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We demonstrate the existence of an anomaly-induced inhomogeneous phase in a class of vectorlike gauge theories without the sign problem, thus disproving the long-standing conjecture that the absence of the 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 an external magnetic field. Our results provide a benchmark for future lattice studies of inhomogeneous phases in dense quark matter.

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, unless a specific mechanism for structure formation is in place. The question under what conditions the ground state

Various nonuniform phases are expected to play an important role for the thermodynamics of quark matter under extreme conditions

In fact, Ref.

In this Letter, we disprove this conjecture. We demonstrate that a class of QCD-like theories free of the sign problem features the nonuniform chiral soliton lattice (CSL) phase

To the best of our knowledge, this is the first time that existence of an inhomogeneous phase in gauge theories amenable to direct lattice Monte Carlo simulation has been shown. Our results can thus serve as a benchmark for future

We consider the class of QCD-like theories where quarks transform in a (pseudo)real representation of the gauge group

In (pseudo)real QCD-like theories, the color generators

Provided the electric charges of the

We shall 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 low-energy 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

For

In magnetic fields

Magnetic fields around the characteristic scale of the theory,

Different regimes of the EFT and the corresponding light d.o.f., depending on the strength of the magnetic field. For

Taking finally into account the discrete symmetries

The

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

Second, the assumption that the external magnetic field satisfies

Finally, in contrast to the chiral perturbation theory of QCD (see Ref.

In the rest of this Letter, we analyze the ground state of the EFT at nonzero baryon number chemical potential,

To bring the EFT into a form more suitable for the analysis, we map the matrix

The ground state is easy to determine by a direct minimization of the Hamiltonian in the chiral limit,

For

To get insight into the phase diagram away from the chiral limit, it is convenient to parametrize the unit four-vector variable in terms of three spherical angles

Restricting first to uniform field configurations, it readily follows that there are two candidate states: the trivial vacuum with

The nonuniform CSL state, found in Ref.

Comparing the energies of the BEC and CSL states leads to the phase diagram in Fig.

Tentative phase diagram in the

According to Fig.

Rewriting this condition as

For theories with small

To summarize, we have constructed a class of counterexamples to the conjecture that in vectorlike 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

Our analysis utilizes low-energy EFT and is thereby model independent. Hence, Fig.

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 QCD—the question of the existence of a nonuniform phase remains open. Assuming that our EFT remains valid in strong magnetic fields, that is, the ground state at zero chemical potential carries a chiral condensate

We are indebted to Naoki Yamamoto for collaboration preceding the present project and for insightful comments. The study of the CSL phase in theories without sign problem was inspired by a discussion with Hiromichi Nishimura. Last but not least, we are also grateful to Thomas Cohen, Gergely Endrődi, Philippe de Forcrand, Simon Hands, Carlos Hoyos, Aleksi Kurkela, Eugenio Megías, Andreas Schmitt, and Igor Shovkovy for fruitful discussions and for asking questions that sharpened our understanding of the subject. This work has been supported by a ToppForsk-UiS Grant No. PR-10614.

Here we

In the limit of vanishing quark mass, this state, sometimes referred to as a “meson supercurrent,” was discussed in Ref.

This argument requires a gauge where

The very last