Anomaly-Induced Inhomogeneous Phase in Quark Matter without Sign Problem

We demonstrate the existence of an anomaly-induced inhomogeneous phase in a class of vector-like gauge theories without sign problem, thus disproving the long-standing conjecture that the absence of sign problem precludes spontaneous breaking of translational invariance. The presence of the phase in the two-color modification of quantum chromodynamics can be tested by an independent nonperturbative evaluation of the neutral pion decay constant as a function of external magnetic field. Our results provide a benchmark for future lattice studies of inhomogeneous phases in dense quark matter.

Introduction.-Self-organization of matter into inhomogeneous patterns is ubiquitous in nature.After all, most natural materials develop crystalline order at sufficiently low temperatures.Yet, in quantum field theory, one usually assumes that the ground state of a given quantum system is uniform; exceptions are often considered exotic and require a specific mechanism for the formation of structure.The question under what conditions the ground state can be nonuniform does not seem to have a general satisfactory answer.
Various inhomogeneous phases are expected to play an important role for the thermodynamics of quark matter under extreme conditions (see Ref. [1] for a review).The predictions of such phases for the phase diagram of Quantum ChromoDynamics (QCD) are, however, mostly based on model calculations and/or on neglecting order parameter fluctuations, which may be crucial for the (in)stability of the phase [2].An exception, shown to exist in the phase diagram of QCD [3,4], is the Chiral Soliton Lattice (CSL) [5].This state is a remarkable manifestation of the chiral anomaly, and requires subjecting quark matter to a strong external magnetic field, or to global rotation [6]; see also Ref. [7] for closely related recent work.
In Ref. [8], an intriguing hypothesis, linking the appearance of nonuniform states in the phase diagram of vector-like gauge theories (hereafter referred to as "QCD-like theories") to the presence of the notorious sign problem [9], was put forward.If true, this would provide a rare example of a no-go theorem for spontaneous breaking of a spacetime symmetry.
In this Letter, we give a counterexample to this conjecture by showing that the CSL or a similar nonuniform state is also present in the phase diagram of a class of QCD-like theories that do not suffer from the sign problem.We start by demonstrating how, in this class of theories, the sign problem can be avoided in presence of a magnetic field.Then the low-energy effective theory (EFT) is constructed based on the knowledge of the symmetries in strong magnetic field.This is finally used to analyze the phase diagram at nonzero baryon chemical potential and magnetic field (and zero temperature).
Absence of sign problem.-Weconsider the class of QCDlike theories where quarks transform in a (pseudo)real representation of the gauge group [10], restricting ourselves for simplicity to two degenerate quark flavors u, d with the common current mass m.Let us denote the Euclidean Dirac operator for a single quark flavor as D i ≡ γ µ D iµ +m−µγ 0 , where i = u, d.Here µ is the quark number chemical potential and the Hermitian Euclidean Dirac matrices γ µ satisfy the charge conjugation property where A a µ are the gluon fields, q i the quark electric charge, and A Q µ represents a background electromagnetic field.In (pseudo)real QCD-like theories, the color generators T a satisfy by assumption T * a = −PT a P −1 .Without loss of generality, the matrix P can be assumed unitary and symmetric for real quarks, and unitary and antisymmetric for pseudoreal quarks [11].For instance, for a theory with the SU(2) gauge group and fundamental quarks ("two-color QCD"), P is given by a Pauli matrix in the color space, P = σ 2 .
Provided the electric charges of the u-and d-quarks satisfy q u = −q d , their respective Dirac operators are related by where K is the operator of complex conjugation.This establishes an antiunitary mapping between the eigenvectors of D u and D d [12].Hence the determinant of the Dirac operator of the theory is real and non-negative, det To conclude, (pseudo)real QCD-like theories with two quark flavors are free of the sign problem in presence of both quark number chemical potential and external electromagnetic field, as long as the two quark flavors have opposite electric charges.
(Pseudo)real theories in magnetic field.-Weshall further assume that the color gauge group and its (pseudo)real quark representation are chosen so that the theory has a confining, chiral-symmetry-breaking vacuum just like QCD.The lowenergy physics of the theory is then dominated by the pseudo-Nambu-Goldstone bosons of its flavor symmetry.
In the limit of zero quark mass ("chiral limit"), (pseudo)real QCD-like theories with N quark flavors feature an enhanced G = SU(2N ) flavor symmetry, which includes as its generators both the electric charge Q and the baryon number B. The chiral condensate in the ground state breaks this spontaneously to H = SO(2N ) in real theories, and to H = Sp(2N ) in pseudoreal theories, resulting in 2N 2 + N − 1 pseudo-Nambu-Goldstone bosons in the real case, and 2N 2 − N − 1 ones in the pseudoreal case.Of these, N 2 −1 are pseudoscalar mesons ("pions"), while the remaining modes, absent in QCD, are scalar diquarks.All the modes together form a single irreducible multiplet with a common mass m π [10].
1. Different regimes of the EFT and the corresponding light degrees of freedom, depending on the strength of the magnetic field.For √ H ΛQCD, the magnetic field can be treated as a perturbation of the ground state of the QCD-like theory.For √ H ΛQCD, the ground state is strongly affected by the field and the low-energy EFT becomes anisotropic.For √ H mπ, the charged degrees of freedom become heavy and decouple from the EFT.(The light charged diquarks d ± , d± are only present in real QCD-like theories.) For N = 2 and with the choice of charges q u = −q d = 0, a uniform external magnetic field reduces the symmetry to Note that G Q is not the usual chiral symmetry of two-flavor QCD: the baryon number B is included as a generator of the "vector" subgroup SU(2) diag .The number of electrically neutral light degrees of freedom, given by the dimension of the coset space G Q /H Q , is three in both real and pseudoreal theories, including the neutral pion π 0 and an electrically neutral diquark-antidiquark pair d 0 , d0 .Low-energy effective theory.-Inmagnetic fields H m 2 π , charged (pseudo)scalars become heavy due to Landau level quantization, and the low-energy physics will be dominated by the electrically neutral modes.In this regime, which we will from now on assume, the low-energy EFT based on the coset space G Q /H Q can be conveniently constructed by using its isomorphism with that of two-flavor QCD.
Magnetic fields around the characteristic scale of the theory, Λ QCD , or stronger, will distort the ground state and make the low-energy EFT anisotropic, breaking the Lorentz group SO(3, 1) down to SO(1, 1) × SO(2).See Fig. 1 for a sketch of the different regimes of the EFT.We aim at finding an effective action for the strong-field regime (C).This can be done by contracting Lorentz indices with projections, g µν and g ⊥µν , of the flat spacetime metric to the two-dimensional subspaces left intact by the magnetic field.Our EFT will also be valid in the moderate-field regime (B), albeit with a reduced predictive power due to the use of a lower spacetime symmetry.
Taking finally into account the discrete symmetries C, P , T , the leading-order effective Lagrangian is given by [13] Here v is a velocity parameter and f π , m π are the pion decay constant and mass, respectively.All the parameters f π , m π , v are given by a priori unknown functions of the magnetic field.The 2 × 2 unimodular unitary matrix field Σ contains the three electrically neutral degrees of freedom.The covariant deriva-tive D µ Σ, specified below in Eq. ( 6), introduces the coupling of diquarks to baryon number chemical potential.The L WZ piece in Eq. ( 3), known as the Wess-Zumino (WZ) term [14], is the contribution of the chiral anomaly.This can also be found following the analogy with two-flavor QCD, by swapping the roles of electric charge and baryon number in the result given in Ref. [3].In Minkowski space, it reads + iC where F Q µν is the electromagnetic field strength tensor and A B µ an external gauge potential that couples to the baryon number current.The overall normalization of the WZ term is not determined by symmetry, but can be fixed by matching the EFT to the underlying QCD-like theory.One then finds [15] where b is the baryon number of a single quark and d is the dimension of the representation of the color gauge group that a single quark flavor transforms in.
A few remarks are in order here.First, while we exploited the analogy with two-flavor QCD, the form of the EFT can as well be obtained by first constructing the EFT for the full coset space G/H and then discarding all charged degrees of freedom.This requires the knowledge of the gauged WZ term in (pseudo)real QCD-like theories though [16].
Second, the assumption that the external magnetic field satisfies H m 2 π , that is lies in the regime (B) or (C), is of little practical limitation.We shall see that interesting physics occurs above certain critical value of the field, which is safely above the (A) regime for light quarks.
Finally, in contrast to the chiral perturbation theory of QCD (see Ref. [17] for a review), the WZ term (4) contributes to the leading order of the EFT.This is due to a modified power counting, whereby the baryon gauge field A B µ counts, just like all derivatives, as O(p 1 ), whereas the magnetic field H counts as O(p 0 ).The latter is required for consistency of the EFT in the (C) regime, and makes the coefficients f π , m π , v functions of H.The resulting EFT may be valid for arbitrarily strong fields as long as the ground state breaks the flavor symmetry by the formation of the chiral condensate, which is supported by the magnetic catalysis phenomenon [13].Note that, in contrast, in the (A) regime f π , m π are true constants (and v = 1) and the dependence of the EFT on H is fully fixed by electromagnetic gauge invariance.
In the rest of this Letter, we analyze the ground state of the EFT at nonzero baryon number chemical potential, µ B , and magnetic field H. Details of excitation spectrum in the various phases in the phase diagram will be reported elsewhere [15].We will fix without loss of generality q u = −q d = 1/2 and b = 1/2; any other choice can be absorbed into a redefinition of H and µ B , respectively.
To bring the EFT into a form more suitable for the analysis, we map the matrix Σ on a unit four-vector, Σ ≡ n 0 + i n • τ , where τ are the Pauli matrices and n 2 0 + n 2 = 1.The nontrivial components of the covariant derivative D µ Σ then read The effective Lagrangian (3) now boils down to where we oriented the magnetic field along the z-axis and set A B µ = (µ B , 0).The ellipsis indicates terms with three derivatives, coming from the first line of Eq. ( 4); being linear in time derivatives, they do not contribute to the Hamiltonian and thus do not affect the structure of the ground state [18].
Chiral limit.-Theground state is easy to determine by a direct minimization of the Hamiltonian in the chiral limit, m π → 0. For any nonzero µ B , there turn out to be two phases.For CH < f 2 π , the ground state is n 0 = n 3 = 0 and n 1 2 + n 2 2 = 1.This describes a Bose-Einstein condensate (BEC) of diquarks, which appears in the phase diagram of (pseudo)real QCD-like theories generally for µ B > m π [10].
For CH > f 2 π , on the other hand, the ground state features a spatially dependent chiral condensate and neutral pion condensate, but no diquark condensate: n 1 = n 2 = 0 and up to an arbitrary translation of the z-coordinate.This corresponds to the CSL state in the chiral limit [4,19].By Eq. ( 5), the CSL state appears in the phase diagram for The competition of the CSL and BEC phases is in a stark contrast to QCD, where in the chiral limit, the CSL state is triggered by arbitrarily weak magnetic fields.In (pseudo)real QCD-like theories, a nonzero critical field is required to overcome the energy gain of diquark BEC.Full phase diagram.-Toget insight into the phase diagram away from the chiral limit, it is convenient to parameterize the unit four-vector variable in terms of three spherical angles, We also introduce dimensionless variables that allow us to scale out trivial dependence of observables on f π and m π , It is easy to see that the ground state has to be independent of time and the transverse coordinates.The task to find the ground state thus reduces to that of minimizing (the spatial average of) the one-dimensional effective Hamiltonian where the primes denotes derivatives with respect to z [20].
The ground state is realized by constant α, hence we are dealing with a one-dimensional system of two variables θ and φ.
Restricting first to uniform field configurations, it readily follows that there are two candidate states: the trivial vacuum with θ = φ = 0 and Hvac ≡ H eff /(f 2 π m 2 π ) = 0, and, for x ≥ 1, the diquark BEC state with The nonuniform CSL state, found in Ref. [4], can be embedded into the present EFT for (pseudo)real QCD-like theories by setting θ = 0.It satisfies where sn is one of Jacobi's elliptic functions and k the corresponding elliptic modulus.This describes a periodic soliton with lattice spacing = 2kK(k)m −1 π , K(k) being the complete elliptic integral of the first kind.The optimum value of k is found by minimization of the average energy density carried by the soliton, and fulfills the condition E(k) being the complete elliptic integral of the second kind.The energy of the CSL state can be cast implicitly as Owing to 0 ≤ k ≤ 1, this state always has a lower energy than the trivial vacuum, but only exists for H ≥ 4/(πx).
Comparing the energies of the BEC and CSL states leads to the phase diagram in Fig. 2. While this was found with simple ansatz stationary states, we have strong, analytical and numerical, evidence based on a variational treatment of the Hamiltonian (12) that no other state of even lower energy exists [15].We can also conclude rigorously that the ground state in the "CSL" region in Fig. 2, whatever it is, has to be nonuniform.
Discussion and summary.-According to Fig. 2, fields with H > 1 are required to generate a spatially modulated ground state.While our EFT is in principle valid for arbitrarily strong fields, it is nevertheless not clear without detailed knowledge of the function f π (H) whether H > 1 can be satisfied for any physical values of H.The dashed lines are the spinodal curves of the first-order transition between the BEC and CSL phases, obtained from the analysis of the excitation spectrum, to be reported in Ref. [15].
Rewriting this condition as H/(4πf π ) 2 > 1/(2d), and recalling that the loop factor 4πf π controls the derivative expansion of the EFT [21], we expect that the critical field for the formation of CSL can be reached in theories with sufficiently large d.Indeed, a one-loop calculation within the chiral perturbation theory of QCD gives, in the chiral limit [22], The same result will hold without modification in all pseudoreal QCD-like theories, since the H-dependent correction to f π comes from a charged pion loop, and is thus insensitive to the presence of electrically neutral diquarks.By Eq. ( 17), the one-loop correction to the critical magnetic field H cr will be negligible for large d, and sufficient accuracy is achieved by treating f π as an H-independent constant.For theories with small d, such as two-color QCD where d = 2, the H-dependence of f π has to be taken into account to see whether a field for which H > 1, in fact, exists.For very strong fields, one can utilize the first-principle calculations of Ref. [13], giving the asymptotic behavior of the neutral pion decay constant in two-color QCD with Equations ( 17) and ( 18) provide complementary information about the function f π (H), consistent with the existence of the CSL phase in two-color QCD.However, to make a firm conclusion about the presence of an inhomogeneous phase in this concrete theory, we would need detailed input on the behavior of f π (H) between the weak-field and strong-field limits.Such input does not seem to be available at the moment.Existing results, based on model calculations (see e.g.Ref. [23]), are not conclusive enough for our purposes.
To summarize, we have constructed a class of counterexamples to the conjecture that in vector-like gauge theories, positivity of the determinant of the Dirac operator (i.e.absence of the sign problem) implies absence of inhomogeneous phases in the phase diagram [8].The nonuniform order is realized by a topological crystalline condensate of neutral pions, and requires a sufficiently strong background magnetic field.The conjecture might still hold under more restrictive assumptions, for instance when full rotational invariance is imposed.
Our analysis utilizes low-energy EFT and is thereby modelindependent.Hence, Fig. 2 represents the mapping of the true phase diagram of (pseudo)real QCD-like theories to the space of the dimensionless variables H and x.Our results are not limited to weak magnetic fields.Consistency of the derivative expansion requires moderate chemical potentials though: using Eq. ( 8) to estimate the gradients involved in the CSL state leads to the bound Hµ B 4πf π , or Hx 4πf π /m π .For theories with a large enough gauge group and its representation on the quark fields, an inhomogeneous phase can be demonstrably realized with moderate magnetic fields controlled by the derivative expansion of the EFT.In the simplest and most well studied QCD-like theory-two-color QCDthe question of the existence of a nonuniform phase remains open.It can, however, be answered by an independent nonperturbative evaluation of the neutral pion decay constant as a function of magnetic field.

FIG. 2 .
FIG. 2. Tentative phase diagram in theH ≡ CH/[fπ(H)] 2 andx ≡ µB/mπ(H) variables.The solid lines denote phase transitions.The dashed lines are the spinodal curves of the first-order transition between the BEC and CSL phases, obtained from the analysis of the excitation spectrum, to be reported in Ref.[15].