectors in the thermodynamic limit are, strictly speaking, ill-defined. Instead one can diagnose symmetrybreaking through an expectation value of local operators. To this end, let us properly choose a set of operatorsˆΦ α ( x , t ), which may be composite (a product of local operators), and consider the following combination. δ g ˆΦ α ( x , t ) ≡ ˆ g ˆΦ α ( x , t )ˆ g † − ˆΦ α ( x , t ) . (3)For unbroken symmetries, Eq. 2 immediately implies that the expectation value of δ h ˆΦ α ( x , t ) vanishes for anychoice of ˆΦ α . Conversely, if the ground state expectation value (cid:104) δ g ˆΦ α ( x , t ) (cid:105) = (cid:88) β (cid:104) ˆΦ β ( x , t ) (cid:105) ( U g − βα is non-vanishing for a choice of ˆΦ α , Eq. 2 must be violated and the symmetry g ∈ G is spontaneously broken.The expectation value (cid:104) ˆΦ α ( x , t ) (cid:105) hence distinguishes ordered phases from disordered phases and are called an order parameter .Just to help grasping these notations, let us briefly discuss the quantum Ising model on the cubic lattice asan example. The Hamiltonian ˆ H = − J (cid:80) (cid:104) x , x (cid:48) (cid:105) ˆ s z ( x )ˆ s z ( x (cid:48) ) − B (cid:80) x ˆ s x ( x ) is written in terms of a spin operatorˆ s α ( x ) on the site x that satisfies [ˆ s α ( x ) , ˆ s β ( x (cid:48) )] = i(cid:15) αβγ δ x , x (cid:48) ˆ s γ ( x ). The internal symmetry of this model is G = { e, g } = Z , where g is the spin rotation about the x axis by the angle π . To diagnose the breaking ofthis symmetry, we set ˆΦ α = ˆ s z and δ g ˆΦ α ( x ) = − s z ( x ). In the ordered phase with (cid:104) ˆ s z (cid:105) (cid:54) = 0, the unbrokensymmetry H is trivial. The coset G/H = Z in this example corresponds to the two degenerate ground states,one with (cid:104) ˆ s z (cid:105) > (cid:104) ˆ s z (cid:105) < discrete and continuous symmetries. When G is continuous (i.e.,when G is a Lie group), the Noether theorem (6) provides a definition of the conserved current ∂ t ˆ q i ( x , t ) + ∇ · ˆ j i ( x , t ) = 0. In turn, the conserved charge ˆ Q i ≡ (cid:90) d x ˆ q i ( x , t ) (4)plays the role the generator of the symmetry, enabling us to represent elements of G connected to the identityas ˆ g = e i (cid:80) i (cid:15) i ˆ Q i . Here and hereafter, the label i distinguishes generators of G . Also, to simplify the notationwe assume 3 + 1 dimensions in this section. By expanding ˆ g in Eq. 3 in the power series of (cid:15) to the linear order,we see that the spontaneous breaking of the symmetry generated by ˆ Q i can be detected by the expectationvalue (cid:104) [ ˆ Q i , ˆΦ α ( x , t )] (cid:105) . (5)Similarly, let us take generators ˆ Q ρ of H . They are unbroken and the expectation values in Eq. 5 all vanish.For continuous symmetries, the coset space G/H becomes a manifold whose dimension is given by the numberof broken generators n BG ≡ dim G/H = dim G − dim H. (6)This number agrees with the number of “flat directions” of fluctuations of order parameters. When the system spontaneously breaks a continuum symmetry G down to H by developing an order parameter,long-wavelength fluctuations of the order parameter give rise to gapless excitations, so-called Nambu–Goldstonemodes (NGMs) or Nambu–Goldstone bosons. They are also referred to as “massless” because the minimum ofthe dispersion relation E k can be identified with the mass of the quasi-particle. In order to justify the usage ofmomentum k , we always assume that at least discrete translation symmetries remain unbroken in every spatialdirection. • Counting Rules of Nambu-Goldstone Modes 3 n Lorentz invariant systems, it has been long known that each flat direction of order-parameter fluctuationsproduces its own soft mode (6). Therefore, the number of NGMs n NGM is always given by the number of brokengenerators in Eq. 6, n NGM = n BG (in relativistic systems) . (7)When the Lorentz invariance is explicitly broken to start with, this is not always the case. Spontaneouslybroken symmetries still imply the appearance of NGMs (18, 19) but the number of NGMs can be fewer thanthe number of broken generators in the absence of the Lorentz symmetry: n NGM ≤ n BG (in non-relativistic systems) , which means that the one-to-one correspondence between flat directions and gapless modes can be lost. Innon-relativistic systems, knowing the symmetry-breaking pattern G → H is not sufficient to predict the numberof NGMs. We need an additional input about the property of ground states, which turns out to be densitiesof globally conserved charges (20, 21).One of the main goal of this review is to explain the general counting rule of NGMs that applies to bothrelativistic and non-relativistic cases. It was conjectured in Ref. (22) and later proved in Refs. (23, 24) thatthe number of NGMs is always given by n NGM = n BG −
12 rank ρ (general) . (8)Here, ρ is a real anti-symmetric matrix defined by ρ ij ≡ − i V (cid:104) [ ˆ Q i , ˆ Q j ] (cid:105) = (cid:88) k f kij (cid:104) ˆ Q k (cid:105) V , (9)where V is the volume of the system and the thermodynamic limit V → ∞ is implicitly assumed. The relation[ ˆ Q i , ˆ Q j ] = i (cid:80) k f kij ˆ Q k is used in the second equality. (The structure constant f kij here includes possible“central extensions” (22, 25).) One can also express ρ ij using the charge density ˆ q i ( x , t ) in Eq. 4. Assumingthe continuous translation symmetry, we get ρ ij = − i (cid:104) [ ˆ Q i , ˆ q j ( x , t )] (cid:105) = (cid:88) k f kij (cid:104) ˆ q k ( x , t ) (cid:105) . (10)The last expression suggests that ρ ij vanishes in Lorentz invariant systems since ˆ q i is the temporal componentof the four vector (ˆ q i , ˆ j i ). Also, comparing the middle expression of Eq. 10 with Eq. 5, we see that ρ ij (cid:54) = 0plays the role of an order parameter characterizing the spontaneous breaking of the generator ˆ Q i . Therefore, ρ ij must vanish if either ˆ Q i or ˆ Q j is unbroken.The real anti-symmetric matrix ρ can always be diagonalized by an orthogonal matrix into the followingform, ρ = λ − λ λ − λ λ m − λ m O . (11)Here, λ (cid:96) ’s ( (cid:96) = 1 , . . . , m ) are assumed to be nonzero and all blank entries are 0. Since the matrix rank isbasis-independent, we get rank ρ = 2 m so that the right-hand side of Eq. 8 is guaranteed to be an integer.In fact, it is possible to formulate a finer counting rule of NGMs. To this end we classify NGMs into twotypes, type A and B (22, 23). In the basis choice of Eq. 11, type-B modes are those associated with pairsof broken generators ( ˆ Q (cid:96) − , ˆ Q (cid:96) ) ( (cid:96) = 1 , , · · · , m ). As it becomes clear shortly, each pair produces only onegapless mode that normally has a quadratic dispersion E k ∝ k ( k ≡ | k | ) in the long-wavelength limit, unlessone fine-tunes parameters. Type-A modes, on the other hand, correspond to remaining broken generators.Each of such generator produces one type-A mode that tends be linearly dispersive ( E k ∝ k ) for small k . Bydefinition, the number of each type of NGMs is given by n A = n BG − rank ρ, n B = 12 rank ρ, (12)which, of course, satisfies n A + n B = n NGM . In particular, NGMs in Lorentz invariant systems are all type A. a) G / H = S (b) G / H = S (c) G / H = S Figure 1
Illustration of the coset space
G/H . The black dot stands for the chosen ground state and orange arrows representfluctuations corresponding to NGMs. (a)
G/H = S for superfluids, which may be viewed as the bottom of the“Mexican hat” potential. (b) G/H = S for ferromagnets. Two coordinates together form a single precession mode. (c) G/H = S for antiferromagnets. Fluctuations into two directions are independent. Let us review the preceding works that led to the general formulae in Eqs. 8 and 12. In 1976, Nielsen andChadha (26) presented a general counting rule of NGMs valid either with or without relativistic invariance.They divided NGMs into two classes, type I and type II, based on the behavior of their dispersion relation.Type I and II modes, respectively, have an energy dispersion proportional to an odd and even power of itsmomentum in the limit of long wavelengths. By examining the analytic property of the correlation function inEq. 5, they showed that n I + 2 n II ≥ n BG , (13)where n I and n II are the number of the type I and type II modes. If n I and n II are replaced by n A and n B ,the inequality is always saturated: n A + 2 n B = n BG . In fact, there exists a trivial noninteracting example in which the equality of Eq. 13 does not hold under afine-tuning of parameters (see appendix A of Ref. (22)).In 2001, Sch¨afer et al (14) proved a theorem, stating that n NGM = n BG as long as (cid:104) [ ˆ Q i , ˆ Q j ] (cid:105) = 0 for anypair of i and j , via a simple analysis on the linear dependence of states { ˆ Q i | GS (cid:105)} . This is indeed the spacialcase (rank ρ = 0) of Eq. 8. They also identified the mechanism of the reduction of n NGM in a concrete field-theoretical model that we review in Sec. 2.4.3 — they found a term in the Lagrangian that makes fluctuationsinto two orthogonal directions canonically conjugate to each other. We will see in Sec. 2.5 that this is actuallythe general case when the reduction of n NGM occurs for internal symmetries.In 2004 , Nambu (27) made similar but insightful observations (without a formal proof): (i) The relation (cid:104) [ ˆ Q i , ˆ Q j ] (cid:105) (cid:54) = 0 makes two would-be NGMs corresponding to ˆ Q i and ˆ Q j canonically conjugate to each other; (ii)The number n NGM reduces by one for every such pair; and (iii) The NGMs associated with (cid:104) [ ˆ Q i , ˆ Q j ] (cid:105) (cid:54) = 0 hasa quadratic dispersion. One can see that these are essentially the same statements as in the above formulae,given the existence of the basis in which the matrix ρ takes the form of Eq. 11.With these preceding studies, the value of the general formulae in Eqs. 8 and 12 lies in the precise for-mulation to the above intuitive, empirical understanding. The question is how to verify them on the generalground and we review a proof in Sec. 2.5. Before going into the proof, let us see how the formulae work by reviewing pedagogical examples.
The simplest example of spontaneous breaking of internal symmetries in condensed matterphysics occurs in Bose-Einstein condensates of interacting bosons. The original symmetry G = U(1), underlyingthe particle-number conservation, is spontaneously broken to the trivial subgroup H = { e } . The coset spaceis G/H is the ring S corresponding to the choice of the U(1) phase of the macroscopic order parameter[Figure 1(a)]. Since there is only one generator in this example, the matrix ρ always vanishes and ( n A , n B ) =(1 , The received date on the published paper is Dec. 26 of 2002. ••
The simplest example of spontaneous breaking of internal symmetries in condensed matterphysics occurs in Bose-Einstein condensates of interacting bosons. The original symmetry G = U(1), underlyingthe particle-number conservation, is spontaneously broken to the trivial subgroup H = { e } . The coset spaceis G/H is the ring S corresponding to the choice of the U(1) phase of the macroscopic order parameter[Figure 1(a)]. Since there is only one generator in this example, the matrix ρ always vanishes and ( n A , n B ) =(1 , The received date on the published paper is Dec. 26 of 2002. •• Counting Rules of Nambu-Goldstone Modes 5 .4.2. Heisenberg spin model.
As a classic example that realizes n NGM < n BG , let us discuss the Heisenbergspin model on the cubic lattice. ˆ H ≡ J (cid:88) (cid:104) x , x (cid:48) (cid:105) ˆ s ( x ) · ˆ s ( x (cid:48) ) . The sum is over nearest neighbours. This model has the full spin rotation symmetry G = SO(3) generated byˆ Q i = (cid:80) x ˆ s i ( x ) ( i = x, y, z ).When J <
0, the ground state develops a ferromagnetic order, which we set (cid:104) ˆ s ( x ) (cid:105) = (0 , , m z ) T with m z >
0. Since ρ xy = − ρ yx = m z and other components of ρ vanish, we see that both ˆ Q x and ˆ Q y arespontaneously broken ( n BG = 2) and that rank ρ = 2. The unbroken symmetry is H = SO(2) generated by ˆ Q z .The coset space G/H is the two sphere S [Figure 1(b)] in accordance with the fact that the magnetization couldpoint in arbitrary direction. Correspondingly, there are two flat directions for fluctuations of the ferromagneticorder parameter. However, as it can be seen by the linearized spin-wave theory (10, 31), there is only one gaplessmode of the precession type, which has a quadratic dispersion. This is in agreement with ( n A , n B ) = (0 , J >
0, in contrast, the ground state has the vanishing magnetization but instead develops a N´eelorder, (cid:104) ˆ s ( x ) (cid:105) = e i k · x (0 , , n z ) T ( n z > k = πa (1 , , a is the lattice constant). This order also breaksthe spin rotation symmetry G = SO(3) down to the subgroup H = SO(2) ( n BG = 2). Although the symmetrybreaking pattern is identical to the ferromagnetic case as far as the internal symmetry is concerned, this timeall components of ρ vanishes and the relativistic result ( n A , n B ) = (2 ,
0) is recovered. Thus there exist twolinear modes and n NGM = 2 [Figure 1(c)]. In this example, the value ρ = 0 is “protected” by an unbrokensymmetry of the time-reversal operation followed by the unit translation.The effective Lagrangian treatment of NGMs in ferromagnets and antiferromagnets can be found, forexample, in Refs. (32, 33, 34). The above spin model may look too unfamiliar to people with high-energybackground. Thus let us discuss an illuminating example of a scalar field theory, describing the Kaon conden-sation in the color-flavor locked phase of quantum chromodynamics (13, 14). The Lagrangian density for atwo-component complex scalar field φ = ( φ , φ ) T reads L ≡ (cid:88) ν ( ∂ ν + iA ν ) φ † ( ∂ ν − iA ν ) φ − V ( φ † φ ) , where V ( φ † φ ) ≡ m φ † φ + λ ( φ † φ ) is a φ -potential, x ν ( ν = 0 , , ,
3) corresponds to ( t, x, y, z ), and A ν is abackground U(1) gauge field which we set A = µ > A = . Clearly, the Lorentz invariance is explicitlybroken by the chemical potential µ (cid:54) = 0. The internal symmetry of the model is G = U(2) that has fourgenerators ˆ Q ν .The (classical) expectation value φ is found by minimizing the potential V µ ( φ † φ ) ≡ V ( φ † φ ) − µ φ † φ modified by the chemical potential. When µ > m , φ can be set ( v, T with v >
0, which breaks thesymmetry G = U(2) into a subgroup H = U(1) generated by ˆ Q − ˆ Q z . The coset space G/H is the threesphere S and n BG = 3. At the tree-level approximation, the expectation value of the Noether charge is givenby (cid:104) ˆ Q ν (cid:105) (cid:39) (cid:90) d xφ † ( i∂ t + µ ) σ ν φ + c.c. = 2 µv V δ ν,z , where σ ν ’s are Pauli matrices. The nonzero components of ρ are ρ xy = − ρ yx = 4 µv (cid:54) = 0 and rank ρ = 2.Now, let us introduce a small fluctuation π a for each broken generator ˆ Q a ( a = x, y, z ). Plugging φ = φ + v ( iπ z , π y − iπ x ) into the Lagrangian and expand it to the quadratic order in π a , one finds a term2 µv ( ∂ t π x π y − ∂ t π y π x ) (14). As a result, π x and π y are paired up to be a single quadratic mode in thelow-energy limit, while π z produces a linearly dispersive mode (13, 14). This is consistent with the prediction( n A , n B ) = (1 ,
1) based of the general counting rules.This result should be compared to the standard Lorentz-invariant case with µ = 0 and m <
0. The sameform of the classical expectation value φ = ( v (cid:48) , T implies the breaking of the three generators ( n BG = 3).(When µ = 0, G and H are enhanced to O(4) and O(3) but G/H remains to be S .) This time we get rank ρ = 0and ( n A , n B ) = (3 , E k = k . Cold atomic systems with spin degrees of freedom offer a nice playground of variety ofintriguing symmetry breaking patterns. For example, the F = 1 spinor Bose-Einstein condensate can realizethe ferromagnetic phase that breaks G = U(1) × SO(3) down to H = U(1) (cid:48) (the prime indicates that thetwo U(1) factors in G and H are different) (35, 36). The nonzero magnetization implies rank ρ = 2, andthe counting rule predicts ( n A , n B ) = (1 , ose-Einstein condensates can be described by the Bogoliubov theory (also called the Gross-Pitaevskii theory),which formally looks very similar to the above model of Kaon condensation. NGMs in this class of models werestudied in Ref. (37). As there already exist several nice reviews (38, 39, 40, 41), here we avoid repeating thedetails. We still add that the spin wave dispersion in the ferromagnetic phase has been measured in a recentexperiment of Rb (42).
Now let us overview the derivation of general formulae discussed in Sec. 2.2 in the effective Lagrangian ap-proach (23, 25, 43). (For a general review of effective field theories, see Refs. (44, 45, 46).) The formulae werealso verified independently in Mori’s projection operator formalism (24) and have been extended to systems atfinite temperatures (47) or out of equilibrium (48).The effective Lagrangian is designed to capture the low-energy, long distance fluctuations of the macroscopicorder of the system. The only input to the effective theory is the symmetry breaking pattern G → H . Oneconstructs the most general Lagrangian by including all terms that respect assumed symmetries (49, 50, 51).To control the number of terms in the Lagrangian, we make two simplifications. (i) Only fluctuations into thedirection of G/H are taken into account. In other words, fluctuations of the amplitude of the order parameters,giving birth to the Higgs modes (52), are neglected. The effective theory is thus the non-linear sigma modelwhose target space is
G/H . (ii) Since the focus is on the the low-energy, long-wavelength dynamics, thederivative expansion that arrange terms in the power series of derivatives is employed. In the absence ofLorentz invariance, the time and space derivatives have to be counted separately. For simplicity we assumethe spatial rotation symmetry; otherwise the power-counting of ∂ x , ∂ y , · · · should be all independent. We alsoassume first that there are no low energy degrees other than NGMs in the system. When they do exist theycan be added later.The general structure of the effective Lagrangian for non-relativistic systems was studied in details in thepioneering work of Leutwyler (32, 53). The advantage of using the Lagrangian formalism is that commutationrelations among fields upon the quantization are an output , not an input, of the theory in contrast to theHamiltonian formalism as in Ref. (54).As a warm-up, let us discuss the effective Lagrangian describing the superfluid phase of interacting bosons(Sec. 2.4.1). The order parameter of the condensate can be written as Φ( x , t ) = (cid:112) n ( x , t ) e iθ ( x ,t ) . Here thefield θ parameterizes the coset space G/H = S and transforms nonlinearly under the broken U(1) symmetryˆ g = e i(cid:15) ˆ Q , θ (cid:48) ( x , t ) = θ ( x , t ) + (cid:15). (14)The effective Lagrangian, invariant under the U(1) symmetry, is thus written in terms of derivatives of θ : L eff = ¯ g ( ∂ t θ ) − g ( ∇ θ ) + · · · . (15)The dots represent terms with more derivatives, which are not important in the low-energy limit. This La-grangian describes a NGM with the linear dispersion relation E k = vk with v = (cid:112) g/ ¯ g .To do the same for a general symmetry breaking pattern G → H , let us introduce so-called Maurer-Cartanforms that serve as building blocks of the effective Lagrangian (49, 50). Let T i ’s be a faithful representation ofgenerators of G , satisfying [ T i , T j ] = i (cid:80) k f kij T k , and let T a ’s be broken generators. Then the Maurer-Cartanone-form ω iµ ( µ = t, x, y, z ) is defined by (cid:88) i ω iµ T i ≡ − iU † ∂ µ U, U ( π ) ≡ e i (cid:80) a π a T a . Nambu–Goldstone fields π a in U ( π ) can be viewed as a local coordinate of the manifold G/H . By the seriesexpansion, we see that ω aµ = ∂ µ π a + 12 (cid:88) b,c f abc π b ∂ µ π c + O ( π ) . (16)For instance, the above example of the superfluid corresponds to the choice π a = θ and T a = 1 so that U = e iθ and ω µ = − iU † ∂ µ U = ∂ µ θ .The transformation rule of Nambu–Goldstone fields π a under g = e i (cid:80) i (cid:15) i T i is defined by gU ( π ) = U ( π (cid:48) ) h g , h g ∈ H. It follows that π a ’s transform linearly, ( π a ) (cid:48) = (cid:80) b ( U h ) ab π b , under unbroken symmetry h ∈ H and nonlinearly[e.g. Eq. 14] under broken symmetry g ∈ G \ H . • Counting Rules of Nambu-Goldstone Modes 7 o the quadratic order in derivatives, the general effective Lagrangian for G → H , symmetric under thespatial translation and rotation, was found to be (25) L eff = − (cid:88) i e i ω it + (cid:88) a,b ¯ g ab ω at ω bt − (cid:88) a,b (cid:88) r = x,y,z g ab ω ar ω br + · · · . (17)Constants e i , g ab , and ¯ g ab must obey the conditions (cid:80) j f jρi e j = 0 and (cid:80) c ( f cρa g cb + f cρb g ac ) = 0 (the samefor ¯ g ). With these constraints, L eff is fully symmetric under G , not only under the unbroken subgroup H . Asfound by Ref. (32), the constant e i is related to the conserved charge density e i = (cid:104) ˆ q i ( x , t ) (cid:105) .This effective Lagrangian is a natural generalization of the well-established result for Lorentz-invariantsystems ( η µν is the Minkowski metric) L eff = (cid:88) a,b (cid:88) µ,ν g ab η µν ω aµ ω bν + · · · , which corresponds to e i = 0 and ¯ g ab = g ab in Eq. 17. This relativistic form reproduces the original Nambu–Goldstone theorem 7.To examine the nature of excitations described in the general effective Lagrangian, let us expand Eq. 17 tothe quadratic order in π by plugging Eq. 16. We get (23) L eff (cid:39) (cid:88) a,b ρ ab ∂ t π a π b + (cid:88) a,b ¯ g ab ∂ t π a ∂ t π b − (cid:88) a,b (cid:88) r = x,y,z g ab ∂ r π a ∂ r π b + · · · . (18)Assuming the form of ρ in Eq. 11, the first term implies that π (cid:96) − and π (cid:96) are conjugate degrees of freedom,producing a type-B NGM with a quadratic dispersion ( E k = k / m with m ∼ ρ ab /g ab ). When the secondterm of Eq. 18 is nonzero, a “gapped partner,” whose mass gap is the order of ρ ab / ¯ g ab (55), may exist for eachtype-B mode, although such a statement for gapped modes requires further consideration on the consistencyof the derivative expansion. The remaining π a fields produce their own linearly dispersive modes (see Ref. (25)for the detailed analysis). These results imply the counting rules in Eq. 12.The effective Lagrangian can be used not only for counting NGMs but also for analyzing interactions ofNGMs. For example, higher order terms in π a ’s in Eq. 17, dropped in the linearized Lagrangian in Eq. 18,describe interactions among NGMs. Other low-energy degrees possibly existing in actual symmetry brokenphases can also be included to Eq. 17 by adding terms that respect all the assumed symmetries. The difference between the two types of NGMs is not limited to the dispersion relation. To clarify this point, letus review the Mermin-Wagner-Coleman theorem (56, 57, 58). The standard understanding of this theorem isthat it prohibits spontaneous breaking of any continuous symmetry in 1+1 dimensions at the zero temperature(equivalently, two spatial dimensions at a finite temperature). Intuitively, the theorem holds because the would-be gapless mode produces an uncontrolled quantum fluctuation in one spatial dimension. Using the effectiveLagrangian in Eq. 18, one can readily evaluate the fluctuation due to a type-A NGM: (cid:104) π ( x , t ) (cid:105) ∼ (cid:90) d d kdω gω − gk ∝ (cid:90) Λ0 dkk d − , which suffers from the infrared divergence when d = 1. The ultraviolet cutoff Λ is not of our interest as weare focusing on the long-wavelength physics. Such an infrared divergence destroys the order parameter for thepostulated broken symmetry.One may think that the situation gets worse for type-B NGMs because they are softer, but this is not thecase. In fact, fluctuation caused by a type-B NGM (cid:104) π ( x , t ) (cid:105) ∼ (cid:90) d d kdω iρω − gk ∝ (cid:90) Λ0 dkk d − (19)converges even when d = 1. This suggests that continuous symmetries can, in fact, be spontaneously broken in1 + 1 dimensions as far as only type-B NGMs appear. This is actually reasonable — according to Eq. 9, the or-der parameter for the broken generators ˆ Q i , ˆ Q j is given by the sum of conserved charges ρ ij = (cid:80) k f kij (cid:104) ˆ Q k (cid:105) /V .Because ˆ Q k commutes with the Hamiltonian, one can choose the physical ground state to be a simultaneouseigenstate of ˆ Q k and the Hamiltonian. Then there would be no quantum fluctuation of the order parameterregardless of the dimensionality of the system. The trivial example is again provided by the quantum ferromag-net in which (cid:104) ˆ S z (cid:105) plays the role of order parameter. Because of this fundamental difference, sometimes thesecases are excluded from the examples of spontaneous broken symmetries. (For example, see the commentaryby Anderson (59)). . Space-time symmetries We move on to the discussion of spontaneous breaking of space-time symmetries. For this class of symmetries,there are several complications at different levels, and it does not seem possible to write down the most generalformula in any useful manner. Here let us instead start with examples to see what we get out of them ingeneral. Throughout this section we consider general spatial dimension d greater than one. Let us begin by spontaneously breaking of translation symmetry. In many respects the spatial translation isunique among all space-time symmetries. Most importantly, NGMs originating from the spontaneously brokentranslation can be treated in essentially the same way as those associated with internal symmetries.
Suppose that the system develops a crystalline order that breaks the continuous translationsymmetry G = R d down to a discrete one in every direction so that H = Z d . The number of broken generators isthus n BG = d . To describe low-energy deformation of the crystal, we introduce the same number of displacementfields u a ( x , t ) that transform nonlinearly under the corresponding translation, u a ( x + (cid:15) , t ) (cid:48) = u a ( x , t ) + (cid:15) a . The theory of elastic medium (60, 61) gives us the effective Lagrangian that describes the acoustic phonons,which reads L eff = 12 m ( ∂ t u ) − λ abcd ε ab ε cd + · · · , (20)where ε ab ≡ ( ∂ a u b + ∂ b u a ) is the (linearized) strain tensor and m is the mass density of the medium. Unbrokenspatial symmetries of crystals reduce independent parameters in the elastic modulus tensor λ abcd (60) (alsoknown as the elastic constant tensor (61)). The Lagrangian in Eq. 20 predicts the existence of d linearlydispersive modes – one longitudinal and ( d −
1) transverse in the isotropic case. The number of NGMs n NGM thus agrees with the number of broken generators n BG .In principle, however, there is no reason not to add the following term to the effective Lagrangian 20, asfar as it respects all assumed symmetries. 12 (cid:88) a,b ρ ab ∂ t u a u b . This term makes momentum operators non-commuting (cid:104) [ ˆ P a , ˆ P b ] (cid:105) = − iρ ab V as indicated by Eq. 9. In fact,it does appear in several actual settings. The classical example is the Wigner crystal of electrons in twodimensions exposed to an external magnetic field B z . In this case ρ ab = eB z as a result of the Aharonov–Bohmeffect. The phonon mode in the Wigner crystal, which is classified as type B in our scheme, has a dispersionproportional to a fractional power of k due to the long-range Coulomb interaction of electrons (62, 63).A more recent example of ρ ab (cid:54) = 0 is provided by the crystal of topological solitons, called Skyrmions, intwo-dimensional ferromagnets (64, 65). The Berry phase action of underlying spins gives rise to ρ xy = 4 πsn sk where s is the spin density and n sk is the Skyrmion number density (66, 67, 68). This term is the origin of theMagnus force acting on Skyrmions (66) and the quadratic dispersion of the phonon in Skyrmion crystals (67).When the thickness of the system is taken into account in three dimension, each Skyrmion becomes a line-likeobject rather than a point-like soliton and the crystalline phase is formed by a lattice of Skyrmion lines. Lowenergy fluctuations of such a system were studied in Refs. (69, 70). It is often the case in condensed matter physics that the translation symmetry is sponta-neously broken together with some internal symmetries. As a trivial setting, let us imagine two completelyindependent orders — a crystalline order that breaks the continuous translation symmetry down to a latticetranslation symmetry R d → Z d , and another order that breaks an internal symmetry G int → H int . In thiscase phonons discussed in Sec. 3.1.1 and NGMs originating from the spontaneously broken internal symmetries(Sec. 2) would simply coexist.As a more nontrivial example, let us consider a system of interacting bosons with a finite density at zerotemperature. When the repulsive interaction is properly designed, the Bose-Einstein condensate may developa spontaneous crystalline order simultaneously breaking the translation symmetry and the U(1) symmetry.Such a possibility was considered in the context of the postulated supersolid phase of He (71, 72). Then anatural question arises: is the superfluid phonon originating from the spontaneous breaking of U(1) symmetry(Sec. 2.4.1) independent from phonons associated with the crystalline order? This question was answeredaffirmatively by the effectively Lagrangian approach (73) and by the mean-field analysis of a Gross-Pitaevskimodel (74, 75). • Counting Rules of Nambu-Goldstone Modes 9 n examples we discussed so far, the straightforward extension of Eq. 8 to space-time symmetries seems towork if the momentum operators are included to the definition of the matrix ρ in Eq. 9. An example found inRef. (76) in which the commutation relations between internal symmetries and momentum operators becomenontrivial also belongs to this class. We will see, however, that space-time symmetries are not that simple ingeneral. Now we move on to other spatial symmetries such as rotations and dilatations. We still assume t (cid:48) = t in Eq. 1. In d -dimensional crystals, not only the translation R d but also the SO( d ) rotation arespontaneously broken by the crystalline order. Thus the number of broken generators is at least d plus d ( d − / ρ vanishes when the crystal is at rest as long as the standardalgebra among the linear momentum and the angular momentum is assumed. Similar phenomenon is knownto occur for spontaneous breaking of conformal symmetries even in the Lorentz-invariant case (77). How canwe account for these missing NGMs?Low and Manohar (78) addressed this puzzle. Let (cid:104) ˆΦ( x ) (cid:105) be the order parameter detecting the breakingof generators T a (i.e., T a (cid:104) ˆΦ( x ) (cid:105) (cid:54) = 0 for broken generators). According to Ref. (78), the number of NGMsoriginating from spontaneous breaking of spatial symmetries is reduced by the number of nontrivial solutionsto the equation (cid:88) a c a ( x ) T a (cid:104) ˆΦ( x ) (cid:105) = 0 , (21)where c a ( x ) is a function of the coordinates x in the directions of unbroken translation. Using their exam-ple (78), let us set d = 2 and consider an order parameter that depends on x but not on y , implying the spon-taneous breaking of the momentum P x and the angular momentum J z . Plugging the relation J z ≈ xP y − yP x among these generators, one gets( c x P x + c z J z ) (cid:104) ˆΦ( x ) (cid:105) = ( c x − c z y ) P x (cid:104) ˆΦ( x ) (cid:105) = 0 , (22)for which c x ( y ) = yc z ( y ) is a nontrivial solution. Such solutions imply redundancies among long wavelengthfluctuations of the order parameter — some Nambu–Goldstone fields π can be eliminated in favor of a gradientof other Nambu–Goldstone fields ∂ µ π (78). This pattern of symmetry breaking, forming a crystalline order inone direction that also implies a breaking of associated rotation symmetry, occurs in smectic phases of liquidcrystals, and the reduction of low-energy modes in this context has been extensively studied (see e.g. (61, 79,80, 81)). More generally, the reduction of independent degrees of freedom for space-time symmetries throughthis mechanism is called the inverse-Higgs effect (78, 82, 83, 84, 85).This, however, fails to explain the situations which there is no unbroken continuous translation symmetries.Also, the relation among generators should be formulated more rigorously in term of conserved charge densitiesin Eq. 4. These points were refined in Ref. (86). The number of independent Nambu–Goldstone fluctuations isreduced by one for every linear combination of Noether densities of globally conserved charges that annihilatesthe ground state: (cid:88) a c a ( x )ˆ q a ( x ) | GS (cid:105) = 0 . (23)Here, coefficients c a ( x ) are allowed to depend on all coordinates. For example, in a 2D crystal that breaksˆ P i = (cid:82) d x ˆ q i ( i = x, y ) and ˆ J z = (cid:82) d x ˆ q z , the exact relation among the conserved charge densities ˆ q z = x ˆ q y − y ˆ q x implies that (ˆ q z − x ˆ q y + y ˆ q x ) | GS (cid:105) = 0. This is an example of Eq. 23 with c z = 1, c x = y , and c y = − x . Abroken generator ˆ Q a produces an independent NGM when it is not involved in any linear combination in Eq. 23.The counting of the independent low-energy fluctuations applicable to a finite temperature was formulated inRef. (87). In the previous section, we explained a general criterion for the appearance of independentNGMs originating from spontaneously broken space-time symmetries. Here, we assume they do exist andexamine their low-energy properties.In general, ordered phases contain many dynamical degrees of freedom apart from NGMs. If other ex-citations are all gapped, they never affect the low-energy physics in any essential manner — one can safely“integrate out” gapped modes that only renormalize parameters. On the other hand, if there exists othergapless modes, they can, in principle, destroy NGMs via interactions. However, the microscopic symmetryof the system encoded in the effective Lagrangian puts severe restrictions on the way NGMs interact among
10 H. Watanabe hemselves and with other degrees of freedom (6). As a result, NGMs are robust and remain well-definedpropagating modes at least in the low-energy, long-wavelength limit. This is consistent with the statementof the Nambu–Goldstone theorem — the very fact that it could predict the appearance of massless particleswithout caring about other degrees of freedom already implies the stability of NGMs. Below we argue thatsuch a nice low-energy property is absent in the case of NGMs originating from spontaneously broken spatialsymmetries, except for the ordinary phonons.To understand the key point through an educative example, let us review the nematic Fermi fluid discussedin Refs. (88, 89). In this example, a circular Fermi surface of spinless fermions in (2 + 1) dimensions sponta-neously deforms into an elliptic shape, breaking the continuous rotation symmetry down to a discrete two-foldrotation. As the translation symmetry remains unbroken, there is no linear combination that satisfies Eq. 23and a Nambu–Goldstone field θ ( x , t ) can be introduced. It transforms nonlinearly under the rotation by anangle (cid:15) , θ (cid:48) ( R x , t ) = θ ( x , t ) + (cid:15), (24)where R = e − i(cid:15)σ y ∈ SO(2) represents the rotation matrix. The fermion field ψ ( x , t ) is a scaler ψ (cid:48) ( R x , t ) = ψ ( x , t ). Using these ingredients, one can write down the following rotation-symmetric Lagrangian L eff = (cid:104) ψ ∗ ( i∂ t + µ ) ψ − m ∇ ψ ∗ · ∇ ψ (cid:105) + (cid:104)
12 ¯ g ( ∂ t θ ) − g ( ∇ θ ) (cid:105) + λ (cid:2) ( ∂ x ψ ∗ ∂ x ψ − ∂ y ψ ∗ ∂ y ψ ) cos 2 θ + ( ∂ x ψ ∗ ∂ y ψ + ∂ y ψ ∗ ∂ x ψ ) sin 2 θ (cid:3) , (25)which can be derived by a mean-field approximation of a spinless electron model (88). The first term representsthe original circular Fermi sea, the second one gives the bare linearly dispersive NGM, and the third one describethe interaction between them.Assuming that the fluctuation θ is small, we may Taylor expand the interaction in the power series of θ .The zero-th order term together with the first term produces an elliptic Fermi surface, i.e., the set of ( k x , k y )’ssatisfying µ = ( m − λ ) k x +( m + λ ) k y . Surprisingly, the interaction term linear in θ [2 λ ( k (cid:48) x k y + k (cid:48) y k x ) ψ ∗ k ψ k θ q inthe Fourier space] does not contain any derivative acting on θ . The coefficient of this term becomes 4 λk x k y inthe limit of q →
0, which does not vanish at most of parts on the Fermi surface. A perturbative calculation withthis non-derivative coupling signals the breakdown of Fermi-liquid theory and overdamping of the NGM (88).Namely, the NGM, after dressed up with particle/hole excitations near the distorted Fermi surface, looses itsparticle nature.The transformation of the θ -field in Eq. 24 is almost identical to that of superfluids in Eq. 14. The crucialdifference lies in the argument of θ (cid:48) ( R x versus x ), which distinguishes the spatial rotation (a space-timesymmetry) from the phase rotation (an internal symmetry). This difference results in the presence/absence ofthe “non-derivative” interaction of NGMs we discussed just now.This scenario is not restricted to the above specific setting and the generalization to wider class of space-time symmetries was performed in Ref. (90). According to the study, a NGM will be overdamped in thepresence of Fermi surface if the NGM originates from spontaneous breaking of a generator ˆ Q a that does notcommute with the momentum operator, i.e., [ ˆ Q a , ˆ P ] (cid:54) = 0 . (26)This is actually the case for almost all of space-time symmetries other than the translation symmetry. Examplesof such situations have been discussed in Refs. (91, 92, 93).To summarize this section, for spontaneously broken spatial symmetries, one has to pay attention to twoadditional subtleties: (i) The number of independent fluctuations may be fewer than the number of brokengenerators. Namely, introducing one field for every broken generators may be redundant. This mismatchcannot be captured by naively extending the counting rule for internal symmetries [Eq. 8] but can be seen,instead, by listing up relations of the form of Eq. 23. (ii) Even when additional low-energy degrees originatefrom spontaneous breaking of spatial symmetries, some of them may not form a propagating mode, beingoverdamped via interactions with other low-energy degrees. In the presence of a Fermi surface in the orderedphase, Eq. 26 provides the criterion for overdamping.
4. When does symmetry breaking occur?
The Nambu–Goldstone theorem assumes spontaneous breaking of a global symmetry and predicts the con-sequence. In this section, we address two complementary questions. (i) Can any symmetry, in principle, bespontaneously broken if the model Hamiltonian is properly designed? (ii) For symmetries that can indeed bebroken in some setting, when do they tend to be broken? • Counting Rules of Nambu-Goldstone Modes 11 .1. Time translation symmetry
When G is an internal symmetry, the answer to the first question always seems to be positive. Let (cid:104) ˆΦ α ( x ) (cid:105) be an order parameter transforming nontrivially under G as in Eq. 1. By modifying the Hamiltonian ˆ H to ˆ H − λ (cid:82) d x (cid:80) α ˆΦ α ( x ) † ˆΦ α ( x ) with a large enough coefficient λ , the order parameter (cid:104) ˆΦ α ( x ) (cid:105) would benon-vanishing in lowest energy states and G would be spontaneously broken.Such a naive argument does not apply to the time translation symmetry t → t (cid:48) = t + (cid:15) generated bythe Hamiltonian itself. In 2012, Wilczek (94) proposed a possibility of a new phase, dubbed “quantum timecrystal,” in which the quantum ground state spontaneously breaks the time translation symmetry down to adiscrete subgroup. If such a phase existed, it would be a temporal analog of ordinary crystals discussed inSec. 3.1.1. However, several follow-up studies (95, 96) clarified that the time-translation symmetry cannot bespontaneously broken in any ground state or thermal equilibrium. The time translation symmetry is the singleknown example of “never-be-broken” global symmetries, and it would be an interesting future work to pindown the general criterion for space-time symmetries of this type.Remarkably, Wilczek’s proposal eventually led to the recent discovery of “discrete time crystals” (97, 98,99, 100, 101, 102) in non-equilibrium, externally driven systems. This exciting topic will be covered in thereview article by Nayak (ARCMP vol. 11). A non-equilibrium time crystal may support diffusive NGMs, whichhas recently been studied in Refs. (103, 104). See also Ref. (105) for spontaneous breaking of more standardsymmetries in quantum open systems. The Lieb-Schultz-Mattis theorem is a theorem imposing a general, non-perturbative constraint on the low-energy spectrum of many-body quantum systems in arbitrary spatial dimension d . It assumes two symmetries:a U(1) symmetry that defines an integer-valued charge ˆ Q via the Noether theorem (Sec. 2.1) and a discretetranslation symmetry that defines a fundamental domain, called “unit cell.” (In other to make ˆ Q integer-valued, one should properly choose the overall factor and the offset in the definition of ˆ Q ). These symmetriescan be just a subgroup of the full symmetry group of the system. Given them, the average U(1) charge perunit cell in a ground state is well-defined even in the thermodynamic limit. We call this quantity the “filling” ν , borrowing the terminology used in electronic systems. The Lieb-Schultz-Mattis theorem states that it ispossible to isolate a unique ground state in energy from other states only when the filling ν is an integer. Inother words, if ν is not an integer, either ground state degeneracy or gapless excitations must exist.This theorem was originally formulated for Heisenberg spin models (Sec. 2.4.2) in 1 + 1 dimensions (106).For this class of models, the theorem may be viewed as a proof of the half of the Haldane conjecture (107),since it predicts gapless excitations for half-integer spin chains, assuming that the ground state is symmetric.Later the theorem was generalized to a wider class of models with two symmetries stated above in arbitraryspatial dimensions (108, 109, 110). Recently the theorem has been refined under an assumption of crystallinesymmetries (111, 112, 113).A ground state degeneracy often occurs as a result of spontaneous symmetry breaking. In d ≥
2, there is amore exotic possibility called “topological” degeneracy (114), which has been the subject of a large number ofrecent studies as it features many intriguing phenomena including “charge fractionalization” (115). A gaplessexcitation does not have to be NGMs either. Therefore we cannot say anything deterministic from this theorembut we can say at least “something” needs to occur.As an application, let us consider a system of bosons with a nonzero density in free space. Since thecontinuous translation symmetry includes arbitrary discrete translation subgroups, one can assume a unit cellof any size, and the filling ν can take any positive value. Therefore, a unique ground state with an excitation gapis prohibited. This explains why the Bose-Einstein condensation (Sec. 2.4.1), exhibiting both a ground statedegeneracy and gapless excitations, is such common for bosonic systems at finite density at zero temperature.
5. Concluding remarks
In this review we focused on NGMs originating from spontaneous breaking of global symmetries. Formulaefor internal symmetries presented in Sec. 2 do not assume the Lorentz invariance and thus are applicable toboth relativistic and nonrelativistic systems. Subtleties of NGMs associated with space-time symmetries areexplained in Sec. 3. Two relevant issues, which have never been covered in existing reviews on this subject,were addressed in Sec. 4.Before closing, let us briefly mention important related topics we could not discuss in this review. At thetree-level approximation, the number of flat directions of order parameter fluctuations can be accidentally largerthan the minimum number protected by spontaneous symmetry breaking. In that case one encounters quasi-
Nambu–Goldstone modes (or pseudo-
Nambu–Goldstone modes) (116). See, for example, Refs. (117, 39) for
12 H. Watanabe heir realization in cold atom systems and Ref. (118) for the counting rule of quasi-NGMs. When spontaneouslybroken symmetries are fermionic , the resulting low-energy, long-wavelength excitations are also fermionic andare called Nambu–Goldstone fermions (119, 120). Refs. (121, 122, 123) investigated Nambu–Goldstone fermionsin nonrelativistic systems with a supersymmetry. When global symmetries are “gauged” and promoted to localones, the Higgs mechanism may eliminate NGMs and produce a mass to gauge fields (124, 125). The Higgsmechanism in a nonrelativistic setting has been studied in Refs. (126, 127, 128). Finally, the notion of globalsymmetries has been generalized to “ q -form” symmetries (129), where q = 0 corresponds to the conventionalclass discussed in this review. In this scheme, photons in electrodynamics are viewed as a type-A NGMassociated with a one-form symmetry (130), and there are also examples of type-B NGMs (131, 132, 133).Developing an extended formalism describing these NGMs originating from spontaneously broken higher-formsymmetries will be an interesting future work.As we have seen, spontaneous symmetry breaking is quite ubiquitous and examples can be found in almostall areas of physics. It is certainly not possible to exhaust all related topics and we stop here. We hope thisreview helps readers with various background to get into this old but everlasting subject. ACKNOWLEDGMENTS
This review is based on what the author learned through collaborations with Tom´aˇs Brauner, Hitoshi Mu-rayama, Ashvin Vishwanath, and Masaki Oshikawa over years.
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Journal of High Energy Physics arXiv:1903.02846 ••