Finite-Density Massless Two-Color QCD at the Isospin Roberge-Weiss Point and the ’T Hooft Anomaly

PHYSICAL REVIEW RESEARCH 2, 033253 (2020)

Finite-density massless two-color QCD at the isospin Roberge-Weiss point and the ’t Hooft anomaly

Takuya Furusawa ,1,2,* Yuya Tanizaki ,3,† and Etsuko Itou 4,5,6,‡ 1Tokyo Institute of Technology, Ookayama, Meguro, Tokyo 152-8551, Japan 2Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan 3Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan 4Department of Physics and Research and Education Center for Natural Sciences, Keio University, 4-1-1 Hiyoshi, Yokohama, Kanagawa 223-8521, Japan 5Department of Mathematics and Physics, Kochi University, 2-5-1 Akebono-Cho, Kochi 780-8520, Japan 6Research Center for Nuclear Physics (RCNP), Osaka University, 10-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

(Received 19 June 2020; accepted 15 July 2020; published 14 August 2020)

We study the phase diagram of two-ﬂavor massless two-color QCD (QC2D) under the presence of quark chemical potentials and imaginary isospin chemical potentials. At the special point of the imaginary isospin

chemical potential, called the isospin Roberge-Weiss (RW) point, two-flavor QC2D enjoys the Z2 center symmetry that acts on both quark flavors and the Polyakov loop. We find a Z2 ’t Hooft anomaly of this system, which involves the Z2 center symmetry, the baryon-number symmetry, and the isospin chiral symmetry. Anomaly matching, therefore, constrains the possible phase diagram at any temperatures and quark chemical potentials at the isospin RW point, and we compare it with previous results obtained by chiral effective field theory and lattice + simulations. We also point out an interesting similarity of two-flavor massless QC2Dwith(2 1)-dimensional quantum antiferromagnetic systems.

DOI: 10.1103/PhysRevResearch.2.033253

I. INTRODUCTION An exception is two-color QCD, i.e., Nc = 2, and we abbreviate it as QC D.1 Because of the pseudo-reality of the Quantum chromodynamics (QCD) is the fundamental the- 2 SU(2) gauge group, the quark determinant can be shown ory of nuclear and hadron physics, and it provides various to be real-valued even at finite baryon chemical potentials. interesting phenomena in extreme conditions [1]. At low Therefore, QC D with an even number of flavors, N ∈ 2Z, temperatures and densities, the fundamental degrees of free- 2 f does not suffer from the sign problem, and it is a good dom, quarks and gluons, are confined inside color-singlet playground to test various ideas of finite-density QCD. QC D hadrons, and at high temperatures, they are liberated and 2 shares some important aspects of strong dynamics of N = 3 form quark-gluon plasma (QGP). As the density increases, we c QCD: low-energy excitations consist of color-singlet hadrons, have nucleon superfluidity, and at ultimately high densities, it and chiral symmetry is spontaneously broken at low tempera- is expected to transform into the superfluid phase of quark tures. Because of this nature, finite-density QC D is gathering matter, called the color-flavor locked (CFL) phase [2,3]. 2 much attention both theoretically and numerically (see, e.g. Since the system is strongly coupled in most regions of Refs. [6–37]). There are other options to obtain the sign- the QCD phase diagram, we should rely on the numerical problem free setup in N 3 QCD with nonzero chemical lattice Monte Carlo simulation in order to obtain concrete c potentials, such as the imaginary chemical potential [38–47] understandings both qualitatively and quantitatively. When or the isospin chemical potential [48–52], and they also are the number of colors N satisfies N 3, this Monte Carlo c c acquiring significant interest. In this paper, we study the simulation is limited to the systems with zero-baryon den- phase structure of QC D combining these aspects from the sities, as the baryon chemical potential produces the sign 2 viewpoint of symmetry. problem [4,5]. Because of this issue, we still have no reliable We have emphasized the similarity between QC D and first-principle computation for finite-density QCD, despite the 2 N 3 with the usefulness of N = 2 for studying finite fact that we are expecting rich dynamics and its importance c c densities, but it is also important to know how two-color inside neutron stars. nuclear matter is different from that of our world. The most important difference is that there are color-singlet diquarks = *[email protected] when Nc 2. The diquark has baryon charge 1, so the †[email protected] superfluid state of finite-density nuclear matter is basically ‡[email protected] described by Bose-Einstein condensation of the diquark. When Nc = 3, the baryons are fermions, the nuclear matter Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) 1Not to be confused with two-“dimensional” QCD, often denoted and the published article’s title, journal citation, and DOI. as QCD2.

2643-1564/2020/2(3)/033253(20) 033253-1 Published by the American Physical Society FURUSAWA, TANIZAKI, AND ITOU PHYSICAL REVIEW RESEARCH 2, 033253 (2020) forms the Fermi surface of nucleons, and the superfluidity is specific boundary condition of matter fields in order to obtain caused by Bardeen-Cooper-Schrieffer (BCS) pairing of two a nontrivial constraint, and this is equivalent to introducing nucleons and its condensation at low densities. Thus, the imaginary chemical potentials. Thus, the phase diagram of = mechanisms of low-density nuclear superfluidity in the Nc QC2D also has a chance to get a severe constraint under the 2 and Nc = 3 cases are different qualitatively, but the color- presence of nonzero imaginary chemical potentials. superconducting states at ultimately high densities behave in a In this paper, we study the phase diagram of massless = similar manner between Nc 2 and Nc 3. Another notable two-flavor QC2D in detail at finite quark chemical potentials difference is the nature of chiral symmetry. For Nc 3QCD and imaginary isospin chemical potentials. This setup does with Nf massless flavors, the flavor symmetry is SU(Nf )L × not suffer from the sign problem, so our predictions can be SU(Nf )R × U(1), and the chiral condensate breaks it down confirmed by lattice Monte Carlo simulations. At the spe- × 2 − θ = to SU(Nf )V U(1) with Nf 1 massless Nambu-Goldstone cial value of the imaginary isospin chemical potential, I (NG) bosons. When Nc = 2, however, the flavor symmetry −iμI L = π/2, this theory has a Z2 symmetry acting on the is enhanced to SU(2Nf ) ⊃ SU(Nf )L × SU(Nf )R × U(1) be- quark flavor and Polyakov loop at the same time, which can cause of the pseudo-reality of SU(2) color [53,54], which is be thought of as the center symmetry. We refer to this special sometimes referred to as Pauli-Gürsey symmetry. Therefore, point as the isospin Roberge-Weiss (RW) point. We find the the chiral-symmetry-breaking pattern is SU(2Nf ) → Sp(Nf ) Z2 anomaly related to this center symmetry and also chiral 2 − − 2 2 − with 2Nf Nf 1 NG bosons, which consist of Nf 1 symmetry, and we discuss its implications to the finite-density 2 − phase diagram at the isospin RW point. mesons and Nf Nf diquarks. Importantly, with small current quark mass, the lightest diquarks have the same mass as Let us summarize our main result by comparing the phase pions. This clearly tells us why Monte Carlo simulations of diagrams of QC2D with the thermal boundary condition and QC D at the isospin RW point. The left panel of Fig. 1 QC2D do not encounter the fake early onset of baryon-number 2 densities, which is currently one of the biggest obstacles to shows the schematic phase diagram of two-flavor QC2Dwith tackling the low-temperature nuclear matter for Nc = 3[4,55– massive quarks with the thermal boundary condition, which 57]. seems to be consistent with all data given by recent lattice The purpose of this paper is to explore the rigorous nature simulations [19,32,34–37]. However, we should note that the definitions of phases are rather ambiguous because we have of the QC2D phase diagram with exactly massless quarks. Even when the numerical lattice simulation is free from the no exact order parameters for the confinement or for chiral sign problem, the chiral extrapolation requires careful, sys- symmetry breaking, so different conventions are used between tematic studies with many computational costs. Moreover, the those papers. Accordingly, the dashed curves in the left panel chiral symmetry of the continuum theory suffers from lattice of Fig. 1 can be the crossover lines. discretization, so we must have good knowledge about the In the right panel of Fig. 1, we show one of the possible nature of the chiral limit in order to understand such numerical scenarios for the phase diagram of two-flavor QC2Dwith simulations. massless quarks at the isospin RW point. In this setup, as It is, however, usually quite difficult to give a rigorous we shall reveal in this paper, we can discuss the spontaneous constraint on the phase diagram. It is widely thought that breaking of the center, chiral, and baryon-number symme- finite-temperature quantum field theories are mapped to clas- tries, and the corresponding order parameters are given by sical statistical systems in one lower dimension. Since classi- the Polyakov loop P, the chiral condensate qq¯ , and the cal phases of matters are classified by the Ginzburg-Landau diquark condensate qq, respectively. We find the mixed paradigm, such descriptions may accept any kind of phase anomaly between these three symmetries, and we conclude transitions depending on the effective coupling constants. This that, at least, one of these symmetries must be spontaneously expectation is correct in many systems, but a crucial exception broken at any temperatures and chemical potentials in order to was found for pure Yang-Mills theories [58]. Because of satisfy the anomaly matching condition. Figure 1 satisfies this the existence of the center symmetry, thermal Yang-Mills requirement, and we shall discuss it in more detail in Sec. V. theory behaves as if it is a quantum matter even at high This paper is organized as follows. In Sec. II we give a brief temperatures, and, surprisingly, their phases are constrained overview of QC2D. We discuss global symmetry, especially by an ’t Hooft anomaly: Anomaly matching requires that the emphasizing details on the discrete symmetry in this section. center symmetry or CP symmetry at θ = π has to be broken at In Sec. III we discuss the anomaly matching condition on any temperatures. Motivated by this discovery, recent studies the perturbative triangle anomaly. This perturbative anomaly elucidate that QCD also enjoys the similar constraint even is useful to understand the phases at T = 0. In Sec. IV though it does not have a good center symmetry [59–62] we discuss the more subtle anomaly, i.e., discrete anomaly, (For other related developments, see, e.g., Refs. [63–91].) related to the Z2 center symmetry at the isospin RW point. Due to the absence of center symmetry, we have to choose a In Sec. V we see how the discrete anomaly constrains the possible phase diagram of massless QC2D at the isospin RW point. In Sec. VI we point out that the discrete anomaly of 2We use the convention for the compact symplectic group, Sp(N ), QC2D at the isospin RW point has a similar structure to that as Sp(1) SU(2) in this paper. Another useful exceptional isomor- of a (2+1)-dimensional [(2 + 1)-D] quantum antiferromag- phism is Sp(2) Spin(5). For details, see Appendix B. We note net. In Sec. VII we summarize the results. We describe our that, in other literature, people sometimes use a different convention, convention on Euclidean Dirac and Weyl spinors in Appendix Sp(2) = SU(2), so there is a factor 2 difference of the arguments A, and useful facts on the compact symplectic group, Sp(N), between these conventions. in Appendix B.

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FIG. 1. (a) A schematic phase diagram of two-ﬂavor massive QC2D with the thermal boundary condition, predicted by theoretical and numerical works. Since this theory has neither the center nor chiral symmetry, the boundaries of phases expressed by dashed curves will be a crossover. (b) A possible phase diagram for two-ﬂavor massless QC2D at the isospin RW point, which we give in this work. Here we discuss the spontaneous symmetry breaking of the center, chiral, and baryon-number symmetries and classify the phases. Each boundary of phases is expected to be a critical line associated with a second-order phase transition.

II. GLOBAL SYMMETRIES OF MASSLESS QC2D because we mainly use Weyl spinors. The convention of Euclidean spinors used is summarized in Appendix A. This section is devoted to an overview of QC D related 2 The motivation to use Weyl spinors is the existence of to its global symmetry. We first give a review of the global larger chiral symmetry. It is known that QC D has chiral symmetry of massless QC D with N Dirac flavors, paying at- 2 2 f symmetry SU(2N )/Z , which is larger than the usual one, tention to incidental details of the global structure. We discuss f 2 [SU(N ) × SU(N ) × U(1) ]/(Z × Z )[53,54], and it how the symmetry is affected by quark chemical potentials, f L f R V Nf 2 is easier to observe this symmetry in the Weyl notation. This μ, and by isospin chemical potentials, μ , for an even number I enlarged symmetry appears due to the pseudo-real nature of of flavors, N ∈ 2Z. We note that QC D suffers from the sign f 2 the SU(2) gauge group. In order to see it, we decompose problem if both μ and μ are present, but we emphasize that I the four-component Dirac spinors into two-component Weyl this does not occur if we introduce the imaginary isospin μ = θ / spinorsasfollows: chemical potential, I i I L, instead. Because of the RW periodicity for θ , we will find the Z center symmetry at the ψ I 2 ψ = , isospin version of RW point, θ = π/2. In particular, for N = D ε ⊗ ε ψ˜¯ I f ( color spin ) (2) 2, we concretely discuss the property of Nambu-Goldstone ψ = ψ˜ ε ⊗ ε T , ψ¯ . bosons as perturbations from the vacuum at μ = μI = 0. D ( color spin )

Here εspin’s denote the invariant tensors of the spacetime sym- / A. SU(2Nf ) Z2 chiral symmetry metry, Spin(4) SU(2) × SU(2), and εcolor is the invariant tensor of SU(2) color. For details, see the discussion around We consider QC D with massless quarks, ψD,i, i = 2 (A7) in Appendix A. An important point is that ψ and ψ˜ 1,...,Nf . The quark kinetic term is given by the Dirac Lagrangian, are left-handed Weyl spinors, and both of them are in the defining representation, 2, of SU(2). The Dirac Lagrangian Nf now becomes ν ψ , γ (∂ν + iaν )ψD,i, (1) D i Nf i=1 ψ γ ν ∂ + ψ D,i ( ν iaν ) D,i ν 3 where a = aν dx is the SU(2) gauge field. Throughout this i=1 paper, when we use Dirac spinors, we put the subscript “D” Nf ν ¯ ν = [ψ¯iσ (∂ν + iaν )ψi + ψ˜iσ (∂ν + iaν )ψ˜i], (3) i=1 3Throughout this paper, we denote the dynamical SU(2) gauge up to integration by parts. The common chiral symmetry, fields with the lower case, aμ, which is locally a Hermitian 2 × 2 SU(Nf )L × SU(Nf )R, rotates ψ and ψ˜ , separately, but it is matrix-valued field. Upper-case letters, A, B,..., are reserved for now evident that we can mix ψ and ψ˜ as they share the same background gauge fields of global symmetries. Lorentz and gauge structures. In other words, one can rewrite

033253-3 FURUSAWA, TANIZAKI, AND ITOU PHYSICAL REVIEW RESEARCH 2, 033253 (2020) the quark kinetic term as chiral symmetry breaking. As an order parameter of chiral ν symmetry breaking, the most natural operator is the quark σ Dν , (4) bilinear operator, which is spin singlet and color singlet. In where D = d + ia is the covariant derivative, and repre- our case, it is given by sents 2Nf left-handed Weyl fermions defined as c c αβ = ≡ ε 1 2 ε α, , β, , , ij i j ( color ) ( spin ) c1 i c2 j (10) ψi = . (5) where c1,2 ∈{1, 2} are SU(2) color indices, α,β ∈{1, 2} are ψ˜ j spin indices, and i, j are SU(2Nf ) flavor indices. In order At the classical level, this Lagrangian enjoys U(2Nf ) symme- to make this operator color and spin singlet, the fermionic try, wave function must be antisymmetric under both color and spin. Because of Fermi statistics, this fermion bilinear is → U , U ∈ U(2Nf ). (6) antisymmetric under the SU(2Nf ) flavor label too: Because of the Adler-Bell–Jackiw (ABJ) anomaly [92,93], the T =− . fermion integration measure is not invariant under the contin- (11) uous Abelian part of U(2Nf ), so the theory is invariant only Now assume that chiral condensate is as symmetric as possi- 4 under SU(2Nf ) symmetry. This enhanced flavor symmetry is ble away from the origin. Then such an example of vacuum sometimes referred to as the Pauli-Gürsey symmetry. expectation values is given by We note that the center element of SU(2) color group gives the gauge identification, ∼− , so the fermion parity can ∝ = 01N . 0 − (12) be absorbed by the gauge transformation. In other words, 1N 0 any color-singlet operators are bosonic in this theory. As a This causes spontaneous chiral symmetry breaking as (for consequence, Z ={±1 }⊂SU(2N ) does not act on any 2 2Nf f details, see Appendix B) gauge-invariant local operators. Therefore, the actual symmetry should be regarded as the quotient SU(2N )/Z . SU(2N ) Sp(N ) f 2 f → f . (13) Let us identify how the common chiral symme- Z2 Z2 × × / try, [SU(Nf )L SU(Nf )R U(1)V] ZNf , is embedded into The broken symmetry forms the coset SU(2Nf )/Sp(Nf ) de- SU(2Nf ). For this purpose, we note that, in the Dirac notation, 2 scribing 2N − Nf − 1 massless NG bosons. They consist of ψD transforms as the defining representations under the chiral f 2 ψ N − ψψ˜ ∼ ψ ψ , symmetry, while D belongs to the conjugate representations. f 1 massless mesons, D,R D L, associated with Therefore, we can summarize the charges of left-handed the common pattern of chiral symmetry breaking, SU(Nf )L × → 2 − spinors, ψ and ψ˜ , under the SU(2) gauge and the ordinary SU(Nf )R SU(Nf )V, and of Nf Nf massless diquarks, chiral symmetries as follows: ψψ and ψ˜ ψ˜ , which appear as NG bosons due to the enhanced chiral symmetry. The explicit form of the chiral effective SU(2) SU(Nf )L SU(Nf )R U(1)V Lagrangian for Nf = 2 will be discussed in Sec. II D. ψ 2 N f 1 1 (7) ψ˜ 21 N f −1 B. Quark and isospin chemical potentials Therefore, (V ,V , eiα ) ∈ SU(N ) × SU(N ) × U (1) is L R f L f R V In this subsection, we consider the case where the Dirac embedded into SU(2Nf ) in the following fashion: flavor Nf is even. We first introduce the quark chemical iα μ e VL 0 potential, , and discuss how the global symmetry is affected. −iα ∗ ∈ SU(2Nf ). (8) 0 e (VR ) After that, we further introduce the isospin chemical potential, μ π / α I . We also comment on the sign problem when we introduce ω = 2 i (2Nf ) , , i Let us put e . Then (VL VR e ) and both quark and isospin chemical potentials. We show that we ω ,ω ,ω−1 iα ( VL VR e ) give the same element of SU(2Nf ). can evade the sign problem when we consider μ ∈ R and Thus, we identify the subgroup as μI ∈ i R. SU(N ) × SU(N ) × U(1) SU(2N ) In order to discuss the effect of chemical potentials to f L f R V ⊂ f . (9) × SU(2Nf ) chiral symmetry, we need to identify the generators ZNf Z2 Z2 TQ and TI of quark-number and isospin symmetries, respec- Here we take the Z2 quotient of both sides to take into account tively, in the Weyl representation , and then we add the the gauge identification. chemical potential terms, Last, let us give a review on the chiral symmetry breaking − σ 0(μT + μ T ) . (14) of this theory in the vacuum [8,9]. When Nf is not too large, Q I I the confining force is sufficiently strong so that we have We should find the subgroup of SU(2Nf ) that preserves these terms. The quark number symmetry is nothing but U(1)V of 4 Let us comment on the discrete part. Writing U(2Nf ) = × × / [SU(Nf )L SU(Nf )R U (1)V] ZNf . Using the embedding N × / [SU(2 f ) U(1)] Z2Nf , this U(1) symmetry is explicitly broken (8)intoSU(2N ), we obtain → f by the ABJ anomaly as U(1) Z2Nf . This discrete subgroup is identified as the center of SU(2N ), so the Z quotient gives 1 0 f 2Nf T = Nf . [SU(2N ) × Z ]/Z SU(2N ). Q − (15) f 2Nf 2Nf f 0 1Nf

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In order to preserve the chemical potential term, U ∈ and the same type of explicit breaking occurs for SU(Nf )L,R † SU(2Nf ) must satisfy U TQU = TQ. It is easy to see that this by the isospin chemical potential, is true if and only if μ SU(N /2) × SU(N /2) × U(1) −−→I f f . × × SU(Nf )L,R (22) SU(Nf )L SU(Nf )R U (1)V Z / U ∈ . (16) Nf 2 ZNf Thus, when both μ and μI are present, we obtain the resulting Thus, we obtain the ordinary chiral symmetry of QCD. Its global symmetry by combining Eqs. (21) and (22). physical interpretation is very clear. The enlarged chiral sym- Last, let us comment on the sign problem. The one-flavor metry SU(2Nf ) exists because we can rotate the quark and Dirac operator at finite μ, D(μ) = γν Dν + μγ0, has the quar- antiquark, ψ and ψ˜ , but the chemical potential introduces the tet pairing of the spectrum {±λ, ±λ∗} if λ ∈ C \ R and has unbalance between them. Therefore, we can no longer rotate the doublet pairing {±λ} if λ ∈ R [94]. This is a consequence between ψ and ψ˜ . of the pseudo-reality of SU(2) and the existence of chiral Next, let us consider about the isospin chemical potential symmetry. Therefore, with the quark mass m, the one-flavor for even Nf . We then classify those Nf quarks into Nf /2 Dirac determinant takes the form generations of the SU(2) doublets and call those doublets the μ + = 2 − λ2 | 2 − λ2|2. up and down sectors, det(D( ) m) (m ) m (23) λ∈R λ∈R u u˜ ψ = , ψ˜ = . (17) d d˜ This shows that the one-flavor Dirac determinant at finite density is real valued but does not have to be positive semidefinite. We have thus identified the isospin symmetry as SU(2) Therefore, we need even number of flavors in order to evade SU(2) ⊗ 1N/2 ⊂ SU(Nf )V, and we denote their generators the sign problem in a Monte Carlo simulation. (Pauli matrices) as τ1,τ2,τ3. Since ψ is in the fundamental When we consider both the quark and isospin chemical representation while ψ˜ is in the antifundamental representa- potentials, this sign problem comes back. The u quark sector tion as we have seen in (8), its Cartan subgroup is generated acquires the chemical potential μ + μI , while the d quark by sector has μ − μI , so the Dirac determinant, τ ⊗ μ + μ + μ − μ + , 3 1Nf /2 det[D( I ) m] det[D( I ) m] (24) TI = −τ ⊗ 1 / 3 Nf 2 again can oscillate between positive and negative values. ⎛ ⎞ 1 / 0 Thus, when we introduce both the quark and isospin chemical Nf 2 (18) ⎜ − / ⎟ potentials, the Dirac flavors must be multiples of four to be ⎜ 0 1Nf 2 ⎟ = ⎝ ⎠. sign-problem free. −1 / 0 Nf 2 In order to avoid this problem within two flavors, we can 01/ Nf 2 consider real quark chemical potential and imaginary isospin μ → μ ∈ The isospin chemical potential introduces the unbalance be- chemical potential, i.e., we replace I i I i R. Then, tween u and d quarks, so we lose the rotation between them. using the pseudo-reality of SU(2), we can show that However, we still have the symmetry that rotates between det[D(μ − iμ ) + m] = det[D(μ + iμ ) + m]∗. (25) u and d˜, so the remnant symmetry is isomorphic to the I I × × / ordinary chiral symmetry [SU(Nf ) SU(Nf ) U (1)] ZNf , Therefore, the Dirac determinant for the d-quark sector is although they act differently on a given basis = (ψ,ψ˜ )T . given by the complex conjugate with that of the u-quark , This isomorphism between the stabilizer groups of TQ and of sector. The product of the Dirac determinants over u d sectors TI comes from the fact that an element S ∈ SU(2Nf ) relates is now positive semidefinite, and it is free from the sign TI and TQ as follows: problem for any even Nf . S†T S = T . (19) I Q C. Roberge-Weiss periodicity for imaginary isospin chemical The explicit form of S exchanges d and d˜, while it leaves potentials u andu ˜. At the massless point, the quark chemical potential The absence of the sign problem for real quark chemical and isospin chemical potential can be interchanged by a chiral potentials, μ, and imaginary isospin chemical potentials, μI = rotation in SU(2Nf ). Therefore, these two chemical potentials μ i I , motivates us to study this setup in more detail. In partic- are equivalent physically at the massless point. ular, we find an isospin version of the Roberge-Weiss (RW) When we introduce both chemical potentials, the symme- periodicity [95] and an interesting symmetry enhancement for try must preserve a specific value of the imaginary isospin chemical potential. Let L be the length of S1 along the x0 direction;5 then it is μT + μ T . (20) Q I I convenient to introduce the dimensionless imaginary isospin Because of the quark chemical potential, chiral symmetry is explicitly broken as × × μ SU(Nf )L SU(Nf )R U(1)V 5Throughout this paper, we call T = 1/L the temperature whether SU(2Nf ) −→ , (21) ZNf the boundary condition is the thermal one or not.

033253-5 FURUSAWA, TANIZAKI, AND ITOU PHYSICAL REVIEW RESEARCH 2, 033253 (2020) chemical potential, θI ,by This effectively flips the sign of the isospin chemical potential, θI →−θI . Thus, at generic values of θI , this is not a sym- θI μI = i . (26) metry. However, at θI = π/2, this is invariant because of the L isospin RW periodicity, In order to see the RW periodicity, let us redefine the Weyl θ = π/ →−π/ ∼ π/ . fermion as I 2 2 2 (35) 0 0 x x We note that the isospin RW periodicity uses the center = exp −iθ T , = exp iθ T . (27) 0 I I L I I L transformation along the x direction. Therefore, this transformation yields a sign factor to the Polyakov loop, With this redefinition, the imaginary isospin chemical potential is removed from the kinetic term as = P 0 . tr(P) tr exp i a0 dx (36) 1 θ S 0 I 0 σ ∂0 − μTQ − i TI = σ (∂0 − μTQ ) , (28) L Therefore, at the isospin RW point, we have found the Z2 symmetry, which acts on both quark flavors and the Polyakov and its information is encoded into the phase acquired by the 1 6 loop at the same time, as follows: fields along S , ⎛ ⎞ ⎛ ⎞ 0 iθ T 0 u d (x + L) =−e I I (x ). (29) ⎜ ⎟ ⎜ ⎟ ⎜d⎟ ⎜u⎟ θ = ⎝ ⎠ → ⎝ ⎠, tr(P) →−tr(P). (37) Naively, this expression tells us that I is a periodic variable u˜ d˜ θ ∼ θ + π − with I I 2 . However, the center element 1ofSU(2) d˜ u˜ color gives the extra identification, This is the same as the ZN center symmetry identified in ZN - θI ∼ θI + π, (30) 7 twisted QCD for Nc = Nf = N [96–105] , so here we call it × and we call this shortened periodicity the isospin RW period- (Z2 )center. Since this (Z2 )center flips the charges of U(1)L,3 icity. U(1)R,3, the global symmetry at the isospin RW point takes Let us now discuss the global symmetry. For simplicity the form of a semidirect product: = of notation, we consider Nf 2 in the following, and the (Z2 )center [U(1)L,3 × U(1)R,3] × U(1)V Gμ,θ =π/ = , generalization for larger even Nf is straightforward. Under the I 2 × μ θ Z2 Z2 presence of and generic values of I , the global symmetry (38)

Gμ,θI is = i.e., Gμ,θI =π/2 (Z2 )center Gμ,θI =0,π/2. Because the × × (Z2 )center symmetry at θI = π/2 acts on the Polyakov loop, it = U(1)L,3 U(1)R,3 U(1)V Gμ,θI deserves special attention, and we shall discuss properties at Z × Z 2 2 the isospin RW point in more detail in later sections. × × ⊂ SU(2)L SU(2)R U(1)V ⊂ SU(4), × (31) Z2 Z2 Z2 D. Chiral effective Lagrangian in two-ﬂavor case where U(1)L/R,3 denote the Cartan subgroups of SU(2)L/R, For N = 2, we try to see how the global symmetry is re- θ f respectively. However, there are two special points of I mod alized in the low-energy chiral Lagrangian [8,9]. We note that π θ = π . The obvious one is I 0mod , in which case the global there is an exceptional isomorphism, SU(4) Spin(6) and symmetry is (21) Sp(2) Spin(5) (see Appendix B), so the chiral symmetry SU(2)L × SU(2)R × U(1)V breaking (13) is realized as Gμ,θ = = . (32) I 0 × Z2 Z2 SO(6) → SO(5). (39) θ = θ = π We note that, for example, not only I 0 but also I has The order parameter ﬁeld, = , can be mapped to the this global symmetry because they are gauge equivalent. six-dimensional unit vectors n ∈ S5 ⊂ R6, i.e., n · n = 1. The Another special point is physical interpretation of this unit vector is as follows: π ⎛ ⎞ ⎛ ⎞ ⎛ √ ⎞ θI = , (33) σ + ˜ / 2 n1 (˜uu dd) √2 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜n ⎟ ⎜π ⎟ ⎜(˜uu − dd˜ )/ 2⎟ which we would refer to as the isospin RW point. In this ⎜ 2⎟ ⎜ 0 ⎟ ⎜ ⎟ π τ ∈ ⎜n ⎟ ⎜π ⎟ ⎜ + ˜ / ⎟ case, let us consider the rotation along 1 SU(2)V, which = ⎜ 3⎟ = ⎜ 1 ⎟ = ⎜ (˜ud du) 2 ⎟. n ⎜ ⎟ ⎜π ⎟ ⎜ ⎟ (40) exchanges u and d quarks, ⎜n4⎟ ⎜ 2 ⎟ ⎜ (˜ud − du˜ )/2i ⎟ ⎝ ⎠ ⎝ ⎠ ⎜ ⎟ u ↔ d, u˜ ↔ d˜. (34) n5 1 ⎝ (ud + d˜u˜)/2 ⎠ n6 2 (ud − d˜u˜)/2i

6Following the standard convention of thermal quantum ﬁeld theories, here we put the (−1) sign for the fermion boundary condi- 7We note that the same/similar boundary conditions play an im- tion. We note, however, that the periodic and antiperiodic boundary portant role to study the ground-state structures of strongly coupled conditions give the same result for the SU(2) fundamental fermions, theories via adiabatic continuity to weakly coupled regions [60,106– because the difference can be absorbed by gauge transformations. 139].

033253-6 FINITE-DENSITY MASSLESS TWO-COLOR QCD AT THE … PHYSICAL REVIEW RESEARCH 2, 033253 (2020)

That is, σ is the sigma meson, which is an isospin singlet, small quark mass, we obtain the effective potential,8 π = (π ,π ,π ) are pions, which form an isospin triplet, and 0 1 2 2 2 Veff 1 2μ 1 2θI = (1,2 ) are diquarks, which are isospin singlets with =−σ − 2 + π 2 + π 2 . 2 2 1 2 baryon charge 1. The leading term of the chiral Lagrangian is fπ mπ 2 mπ 2 Lmπ given as (49) Setting π = 0, we can analyze the minima of this potential 2 2 σ = − 2 fπ fπ 2 2 2 by substituting 1 . We can ﬁnd the second-order ∂ν n · ∂ν n = (|∂ν σ| +|∂ν π| +|∂ν | ), (41) 2 2 phase transition at μ = μc = mπ /2[8,9], where σ =1for μ<μc. So diquark condensation starts to appear at half of the where fπ denotes the pion decay constant. pion mass. In the chiral limit, this critical value goes to zero, When we add a Dirac mass, m > 0, it gives the explicit and the diquark condensate and chiral condensate describe the breaking term, same symmetry-breaking pattern at μ = 0.

2 − fπ mσ, (42) III. PERTURBATIVE ’T HOOFT ANOMALY OF MASSLESS QC D where is a strong scale ( fπ and are of the same order). In 2 this case, the vacuum is chosen to be In this and next sections, we study the nature of ’t Hooft anomalies in massless QC2D. An ’t Hooft anomaly can be σ=1, (43) characterized as an obstruction of gauging global symmetries, and, importantly, this anomaly is invariant under the renor- and other fields√ are zero. Pions and diquarks acquire the same malization group (RG) flow [140–142]. Because of this RG = mass mπ m , which obeys the Gell-Mann-Oakes-Renner invariance, ’t Hooft anomaly provides a useful constraint on relation. low-energy dynamics of strongly coupled systems, and this Next, we introduce the quark chemical potential. The quark is called the anomaly-matching condition. In this section, , → iα , , ˜ → number rotation acts as (u d) e (u d) and (˜u d) we compute the ’t Hooft anomaly for infinitesimal chiral −iα , ˜ e (˜u d), so n transforms as transformations, i.e., a perturbative anomaly. This anomaly is already useful to constrain the properties of QC D dynamics σ → σ, π → π, → ατ . 2 exp (2i 2 ) (44) at the zero temperature (T = 0orL =∞). In the next section, we shall discuss a more subtle ’t Hooft anomaly. Under the presence of quark chemical potential, we replace 2 ∂0 by (∂0 + 2μτ2 ).Sotheterm|∂0| now becomes A. Perturbative anomaly of SU(2Nf )forμ = 0 T ([∂0 + 2μτ2]) ([∂0 + 2μτ2]) Let us compute the perturbative anomaly of SU(2Nf ) chiral symmetry. Since the discrete factor does not affect the =|∂ |2 + μ ∂ · τ − · τ ∂ − μ 2 2. 0 2 ( 0 2 2 0 ) (2 ) computation of the perturbative anomaly, we will not address (45) it in this section. Its subtle effect is taken into account in the next section and essential in the computation of a discrete Therefore, at nonzero quark chemical potential, the vacuum 1 anomaly. manifold is S given by To see the existence of an anomaly, we introduce the SU(2N ) background gauge field A and replace the quark 2 = . f 1 (46) kinetic term as ν ν The pions and sigma meson acquire the mass 2|μ| [8,9]. σ Dν ⇒ σ (Dν + iAν ) . (50) The diquark field is singlet under isospin chiral symmetry, In this way we can compute the partition function Z[A] with SU(2)L × SU(2)R, and the pattern of symmetry breaking is the SU(2Nf ) background gauge field A, but this partition function does not have to be gauge invariant for A.The SU(2)L × SU(2)R × U(1)V SU(2)L × SU(2)R → . (47) anomaly can be explicitly computed by the Fujikawa method Z × Z Z 2 2 2 [143–145], but it can also be obtained in an ad hoc way The isospin chemical potential does the same job. Instead of with the Stora-Zumino descent procedure [146,147] since the replacing ∂0, we have to replace π π ∂ 1 ⇒ ∂ + μ τ 1 . 8For simplicity of the expression, here we pick up only Matsubara 0 π [ 0 2 I 2] π (48) 2 2 zero modes of mesons and diquarks. We note that, in this truncation, the isospin RW periodicity (30) is violated. This can be fixed by In particular, we are interested in the case with the imaginary reinstating all the Matsubara frequencies for π1,2. As a consequence, μ = θ / ± 2π isospin chemical potential I i I L. This increases the charged pions π = (π1 ± iπ2 ) of Matsubara frequency ωn = n π ,π L energy for the ( 1 2 ) direction unlike the case of the real get the mass, |ωn ± 2θI /L|,andθI → θI + π gives the level crossing chemical potential. of Matsubara modes, respecting the isospin RW periodicity. As they When we introduce the quark chemical potential and imag- acquire positive energies, the consequence is unaffected by these inary isospin chemical potential under the presence of the details.

033253-7 FURUSAWA, TANIZAKI, AND ITOU PHYSICAL REVIEW RESEARCH 2, 033253 (2020) anomaly must satisfy the Wess-Zumino (WZ) consistency according to (8). Substituting this expression into (51), we condition [148]. Here we choose to use the anomaly descent can obtain the six-dimensional form for the Stora-Zumino procedure. Let us start from the six-dimensional Abelian procedure, anomaly: 2 2 3 = 3 − 3 tr FA tr FL FR 2π 24π 2 24π 2 2 × tr F 3 , (51) 3!(2π )3 A 2 + F ∧ tr F 2 − F 2 . (55) 8π 2 V L R where FA = dA + iA ∧ A, and the factor 2 in front comes from the number of color. This gives the five-dimensional parity Let us now discuss how we can match the anomaly from anomaly, characterized by the Chern-Simons action, spontaneous symmetry breaking. , A typical example of anomaly-free subgroups is the vector- 2CS5[A] (52) like subgroup, which leads to the standard chiral symme- = 1 3 try breaking of QCD with Nc 3, SU(Nf )L × SU(Nf )R → where d(CS5[A]) 24π 2 tr(FA ). This topological action com- pletely specifies the perturbative anomaly of Z[A]; the system SU(Nf )V. Indeed, once we assume this spontaneous breaking Z[A] as the boundary of five-dimensional Chern-Simons the- pattern, we can match both terms of the anomaly (55) at once. ory, However, QC2D has the massless diquarks, which start to condense immediately after introducing nonzero μ.This condensation breaks U(1) spontaneously, so the second term Z[A]exp 2i CS [A] , (53) V 5 of (55) is already matched by the associated NG boson. Therefore, SU(Nf )L × SU(Nf )R has to be broken so as to is gauge invariant, because the boundary term for gauge match only the first term of (55). This requirement opens a variations of 2 CS [A] cancels the anomaly of Z[A]. 5 new possibility to saturate the anomaly. Indeed, when Nf is When Nf is not too large, it is natural to expect that even, one of the possible patterns is anomaly matching is satisfied by chiral symmetry breaking, × × SU(2Nf ) → H. If we further assume that anomaly matching SU(Nf )L SU(Nf )R U(1)V is satisfied only by NG bosons, H must be anomaly free. There → Sp(Nf /2)L × Sp(Nf /2)R. (56) are two important anomaly-free subgroups of SU(2Nf ), which = 9 = are H SO(2Nf ) and H Sp(Nf ). Indeed, for QC2D, This breaking pattern is indeed found in the analysis of we are expecting the spontaneous breaking pattern to be chiral effective model [8,9]. The number of NG bosons is → 2 − − SU(2Nf ) Sp(Nf ), as we have reviewed in Sec. II A.We Nf Nf 1. can match the anomaly with this symmetry-breaking pattern Especially when Nf = 2, since SU(2) = Sp(1), it shows no using the WZ term, and its explicit form can be found, e.g., in chiral symmetry breaking at finite densities, and we have only Ref. [149]. U(1)V/Z2 → 1, as we have discussed in Sec. II D. We note 3 = 3 = that this is consistent with (55), because tr(FL ) tr(FR ) = × × 0forNf 2. Within the chiral effective description, the B. Perturbative anomaly of SU(Nf )L SU(Nf )R U(1)V π for μ>0 vacuum manifold (46) can be parametrized by the 2 periodic scalar field, ϕ, where We move on to the discussion of the perturbative anomaly iϕ for μ>0. The perturbative anomaly matching at T = 0 and 1 + i2 = e . (57) finite μ has been considered in Ref. [150]forN = 3, and we c Under the presence of background gauge fields, we can write discuss it here in the context of QC D. 2 the axion-like coupling, As we have seen in (21), the SU(2Nf ) chiral symmetry × × i is explicitly broken to SU(Nf )L SU(Nf )R U(1)V by the ϕ ∧ 2 − 2 . tr FL FR (58) presence of μ. Let us denote (FL, FR, FV ) as the gauge-field 8π 2 N × N × strengths of SU( f )L SU( f )R U(1)V; then they are em- It is exactly the term that matches the second term of the N F bedded into the SU(2 f ) field strength, A, as follows: anomaly (55), since ϕ has the quark charge 2, i.e., the baryon charge 1. FL + FV 0 FA = , (54) 0 −(FR + FV ) IV. DISCRETE ANOMALY OF MASSLESS TWO-FLAVOR

QC2D AT ISOSPIN RW POINT

9 In this section, we discuss a more subtle anomaly related to Spontaneous breaking, SU(2Nf ) → SO(2Nf ), occurs if the gauge group is strictly real instead of pseudo-real, because the chiral con- the discrete factor of the global symmetry. The discussion here densate, , is then in the two-index symmetric representation [54]. is a crucial step to ﬁnd a rigorous constraint on the phase di- Because of the exceptional isomorphism, Lie(SO(6)) = Lie(SU(4)), agram of QC2D at the isospin RW point. Perturbative anoma- however, we cannot immediately say SO(6) ⊂ SU(6) is anomaly lies, discussed in the previous section, are very powerful for = free, so one may wonder if Nf = 3 can be special. But this does restrictions on the massless spectrum at T 0, but they do not happen, fortunately. For Nf = 3, is in 6 representation of not provide useful constraints on the phase diagram. This can SU(6), which is in the two-index antisymmetric representation of be seen from the well-known fact that the high-temperature SU(4) [= Spin(6)] and does not have the triangle anomaly. phase of QCD with fundamental quarks is a trivial phase: The

033253-8 FINITE-DENSITY MASSLESS TWO-COLOR QCD AT THE … PHYSICAL REVIEW RESEARCH 2, 033253 (2020) vacuum is unique, quark excitations are gapped by Matsubara this basis, the Z2 one-form gauge invariance is implemented frequencies (πT ), and gluon excitations are also gapped as because of the electric (∼gT ) and magnetic (∼g2T ) masses. U → U − , When we introduce a specific flavor-twisted boundary con- AV AV (62) dition, however, it is no longer the case. The high-temperature D → D − . phase is also nontrivial as it is doubly degenerate by the RW AV AV (63) phase transition [95], for example. Similar vacuum degener- Therefore, one-form gauge invariant combinations of their acy is now found in other systems and understood as con- field strengths are given by sequences of anomaly matching of subtle discrete anomalies = U + , = D + , [58–62,81] rather than perturbative anomalies. In this section, FU dAV B FD dAV B (64) we, therefore, compute the discrete anomaly for massless and we can identify the gauge field AB for the baryon-number two-flavor QC2D at the isospin RW point. (not quark-number) symmetry as [76] In order to see the discrete anomaly in our system, we have = + = U + D + . to introduce the background gauge field for Gμ,θI =π/2 in (38) dAB FU FD dAV dAV 2B (65) very carefully, as we can easily miss such an anomaly just This baryon-number gauge field, A , satisfies the canonical by getting an extra factor 2. For this purpose, it is convenient B geometric normalization of the U(1) gauge field. For later to rewrite the global symmetry to eliminate the redundancy use, we mention that the (Z ) symmetry acts on the as much as possible. Indeed, the symmetry group at generic 2 center μ θ background gauge fields as values of and I , Gμ,θI in (31), can be written as U D A , →−A , , A ←→ A , B → B. (66) U × D L 3 L 3 V V U(1)V U(1)V Gμ,θ = × U(1) , . (59) I L 3 Especially, we note that AB is unchanged under (Z2 )center. Z2 Let all the background gauge field be three-dimensional U,D ones so that we discuss an anomaly present even in the high- Here U(1)V are the vector-like U(1) symmetries acting on u and d quarks, respectively. They act on the quark fields as temperature limit. To understand how the background gauge ⎛ ⎞ fields violate the (Z2 )center symmetry, recall the derivation ⎛ ⎞ U ⎛ ⎞ iαL,3+iα u e V u on the (Z2 )center symmetry in Sec II C.Thekeystepin D ⎜ −iα , +iα ⎟ ⎜d⎟ ⎜ e L 3 V ⎟⎜d⎟ the discussion is translating the isospin imaginary chemical ⎜ ⎟ → ⎜ ⎟⎜ ⎟, ⎝ ⎠ − αU ⎝ ⎠ potential into the twisted boundary condition [see Eq. (85)]. u˜ ⎝ i V ⎠ u˜ e When the background gauge fields are turned on, however, the ˜ −iαD ˜ d e V d fermion path integral measure generates the following phase (60) factor under this operation: α αU αD i L,3 i V i V U where e , e , and e belong to U(1)L,3,U(1)V, and D i 1 U(1) , respectively. These three U(1) transformations do not S = A , ∧ dA , V π L 3 B (67) have any overlaps, which is why we have only one Z2 quotient 2 3 2 10 in (59) rather than two. which one can readily show via the standard Fujikawa method As discussed around (38), the model acquires the (Z2 )center for chiral gauge theories [152]. Thus, we must take into ac- symmetry at the isospin RW point. This symmetry flips the count the extra phase factor S when we consider the (Z2 )center sign of the U(1)L,3 charge while it exchanges the charges of symmetry in the presence of the background gauge fields. U D U(1)V and U(1)V. We try to find the Z2 anomaly by two steps: The generated phase factor S is actually responsible for We first introduce the background gauge field for Gμ,θ in a I the ’t Hooft anomaly of QC2D at the isospin RW point. Let gauge-invariant way, and then we will observe the violation us perform the (Z2 )center transformation (66). This keeps the of (Z2 )center [81]. Our discussion is analogous to the parity original action unchanged, but due to the generated phase anomaly of three-dimensional Dirac fermions. factor, it changes the partition function as follows: μ,θ Introducing the background gauge field for G I , the gauge A , , → , i [AL,3 AB]. structure becomes Z[AL,3 AB] Z[AL,3 AB]e (68)

× U × D Here A[AL,3, AB] takes the form SU(2)gauge U(1)V U(1)V × U(1)L,3. (61) Z2 1 A[A , , A ] =− A , ∧ dA . (69) L 3 B 2π L 3 B We denote the background gauge fields for the U(1)L,3, U D U D U(1)V, and U(1)V symmetries as AL,3, AV, and AV. Because We have no local counterterm to cancel this anomaly without of the Z2 quotient, we also need to introduce a Z2 two-form breaking the Gμ,θI gauge invariance. To recover the (Z2 )center gauge field B and must postulate the invariance under one- symmetry keeping the Gμ,θI gauge invariance, we have to → + form gauge transformation, B B d [151]. Note that in attach QC2D onto the four-dimensional bulk mixed term, with angle π: dAL,3 dAB S = iπ ∧ . π π (70) 10 4 2 2 Recall that the previous expression (31) has two Z2 quotients because of an extra overlap between quark-number and isospin Therefore, we find the anomaly for Gμ,θI =π/2, and its classi- symmetries. fication is Z2 because it is saturated by the four-dimensional

033253-9 FURUSAWA, TANIZAKI, AND ITOU PHYSICAL REVIEW RESEARCH 2, 033253 (2020) mixed term. We would like to emphasize that the anomaly T = 0, perturbative anomalies in Sec. III require the existence is present on the entire (μ, T ) phase diagram at θI = π/2. of massless excitations. At T = 0, those perturbative con- Let us comment on another understanding of the anomaly. straints no longer exist. At the isospin RW point, θI = π/2, When we compactify the imaginary time direction, we find we have the discrete anomaly discussed in Sec. IV, and this an infinite tower of three-dimensional Dirac fermions, whose discrete anomaly gives the constraint on possible phase dia- real masses are given by background Polyakov-loop phases grams even at nonzero T [59–61]. In this section, we discuss and Matsubara frequencies. It is known that, in this setup, the possible phase structure consistent with these anomalies, we can make the system gauge invariant, e.g., by taking the taking into account the analytic results of the chiral effective Pauli-Villars (PV) regularization. However, such PV regula- theory and the numerical results of lattice simulations. tors induce Chern-Simons terms [153,154], and this is crucial The ’t Hooft anomaly-matching argument [140–142] states for the parity anomaly. In our case, we introduce the isospin that the ’t Hooft anomaly is preserved under the renormal- chemical potential, θI = π/2, which is nothing but the zeroth ization group flow so that its low-wavelength effective field component of the background gauge fields, theory must reproduce the same ’t Hooft anomaly. As a π 1 π 1 corollary of the anomaly matching, QC2D must show spon- U = , D =− . AV,ν=0 AV,ν=0 (71) taneous symmetry breaking, topological order,orconformal 2 L 2 L behavior, and a scenario with a unique gapped vacuum is Because of the shift of real masses between u- and d-quark ruled out. Since the discussion depends only on the symmetry sectors, the Chern-Simons term is induced by the loop effect, consideration, it is applicable to massless QC2D with arbitrary or by the fermion measure, and it ends up with temperatures T = 1/L and chemical potentials μ, when we θ = π/ i 1 introduce the isospin imaginary chemical potential, I 2. S = A , ∧ dA . (72) 2 2π L 3 B In order to match the anomaly, we assume that spontaneous 3 symmetry breaking happens to the anomaly-free subgroups. Its term coincides with Eq. (67). That is, at any temperatures and chemical potentials, massless Moreover, we can make the connection of this loop- QC2D at an isospin RW point at least spontaneously breaks induced Chern-Simons term and the perturbative anomaly one of the following: discussed in Sec. III (see also Refs. [81,155]). Our contents (1) Center symmetry [(Z2 )center → 1] of gauge fields can be embedded into the SU(4) chiral gauge (2) Chiral symmetry [U(1)L,3 → 1] field in Sec. III as (3) Baryon-number symmetry [U(1) /Z → 1]. V 2 The anomaly itself allows other possibilities, but the anal- FA = diag FU + dAL,3, FD − dAL,3, −FU, −FD , (73) ysis from chiral effective Lagrangian and lattice simulations where FU,D are given by (64) and the one-form gauge invari- [19,32,34–37] supports this assumption. The expected phase ance is already taken into account. The six-dimensional form diagram is given in the right panel of Fig. 1. We now look of the Stora-Zumino descent procedure now becomes into its details and see how the anomaly matching condition is π satisfied in each phase. 2 3 1 2 tr F = dA , ∧ dA ∧ (F − F ) 3!(2π )3 A (2π )2 L 3 B U D

1 2 A. Chiral symmetry breaking at μ = 0 + (dA , ) ∧ dA . (74) π 2 L 3 B (2 ) Let us first discuss the case μ = 0. When the temperature The significant term for us is the first one on the right-hand T is not so large, we expect that chiral symmetry breaking side. The effect of the imaginary isospin chemical potential is realized. We note, however, that chiral symmetry breaking can be expressed as the imaginary-time integration along S1 does not have to be caused by the common chiral condensate, x0 of the background gauge fields: ψψ˜ , at the massless point: The diquark condensate, ψψ, plays π π the same role because of the Pauli-Gürsey symmetry, SO(6). − = U − D = − − = π, = (FU FD) AV AV (75) At T 0, the isospin chemical potential can be eliminated, D2 S1 2 2 so the system is the same with the usual vacuum. In this case, where D2 is the two-dimensional disk whose boundary is the as we have reviewed in Sec. II, chiral symmetry breaking imaginary-time circle S1. Therefore, this replacement on third occurs as component in the first term of (74)gives SO(6) → SO(5). (77) 1 1 × dA , ∧ dA , (76) 2 2π L 3 B The target space of the nonlinear sigma model is n ∈ S5, which is equivalent to the bulk term (70). This is the which is defined as (40), and the unit vector n consists of characterization of the induced Chern-Simons term (67)via mesons (σ,π0,π1,π2 ) and diquarks (1,2 ). the descent procedure. Let us now consider nonzero temperatures, T = 0, and then the imaginary isospin chemical potential affects the sym- V. PHASE DIAGRAM OF MASSLESS TWO-FLAVOR QC2D metry. Because of the isospin chemical potentials, the SO(6) AT THE ISOSPIN RW POINT chiral symmetry is explicitly broken as

We can now discuss the constraint on the phase dia- θ −→I . gram of QC2D by using the anomaly-matching condition. At SO(6) SO(4) (78)

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Indeed, within chiral effective Lagrangian (49), the imaginary (Z2 )center transformation, and then π π isospin chemical potential introduces the mass to 1 and 2: i Stop → (dϕ − AB) ∧ (−dAL,3) f 2m2 2θ 2 4π = π π I π 2 + π 2 . Veff 1 2 (79) 2 Lmπ i = S + A ∧ dA , , (mod 2πi). (85) top 2π B L 3 Therefore, SO(4) chiral rotations act on (σ,π0,1,2 ) ∈ 3 4 By integration by parts, this extra phase is nothing but the S ⊂ R , as we can set π1 = π2 = 0 to minimize the potential. Spontaneous symmetry breaking occurs as anomaly (69). Besides, the mixed anomaly predicts that, when = π we take x1,x2 dAB 2 , this topological coupling induces the SO(4) → SO(3). (80) one-dimensional anomalous term: μ = i We should note that, at 0, this symmetry breaking can AL,3. (86) 2 3 be regarded both as chiral symmetry breaking, U(1)L,3 → x 1, and as baryon-number symmetry breaking, U(1)V/Z2 → In other words, a half vortex of the diquark supports a one- × / ⊂ 1. Indeed, U(1)L,3 U(1)V Z2 SO(4), and these two dimensional theory with the (Z2 )center U(1)L,3 anomaly. symmetry-breaking patterns are identical up to an SO(4) This anomaly is equivalent to that in the quantum mechanics chiral transformation. on a circle with the antiperiodic boundary condition. It indi- Let us now ask how the anomaly is matched in this phase. cates a fermion zero mode bound on the superﬂuid vortex. We note that this phase hosts a topological soliton because

3 C. Quark gluon plasma π3(SO(4)/SO(3)) = π3(S ) = Z. (81) At sufﬁciently high temperatures, the perturbative potential The topological current takes the same form with that of the for the Polyakov loop prefers the center-broken phase by hav- Skyrmion current [156], and the effect of the discrete gauge ing nonzero expectation value, P = 0[157]. At the isospin ﬁelds has been discussed in Ref. [76]. An only difference from RW point, the high-temperature effective potential is given as usual Skyrmions is that the corresponding U(1) symmetry [61,105](seealsoRef.[158]) is the isospin symmetry U(1) , /Z ⊂ SU(2) /Z , not the V 3 2 V 2 2 1 1 (−1)n baryon-number symmetry. As it couples to the imaginary V (P) =− [|tr(Pn )|2 − 1] + θ eff π 2L4 n4 4π 2L4 n4 isospin chemical potential, I , we can reproduce the correct n1 n1 discrete anomaly (69) (see Ref. [81] for details). × [e2nμLtr(P2n ) + e−2nμLtr(P−2n )]. (87)

B. Baryon superﬂuidity at μ>0 The ﬁrst term on the right-hand side is the gluon contribution, and the second term comes from the quark contribution with Assuming the chiral symmetry breaking SO(6) → SO(5) the imaginary isospin chemical potential, θ = π/2. It is im- in vacuum, there are massless diquarks, , , in massless I 1 2 portant to note that this potential respects the center symmetry, QC D. Therefore, soon after introducing the chemical poten- 2 P →−P, as we have reviewed in Sec. II C. In order to get an tial, μ = 0, they start to condense as the effective potential insight of this potential, let us pick up the n = 1termsinEq. (49) takes the form (87). We then see that it is minimized by P =±1 . Therefore, 2 f 2m2 2μ 2 we get the symmetry breaking, =− π π 2 + 2 . Veff 1 2 (82) 2 mπ (Z2 )center → 1. (88)

The ground state at μ = 0 breaks the baryon number symme- This corresponds to quark-gluon plasma (QGP) at θI = π/2. try spontaneously, Locally, the three-dimensional effective theory in the QGP phase is the three-dimensional pure Yang-Mills theory, which / → . U(1)V Z2 1 (83) is believed to be gapped. Thus, we have no massless degrees ϕ of freedom in the QGP phase, and the anomaly must be We parametrize our vacuum manifold as + i = ei .We 1 2 reproduced in a different way from the above two phases. The note that we encounter the second-order phase transition in critical point is that the degenerated vacua belong to different the limit μ → 0 as other NG bosons reduce their mass, which symmetry-protected topological (SPT) phases protected by behaves as 2|μ|. the U(1) , × U(1) symmetry. Under the presence of back- In order to match the anomaly (69), a vortex of the diquark L 3 B ground gauge fields, the effective actions for the two vacua must carry a nontrivial quantum number of U(1)L,3. Let us see SQGP1 and SQGP2 differ by the factor this explicitly. When we turn on the background gauge fields, AB and AL,3, we introduce the topological coupling, i S − S =− A , ∧ dA . QGP1 QGP2 π L 3 B (89) i 2 3 Stop = (dϕ − AB) ∧ dAL,3. (84) 4π Since the (Z2 )center transformation exchanges two vacua, we can reproduce the anomaly as follows: We note that this is gauge invariant for AB and AL,3, because ϕ − i (d AB) is the minimal coupling and dAL,3 is the field S → S = S + A , ∧ dA . QGP1 QGP2 QGP1 π L 3 B (90) strength. In order to see the Z2 anomaly, we perform the 2 3

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A physical consequence is that the high-temperature do- unconventional quantum critical point between the Néel and main wall supports Jackiw-Rebbi gapless excitations. At μ = valence bond solid (VBS) phases in (2 + 1)-D antiferromag- 0, this high-temperature domain wall is studied in Ref. [83], nets, which is a representative example of phase transitions and it supports massless two-flavor Schwinger model, which beyond the Landau-Ginzburg paradigm [159–161].12 It is is equivalent to the SU(2) level-1 WZW conformal field also attracting attention as a condensed matter application of theory at low energies (see also Refs. [77,78,80]). This means the (2 + 1)-D dualities [161,164,165](seeRef.[166]fora that the domain walls are charged under the chiral symmetry, review). which partially explains why we can naturally expect the To clarify the relation between two-flavor QC2Datthe direct transitions between the deconfined phase and the chiral- isospin RW point and easy-plane CP1 model, let us more symmetry-breaking phase. Besides, the massless Schwinger closely look at the global symmetry of (91), which is given model predicts that the correlation function shows an alge- by braic decay along the wall, while it decays exponentially in × the other directions. This prediction can be tested in a lattice (Z2 )C [O(2)spin U(1)M] simulation of our model with a twisted boundary condition by ⊂ (Z2 )C [SO(3)spin × U(1)M]. (92) (Z2 )shift. We note that, at the large chemical potential with in- Here (Z2 )C is the charge-conjugation symmetry, O(2)spin termediate temperatures, the lattice simulation predicts the is the remnant spin symmetry, and U(1)M is the magnetic existence of a deconfined BCS phase [36]. This phase breaks symmetry. We note that O(2) Z2 U(1), so the O(2)spin symmetry is composed of the continuous U(1) transformation, the center symmetry, (Z2 )center → 1, and the baryon-number symmetry, U(1) /Z → 1, simultaneously. Although each iα V 2 U(1)spin: φ1 → e φ1,φ2 → φ2, b → b, (93) symmetry breaking is sufficient to match the anomaly, the anomaly matching does not prohibit further symmetry break- and the discrete Z2 transformation, ing. Therefore, the deconfined BCS phase is also consistent (Z ) : φ ←→ φ , b → b. (94) with the anomaly-matching condition. 2 spin 1 2 A more interesting symmetry of (91) is the magnetic symme- = 1 VI. SIMILARITY TO (2 + 1)-D QUANTUM try, U(1)M, whose conserved current is given by JM 2π db. ANTIFERROMAGNETS This symmetry does not acts on the fields in the Lagrangian, but acts on a monopole operator Mb for the dynamical gauge In this section, we consider a possible and exciting con- field b [167]as13 nection between two-flavor QC2D and quantum spin systems. iβ As we have discussed in the previous sections, two-flavor Mb → e Mb. (95) massless QC D must break some of its symmetries for any 2 This symmetry structure motivates us to make a correspon- temperatures and quark chemical potentials at the isospin dence between two-flavor QC D at the isospin RW point and RW point. This situation is nothing but the persistent order 2 the easy-plane CP1 model as discussed in condensed-matter contexts, and we will try to make their connection as concrete as possible. (Z2 )center U(1)L,3 ⇔ O(2)spin, The form of the mixed anomaly (69) indeed takes almost (96) U(1) /Z ⇔ U(1) . the same form of the ’t Hooft anomaly of antiferromagnetic V 2 M spin systems in (2 + 1) dimensions [67]. At low energies, that So far, we have checked that the group structures are indeed system can be described by the three-dimensional easy-plane the same. CP1 model: The vital point here is that not only the group structures but also the ’t Hooft anomaly has the same form under this 3 | − φ|2 + |φ|2 + λ|φ|4 + λ φ†σ φ 2 , d x[ (d ib) r EP( z ) ] (91) correspondence. To see this, let us introduce the U(1)spin gauge field Aspin, and then the kinetic term is replaced as where we introduced a two-component complex scalar φ = follows: (φ ,φ )T and a dynamical noncompact U(1) gauge field b.11 1 2 | − φ|2 ⇒| − − φ |2 +| − φ |2. We schematically suppress the Maxwell term for b because (d ib) (d ib iAspin ) 1 (d ib) 2 (97) the gauge coupling flows to the strong coupling limit in three Under the presence of Aspin, we should modify the (Z2 )spin dimensions. While the first three terms respect the SO(3)spin † transformation to keep this kinetic term invariant: spin symmetry for the spin vector φ σaφ, the last term explicitly breaks the spin symmetry down to O(2)spin and is called φ1 ←→ φ2, b → b + Aspin, Aspin →−Aspin. (98) the easy-plane potential. For λEP > 0, the spin vector prefers the xy plane in the spin space, while the spin vector tends to be aligned along the z axis for λ < 0. This model describes the EP 12To be precise, we have to add a monopole operator to the nth power to take into account the discreteness of the VBS order parameter when we consider the Néel-VBS transition on the rect- 11“Noncompact” means that we do not perform the path integral angular (n = 2), honeycomb (n = 3), and square lattices (n = 4) over the monopole configurations. Consequently, the model has the [160,162,163]. U(1) magnetic symmetry, which plays an important role in the 13The VBS order parameter in the antiferromagnets is realized as following discussion. the monopole operator [159].

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FIG. 2. Illustrations of the phases of the easy-plane CP1 model. (a) In the easy-axis Néel phase, the microscopic spins are ordered in a staggered manner and point in the ±z directions. (b) In the easy-plane Néel phase, the spins form an antiferromagnetic order, preferring the xy plane. (c) In the valence bond solid phase, the spins are paired with their nearest neighbors to be spin-singlet. Green ellipses indicate the spin-singlet pairs.

Let us see how it affects the partition function when we also realization of the VBS phase quickly. In terms of quantum gauge the U(1)M symmetry. The minimal coupling of the magnets, the microscopic 1/2 spins are paired with their U(1) gauge field, A , is given by the topological coupling, nearest neighbors to be spin-singlet and break the Z lattice M M 4 i rotational symmetry, as shown in Fig. 2(c). Note that the Z4 A ∧ db. π M (99) anisotropy is expected to be dangerously irrelevant around 2 the critical point [159,160], and the rotational symmetry is 1 Because of this topological term, (100), the action is affected enlarged to U(1), which is regarded as U(1)M in the CP by the (Z2 )spin transformation and acquires the overall phase model. In Table I we make the correspondence of phases i between these models according to this discussion. A ∧ dA . We note that the charge-conjugation symmetry does not π M spin (100) 2 3 enter the anomaly. However, since it has the structure of This is the signature of the [Z2 U(1)]spin × U(1)M anomaly. the semidirect product, the correspondence of symmetries We here emphasize that this anomaly is identical with the between these models looks to be slightly different when anomaly (69)ofQC2D at the isospin RW point. we combine it with (Z2 )spin symmetry. Let us denote this The identical structures of the global symmetry and ’t combined symmetry as (Z2 )C+spin, and then the symmetry Hooft anomaly motivate us to translate the QGP, chiral sym- looks like metry breaking, and baryon superfluid phases in QC Din 2 [(Z )C+ U(1) ] × U(1) . (101) termsoftheCP1 model. The corresponding phases of the 2 spin M spin 1 CP model are accessible by tuning the parameters r and Therefore, the roles of U(1)M and U(1)spin maybeinter- λEP. Varying r, we encounter a phase transition at r = rc changed depending on which Z2 symmetry is chosen for the which separates Néel phases (r < rc) and the VBS phase correspondence. This (Z2 )C+spin transformation acts on the (r > rc). For r < rc, the complex scalar condenses so that dynamical and background fields as † the spin vector φ σ φ acquires a nonzero expectation value ∗ a φ ←→ φ , b →−b − A , A → A , A →−A . and breaks the spin symmetry. When λ = 0, we obtain 1 2 spin spin spin M M EP (102) the SO(3) -symmetric Néel phase characterized by the spin We can indeed check that we obtain the same anomaly (100). symmetry-breaking pattern SO(3) → SO(2) . Neverthe- spin spin Therefore, the easy-plane Néel and VBS phases in Table I less, turning on the easy-plane potential alters the symmetry- should also be exchanged when we respect (Z )C+ .Inview breaking pattern in the Néel phase. (Z ) ⊂ O(2) is 2 spin 2 spin spin of the anomaly, both choices are equally good to make up the broken for a negative λ because the spin vector points in EP correspondence. the ±z direction to reduce the ground state energy. This phase Here we have clarified that QC D at the isospin RW point is known as the easy-axis Néel phase, which corresponds to 2 has a very similar structure with the easy-plane CP1 model. the QGP phase in two-flavor QC D. On the other hand, a 2 They share not only the same symmetry group, but also the positive λ confines the spin vector in the xy plane in the EP same ’t Hooft anomaly, which is highly nontrivial. We are spin space, breaking U(1) spontaneously. This phase is spin tempted to ask if their similarity extends to the dynamics, called the easy-plane Néel phase and the remnant of the chiral symmetry-breaking phase in two-flavor QC2D. We illustrate these phases in Figs. 2(a) and 2(b). TABLE I. Correspondence of the phases and order parameters in 1 On the other hand, for r > rc, the complex scalar is gapped two-flavor QC2D and the easy-plane CP model. and is integrated out. The effective theory in this phase is thus QC D at isospin RW point Easy-plane CP1 model the three-dimensional Maxwell theory, where the U(1)M is 2 spontaneously broken. A photon is understood as a Nambu- † Quark-gluon plasma, P Easy-axis Néel, φ σzφ Goldstone boson associated with this symmetry breaking. σ + π φ∗φ Chiral symmetry breaking, i 0 Easy-plane Néel, 2 1 Therefore, the VBS phase is regarded as baryon superfluid Baryon superfluid, Valence bond solid, Mb in the QC2D language. We also comment on a microscopic

033253-13 FURUSAWA, TANIZAKI, AND ITOU PHYSICAL REVIEW RESEARCH 2, 033253 (2020) i.e., the nature of phase transitions. The (2 + 1)-D easy-plane simulations, we argue that this constraint is indeed satisfied, CP1 model is now providing a typical example of quan- and we comment on how each phase matches this anomaly, tum phase transitions, called deconfined quantum criticality. as shown in Fig. 1. In future lattice simulations near the The phase transition point between the easy-plane Néel and massless limit, we expect that the hadronic phase would VBS phases is believed to acquire the emergent symmetry shrink, and the superfluidity would appear at low temperature, SO(4) ⊃ U(1)spin × U(1)M, which can rotate the Néel and once we turn on a finite chemical potential. Moreover, these VBS order parameters. Furthermore, the emergent symmetry symmetry-broken phases support defect excitations, and there is enlarged to SO(5) ⊃ (Z2 )spin × SO(4) when the SO(3)spin are topological zero modes localized on those defects because 14 symmetry recovers (i.e., λEP = 0). Recall that the SO(4) of the anomaly. / symmetry-mixing U(1)L.3 and U(1)V Z2 does appear in Interestingly, the discrete anomaly for massless QC2Dat chiral-symmetry-breaking phase at μ = 0. Thus, this scenario the isospin RW point is very similar to that of (2 + 1)-D may indicate an emergence of an intriguing SO(5) symmetry quantum antiferromagnetic systems. We propose an explicit that rotates the SO(4) chiral condensate (σ, π, 1, and 2) correspondence between these two systems based on the and the Polyakov loop P. Therefore, it sounds like a very consideration on their symmetries and anomalies. It would reasonable question in this context if the similar enhancement be an interesting future study to consider if this kinematic of symmetry occurs inside the phase diagram of QC2D. We similarity extends to the similarity of dynamics between these note that this depends on the dynamics of QC2D, so we theories. must go beyond the kinematic analysis based on symmetry and anomalies to answer this question. We shall leave it for possible future studies. ACKNOWLEDGMENTS The authors thank Y. Nishida for useful discussions. T.F. VII. SUMMARY was supported by the RIKEN Junior Research Associate Program when we started this work and by Japan Society In this paper, we have studied the phase diagram of two- for the Promotion of Science (JSPS) KAKENHI Grant No. flavor massless QC D at the isospin RW point based on the 2 JP20J13415 since April. E.I. was supported by the HPCI- ’t Hooft anomaly matching. We note that this setup does not JHPCN System Research Project (Project ID: jh200031) and suffer from the sign problem, because the Dirac determinant by Grants-in-Aid for Scientific Research through Grant No. over the u-quark sector is related to that of d-quark sector by 19K03875, which were provided by JSPS and by Iwanami- complex conjugation. Therefore, we can compare our study Fukuju Foundation (JSPS). Discussions during the workshop with numerical lattice simulations at finite quark chemical “CPN models: recent development and future directions” held potentials with imaginary isospin chemical potentials. at Keio University were useful to initiate this work. We first gave a careful review of the global symmetry of massless QC2D, especially paying attention to its global nature, such as the discrete symmetry and possible quo- APPENDIX A: CONVENTION OF γ MATRICES tients by discrete factors. For the computation of perturbative AND SPINORS anomalies, such details are not essential. On the other hand, γ the careful treatment of the discrete parts becomes crucial We here summarize the convention of Euclidean ma- = δ = when we discuss the more subtle global anomaly. Especially, trices in this paper. The flat space metric is gμν μν + , + , + , + when the imaginary isospin chemical potential takes the spe- diag( 1 1 1 1). The Weyl representation of the Eu- θ =−μ = π/ clidean gamma matrices is cial value, I i I L 2, two-flavor QC2D enjoys the (Z2 )center symmetry, which acts on both the quark flavors and μ μ 0 σ the Polyakov loop. Because of this fact, we concentrate on γ = μ , (A1) σ 0 studying the phase diagram at the isospin RW point, θI = π/2 in this paper. μ μ μ † After studying the perturbative anomaly to get an in- where σ = (1, iσ1, iσ2, iσ3) and σ = (σ ) = sight on T = 0, we computed the discrete mixed ’t Hooft (1, −iσ1, −iσ2, −iσ3). In the dotted and undotted spinor σ μ σ μ αβ˙ anomaly, which involves the center symmetry (Z2 )center,the notation, spin indices are assigned as ( )αβ˙ and ( ) , α,α,...,∈{ , } baryon-number symmetry U(1)V/Z2, and the isospin chiral where ˙ 1 2 . We describe the undotted spinor as ¯ symmetry, U(1)L,3. This is the main result of this paper. ψα and the dotted spinor as ψ˜β˙ , and their conjugate fields are This anomaly provides a meaningful constraint on the phase 15 ψ¯α˙ and ψ˜β , respectively. These are Weyl fermions, and the diagram of massless two-flavor QC2D. In order to satisfy the anomaly matching condition, the phase diagram of massless θ = π/ QC2Dat I 2 basically has to break one of the above three symmetries spontaneously at any temperatures T and quark chemical potentials μ. Combined with the study of 15Let us comment on the physical interpretation about the symbols chiral effective Lagrangian and the numerical results of lattice of spinors. Both ψα and ψ˜α denote left-handed Weyl fermions under the Lorentz transformation, i.e., in the (2, 1) representation of Spin(4) SU(2) × SU(2). In the context of QCD, however, we 14 αβ See Refs. [168,169] for theoretical studies, Ref. [170] for numer- usually interpret ψα as the left-handed quark (= ψD,L), and ε ψ˜β as = ψ ical observation, and Ref. [161] for recent theoretical development. the antiparticle of right-handed quark ( D,R ).

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Dirac fermions are defined as16 As we explain in Sec. II, we can now rotate ψ and ψ˜ as ψ a global symmetry because they share the same color and ψ = α , ψ = εαβψ˜ ψ¯ . D α˙ β˙ ¯ D β α˙ (A2) Lorentz structures, so it is now easy to see that the chiral sym- ε ψ˜β˙ metry is extended as SU(2Nf ) ⊃ [SU(Nf )L × SU(Nf )R × / In this convention, the Dirac Lagrangian can be written as U(1)V] ZNf [53,54]. μ μ αα˙ ¯ μ ββ˙ ψ γ ∂μψ = ψ¯α (σ ) ∂μψα + ψ˜ (σ ) ∂μψ˜β , (A3) D D ˙ β˙ APPENDIX B: SYMPLECTIC GROUP, Sp(N), AND up to the integration by parts. The Dirac mass, or chiral VACUUM MANIFOLD, SU(2N)/Sp(N) condensate, can be written as We here summarize properties of the compact symplectic αβ α˙ β˙ ¯ group, Sp(N), and its relation to SU(2N)[54], which are ψ ψD = ε ψ˜β ψα + ε ψ¯α˙ ψ˜β˙ . (A4) D important for understanding the chiral symmetry breaking of ε Below we omit spin indices and the tensor for simplicity QC2D. when the means of contraction is evident. Let us start with the definition of Sp(N). It is convenient When we consider the vector-like SU(Nc ) gauge theory, ψ to introduce the noncompact symplectic group Sp(2N, C), belongs to the defining representation, Nc, and ψ˜ belongs to defined by its conjugate representation, N , so that ψ transforms as N . c D c Sp(2N, C) ={M ∈ GL(2N, C) | MT M = }, (B1) The Dirac Lagrangian with minimal coupling is given as N N ψ γ μ ∂ + ψ where D ( μ iaμ) D = 01N = ψ¯ σ μ ∂ + ψ + ψ˜¯ σ μ ∂ − ψ.˜ N (B2) ( μ iaμ) ( μ iaμ) (A5) −1N 0

When we consider Nf Dirac ﬂavors, the list of charges is the symplectic form on C2N . We can show that det(M) = T under SU(Nc ) gauge symmetry and global chiral symmetry, 1 from the condition M N M = N , and thus Sp(2N, C) ⊂ × × / [SU(Nf )L SU(Nf )R U(1)V] ZNf , can be summarized as SL(2N, C). This is a simple Lie group, which is noncompact follows: and simply connected. Now, the compact symplectic group,

SU(Nc )SU(Nf )L SU(Nf )R U(1)V Sp(N), is defined by ψ Nc N f 1 1 (A6) Sp(N) = Sp(2N, C) ∩ SU(2N), (B3) ψ˜ N 1 N −1 c f T that is, if and only if U ∈ SU(2N) satisfies U NU = N , We note that this fits into the standard convention in super- U ∈ Sp(N). Since unitary matrices satisfy U T = (U ∗)−1, T symmetric QCD. the condition UNU = N is also equivalent by taking = −1 T −1 When Nc 2, the defining representation can be identified the complex conjugation of (U ) NU = N . In other with its conjugate representation, 2 2, by the pseudo-reality words, U ∈ Sp(N) ⇔ U ∗ ∈ Sp(N)forU ∈ SU(2N). of SU(2). Because of this, it is more convenient to take ψ˜ An important property of Sp(N) as a subgroup of SU(2N) in the defining representation, and the Dirac fermion can be emerges by considering the antisymmetric two-index repre- represented as sentation of SU(2N). More concretely, we define a (2N × 2N) ψ matrix-valued bosonic field, ij, using the fermionic field i ψ = , D ¯ (A7) in the defining representation of SU(2N)as (εcolor ⊗ εspin )ψ˜ ij = i j. (B4) where εcolor (= iτ2 ) is the invariant tensor of the SU(2) color space, and we also denote the invariant tensor of Spin(4) By the anticommutativity of fermions, T =− , and this SU(2) × SU(2) as εspin in order to avoid confusion. Through- belongs to the two-index antisymmetric representation. Under out the main text, we use this convention for Weyl and Dirac the SU(2N) transformation, U → U , spinors. We readily find that the Dirac Lagrangian becomes → U U T . (B5) ψ γ μ ∂ + ψ D ( μ iaμ) D Let us pick a specific point, = ψ¯ σ μ ∂ + ψ + ψ˜¯ σ μ ∂ + ψ,˜ ( μ iaμ) ( μ iaμ) (A8) = 0 ≡ N . (B6) T T because (εcolor ) a (εcolor ) =−a for SU(2) gauge field a. Then the stabilizer subgroup of 0 in SU(2N) is given by When we consider Nf Dirac flavors, the charge table (A6)is {U ∈ SU(2N) |U U T = }=Sp(N). (B7) modified as 0 0 As a consequence, when the bosonic field condenses as SU(2) SU(N ) SU(N ) U(1) f L f R V = , the spontaneous breaking pattern is SU(2N) → ψ 2 N 1 1 (A9) 0 f Sp(N). The vacuum manifold of this SSB is given by the ψ˜ 21 N − f 1 symmetric space T SU(2N)/Sp(N) {U 0U |U ∈ SU(2N)}, (B8) 16When we use the Dirac spinor, we always put the subscript “D” which is a connected subspace of (2N × 2N) antisymmetric throughout the paper. matrices with determinant 1. Its dimension is 2N2 − N − 1.

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Since we especially pay attention to the two-flavor case SO(6). Indeed, the two-index antisymmetric representation of in this paper, it would be useful to closely look at the case SU(4) is nothing but the defining representation of SO(6), i.e., 6 N = 2. In this case, it is convenient to use the exceptional ∈ R .If 0 = 0, this means the spontaneous breaking isomorphisms, SU(4) → Sp(2), , . SO(6) SO(5) (B10) Spin(6) SU(4) Spin(5) Sp(2) (B9) Z2 Z2 Since is in a two-index representation of SU(4), we can which fits the general argument by putting N = 2. The vac- 5 regard it as in a representation of SU(4)/Z2 Spin(6)/Z2 uum manifold is given by SO(6)/SO(5) S .

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