Phases of circle-compactified QCD with adjoint fermions at finite density

TL;DR

This paper investigates how chemical potential influences confinement, mass gap, and phase transitions in adjoint QCD with one spatial compact dimension, revealing alternating phases and novel superfluid phenomena.

Contribution

It introduces a detailed analysis of phase structure and nonperturbative effects in adjoint QCD at finite density with a compact dimension, including new insights into superfluid transitions and BEC-BCS crossover.

Findings

Alternating center-symmetric and center-broken phases with increasing chemical potential.

Photon mass gap in the magnetic bion plasma grows with chemical potential.

Possible superfluid transition leading to a nonperturbative energy gap for quarks.

Abstract

We study chemical-potential dependence of confinement and mass gap in QCD with adjoint fermions in spacetime with one spatial compact direction. By calculating the one-loop effective potential for the Wilson line in the presence of chemical potential, we show that a center-symmetric phase and a center-broken phase alternate when the chemical potential in unit of the compactification scale is increased. In the center-symmetric phase we use semiclassical methods to show that photons in the magnetic bion plasma acquire a mass gap that grows with the chemical potential as a result of anisotropic interactions between monopole-instantons. For the neutral fermionic sector which remains gapless perturbatively, there are two possibilities at non-perturbative level. Either to remain gapless (unbroken global symmetry), or to undergo a novel superfluid transition through a four-fermion interaction (broken global symmetry). If the latter is the case, this leads to an energy gap of quarks proportional to a new nonperturbative scale $L^{-1}\exp[-1/(g^4 \mu L)]$, where $L$ denotes the circumference of $S^1$, the low-energy is described as a Nambu-Goldstone mode associated with the baryon number, and there exists a new type of BEC-BCS crossover of the diquark pairing as a function of the compactification scale at small chemical potential.

Equations

$$\text{U}(2{)}_f\times [\text{SU}(2){]}_c$$

$$\stackrel{\text{anomaly}}{\to }$$

$$\stackrel{\mu \ne 0}{\to }$$

$$\stackrel{\text{Adj. Higgs}}{\to }$$

$$\stackrel{⟨\chi ⟩=0,\pi }{\to }$$

$$\stackrel{⟨\psi \psi ⟩\ne 0}{\to }$$

$$\mathcal{L}=\mathrm{tr}\left\{\frac{1}{2g^2}{F}_{\alpha \beta }^2+2{\overline{\Psi }}_f[/D(\mu )+m]{\Psi }_f\right\},$$

$$\Omega \equiv \mathcal{P}\exp \left(i\oint \mathrm{d}{x}^3{A}_3\right).$$

$$A_3=\mathrm{diag}(a_1,a_2,\dots ,a_{N_c}),\sum _{\mathtt{a}=1}^{N_c}{a}_{\mathtt{a}}=0$$

$$\hat{\mu }\equiv L\mu ,\hat{m}\equiv Lm\text{and}{\hat{a}}_{\mathtt{a}}\equiv L{a}_{\mathtt{a}}.$$

$$V(\Omega ;\mu )$$

$$V_{\mathrm{YM}}(\Omega )$$

$$V_{\mathrm{F}}(\Omega )$$

$$=\frac{2}{{\pi }^2{L}^4}\sum _{\mathtt{a},\mathtt{b}=1}^{N_c}{F}_2(\hat{m},{\hat{a}}_{\mathtt{a}}-{\hat{a}}_{\mathtt{b}})$$

$$F_2(\hat{m},\hat{a})\equiv {\hat{m}}^2\sum _{n=1}^{\infty }\frac{1}{n^2}{K}_2(n\hat{m})\cos (n\hat{a}).$$

$${\Gamma }_q(\Omega ;\mu )$$

$$\delta V_{\mathrm{F}}(\Omega ;\mu )=-\frac{1}{V_{{\mathbb{R}}^3}L}\left\{{\Gamma }_q(\Omega ;\mu )-{\Gamma }_q(\Omega ;0)\right\}$$

$$\displaystyle\delta V_{\rm F}(\Omega;\mu)=-\frac{1}{2\pi L^{4}}\bigg{\{}\sum_{\mathtt{a},\mathtt{b}=1}^{N_{c}}F_{3}(\hat{\mu},\hat{m},\hat{a}_{\mathtt{a}}-\hat{a}_{\mathtt{b}})-F_{3}(\hat{\mu},\hat{m},0)\bigg{\}}$$

$$F_3(\hat{\mu },\hat{m},\hat{a})$$

$$\displaystyle\quad\times\bigg{[}\frac{1}{3}\hat{\mu}^{3}-\hat{\mu}\big{\{}(\hat{a}+2n\pi)^{2}+\hat{m}^{2}\big{\}}$$

$$\displaystyle\quad+\frac{2}{3}\big{\{}(\hat{a}+2n\pi)^{2}+\hat{m}^{2}\big{\}}^{3/2}\bigg{]}\,,$$

$$n_q$$

$$\displaystyle=\frac{N_{f}^{\rm D}}{2\pi L^{3}}\big{\{}2F^{\prime}_{3}(\hat{\mu},\hat{m},2\hat{a})+F^{\prime}_{3}(\hat{\mu},\hat{m},0)\big{\}}$$

$$F_3^{\prime }(\hat{\mu },\hat{m},\hat{a})$$

$$\displaystyle\quad\times\theta\big{(}\hat{\mu}^{2}-\hat{m}^{2}-\left(\hat{a}+2n\pi\right)^{2}\big{)}\,.$$

$$\displaystyle\text{KK}_{\ell}\equiv\left\{(p_{1},p_{2},p_{3})\,\bigg{|}\;p_{3}=\frac{\ell\pi}{L},\;\sqrt{\bm{p}^{2}+m^{2}}\leq\mu\right\}$$

$$g_3^2\equiv \frac{g^2}{L}.$$

$$S$$

$$\displaystyle\delta S=G\int\!\!\mathrm{d}^{3}x\;\Big{[}\cos\chi\cdot\det\limits_{1\leq I,J\leq 2N_{f}^{\rm D}}\;(\psi_{I}^{\rm T}\sigma^{2}\psi_{J})+\text{h.c.}\Big{]}$$

$$\begin{array}{rl}G& \sim g^{-4}{e}^{-4{\pi }^2/g^2}{L}^{4N_f^{\mathrm{D}}-3}\\ & =g^{-4}{e}^{-4{\pi }^2/g^2}L\text{for}{N}_f^{\mathrm{D}}=1,\end{array}$$

$$\displaystyle\!\mathcal{B}(x)\sim G^{2}e^{2i\chi(x)}\!\!\!\!\int\!\!\!\mathrm{d}^{3}y\,e^{-V_{\rm C}(y)}\Big{\langle}\!\det\!\big{(}\psi\psi(y)\big{)}\det\!\big{(}\psi^{\dagger}\psi^{\dagger}(0)\big{)}\Big{\rangle}$$

$$V_{\mathrm{C}}(y)\equiv \frac{4\pi }{g_3^2}\frac{1}{|y|}.$$

$$\displaystyle\Big{\langle}\!\det\!\big{(}\psi\psi(x)\big{)}\det\!\big{(}\psi^{\dagger}\psi^{\dagger}(0)\big{)}\Big{\rangle}$$

$$\propto$$

$$\displaystyle\times\big{\{}(\partial_{1}D_{-})^{2}+(\partial_{2}D_{-})^{2}+[(\partial_{4}+\mu)D_{-}]^{2}\big{\}}\,,$$

$$[-{\partial }_1^2-{\partial }_2^2-({\partial }_4-\mu {)}^2]D(x;\mu )=\delta (x).$$

Full text

††preprint: RIKEN-QHP-256

I Introduction

Semiclassical analysis is a powerful tool to study quantum systems nonperturbatively. In QCD, instantons have long been a subject of intensive research Gross et al. (1981); Schafer and Shuryak (1998); Diakonov (1996); *Diakonov:2002fq; Vandoren and van Nieuwenhuizen . They played crucial roles in phenomenological models of spontaneous chiral symmetry breaking, as well as the U(1) problem and the strong CP problem. In dense quark matter, instanton effects are semiclassically reliable as it has been been widely recognized Shuryak (1982); Rapp et al. (1998); Alford et al. (1998); Carter and Diakonov (1999); Rapp et al. (2000); Son et al. (2001); Schafer (1998); *Schafer:2002ty.

Our understanding of semiclassical aspects of non-Abelian gauge theories including QCD was significantly advanced with two related progress:

• •

KvBLL monopole-instantons on $\mathbb{R}^{3}\times S^{1}$ Lee and Yi (1997); Lee and Lu (1998); Kraan and van Baal (1998a, b),

• •

Semiclassically calculable domains on $\mathbb{R}^{3}\times S^{1}$ in (non-supersymmetric) QCD-like and Yang-Mills theories Unsal (2008, 2009); Shifman and Unsal (2008); Unsal (2009); Argyres and Unsal (2012).

Monopole-instantons carry fractional topological charge, and obey the Nye–Singer index theorem Nye and Singer (2000); Poppitz and Unsal (2009) (which is a refinement of the Atiyah–Singer index). The sum of the fermion zero modes of the monopole-instantons is equal to the one of instanton in four dimensions (4d), hence, they are more relevant for chiral symmetry breaking. In sharp contrast to 4d instantons, monopole-instantons on $\mathbb{R}^{3}\times S^{1}$ explicitly depend on the background holonomy (Wilson line) field. Therefore, monopole-instantons provide a concrete connection between center-symmetry realization (confinement in a thermal context) and chiral symmetry realization.

In ${\cal N}=1$ supersymmetric Yang-Mills (SYM) theory on $S^{1}\times\mathbb{R}^{3}$, the ensemble of monopole-instantons generates the gluino condensate Davies et al. (1999); *Davies:2000nw. The applicability of semi-classics to non-supersymmetric QCD-like theories was shown in Refs. Unsal (2008); *Unsal:2007jx; Shifman and Unsal (2008); Unsal and Yaffe (2008); Meisinger and Ogilvie (2010); Argyres and Unsal (2012); Ogilvie (2012). (Also see Ref. Diakonov et al. (2004).) In such theories gauge symmetry “breaking” occurs at small $S^{1}$ (at weak coupling) due to the Hosotani mechanism Hosotani (1983); *Hosotani:1988bm. In SYM as well as in adjoint QCD, in the Abelian regime, topologically neutral molecules of monopoles called magnetic bions form and generate a bosonic potential that engenders a mass gap of gluons and confines quarks Unsal (2008); *Unsal:2007jx; Argyres and Unsal (2012).

In this work, we study phases of compactified adjoint QCD on $\mathbb{R}^{3}\times S^{1}$ at nonzero quark chemical potential $\mu$. It should be noted that $S^{1}$ in this paper is a compactified spatial direction along which fermions obey the periodic boundary condition (PBC); the imaginary-time direction is infinite and the system is at zero temperature. This setup must not be confused with previous studies of a holonomy potential for thermal $S^{1}$ at $\mu\neq 0$ Korthals Altes et al. (2000); Myers and Ogilvie (2009); Hands et al. (2010); Bruckmann et al. (2013); Liu et al. (2016).

This theory is semiclassically calculable at small $S^{1}$ for any value of $\mu$.111At large $\mu$ and for values of $S^{1}$ larger than the strong length scale, there are sub-sectors of the theory amenable to the weak coupling treatment, however, there are sectors of the theory which are still strongly coupled, in particular, the gauge sector. For generic colors $(N_{c})$ and Dirac flavors $(N_{f}^{\rm D})$ we compute the perturbative one-loop potential for the Wilson line holonomy. We find that turning on a small $\mu$ weakens gauge symmetry breaking, while larger $\mu$ induces an infinite number of oscillatory transitions between a gauge-symmetry-restored phase and a broken phase. In the specific case of $N_{c}=2$ and $N_{f}^{\rm D}=1$, we show that perturbatively induced four-fermion operators222The monopole operators which also possess four fermionic zero modes are nonperturbatively weak in the semiclassical domain. that have so far been neglected in studies of the Hosotani phase can play a pivotal role in generating diquark condensation and drive the system into a superfluid phase with broken $\text{U}(1)_{B}$ baryon number symmetry. Our calculations are performed in a theoretically controlled setting thanks to the small running coupling and complete Abelianization of the gauge group at small $S^{1}$. This is a unique opportunity in which one can derive both continuous symmetry breaking and confinement from first principles analytically, providing a valuable laboratory to study the strongly coupled QCD vacuum.

As a quick guide to readers, we summarize a cascade of symmetry breaking in the presence of $\mu$ for $N_{c}=2$ and $N_{f}^{\rm D}=1$ when $\mu$ is small compared to the compactification scale $L^{-1}$:

[TABLE]

Here $[...]_{c}$ denotes the unbroken part of the gauge group and $\chi$ is the dual photon field. In (2)–(5) we have factored a common $\mathbb{Z}_{2}$ group to avoid double counting. Detailed explanations will follow later.

In Fig. 1 we present a schematic phase diagram of adjoint QCD with the aim to illustrate how our analysis in this paper in the small-$S^{1}$ regime should fit in the global phase diagram from small to large $S^{1}$. Adjoint QCD on $\mathbb{R}^{4}$ is believed to exhibit a relativistic analogue of the BEC-BCS crossover Chen et al. (2005); Zwerger (2012) when the chemical potential is swept across $\Lambda_{\text{QCD}}$ Kogut et al. (2000); Hands et al. (2000); Splittorff et al. (2001); Hands et al. (2001); Nishida and Abuki (2005); Sun et al. (2007); He (2010); Kanazawa et al. (2011); Kanazawa (2013):

• $\checkmark$

At $\mu=0$ and large $L$, the center symmetry is preserved, while the flavor symmetry is believed to be spontaneously broken as $\text{SU}(2)_{f}\to\text{SO}(2)_{f}$ for $N_{f}^{\rm D}=1$ Peskin (1980). The physical spectrum contains a light diquark that undergoes a Bose condensation for $\mu>m_{\pi}/2$ and breaks $\text{U}(1)_{B}$. This onset of superfluidity is well described by chiral perturbation theory Kogut et al. (2000).

• $\checkmark$

In the asymptotic region $\mu\gg\Lambda_{\text{QCD}}$, on $\mathbb{R}^{4}$, quarks acquire a large energy gap $\Delta\sim\mu\,e^{-1/g}$ from the perturbative one-gluon exchange at the Fermi surface Son (1999); Schafer and Wilczek (1999) and the low-energy effective theory is anisotropic Yang-Mills theory with a confining scale $\Lambda^{\prime}$ that goes down to zero as $\mu\to\infty$ due to medium effects Rischke et al. (2001). The compactification $L<\infty$ acts on gluons as a thermal scale and triggers a deconfinement transition at $L^{-1}\sim\Lambda^{\prime}$. This is why the phase transition line must emanate from the top-right corner of the phase diagram in Fig. 1.

• $\checkmark$

At $\mu=0$, if $L\Lambda_{\text{QCD}}\ll 1$, the system enters a center-symmetric weakly-coupled regime where the discrete chiral symmetry remains broken, while the continuous chiral symmetry is restored Unsal (2008). This is indicated by a red line in the bottom-left corner of Fig. 1.

• \normalsize$\checkmark$⃝

The above three domains are well understood by now. In this paper, we venture into the novel regime of small $S^{1}$ and any $\mu$. Among other things we uncover, in addition to a series of confinement–deconfinement transitions, we find that a novel mechanism which can engender superfluidity even for arbitrarily small $L$ and $\mu$, as long as $\mu\neq 0$, is a logical possibility. The condition under which this possibility may be realized will be carefully examined later. The estimated nonperturbative gap of quarks is of order $\Delta\sim L^{-1}e^{-1/(g^{4}\mu L)}$.333It is intriguing to draw a comparison between dense QCD, the NJL model, and our case. In dense QCD the gap is $\sim e^{-1/g}$ due to unscreened magnetic gluons Son (1999). In the NJL-type model with a point-like four-fermion interaction the gap is generally $\sim e^{-1/g^{2}}$ Buballa (2005). The dependence $e^{-1/g^{4}}$ found in this work differs from either of them.

Now, if the adiabatic continuity of center symmetry at $\mu=0$ from small to large $S^{1}$ (conjectured in Refs. Unsal (2008); *Unsal:2007jx; Shifman and Unsal (2008); Argyres and Unsal (2012) and tested numerically in Refs. Cossu and D’Elia (2009); Cossu et al. (2014)) holds, then it follows that a double crossover should emerge on the phase diagram (Fig. 1), uniting two weakly-coupled BCS superfluids: one at small $S^{1}$ and the other at large $S^{1}$ and $\mu\gg\Lambda_{\text{QCD}}$. It is highly nontrivial that such a continuity may exist, because the two phases have distinct physical mechanisms for quark pairing: two-charged-boson exchange interaction on one hand (at small $S^{1}$; see Fig. 11 below) and the one-gluon-exchange interaction on the other hand.

This paper is organized as follows. In Sec. II we examine the perturbative holonomy potential and reveal a complex phase diagram at $\mu\neq 0$ and nonzero masses for $N_{c}=2$ and $3$. The quark number density is also computed. In Sec. III we analyze how the monopole-binding interaction due to fermion-zero-mode exchange is modified at $\mu\neq 0$. It is shown for $N_{c}=2$ that the chemical potential renders the inter-monopole potential strongly anisotropic, favoring a temporal separation. A fermionic low-energy effective theory is also examined and the conditions under which spontaneous $\text{U}(1)_{B}$ breaking occurs at an arbitrarily small chemical potential are derived. The fermion gap $\Delta$ is shown to be proportional to $e^{-1/(g^{4}\mu L)}$, which is smaller than any finite order of the semiclassical expansion in powers of $e^{-S_{\rm m}}$ where $S_{\rm m}=8\pi^{2}/(g^{2}N_{c})=4\pi^{2}/g^{2}$. The effective Lagrangian of the Nambu–Goldstone mode associated with the $\text{U}(1)_{B}$ breaking is also derived. We conclude in Sec. IV. Technical details on the derivation of the perturbative potential and properties of a free fermion propagator at $\mu\neq 0$ are summarized in Appendix A and B, respectively.

II Hosotani mechanism with chemical potential

We consider $\text{SU}(N_{c})$ gauge theory on $\mathbb{R}^{3}\times S^{1}$ with $N_{f}^{\rm D}$ flavors of 4-component Dirac fermions $\Psi_{f}=\Psi^{\textsf{A}}_{f}t^{\textsf{A}}$ in the adjoint representation, obeying PBC on $S^{1}$. We will use $f,g,\dots$ to label flavors $\{1,2,\dots,N_{f}^{\rm D}\}$ and $\textsf{A},\textsf{B},\dots$ to label the adjoint colors $\{1,2,\dots,N_{c}^{2}-1\}$. (The fundamental colors $\{1,2,\dots,N_{c}\}$ will be labeled by $\mathtt{a},\mathtt{b},\dots$.) The generators are normalized as $\operatorname{tr}(t^{\textsf{A}}t^{\textsf{B}})=\delta^{\textsf{A}\textsf{B}}/2$. The circumference of $S^{1}$ is denoted by $L$. The asymptotic freedom requires $N_{f}^{\rm D}<2.75$ and this is hereafter assumed unless stated otherwise. The Lagrangian reads

[TABLE]

where $F_{\alpha\beta}=F^{\textsf{A}}_{\alpha\beta}t^{\textsf{A}}$ and $\not{D}(\mu)=\not{D}-\mu\gamma_{4}$. In this paper we will always work in Euclidean spacetime, using $\alpha,\beta,\dots$ to denote the entire four directions $\{1,2,3,4\}$ while reserving $i,j,\dots$ for non-compact directions $\{1,2,4\}$ only. The compactified spatial direction is $x^{3}$ in our setting.

For $N_{f}^{\rm D}\geq 1$ and $\mu=0$ the gauge symmetry is known to be dynamically broken to the maximal torus $\text{U}(1)^{N_{c}-1}$ by quantum effects of periodic fermions if both $L\Lambda_{\text{QCD}}$ and $Lm$ are sufficiently small Hosotani (1989); Unsal and Yaffe (2010); Unsal (2008, 2009). In this section we examine how the chemical potential term $\mu\Psi^{\dagger}\Psi$ influences gauge symmetry breaking at small $S^{1}$, putting aside the issue of photon mass gap and fermion bilinear condensation. This treatment is justified because the latter effects are nonperturbatively small while gauge symmetry is broken at one loop.

II.1 Perturbative potential for the Wilson line

We consider a uniform background gauge field that is zero in $\mathbb{R}^{3}$ directions and nonvanishing along $S^{1}$. The holonomy is given by

[TABLE]

We employ the gauge in which $\Omega$ is diagonal,

[TABLE]

and define dimensionless variables for later convenience:

[TABLE]

The object to be calculated is the one-loop effective potential for the holonomy at $\mu\neq 0$. It consists of several pieces,

[TABLE]

The first term is the contribution of the gauge fields and ghosts Gross et al. (1981)

[TABLE]

with $F_{1}(\hat{a})\equiv[\hat{a}]^{2}(2\pi-[\hat{a}])^{2}$ where $0\leq[\hat{a}]\leq 2\pi$ is equal to $\hat{a}$ mod $2\pi$ . $V_{\rm F}(\Omega)$ is the potential induced by a single adjoint Dirac fermion with PBC at $\mu=0$ Meisinger and Ogilvie (2002); Unsal and Yaffe (2010),

[TABLE]

with

[TABLE]

What remains is to compute $\delta V_{\rm F}(\Omega;\mu)$, which controls the $\mu$-dependence of the effective potential. In terms of the quark determinant

[TABLE]

we have

[TABLE]

where $V_{\mathbb{R}^{3}}$ formally denotes the infinite volume of the spacetime in $x^{1,2,4}$ directions. Technical details of the evaluation of $\Gamma_{q}$ is given in Appendix A. The result is

[TABLE]

with

[TABLE]

where $\theta(x)$ is the Heaviside step function. It is immediately clear from (18) that $\delta V_{\rm F}(\Omega;\mu)$ vanishes when $\mu<m$. This is expected because $\mu$ below the energy gap (mass of adjoint fermion) has no physical effect at zero temperature.

Collecting (11), (12), (13) and (17) we obtain the one-loop effective potential. Next we proceed to numerical analysis of the phase diagram for $N_{c}=2$ and $3$ based on minimization of $V(\Omega;\mu)$.

II.2 $N_{c}=2$

For $N_{c}=2$ the holonomy can be parametrized as $\Omega=\operatorname{diag}(e^{i\hat{a}},e^{-i\hat{a}})$. In numerical minimization of the potential $V(\Omega;\mu)$ we encounter the following two phases.

[TABLE]

Let us summarize the bosonic spectrum in each phase.

• •

In the first phase with broken center symmetry, the $A_{3}$ component of gluons acquires a mass of $\mathcal{O}(g/L)$ (called the electric screening mass in the case of thermal $S^{1}$) while the other components $A_{1,2,4}$ are perturbatively massless and interact strongly with periodic fermions. In the limit $L\to 0$ with fixed $\mu$ the system would reduce to three-dimensional adjoint QCD at finite $\mu$.

• •

The second phase entails Abelianization of the gauge group due to the Hosotani mechanism; similarly to $W$ bosons in the electroweak theory, $A_{1,2,4}^{\textsf{A}=1,2}$ acquire a mass $m_{\rm W}=\pi/L$ by eating $A_{3}^{\textsf{A}=1,2}$; the “Higgs mode” $A_{3}^{\textsf{A}=3}$ gains a mass $m_{\rm H}\sim g/L$; The photons $A_{1,2,4}^{\textsf{A}=3}$ are massless to all orders in perturbation theory.

In Fig. 2 the holonomy potential for $N_{f}^{\rm D}=1$ in the chiral limit is plotted. As $\hat{\mu}$ increases, the center-symmetric phase is turned into a center-broken phase via a first-order transition. Namely $\mu$ counteracts the Hosotani mechanism in this parameter range.444This is in contrast with the case of an external magnetic field Anber and Ünsal (2014), which enlarges the center-symmetric phase.

To explain why, let us look at the quark number density

[TABLE]

with

[TABLE]

The first term in (19) represents the density of $\Psi^{\textsf{A}=1,2}$, which are charged under the unbroken $\text{U}(1)$ gauge symmetry. The second term in (19) represents the density of neutral fermions $\Psi^{\textsf{A}=3}$; since it does not depend on the holonomy it can be ignored for the moment.

As (20) implies, due to compactification, the Fermi sea becomes a union of multiple disks in momentum space, labeled as

[TABLE]

with $\ell\in\mathbb{Z}$. (Precisely speaking, this $p_{3}$ is the eigenvalue of $i\partial_{3}+A_{3}$ rather than $i\partial_{3}$.) Such a discrete structure of energy levels is analogous to the Landau levels in a magnetic field. As shown in Fig. 3, charged fermions occupy either $\text{KK}_{\ell=\text{even}}$ or $\text{KK}_{\ell=\text{odd}}$ with $|\ell|\leq\sqrt{\hat{\mu}^{2}-\hat{m}^{2}}/\pi$ depending on the holonomy. Namely, in the center-symmetric phase $(\hat{a}=\pi/2)$ the holonomy dynamically changes the boundary condition of charged fermions along $S^{1}$ from periodic to anti-periodic. We note that every time $\mu$ reaches a new $\text{KK}_{\ell}$ level, a second-order phase transition occurs; the quark number susceptibility jumps. Indeed there are an infinite number of second-order transitions as a function of $\mu$.555This phenomenon, analogous to the de Haas–van Alphen effect in solid state physics, has been studied in NJL models with spatial compactification Vshivtsev et al. (1998).

Let us examine what happens at small $\hat{\mu}$. When $\hat{m}\leq\hat{\mu}<\sqrt{\pi^{2}+\hat{m}^{2}}$, $\mu$ lies between $\text{KK}_{0}$ and $\text{KK}_{\pm 1}$. Hence $\text{KK}_{0}$ forms the charged Fermi sea in the center-broken phase. On the other hand, no Fermi sea exists yet in the center-symmetric phase. This means that the pressure in the center-broken phase is higher, by the amount of the degenerate Fermi sea, implying that center symmetry breaking is favored by $\delta V_{\rm F}(\Omega;\mu)$ for $\hat{\mu}$ in this range. This explains the first-order transition in Fig. 2.

Interestingly, this is not the whole story. As $\hat{\mu}$ increases further, many $\text{KK}_{\ell}$ join the Fermi sea and the competition between $\text{KK}_{\ell=\text{even}}$ and $\text{KK}_{\ell=\text{odd}}$ becomes quite subtle. In Fig. 4 we plot the difference of $F_{3}$, (18), in the two vacua at $\hat{m}=0$.

It shows a growing oscillation in $\hat{\mu}$ with period $\approx 2\pi$. At small $\hat{\mu}$ the oscillation starts with a negative value, implying that the center-broken vacuum is favored by medium. As $\hat{\mu}$ increases, however, the zero is crossed infinitely many times and consequently it leads to an endless alternation of the two vacua. In Fig. 5 we show the phase diagram of center symmetry, showing the presence of infinitely many center-symmetric phases.

One can tune $\hat{\mu}$ to bring the system into a center-symmetric phase, however large the mass $\hat{m}$.

It should be noted that the center-symmetric phases at $\hat{\mu}>1$ are different from the well-studied center-symmetric phase at $\hat{m}\sim\hat{\mu}\sim 0$ in that there is a Fermi sea of charged fermions $\Psi^{\textsf{A}=1,2}$ in the former but not in the latter.

The stripe structure of the phase diagram in Fig. 5 is reflected in the $\mu$-dependence of other observables as well. Figure 6 plots the quark density $n_{q}$ (divided by $\mu^{2}$ to make the stratified structure clearer). The lines of center-symmetry-changing first-order transitions are clearly visible. In addition, $m_{\rm H}$ (the mass of $A_{3}^{\textsf{A}=3}$) oscillates with $\hat{\mu}$, but it stays at $\mathcal{O}(g/L)$ and never vanishes because the transitions are first order.

II.3 $N_{c}=3$

For $N_{c}=3$ the holonomy can be parametrized as \Omega=\operatorname{diag}\big{(}e^{i\hat{a}_{1}},e^{i\hat{a}_{2}},e^{-i(\hat{a}_{1}+\hat{a}_{2})}\big{)}. In numerical minimization of the potential $V(\Omega;\mu)$ with respect to $\hat{a}_{1}$ and $\hat{a}_{2}$ we found three phases below.

$(\hat{a}_{1},\hat{a}_{2})$ at the minimum $\mathbb{Z}_{3}$ center gauge symmetry

$(0,0)$, $\pm\left(\frac{2\pi}{3},\frac{2\pi}{3}\right)$ broken $\text{SU}(3)$ ​​

$\pm\left(\frac{\pi}{3},\frac{\pi}{3}\right),\pm\left(\frac{\pi}{3},-\frac{2\pi}{3}\right),\pm\left(-\frac{2\pi}{3},\frac{\pi}{3}\right)$ broken $\text{SU}(2)\times\text{U}(1)$ ​​

$\left(0,\pm\frac{2\pi}{3}\right),\left(\pm\frac{2\pi}{3},\mp\frac{2\pi}{3}\right),\left(\pm\frac{2\pi}{3},0\right)$ unbroken $\text{U}(1)\times\text{U}(1)$ ​​

The phase in the second row, often called the split (or skewed) phase, was found in Ref. Myers and Ogilvie (2008) in a deformed $\text{SU}(3)$ Yang-Mills theory and has been studied in a variety of setups Myers and Ogilvie (2009); Cossu and D’Elia (2009); Cossu et al. (2014); Misumi and Kanazawa (2014). The other two phases are natural generalizations of those for $N_{c}=2$.

As for $N_{c}=2$, we again find that the addition of $\hat{\mu}$ destabilizes the center-symmetric vacuum realized at $\hat{\mu}=0$. Figure 7 displays the evolution of the holonomy potential for increasing $\hat{\mu}$. We observe that each phase in the above table is realized successively (from bottom to top) for increasing $\hat{\mu}$. An explanation in terms of $\text{KK}_{\ell}$ levels becomes rather cumbersome and is not as useful as for $\text{SU}(2)$.

When $\hat{\mu}$ is further increased, the three phases periodically alternate and there are an infinite number of first-order transitions. The phase diagram for $N_{f}^{\rm D}=1$ is displayed in Fig. 8. Again we find that a center-symmetric phase can be realized for an arbitrarily large fermion mass by tuning $\mu$ judiciously. The global features of Fig. 8 are similar to Fig. 5 except for the presence of the split phase.

The quark density $n_{q}/\mu^{2}$ is plotted in Fig. 9. It increases monotonically with $\hat{\mu}$ and exhibits tiny jumps along the first-order transition lines in Fig. 8.

It is tempting to conjecture that, as we increase $N_{c}$, the phase diagram will contain a variety of more exotic phases with partial center symmetry breaking, such as those found in Refs. Myers and Ogilvie (2009); Unsal and Yaffe (2010); Bringoltz et al. (2012). However, when $L$ is made larger, all such center-breaking phases must somehow disappear because adjoint QCD on $\mathbb{R}^{4}$ is a strongly-coupled confining theory (cf. Fig. 1). Although revealing the phase structure at intermediate $L$ is beyond the realm of weak-coupling method in this paper, this can be studied in lattice simulations in principle as the path-integral measure of adjoint QCD is positive semi-definite even at $\mu\neq 0$ Alford et al. (1999).

III Low-energy effective theory and symmetry breaking

III.1 General discussion

In the previous section we have shown by using the one-loop effective potential of holonomy that there are three phases (below we assume $N_{c}=2$ for simplicity):

• •

Phase I: Center-broken phase with $\Omega=\pm\mathbbm{1}_{2}$

• •

Phase II: Center-symmetric phase at $\hat{\mu}>\pi$

• •

Phase III: Center-symmetric phase at $\hat{\mu}<\pi$ (the small square next to the origin in Fig. 5)

Phases II and III are similar in terms of center symmetry, but it is useful to distinguish them for later purposes. In the following, we aim to analyze the low-energy dynamics in each phase, with a focus on Phase III. Before doing so, it may be useful to remind global symmetries of adjoint QCD as it has unique features absent in ordinary QCD. The classical massless Lagrangian of this theory with $N_{f}^{\rm D}$ Dirac flavors at $\mu=0$ is invariant under $\text{U}(1)_{A}\times\text{SU}(2N_{f}^{\rm D})$ symmetry due to reality of the adjoint representation Peskin (1980); Leutwyler and Smilga (1992), with $\text{U}(1)_{B}\subset\text{SU}(2N_{f}^{\rm D})$ the baryon number symmetry acting as $\Psi\to e^{i\varphi}\Psi$. Chiral anomaly reduces $\text{U}(1)_{A}$ down to $\mathbb{Z}_{4N_{c}N_{f}^{\rm D}}$. In $\mathbb{R}^{4}$ (i.e., $L\to\infty$), if $N_{f}^{\rm D}$ is below the conformal window, the continuous part of the flavor symmetry will be spontaneously broken by chiral condensate $\langle\operatorname{tr}\overline{\Psi}\Psi\rangle\neq 0$ as $\text{SU}(2N_{f}^{\rm D})\to\text{SO}(2N_{f}^{\rm D})$ Peskin (1980); Leutwyler and Smilga (1992). The chemical potential $\mu\neq 0$ spoils the symmetry between fermions and antifermions and breaks the flavor symmetry explicitly as $\text{SU}(2N_{f}^{\rm D})\to\text{U}(1)_{B}\times\text{SU}(N_{f}^{\rm D})_{L}\times\text{SU}(N_{f}^{\rm D})_{R}$ for $N_{f}^{\rm D}\geq 2$ and $\text{SU}(2N_{f}^{\rm D})\to\text{U}(1)_{B}$ for $N_{f}^{\rm D}=1$ Kogut et al. (2000). From here on, we will focus our attention on the minimal $N_{f}^{\rm D}=1$ case, so that the non-anomalous global symmetry that can be spontaneously broken at $\mu\neq 0$ is $(\mathbb{Z}_{8})_{A}\times\text{U}(1)_{B}$, as noted earlier in (3).

Let us begin with Phase I in the list above. This phase, having intact $\text{SU}(2)$ gauge symmetry, bears similarity to the deconfined phase of thermal QCD, except that fermions obey PBC along $S^{1}$ here. In the limit $L\to 0$ all modes other than $\text{KK}_{0}$ will decouple and we end up with adjoint QCD in $\mathbb{R}^{3}$ with $\mu$, with a dimensionful coupling666Notice that the scale of $\mu$ is extremely high in this dimensionally reduced theory: $\mu/g_{3}^{2}=\hat{\mu}/g^{2}\sim 1/g^{2}\gg 1$.

[TABLE]

This is a strongly coupled theory and not amenable to semiclassical analysis, but we can guess the physics qualitatively. Because there is a nonzero density of fermions for $\mu>m$ and there is an attractive channel in the color interaction mediated by $\text{SU}(2)$ gluons, the BCS mechanism will most likely trigger diquark condensation $\langle\operatorname{tr}\Psi^{\rm T}\Psi\rangle$ that breaks $\text{U}(1)_{B}$ spontaneously and generates a pairing gap for fermions. While this scenario sounds plausible, a serious study must incorporate the pairing of fermions and the nonperturbative mass gap of gluons simultaneously. This is an interesting open problem.

Next we consider Phase II. At energy scales far below $m_{\rm W}=\pi/L$ and $m_{\rm H}=\mathcal{O}(g/L)$, one can adopt a description in terms of compact $\text{U}(1)$ gauge theory with fermions $\Psi$. At $\mu=0$, charged fermions $\Psi^{\pm}\equiv(\Psi^{\textsf{A}=1}\pm i\Psi^{\textsf{A}=2})/\sqrt{2}$ can also be integrated out due to their large “screening mass” $\pi/L$ Unsal (2008, 2009). However, at $\mu>\sqrt{(\pi/L)^{2}+m^{2}}$ (or $\hat{\mu}>\pi$ in the chiral limit), there is a Fermi surface of $\Psi^{\pm}$ on top of that of neutral $\Psi^{\textsf{A}=3}$, so the charged fermions cannot be dropped from the long-distance effective description.777This situation is analogous to center-stabilized QCD with fundamental fermions in $\mathbb{R}^{3}\times S^{1}$ Shifman and Unsal (2008); Poppitz and Sulejmanpasic (2013). The attractive Coulomb interaction between $\Psi^{+}$ and $\Psi^{-}$ will destabilize their Fermi surface and engender a condensate $\langle\Psi^{+}\Psi^{-}\rangle\neq 0$ that breaks $\text{U}(1)_{B}$ symmetry spontaneously. (Note that $\Psi^{+}\Psi^{-}$ is charge neutral and does not break $\text{U}(1)$ gauge symmetry.)

In Phase III the long-distance theory comprises a massless photon and neutral fermions $\Psi^{\textsf{A}=3}$ at finite $\mu$. It naively appears as though $\text{U}(1)_{B}$ symmetry is unbroken, but as will be shown in Sec. III.3, it could be spontaneously broken through an effective interaction mediated by heavy charged particles. Recalling the intuitive pictures for Phase I and II above, we conjecture that actually $\text{U}(1)_{B}$ is broken everywhere in the phase diagram (Fig. 6).888This is a hypothesis at exactly zero temperature. As soon as the imaginary-time direction is compactified, the true symmetry breaking would be replaced with a quasi-long-range order Mermin and Wagner (1966); *Hohenberg:1967zz; *Coleman:1973ci.

III.2 Monopoles, bions and semiclassical confinement

Let us focus on Phase III for $N_{c}=2$ and $N_{f}^{\rm D}=1$. We assume $m=0$ for simplicity and switch to the two-component notation $\Psi^{\textsf{A}=3}=L^{-1/2}\begin{pmatrix}\psi_{1}\\ \sigma^{2}\psi_{2}^{*}\end{pmatrix}$, thereby ensuring that $\psi_{1,2}$ possess the canonical dimension of spinors in three dimensions. The tree-level effective theory at small $L$ reads

[TABLE]

where $\sigma^{1,2,4}\equiv(\sigma^{1},\sigma^{2},-i\mathbbm{1}_{2})$ and $g_{3}$ is defined in (22). Notice that $\mu\neq 0$ breaks the global $\text{SU}(2)_{f}$ symmetry down to $\text{U}(1)_{B}$ that rotates $\psi_{1}\to e^{i\varphi}\psi_{1}$ and $\psi_{2}\to e^{-i\varphi}\psi_{2}$.

At a nonperturbative level, we also have instanton-monopoles, reflecting the compact nature of the $\text{U}(1)$ gauge group. At first sight, Polyakov’s mechanism of confinement due to Debye screening by monopoles Polyakov (1977) seems to apply, but this is not the case here: monopoles are accompanied by fermionic zero modes999For the index theorem at finite $\mu$, see Refs. Gavai and Sharma (2010); Kanazawa et al. (2011); Bruckmann et al. (2013). and do not generate a mass gap for photons Affleck et al. (1982).101010The fermionic zero modes of monopoles are absent if one starts from a genuine compact $\text{U}(1)$ theory rather than breaking a non-Abelian group with a Higgs mechanism. In such a case the dynamics of monopoles at long distances can be different from that in $\mathbb{R}^{3}\times S^{1}$; see e.g., Refs. Ioffe and Larkin (1989); *Marston:1990bj; *Murthy1991; *Kleinert:2002uv; *Herbut:2003bs; *Herbut2003PRB; *Hermele:2004hkd.111111When $m\neq 0$, fermionic zero modes can be soaked up by the mass term and monopoles are allowed to yield a bosonic potential Unsal and Yaffe (2010); Poppitz et al. (2012). Similarly, if the difermion condensate $\langle\psi\psi\rangle$ forms dynamically, it would lift the fermionic zero modes and induce a monopole-induced potential. As we will see in Sec. III.3, indeed such a condensate can form, but it occurs only at a super-soft scale and can be safely ignored at the scale of average monopole-monopole separation $\sim L\exp(S_{m}/3)$. The effective theory at length scales $\gg L/g$ gains an additional piece that accounts for this effect Unsal (2008); *Unsal:2007jx; Argyres and Unsal (2012),

[TABLE]

where the scalar field $\chi(x)$ is a dual photon with $2\pi$-periodicity, related to the original field via Abelian duality relation as $F_{ij}\sim\varepsilon_{ijk}\partial_{k}\chi$. The monopole operator (’t Hooft type vertex for the monopole) (24) evidently respects the $\text{SU}(2)_{f}$ flavor symmetry. One can also check that (24) is invariant under the anomaly free discrete subgroup $(\mathbb{Z}_{8})_{A}$, which acts as $\psi\to e^{i2\pi/8}\psi$, and $\chi\to\chi+\pi$. The factor $G$ can be extracted from the monopole measure, see Sec. 4 of Ref. Argyres and Unsal (2012).121212Normalization of the fermion kinetic term in (23) is different from that in Ref. Argyres and Unsal (2012). Up to an $\mathcal{O}(1)$ numerical factor,

[TABLE]

where $g=g(L^{-1})$ is the renormalized coupling at the scale $L^{-1}$.

Magnetic bion formation at $\mu\neq 0$

In order to see mass gap generation for photons, one needs to go to the next order in the semiclassical expansion. The point is that due to the compactness of the $A_{3}$ holonomy there is an additional class of monopole-instanton (called KK monopoles) in $\mathbb{R}^{3}\times S^{1}$ Lee and Yi (1997); Lee and Lu (1998); Kraan and van Baal (1998a); *Kraan:1998sn.

There exists a topologically neutral, but magnetically charged combination of the BPS-monopole with KK-anti-monopole called magnetic bion Unsal (2008); *Unsal:2007jx, which induces the operator $(\cos 2\chi)$. Note that this operator is invariant under the action of discrete chiral symmetry $\chi\to\chi+\pi$. In some respect, this phenomenon is similar to the formation of instanton–anti-instanton molecules in thermal QCD Ilgenfritz and Shuryak (1994); Schafer et al. (1995), and in others it differs from it as the magnetic bion has still a topological quantum number for its magnetic charge. The physics of multi-instanton correlation in QCD with chemical potential has been studied in, e.g., Refs. Carter and Diakonov (1999); Schafer (1998); Rapp et al. (2000). In quark matter at high density, instantons of large sizes are exponentially suppressed due to the Debye screening of color-electric fields inside instantons Shuryak (1982). In contrast, this is of no concern here because the medium of $\Psi^{{\textsf{A}}=3}$ is transparent for monopoles.

Below, we would like to explicitly calculate the effect of chemical potential $\mu$ on the magnetic bion formation. The operator ${\cal B}(x)$ of a magnetic bion involves a connected correlator of two monopole operators Unsal (2008); *Unsal:2007jx; Argyres and Unsal (2012):

[TABLE]

with the repulsive Coulomb potential

[TABLE]

At distances $\gg L$ the correlator of ’t Hooft vertices in (26) can be computed with a free fermion propagator $S_{\rm F}(x;\mu)=\langle\psi_{1}(x)\psi_{1}^{\dagger}(0)\rangle$ and $S_{\rm F}(x;-\mu)=\langle\psi_{2}(x)\psi_{2}^{\dagger}(0)\rangle$, resulting in

[TABLE]

where $D_{\pm}(x)\equiv D(x;\pm\mu)$ is a bosonic propagator that solves the Klein-Gordon-type equation

[TABLE]

The explicit form of $D(x;\mu)$ and its properties are summarized in Appendix B. In the above we used (76) there. At $\mu=0$, $D\propto|x|^{-1}$ and the correlator (28) is just proportional to $|x|^{-8}$. However, once $\mu\neq 0$, there is anisotropy in space ($x^{1,2}$) and imaginary time ($x^{4}$); the weight (28) exhibits a Friedel-type oscillation in spatial directions, reflecting the presence of a sharp Fermi surface, which tends to render bions anisotropic.

Recalling that the three-dimensional Coulomb interaction is governed by the dimensionful coupling $g_{3}^{2}\equiv g^{2}/L$, it naturally follows that the strength of the bion deformation is measured by the ratio

[TABLE]

If we fix $\mu$ and let $L\to 0$, then $\widetilde{\mu}\sim\mu L\log\frac{1}{L}\to 0$, i.e., the effect of $\mu$ on bions inevitably disappears. In order to see a nontrivial deformation of bions one has to increase $\mu$ in the course of compactification so that $\mu\gtrsim\frac{1}{L\log\frac{1}{L}}$. At the same time, the condition $\mu<\frac{1.98}{L}$ must also be satisfied, in order to stay inside Phase III (cf. Fig. 2).

To simplify the $\widetilde{\mu}$-dependence of the weight we make all variables dimensionless in units of $g_{3}^{2}$:

[TABLE]

Using (25) we find that the bion amplitude is given by

[TABLE]

with a dimensionless function $W$ defined by

[TABLE]

The prefactors in (35) are in precise agreement with Refs. Anber and Poppitz (2011); Argyres and Unsal (2012) for the case of two Weyl fermions.

All the $\widetilde{\mu}$-dependence is now encapsulated in $w$. As displayed in Fig. 10(a), $w$ dramatically changes its profile as a function of $\widetilde{\mu}$. At $\widetilde{\mu}=0$, $w$ is isotropic with a sharp peak along the circle $\sqrt{\rho^{2}+\tau^{2}}=\pi/2\simeq 1.57$. This implies that the typical size of a magnetic bion is $\pi/(2g_{3}^{2})=\pi L/(2g^{2})$, as noted in Refs. Anber and Poppitz (2011); Argyres and Unsal (2012). As $\widetilde{\mu}$ grows, the weight of $w$ gradually moves to higher $\tau$, with a peak position at $\tau\sim 3$. At $\widetilde{\mu}=10$, $w$ is further focused on the $\tau$ axis with a quite narrow range of $\rho\ll 1$. This means that the magnetic bion tends to be rigidly oriented in the $x^{4}$ direction. One can also see a ripple-like pattern in the rightmost plot of Fig. 10(a), caused by a Friedel oscillation of $\widetilde{D}$.

Our numerical analysis indicates that the global maximum of $w$ is always located on the $\tau$ axis, so let us examine the behavior of $w$ at $\rho=0$ more closely. Using $\partial_{\rho}\widetilde{D}\to 0$ as $\rho\to 0$, one gets $w(0,\tau;\widetilde{\mu})=e^{-4\pi/\tau}[(\partial_{\tau}-\widetilde{\mu})\widetilde{D}_{+}]^{2}[(\partial_{\tau}+\widetilde{\mu})\widetilde{D}_{-}]^{2}$. Substituting the analytic form (80) of $\widetilde{D}$ in Appendix B one obtains

[TABLE]

for $\tau>0$. It shows that the peak of $w$ evolves from $\tau=\pi/2$ at $\widetilde{\mu}=0$ to $\tau=\pi$ at $\widetilde{\mu}\gg 1$. This behavior is illustrated in Fig. 10(b).

In summary, we have found two features of magnetic bions at $\mu\gtrsim\frac{1}{L\log\frac{1}{L}}$. First, the monopole–anti-monopole pair inside a bion is predominantly oriented in the imaginary-time direction. This is due to the fact that fermions with $\mu$ favor hopping in the temporal direction, which has been observed in the context of instantons in dense matter as well too Schafer (1998); Rapp et al. (2000). Secondly, the bion has a typical size $\pi/g_{3}^{2}=\pi L/g^{2}$. This is twice larger than that at $\mu=0$.

Semiclassical confinement at $\mu\neq 0$

If we replace the integral of $w$ with its peak value, we obtain a crude estimate

[TABLE]

It has an interesting implication for the dual photon mass. Let us recall that in addition to (35) there are also operators associated with anti-bions. Their sum reads

[TABLE]

This potential has minima at $\chi=0$ and $\pi$ associated with the spontaneous breaking of discrete chiral symmetry. (Note that an order parameter for the discrete chiral symmetry is $e^{i\chi}$ and the two vacua correspond to $\langle e^{i\chi}\rangle=\pm 1$.) At these points the dual photon $\chi$ acquires a nonperturbative mass gap, given by

[TABLE]

This means that the perturbatively massless photon is Debye-screened by the plasma of magnetic bions Unsal (2008); *Unsal:2007jx. The fact that $W(\widetilde{\mu})$ asymptotically increases with $\widetilde{\mu}$ implies that $\mu$ tends to enhance nonperturbative effect on the gauge field.

Using (39) and the one-loop $\beta$ function for $g$,

[TABLE]

one finds

[TABLE]

with a renormalization-group invariant scale $\Lambda$. The mass gap for the dual photon implies that the fundamental Wilson loop in $\mathbb{R}^{3}$ obeys area law; quarks are permanently confined. The string tension is given by $T\sim\frac{g^{2}}{L}M_{\chi}$ Polyakov (1977). The $L$ dependence follows from (42) and (43) as

[TABLE]

This estimate is valid for $\mu$ in the range

[TABLE]

III.3 Superfluidity

Generically a fermionic system at finite density can be unstable toward pair condensation if there is an attractive channel in their interaction. It is therefore important to ask what interactions can happen for $\psi_{1,2}$ in the effective theory (23). Although they do not couple to photons by themselves, they interact through two mechanisms of different physical origins. The first one is mediated by monopole-instantons. As was shown in the previous section the dual photons are screened at distances beyond $1/M_{\chi}$. As $\chi$ settles in one of the two minima $\{0,\pi\}$ of the potential (40),131313This breaks the anomaly-free axial $\mathbb{Z}_{8}$ down to $\mathbb{Z}_{4}$ spontaneously. one obtains from (24) a four-fermion operator

[TABLE]

Note that the 4d instanton would induce an 8-fermion operator, with much weaker coefficient (of order $G^{2}$). In this sense, the monopole-instantons are much more important than 4d instantons.

Yet another kind of interaction stems from integrating out heavy $\Psi^{\pm}$ and $W^{\pm}$ that are charged under $\text{U}(1)$. As an example, we show in Fig. 11 an effective interaction in the diquark channel where the 4-component Dirac spinor in the Cartan subalgebra $\Psi^{\textsf{A}=3}$ is notated by $\Psi$.

There are a myriad of operators that can be perturbatively generated this way, e.g., $(\overline{\Psi}\gamma^{3}\Psi)^{2}$, $(\overline{\Psi}\gamma^{i}\Psi)^{2}$ and $(\overline{\Psi}\gamma^{3}\gamma^{i}\gamma^{j}\Psi)^{2}$, to name but a few. They are all invariant under $\text{U}(1)_{A}$ and hence do not couple to the dual photon. Here we do not systematically enumerate all possible forms of the interaction, but instead try to capture an essential physical outcome by considering the minimal interaction in the (pseudo-)scalar channel

[TABLE]

with a charge-conjugation matrix $C=\operatorname{diag}(-\sigma^{2},\sigma^{2})$ and $\gamma_{5}=\operatorname{diag}(\mathbbm{1}_{2},-\mathbbm{1}_{2})$. This interaction, being perturbatively induced, preserves $\text{U}(1)_{A}\times\text{SU}(2)_{f}$ Zhang et al. (2010). The coupling $H$ should be $\propto L$ according to dimensional analysis, and it must be multiplied by $g^{4}$ because there are at least four bare vertices in the process of Fig. 11. Thus

[TABLE]

Since $G$ [cf. (25)] is proportional to $e^{-4\pi^{2}/g^{2}}\ll 1$, (48) means that $|H|\gg G$.

The sign of $H$ is not fixed by symmetries alone. In this regard we would like to note that the process of Fig. 11 is of the same kind as the Van der Waals interaction in QED, which originates from the two-photon exchange and is attractive because it arises at second order in perturbation theory Landau and Lifshits (1991). Similar long-range force between color-singlet hadrons in QCD, originating from the two-gluon exchange process, is also known to be attractive Fishbane and Grisaru (1978); Appelquist and Fischler (1978); Fujii and Kharzeev (1999). Therefore in the following analysis we will assume $H>0$ and pursue its physical consequences, leaving a microscopic verification to future work. We remark that the phase diagram shown earlier as Fig. 1 was also based on the assumption that $H>0$.

Fermionic effective theory

Incorporating both types of interactions [(46) and (47)], and dropping the quark-antiquark vertices because they are not relevant to the BCS instability, we end up with the fermionic effective theory

[TABLE]

where the sign in front of $G$ depends on whether $\langle\chi\rangle=0$ or $\pi$. Analogous four-fermion models in three and four dimensions have been studied before Nambu and Jona-Lasinio (1961a); *Nambu:1961fr; Rosenstein et al. (1989); *Rosenstein:1990nm. We note that while the models in Refs. Nambu and Jona-Lasinio (1961a); *Nambu:1961fr; Rosenstein et al. (1989); *Rosenstein:1990nm were introduced on phenomenological grounds, (49) is a faithful description of the microscopic dynamics of adjoint QCD in the circle-compactification limit.

A remark on the parity of the ground sate is in order. Because the second line of (49) does not distinguish between $0^{+}$ and $0^{-}$ diquarks, one shall look at the monopole-induced operator. With the help of the relation

[TABLE]

we observe that interaction in the $0^{+}$ channel is stronger than $0^{-}$ for $\langle\chi\rangle=0$. However the opposite happens in the other vacuum $\langle\chi\rangle=\pi$. Thus, after all, the effective theory (49) per se does not uniquely determine the parity of the ground state, and prompts us to add a small but nonzero external source field to resolve the ambiguity. If one inserts to the action a real mass term $m\overline{\Psi}\Psi$ with an arbitrarily small $m$, the ambiguity is resolved.141414In QCD-like theories at finite density Kogut et al. (1999); Son and Stephanov (2001); Kogut et al. (2000), the parity of the diquark or pion condensate was automatically determined because the parity of the ground state at zero chemical potential was fixed by a real mass term. There are two ways to check this. First, one can invoke QCD inequalities generalized to $\mu\neq 0$ Kogut et al. (1999); Cherman and Tiburzi (2011); Kanazawa et al. (2011) to argue that $\Psi^{\rm T}C\gamma_{5}\Psi$ is the lightest bilinear field.151515This observation lends support to our strategy to consider a simplified effective theory (47) from the very beginning. We stress that this argument works equally in both $\mathbb{R}^{4}$ and $\mathbb{R}^{3}\times S^{1}$. The second argument to check parity only works in the semiclassical small-$L$ regime. Namely, when the above sources are present in the action, they will absorb the fermionic zero modes inhabiting the monopoles and allow for a bosonic potential $\sim m^{2}\cos\chi$ to be generated on top of the bion-induced potential (40). This potential, however small in magnitude, can and do lift the degeneracy between $\chi=0$ and $\pi$, and consequently fix the parity of the ground state.

From the consideration above, one may assume that the interaction is stronger for the $0^{+}$ channel than for $0^{-}$, tacitly assuming the presence of a suitable infinitesimal external field in the action. Now, since the monopole-induced interaction is nonperturbatively small, it can be safely dropped, and we arrive at a simpler theory that sufficiently serves our purpose of probing the ground state of the fermionic sector:

[TABLE]

Gap equation

We solve the theory in the mean-field approximation following the standard procedure Hatsuda and Kunihiro (1994); Buballa (2005). This is expected to be accurate, given the smallness of the coupling $H\mu\propto g^{4}\hat{\mu}\ll 1$. The Hubbard–Stratonovich transformation applied to (51) with an auxiliary complex field $\Delta(x)$ yields

[TABLE]

With a suitable $\text{U}(1)_{B}$ rotation, one can take $\Delta\geq 0$ without loss of generality. In the Nambu–Gor’kov basis $(\Psi,C\overline{\Psi}^{\rm T})$ we have for the energy density

[TABLE]

The final integral with a prime is to be regularized with a cutoff. Then the gap equation $\partial U(\Delta)/\partial\Delta=0$ for a nontrivial solution $\Delta\neq 0$ is given by

[TABLE]

As $\Delta\to 0$ the first term yields IR divergence at the Fermi surface. Retaining only the first term, and imposing a momentum cutoff $\Lambda^{\prime}\,(>\mu)$, one obtains

[TABLE]

or

[TABLE]

Physically, this means that fermions acquire an energy gap $\Delta$ and the $\text{U}(1)_{B}$ symmetry is spontaneously broken by the condensate $\langle\Psi^{\rm T}C\gamma_{5}\Psi\rangle\neq 0$; the system is in a superfluid phase.161616Since $\Psi$ does not couple to the $\text{U}(1)$ photon, this phase is not a superconductor. We stress that the parity of the diquark condensate is the same at small $L$ and large $L$ [recall our remarks below (50)]. Note that in this analysis the density of fermions need not be high; the computation in this section is valid for any $\mu>0$ as long as we are in Phase III ($\mu<1.98/L$), which means that superfluidity may take place at arbitrarily small $\mu$. This is a remarkable result.

Plugging (48) and $\Lambda^{\prime}\sim L^{-1}$ into (56), we gain the crude estimate

[TABLE]

Notably, this ultrasoft scale is far smaller than the dual photon mass $M_{\chi}\propto L^{-1}\exp(-4\pi^{2}/g^{2})$ and is invisible at any finite order of the semiclassical expansion. Since the extent of the Cooper pair $1/\Delta$ far exceeds the typical bion size, the diquark condensate is not expected to modify the fermion-binding mechanism of bions in Sec. III.2.

The various length scales in Phase III may be summarized as follows:

[TABLE]

($r_{\rm m}$: monopole size; $d_{\psi\text{-}\psi}$: interquark distance; $r_{\rm b}$: bion size; $d_{\rm m\text{-}m}$: intermonopole distance; $d_{\rm b\text{-}b}$: interbion distance; $1/M_{\chi}$: inverse Debye screening length; $1/\Delta$: Cooper-pair size; $S_{\rm m}=4\pi^{2}/g^{2}$: the monopole action).

Low-energy effective theory for superfluid phonons

In the far-infrared limit at energies below $\Delta$, the effective degree of freedom is phonon, i.e., the Nambu–Goldstone mode associated with the baryon number. It emerges as a phase fluctuation of the condensate and can be introduced as $\Delta\to\Delta e^{2i\phi(x)}$. The low-energy effective theory of phonons can be derived by expanding the functional determinant of $\Psi$ in powers of the derivative of $\phi$ (the so-called gradient expansion Eguchi (1976)). The result is

[TABLE]

with

[TABLE]

The factor of 2 in the denominator of $v^{2}$ reflects the dimensionality of space. The effective theory (62) can also be derived from the equation of state $P=\mu^{3}/(6\pi)$ on the basis of symmetry arguments along the lines of Ref. Son . Equation (63) should be contrasted with $f^{2}\sim\mu^{2}$ and $v^{2}=1/3$ for the Nambu–Goldstone modes in high-density quark matter in four dimensions Son and Stephanov (2000a); *Son:2000tu; Beane et al. (2000); Fukushima and Iida (2005).

IV Conclusion

In this paper we extended the bion mechanism of confinement Unsal (2008); *Unsal:2007jx to nonzero quark chemical potential $\mu$ in adjoint QCD with spatial compactification. In the first part, we performed a perturbative analysis of Wilson line potential (whose result gives a realization of Hosotani mechanism) in the presence of $\mu$ and revealed a rich phase diagram in the space of $\mu$ and $m$. In the second part we studied the $\mu$-dependence of semiclassical configurations (monopoles and their molecules called magnetic bions) in the center-symmetric phase. In addition to the Coulomb interaction, monopole-instantons also talk to each other via a fermion zero mode exchange, which at $\mu\neq 0$ is modified due to the anisotropic hopping amplitude of fermions. Consequently, bions favor a temporal orientation and its amplitude grows with $\mu$, leading to a larger mass gap of photons and a greater string tension. Intriguingly, neutral massless fermions that are free in the leaing-order perturbation theory may exhibit novel superfluidity triggered by the combination of perturbatively induced four-fermion operators and nonperturbatively induced monopole operators (’t Hooft vertex). The analysis in this paper remains valid at any $\mu\neq 0$ as long as the compactified direction $S^{1}$ is small enough. It would be interesting to examine the dimensional crossover from small to large $S^{1}$ (Fig. 1) in future lattice simulations, to clarify how the BEC-BCS crossover region in $\mathbb{R}^{4}$ is connected to the exotic phase structure in $\mathbb{R}^{3}\times S^{1}$ found in this work. Dimensional crossover of an interacting Fermi gas has been actively investigated in the condensed matter physics community Valla et al. (2002); *PhysRevA.67.031601; *PhysRevLett.106.105304; *PhysRevA.84.053624; *PhysRevLett.108.045302; *PhysRevA.88.023612; *PhysRevLett.112.045301; *Toniolo2017 and it is of great interest to see what new physics will emerge for the case of relativistic quark matter.

Acknowledgements.

We are grateful to A. Cherman and T. Sulejmanpasic for valuable comments on the manuscript. T. K. was supported by the RIKEN iTHES project. N. Y. was supported by JSPS KAKENHI Grant No. 16K17703 and MEXT-Supported Program for the Strategic Research Foundation at Private Universities, “Topological Science” (Grant No. S1511006). M. Ü. was supported by the DOE under Grants No. DE-SC0013036.

Appendix A One-loop effective potential

Below we will outline the derivation of the formula (17). Let us momentarily put the system in a box of linear extent $L_{\perp}\times L_{\perp}\times L\times\beta$ with $\beta$ the inverse temperature and $L_{\perp}$ the length in the $x^{1,2}$ directions. The Euclidean Dirac operator in the background (9) reads

[TABLE]

where $D_{3}^{\text{ad}}$ is the adjoint covariant derivative that acts on a matrix test field $v=(v_{\mathtt{a}\mathtt{b}})$ as

[TABLE]

Since each component of $v$ except for its trace can be regarded as independent, we get Gross et al. (1981)

[TABLE]

where $p_{\perp}^{2}\equiv p_{1}^{2}+p_{2}^{2}$,  $\bm{p}^{2}\equiv p_{\perp}^{2}+p_{3}^{2}$,  $\displaystyle p_{3}=\frac{2n\pi}{L}(n\in\mathbb{Z})$,  $\displaystyle p_{4}=\frac{(2\ell+1)\pi}{\beta}(\ell\in\mathbb{Z})$,  ${\cal E}\equiv\sqrt{\bm{p}^{2}+m^{2}}$ and

[TABLE]

The summation over $p_{4}$ can be taken with the standard formula Kapusta and Gale (2006)

[TABLE]

where an $\omega$-independent divergence has been subtracted. We thus obtain

[TABLE]

and

[TABLE]

The remaining integral may be done with the formula

[TABLE]

with the result

[TABLE]

Substituting $\displaystyle p_{3}=\frac{2n\pi}{L}(n\in\mathbb{Z})$ into this equation, we finally arrive at (17).

Appendix B Free propagators with $\mu\neq 0$ in three dimensions

We solve for the Euclidean propagators in $2+1$ dimensions in the presence of $\mu$. The fermion propagator $S_{\rm F}$ solves the equation

[TABLE]

The boson propagator solves the Klein-Gordon equation

[TABLE]

They are related through

[TABLE]

It is easy to verify that

[TABLE]

Thus one can assume $\mu>0$ without loss of generality. Writing $r\equiv\sqrt{x_{1}^{2}+x_{2}^{2}}$ , we have

[TABLE]

An alternative expression which has no singularity at $r=0$ can also be obtained by integrating over $p_{4}$ first, with the result

[TABLE]

These formulas give access to limiting behaviors of $D$ in some cases of interest:

• •

For $\mu=0$,

[TABLE]

• •

For $r=0$,

[TABLE]

• •

For $x_{4}=0$,

[TABLE]

• •

For $\mu r\gg 1$ with any $x_{4}$,

[TABLE]