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Springer

Received: June 8, 2020 Accepted: July 7, 2020 Published: August 27, 2020

Andrew Baumgartner Department of Physics, University of Washington, 3910 15th Ave NE, Seattle, WA, 98195-1560, U.S.A.

E-mail: [email protected] Abstract: We examine the vacuum structure of QCD3 with flavor group U (f )×U (Nf −f ) in the limit N → ∞ with g 2 N =fixed. We find that, generically, the resolution of critical points into a series of first order pahse transitions persists at special locations in the phase diagram. In particular, the number of Grassmannians that one traverses and their locations in the phase diagram is a function of f . Keywords: 1/N Expansion, Chern-Simons Theories, Field Theories in Lower Dimensions ArXiv ePrint: 2005.11339

c The Authors. Open Access, Article funded by SCOAP3 .

https://doi.org/10.1007/JHEP08(2020)145

JHEP08(2020)145

Flavor broken QCD3 at large N

Contents 1 Introduction

1

2 Symmetry breaking in QCD3 and vacuum structure at large N 2.1 Symmetry breaking with unequal masses 2.2 Vacuum structure at large N

3 4 5 9 9 10 10 11 13 16

4 Scalar potentials and double condensation

17

5 Conclusion

19

A Numerical evidence for the phase diagram

20

1

Introduction

In the past couple years a large body of work on 2+1 dimensional gauge theories has revealed a panoply of fascinating physics. Partially spurred by advances in Chern-Simons matter theories [1–6], higher spin gravity [7, 8], emergent gauge field [9] and supersymmetric dualities [10, 11], this research program has shed light on important concepts in particle, string and condensed matter physics (see [12–36] for a subset of this work). A large portion of these insights stem from the existence of the non-Abelian Chern-Simons term:   Z k 2i SCS = Tr A ∧ dA + A ∧ A ∧ A (1.1) 4π 3 where the level k is quantized to ensure gauge invariance and Tr refers to trace over the gauge group (we will be primarily concerned with SU (N )). This term generically dominates the dynamics at low energies due to the single derivative. Equation (1.1) has a long and storied history in theoretical and mathematical physics (see the review [37] and the references therein), but has recently garnered attention due to its ubiquity in emergent 2+1 d gauge theories in condensed matter physics. Indeed, eq. (1.1) is unavoidable in any planar theory of fermions since fermions of mass m dynamically generate a Chern-Simons term with level sgn(m)/2 at one loop.

–1–

JHEP08(2020)145

3 Vacuum structure with an explicitly broken flavor group 3.1 Leading order 3.2 Next-to-leading order 3.2.1 A lack of double condensation 3.2.2 Single massive flavor 3.2.3 Double massive flavors 3.2.4 Matching onto the finite N solution

An interesting story arises when one studies eq. (1.1) coupled to massless fermions. One finds the surprising result that for certain parameter regimes (namely when the level k ≥ Nf /2 where Nf is the number of fermions)1 there exists a duality [43]:   Nf SU (N )k with Nf ψ ↔ U +k with Nf φ (1.2) 2 −N

N

Between these critical points exists a NLσM gi