Cascade of phase transitions in a planar Dirac material
PPrepared for submission to JHEP
Cascade of phase transitions in a planar Diracmaterial
Takuya Kanazawa, a Mario Kieburg b and Jacobus J.M. Verbaarschot c a Research and Development Group, Hitachi, Ltd.,Kokubunji, Tokyo 185-8601, Japan b School of Mathematics and Statistics, University of Melbourne,Parkville, Melbourne VIC 3010, Australia c Department of Physics and Astronomy, Stony Brook University,Stony Brook, NY 11794, U.S.A.
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We investigate a model of interacting Dirac fermions in dimensionswith M flavors and N colors having the U( M ) × SU( N ) symmetry. In the large- N limit,we find that the U( M ) symmetry is spontaneously broken in a variety of ways. In thevacuum, when the parity-breaking flavor-singlet mass is varied, the ground state undergoesa sequence of M first-order phase transitions, experiencing M + 1 phases characterizedby symmetry breaking U( M ) → U( M − k ) × U( k ) with k ∈ { , , , · · · , M } , bearing a closeresemblance to the vacuum structure of three-dimensional QCD. At finite temperature andchemical potential, a rich phase diagram with first and second-order phase transitions andtricritical points is observed. Also exotic phases with spontaneous symmetry breaking ofthe form as U(3) → U(1) , U(4) → U(2) × U(1) , and U(5) → U(2) × U(1) exist. For a largeflavor-singlet mass, the increase of the chemical potential µ brings about M consecutivefirst-order transitions that separate the low- µ phase diagram with vanishing fermion densityfrom the high- µ region with a high fermion density. a r X i v : . [ h e p - t h ] F e b ontents √ π < | (cid:101) g | < (cid:101) g tri2 ( γ tri2 < γ < ) 124.3.2 Low temperature regime with (cid:101) g tri2 < | (cid:101) g | < (cid:101) g cr2 ( γ cr2 < γ < γ tri2 ) 134.3.3 Low temperature regime with | (cid:101) g | > (cid:101) g cr2 ( γ < γ cr2 ) 15 T and µ
227 Conclusions and outlook 25A Phase diagram of the effective potential and a toy model 26
A.1 Saddle point equation and its asymptotic solutions 27A.2 Local extrema of g ( e ) and implications on the possible phases 27A.3 Cascade of phase transitions 28A.3.1 No-go statement for second order phase transitions 30A.4 Phase diagram of the toy model for M = 2 M > B Some computations of Sec. 3.2 35C Comment on the role of bosonic fluctuations 37
Dirac fermions play a central role in physics – not only in elementary particle physicsbut also in condensed matter physics [1–4]. Interactions of Dirac fermions are essential indetermining the ground state of various physical systems, and models with quartic inter-actions have been studied for decades in a variety of fields. For instance, in nuclear and– 1 –adron physics, the Nambu-Jona-Lassinio (NJL) model [5, 6] is famous as a phenomeno-logical effective theory of QCD [7, 8]. Lower-dimensional four-fermion models such as theGross-Neveu model in dimensions [9] have also played a pivotal role in advancing ourunderstanding of phenomena like dynamical symmetry breaking, asymptotic freedom anddimensional transmutation. Recently there are renewed interests in Dirac fermions in dimensions. They appear in some condensed matter systems [10–20] and understandingthe effects of interactions is therefore imperative. Historically, four-fermion models of Diracfermions in dimensions have been thoroughly studied both analytically [21–35] andby numerical simulations [36–43], and intriguing features such as superfluidity, Kosterlitz-Thouless transitions, non-Gaussian Ultra-Violet (UV) fixed points and magnetic catalysishave been elucidated. These studies have provided a tractable avenue for understandingnonperturbative aspects of (2 + 1) -dimensional strongly coupled gauge theories, includingQED and QCD as prominent examples.Recently QCD has experienced a flurry of revived attention [44–52]. In [49] the presentauthors have proposed a new random matrix theory (RMT) which, when random matrixelements are integrated out, reduces to a four-fermion model that spontaneously breakssymmetries in exactly the same way as does QCD with a Chern-Simons term [44], thusextending the previous work [53]. Although RMT is a zero-dimensional theory with nogauge interactions, it provides exact descriptions of the low-lying Dirac spectrum owing tothe universality of the microscopic domain [54–58].In this work, we study thermodynamics and symmetry breaking of an unconventionalinteracting model of Dirac fermions in dimensions at finite temperature and chemicalpotential in the large- N limit, where N denotes the number of “colors.” Each fermioncomes in M different flavors. This model can be viewed as a generalization of the RMTproposed in [49]. The model has three key ingredients: a repulsive interaction, an attractiveinteraction, and a flavor-symmetric parity-breaking mass term. Their interplay leads to asurprisingly rich phase diagram. At zero temperature and zero density, the model exhibitsa spontaneous symmetry breaking patterns U ( M ) → U ( M − k ) × U ( k ) with various k and experiences a sequence of first-order phase transitions, bearing a close resemblanceto three-dimensional QCD [44, 51]. The model reduces to a sigma model on a complexGrassmannian at low energy. At nonzero temperature or chemical potential, there appeareven more exotic phases where the symmetry is broken as U (3) → U (1) × U (1) × U (1) ,U (4) → U (2) × U (1) × U (1) , and U (5) → U (2) × U (2) × U (1) , to name but a few. All thesepatterns show up in a single model with a few adjustable parameters.The present work is structured as follows. In section 2, the model is defined and thethermodynamic potential is derived. In section 3, the ground state at zero temperatureand density is analyzed. In section 4 the effect of nonzero temperature is considered. Insection 5, a nonzero chemical potential is introduced, and the fermion number density iscalculated. In section 6, phases at nonzero temperature and density are studied. It is shownthat the phase structure changes dramatically, depending on the interaction strength andthe flavor-singlet mass. We conclude in section 7, and technical details are worked outin several appendices. Throughout this article we will work in the natural units where (cid:126) = c = k B = 1 and with Einstein’s summation convention where we sum over repeated– 2 –ndices. We consider a system of two-component Dirac fermions ψ isα in dimensions. Here α = 1 , are spinor indices, i = 1 , · · · , N are color indices and s = 1 , · · · , M are flavorindices. The Lagrangian in the Euclidean spacetime is given by L = ψ is ( σ ν ∂ ν + κ − µσ ) ψ is + g N ( ψ is ψ is ) − g N ( ψ is ψ is (cid:48) )( ψ js (cid:48) ψ js ) , (2.1)where σ ν = ( σ , σ , σ ) are the Pauli matrices in spinor space. The couplings have dimen-sions [ g ] = [ g ] = − / . The Lagrangian L is invariant under U (1) × SU ( N ) × SU ( M ) transformations of ψ . The mass term κψψ breaks parity symmetry, and µ is the baryonchemical potential. The four-fermion interactions of the form (2.1) arise in the random ma-trix model proposed in [49] which also gives the sign of the interaction terms. We underlinethat these signs are essential for the results of the present work.To rephrase the four-fermion terms in two quadratic ones, we perform the Hubbard-Stratonovich transformation and obtain Z = (cid:90) D ( ψ, ψ, φ, Φ) exp (cid:18) − (cid:90) β d τ (cid:90) d x (cid:101) L (cid:19) (2.2)with β = 1 /T the inverse temperature and the Lagrangian (cid:101) L = ψ is ( σ ν ∂ ν + κ − µσ + 2 ig φ + 2 g Φ) ss (cid:48) ψ is (cid:48) + N ( φ + Tr Φ ) , (2.3)where φ is a scalar field and Φ is a Hermitian M × M matrix field, i.e., Φ † = Φ . Fermionscan now be integrated out, yielding Z = (cid:90) D φ (cid:90) D Φ det N ( σ ν ∂ ν + κ − µσ + 2 ig φ + 2 g Φ) exp (cid:26) − N (cid:90) d τ d x ( φ + Tr Φ ) (cid:27) . (2.4)Next, we introduce a shifted field Φ (cid:48) ≡ Φ + ig g φ M + κ g M to obtain Z = (cid:90) D φ (cid:90) D Φ (cid:48) det N ( σ ν ∂ ν − µσ + 2 g Φ (cid:48) ) × exp (cid:34) − N (cid:90) d τ d x (cid:40) φ + Tr (cid:18) Φ (cid:48) − ig g φ M − κ g M (cid:19) (cid:41)(cid:35) . (2.5)Assuming that the condition g > M g is fulfilled, the integral over the φ field can becarried out and leads to the result Z ∝ (cid:90) D Φ (cid:48) det N ( σ ν ∂ ν − µσ + 2 g Φ (cid:48) ) × exp (cid:20) − N (cid:90) d τ d x (cid:26) g g − M g (cid:0) Tr Φ (cid:48) − m (cid:1) + Tr Φ (cid:48) (cid:27)(cid:21) (2.6)– 3 –ith m ≡ g κ g . (2.7)After substituting this into (2.4) and the shifted field Φ (cid:48) ≡ Φ + ig g φ M + κ g M we arriveat (2.6). Alternatively, the Gaussian integral over φ in equation (2.4) can also be evaluatedfrom the saddle point equation in φ with the saddle point φ = ( ig /g ) Tr Φ .In the large- N limit the partition function is dominated by saddle points of the effectivepotential V eff (Φ (cid:48) ) = g g − M g (cid:0) Tr Φ (cid:48) − m (cid:1) + Tr Φ (cid:48) − TL log det( σ ν ∂ ν − µσ + 2 g Φ (cid:48) ) (2.8)where L is the linear extent of the plane. Assuming a constant field Φ (cid:48) ( τ, x , x ) = Φ (cid:48) wefind V eff (Φ (cid:48) ) = g g − M g (cid:0) Tr Φ (cid:48) − m (cid:1) + Tr Φ (cid:48) − T (cid:90) d p (2 π ) ∞ (cid:88) n = −∞ tr log[ ip σ + ip σ + ( iω n − µ ) σ + 2 g Φ (cid:48) ] , (2.9)where ω n = (2 n + 1) πT and tr is the trace over the spinor and flavor indices. Next, weperform the diagonalization Φ (cid:48) = U EU † with E = diag( E , · · · , E M ) and combine termswith n ≥ and n < to get V eff ( E ) = g g − M g (cid:32) M (cid:88) k =1 E k − m (cid:33) + M (cid:88) k =1 E k − T M (cid:88) k =1 (cid:90) d p (2 π ) (cid:34) ∞ (cid:88) n = −∞ log (cid:40) β ω n + β (cid:18)(cid:113) p + 4 g E k + µ (cid:19) (cid:41) + ∞ (cid:88) n = −∞ log (cid:40) β ω n + β (cid:18)(cid:113) p + 4 g E k − µ (cid:19) (cid:41) (cid:35) . (2.10)We have included a factor β in the argument of the logarithm which just amounts to anoverall normalization constant. Finally, we use the standard formula for summation overMatsubara frequencies [59, 60] ∞ (cid:88) n = −∞ log (cid:18) β ω n + z β ω n (cid:19) = z + 2 log(1 + e − z ) − (2.11)to obtain V eff ( E ) = g g − M g (cid:32) M (cid:88) k =1 E k − m (cid:33) + M (cid:88) k =1 E k − M (cid:88) k =1 (cid:90) d p (2 π ) (cid:40)(cid:113) p + 4 g E k + T log (cid:20) − β (cid:16) √ p +4 g E k + µ (cid:17) (cid:21) + T log (cid:20) − β (cid:16) √ p +4 g E k − µ (cid:17) (cid:21) (cid:41) . (2.12) This change of variables yields a Jacobian (cid:81) ≤ i
T > and µ = 0 can be evaluated analytically. Absorbing Λ in T → Λ / T , we have V eff ( E )Λ = (cid:101) g (cid:101) g − M (cid:101) g (cid:32) M (cid:88) k =1 e k − (cid:101) λ (cid:33) + M (cid:88) k =1 (cid:40) e k + 43 π | (cid:101) g e k | − π (1 + 4 (cid:101) g e k ) / + T π (cid:20) | (cid:101) g e k | T Li (cid:16) − e − | (cid:101) g e k | /T (cid:17) + Li (cid:16) − e − | (cid:101) g e k | /T (cid:17)(cid:21) (cid:41) , (4.1)where Li s ( z ) = ∞ (cid:88) k =1 z k k s is the polylogarithm function. In the notation of (3.6), we find (cid:98) V eff ( e ) = γ (cid:32) M (cid:88) k =1 e k − λ (cid:33) + M (cid:88) k =1 (cid:26) γ e k + | e k | − (1 + e k ) / + 6 T (cid:20) | e k | T Li (cid:16) − e −| e k | /T (cid:17) + Li (cid:16) − e −| e k | /T (cid:17) (cid:21)(cid:27) . (4.2)Thus, the saddle point equation becomes − s = g ( e ) with g ( e ) = 2 γ e + 3 e (cid:16) | e | − (cid:112) e (cid:17) + 6 T e log (cid:16) e −| e | /T (cid:17) (4.3)and s = γ ( (cid:80) Mk =1 e k − λ ) the same as before. The possible shapes are depicted in the insetsof figure 6.Depending on the temperature and γ we can distinguish 4 different domains dependingon the maximum number of different solutions of g ( e ) = − s , see insets in figure 6. Thedomains are separated by the following curves: This series is convergent for | z | < . The values for | z | ≥ are defined by analytic continuation. – 10 – ( e ) e 00 g ( e ) e 00 g ( e ) e00 g ( e ) e ( γ , T cr ) γ , T tri III IIIIV γ T Figure 6 . Phase diagram of g ( e ) . For each region, labeled by Roman numerals, we included aninset with the shape of g ( e ) = v (cid:48) ( e ) . i) The vanishing of the slope at e = 0 (blue line in figure 6), g (cid:48) ( e = 0) = − γ + 6 T log 2 = 0 . (4.4)Because the asymptotic behavior of g ( e ) ≈ γ e , if the slope at 0 is negative g ( e ) cannot be a monotonic function, and the equation g ( e ) = − s can have three solutionsfor T < (3 − γ ) / (6 log 2) .ii) The curve in the ( γ , T ) plane (red curve in figure 6) with g ( e ) = g (cid:48) ( e ) = 0 (4.5)separates region II and region IV. At those points a minimum of g ( e ) touches the e -axes, and because g ( e ) is an odd function, the equation g ( e ) = − s can have 5possible real solutions in the region IV.iii) The curve in the ( g , T ) plane when a minimum and a maximum of g ( e ) coincide(green curve in figure 6) is given by g (cid:48) ( e ) = g (cid:48)(cid:48) ( e ) = 0 . (4.6)It indicates definitely a phase transition that splits the region above curve i) and ii)into regions I and II. In region I the function g ( e ) increases monotonously, while inregion II the equation g ( e ) = − s can have at most three solutions despite g ( e ) hastwo local minima and two local maxima.– 11 –t the tricritical point, the potential which had three minima in the region V, joins thepotential in regions I and II, with one and two minima, respectively. Because the potentialis even, this has to happen at e = 0 . The condition for the tricritical point is thus g (cid:48) ( e = 0) = − γ + 6 T log 2 = 0 , (4.7) g (cid:48)(cid:48) ( e = 0) = −
32 + 34 T = 0 , (4.8)which is solved by ( γ tri2 , T tri ) = (cid:18)
32 (1 − log 2) , (cid:19) . (4.9)A second special point in the ( γ , T ) plane is the point on the curve g (cid:48) ( e ) = g (cid:48)(cid:48) ( e ) = 0 where d g d T (cid:12)(cid:12)(cid:12)(cid:12) γ cr2 ,T cr = 0 . (4.10)This point is at γ cr2 = 0 . with T cr = (3 − γ ) / (6 log 2) . For γ < γ cr2 the system alwaysexperiences a cascade of phase transitions when varying λ . At sufficiently high temperatures and fixed γ > , see region II in figure 6, the curve g ( e ) shows a “wiggle” (a local maximum followed by a minimum) for large | e | . Taking intoaccount the arguments of [49] and the discussion in Appendix A.3, we expect a cascade ofphase transitions for sufficiently large | (cid:98) λ | . This is indeed observed numerically, see the plotsin figures 9 as well as 10 for M = 2 , , . It shows as a strip which obeys approximately alinear relation between T and λ . The cascade of symmetry breaking patterns are those ofU ( M ) → U ( j ) × U ( M − j ) where j = 0 , , . . . , M − changes by .In Appendices A.3 and A.3.1 we have argued that all phase transitions for a locallydouble well shaped potential have to be of first order for M ≥ . For M = 2 , a secondorder phase transition is possible, but our numerics confirm that at high temperature alltransitions are first order. As we will see in the next section, a second order phase transitiondoes occur for M = 2 at lower temperatures. At low temperature
T < (3 − γ ) / (6 log 2) and γ < (region III in figure 6) we find a g ( e ) in the shape of a wiggle, this time about the origin. When increasing the temperaturewe encounter three scenarios depending on the value of γ which will be discussed in thenext three subsections. √ π < | (cid:101) g | < (cid:101) g tri2 ( γ tri2 < γ < ) For γ tri2 < γ < , the function g ( e ) becomes a strict monotonously increasing functionfor T > (3 − γ ) / (6 log 2) , as we already have seen at T = 0 for γ > (or | (cid:101) g | < √ π ).The second order phase transition on the curve T = (3 − γ ) / (6 log 2) is at λ = 0 . For– 12 – .3 0.2 0.1 0.0 0.1 0.2 0.30.10.20.30.40.50.6 TCPTCP (cid:101) λT (a) (b) Figure 7 . (a) The phase diagram for M = 2 at µ = 0 with (cid:101) g = 1 and (cid:101) g = 3 in the large- N limit. The magnitude of | e − e | is plotted. (b) A simplified sketch of (a). There are two tricriticalpoints (TCP). The dashed line denotes a second-order transition and the thick solid line a first-ordertransition. We have omitted the high temperature phase diagram where a strip of first order phasetransitions starts at about T = 1 . and (cid:101) λ = 6 . . the parameters of figure 7 this gives a critical temperature of T = 0 . . For M = 2 the transition remains of second order until the tricritical point in the ( γ , T ) plane. Thelocation of this point can be extracted numerically. For (cid:101) g = 3 ( γ = 0 . ) it is at ( (cid:101) λ, T ) = ( ± . , . . There is another second order phase transition point in the ( γ , T ) plane at T = 1 . (not displayed in figure 7). This is the starting point of a stripof two close first order phase transitions in the ( (cid:101) λ, T ) plane and begins at (cid:101) λ = 6 . . Thefirst order transitions are from a phase with unbroken flavor symmetry to a phase withU (1) × U (1) breaking and back to the unbroken phase.Adopting the order parameter (cid:80) Mk =2 ( − k ( e k − − e k ) (assuming e ≥ e ≥ · · · e M ) thephase diagram for M = 2 in the ( (cid:101) g , T ) plane is mapped out in figure 7. The second orderline extends from λ = − λ tri to λ = λ tri . We have omitted the high temperature regime inthis figure which contains the strip of first order phase transitions. It will be discussed inmore detail in the next subsection.As is discussed in Appendix A.3.1 for M ≥ there is no line of second order phasetransitions in the ( (cid:101) λ, T ) plane and the only second order point at (cid:101) λ = 0 .Moreover, we expect a cascade of phase transitions between phases with the symmetrybreaking pattern U ( M ) → U ( j ) × U ( M − j ) with j = 0 , , . . . , M , as in the high temperaturephase, the system runs through all possible j from j = 0 to j = M when increasing (cid:98) λ . Wehave corroborated this by numerical minimization of the potential (4.1) for M = 3 and M = 4 (see figure 8) where in both cases we have chosen (cid:101) g = 3 < | (cid:101) g tri2 | . Again we did notconsider the high temperature phase. (cid:101) g tri2 < | (cid:101) g | < (cid:101) g cr2 ( γ cr2 < γ < γ tri2 ) In this regime we have a richer phase diagram which is mapped out in figure 9 using e − e as an order parameter. The most notable feature is that the strip with the cascade of phase– 13 – .6 0.4 0.2 0.0 0.2 0.4 0.60.10.20.30.40.50.6 CP (cid:101) λT (a) (b) CP (cid:101) λT (c) (d) Figure 8 . (a) The phase diagram for M = 3 at µ = 0 with (cid:101) g = 1 and (cid:101) g = 3 in the large- N limit.The value plotted is e − e + e , where the ordering e ≥ e ≥ e is assumed. (c) The phasediagram for M = 4 at µ = 0 with (cid:101) g = 1 and (cid:101) g = 3 in the large- N limit. The value plotted is e − e + 2 e − e , where the ordering e ≥ e ≥ e ≥ e is assumed. The diagrams (b) and (d)are simplified versions of figures (a) and (c). There is a critical point (CP) at which all first-ordertransition lines meet. Again we have omitted the high temperature regime. transitions is split into two pieces. The strips end in second order points that are visible inthe ( γ , T ) plane as the two transitions from region I to region II. The two parts join eachother at (cid:101) g = (cid:101) g cr2 . The function g ( e ) is shown in figure 9 for three different temperatures, T = 0 . corresponding to the lower part of the strip (green dotted curve), T = 0 . inbetween the two strips (blue solid curve) and T = 2 . corresponding to the upper part ofthe strip (red dashed curve).The transition between region III and region IV is first order. Since the curve separatingthe regions III and IV describes two minima of v ( e ) coalescing with the minimum at e = 0 ,one would expect a second order transition, and it may be accidental that the position of thefirst order transition is located on this curve (it could also be that our numerically accuracyis not sufficient). In the region IV we have three first order transitions as a function of λ while there are only two transitions in the region II which become second order at anintermediate value of T . – 14 – (cid:101) λT (cid:101) λT (a) (b) M = 2 M = 3 e g ( e ) (c) Figure 9 . (a) The phase diagram for M = 2 (a) and M = 3 (b) at µ = 0 with (cid:101) g = 1 and (cid:101) g = 3 . in the large- N limit. The magnitude of | e − e | is plotted. The strip of first order transitionsfor high temperature is interrupted roughly between (cid:101) λ = 1 and (cid:101) λ = 2 , but is present close to thebroken phase around the origin and at high temperatures. In figures (c) we show a log-log-plotof the function g ( e ) where γ = 3 π/ (2 (cid:101) g ) ≈ . for three different temperatures T = 0 . (greendashed curve), T = 0 . (blue solid curve), and T = 2 . (red dashed curve). The remnant of thestrip close to the bulk of phase transitions at the origin can be explained by the existence of a“wiggle” of g ( e ) which briefly dissolves for larger temperature and reappears anew. For smaller (cid:101) g (larger γ ) the high temperature strip of phase transitions is completely separated from the brokenphase near the origin, cf. figures 7 and 8. | (cid:101) g | > (cid:101) g cr2 ( γ < γ cr2 ) For these values of (cid:101) g or γ , the system no longer enters region I with increasing temperatureso that the strip with the cascade of first order phase transitions is no longer interrupted.We have numerically analyzed this regime for M = 2 , and at (cid:101) g = 5 in figure 10.The cascade of phase transitions at high temperature has been visualized in figure 10(d)where we have plotted the actual solutions e k at the global minimum of the potential (4.1).To understand the nature of the two phases above and below this strip, we interpret the e k as the effective masses of the fermions of the theory. As shown in figure 10, the low- T region is characterized by a large value of | e k | implying that the effective masses areheavy. In contrast, in the high- T region above the strip all | e k | drop nearly to zero, making– 15 – .0 0.5 1.0 1.5 2.0 2.50.600.650.700.750.800.850.90 M = 2 (cid:101) λT (a) M = 3 (cid:101) λT (b) M = 4 (cid:101) λT (c) { e k } M = 2 , (cid:101) λ = 0 . M = 3 , (cid:101) λ = 1 M = 4 , (cid:101) λ = 2 T T T (d)
Figure 10 . Figures (a), (b) and (c) show the same plots as in figures 7 and 8 but with (cid:101) g = 5 .All phase transitions are first order. (d) The minimum of V eff ( E ) with (cid:101) g = 1 and (cid:101) g = 5 at µ = 0 .As the temperature rises, the eigenvalues drop sequentially through M first order transitions. the fermions almost massless. The large bare mass κψψ of the constituent fermions isdynamically canceled by interactions. This cancellation proceeds step by step across thestrip. For a large fixed (cid:101) λ , as the temperature goes up, there are M first-order transitions;across each transition one of the M species of fermions becomes light. After all transitionsare traversed, all M fermions become light.In the same way as in the high temperature regimes, one can depict the phase transitionsin the low T and low (cid:101) λ phase where we also find a cascade of phase transitions. A newphenomenon shows up for | (cid:101) g | > | (cid:101) g cr2 | for parameter values in region IV. When zoominginto figure 10(b) there is a large region where the symmetry breaking pattern is U (3) → – 16 – .00 0.05 0.10 0.15 0.20 0.250.7200.7250.7300.7350.7400.745 M = 3 (cid:101) λT (a) (cid:101) λ = 0 . T { e k } (b) Figure 11 . (a) The phase diagram for M = 3 at µ = 0 with (cid:101) g = 1 and (cid:101) g = 5 in the large- N limit. The plotted observable is Min ( | e − e | , | e − e | , | e − e | ) . Within the red triangle the three e k differ from one another, indicating spontaneous symmetry breaking U (3) → U (1) . (b) The (cid:101) T -dependence of { e k } at (cid:101) λ = 0 . . There is a range of T in which the three e k are all different. U (2) × U (1) (namely, two of the three e k coincide). Yet, in a tiny region, shown in figure 11,all e k are mutually distinct and break the symmetry as U (3) → U (1) × U (1) × U (1) . In thisphase, one of the bosons is very light but the other two are heavy. Indeed when | (cid:101) g | > | (cid:101) g cr2 | (or | γ | < | γ tri2 | ), we find a different kind of transition in the shape of the function g ( e ) in theregion IV (see insets in figure 6). One of the consequences is the occurrence of exotic phasescorresponding to the symmetry breaking patterns U ( M ) → U ( j ) × U ( k ) × U ( M − j − k ) because g ( e ) has three positive slopes so that the potential v ( e ) has three minima, seeAppendix A.2. The zero-temperature potential at µ > can be readily found from (2.12) as V eff ( E ) = V eff ( E ) (cid:12)(cid:12)(cid:12) µ =0 − M (cid:88) k =1 (cid:90) d p (2 π ) (cid:18) µ − (cid:113) p + 4 g E k (cid:19) Θ (cid:18) µ − (cid:113) p + 4 g E k (cid:19) = V eff ( E ) (cid:12)(cid:12)(cid:12) µ =0 − π M (cid:88) k =1 ( µ − | g E k | ) ( µ + 4 | g E k | )Θ( µ − | g E k | ) (5.1)where V eff ( E ) (cid:12)(cid:12)(cid:12) µ =0 is as given in (3.2) and Θ( x ) is the Heaviside step function. In dimen-sionless units where Λ is absorbed in µ → Λ / µ we have V eff ( E )Λ = (cid:101) g (cid:101) g − M (cid:101) g (cid:32) M (cid:88) k =1 e k − (cid:101) λ (cid:33) + M (cid:88) k =1 (cid:26) e k + 43 π | (cid:101) g e k | − π (1 + 4 (cid:101) g e k ) / − π ( µ − | (cid:101) g e k | ) ( µ + 4 | (cid:101) g e k | )Θ( µ − | (cid:101) g e k | ) (cid:27) . (5.2)– 17 – ( e ) e 00 g ( e ) e00 g ( e ) e00 g ( e ) e γ , μ tri ( γ , μ cr ) IV IIIIII γ μ Figure 12 . Phase diagram of the derivative of the potential, v (cid:48) ( e ) = g ( e ) , in the ( γ , µ ) plane. Theinsets show the different shapes of the function g ( e ) . For analytical considerations, we adopt again the notation of (3.6), where the potentialbecomes (cid:98) V eff ( e ) = γ (cid:32) M (cid:88) k =1 e k − λ (cid:33) + M (cid:88) k =1 (cid:26) γ e k + | e k | − (1 + e k ) / −
12 ( µ − | e k | ) ( µ + 2 | e k | )Θ( µ − | e k | ) (cid:27) . (5.3)with the saddle point equation − s = g ( e ) with g ( e ) = 2 γ e + 3 e (cid:16) | e | − (cid:112) e (cid:17) + 3 e ( µ − | e | )Θ( µ − | e | ) (5.4)and s = γ ( (cid:80) Mk =1 e k − λ ) .The insets in figure 12 show the different shapes of the function g ( e ) in the ( γ , µ ) plane. Taking into account the Heaviside Θ function, in a similar way as at µ = 0 andnonzero temperature, the regions are separated by the following three curves:i) The vanishing of the slope at e = 0 , g (cid:48) ( e = 0) = 2 γ + 3 µ − , (5.5)see blue line in figure 12. – 18 –i) The curve (red curve in figure 12) defined by g ( µ ) = 0 , (5.6)has the explicit solution is given by γ = 32 (cid:16)(cid:112) µ − µ (cid:17) . (5.7)iii ) The third curve is given by lim ε → + g (cid:48) ( µ − ε ) = 0 . (5.8)The limit is introduced so that the Heaviside Θ function is equal to one (green curvein figure 12). This equation can be solved explicitly with the solution given by γ = 3(1 + 2 µ )2 (cid:112) µ − µ. (5.9)From that expression we obtain the critical point d γ d µ = 3 µ (3 + 2 µ )2(1 + µ ) / −
32 = 0 , (5.10)which is solved by µ cr = 0 . with a corresponding value of γ given by γ cr2 = 1 . . (5.11)This point is indicated by the yellow point in figure 12.These three curves partition the ( γ , µ ) plane into four regions which are anew referredto by Roman numerals, see figure 12. The curves meet at the tricritical point where theminima of the potential coalesce. Combining γ + 3 µ = 3 with γ = 3( (cid:112) µ − µ ) wefind that the tricritical point is at µ tri = 0 and γ tri2 = 3 / (blue point in figure 12).Since the shapes of g ( e ) at nonzero µ and T = 0 are similar to the shapes of g ( e ) for µ = 0 and nonzero T , we expect a similar phase diagram where we again can distinguishthree regions depending on the value of γ relative to γ cr2 and γ tri2 . Especially, we see acascade of phase transitions between phases of the form U ( M ) → U ( j ) × U ( M − j ) . Wehave checked this numerically for M = 2 and M = 3 with (cid:101) g = 3 ( γ = 0 . ), see figure 13(a) and (b). As is the case at nonzero T and µ = 0 for γ < γ cr2 the strip of first order phasetransitions is connected. Both inside this strip and in the region around ( µ, (cid:101) λ ) = (0 , weobserve a cascade of first order phase transitions.When increasing the chemical potential at fixed | (cid:101) g | > (cid:101) g cr2 (or γ < γ cr2 ) we enter regionIV, where v ( e ) has three minima. This opens the possibility of phases with the symmetrybreaking pattern U ( M ) → U ( j ) × U ( k ) × U ( M − j − k ) . This has been indeed observedby us for M = 3 , see figure 14. The region where this kind of symmetry breaking patternhappens is very narrow as it has been the case for finite temperature. Note the similarityof figure 14 to figure 11. We repeated the same analysis for M = 4 and M = 5 . For M = 4 – 19 – .0 0.2 0.4 0.6 0.8 1.0 1.20.00.20.40.60.81.01.21.41.6 M = 2 (cid:101) λµ (a) M = 3 (cid:101) λµ { e k } { e k } M = 2 , (cid:101) λ = 0 . M = 3 , (cid:101) λ = 1 µ µ (c) Figure 13 . The T = 0 phase diagram at nonzero chemical potential for (a) M = 2 and (b) M = 3 with (cid:101) g = 1 and (cid:101) g = 3 . The value plotted is | e − e | in (a) and e − e + e in (b)with e ≥ e ≥ e assumed. All phase transitions are first order. (c) The minimum of V eff ( E ) with (cid:101) g = 1 and (cid:101) g = 3 at T = 0 . As the chemical potential increases, the eigenvalues drop sequentiallythrough M first-order transitions. we found an exotic phase in which U (4) is broken down to U (2) × U (1) × U (1) . For M = 5 we found a phase with the breaking pattern U (5) → U (2) × U (2) × U (1) .When γ < γ cr2 the strip of first order phase transitions is connected to the brokenregion around the origin of the ( λ, µ ) plane. For γ cr2 < γ < γ tri2 the strip is interruptedexactly as in the finite T and µ = 0 case. The part that is connected to the broken regionabout the origin disappears for γ < γ tri2 .The strip divides the U ( M ) -symmetric part of the phase diagram into two regions.The qualitative distinction between these two regions is clear from the behavior of the { e k } (figure 13). They start with a large value at low (cid:101) µ and then successively drop to a smallvalue as (cid:101) µ increases.At nonzero chemical potential the region between the tricritical point and γ = 3 / ,which gives a second order phase transition for M = 2 at low temperature, is absent.Therefore, we do not expect second order transitions at µ (cid:54) = 0 and T = 0 , even for M = 2 .Only at the end points of the cascades of phase transitions where all first order transitionsrun together, we expect second order phase transitions.Finally, we consider the fermion number density n . Since it is proportional to N it is– 20 – .00 0.02 0.04 0.06 0.08 0.10 0.120.760.780.800.820.840.860.880.900.92 M = 3 (cid:101) λµ (a) (cid:101) λ = 0 . µ { e k } (b) Figure 14 . (a) The T = 0 phase diagram for M = 3 with (cid:101) g = 1 and (cid:101) g = 3 . The plottedobservable is Min ( | e − e | , | e − e | , | e − e | ) . Within the red triangle the three e k differ from oneanother, indicating spontaneous symmetry breaking U (3) → U (1) . (b) The µ -dependence of { e k } at (cid:101) λ = 0 . . There is a range of µ in which the three e k are all different. = 0= 0.2= 0.6 M = 2 µnN Λ = 0= 0.3= 0.8 M = 3 µnN Λ Figure 15 . Fermion number density at T = 0 with (cid:101) g = 1 and (cid:101) g = 3 . useful to divide it by N . In dimensionless units, we have nN Λ = 14 π M (cid:88) k =1 ( µ − (cid:101) g e k )Θ( µ − | (cid:101) g e k | ) . (5.12)In figure 15 we show the µ -dependence of the fermion number density. It jumps atfirst-order transitions. The plots for (cid:101) λ = 0 . M = 2) and (cid:101) λ = 0 . M = 3) reveal that thetwo U ( M ) -symmetric phases at small µ and large µ are physically quite distinct, as wassuggested in figure 13 as well. At small µ , fermions are rather heavy and the number densityis quenched to zero. In contrast, at large µ , fermions are nearly massless and their densityis enhanced: the large bare mass originating from κψψ in the Lagrangian is dynamicallyscreened by interactions. – 21 – Phases at nonzero T and µ Finally, in this section we investigate the effect of simultaneously nonzero T and µ . From(2.12) the potential in this case is found to be V eff ( E )Λ = (cid:101) g (cid:101) g − M (cid:101) g (cid:32) M (cid:88) k =1 e k − (cid:101) λ (cid:33) + M (cid:88) k =1 (cid:40) e k + 43 π | (cid:101) g e k | − π (1 + 4 (cid:101) g e k ) / + T π (cid:34) | (cid:101) g e k | (cid:101) T Li (cid:16) − e − | (cid:101) g ek | + µT (cid:17) + Li (cid:16) − e − | (cid:101) g ek | + µT (cid:17) + 2 | (cid:101) g e k | T Li (cid:16) − e − | (cid:101) g ek |− µT (cid:17) + Li (cid:16) − e − | (cid:101) g ek |− µT (cid:17) (cid:35)(cid:41) . (6.1)To speed up numerical minimization, we used the Taylor expansion of Li ( z ) and Li ( z ) around z = − up to 11th order to compute values for − ≤ z < , and then used functionalidentities relating Li s ( z ) to Li s (1 /z ) [66, 67] to compute values for z < − .De novo we use the notation of (3.6) in which the potential reads (cid:98) V eff ( e ) = γ (cid:32) M (cid:88) k =1 e k − λ (cid:33) + M (cid:88) k =1 (cid:26) γ e k + | e k | − (1 + e k ) / + 3 T (cid:20) | e k | T Li (cid:16) − e − ( | e k | + µ ) /T (cid:17) + Li (cid:16) − e − ( | e k | + µ ) /T (cid:17) + | e k | T Li (cid:16) − e − ( | e k |− µ ) /T (cid:17) + Li (cid:16) − e − ( | e k |− µ ) /T (cid:17) (cid:21)(cid:27) . (6.2)with the saddle point equation − s = g ( e k ) with g ( e ) = 2 γ e − e (cid:112) e + 3 T e log (cid:16) (cid:104) eT (cid:105) + 2 cosh (cid:104) µT (cid:105)(cid:17) (6.3)and s = γ ( (cid:80) Mk =1 e k − λ ) . The possible four shapes of the function g ( e ) , see figure 16,are basically smoothened versions of the zero temperature but finite chemical potentialsetting, compare with figure 12. This also implies that the discussion of the phase diagramin the ( (cid:101) λ, µ ) -plane looks essentially the same with one exception, namely the onset of asecond order phase transition for M = 2 which results from the finite temperature picturein section 4.The phase transitions can again be understood via a Taylor expansion of the function g ( e ) about the origin, i.e., g ( e ) ≈ (cid:16) γ + 3 T log (cid:104) (cid:16) (cid:104) µT (cid:105)(cid:17)(cid:105) − (cid:17) e + 32 (cid:18) T (1 + cosh [ µ/T ]) − (cid:19) e + 18 (cid:18) cosh [ µ/T ] − T (1 + cosh [ µ/T ]) + 3 (cid:19) e + o ( e ) . (6.4)The coefficient of the linear term determines the plane (blue dotted plane in figure 16) thatseparates region III from regions I and IV, which is explicitly given by γ ( T, µ ) = 32 (cid:16) − T log (cid:104) (cid:16) (cid:104) µT (cid:105)(cid:17)(cid:105)(cid:17) . (6.5)– 22 – igure 16 . Three-dimensional phase diagram of g ( e ) in the ( γ , µ, T ) space. The insets show thepossible shapes of the derivative of the potential g ( e ) , see (6.3), at finite chemical potential andtemperature. The intersections of the blue surface with the µ = 0 and T = 0 planes are given by theblue curve in figures 6 and 12, respectively. For M = 2 and λ = 0 , the part of this surfacebetween regions I and III allows second order phase transitions. It continues to exist fornot too large values of the chemical potential. When γ < γ ( T, µ ) the phase will have thesymmetry breaking pattern U (2) → U (1) × U (1) , and for γ > / the flavor symmetryremains unbroken at low temperature and chemical potential.There are two additional planes that divide the ( µ, T, γ ) space. As is the case at zerochemical potential, the first one is given by (green dotted surface in figure 16) g (cid:48) ( e ) = g (cid:48)(cid:48) ( e ) = 0 . (6.6)On this surface, that separates regions I and II, the extrema of g ( e ) in region II join so that g ( e ) in region I becomes monotonous.The second plane is given by the equation (red dotted plane in figure 16) g ( e ) = g (cid:48) ( e ) = 0 . (6.7)On this plane the minimum of g ( e > touches the e -axis so that the correspondingpotential will have three minima in region IV.The tricritical points at T tri = 1 / for µ = 0 becomes now a tricritical curve (see blackcurve in figure 16 for finite µ , namely when the cubic term in g ( e ) is also vanishing, whichis at µ tri = T tri arccosh (cid:20) − T tri T tri (cid:21) ⇒ γ tri2 = 32 (cid:18) − T tri log (cid:20) T tri (cid:21)(cid:19) . (6.8)There are bounds for the location of this curve, particularly T tri ∈ [0 , / , γ tri2 ∈ [3 / − log 2) , . and µ tri ∈ [0 , . (the latter number is an approximation for the maximum of– 23 – .0 0.2 0.4 0.6 0.8 1.00.10.20.30.40.50.6 T Tµ µ (cid:101) λ = 0 . (cid:101) λ = 0 . (cid:101) λ = 0 . µT Figure 17 . The M = 3 phase diagram in the large- N limit with (cid:101) g = 1 and (cid:101) g = 3 for various (cid:101) λ .The plotted observable is e − e + e with e ≥ e ≥ e assumed. the right hand side of (6.8)). Whenever µ < µ tri for M = 2 the system experiences a secondorder phase transition at γ = γ (see eq. (6.5)). Larger values of M remain untouchedand all phase transitions are of first order apart from the critical points where first ordertransition lines end. Also for suitably large (cid:101) λ all second order phase transitions will vanishand what remains are first order transitions.In region IV at fixed γ the potential will have three minima and we can again expectexotic phases of the form U ( M ) → U ( j ) × U ( k ) × U ( M − j − k ) . They will certainly appearonly for small µ since this has been already the case for either T = 0 or µ = 0 .For suitably large γ , µ and T , we find either a strictly monotonous g ( e ) (region I) orone which has local maxima and minima symmetrically about the origin in two separatequadrants (region II). The latter signals again the existence of a strip of cascades of phasetransitions in the high T and µ region. Yet, the shape of g ( e ) in region II can be also foundfor a small region for γ < γ tri2 which will show itself as a remaining appendix of this stripof phase transitions at the phase region about the origin, cf. figure 9.For simplicity of exposition we limit our numerical analysis to M = 3 . Our mainresults are summarized in figure 17 (for (cid:101) g = 3 ) and figure 18 (for (cid:101) g = 5 ). Figure 17 showsthat the phase structure depends on (cid:101) λ in a nontrivial way. At small (cid:101) λ , there is a largeregion at low T and low µ in which U (3) is spontaneously broken to U (2) × U (1) . Along the– 24 – Tµ µ (cid:101) λ = 0 . (cid:101) λ = 0 . Figure 18 . Same as figure 17 but with (cid:101) g = 5 . boundary of this phase, there is a narrow strip in which U (3) is broken to U (1) × U (1) × U (1) (cf. figure 14). As (cid:101) λ increases, this strip gradually disappears. At (cid:101) λ = 0 . there are threesymmetry-broken phases: they have the same symmetry (U (2) × U (1) ) but are separatedby first-order phase transitions. As λ increases further, the symmetry gets restored in thelow- T low- µ region but remains broken in the cold dense region.At stronger coupling a qualitatively new feature emerges. In figure 18 we observe thatthe symmetry-broken phase forms a thin annulus, separating the low- T low- µ region fromthe high- T high- µ region. This annulus never disappears even at very large λ , although itis shifted to higher T and µ gradually. By monitoring the behavior of | e k | we found thatthe U (3) -symmetric phase below the annulus is characterized by very heavy fermions, whilethe other U (3) -symmetric phase above the annulus is characterized by massless fermions.Although these two phases cannot be distinguished by symmetries, they host quite differentphysics. In the present article, we investigated various aspects of Dirac fermions with nonstandardquartic interactions in two spatial dimensions. We showed within the mean-field approxi-mation that the model experiences a cascade of phase transitions when the flavor-symmetricparity-breaking mass is varied, in a way quite analogous to the behavior of QCD [44, 51].At nonzero temperature and chemical potential we provided analytical and numerical ar-guments that show how a complicated phase diagram embellished by exotic symmetrybreaking patterns emerges. In particular, we showed (in figures 10, 13 and 18) that, atstrong coupling, the low- ( µ, T ) phase with heavy bosons is separated from the high- ( µ, T ) phase with almost massless bosons by a series of M phase transitions, through which M species of fermions become light one after another. At finite temperature there is a subtletyabout symmetry breaking due to enhanced infrared singularities and we gave a speculativecomment on this. Summarizing above, our results shed light on previously unnoticed noveldynamics of Dirac fermions in dimensions and have potential implications for planargauge theories as well as planar condensed matter systems.– 25 –here are several directions in which this work can be extended. First, the presentanalysis in the large- N limit could be generalized to incorporate finite- N corrections. Fluc-tuations of bosonic fields can be conveniently included by employing methods such as thefunctional renormalization group [68]. At finite N , the Jacobian (the squared Vander-monde determinant) associated with the diagonalization of the matrix field can no longerbe neglected and will affect ground state properties. Secondly, it would be interesting tosee what happens if our assumption g > M g is relaxed. Thirdly, various topologicalexcitations arise our model. For instance, in the phases depicted in figures 11 and 14, π ( U (3) / U (1) ) = Z × Z , implying there are two kinds of Skyrmions. Fourthly, while wehave only considered fermion-anti-fermion condensates, a di-fermion condensate may format high density [69, 70]. The competition of two kinds of condensates may be an interest-ing subject of research. Finally, it would be challenging but quite important to take intoaccount the possibility of an inhomogeneous condensate that spontaneously breaks trans-lation symmetry. While the existence of such a condensate has been firmly established insome (1 + 1) -dimensional models at finite density [71–73], the situation is elusive in higherdimensions [74, 75]. Acknowledgments
This work was in part supported (JV) by U.S. DOE Grant No. DE-FAG-88FR40388.
A Phase diagram of the effective potential and a toy model
In this Appendix, we outline the general strategy for analyzing the phases for an effectivepotential of the general structure considered in the main body of the text given by the sumof a confining ( γ > ) harmonic collective potential and confining “single-particle” termswith potential v ( x ) , V ( e , · · · , e M , λ ) = γ M (cid:88) j =1 e j − λ + M (cid:88) j =1 v ( e j ) . (A.1)Generically, we assume that v ( x ) has the shape of a double well potential that increasesfaster than linear for large | e | (i.e. lim | e |→∞ v ( e ) / | e | = ∞ ) ). Such potentials show asimilar behavior such as the cascade of phase transitions and the kind of symmetry breakingpatterns we have found in the physical system. Moreover, the mechanism when and howthe system experiences a second order phase transition is very similar for different v ( e ) .In the last part of this Appendix, we illustrate the general arguments with the propertiesof a much simpler toy model (indicated by the subscript tm) with the confining potential v tm ( e j ; T ) = ( e j + T − . (A.2)This model is motivated by the analysis in section 4 and its numerical observations. It isessentially a truncation of the expansion of the single particle part of the potential (4.2) tofourth order, which is expected to capture some essential part of the physics in the vicinity– 26 –f a second-order phase transition. Regardless of its simplified form, this toy model alreadyexhibits generic features for general v ( e ) . Hence, we would like to underline that mostconclusions apply for a more general confining potential v ( e ) . A.1 Saddle point equation and its asymptotic solutions
What has to be studied are the M saddle point equations − s = − γ M (cid:88) j =1 e j − λ = g ( e k ) = v (cid:48) ( e k ) . (A.3)The extrema of the potential V ( e, λ ) are determined by the intersections of g ( e ) with − s ,which also select the possible phases, especially, which symmetry breaking patterns, thesystem can exhibit as a function of λ and the parameters of the potential. Note that s canhave different values for the same values of the parameters.One particular ingredient is the asymptotic behavior of the solutions of (A.3) for large | λ | . The asymptotic super-linear growth of v ( e ) implies also an asymptotic growth of | g ( e ) | .Particularly we have three cases to consider where the asymptotic value lim e →±∞ g ( e ) e = c ± (A.4)can be either vanishing ( c ± = 0 ), be finite ( < c ± < ∞ ), or diverge ( c ± ( e ) = ∞ ).Depending on which case the asymptotic solution for e j becomes unique and takes the form e k ≈ λM − γ M g (cid:18) λM (cid:19) , c sign( λ ) = 0 , γ λ γ M + c sign( λ ) , < c sign( λ ) < ∞ ,g − (2 γ λ ) , c sign( λ ) = ∞ , (A.5)for all k = 1 , . . . , M . The function g − is the inverse of g in the asymptotic regime; forinstance when g ( e ) grows like e L , the inverse is essentially e /L . In the physical system inthe main text we have the situation of a finite c + = c − while the toy model (A.2) leads to c sign( λ ) = ∞ . An asymptotic behavior with c + and c − in a different class is possible, butwe do not consider that in the present work.Regardless which case the asymptotic satisfies, we obtain the same conclusions for theglobal minimum of the potential. First, all e k are degenerate for suitably large | λ | . Second,the modulus of the auxiliary parameter s in (A.3) also grows asymptotically, and its signis the opposite of λ and e k . As a physical conclusion we find that for suitably large | λ | wealways have a solution with all e k equal which has unbroken flavor symmetry. A.2 Local extrema of g ( e ) and implications on the possible phases The solutions of the saddle point equation can be either minima or maxima for V ( e, λ ) .For a saddle point with g (cid:48) ( e k ) > for all e k the potential has certainly a minimum, not– 27 –ecessarily the global one we are looking for. This follows from the fact that the Hessianat the saddle point is given by H = { ∂ e k ∂ e j V ( e, λ ) = 2 γ + g (cid:48) ( e k ) δ jk } j,k =1 ,...,M (A.6)The Hessian is positive definite if the determinants det( H jk ) j,k =1 ,...,n = det { γ + g (cid:48) ( e k ) δ jk } j,k =1 ,...,n > , for all n ≤ M. (A.7)The term proportional to γ is of rank which simplifies the evaluation of the determinantdrastically and it is equal to det( H jk ) j,k =1 ,...,n = (cid:32) γ n (cid:88) l =1 g (cid:48) ( e l ) (cid:33) n (cid:89) k =1 g (cid:48) ( e k ) , for all n ≤ M. (A.8)In general, we may have a solution with L different e k . At most one of the e k may have g (cid:48) ( e k ) < . The reason is that the term of the Hessian that is proportional to γ is of rankone. A rank one addition can maximally switch one eigenvalue of a matrix from positive tonegative and vice versa, regardless how large its prefactor is. Let us label the intersectionwith g (cid:48) ( e k ) < as e M . Then all subdeterminants up to n = M − are positive, and thecondition for the positive definiteness of the Hessian matrix is given by the positivity of itsdeterminant, det { γ + g (cid:48) ( e k ) δ jk } j,k =1 ,...,M > ⇔ γ M (cid:88) l =1 g (cid:48) ( e l ) < . (A.9)If the solution with L different e k is a global minimum, this would result in the symmetrybreaking pattern U ( M ) → U ( j ) × · · · × U ( j l ) with (cid:80) Ll =1 j l = M . The unbroken symmetryassociated with e k with g (cid:48) ( e k ) < can only be a U (1) factor. However, not all of thesesaddle points are global minima of V ( e, λ ) which is the hard part of the analysis.The simplest case is M = 2 . Then the only possible flavor symmetry breaking patternis U (2) → U (1) × U (1) , that means a transition from a solution with e = e to a solutionwith e (cid:54) = e . This transition can only be of second order if the solutions join continuouslyto the point where the determinant of the Hessian vanishes, i.e. for parameter values with g (cid:48) ( e ) = g (cid:48) ( e ) = 0 (or at g (cid:48) ( e ) = g (cid:48) ( e ) = − γ , but that is not allowed for γ > ). Thereis no second order phase transition when g ( e ) increases monotonically. More generally, thephase transition is first order when the global minimum is not a continuous function of λ .As is shown for the potential in the main text and for the toy model (A.2) a secondorder phase transition is realized for M = 2 . However, as we will show below, for M > the phase transition for a potential of the form (A.1) is always first order. A.3 Cascade of phase transitions
In this subsection, we discuss the phases of the general potential (A.1) with v ( e ) havinglocally the shape of a confining (not necessarily symmetric) double well. The notion “locally”means that there is region for s bounded by a local maximum and local minimum of g ( e ) – 28 –here the saddle point equation (A.3) has only three solution when fixing s . The integralof g ( e ) in this region looks like a double well potential.For the solutions of the saddle point equations we use the Ansatz diag( e ) = diag( X M − j , X j ) ,with X < X without restriction of generality. The two variables X and X must satisfythe equations − γ (( M − j ) X + jX − λ ) = g ( X ) and − γ (( M − j ) X + jX − λ ) = g ( X ) . (A.10)These equations can be solved for a real (cid:98) j ∈ R . We would like to highlight the differenceof (cid:98) j ∈ R and j = 0 , . . . , M ; while the former is real and can take an optimal positionminimizing V (cid:98) j ( X , X , λ ) = γ (( M − (cid:98) j ) X + (cid:98) jX − λ ) + ( M − (cid:98) j ) v ( X ) + (cid:98) jv ( X ) (A.11)the latter can be only an integer and only approximately minimizes the potential. Thesaddle point equation for (cid:98) j is given by ∂V (cid:98) j ∂ (cid:98) j = 2 γ ( X − X )(( M − (cid:98) j ) X + (cid:98) jX − (cid:98) λ ) + v ( X ) − v ( X ) , (A.12)which yields a unique minimum for (cid:98) j in terms of X and X . This follows from the secondderivative in (cid:98) j which is always positive when X (cid:54) = X , ∂ V (cid:98) j ∂ (cid:98) j = 2 γ ( X − X ) > . (A.13)Note that for X = X the saddle point equation for (cid:98) j is satisfied trivially.The minimizer (cid:98) j will generally not lie on one of the integers j = 0 , . . . , M . Yet, theconvexity of V (cid:98) j ( X , X , λ ) in (cid:98) j shows that only those closest to (cid:98) j minimize the potential V j ( X , X , (cid:98) λ ) with j = 0 , . . . , M .The saddle point equations (A.10) are invariant under (cid:98) j → (cid:98) j + δ (cid:98) j, λ → λ + ( X − X ) δ (cid:98) j. (A.14)as is the saddle point equation (A.12) for (cid:98) j . We therefore must have d (cid:98) j d (cid:98) λ = 1 X − X > . (A.15)Since λ is a function of (cid:98) j , the saddle point solutions are functions of (cid:98) j only and increasing λ will increase (cid:98) j . For the discretized version j = 0 , . . . , M the solutions X and X getan explicit dependence on (cid:98) λ though it has only limited impact as j tries to be as closeas possible to (cid:98) j . The convexity of the potential as a function of (cid:98) j also tells us that therecan be only phase transitions from the phase with flavor symmetry U ( j ) × U ( M − j ) tothe phase with flavor symmetry U ( j + 1) × U ( M − j − or U ( j − × U ( M − j + 1) for j = 1 , . . . , M − . We still could have a phase transition between a solutions with all e k the same. As we will see below, in the toy model, this happens when T > . Also in the– 29 –hysical system at finite temperature and/or at finite chemical potential there is a regionwhere the system may experience such a direct transition from all e k equal and negative toall e k equal and positive, see sections 4, 5 and 6.In summary, the system runs through all phases corresponding to U ( M ) → U ( j ) × U ( M − j ) from j = 0 , . . . , M as the real minimizing set ( (cid:98) j, X ( (cid:98) j ) , X ( (cid:98) j )) will depend con-tinuously on (cid:98) λ . The kinks and, hence, phase transitions only originate from the discretenessof j rather than the continuous variable (cid:98) j → j implying X , ( (cid:98) j ) → X , ( j, λ ) .Let us underscore the following. What is locally required for the discussion above isa potential with two minima and the validity of our assumption that the bipartite Ansatzis valid. However, we could not exclude the possibility that the flavor symmetry is brokenaccording to U ( M ) → U ( j ) × U ( M − j − × U (1) when the potential has a minimum withthree different e k , two with g (cid:48) ( e k ) > and one with g (cid:48) ( e k ) < . This possibility has to beinvestigated case by case. A.3.1 No-go statement for second order phase transitions
The question that remains is whether any of these phase transitions are of second order.The transition from j to j + 1 for j (cid:54) = 1 and j (cid:54) = M necessarily has to be of first order,because if X = X is at a second order phase transition point, we would have a transitionfrom a state with j of the e k at X to a state with all of the e k at X as the positivitycondition (A.7) does not allow any other possibility. The only exceptions are the transitionsfrom j = 0 to j = 1 and from j = M − to j = M . We now consider the latter, while theformer can be worked out in the same way.As before, the arguments below apply to a potential v ( e ) that has locally the form of adouble well potential and which is super-linearly increasing for large and small e , this meansits derivative g ( e ) = v (cid:48) ( e ) has the shape of a wiggle. For a second order phase transition tooccur two solutions of the saddle-point equations have to coalesce. This can only happen atthe point e with g (cid:48) ( e ) = 0 . Since we are considering the solution M − → M , we studythe behavior of the potential around e k = e , for k = 1 , · · · , M . At this point we have γ ( M e − λ ) + M v (cid:48) ( e ) = 0 ,v (cid:48)(cid:48) ( e ) = 0 . (A.16)Although this point is a solution of the the saddle-point equations it does not have to bea global minimum. Below we will show that for M > there is one direction in which thepotential decreases. We use the Ansatz ( e , · · · , e M ) = ( e + x , · · · , e + x (cid:124) (cid:123)(cid:122) (cid:125) M − , e + x ) . (A.17)Because the potential is homogeneous in the x k the derivatives of the potential with respectto x and x also vanish for this Ansatz. Since the determinant of the Hessian vanishes at e , there is at least one direction in which the second order fluctuations vanish. To find adecreasing direction, we thus have to Taylor expand the potential at least to third order V = V ( e , · · · , e ) + γ (( M − x + x ) + 13! g (cid:48)(cid:48) ( e )(( M − x + x ) + · · · . (A.18)– 30 –n the direction of vanishing second derivative, given by x = − ( M − x , (A.19)the third order term behaves as ( M − − ( M − g (cid:48)(cid:48) ( e ) x = − M ( M − M − g (cid:48)(cid:48) ( e ) x . (A.20)For the assumed shape of the potential we have g (cid:48)(cid:48) ( e ) > (minimum of g ( e ) at e ) so thatgenerally the third order term becomes negative. One exception is M = 2 . In that casethe fourth order term of the expansion is positive. So for M = 2 the point e = e = e can be a global minimum. For M > , we always have a decreasing direction excluding thepossibility of a second order phase transition.When g (cid:48)(cid:48) ( e ) = 0 , we need to expand to higher orders. For a local double well shapeof v ( e ) or local wiggle shape of its derivative g ( e ) the first non-vanishing derivative of v ( e ) must be even. For L = 3 , , . . . , we obtain M − L ! g ( L − ( e ) x L + 1 L ! g ( L − ( e ) x L = ( M − (cid:2) − ( M − L − (cid:3) x L L ! g ( L − ( e ) < (A.21)because it must be g ( L − ( e ) > if we consider the transition from j = M − to j = M as g ( e ) has to have a minimum at e .In summary, we can say that for M ≥ the system always experiences a cascade of firstorder phase transitions. The only requirement is that the potential has locally the shapeof a double well or its derivative g ( e ) has the shape of a “wiggle” meaning a maximumfollowed by a minimum and then growing again. This is the situation also for the physicalsystem in the main text. There are surely more complex situations when the potential v ( e ) has more than two minima in a restricted region so that the saddle point equation (A.3)has more than three real solutions. For instance this happens in the middle temperatureregime discussed in section 4. However, generally the mechanism for the phase transitionis similar. A.4 Phase diagram of the toy model for M = 2 Let us illustrate the phase diagram for M = 2 with the toy model (A.2), where we have V ( e , e ) = γ ( e + e − λ ) + v tm ( e ) + v tm ( e ) (A.22)with v tm ( e ) = ( e + T − . (A.23)The saddle point equations are given by − γ ( e + e − λ ) = g tm ( e ) − γ ( e + e − λ ) = g tm ( e ) , (A.24)– 31 –here the derivative of the potential is defined as g tm ( e ) = v (cid:48) tm ( e ) = 4 e ( e + T − . (A.25)This function can have only two distinct shapes depending on the temperature T . For T ≥ ,it is monotonously increasing so that all e k need to be equal, and the flavor symmetry isnot broken in this case. For T < , the function develops a local minimum and maximum.Therefore, equation (A.3) can exhibit three solutions for a fixed s , two at e (+) > > e ( − ) with g tm ( e ( ± ) ) > , and one at e (0) with g (cid:48) tm ( e (0) ) < . In agreement with the asymptoticanalysis in subsection A.1, for sufficiently large or small λ only one solution exists when all e k are the same.In the toy model (A.2), we can have a transition from a broken phase with e (cid:54) = e to aphase with unbroken flavor symmetry with e = e . At a second order transition curve wetherefore must have e = e = e while the determinant of the Hessian must vanish. At thispoint we also must have that g (cid:48) tm ( e ) = 0 . To determine the possible critical behavior wecalculate the Hessian at the saddle point in terms of X = ( e + e ) / and ∆ = ( e − e ) / and simplify it with the difference of the two saddle point equations, g tm ( e ) − g tm ( e ) = 4( e − e )( e + e + e + e − T ) = 0 . (A.26)On the branch e = e = e the Hessian reduces to det H = (12 e + 4( T − + 4 γ (12 e + 4( T − , (A.27)which vanishes when g (cid:48) tm ( e ) = 0 , i.e. at e = ± (cid:114) − T . (A.28)The value of λ at the second order transition follows from the saddle point equations (A.24)for ee , → e , λ = ± γ (cid:20) γ −
23 (1 − T ) (cid:21) (cid:114) − T . (A.29)The line of second order phase transitions in the ( λ, T ) plane may end in a tricriticalpoint. At this point a second zero of the determinant of the Hessian vanishes. To determineit, we consider the branch e + e + e e − T = 0 of the difference of the saddle pointequations, see (A.26), where the determinant of the Hessian in terms of ∆ is given by det H = 32∆ (8∆ + γ − − T )) . (A.30)The tricritical point, when ∆ changes from ∆ = 0 to ∆ (cid:54) = 0 , is located at T = 1 − γ , (A.31)and λ given by (A.29). – 32 – igure 19 . The trajectories (red and blue curves) of the two solutions e and e as functions of λ for M = 2 , γ = 1 and various T . The tri-critical point is at T = 5 / . To highlight the way howthe two solutions e = e = e ( − ) and e = e = e (+) depend on λ we also show the function g tm ( e ) .The dashed curves indicate the location of the phase transition. The dashed lines parallel to the λ axis are the “ridges” of the local minimum and maximum of g ( e ) given by (A.28), while the dashedcurves parallel to e are given by (A.29). If e = e = e is a global minimum, the line of second order phase transitions has toend at T = 1 − γ / for γ < . In particular, at zero temperature the phase transitionhas to be of first order for γ < . The phase diagram will look like the sketch in figure 20.There is a region for − γ / < T < where a second order phase transition happens. For T < − γ / we only have a first order phase transitions. To determine if e = e = e is indeed a global minimum, we need explicit expressions for the solutions and substitutethem in the potential which will be done in the remainder of this subsection.We will solve the saddle point equations in terms of X = ( e + e ) / and ∆ = ( e − e ) / .From the difference of the two saddle point equation (A.26) we see that we can have twodifferent cases ∆ = (cid:40) , ±√ − T − X , only when X < − T . (A.32)The solution ∆ = 0 always exists and is even a solution for general g ( e ) . It correspondsto unbroken flavor symmetry. The non-trivial solution for ∆ obviously represents the case– 33 – (2) → U (1) × U (1) . The potential in terms of ∆ and X reads as follows V ( X, ∆ , λ, T ) = − X + 8(1 − T ) X + γ (2 X − λ ) + 2(∆ − T + 3 X ) , (A.33)which implies that ∆ = 1 − T − X is the global minimum whenever this solution isallowed.The resulting saddle point equations for X of the two cases of ∆ differ and are givenby − γ (2 X − λ ) = 4 X ( X + T − ⇔ γ λ = 2 X + 2( γ + T − X (A.34)for ∆ = 0 , and − γ (2 X − λ ) = 4 (cid:18) X ± (cid:113) − T − X (cid:19) (cid:34)(cid:18) X ± (cid:113) − T − X (cid:19) + T − (cid:35) (A.35)for ∆ = ± (cid:112) − T − X . The latter one can be simplified by adding the two equationswith signs ± which gives − γ (2 X − λ ) = 4[2 X +6 X (1 − T − X )+2( T − X ] ⇔ γ λ = − X +2[ γ +2(1 − T )] X . (A.36)Using eq. (A.29) for λ at the saddlepoint, this equation can be factorized as (cid:32) X − (cid:114) − T (cid:33) (cid:32) − X − X (cid:114) − T −
43 (1 − T ) + 2 γ (cid:33) = 0 . (A.37)The solutions of the quadratic equation are given by X ± = 12 (cid:18) ± (cid:114) γ − (cid:112) (1 − T ) / (cid:19) . (A.38)One can easily see that the solution X + has the lower potential. Next compare the differenceof the potential for X = (cid:112) (1 − T ) / and X + . With some work one can show that ∆ V = 24(1 − T ) / ( γ − − T )) (cid:18)(cid:114) γ − √ − T (cid:19) . (A.39)This is negative for T > − γ / which shows that the solutions with the second orderphase transition is the global minimum. As we have seen before from the analysis of theHessian, T = 1 − γ / is the tricritical temperature.Summarizing, there is a second order phase transition whenever ( X, ∆ , | λ | ) = (cid:32) sign( λ ) (cid:114) − T , , γ (cid:20) γ −
23 (1 − T ) (cid:21) (cid:114) − T (cid:33) with 1 ≥ T ≥ − γ . (A.40)The point T = 1 − γ / is a tricritical point where the transition changes into a first orderphase transition, we have sketched it in figure 20.– 34 – Δ| ≠ 0 Δ = 0 Figure 20 . Sketch of the phase diagram of the toy model for M = 2 . The solid curve denotes afirst-order phase transition and the dashed curve a second-order phase transition. The open circlemarks a tri-critical point whose exact location is given by (A.40) at T = 1 − γ / . A.5 Phase diagram of the toy model for
M > The situation for larger values of M is more complicated. The saddle point equation have M solutions, most of them complex. However, we find a substantial number of real solutionswith different values of (cid:80) k e k . A necessary condition for a global minimum is that theHessian is positive definite, but to uniquely identify the solution we have to substitute it inthe potential. For example, for M = 3 , excluding permutations, we find three real solutionswith a positive definite Hessian for a significant range of parameters. We also have a saddlepoint with all three e k different but this had never been a minimum for the values of theparameters we have analyzed.However, as we argued in previous sections, the global minimum of the potential canhave at most two different e k . Using this as an Ansatz, this substantially simplifies thesaddle point equations. We still have that a second order phase transition can only happenat the minimum of g tm ( e ) which is at (cid:112) (1 − T ) / . The difference of the two saddle-point equation is again given by (A.26). The determinant of the Hessian on the branch e + e + e e − T can have only other zeros in addition to ∆ = 0 when j = M . Inthat case the location of the tricritical point is given by T = 1 − γ M . (A.41)In principle these equation can be solved analytically, and by evaluating the potential atall minima it is possible to determine whether or not this Ansatz yields a global minimum.However, as was shown in A.3.1 2nd order phase transitions are not possible for M > andthis Ansatz with j = M cannot be the true minimum. B Some computations of Sec. 3.2
In this section we solve the saddle point equations g ( e k ) = − s, k = 1 , · · · , M (B.1)– 35 –ith g ( e ) = 2 γ e + 3 e ( | e | − (cid:112) e ) (B.2)and s = γ (cid:32) M (cid:88) k =1 e k − λ (cid:33) . (B.3)In the first part of this appendix, we calculate the solution with all e k equal in which casewe can find the exact solution. In the second part, we compute the solution as a functionof s for the case that there are three different solutions.When all e k take the same value we look for the solution e closest to the origin when s and e have opposite signs. The saddle point equation simplifies to γ λ = 2( γ + M γ ) e + 3 e (cid:18) | e | − (cid:113) e (cid:19) (B.4)which can be rephrased as x − γ λ − / γ + M γ x − x + 32( γ + M γ ) = 0 with e = x − x − . (B.5)Its solution is x = − a + 2 / (1 + 3 a ) (cid:16)(cid:112) b − a ) − b (cid:17) / + (cid:16)(cid:112) b − a ) − b (cid:17) / × / (B.6)with a = − γ + M γ ) (cid:18) γ λ + 1 (cid:19) and b = 27 (cid:18) γ + M γ ) + a + 2 a (cid:19) . (B.7)For γ < / the function g ( e ) develops a local minimum and maximum. Thus, thederivative of g ( e ) with respect to e has to vanish at these points. Its symmetry tells us thatthere is one zero of g (cid:48) ( e ) for e > and one for e < . Indeed, the equation g (cid:48) ( e ) = 0 can berewritten as x + x − γ = 0 (B.8)with | e | = ( x − x − ) / and x > . The corresponding solution is given by e min = max (cid:26) x − x − , (cid:27) , (B.9) x = (cid:16) /γ + (cid:112)
12 + (27 /γ ) (cid:17) / / − (2 / / (cid:16) /γ + (cid:112)
12 + (27 /γ ) (cid:17) / , (B.10)– 36 –hich is the local minimum of g ( e ) . The local maximum lies at − e min . Hence, for a fixed s ∈ [ g ( e min ) / , g ( − e min ) / we find three solutions for g ( e ) = − s .Two solutions of g ( e ) = − s , that we denote by e ( − ) ( s ) < < e (+) ( s ) , come with apositive slope g (cid:48) ( e ( ± ) ( s )) > . The two solutions e (+) ( s ) and e ( − ) ( s ) satisfy the relation e ( − ) ( s ) = − e (+) ( − s ) due to the symmetry of g ( e ) . Thus, it is enough to state the solutionfor e (+) ( s ) = ( x + − x − ) / with x + = − a + + 2 / (1 + 3 a ) (cid:16)(cid:113) b − a ) − b + (cid:17) / + (cid:16)(cid:113) b − a ) − b + (cid:17) / × / (B.11)and a + = 12 γ (cid:18) s − (cid:19) and b + = 27 (cid:18) γ + a + + 2 a (cid:19) . (B.12)This can be derived by solving the cubic equation ˜ x + 2 s − / γ ˜ x − ˜ x + 32 γ = 0 (B.13)in ˜ x > which is equivalent to g ( e ) = − s for e > . The correct solution can be selectedby the special case s = 0 which should yield ˜ x = 3 / (2 γ ) as can be readily checked for theequation in g ( e ) = 0 .At the third solution e (0) ( s ) ∈ ] − e min , e min [ , the function g ( e ) has a negative slope. For s ∈ [0 , g ( − e min ) / , the solution e (0) ( s ) = ( x − x − ) / has the form x = − a + − e − iπ/ / (1 + 3 a ) (cid:16)(cid:113) b − a ) − b + (cid:17) / − e iπ/ (cid:16)(cid:113) b − a ) − b + (cid:17) / × / (B.14)with a + and b + as in (B.12), since its limit for s = 0 should be x + = 1 . For s ∈ [ g ( e min ) / , ,we can use again the symmetry of g ( e ) = − g ( − e ) , meaning the solution is then e (0) ( s ) = − e (0) ( − s ) . C Comment on the role of bosonic fluctuations
In the main text we have explored the pattern of symmetry breaking in the large- N limit.In this limit the fluctuations of the bosonic fields are completely negligible, but this is nolonger the case at finite N . Actually the celebrated Coleman-Mermin-Wagner-Hohenberg(CMWH) theorem [76–78] stipulates that continuous symmetries cannot be spontaneouslybroken at nonzero temperature in dimensions in the absence of long-range interactions.Thus any symmetry-breaking condensate at T = 0 must disappear as soon as nonzerotemperature is turned on. No massless Nambu-Goldstone modes can appear; in fact theyacquire nonzero masses, as demonstrated explicitly for O ( N ) -invariant models in [79–83].– 37 –n our model, the ground state has to be E ∝ M everywhere on the phase diagram at T > . The second-order phase transition line in figure 7 will be wiped out at finite N andbecomes a crossover. As long as N (cid:29) , the first-order transitions in figures 7, 8, and 10may well persist. However, if thermal fluctuations were so strong that the critical point at λ = 0 (present in figures 8) is destroyed, then M first-order transition lines emanating fromthe T = 0 axis would probably end at M distinct critical points.From the viewpoint of the CMWH theorem it may seem that there is no point in talkingabout symmetry breaking for T > . However this is not the case. Preceding analyses [79–83] have shown that the mass of the would-be Nambu-Goldstone modes m NG is of order F exp( − cF /T ) , where c is an O(1) pure number and F is a square of the “pion decayconstant”, also known as spin stiffness in the literature of quantum magnets. It is well knownthat F ∝ N in the large- N limit [84], so we have parametrically m NG ∼ N exp( − N/T ) which gets exponentially small at low temperatures or large N . In experiments or numericalsimulations, the correlation length ∼ m − can easily exceed the system size. The system isthen virtually indistinguishable from a genuine symmetry-broken phase. The interactionsof the would-be Nambu-Goldstone modes get weaker and weaker as the energy scale goesdown, but start to increase at the scale ∼ m NG . As far as physics at length scales (cid:28) m − is concerned, it is perfectly sensible to adopt a description based on spontaneously brokensymmetry. A detailed discussion on the consistency between the CMWH theorem and theutility of low-energy effective theories of Nambu-Goldstone modes can be found in [85].To go beyond the mean-field analysis of this paper we must employ nonperturbativemethods such as Monte Carlo simulations on a lattice, which seems feasible since the sta-tistical weight (2.6) is real and nonnegative for even N . References [1] O. Vafek and A. Vishwanath,
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