A pedagagical introduction to the Lifshitz regime

We give an elementary and pedagogical review of the phase diagrams which are possible in Quantum ChromoDynamics (QCD). Currently, the emphasis is upon the appearance of a critical endpoint, where disordered and ordered phases meet. In many models, though, a Lifshitz point also arises. At a Lifshitz point, three phases meet: disordered, ordered, and one where spatially inhomogeneous phases arise. At the level of mean field theory, the appearance of a Lifshitz point does not dramatically affect the phase diagram. We argue, however, that fluctuations about the Lifshitz point are very strong in the infrared, and significantly alter the phase diagram. We discuss at length the analogy to inhomogenous polymers, where the Lifshitz regime produces a bicontinuous microemulsion. We briefly mention the possible relevance to the phase diagram of QCD.


I. INTRODUCTION
Experiments at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) have demonstrated conclusively that a new state of matter is produced in the collisions of heavy ions at high energy. In the central region, the system behaves like a Quark-Gluon Plasma (QGP), at high temperature and very small baryon chemical potential.
The central region was studied originally because it is (almost) free of baryons. This is useful, because in themodynamic equilibrium, it is possible to compare with the results of numerical simulations on the lattice.
It is natural to ask what happens as one goes down in energy. In that case, even in the central region the temperature decreases, and the baryon (or quark) chemical potential becomes significant. In the collisions of heavy ions it will never be possible to go to very low temperature, but clearly the phase diagram, as a function of the temperature T , and the quark chemical potential, µ, is probed.
At low µ and nonzero T , numerical simulations on the lattice indicate that while there is no true phase transition, there is a large increase in the pressure in a relatively narrow region of temperature [1]. That is, there is a crossover, which appears to be associated with the chiral transition.
This need not remain true as the chemical potential increases. It is plausible that at increasing µ, a line of first order transitions appears. If so, the line of first order transitions must end in a critical endpoint [2][3][4]. This is a true critical point, which in thermodynamic equilibrium exhibits infinite correlation lengths.
In this paper we discuss one appears to be a footnote to the phase diagram: the appearance of spatially inhomogenous phases. In nuclear matter this is known as pion and kaon condensates. The appearance of such phases is difficult to derive, even in mean field theory.
Nevertheless, although these phases are certainly important, naively one wouldn't expect such condensates to dramatically affect the phase diagram. This is true at the level of mean field theory. We show, however, that fluctuations dramatically affect the phase diagram [5]. In mean field theory, three phases meet at what is known as a Lifshitz point. In three spatial dimensions, the fluctuations at a Lifshitz point are so strong that they completely wipe out the Lifshitz point, leaving only a Lifshitz regime.
It is possible that the critical endpoint is completely wiped out, leaving only a line of first order transitions. In this case, while infrared fluctuations can be strong in the infrared, they remain finite at all points in the phase diagram.
While our arguments are qualitative, they are rather general. We also discuss the close analogies between the phase diagram of QCD and that in inhomogenous polymers [6,7].
There, what we call the Lifshitz regime is known as bicontinuous microemulsions, and is of practical importance.

II. MEAN FIELD THEORY: TRICRITICAL POINTS
We first review the standard theory of how a critical endpoint can arise. Consider a scalar field φ, which I assume for simplicity to be single component. It is trivial to immediate to the case where φ transforms under some global symmetry group G. We take as the Lagrangian If we consider the behavior at nonzero temperature in four space time dimensions, then static correlations functions are determined by correlation functions in three spatial dimensions. In that case, φ has dimensions of √ mass, so λ has dimensions of mass, while κ is dimensionless.
Thus in the sense of the renormalization group, κ is a marginal operator, and should be included.
Let us begin with the case where λ is positive. Then we have the standard phase diagram.
The theory is invariant under a global symmetry of Z(2), φ → −φ. When m 2 is positive the expectation value φ = 0, and one is in the symmetric phase. For negative m 2 φ = 0, which is the broken phase. There is a second order phase transition for m 2 = 0, which is a second order phase transition. By the renormalization group, the behavior is controlled by the universality class of a Z(2) invariant theory, such as the Ising model. For other models the universality class is that of the symmetry group G.
It is also possible to consider negative quartic couplings, where λ < 0. To ensure that the potential is bounded from below, we have to assume that the hexatic coupling κ is positive.
Then one has a first order transition from the symmetric to the broken phase. It is possible to determine in detail where it occurs: there transition is from φ = 0 to some nonzero φ 0 ; by the symmetry, it is to ±φ 0 . This is determined by the potential being degenerate with If the symmetry is not exact, one adds a term which breaks the symmetry, such as hφ. If the background field h is small, the position of the transitions should be near that for h = 0.
The most dramatic change is that the line of second order transitions becomes a line for crossover. That is, when h = 0 the field always has a nonzero expectation value, φ = 0, so the theory is always in a broken phase.
For sufficiently negative values of λ, though, the first order transition must remain. Thus the line of first order transitions persists. This implies that it terminates in a critical endpoint, precisely as for the liquid-gas transition in water. The critical field is φ− φ . If the underlying theory has a larger symmetry of G, the universality class of the critical endpoint remains as in the Ising model, Z(2). This is because the field develops an expectation value along some direction, and the critical field is still φ − φ , for that particular direction.

III. MEAN FIELD THEORY: LIFSHITZ POINT
Next we consider a more general Lagrangian, This is an effective Lagrangian, so it is possible to have terms involving higher derivatives.
Because of causality, this is not possible for derivatives with respect to time: these can only be to quadratic order. For spatial derivatives, however, it is possible to consider terms which are of higher order. We then include term with four spatial derivatives, ∼ (∂ 2 i φ) 2 ; by dimensionality, this coefficient must have dimensions of ∼ 1/M 2 , where M is some mass scale which arises by constructing the effective theory.
To ensure the theory has a stable vacuum, this coefficient must be positive. This implies that the usual term, with two spatial derivatives, can have a coefficient, Z, which is negative. This is a circumstance which may be unfamiliar. The theory can be either in the symmetric or disordered phase.
Consider first the symmetric phase, where m 2 > 0. The dispersion relation is plotted in Fig. Figure 2. There is no condensate, but clearly the minimum of the propagator is at a nonzero momentum, k 0 . We can choose this direction to be along some direction, say where The first condition is only satisfied if Z is negative. As can be seen from the effective mass, having Z < 0 tends to drive the effective mass negative, but if m 2 is sufficiently large, we can still remain in the symmetric phase.
It is notable that in Eq. 3, the terms which are quadratic in the transverse momenta, k 2 ⊥ , vanish identically. This is due to the spontaneous breaking of the rotational symmetry: the propagator has a minimum about some nontrivial value, and we choose a direction about which to expand. This is also why there are terms ∼ k z k 2 in the inverse propagator.
As Z becomes more negative, eventually we are driven into a phase when locally the symmetry is broken, with φ = 0. Since to lowest order the kinetic term is negative, though, this is a qualitatively different state. In detail, the nature of this state depends intricately upon the symmetry group. We chose to discuss the very simplest possibility, where now φ has two components. In that case, we assume that along some arbitrary direction, which we choose to beẑ, that there is a spiral: We then have two parameters to determine, p 0 and φ 0 . The kinetic terms contribute 1 2 Minimizing with respect to k 0 gives This the lowest energy state with k 0 = 0 when Z < 0. Using this value for k 0 , the value of the condensate is determined by the usual equation, Minimizing this potential gives the usual value for the condensate, In this spatially homogeneous phase, while φ = 0 locally, it is not globally. This is obvious even for one condensate oriented in a given direction, as when we integrate over z, clearly φ(z) will average to zero.
Further, this state is itself unstable, as we show later. Even without computation, this can be guessed. There is nothing special about theẑ direction, and fluctuations will tend to disorder the theory. It is natural to expect that instead there is a series of patches, whose width is determined by the underlying dynamics of the theory. We shall discuss this later.
Without going into the details, we can understand the nature of the phase diagram in mean field theory, which we illustrate in When m 2 is negative, one goes from a broken phase, to one which is spatially inhomogenous, as Z becomes negative. At the level of mean field theory, the free energy is given by T c . By Eq. 4, there are then terms linear in T − T c in the potential, and so the free energy.
This is the sign of a first order transition, as then derivative of the free energy, with respect to temperature, is discontinuous. This is also obvious physically. In a typical first order transition the theory jumps from one phase to another, and the masses are discontinuous. In this case the masses are continuous, but the structure of the theory is completely different, as one goes from a homogeneous ground state, to a ground state dominated by patches of spatially inhomogeneous condensates.

IV. ANISOTROPIC FLUCTUATIONS AND THE PHASE DIAGRAM
The phase diagram changes dramatically once fluctuations are included. The basic physics can be understood from the propagator in the symmetric phase, Eq. 3. Because the minimum is at nonzero momentum, the ground state spontaneously breaks Lorentz symmetry, and the fluctuations are anisotropic. To one loop order, there is a contribution to the mass term, For small effective masses, the dominant contribution is from k z − k 0 ∼ m eff , and the anistropic propagator makes the theory effectively one dimensional. This is the origin of the term ∼ M/m eff . The integral over transverse fluctuations, k ⊥ , is cutoff by the higher order terms in the propagator, proportional to the mass scale associated with the higher derivative terms, ∼ M .
The effective reduction to one dimension produces the factor of 1/m eff . This implies that while in mean field theory there is a second order transition as m eff → 0, this is not consistent with fluctuations. This does not preclude a phase transition from occurring: for a fixed, negative value of Z, one is clearly in a symmetric phase for large, positive m 2 , and in a spatially inhomogeneous phase for negative m 2 . Thus a phase transition must happen, but it will do so through a first order transition, jumping from one value of m 2 eff to another. In condensed matter physics this was first pointed out by Brazovski [8,9]. There it is often referred to as a fluctuation induced first order transition, but it is rather different from, This is the natural generalization of the propagator in the symmetric phase, Eq. 3. The coefficients c i are to be determined. The important point is that to quadratic order in the fluctuations, only the longitudinal momenta enter.
Consider again a tadpole diagram in this phase. For simplicity we set c 1 = c 3 = 0, where Λ IR is an infrared cutoff. Because of this divergence, there is no true long range order.
This is exactly analogous to the smectic-C phase of liquid crystals: these are systems which are ordered in one direction, but act as a liquid in the transverse directions [10].
The lack of long range order is to be expected. After all, we had assumed that the theory had broken the three dimensional to one dimensional symmetry. As commented, we expect that the theory will form patches of one dimensional structure. The interaction between the patches is controlled by the logarithmic infrared divergences above. Nevertheless, there can be a large separation of scales.

V. ISOTROPIC FLUCTUATIONS
We now consider the effect of fluctuations near the Lifshitz point. At the Lifshitz point, the effective theory is given by Although it will not enter into our considerations at nonzero temperature, which are governed by static correlation functions, we stress that the time derivative is customary, of quadratic order.
In momentum space the propagator at the Lifshitz point is critical dimension is eight, when the renormalization of the coupling constant, develops a logarithmic divergence. Similarly, the low critical dimension if four, when the shift in the mass squared, is logarithmically divergent. This implies that in less than four dimensions, that there are power like infrared divergences, and it cannot be possible to reach the Lifshitz point. It is useful to consider the analogy to a spin system in two (or fewer) dimensions. Begin in the symmetric phase, and tune the mass to decrease. Then unlike in more than two dimensions, it is not possible to tune the mass to vanish: a nonzero mass will be generated non-perturbatively. Conversely, in mean field theory, it is only possible to go to a spatially inhomogeneous phase when Z is negative. It is possible, however, that due to strong non-perturbative fluctuations, that the theory develops a spatially inhomogeneous phase even if Z is positive, but small.
That is, to avoid the instability of a Lifshitz point, everywhere along the line of first order transitions either Z or m 2 are nonzero. It is possible to have an isolated point where Z = 0, but then m 2 must be nonzero. We suggest the following: there is a region, which we term the "Lifshitz regime", where Z is small, and m 2 is nonzero. The possible phase diagram is illustrated in Figure 5.

VI. LIFSHITZ REGIME IN INHOMOGENEOUS POLYMERS
There is a known example of a would be Lifshitz point in inhomogeneous polymers [6,7,[11][12][13]. Consider first the example of a mixture of oil and water, which separate into droplets of either oil or water. By adding a surficant, however, the interface tension between the phases changes, and other phases emerge. A more controlled example is given by mixing two different types of polymers, formed of monomers of type A and of type B, which also separate. To this A-B diblock copolymers, which are long sequences of type A polymer, followed by type B. These A-B copolymers localize at the boundaries between phases with only A or B polymers: the part with type A sticks into the part with type A, and similarly for type B. The result is that adding copolymer decreases the surface tension between the A and B phases.
The phases are then the following. At very high temperature A and B polymers mix, which is then a symmetric phase. At low temperature, a mixture of A and B polymers separate into regions with only A or B homopolymers, which is the broken phase. By adding the copolymer, one can obtain a lamellar phase, where A and B regions forms stripes. This is like a smectic liquid crystal, albeit without orientational order.
Mean field theory predicts the existence of a Lifshitz point, where these three phases meet. In contrast, both experiment and numerical simulations with self consistent field theory indicate that there is no Lifshitz point [6,7,[11][12][13]: see, e.g., Fig. 3 of Ref. [6]. In terms of an effective theory, the surface tension is proportional to the wave function renormalization, Z. Thus the bicontinuous microemulsion is a region where Z is very small and m 2 is nonzero. This is what we call the Lifshitz regime.

VII. RELATION TO QCD
We conclude by briefly discussing the possible relevance to QCD. In general, there are two possible instabilities: either the quartic coupling constant, λ, can become negative. This generates the usual critical endpoint suggested previously.
It is also possible that the wave function renormalization constant for the quadratic spatial derivatives, Z, becomes negative. This generates a Lifshitz point.
At present the relationship between the two can only be studied by using effective models.
In the simplest Nambu Jona-Lasino model, it is found that the two points coincide. This can be understood as following. Starting with Bosonizing this through introducing σ = qq, at one loop order we need to evaluate tr log( ∂ + σ) ≈ d 1 σ 4 + (∂σ) 2 + . . . , for some constant d 1 . We only indicate the first term in this expansion, as there are an infinite series of term involving higher powers of derivatives and factors of σ.
What is found, however, is that the coefficient of the first two terms are tied together.
While surprising at first, this can be understood through a simple scaling argument: we can rescale both length, ∂ → κ ∂, and σ → κσ. The one loop determinant is invariant under this scaling, and so any expansion must respect it as well.
The first coefficient, σ 4 , controls the location of where the quartic coupling becomes negative. The second coefficient, (∂σ) 2 , determines when Z becomes negative. This explains why the critical endpoint and the Lifshitz point coincide in the simplest NJL model. See, for example, Fig. 6 of Buballa and Carignano [14]. This is also seen in solutions of Schwinger-Dyson equations [15].
This equality fails when a more complicated model is considered. In the simplest NJL model, This is special to models where the sigma mass is twice the constituent quark mass, m σ /m qk = 2. Carignano, Buballa, and Schaefer [16] showed that in a quark-meson model, where one can allow m σ /m qk = 2, that the Lifshitz and critical endpoints separate.
There are then two possibilities: in the plane of temperature T and quark (or baryon) chemical potential µ, the first singularity which one meets can be either the critical endpoint, or the (would be) Lifshitz point. Since one can only use effective models, clearly a definitive answer cannot be given.
What can be suggested is the following. Since the simplest NJL model indicates that the critical endpoint and (would be) Lifshitz point are near one another, it suggests the following. The critical endpoint is dominated by a single massless mode, which exhibit true infrared fluctuations in the infinite volume limit. The Lifshitz point has fluctuations which are large, but finite, in the infinite volume limit.
For the case of heavy ion collisions, which occur over a finite region of space and time, it is clearly a challenge to distinguish between the two types of infrared fluctuations. This is not as difficult as it may seem. For the critical point, there is a single massless mode, due to the σ meson, Quark matter (21) A proposed phase diagram for QCD is given in Figure 6. The example of NJL models, however, demonstrates that the existence of the Lifshitz point cannot be ignored. Because