Critical Fluctuations at a Many-Body Exceptional Point

PHYSICAL REVIEW RESEARCH 2, 033018 (2020)

Critical ﬂuctuations at a many-body exceptional point

Ryo Hanai 1,2,* and Peter B. Littlewood1,3 1James Franck Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA 2Department of Physics, Osaka University, Toyonaka 560-0043, Japan 3Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA

(Received 15 September 2019; revised 14 June 2020; accepted 17 June 2020; published 6 July 2020)

Critical phenomena arise ubiquitously in various contexts of physics, from condensed matter, high-energy physics, cosmology, to biological systems, and consist of slow and long-distance fluctuations near a phase transition or critical point. Usually, these phenomena are associated with the softening of a massive mode. Here, we show that a non-Hermitian-induced mechanism of critical phenomena that does not fall into this class can arise in the steady state of generic driven-dissipative many-body systems with coupled binary order parameters such as exciton-polariton condensates and driven-dissipative Bose-Einstein condensates in a double-well potential. The criticality of this “critical exceptional point” is attributed to the coalescence of the collective eigenmodes that convert all the thermal-and-dissipative-noise-activated fluctuations to the Goldstone mode, leading to anomalously giant phase fluctuations that diverge at spatial dimensions d 4. Our dynamic renormalization group analysis shows that this gives rise to a strong-coupling fixed point at dimensions as high as d < 8 associated with a universality class beyond the classification by Hohenberg and Halperin, indicating how anomalously strong the many-body corrections are at this point. We find that this anomalous enhancement of many-body correlation is due to the appearance of a sound mode at the critical exceptional point despite the system’s dissipative character.

DOI: 10.1103/PhysRevResearch.2.033018

I. INTRODUCTION many-body phenomena associated with the divergence of length scales and timescales near a phase transition or critical Understanding and manipulating dissipation effects in point [14]. Critical points (CPs) and EPs have a crucial feature open quantum systems [1] is increasing in importance, due to in common: the occurrence of a gap closure [15], where in the its crucial role in designing new optical devices and perform- latter the energy difference between the two eigenstates van- ing quantum computation. Particularly, intense interest has ishes as they coalesce. This raises the issue whether EPs can recently emerged to the study of “exceptional points (EPs)” possess similar properties to CPs such as critical fluctuations [2–4] that can arise in these dissipative devices. An EP is a when EPs occur in a many-body context. Indeed, some EPs point where two (or more) eigenstates that characterize the of a non-Hermitian many-body Hamiltonian are shown to be dynamics coalesce owing to the non-Hermitian nature of the marked as a quantum critical point [16–18]. However, these system, such that they lose their completeness, leading to a were found only in the lowest-energy eigenstate, limiting to spectral singularity. It turns out that this singularity gives rise situations where the thermal or dissipation-induced noise is to a number of counterintuitive phenomena in the vicinity negligible or can be excluded by postselection. of the EP, such as loss-induced transmission [5], unidirec- Here, we propose a non-Hermitian-induced critical phe- tional invisibility [6], enhanced quantum sensitivity [7,8], and nomenon activated by the thermal and dissipative noise that chiral behavior [9]. These concepts have been proven to be occurs at a many-body EP, which can arise in generic driven- applicable to a rich variety of many-body systems such as dissipative quantum many-body systems composed of cou- superconductivity [10], atomic gases [11], spin chains [12], pled binary order parameters. We illustrate this phenomenon and correlated materials [13], where EPs arise in the spectral by analyzing a two-component driven-dissipative condensate, properties of these systems. where examples in scope include exciton-polariton conden- One of the most striking phenomena that arise in many- sates [19] (composed of excitons and photons) and driven- body systems are critical phenomena, which are collective dissipative Bose-Einstein condensates (BECs) in a double- well potential [20–22]. Strikingly, our analysis reveals that our many-body EP (which we refer to as critical EP, CEP) exhibits *[email protected] anomalously giant phase fluctuations that diverge at spatial dimensions d 4, which is to be compared to the conventional Published by the American Physical Society under the terms of the case where the divergence only happens at d 2[23,24]). Creative Commons Attribution 4.0 International license. Further We also find the emergence of a sound mode despite the pres- distribution of this work must maintain attribution to the author(s) ence of the dissipation, which turns out to anomalously en- and the published article’s title, journal citation, and DOI. hance many-body correlation effects that become relevant at

2643-1564/2020/2(3)/033018(16) 033018-1 Published by the American Physical Society RYO HANAI AND PETER B. LITTLEWOOD PHYSICAL REVIEW RESEARCH 2, 033018 (2020)

(a) (b) (a) (b) (c)

(d) (e) (f) FIG. 2. Non-Hermitian phase transition and the critical exceptional point (CEP). (a) Schematic phase diagram of driven- dissipative many-body systems with coupled binary order parameters, in terms of the generic input parameters (α1,α2 ). The solid line is the phase boundary of the non-Hermitian-induced phase transition, which exhibits an end point at the critical exceptional point (CEP). (b) Demonstration of the appearance of the CEP for systems described by the coupled driven-dissipative Gross-Pitaevskii equations. At a small decay rate κ/g < 1, as the density of the loss FIG. 1. Difference between the conventional critical point (CP) component n =|0|2 increases, the steady state exhibits a phase and the critical exceptional point (CEP). (a)–(c) Schematic explana- l0 l transition signaled by a discontinuity in the condensate emission tion of the criticality of the CP in conventional equilibrium cases [the energy E. The CEP found at κ/g = 1, represented by the star, marks figure describes the case of U(1)-symmetry-breaking transition). As the end point of the phase boundary. The discontinuity is absent at one sweeps the parameter α to the CP, the massive longitudinal mode κ/g > 1. We set ωl/g =−1.9,ωg/g =−2, and Ug/g = 0.1. δϕ itself softens at the CP caused by the flattening of the free-energy landscape V (where ϕ is the order parameter). Note how the two eigenmodes, the Goldstone mode δϕ⊥ and the longitudinal mode δϕ, are always orthogonal. (d)–(f) Schematic explanation of the thus usually gives a diffusive mode [27,28]) that turns out to criticality of the CEP. In the non-Hermitian case, the two eigenmodes lead to the anomalous enhancement of many-body correlation are not necessarily orthogonal. As a result, while the Goldstone’s effects. δϕ theorem ensures the Goldstone mode ⊥ to be an eigenmode, the It is worth noting that our CEP is found in the δϕ other eigenmode + is not pointing to the longitudinal direction. At steady state that characterizes the long-time behavior of the the CEP, where the two eigenmodes coalesce to the Goldstone mode, driven-dissipative many-body system, in stark contrast to anomalously giant phase fluctuations occur which diverge at d 4, Refs. [16–18] where the lowest-energy state (that exhibits as well as anomalously enhanced many-body correlation effect. [We criticality) needs to be postselected after measurement since briefly note that the free energy drawn in (d)–(f) is schematic.] the system would eventually approach a trivial state such as infinite temperature or zero-particle state by the heating or dimensions as high as d < 8 (which is to be compared to, e.g., particle loss processes. We also remark that, since the critical the Ising model or XY model where the many-body correla- fluctuations of our system are activated by the thermal and tion effect is relevant at d < 4). We demonstrate this physics dissipative noise, our CEP can be regarded as semiclassical by performing a dynamic renormalization group analysis [25] and dynamical critical phenomena, which is a different class to identify a strong-coupling fixed point in the vicinity of of criticality from the quantum and static critical phenomena dc = 8 dimensions associated with a new universality class at many-body EPs found in the previous studies [16–18]. beyond the classification by Hohenberg and Halperin [25]. The discovered mechanism is generic; the CEP should These anomalous critical fluctuations are caused by a fun- arise as long as the many-body system is (1) driven dissipa- damentally different mechanism of criticality from the con- tive, (2) composed of two components, and (3) exhibits spon- ventional critical phenomena. At conventional CPs, the criti- taneous symmetry breaking since the rise of our CEP is ulti- cal fluctuations arise due to the softening of the longitudinal mately due to the nonorthogonality of collective eigenmodes. mode, originated from the flattening of the free-energy land- Our previous work [29] has shown that driven-dissipative sys- scape [Figs. 1(a)–1(c)]; longitudinal and transverse fluctua- tems composed of two components exhibit a non-Hermitian tions are separable. In the driven-dissipative case, on the other phase transition with an end point of its phase boundary hand, these collective eigenmodes need not be orthogonal marked by a many-body EP (as schematically shown and because of the non-Hermitian nature of the system [Figs. 1(d) demonstrated in Fig. 2), proposed as an interpretation of and 1(e)]. We show here that these collective eigenmodes the phase transition observed in some polariton experiments coalesce (i.e., they become collinear) at the CEP [Fig. 1(f)]; in the U(1)-broken phase [30–35]) (the so-called “second hence, thermally activated fluctuations are all converted to threshold”). The two-component nature offers two branches of the Goldstone mode. This peculiar property turns out to give eigenstates that the system can condense into, letting us define rise to the anomalously giant phase fluctuations, which is two distinct phases of matter not distinguished by symmetry, reminiscent of the noise-induced pattern formation due to the analogous to liquid and gas in an equilibrium system. Because nonorthogonality of eigenstates discussed in Ref. [26]. We of the driven-dissipative feature, the CEP arises as a point also find that this mode coalescence gives rise to a sound mode where the two phases coalesce, making that the end point of (despite the dynamics is overdamped by the dissipation and the phase boundary [29]. (See Appendix A for details.)

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T T Due to the generality of its mechanism, it should be pos- with = ( l, g) , η = (ηl,ηg) , and sible to implement the CEP in generic driven-dissipative sys- A (∇2 ) tems by engineering the system to be binary, e.g., by preparing GP a double-well potential or checkerboard lattice structure of ω − κ − ∇2 = l i Kl g , gain and loss components. Considering the rapid experimental 2 2 g ωg + iP − K˜g∇ + U˜g| g| development in various driven-dissipative many-body plat- forms, such as ultracold atoms [36], circuit QED [37,38], where α = l (g) labels the loss (gain) component with its net photon BEC [39], plasmonic-lattice polariton BEC [40], and one-body decay (gain) rate given by κ(P). Here, ωl(g) and strongly interacting photons [41], it seems promising to real- l(g)(r, t ) are the energy and macroscopic wave function of ize the CEP in these systems. An exciton-polariton condensate the loss (gain) component, respectively. g is the intercompo- [19] and a plasmonic-lattice polariton BEC [40] seem espe- nent coupling and K˜g = Kg − iDg, U˜g = Ug − ivg are com- cially promising since they are both composed of two complex coefficients, where Dg, Ug, vg are the diffusion constant, ponents (i.e., photons and excitons for the former and surface repulsive interaction strength, and two-body loss rate that lattice resonance mode and the dye molecule excitation modes gives nonlinear saturation of the gain component, respectively, for the latter) and thus meet all three criteria raised above. and Kl(g)determines the kinetics of the loss (gain) component. Especially in the polariton system, both the first-order-like We have assumed in this model that only the gain components phase transition [30–35]) [corresponding to the solid line in interact and have a nonlinear saturation, which, however, does Fig. 2(a)] and the crossover behavior [42] has already been not affect the critical properties of the CEP. White noise observed [29]. This makes us strongly expect that the realiza- ηα (r, t ) that originates from thermal fluctuations and dissi- ∗ d tion of the CEP can be achieved with the current experimental pation is characterized by ηα (r, t )ηβ (r , t )= αδαβδ (r − techniques, by tuning the pump power and detuning. (See r )δ(t − t ), and ηα (r, t )=ηα (r, t )ηβ (r , t )=0, where the discussion in the Supplemental Material in Ref. [29]for the noise level α is determined from the sum of the one-body further details on comparison to polariton experiments.) gain and loss rates of respective components [49]. In general, The appearance of the CEP does not rely on the fact the the noise may not satisfy the dissipation-fluctuation theorem system is composed of quantum particles; analogous phenom- in driven-dissipative systems. ena may appear even in classical many-body systems as well, We stress that we are interested in the collective excitation when the system is intrinsically driven out of equilibrium, properties of the steady state (enabled to stabilize thanks as in active matter systems [43]. Indeed, recently, we have to the nonlinear saturation effect), in stark contrast to the found the emergence an analogous point [44] in a generalized majority of problems discussed in the field of non-Hermitian Vicsek model of flocking [45,46] and Kuramoto model of syn- physics where transient dynamics of the excited states are chronization [47] with nonreciprocally interacting (classical) often discussed [50]. Despite its similarity, the consequences agents. There, the CEP was found to arise as a phase transition are crucially different [29] where it exhibits a first-order-like point to a time crystal phase. A similar point has been found phase transition associated with a critical point as shown and in the neural nets of synaptically coupled excitatory and demonstrated in Fig. 2. (See also the discussion below and in inhibitory neurons in the neocortex [48] (where the CEP is Appendix A.) We will see in the following that anomalous phrased as the Bogdanov-Takens point). We expect to find critical phenomena arise at this point, which are absent in more in the near future. the conventional EPs. We also emphasize that this stochastic equation of motion incorporates beyond-mean-field effects, due to the white noise that continuously fluctuates the con- II. NOISY DRIVEN-DISSIPATIVE GROSS-PITAEVSKII densate about the steady state. EQUATION FOR BINARY CONDENSATES Before going into the analysis of critical fluctuations, we first give a brief review on the steady-state properties of Below, we consider a driven-dissipative, repulsively in- this system in the absence of noise ηα (r, t )[29]. At low teracting BEC composed of two components. Examples of pump power P(< Pth) where the gain is not large enough to such systems include exciton-polariton condensates, driven- compensate the loss, the macroscopic wave function dies out dissipative condensates in a double-well potential, plasmonic- to a normal state, i.e., α (t →∞) = 0. On the other hand, lattice polariton condensates, etc. To our knowledge, this is the at a sufficiently high pump power P > Pth, the amplitude of simplest system that exhibits a non-Hermitian-induced phase the macroscopic wave function starts to grow until the non- transition associated with the rise of the CEP [29]. linear saturation makes the system approach a uniform steady Our goal is to reveal the dynamic critical properties of 0 0 −iEt state. The macroscopic wave function α (t ) ≡ αe then the CEP. For a one-component case, it has been shown follows the relation [49] by coarse graining the Keldysh partition function [15] 0 = 0,0 0 , that the critical properties of a driven-dissipative condensate E α AGP l g αβ β (2) can be captured by taking the noise average of the noisy β=l,g driven-dissipative Gross-Pitaevskii (GP) equation [27,28]. where E is the (real) condensate emission energy. This steady- It is straightforward to extend this discussion to our two- 0 state condition determines the amplitude of α for a given component case [29], to show that the stochastic equation of setup. We emphasize that the above equation is essentially motion to consider is different from conventional non-Hermitian quantum mechan- ics problems [50], where the nonlinear effect (that makes it 2 i∂t (r, t ) = AGP(∇ )(r, t ) + η(r, t ), (1) possible to reach a steady state) is present and the emission

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2 energy E, corresponding to the eigenvalue, is necessarily where γ = sl − sg, v = [2(sgνlg + slνgl ) + (sl + sg)(νgg − 2 real because the system is in the steady state. As detailed νll )]/2, and D = (νgg + νll )/2. We remark that γ and v in Appendix A, the above steady-state restriction allows us can take a negative value, which may cause a dynamical to classify the solutions into two types [29], that satisfy instability, not uncommon for systems with loss and gain. In E = E− and E = E+, respectively, where E± are the two this paper, we assume that those situations are avoided by eigenvalues of the matrix AGP. These physically correspond having a large enough nonlinear saturation vg and diffusion 2 to condensation into the lower and upper branch, respectively. constant Dg in the gain component, to have γ,v 0. (See Crucially, because of the non-Hermitian nature of AGP,the the discussions in Appendix B.) two eigenvectors of AGP need not be orthogonal, giving rise When the system is away from the CEP, i.e., γ>0, to a point where the two eigenvectors coalesce: the CEP. This the eigenmodes are given by the diffusive Goldstone mode 2 corresponds to the coalescence of the two solutions. The CEP ω−(k) ∝−ik [27,28] and a relaxational mode ω+(k) =−iγ . is proven to mark the end point of the first-order-like phase Since the relaxational mode would be gapped away as we boundary that can occur between the two types of solutions further coarse grain the system and thus play no role in the ef- [29], in a similar manner to the critical point of a liquid and fective low-energy physics, the physics is essentially the same gas where the two phases also coalesce. This is demonstrated as the one-component case and recovers the dynamic scaling in Fig. 2(b). The CEP is found at the parameter that satisfies behavior [49,51,53–55] known to obey the KPZ scaling [52]. κ = ω = ω + |0|2 g and l g Ug g . (See Appendix A for details.) The situation is dramatically different at the CEP γ = 0. In We note that there are only two parameters to tune, which is this case, we find both eigenmodes ω±(k)tobegapless,which the same number of parameters for the conventional EP and interestingly are sound modes, critical point of liquid-gas phase diagram to be realized. ω =± | |− 2, For the investigation of criticality, it is useful to rewrite ±(k) v k iDk (5) the GP equation (1) in terms of the noise-activated ampli- δ| , | δθ , showing that both components play a role and thus modify the tude α (r t ) and phase fluctuations α (r t ) by rewrit- scaling properties. ing the macroscopic wave function as the deviation from 0 −iEt 0 A crucial observation is that this gap closure is associated the steady state α (r, t ) = [α + δα (r, t )]e = [|α|+ θ 0+δθ , − θ 0 with the coalescence of the collective modes, which is cru- δ| , | i( α α (r t )) iEt 0 =|0 | i α α (r t ) ]e e , where α α e . By inte- cially different from just being degenerate. The eigenmodes grating out δ|α (r, t )| up to linear order and further assuming in the uniform limit k → 0aregivenby that the amplitude dynamics is overdamped [51], we arrive at δθ 1 δθ s /s a Kardar-Parisi-Zhang (KPZ) type [52] stochastic equation of l ∝ , l ∝ l g (6) motion (see Appendix B) δθg 1 δθg 1 2 for the corresponding eigenenergies ω−(k) and ω+(k), respec- ∂ δθα (r, t ) = Wαβ (∇)δθβ (r, t ) + tα ( δθ(r, t )) t tively. The former in-phase mode, i.e., the Goldstone mode, is β=l,g ensured to be gapless by the global symmetry under the trans- α , → , iφ + λβγ (∇δθβ (r, t )) · (∇δθγ (r, t )) formation α (r t ) α (r t )e . These modes coalesce at = β,γ=l,g the CEP sl sg as schematically described in Figs. 1(d)–1(f), giving rise to the gap closure. This gap-closing mechanism + ξα (r, t ), (3) is fundamentally different from that in the conventional CPs, with where the longitudinal mode itself softens by the flattening of the free energy but is still orthogonal to the Goldstone mode − + ν ∇2 + ν ∇2 ∇ = sl ll sl lg , [Figs. 1(a)–1(c)]. W ( ) 2 2 −sg + νgl∇ sg + νgg∇ The rise of the sound mode at the CEP [Eq. (5)] is one of the outcomes of this mode coalescence. Noting that Wˆ (k = 0) and δθ(r, t ) = δθl(r, t ) − δθg(r, t ). Here, we have retained is at an exceptional point at the CEP, the linear |k| dependence the most relevant nonlinear terms and the real parame- in the dispersion originates from the square-root behavior α ters sα,ναβ,λβγ are determined from the parameters used in Eq. (5), a typical property seen in the vicinity of the in AGP, where its explicit form is given in Appendix B. exceptional point [7,8]. In fact, the appearance of the sound ξα (r, t ) is a white noise for the phases, characterized by mode plays a crucial role in exhibiting anomalously large d ξα (r, t )ξβ (r , t )=σαβδ (r − r )δ(t − t ) and ξα (r, t )=0. many-body correlation effects, as we show in the next section. This peculiar collective mode coalescence at the CEP gives III. ANOMALOUSLY GIANT PHASE FLUCTUATIONS rise to anomalously giant phase fluctuations. To see this in a transparent way, it is useful to triangularize the kernel AT THE CEP † T W (k) by using an orthogonal basis U (k) ≡ (u⊥(k), u(k)) We show below that an anomalous critical behavior ap- [instead of diagonalizing W (k), which is ill defined at k → 0 pears at the CEP. At the CEP, we find sl = sg = κ,asshown at the CEP] as in Appendix B. Let us start with a linearized theory (i.e., −iω− k ζ tα = λαβ = 0). By solving the secular equation det[−iω1 − ¯ = † = ( ) . W (k) U (k)W (k)U (k) − ω W (k)] = 0, the eigenenergies are given by 0 i +(k) Here, we have chosen one basis vector as the Goldstone mode 1 2 2 ω± = − γ + ± −γ 2 + 2 , =−ω (k) 2 [ i( 2Dk ) 4v k ] (4) W (k)u⊥(k) i −(k)u⊥(k), and the other basis, satisfying

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W (k)u(k) =−iω+(k)u(k) + ζ u⊥(k), is chosen as the lon- Ising model, where the correlation function in the vicinity of gitudinal direction that is perpendicular to the Goldstone the critical point is given by mode u(k) · u⊥(k) = 0. Importantly, the off-diagonal piece σ m m 2 ζ = sl + sg arises from the non-Hermitian nature of W , which δφ δφ − ∼ + + , (k) ( k) 2 1 2 O 2 (10) converts the longitudinal fluctuations to the Goldstone mode. νk νk νk ¯ 0 ,ω = − ω − The Green’s function in this basis Gss (k ) ([ i 1 where δφ are the fluctuations of the Ising spins, m is the −1 W¯ (k)] ) is given by ν σ ∼ ss damping gap, is the diffusion constant, and kBT is the i −ζ thermal noise strength (T is the temperature). m characterizes ω−ω ω−ω ω−ω G¯ 0(k,ω) = − (k) [ − (k)][ + (k)] . (7) the distance from the critical point. Here, we have expanded 0 i ω−ω+ (k) the correlator in terms of m, which is justified at the short- Here, we find that the off-diagonal, non-Hermitian- wavelength regime√ compared to the correlation length, i.e., −1 0 |k|ξ ∼ m/ν. However, the fluctuations are dominated induced component G¯ ⊥ exhibits a peculiarly strong singularity at the CEP, involving two gapless poles in the long-wavelength regime longer than the thermal de at ω = ω±(k). As a result, noting that the phase Broglie wavelength [which is the wavelength that makes the σ/ ν 2 | | < λ−1 ∼ fluctuations are expressed in terms of Green’s functions √leading factor of Eq. (10), ( k ), order unity] k ∼ T σ/ν δθ¯ ,ω = ¯ 0 ,ω ξ¯ ,ω . Thus, the enhancement of the correlation is realized as s=⊥, (k ) s=⊥, Gss (k ) s (k ) [where δθ¯ ,ω = δθ ,ω ξ¯ ,ω = when there exists a momentum window that satisfies both, i.e., s(k ) α=g,l Usα (k) α (k ) and s(k ) λ < ξ ξ ,ω T ∼ ,or α=g,l Usα (k) α (k )], the most strongly diverging term 0 < σ. in the equal-time correlation function involves two G¯ ⊥’s, m ∼ (11) δθα(=l,g)(r)δθβ(=l,g)(r ) Physically, this means that the critical fluctuations appear when the damping rate m is small enough for the thermal noise c ∼ dk kd−1eik·(r−r ) to populate the gapped mode before it damps. Note how this critical region scales linearly in terms of the noise strength σ . 0 ∞ ω In the vicinity of the CEP, on the other hand, the correlator d 0 0 × G¯ ⊥(k,ω)σG¯ ⊥(−k, −ω) (8) is expanded in terms of the distance from the CEP γ as −∞ 2π γ γ 2 c A − · − A δθα (k)δθβ (−k)∼ 1 + + O . (12) ∼ dk kd 1eik (r r ) , (8) 4 2 2 4 k Dk Dk 0 k ξ which diverges for d 4, implying the absence of long-range Therefore, the correlation length CEP (which sets the wave- order at the CEP. Here, “∼” means that we have retained the length that√ justifies the above expansion) is estimated as ξCEP ∼ D/γ , while the effective thermal de Broglie length most strongly diverging term and − [that makes A/k4 > O(1) at |k| ∼< (λCEP ) 1] is estimated as 2 T κ σ λCEP ∼ −1/4 A ≡ , (9) T A . Following the same logic as above, we can 2 λCEP < ξ Dv estimate the critical region as T ∼ CEP,or σ = σ + σ − σ − σ / with longitudinal noise ( ll gg lg gl ) 2. (We κ σ ζ = κ < D have used the relation 2 that holds at the CEP.) c is an γ ∼ γc = . (13) v ultraviolet cutoff. The obtained phase fluctuations are anoma- √ lously giant, in the sense that the phase fluctuations in the Thus, the critical region scales as γ ∼ σ, in contrast to the c δθ 2∼ c d−1 −2 conventional case ( ) 0 dk k k diverges only for linear scaling for conventional critical points [Eq. (11)]. This d 2, as stated in the Mermin-Wagner-Hohenberg theorem difference attributes to the collective mode coalescence to the [23,24]. As is clear from this structure, the giant fluctuations Goldstone mode at the CEP. Since, in the vicinity of the CEP, are activated by the longitudinal noise σ that gets converted the gapped mode is almost collinear to the Goldstone mode, it to the Goldstone mode through the non-Hermitian-induced is easier for the noise to excite this mode compared to that in mixing ζ = 2κ. the conventional case. It is interesting to compare this result to an O(N) model in a static random field studied by Imry and Ma [56], where IV. STRONG-COUPLING FIXED POINTS AT SPATIAL the correlation function of the transverse magnetization also DIMENSION d < dc = 8 diverges at d 4. In that case, the anomalous fluctuations emerge from coupling between a static random field and We now put back the nonlinear terms to analyze the the Goldstone mode, causing the system to separate into dynamic critical behavior of the CEP. Following the stan- domains. Our case may be viewed as a dynamical extension of dard procedure of the dynamic renormalization group method this discussion, where the coupling between the longitudinal [25], we first compute the perturbative correction from the white noise and the Goldstone mode triggers the anomalous nonlinear couplings and then rescale space, time, and phase fluctuations. fluctuations according to

Before closing this section, we estimate how close to l zl χα l r → e r, t → e t,δθα → e δθα (14) the CEP one should tune the parameters to see the above enhancement of the phase fluctuations. Let us consider first to formulate the flow equations. Finding the fixed point of the the conventional critical phenomena, say of a (relaxational) flow equations provides the universal scaling features of the

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CEP, such as the appearance of the linear dispersion at the CEP [i.e., the 2 − term ±v|k| in Eq. (5)]. This makes the diffusion term −iDk χα +χβ t t δθα (r, t )δθβ (r , t )=|r − r | fαβ , (15) dangerously irrelevant, i.e., D → 0, which makes the system | − |z r r infinitely sensitive to the noise that activates the fluctuations where fαβ (x) is a scaling function. [see Eq. (9)]. This singular sensitivity turns the effective In our two-component case, however, not all of the pa- nonlinear coupling relevant even at dimensions where the rameters can be fixed simultaneously under renormalization. nonlinear couplings λ⊥⊥ and t themselves are irrelevant For instance, the rescaling (14) changes κ, v, and D as κ → because the factor D5 in the denominator of Eq. (19)alsoflows κezl , v → ve(z−1)l , and D → De(z−2)l , respectively, but when to zero as approaching the fixed point. we require, as usual, the lowest-order kinetic term (i.e., the At spatial dimensions close to the upper critical dimension velocity v) to be fixed, κ flows to infinity as l →∞while d = dc − = 8 − , we find a strong-coupling fixed point 2 2 D → 0. Here, we require A = κ σ/(Dv ), γ˜ ≡ γ/D, and v at (γ ˜∗,∗) ≈ (0,(32 × 8)/(5C8)), associated with a new to be fixed since the correlation function within the linearized universality class with critical exponents theory is determined solely by these parameters in the vicinity → χ ≈ χ − , z = z . (21) of the CEP in the limit D 0, as seen in Eq. (12). We have G 10 G implicitly assumed here that the strong-coupling fixed point We briefly note that, as a consequence of the two-component is not very far away from the Gaussian fixed point (γ∗ = = λγ = nature, may take either sign. Since cannot change its tα∗ αβ∗ 0), which we will justify by restricting ourselves / = = to spatial dimensions close to the upper critical dimension sign during the flow because d dl 0at 0 as seen d . The above assumption also requires χ = χ (≡ χ ). In the in Eq. (20), the system can only flow to the above-obtained c l g strong-coupling fixed point when the bare parameter (l = 0) linearized theory, we get the roughening exponent χ = χG = (4 − d)/2 and the dynamic exponent z = z = 1. is positive, which is determined by the balance between the G diffusion rate of the pump and the nonlinear loss rate of the The flow equations within the one-loop order read as (see = < Appendix C for derivation) field [see Eq. (B22)]. Otherwise, when (l 0) 0, the flow direct toward →−∞, implying the existence of another γ d ˜ C8 C10 distinct phase of matter, which will be explored in the future = 2 − + γ˜ γ,˜ (16) dl 32d 16d work. dv As a final remark, we ask the following: How close to the = (z − 1)v, (17) dl CEP should we be for the nonlinear many-body correlation effect to be detectable? We attempt to answer this by observ- dA C C = 4 − d − 2χ − 8 + 10 γ˜ A, (18) ing that our CEP flow equations (16)–(20) are derived under dl 32d 16d the assumption that we are in the vicinity of the CEP, where where we have retained only the most relevant coupling. we can expand the valuables in terms ofγ ˜ (l). This is only = / π d d−i γ l −2 < Here, Ci (Sd (2 ) ) c with Sd the surface area of a justified when ˜ ( ) c 1. d-dimensional sphere. Since D is (dangerously) irrelevant and First, if the bare quantities are in the regimeγ ˜ (l = −2 > γ κ flows to infinity, the most relevant coupling is the term that 0) c 1, since Eq. (16) makes ˜ (l) to be monotonically has the lowest order on D and the highest order on κ, which increasing function in terms of l, there is no region that is turns out to be appropriate to use our CEP flow equations (16)–(20). Thus, σ we would not see any CEP physics in this regime. This is con- t 2 2 ≡ (tv + 4κ λ ). (19) 5 ⊥⊥ sistent with the critical region (13) estimated in Sec. III, when D CEP −1 1/4 √ we estimate the ultraviolet cutoff to be c ∼ (λ ) ∼ A . = − T Here, t 2(tl tg) is the massive out-of-phase nonlinear-√ This is reasonable because the shortest length scale that enters λ = λl + λl + λl − λg + λg + λg / the physics in this semiclassical treatment is the shortest ity and ⊥⊥ [( ll lg gg) ( ll lg gg)] (2 2) is the KPZ-type nonlinear coupling that converts the two length scale of fluctuations occupied by the white noise, which λCEP incoming Goldstone modes into the longitudinal mode. The is the effective de Broglie length T . γ = −2 < effective coupling follows the flow equation Next, we consider the critical region ˜ (l 0) c 1. In this case, our CEP flow equations (16)–(20) are valid in the d 5C8 5C10 = γ = 8 − d − + γ˜ , (20) initial flow but stop being justified at l l0 where ˜ (l) has dl 32d 16d grown toγ ˜ (l0 )c = 1. Ignoring (l) for simplicity, we get γ˜ (l) e2l γ˜ (l = 0), which gives indicating that is relevant at dimensions as high as d dc = 1 8, which is to be compared to the conventional critical point l0 − e = [˜γ (l = 0)] 2 c. (22) in the Ising model with dc = 4[15] or the KPZ scaling with d−8 < = dc = 2[52]. In the case where (l) c ∼ 1 in the CEP region l , We remark that the upper critical dimension dc = 8, which [0 l0], the nonlinear many-body correlation effects can be characterizes the strength of many-body correlation effects, is safely neglected, but otherwise the non-Gaussian fluctuations higher than that obtained from the trivial scaling, which are potentially get dominated. The latter region can be estimated dc = 1 and 3 for λ⊥⊥ and t, respectively. This anomalously as enhanced many-body correlation effect is again due to the d−8 d−8 (8−d)l0 (l0 ) e (l = 0) coalescence of collective eigenmodes to the Goldstone mode. c c − 8−d As we have discussed earlier, the mode coalescence leads to (l = 0)[˜γ (l = 0)] 2 1 (23)

033018-6 CRITICAL FLUCTUATIONS AT A MANY-BODY … PHYSICAL REVIEW RESEARCH 2, 033018 (2020) or critical phenomenon at the CEP, which is the anomalous enhancement of the many-body correlation effects, where 2 < − γ ∼ D 8 d (24) we find a stable strong-coupling fixed point just below the upper critical dimension dc = 8 associated with a universality in terms of the bare quantities. Thus, the critical region where class beyond the classification by Hohenberg and Halperin the non-Gaussian fluctuations can be detected is given by the [25]. We expect the dynamic scaling law to be different regions where the bare quantities satisfy both Eqs. (13) and from any known scaling in the physical dimensions as well (24). Notice the low power on the left-hand side that makes since the origin of the criticality is fundamentally different. it easier to get into the non-Gaussian regimes with small The signatures of these anomalous critical fluctuations at the nonlinearities, which is to be compared to the conventional CEP should be observable through the measurement of the Ising model case where this power is replaced by 2/(4 − d) phase-phase correlator [Eq. (15)], which is experimentally [57]. This is originated from the anomalously high upper accessible in, e.g., polariton condensates using the Michelson interferometer [58]. We also believe an anomalous signal critical dimension dc = 8. would appear in various response functions as well, although we have not studied them thoroughly, remaining as our future V. SUMMARY work. In this work, we have implicity restricted ourselves to To summarize, we have proposed a mechanism for the situations where the amplitude of the macroscopic wave occurrence of the dynamic critical phenomena that arise at | 0|, | 0| functions in the steady state l g is large enough such the CEP, originated from the coalescence of the collective that the amplitude fluctuation dynamics is overdamped. It is eigenmodes to the Goldstone mode. We showed that this interesting to ask how the critical fluctuations would change peculiar property gives rise to anomalously giant phase fluc- their character when this restriction is lifted. tuations that diverge at d 4. It also leads to the appearance of a sound mode that anomalously enhances the many-body ACKNOWLEDGMENTS correlation effects, which survive at exceptionally high spatial dimensions (d < 8). We thank S. Diehl, V. Vitelli, M. Fruchart, and A. Edelman Of course, our (= 8 − d) expansion performed in this for critical reading of the manuscript and giving helpful study cannot be directly applied to the physical dimensions comments. We acknowledge A. Galda for fruitful discussions. d = 1, 2, 3. Direct numerical simulation of the stochastic GP R.H. was supported by a Grant-in-Aid for JSPS fellows (Grant equation (1) at the CEP to extract the critical exponents at No. 17J01238). Work at Argonne National Laboratory is sup- these dimensions d = 1, 2, 3 is under progress. However, our ported by the U. S. Department of Energy, Office of Science, analysis clearly demonstrates the key aspect of this dynamical BES-MSE under Contract No. DE-AC02-06CH11357.

APPENDIX A: STEADY-STATE SOLUTION Unless dynamically unstable, the coupled driven dissipative Gross-Pitaevskii (GP) equation (1) approaches a steady state at →∞ |0|2 long-time limit t due to the nonlinear saturation term (vg g ). In the absence of noise, the steady state is described by 0 0 −iEt 0 the ansatz α (t ) = αe , which reduces Eq. (1) to the steady-state condition for α: 0 0 ω − κ 0 l 0 0 l l i g l E = A , = 2 2 . (A1) 0 GP l g 0 ω + 0 + − 0 0 g g g g Ug g i P vg g g

0 Here, E is the emission energy of the condensate that needs to be real such that it is consistent with our assumption that α is a steady-state solution. Diagonalizing this eigenvalue problem gives 0 0 E− 0 − = − , E 0 0 (A2) + 0 E+ + where the order parameter is transformed according to 0 0 − = U l , (A3) 0 0 + g

−1 with U = (u−, u+). Here, 1 02 02 ± = ω + ω + − κ − + ± , E 2 l g Ug g i P vg g (A4) = δ˜2 + 2 − κ + − 02 2 − δ˜ κ + − 02 4g P vg g 2i P vg g (A5)

033018-7 RYO HANAI AND PETER B. LITTLEWOOD PHYSICAL REVIEW RESEARCH 2, 033018 (2020)

are the eigenvalues of AˆGP with the corresponding eigenvectors given by −ϕ g u− = 2 , u+ = (A6) −g −ϕ 2 for E− and E+, respectively. We have introduced the complex detuning ϕ = ω − ω − 02 − κ + − 02 l g Ug g i P vg g (A7) and the effective detuning δ˜ = Reϕ. Equation (A2) shows that the steady-state solution can be classified into two types [29] (other than the trivial solution 0 0 0 0 0 − = + = 0), namely, the “−” solution with (E = E−,− = 0,+ = 0) and the “+” solution with (E = E+,+ = 0 0,− = 0). These solutions correspond to a macroscopically occupied state into the lower and upper branch, respectively. This discussion shows that, unlike in similar non-Hermitian systems where the transient dynamics is discussed, a superposition of the two eigenstates is forbidden due to the steady-state constraint. This is reasonable since the simultaneous occupation of two eigenmodes will cause beating, which is obviously not a steady state. 0,0 Crucially, the matrix AGP( l g ) is a non-Hermitian matrix, which makes the eigenvectors u± [Eq. (A6)] nonorthogonal, giving rise to an exceptional point where they become collinear. This means that the two solution types coalesce at this point, in quite the same manner to the critical point of the liquid-gas phase diagram where the liquid and gas solutions coalesce at that point. As proven in Ref. [29], this point indeed marks the end point of a first-order-like phase boundary between the two types of solutions (see Fig. 2). Since this point exhibits critical fluctuations as shown in the main text, which we dub the critical exceptional point (CEP). The CEP is found at the parameter that satisfies

E = E− = E+. (A8) For this to be true, = 0 needs to be satisfied. In such case, the steady-state constraint requires E to be real, giving P − κ 02 = , g (A9) vg |0| which sets the amplitude of the gain component g in the steady state. It is clear from this expression that the pump power needs to be P >κsuch that we realize the CEP. To obtain the condition to realize the CEP, we look for parameters that make vanish. For the imaginary part of 2 to vanish, we need to set to an effective zero detuning: (P − κ)U δ˜ = ω − ω − 02 = ω − ω − g = . l g Ug g l g 0 (A10) vg Substituting Eqs. (A9) and (A10) back to , we get the condition κ = g. (A11) We emphasize that these conditions require only two parameters to fine tune, i.e., conditions (A10) and (A11), where Eq. (A9) is automatically satisfied in the steady state at the CEP.

APPENDIX B: NOISY DRIVEN-DISSIPATIVE GROSS-PITAEVSKII EQUATION TO THE KARDAR-PARISI-ZHANG EQUATION Here, we derive the Kardar-Parisi-Zhang (KPZ) type stochastic equation of motion (3) from the coupled driven-dissipative GP equation , ω − κ − ∇2 , η , ∂ l(r t ) = l i Kl g l(r t ) + l(r t ) . i t 2 2 (B1) g(r, t ) g ωg + iP − (Kg − iDg)∇ + (Ug − ivg)| g(r, t )| g(r, t ) ηg(r, t ) We rewrite the GP equations (B1) in terms of the amplitude and phase fluctuations around the uniform steady state. Here, we 0 0 −iEt 0 i[θ +δθα (r,t )] −iEt define the amplitude and phase fluctuations by α (r, t ) = [α + δα (r, t )]e = [Mα + δ|α (r, t )|]e α e , where 0 0 iθ 0 2 α = Mαe α . Assuming that the amplitude fluctuations are small such that the terms with O((δ|α|) ) are negligible, we obtain − 0∂ δθ , = ω − δ| , |+ 0{ θ + δθ , − δθ , − θ } Ml t l(r t ) ( l E ) l(r t ) gMg cos[ 0 l(r t ) g(r t )] cos 0 + θ + δθ , − δθ , δ| , |+ 0 ∇θ , 2 + η , , gcos[ 0 l(r t ) g(r t )] g(r t ) KlMl ( l(r t )) l(r t ) (B2) ∂ δ| , |=−κδ| , |− 0∇2θ , − 0{ θ + δθ , − δθ , − θ } t l(r t ) l(r t ) KlMg l(r t ) gMg sin[ 0 l(r t ) g(r t )] sin 0 − θ + δθ , − δθ , δ| , |+η , , gsin[ 0 l(r t ) g(r t )] g(r t ) l (r t ) (B3)

033018-8 CRITICAL FLUCTUATIONS AT A MANY-BODY … PHYSICAL REVIEW RESEARCH 2, 033018 (2020)

− 0∂ δθ , = 0{ θ + δθ , − δθ , − θ }+ θ + δθ , − δθ , δ| , | Mg t g(r t ) gMl cos[ 0 l(r t ) g(r t )] cos 0 gcos[ 0 l(r t ) g(r t )] l(r t ) + ω + 0|2 − δ , |− 0∇2θ , + 0 ∇θ , 2 + η , , g 3Ug g E g(r t ) DgMg g(r t ) KgMg ( g(r t )) g(r t ) (B4) ∂ δ| , |=− 0∇2θ , + 0{ θ + δθ , − δθ , − θ } t g(r t ) KgMg g(r t ) gMl sin[ 0 l(r t ) g(r t )] sin 0 + θ + δθ , − δθ , δ| , |+ − 02 δ| , |− 0 ∇θ , 2 + η , , gsin[ 0 l(r t ) g(r t )] l(r t ) P 3vg g g(r t ) DgMg ( g(r t )) g (r t ) (B5) θ = θ 0 − θ 0 η , = η , + η , where 0 l g and α (r t ) α (r t ) i α (r t ). Following Ref. [51], we further assume that the amplitude ﬂuctuations are overdamped [i.e., we neglect the time derivative in the left-hand side of Eqs. (B3) and (B5)]. This brings us to the desired form 2 α ∂t δθα (r, t ) = Wαβ (∇)δθβ (r, t ) + tα[δθl(r, t ) − δθg(r, t )] + λβγ (∇δθβ (r, t )) · (∇δθγ (r, t )) + ξα (r, t ), (B6) β=l,g β,γ=l,g with − + ν ∇2 + ν ∇2 ∇ = sl ll sl lg , W ( ) 2 2 −sg + νgl∇ sg + νgg∇

3 where we have neglected the higher-order massive term O((δθl − δθg) ) that does not contribute to the critical properties within the one-loop order. To obtain the explicit expression of the parameters in the coupled KPZ-type equations (B6) in terms of those in the coupled driven-dissipative GP equations (B1) (which is crucial for determining the stability condition), we ﬁrst need to solve the steady- state condition (A1). This requires numerical computation in general, but for the special case where the “effective detuning” δ˜ = ω − ω − |0|2 δ˜ = l g Ug g is “on resonance,” 0, an analytic form can be obtained. This includes the critical exceptional point δ˜ = κ = = − |0|2 (CEP), our main focus of this paper, which is found at 0 and g P vg g (see Appendix A). Plugging these into Eqs. (B2)–(B5), we obtain the explicit form of the parameters in Eq. (B6). Here, we provide the list of the parameters for the “−” solution: g2 g2 U g2 − κ2 s = , s = 2κ − − g , (B7) l κ g κ v g 2 − 2 − κ2 κ − κ0 2 2 g 2vg g Kl g − κ Kl νll = ,νlg = , κ02 02 2vg g 2vg g K (U κ + v g2 − κ2 ) K U (K − K ) g2 − κ2 K κ g2 − κ2 U g2 − κ2 ν =− l g g ,ν= D + g g + g l + l κ − g , gl gg g 2 2 2 vgκ vg 2vg|g| 2g vg|g| vg (B8)

K (κ − 2v | |2 )[g2 − κ(κ − 2v | |2 )] K (−κ + v | |2 )[g2 − κ(κ − 2v | |2 )] λl = l g g g g ,λl = l g g g g , ll 2κ| |2 lg 2κ| |2 2vg g vg g K (g2 − κ2 + 2v κ| |2 − 2v2| |4) λl = l g g g g , gg 2| |4 2vg g 1 λg = g4v (K − K )κ + 2K v 02 + g2κ (2K − K )v κ2 − K U κ g2 − κ2 ll 2 3κ|0|4 g g l l g g l g g l g 2g vg g + − + 2κ02 + 2 − κ202 + 304 2( 3Kl Kg)vg g 2KlUgvg g g 2Klvg g − κ2 κ − 2 − κ202 κ2 − κ02 + 204 , Kl vg Ug g g 4vg g 2vg g 1 λg =− 2 + κ − κ + 2 − κ2 − κ + 02 2 − κ κ − 02 lg Kl g vg ( vg Ug g ) vg g g 2vg g 2 3κ04 2g vg g 2 2 2 2 02 02 + g vgκ g Kg + Dgvg g − κ − Kgκ κ − 2vg , g g 1 λg = 2 − κ2 + κ02 + 2 − κ202 + 04 − 204 gg Kg(g ) 2Kgvg g 2Dgvg g g 2DgUgvg g 2Kgvg g 04 2vg g

033018-9 RYO HANAI AND PETER B. LITTLEWOOD PHYSICAL REVIEW RESEARCH 2, 033018 (2020) κ(κv − U g2 − κ2 + K − 1 + g g g2 − κ2 + 2v κ02 − 2v204 , (B9) l v g2 g g g g g 2 2 2 2 κ κ + 2 − κ2 κ − 0 − 2 κ + 2 − κ2 g g − κ 2 (Ug vg g ) vg g g (2Ug vg g ) tl = , tg = , (B10) κ02 2κ02 2vg g 2vg g g ξ r, t = η r, t , l( ) κ l ( ) 1 ξ , = κ2 + 2 − κ2 η , − 2 + κ − κ + 2 − κ2 η , g(r t ) ( Ug vg g ) l(r t ) [g vg ( vg Ug g )] l (r t ) 02 gκvg g − κ η , − κ η , . g vg g(r t ) g Ug g(r t ) (B11) Especially at the CEP g = κ, we ﬁnd

sl = sg = κ, (B12)

KlUg KgUg νll = νlg = 0,νgl =− ,νgg = Dg + , (B13) vg vg κ κ κ λl =− − + ,λl = + ,λl = − + , ll Kl 2 lg 2Kl 1 gg Kl 1 02 02 02 vg vg vg g g g K κ 2K κ D U κ λg = g ,λg =− g ,λg = g g + − + , ll 2 lg 2 gg Kg 1 2 (B14) 0 0 vg 0 vg g vg g vg g Ugκ tl = 0, tg =− , (B15) vg η (r, t ) −vgη (r, t ) + Ug[η (r, t ) − η (r, t )] ξ (r, t ) = l ,ξ(r, t ) = g l g . (B16) l 0 g 0 l vg g As discussed in the main text, since the collective mode is given by [Eq. (2)]

1 2 2 ω±(k) = [−i(γ + 2Dk ) ± −γ 2 + 4v2k ], (B17) 2 2 the stability condition can be determined by γ = sl − sg, v = [2(sgνlg + slνgl ) + (sl + sg)(νgg − νll )]/2, D = (νll + νgg )/2 0. For the above-obtained “−” solution, we get 2g2 2U g2 − κ2 γ = − 2κ + g 0, (B18) κ vg 1 2 = κ5 κ − 2 2 − κ2 + 2 2 − κ2 v Kl 2Ugvg Ug g vg g 2κ2 302 2g vg g − 4 κ κ + 2 − κ2 + κ2 + 2 2 − κ202 g vg Kg (Ug vg g ) Kl Ug 2vg g g + 2κ2 2κ 2 − κ2 + 2 2 − κ2 κ − 02 − κ κ + 02 g Kl[Ug g vg g 2vg g Ugvg 2vg g + κ − 2 − κ2 02 + κ 2 − κ2 − 2 2 − κ202 + κ κ + 02 , vg 2Dgvg(vg Ug g ) g Kg vg g 2Ug g g Ug ( 2vg) g (B19) 1 3 2 2 D = Klκ (Ugκ + vg g − κ ) 2κ 202 4g vg g + 2 κ 2 − κ2 + + 02 − κ2 + 2 − κ2 κ − 02 . g vg Kg g 2 KgUg Dgvg g Kl Ug 2vg g 2vg g (B20) Especially, at the CEP, κ(D v + K U − K U ) v2 = g g g g l g , (B21) vg 1 KgUg D = Dg + > 0. (B22) 2 vg

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2 This shows that the stability condition v > 0 is satisﬁed when the nonlinear saturation vg and the diffusion constant of the gain component Dg are large enough. √ = − λ = λl + λl + For completeness, we also√ provide the explicit form of the nonlinear couplings t 2(tl tg) and ⊥⊥ [( ll lg λl − λg + λg + λg / gg) ( ll lg gg)] (2 2) at the CEP: √ 2Ugκ t = , (B23) vg DgUg + (Kl − Kg)vg λ⊥⊥ =− √ . (B24) 2 2vg Note how the sign of the KPZ-type coupling λ⊥⊥ depends on the details of the parameters in the original GP equation. This makes it possible for the effective coupling to take either sign, √ 2 2 3 16 2(Kl − Kg)Ugvg[DgUg + (Kl − Kg)vg] U + v κ = g g , (B25) 5 (KgUg + Dgvg) leading to distinct phases of matter, as we have discussed in the main text.

APPENDIX C: DYNAMIC RENORMALIZATION GROUP We extend the perturbative dynamic renormalization group method [25] (see also Ref. [59] for the case on the KPZ equation) to our coupled KPZ-type equation of motion (1). This is performed conveniently in the in-phase and out-of-phase bases, δθ˜s(k,ω) = Usαδθα (k,ω), ξ˜s(k,ω) = Usαξα (k,ω)(C1) α=g,l α=g,l with † † T 1 1 −1 U ≡ U (k = 0) = (u⊥(k = 0), u(k = 0)) = √ , (C2) 2 11 transforming the kernel as − 2 ζ − 2 ˜ = U U † = D⊥⊥k D⊥k , W (k) W (k) − 1 2 2 −γ − 2 (C3) ζ v k Dk with ζ = sl + sg, D⊥⊥ = (νll + νgg + νlg + νgl )/2, D = (νll + νgg − νlg − νgl )/2, and D⊥ = (−νll + νgg + νlg − νgl )/2. The equation of motion (1) is expressed in this basis as (s =⊥, )

∞ dω −iωδθ˜s(k,ω) = W˜ss (k)δθ˜s (k,ω) + ts δθ˜(k ,ω )δθ˜(k − k ,ω− ω ) −∞ 2π s k ∞ dω − λs · − δθ˜ ,ω δθ˜ − ,ω− ω + ξ˜ ,ω . ss [k (k k)] s (k ) s (k k ) s(k ) (C4) −∞ 2π s,s k We briefly note that the in-phase and out-of-phase bases used here are slightly different from the triangular basis introduced in the main text, where the present basis U † is momentum independent. The former is more convenient since the massive nonlinear term only involves the out-of-phase fluctuations δθ˜(k,ω), but we emphasize that the two representations are identical in the uniform limit k → 0. A simple power-counting analysis tells us that the diffusion constants D⊥⊥, D⊥, and D are irrelevant when the velocity v is fixed because the former are coefficients of higher spatial derivatives. However, the average of D⊥⊥ and D, i.e., D = (D⊥⊥ + D)/2 = (νll + νgg )/2, turns out to be dangerously irrelevant, in the sense that D still affects the critical properties, as we have discussed in the main text. From here on, we ignore the parts that do not affect the critical properties, by setting D⊥⊥ = D = D and D⊥ = 0. Equation (C4) can be rewritten in terms of Green’s function G˜ 0 as

∞ dω δθ˜ ,ω = ˜ 0 ,ω ξ˜ ,ω + ˜ 0 ,ω δθ˜ ,ω δθ˜ − ,ω− ω s(k ) Gss (k ) s (k ) Gss (k )ts (k ) (k k ) −∞ 2π s s k ∞ ω s d − ˜ 0 ,ω λ · − δθ˜ ,ω δθ˜ − ,ω− ω , Gss (k ) ss [k (k k)] s (k ) s (k k ) (C5) −∞ 2π s,s,s k

033018-11 RYO HANAI AND PETER B. LITTLEWOOD PHYSICAL REVIEW RESEARCH 2, 033018 (2020)

σ FIG. 3. Diagrammatic representation of nonlinear corrections. (a) The self-energy s1s2 . (b) The noise kernel correction s1s2 .(c)The three-point vertex correction to the KPZ-type coupling λs1 and (d) the massive out-of-phase coupling t . Here, the solid line is the Green’s s2s3 s ˜ 0 σ λs function Gss , the open circle represents the longitudinal noise strength , the solid circle is the KPZ-type nonlinear coupling ss , and the solid square represents the massive out-of-phase coupling ts.

− ˜ 0 ,ω = − ω − ˜ ,ω 1 where the nonperturbative Green’s function Gss (k ) ([ i 1 W (k )] )ss is given by 1 ω − γ − 2 − κ ˜ 0 ,ω = i Dk 2 . G (k ) 2 2 2 (C6) [ω − ω−(k)][ω − ω+(k)] v k /(2κ) iω − Dk

Here, we have used the relation ζ = 2κ that holds at the CEP. By iteratively substituting δθ˜s(k,ω) into the right-hand side and taking the noise average, the perturbative corrections to the linearized theory can be conveniently computed by diagrammatic techniques. The obtained corrections are formalized into a dynamic renormalization group by integrating out only the fast modes −l with ce < |k| <c and then rescaling the valuables according to l zl χl r → e r, t → e t,δθ˜s → e δθ˜s. (C7)

δθ˜ ,ω ≡ ˜ ,ω ξ˜ ,ω The self-energy ss , deﬁned as the correction to the Green’s function [where s(k ) s Gss (k ) s (k )] by −1 − ˜ ,ω = ˜ 0 ,ω 1 − ,ω Gss (k ) ([G (k )] )ss ss (k ), is given diagrammatically by Fig. 3(a) within the one-loop order. Here, we have dropped the terms that are obviously less relevant than the ones retained, by using the following properties at the CEP: (1) 0 The most relevant propagator is the non-Hermitian-induced off-diagonal component G˜ ⊥, hence, for the KPZ-type nonlinear λs ˜ 0 couplings ss , the diagrams associated with G⊥’s are the most relevant. (2) For the massive out-of-phase coupling ts,thetwo 0 0 0 outgoing lines should be labeled “.” Thus, they may only be connected to propagators G˜ and G˜ ⊥. (3) Since G˜ is more 0 0 relevant than G˜ ⊥ (because κ →∞at the ﬁxed point), for massive nonlinearity ts, the diagrams associated with G˜ ’s are the most relevant. Computing the self-energy shown in Fig. 3(a) at ω = 0 (which is all we need for our purpose), given by ∞ ω d s k k ,ω = = σ −λ 1 − λ + · − + s1s2 (k 0) 4 ( ⊥⊥) ⊥s q q −∞ 2π 2 2 2 q k 0 k 0 k 0 k × − q − · k G˜ ⊥ q + ,ω G˜ ⊥ −q + , −ω G˜ ⊥ −q − , −ω 2 2 2 2 k 0 k 0 k 0 k + ts (−λ⊥ ) − q − · k G˜ q + ,ω G˜ −q + , −ω G˜ ⊥ −q − , −ω 1 s2 2 2 2 2 k k k k k s1 0 0 0 + (−λ )t q + · − q + G˜ ⊥ q + ,ω G˜ ⊥ −q + , −ω G˜ −q − , −ω ⊥⊥ s2 2 2 2 2 2 0 k 0 k 0 k + t t G˜ q + ,ω G˜ −q + , −ω G˜ −q − , −ω , (C8) s1 s2 2 2 2

033018-12 CRITICAL FLUCTUATIONS AT A MANY-BODY … PHYSICAL REVIEW RESEARCH 2, 033018 (2020) we obtain C6 3C8 2 ⊥⊥(k,ω = 0) = − γ˜ ⊥Dk , (C9) 4d 8d C4 C6 2 ⊥(k,ω = 0) = − γ˜ ⊥ + O(k ), (C10) 4 4 2 C6 3C8 v 2 ⊥ k,ω = = − + γ k , ( 0) ˜ κ (C11) 2d 4d 2 C4 C6 C8 C10 2 (k,ω = 0) = − γ˜ − + γ˜ Dk , (C12) 4 4 16d 8d where σ t 2 2 = (tv + 4κ λ ), (C13) D5 ⊥⊥ κ2λ ⊥⊥ 2 2 = (tv + 4κ λ ), (C14) D3v4 ⊥⊥ κ2λ ⊥⊥ 2 2 ⊥ ⊥ = (t⊥v + 4κ λ⊥⊥), (C15) D4v2 σ t 2 2 = (tv + 4κ λ ), (C16) D3v2 ⊥⊥ σ t 2 2 ⊥ = (t⊥v + 4κ λ ). (C17) D3v2 ⊥⊥ Here, we have restricted ourselves to the vicinity of the CEP γ = 0 where we expanded the Green’s function in terms of γ , and have retained the lowest-order correction in terms of D since it is a (dangerous) irrelevant parameter that ﬂows to zero as l →∞. Since the dressed Green’s function G is given by − ω + 2 − − κ − −1 ,ω = i Dk ⊥⊥(k) 2 ⊥(k) G (k ) 1 2 2 − − ω + γ + 2 − κ v k ⊥(k) i Dk (k) 2 − ω + + 2 − κ + κ ≡ i (D D⊥⊥)k 2( ) , 1 2 2 2 2 (C18) + − ω + γ + γ + + 2κ (v v )k i ( ) (D D )k these self-energy corrections give the nonlinear correction ( v, κ, D, γ) to the valuables (v,κ,D,γ). The correction to the diffusion constant is given by D = ( D⊥ + D)/2. Now, let us examine which effective couplings are the most relevant. Since the transformation (C7) changes the parameters to κ → κ zl , → (z−1)l , → (z−2)l ,σ→ σ (z−d−2χ )l ,λs → λs (z−2+χ )l , → (z+χ )l , e v ve D De e ss ss e ts tse (C19) the effective couplings change as

→ e(8−d)l , (C20)

(6−d)l (,⊥) → e (,⊥), (C21)

(4−d)l (,⊥) → e (,⊥). (C22)

This tells us that the effective coupling is more relevant than all other effective couplings, with upper critical dimension dc = 8. We note that the most relevant coupling in originates from the third and fourth diagrams in Fig. 3(a). The third diagram 0 0 0 involves both G˜ ⊥ and G˜ , while the last only includes G˜ , which means that for the latter nonlinear correction is solely 0 determined by the dynamics of the out-of-phase motion. However, note that G˜ in the triangular basis introduced in the main text (u⊥(k), u(k)) (i.e., the transverse and longitudinal basis) is given by | | 0 0 iv k 0 G˜ (k,ω) = G¯ (k,ω) + G¯ ⊥(k,ω). (C23) 2κ The second term converts the longitudinal ﬂuctuations to the transverse (Goldstone) mode, and has the strongest singularity. This contributes to the resulting singular form of effective coupling ∝ D−5, which is the origin of the anomalously high upper critical dimension.

033018-13 RYO HANAI AND PETER B. LITTLEWOOD PHYSICAL REVIEW RESEARCH 2, 033018 (2020)

Below, we restrict ourselves to spatial dimensions close to the upper critical dimension dc = 8, where is the only relevant term. In such case, the self-energy reduces to C8 C10 2 (k,ω = 0) =− + γ˜ Dk , (C24) 16d 8d and ⊥⊥ = ⊥ = ⊥ = 0, which gives C C D = 8 + 10 γ˜ , (C25) 32d 16d and γ = κ = v = 0. The ﬂow equations are thus given by dγ = zγ, (C26) dl dκ = zκ, (C27) dl dv = (z − 1)v, (C28) dl dD C C = z − 2 + 8 + 10 γ˜ D. (C29) dl 32d 16d

σ λs , The noise kernel correction ss and the three-point vertex corrections ss ts can be analyzed similarly by computing the diagrams shown in Figs. 3(b)–3(d), respectively. It turns out that these diagrams only give nonlinear effective couplings that are less relevant than , with the maximum upper critical dimension being dc = 6. Ignoring such less-relevant terms, the ﬂow equations for the noise kernel and the three-point vertices are given by

dσss = (z − d − 2χ )σ , (C30) dl ss λs1 d s s 2 3 = (z − 2 + χ )λs1 , (C31) dl s2s3 dt s = (z + χ )t . (C32) dl s Combining the ﬂow equations for γ, κ, v, D,σ,λ⊥⊥, and t [Eqs. (C26)–(C32)], we obtain the ﬂow equations forγ ˜ = γ/D 2 2 [Eq. (16)], A = κ σ/(Dv )[Eq.(18)], and the effective coupling [Eq. (20)] presented in the main text.

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