Replica Symmetry Breaking and Phase Transitions in a PT Symmetric Sachdev-Ye-Kitaev Model
Antonio M. García-García, Yiyang Jia, Dario Rosa, Jacobus J. M. Verbaarschot
RReplica Symmetry Breaking and Phase Transitions in a PT SymmetricSachdev-Ye-Kitaev Model
Antonio M. Garc´ıa-Garc´ıa, Yiyang Jia ( 贾 抑扬 ), Dario Rosa,
3, 4 and Jacobus J. M. Verbaarschot Shanghai Center for Complex Physics, School of Physics and Astronomy,Shanghai Jiao Tong University, Shanghai 200240, China ∗ Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794, USA † Center for Theoretical Physics of Complex Systems,Institute for Basic Science(IBS), Daejeon 34126, Korea Department of Physics, Korea Advanced Institute of Science and Technology,291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea ‡ (Dated: February 22, 2021)We show that the low temperature phase of a conjugate pair of uncoupled, quantum chaotic,nonhermitian systems such as the Sachdev-Ye-Kitaev (SYK) model or the Ginibre ensemble ofrandom matrices are dominated by replica symmetry breaking (RSB) configurations with a nearlyflat free energy that terminates in a first order phase transition. In the case of the SYK model,we show explicitly that the spectrum of the effective replica theory has a gap. These features arestrikingly similar to those induced by wormholes in the gravity path integral which suggests a closerelation between both configurations. For a non-chaotic SYK, the results are qualitatively different:the spectrum is gapless in the low temperature phase and there is an infinite number of second orderphase transitions unrelated to the restoration of replica symmetry. The study of nonhermitian effective Hamiltonians hasa long history [1, 2]. Perhaps the best known example isthe effective Hamiltonian that describes resonances witha finite width, for example the one that enters in the cal-culation of the S -matrix of open quantum systems suchas quantum dots [3] or compound nuclei [4]. Another ex-ample is the Euclidean QCD Dirac operator at nonzerochemical potential, which is nonhermitian with spectralsupport on a two-dimensional domain of the complexplane [5]. In hermitian theories, a phase transition mayarise due to the formation of a gap. This may also happenfor nonhermitian systems when the domain of eigenvaluessplits into two or more pieces. However, another mecha-nism is possible. Because of the nonhermiticity, the ac-tion is generally complex, and the saddle point with thelargest real part of the free energy may get nullified afterensemble averaging. In QCD at nonzero baryon chemicalpotential, the pion condensation phase is nullified so thatthe phase transition to nonzero baryon density becomesvisible [6]. The conclusion is that the phase diagram canbe altered dramatically by the nullification of the leadingsaddle point [6, 7].A second point we wish to make is about the nature ofquenched averages in nonhermitian theories. Althoughalternatives are possible [4, 8, 9], quenched averages areoften carried out by means of the replica trick [10]. How-ever, because of Carlson’s theorem [11], a naive applica-tion of the replica trick is not guaranteed to work [12].The best known example of the failure of the replicatrick is in the calculation of the quenched free energyof the Sherrington-Kirpatrick model [13], a toy model forspin glasses, which in the low temperature limit yields anegative entropy [13]. This inconsistency was ultimatelyresolved by postulating a ground state that breaks the replica symmetry [14, 15]. The problems with the replicatrick are more dramatic for nonhermitian theories as wasfirst demonstrated for QCD at nonzero chemical poten-tial µ [16]. In this case, the n replica (or n flavor) parti-tion function is given by Z n = (cid:104) det n D ( µ ) (cid:105) , (1)where the averaging is over gauge field configurationsweighted by the Euclidean Yang-Mills action. It wasshown that the quenched approximation, where the de-terminant is put to unity, is not given by lim n → Z n butrather by lim n → (cid:104) det n ( D ( µ ) D † ( µ )) (cid:105) . (2)Because the disconnected part of the partition functionis nullified due to the phase of the fermion determinant,this partition function is dominated by replica symmetrybreaking (RSB) configurations which in this context arereferred to as Goldstone bosons of a quark and a con-jugate quark. A similar RSB mechanism has been iden-tified in the context of random matrix theory, for bothhermitian [17, 18] and nonhermitian [19] random matrixensembles.The possibility of a partition function where the con-nected part dominates the disconnected part has recentlyreceived a great deal of attention in the analysis of worm-hole solutions in Jackiw-Teitelboim (JT) [20, 21] grav-ity and related theories [22–32]. The existence of thesesolutions in Lorentzian signature was first observed in[33] with the discovery of a low temperature traversablewormhole phase in a near AdS background deformedby weakly coupling the two boundaries. As temperatureincreases, the system eventually undergoes a first orderwormhole to black hole transition. By adding complex a r X i v : . [ h e p - t h ] F e b sources, it is possible to find [30] Euclidean wormholessolutions of JT gravity that undergo a similar transitionat finite temperature.Interestingly, a replica calculation [26] of the quenchedfree energy in JT gravity found that, in the low tem-perature limit, the contribution of replica wormholes isdominant. Likewise, the evaluation of the von Neum-mann entropy by the replica trick [27–29, 34] revealedthe existence of additional RSB saddle points, wormholesconnecting different copies of black holes in this context.These wormhole configurations are crucial to make theprocess of black hole evaporation consistent with unitar-ity [35].A natural question to ask is whether these replicawormholes have a field theory analogue. RSB configura-tions have indeed been explored in the SYK model withreal couplings [36, 37]. However, there is no evidencethat they dominate the partition function.In this paper, we answer the question posed in theprevious paragraph affirmatively by identifying a pair ofnonhermitian random Hamiltonians whose sum is PT-symmetric where RSB configurations are the leading sad-dle points of the action in the low temperature phase. Weconsider a nonhermitian version of the SYK model andshow that it has a phase transition from a phase domi-nated by the disconnected part of the partition functionto a phase dominated by the connected part, namely, aphase dominated by RSB configurations.The q -body SYK Hamiltonian [38–41] is defined by H SYK = ( i ) q/ (cid:88) α < ··· <α q J α ··· α q χ α · · · χ α q (3)where the χ α represent N Majorana fermions, satisfy-ing the anti-commutation relations { χ α , χ β } = δ αβ , andthe J α ··· α q are real couplings, sampled from a Gaus-sian distribution having a vanishing mean value and avariance proportional to 1 /N . The coupled SYK modelintroduced by Maldacena and Qi (MQ) in [33] consistsof a Right (R) SYK model and a Left (L) SYK modeleach with N/ iµ (cid:80) N/ k =1 χ Rk χ Lk . Although the left and right couplings,denoted by J L ( R ) α ··· α q are chosen to be the same as in theMQ model, it is also possible, as was noted by the sameauthors, to take them different. One remarkable obser-vation was made: the solution that couples the right andleft SYK continues to exist in absence of an explicit cou-pling ( µ = 0) provided that (cid:104) J L J R (cid:105) > (cid:104) J L J L (cid:105) = (cid:104) J R J R (cid:105) (4)where (cid:104) . . . (cid:105) stands for ensemble average. Since the covari-ance matrix is no longer positive, this cannot be realizedby real-valued J L and J R . However, this can be achievedfor the complex couplings J L = J + ikK, J R = J − ikK (5) with J , K independent real Gaussian stochastic variableswith the same variance and zero mean.Before continuing, let us analyze the quenched free en-ergy of the single-site SYK we have just introduced: (cid:104) log Z (cid:105) = (cid:104) log | Z |(cid:105) + i (cid:104) arg Z (cid:105) , (6)where Z is the partition function for a specific realizationof the couplings. Since the phase of the partition func-tion does not have a preferred direction, we expect that (cid:104) arg Z (cid:105) = 0. We conclude (cid:104) log Z (cid:105) = 12 (cid:104) log ZZ ∗ (cid:105) (7)for a theory where Z and Z ∗ have equal probability.In particular, the quenched free energy is given by thereplica limit 12 lim n → (cid:28) ( ZZ ∗ ) n − n (cid:29) . (8)Therefore, we arrive naturally at a system of two conju-gate SYK Hamiltonians only coupled through the prob-ability distribution. The Hamiltonian corresponding to ZZ ∗ is given by H SYK ⊗ ⊗ H † SY K , (9)which is exactly the two-site Hamiltonian proposed in[33] with the explicit coupling turned off. This Hamil-tonian is P T symmetric [1], where the P operator inter-changes the L and R spaces and T is the tensor product oftwo copies of the time reversal operator for the standardSYK model [42–44].Next, we calculate the partition function (cid:104) ZZ ∗ (cid:105) forlarge N . We expect that in this limit (cid:104) ( ZZ ∗ ) n (cid:105) = (cid:104) ZZ ∗ (cid:105) n (10)so that the replica limit (8) requires only the computa-tion of (cid:104) ZZ ∗ (cid:105) . We consider a pair of nonhermitian SYKmodels with couplings (5) for k = 1. The spectral densityis given by a disk with radius E in the complex plane.Although the eigenvalue density is rotationally invariant,it is not constant as is the case for the large N limit of theGinibre [45, 46] ensemble or random matrices (see Fig.1). However, we expect that eigenvalue correlations arein the universality class of the Ginibre model. If the aver-aged eigenvalue density is denoted by ρ ( z ), the two-levelcorrelation function is given by ρ ( z , z ) = R c ( z , z ) + δ ( z − z ) ρ ( z ) + ρ ( z ) ρ ( z ) . where R c ( z , z ) is the averaged connected two-pointcorrelation function not including the self-correlations.The partition function is given by (cid:104) ZZ ∗ (cid:105) = (cid:104) Z (cid:105)(cid:104) Z ∗ (cid:105) + (cid:104) ZZ ∗ (cid:105) c (11)where (cid:104) ZZ ∗ (cid:105) c = (cid:82) d z d z ρ c ( z , z ) e − β ( z + z ∗ ) , ρ c ( z , z ) = R c ( z , z ) + δ ( z − z ) ρ ( z ) and EigenvaluesCircle - - - - - - - - - - - - xy FIG. 1. The eigenvalue density, obtained from exact diagonal-ization, for one realization of the q = 4 , k = 1 nonhermitianSYK model with N/ (cid:104) Z (cid:105) = (cid:82) d zρ ( z ) e − βz . Because the eigenvalue densityhas rotational invariance, we can use the mean valuetheorem to show that the partition function is indepen-dent of β and given by the normalization of ρ ( z ), whichwe denote by D , (cid:104) Z (cid:105) = D .To evaluate the second term of (11), we use the sumrule (cid:90) d z ( R c ( z , z ) + δ ( z − z ) ρ ( z )) = 0 , (12)and the fact that the correlations are short-range [45, 46]with the connected correlator taking the universal form R c ( z , z ) = R unv2 c ( (cid:112) ρ (¯ z )( z − z )) ρ (¯ z ) , (13)where ¯ z = ( z + z ) /
2. We thus have that | z − z | < / √ D region gives the dominant contribution and wecan Taylor expand the exponent in (11) in powers of β Im( z − z ). The zero order term vanishes because ofthe sum rule (12), the linear term vanishes because theprobability distribution is even under complex conjuga-tion. After performing the integral over z − z , we obtainthe connected partition function (cid:104) ZZ ∗ (cid:105) c = β (cid:104) ζ (cid:105) (cid:90) | ¯ z | 1, where we also find a firstorder phase transition with T c ∼ k for small k .We have thus observed that the leading exponent ofthe disconnected part of the partition function is nulli-fied by the phase of the Boltzmann factor so that thecontribution due to the connected part of the two-pointcorrelation function becomes dominant. The free energybehaves as if the system has a gap. Both features aretypical of RSB configurations.In order to make this connection more explicit, we showthat these results can also be obtained by solving theSchwinger-Dyson equations, in the Σ G formulation of thetwo-site SYK model [33, 47] which is equivalent to per-forming the replica trick and then solving the model inthe saddle point approximation. For T > T c and k = 1,the solution with the free G and Σ is dominant so thatonly the kinetic term of the Lagrangian remains. As aconsequence, the free energy is − T log 2 / T < T c , a non-trivial RSB solution becomes dominant which results ina constant free energy up to exponentially small correc-tions. Similar results can be derived for k < 1, where,in agreement with the previous analytical calculation, wehave also found T c ∼ k . Indeed, this feature is shared byboth Euclidean [30] and traversable [33, 48] wormholes.Details of this and the previous analytical calculation willbe given elsewhere [49]. Data pointsSinh fit 100 300 500 700 900 τ - - - - G LR ( τ ) T G LR ( 0, T ) Data pointsQuadratic fit k E g ( k ) FIG. 3. Top: G LR ( τ ) from the solution of the Schwinger-Dyson equations for the SYK model (9) with q = 4, T =0 . k = 0 . G LR (0), versus temperature for k = 0 . 5. Bottom: Theenergy gap E g for T = 0 . (cid:28) T c as a function of k fromthe fit (17). For traversable wormholes [33], the spectrum isgapped. Physically, it is related to the interaction-driventunneling between the left and right sites. The existenceof the energy gap can be demonstrated [33] directly fromthe effective boundary gravity action or from the expo-nential decay of the left-right Green’s function, G LR ( τ ),of the two-site SYK model for low temperatures. Westudy whether a similar gap exists in the two-site non-hermitian SYK. We stress the gap in this case is not aproperty of the microscopic Hamiltonian (9) but only ofthe resulting replica field theory after ensemble average.Since G LR ( β/ − τ ) = − G LR ( τ − β/ 2) we employ theansatz [33] G LR ( τ ) ∼ sinh E g ( β/ − τ ) , (17) T - - - N dF ( T ) d T FIG. 4. Derivative of the free energy for the nonhermitian q = 2 , k = 1 SYK model. On the way to T = 0, the systemundergoes infinitely many second order phase transitions. For T > /π , it becomes a constant which is a typical feature offree fermions. where the gap E g is a fitting parameter. The fit is ex-cellent except for very small times [50], see Fig. 3. Thisreinforces the picture that RSB configurations mediatetunneling between the two sites even though there is nodirect coupling term in the Hamiltonian. We note that G LL and G RR show a similar decay. In Fig. 3, we alsoshow that the gap E g depends quadratically on k . Itwould be interesting to understand this exponent fromthe gravity side. We propose G LR (0) as the order pa-rameter of the transition since a non-vanishing G LR isa distinctive feature of RSB configurations. Results de-picted in in Fig. 3, confirm that G LR (0) remains almostconstant in the wormhole phase and vanishes for T > T c .The studied q = 4 SYK model is quantum chaotic [41].We expect very similar results in others quantum chaoticsystems such as q > q = 2 nonher-mitian SYK model which admits an explicit analyticalsolution of the Schwinger-Dyson equations. It can alsobe solved, see [49] for details, by mapping it onto a modelof free fermions. The free energy can be expressed, seealso [49], as a sum over Matsubara frequencies. Eachtime a new Matsubara frequency enters the sum, by low-ering the temperature, a second order phase transitionoccurs. Kinks in − dF/dT , depicted in Fig. 4, indicatethe positions of the critical temperatures. The propaga-tor G LR ( τ ) can be also expressed as a finite sum overMatsubara frequencies so it does not depend exponen-tially on τ . Because of the absence of a gap, there isno direct relation between RSB and wormholes whichfurther suggests that the physics is qualitatively differ-ent from the quantum chaotic case. Further research isneeded to delimit the importance of quantum chaos inRSB.In summary, we have provided evidence that RSBconfigurations dominate the low temperature phase ofthe partition function of pairs of random non-hermitian,quantum chaotic systems whose sum is PT-symmetric.These field theory configurations mimic the contributionof wormholes in the gravitational path integral. 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