Functional Quantum Monte Carlo Method

Itinerant Quantum Critical Point with Fermion Pockets and Hot Spots

Zi Hong Liu,1, 2 Gaopei Pan,1, 2 Xiao Yan Xu,3 Kai Sun,4 and Zi Yang Meng1, 5, 6, 7 1Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China 3Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China 4Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA 5Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China 6CAS Center of Excellence in Topological Quantum Computation and School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China 7Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China (Dated: August 13, 2019) Metallic quantum criticality is among the central theme in the understanding of correlated electronic systems, and converging results between analytical and numerical approaches are still under calling. In this work, we develop state-of-art large scale quantum Monte Carlo simulation technique and systematically investigate the itinerant quantum critical point on a 2D square lattice with antiferromagnetic spin fluctuations at wavevector Q = (π, π) – a problem that resembles the Fermi surface setup and low-energy antiferromagnetic fluctuations in high-Tc cuprates and other critical metals, which might be relevant to their non-Fermi-liquid behaviors. System sizes of 60 × 60 × 320 (L × L × Lτ) are comfortably accessed, and the quantum critical scaling behaviors are revealed with unprecedingly high precision. We found that the antiferromagnetic spin fluctuations introduce effective interactions among fermions and the fermions in return render the bare bosonic critical point into a new universality, different from both the bare Ising universality class and the Hertz-Mills-Moriya RPA prediction. At the quantum critical point, a finite anomalous dimension η ∼ 0.125 is observed in the bosonic propagator, and fermions at hot spots evolve into a non-Fermi-liquid. In the antiferromagnetically ordered metallic phase, fermion pockets are observed as energy gap opens up at the hot spots. These results bridge the recent theoretical and numerical developments in metallic quantum criticality and can be served as the stepping stone towards final understanding of the 2D correlated fermions interacting with gapless critical excitations.

I. INTRODUCTION mentum ultra-size quantum Monte Carlo (EQMC) [41], it now becomes possible to explore larger system sizes than those han- dled with conventional determinantal quantum Monte Carlo, In the study of correlated materials, quantum criticality and consequently allowing us to access the genuine scaling in itinerant electron systems is of great importance and in- behaviors in the infrared (IR) limit for itinerant quantum criti- terests [1–8]. It plays a vital role in the understanding of cality. anomalous transport, strange metal and non-fermi-liquid be- Although a lot intriguing results and insights have been ob- haviors [9–13] in heavy-fermion materials [14, 15], Cu- and tained, for the search of novel quantum critical points beyond Fe-based high-temperature superconductors [16–18] as well the Hertz-Millis-Moriya theory, a major gap between theory as the recently discovered pressure-driven quantum critical and numerical studies still remain. So far, in QMC simula- point (QCP) between magnetic order and superconductiv- tions, among all recently studied itinerant QCPs, either the ity in transition-metal monopnictides, CrAs [19], MnP [20], Hertz-Millis mean-field scaling behavior is found [33, 40], or CrAs1−xPx [21] and other Cr/Mn-3d electron systems [22]. unpredicted exponents, deviating from existing theories, are However, despite of extensive efforts in decades [1–9, 23–30], observed [13], while theoretically proposed properties beyond itinerant quantum criticality is still among the most challenging the Hertz-Millis-Moriya scaling behaviors still remain to be subjects in condensed matter physics, due to its nonperturba- numerically observed and verified. tive nature, and many important questions and puzzles remain In this paper, we aim at improving the convergence between open. theoretical and numerical studies by focusing on itinerant The recent development in sign-problem-free quantum QCPs with finite ordering wave vector Q , 0, e.g. charge/spin

arXiv:1808.08878v3 [cond-mat.str-el] 12 Aug 2019 Monte Carlo techniques has paved a new path way towards density waves (CDW/SDW). One key question in the study of sharpening our understanding about this challenging prob- these QCPs is about the universality class, i.e. whether all lem (see a concise commentary in Ref. [31] and a review in these types of QCPs, e.g. commensurate and incommensurate Ref. [32] that summarize the recent progress). Via coupling a CDW/SDW QCPs, belong to the same universality class or Fermi liquid with various bosonic critical fluctuations, a wide not. To the leading order, within the random phase approxi- variety of itinerant quantum critical systems have been stud- mation (RPA), as long as the ordering wave vector Q is smaller ied, such as Ising-nematic [33, 34], ferromagnetic [13], charge than twice the Fermi wavevector 2kF , or more precisely, as we density wave [35], spin density wave [36–41] and interaction- shift the Fermi surface (FS) by the ordering wave vector Q driven topological phase transitions and gauge fields [42–47]. in the momentum space, the shifted FS and the original one With the fast development in QMC techniques, in particular the shall cross at hot spots, instead of tangentially touching with self-learning Monte Calro (SLMC) [48–54] and elective mo- or overrun each other, the same (linear ω) Landau damping 2 and critical dynamics is predicated regardless of microscopic momentum ultra-size QMC (EQMC), both with self-learning details, implying the dynamic critical exponent z = 2. For a updates to access much larger systems sizes beyond existing 2D system, this makes the effective dimensions d + z = 4, co- efforts. The more conventional DQMC technique allow us inciding with the upper critical dimension. As a result, within to access system sizes up to 28 × 28 × 200 for L × L × Lτ the Hertz-Millis approximation, mean-field critical exponents for 2D square lattice, while EQMC can access much larger shall always be expected, up to possible logarithmic correc- sizes (60 × 60 × 320) to further reduce the finites-size effect tions, and thus all these QCPs belong to the same universality and confirms scaling exponents with higher accuracy. These class [1–3, 55]. two efforts (1) and (2) allow us to access the metallic quan- On the other hand, more recent theoretical developments tum critical region and to reveal its IR scaling behaviors with point out that this conclusion becomes questionable once great precision, where we found a large anomalous dimensions higher order effects are taken into account. In particular, two significantly different from the Hertz-Millis theory prediction, different universality classes need to be distinguished, depend- and we also observed that the Fermi surface near the hotspots ing on whether 2Q coincides with a lattice vector or not, which rotates towards nesting at the QCP, as predicted in the RG anal- will be dubbed as 2Q = Γ and 2Q , Γ to demonstrate that ysis [23, 26, 28, 29]. And quantitative comparison between 2Q, mod a reciprocal lattice vector, coincides or not with the theory and numerical results is also performed. These results Γ point. Among these two cases, 2Q = Γ (e.g. antiferromag- bridge the recent theoretical and numerical developments and netic QCP with Q = (π, π)) is highly exotic. As Abanov and are the precious stepping stone towards final understanding of coworkers pointed out explictly, in this case the Hertz-Millis the metallic quantum criticality in 2D. mean-field scaling will breaks down and a nonzero anomalous dimension shall emerge [23]. In addition, the critical fluctuations will also change the fermion dispersion near the hot II. MODEL AND METHOD spots, resulting in a critical-fluctuation-induced Fermi surface nesting: i.e. even if one starts from a Fermi surface with- A. Antiferromagnetic fermiology out nesting, the RG (renormalization group) flow of the Fermi velocity will deform the Fermi surface at hotspots towards nest- The square lattice AFM model that we designed are ing [23]. This Fermi surface deformation will further increases schematically shown in Fig.1(a) with two fermion-layers and the anomalous dimension and make the scaling exponent de- one Ising-spin-layer in between. Fermions are subject to intra- viate even further from Hertz-Millis prediction [23, 24], and layer nearest, second and third neighbor hoppings t1, t2 and even modifies the dynamic critical exponent z, as pointed out t3, as well as the chemical potential µ. The Ising spin layer explicitly by Metlitski and Sachdev and others [26, 28, 29]. z is composed of Ising spins si with nearest neighbor antiferro- For 2Q , Γ, on the other hand, these exotic behaviors are not magnetic coupling J (J > 0) and a transverse magnetic field h expected, at least up to the same order in the 1/N expansion along sx. Fermions and Ising spins are coupled together via an and thus presumably, they follows the Hertz-Millis mean-field inter-layer onsite Ising coupling ξ. The Hamiltonian is given scaling relation. as On the numerical side, a QCP with 2Q , Γ was recently studied [40, 41] and the results are in good agreement with H = Hf + Hb + Hf b (1) the Hertz-Millis-Moriya theory. For the more exotic case with 2Q = Γ, the numerical result is less clear because in QMC where simulations, a superconducting dome usually arises and covers Õ Õ − † − † the QCP [38, 39, 56]. Outside the superconducting dome, at Hf = t1 ci,λ,σcj,λ,σ t2 ci,λ,σcj,λ,σ some distance away from the QCP, mean-field exponents are hiji,λ,σ hhijii,λ,σ observed to be consistent with the Hertz-Millis-Moriya theory. Õ † Õ − t3 ci,λ,σcj,λ,σ + h.c. − µ ni,λ,σ (2) However, whether the predicted anomalous (non-mean-field) hhhijiii,λ,σ i,λ,σ behaviors [23, 26, 28, 29] will arise in the close vicinity of the Õ Õ z z − x QCP remains an open question, which requires the suppression Hb =J si sj h si (3) of the superconducting order. In addition, due to the divergent hiji i Õ length scale at a QCP, to obtain reliable scaling exponents, − z z z Hf b = ξ si σi,1 + σi,2 , (4) large system sizes is necessary to overcome the finite-size i effect. z 1 ( † − † ) In this manuscript, we perform large-scale quantum Monte and σi,λ = 2 ci,λ,↑ci,λ,↑ ci,λ,↓ci,λ,↓ is the fermion spin along Carlo simulations to study the antiferromagnetic metallic z. quantum critical point (AFM-QCP) with 2Q = Γ. In this Hb describes a 2D transverse-field Ising model and has a study, two main efforts are made in order to accurately ob- phase diagram spanned along the axes of temperature T and tain the critical behavior in the close vicinity of the QCP. (1) h. As shown in Fig.1 (c) (light blue line), at h = 0, the We design a lattice model that realizes the desired AFM-QCP system undergoes a 2D Ising thermal transition at a finite with superconducting dome greatly suppressed to expose the T. Gradually turning on a finite h, the system experiences quantum critical regions and (2) we employ the determinan- the same AFM 2D Ising transition with a lower transition tal quantum Monte Carlo (DQMC) as well as the elective temperature, until h = hc (3.04438(2)), where the transition 3 a b c 1 fermion site Ising spin coupling ’ Pure Boson h =3.044(3) K4 K1 c Ising site fermion DQMC hc=3.32(2) 0.8 EQMC hc=3.355(5) -t1 ’ K2 K3 -t2 Q Q -t3 1 4 0.6

λ=2 T SDW metal Fermi liquid ky 0.4 Q Q3 ’ 2 h K1 K4 0.2 J K K’ 0 λ=1 3 2 2.6 2.8 3 3.2 3.4 3.6 3.8 kx h

FIG. 1. (a) Illustration of the model in Eq. (1). Fermions reside on two of the layers (λ = 1,2) with intra-layer nearest, 2nd and 3rd neighbor z hoppings t1, t2 and t3. The middle layer is composed of Ising spins si , subject to nearest-neighbor antiferromagnetic Ising coupling J and a transverse magnetic field h. Between the layers, an onsite Ising coupling is introduced between fermion and Ising spins (ξ). (b) Brilliouin zone (BZ) of the model in Eq. (1). The blue lines are the fermi surface (FS) of Hf and Qi = (±π, ±π), i = 1, 2, 3, 4 are the AFM wavevectors, 0 and the four pairs of {Ki, K i }, i = 1, 2, 3, 4 are the position of the hot spots (red dots), each pair is connected by a Qi vector. The folded FS (gray lines), coming from translating the bare FS by momentum Qi. The green patches show the k mesh built around hot spots, number of momentum points inside each patch is denoted as Nf . (c) Phase diagram of model Eq. (1). The light blue line marks the phase boundaries of the pure bosonic model Hb, with a QCP (light blue dot) at hc = 3.044(3) [57, 58] with 3D Ising universality. After coupling with fermions, the QCP shifts to higher values. The green solid dot is the QCP obtained with DQMC (hc = 3.32(2)). The violet solid dot is the QCP obtained from EQMC (hc = 3.355(5)), although the position of the QCP shifts, as it is an nonuniversal quantity, the scaling behavior inside the quantum critical region is consistent between DQMC and EQMC. The EQMC scheme can comfortably capture the IR physics of AFM-QCP, with much larger system sizes, 60 × 60 × 320, compared with that in DQMC with 28 × 28 × 200. The procedure of how the phase boundary is determined is shown in the Sec. I in the Appendix. turns into a T = 0 quantum phase transition in the 3D Ising namic spin susceptibility universality class [57, 58]. Such an antiferromagnetic order has a wave vector Q = (π, π), as denoted by the Qi with 1 Õ ¹ β χ(T, h, q, ω ) = dτeiωnτ−iq·ri j hsz(τ)sz(0)i. (5) i = 1, 2, 3, 4 in Fig.1(b). n L2 i j ij 0 The fermions in Hf experience the AFM fluctuations in Hb via the fermion-spin coupling Hf b. As shown in the Fig.1 (b), the original FS and FSs from zone folding [shifted by the In principle, near the QCP, the functional form of the dynamic spin susceptibility is complicated and hard to write down ex- ordering wave vector Qi (i = 1, 2, 3, 4)] form Fermi pockets and the so-called hot spots, which are crossing points between plicitly. However, in the quantum critical region, as we set 0 h = hc, the scaling functional form of χ(T, hc, q, ωn) can be the original and the folded FSs labeled as Ki and Ki with i = 1, 2, 3, 4. described by the following asymptotic form In the simulation, we set t = 1.0, t = −0.32, t = 0.128, 1 2 3 1 J = 1, µ = −1.11856 (electron density hni,λi ∼ 0.8), the χ(T, hc, q, ωn) = , (6) 2 2 2 a /2 coupling strength ξ = 1.0 and leave h as control parameters. ctT + (cq |q| + cωω ) q The parameters are chosen according to Ref. [59], such that deep in the AFM phase of Ising spins, the FS exhibits four where aq = 2−η = 1.964(2) is the universal critical exponents big Fermi pockets and four pairs of hot spots (hot spot number of the 3D Ising universal class, and ct , cq and cω are non- Nh.s. = 8 × 2 = 16 where the factor 2 comes from two fermion universal coefficients. This scaling form [Eq. (6)] explicitly layers) as a result of band folding due to AFM ordering. Such respects the emergent Lorentz symmetry at the Ising critical an antiferromagnetic fermiology is summarized in Fig.1 (b). point. Without fermions, we can employ the standard path-integral scheme to map the 2D transverse-field Ising model to a (2+1)D classical Ising model [60]. To solve this anisotropic 3D Ising B. Ising scaling for the bare boson model model with Monte Carlo simulations, we performed Wolff [61] and Swendsen-Wang [62] cluster updates to access sufficiently Before presenting our results about the itinerant AFM-QCP, large system sizes and low temperature. The dynamic sus- we first discuss the QCP in the pure boson limit without ceptibility are shown in Fig.2. To explore the momentum −1 fermions, which serves as a benchmark for the non-trivial itin- dependence of the susceptibility, we plot χ (T, hc, q, ω = −1 erant quantum criticality. It is known that the pure boson QCP 0) − χ (T, hc, q = 0, ω = 0), where the momentum q is mea- (Hb) belongs to the (2+1)D Ising universality class [13, 57, 58]. sured from Q = (π, π) and the substraction is to get rid of the This can be demonstrated numerically by calculating the dy- finite temperature background, such that the following scaling 4

C. DQMC and EQMC L = 48, β = 12 0 0)) L = 60, β = 15 , L , β To solve the problem in Eq. (1) we employ two complemen- 0 = 72 = 18

( aq 1 aq ln(|q|) + ln(cq) tary fermionic quantum Monte Carlo schemes.

− -1 2 χ The ﬁrst one is the standard DQMC [13, 63–65] with −

0) -2 aq = 1.96 self-learning Monte Carlo update scheme (SLMC) [48–54] ,

q to speedup the simulation. In SLMC, we ﬁrst perform the ( 1

− -3 standard DQMC simulation on the model in Eq. (1), and then χ train an effective boson Hamiltonian that contains long-range ln( (a) -4 two-body interactions both in spatial and temporal directions. -2 -1.5 -1 -0.5 0 The effective Hamiltonian serves as the proper low-energy ln(|q|) description of the problem at hand with the fermion degree of freedom integrated out. We then use the effective Hamil- -2 L = 20, β = 20 tonian to guide the Monte Carlo simulations, i.e., we will L 0)) = 20, β = 30

, perform many sweeps of the eﬀective bosonic model (as the L 0 = 20, β = 40

( -3

1 ( ) L = 20, β = 50 computational cost of updating the boson model is O βN , dra- −

χ L = 20, β = 60 matically lower than the update of fermion determinant which -4 a − q 3 aq ln(ω) + ln(cω) ) 2 scales as O(βN ), and then evaluate the fermion determinant a = 1.96 , ω q -5 of the original model in Eq. (1) such that the detailed balance 0 (

1 of the global update is satisﬁed. As shown in our previous −

χ -6 works [40, 49, 50, 53, 54], the SLMC can greatly reduce the ln( (b) autocorrelation time in the conventional DQMC simulation -7 -2.5 -2 -1.5 -1 -0.5 0 and make the larger systems and lower temperature accessible. ln(ωn) The other method is the elective momentum ultra-size quantum Monte Carlo (EQMC) [41]. EQMC is inspired by the FIG. 2. (a) Momentum dependence of the χ(q, ω = 0) at h = hc, awareness that critical fluctuations mainly couples to fermions for the bare boson model Hb. The system sizes are L = 48, 60, 72 near the hot spots. Thus, instead of including all the fermion respectively and to achieve quantum critical scaling, β ∝ L is applied. degrees of freedom, we ignore fermions far away from the aq The line goes through the data points is aq ln(|q|) + 2 ln(cq), with hot spots and focus only on momentum points near the hot aq = 2 − η = 1.96. (b) Frequency dependence of χ(q = 0, ω) at spots in the simulation. This approximation will produce dif- h = hc, for the bare boson model Hb. The system sizes is L = 20 ferent results for non-universal quantities compared with the with increasing β = 20, 30, 40, 50 and 60. The line goes through the aq original model, such as h or critical temperature. However, data points is a ln(ω) + ln(c ), with a = 2 − η = 1.96. c q 2 ω q for universal quantities, such as scaling exponents, which are independent of microscopic details and the high energy cut- off, EQMC has been shown to generate consistent values with relation is expected at low T those obtained from standard DQMC [41]. In EQMC, because to a local coupling (in real space) becomes non-local in the momentum basis, one can no longer / −1 −1 aq 2 aq use the local update as in standard DQMC, as that would cost χ (T, hc, q, ω = 0) − χ (T, hc, q = 0, ω = 0) = c |q| . q · ( 3) (7) βN O βNf computational complexity. Fortunately, there are cumulative update scheme in the SLMC developed re- As shown in Fig.2 (a) such a scaling relation is indeed cently [49, 50]. Such cumulative update is global move of / ∝ observed with L = 48, 60 and 72 with β = 1 T L. The the Ising spins and gives rise to the complexity O(βN3) for power-law divergence of the (2 + 1)D Ising quantum critical f computing the fermion determinant. Since N can be much susceptibility with power a = 2 − η = 1.96 is clearly revealed f q smaller than N, speedup of the order ( N )3 ∼ 103 of EQMC (the 3D Ising anomalous dimension η = 0.04.) Nf over DQMC, with N ∼ 10, can be easily achieved. Similar scaling relation is also observed in the frequency Nf dependence as shown in Fig.2 (b), where we plot In the square lattice model, as shown in Fig.1 (b), the AFM wave vectors Qi connect 4 pairs of hot spots (Nh.s. = 8 in one layer and Nh.s. = 16 in two). In the IR limit, only a /2 −1 −1 q aq fluctuations connecting each pair of hot spots are important to χ (T, hc, 0, ω) − χ (T, hc, 0, ω = 0) = cω ω (8) the universal scaling behavior in the vicinity of QCP [7,9, 23– 26, 66]. Hence, to study this universal behavior, we draw for L = 20 with increasing β. The expected power-law decay one patch around each Kl and keep fermion modes therein, with the small anomalous dimension aq = 2 − η = 1.96 is and neglect other parts of the BZ. In this way, instead of the clearly obtained. Hence, the data in Fig.2 (a) and (b) confirm original N = L × L momentum points, EQMC only keeps the QCP for the pure boson part Hb in Eq. (1) belongs to the Nf = Lf × Lf momentum points for fermions inside each (2 + 1)D Ising universality class. patch. Here, L and Lf denotes the linear size of the original 5 lattice and the size of the patch, respectively. III. RESULTS

A. Non-Fermi liquid

As we emphasized above, the dramatically improved mo- (a) 3 (b) 3 mentum resolution in EQMC enables us to study the fermionic modes on the FS more precisely. We studied the fermion self- 2 2 energies in the AFM-metal phase and at the AFM-QCP. The High 1 1 results are shown in Fig.4. 0 ky ky 0 In the AFM-metal phase, although the bands are folded -1 -1 accroding to Fig.1 (b), the system remains a Fermi liquid, -2 -2 with a band gap opening up at hot spots. Such expectations -3 -3 are revealed in Fig.4 (a) and (c). The Matsubara-frequency -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 (Σ( )) kx kx dependence of the Im k, ω either goes to zero linearly (on (c) 3 (d) 3 the pockets) or diverges (at the hot spot). Near the AFM-QCP, 2 2 however, the situation is very different. Fermions at the hot 1 1 spots shows non-Fermi liquid behavior, namely, as shown in Fig.4 (b) and (d), Im (Σ(k, ω)) goes to a small constant at low ω, ky 0 ky 0 and no sign of either vanishing or diverging are observed. The -1 -1 fermions away from the hot spots remains in Fermi-liquid like. -2 Low -2 Once again, DQMC and EQMC simulations give consistent -3 -3 results with the same qualitative behavior. -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 kx kx It is worthwhile to point out, at the QCP, for fermions at the hot spot, a finite imaginary part in fermion self-energy is FIG. 3. Ferm surface obtained from DQMC (panels (a) and (b)) observed, which doesn’t seem to decay to zero as we reduce and EQMC (panels (c) and (d)). Here we show the Fermi surface the frequency. This behavior (a constant term in the imaginary by plotting the fermion spectrum function at zero energy A(k, ω = part of the self-energy) is not yet theoretically understood. ) ( / ) ∼ ( ) 0 utilizing the standard approximation G k, β 2 A k, ω = 0 . However, it is consistent with similar QMC studies, where (a) and (c), FS in the AFM ordered phase (h < hc), where Fermi pockets are formed from zone folding. DQMC and EQMC results such a finite or constant term always seem to emerge near are consistent with each other, while EQMC (with system size L = 60) itinerant QCPs [13, 40, 41]. gives much higher resolution in comparison with DQMC (L = 28). (b) and (d), similar comparison at the QCP (h = hc). B. Universality class and critical exponents

In our previous work on triangle lattice AFM-QCP [40] with 2Q , Γ (in fact in that case one has 3Q = Γ), the bosonic susceptibilities χ(T, hc, q, ω) [as defined in Eq. (5) and Eq. (6)] DQMC and EQMC are complementary to each other, the close to the QCP, revealed with 30 × 30 × 600 (L × L × Lτ) former provides unbiased results with relatively small systems from DQMC, fits to the form of and the latter, as an approximation, provides results closer to the QCP with finite size effects better suppressed. One other χ(T, hc, q, ωn) benefit of EQMC is that it provides much higher momentum 1 = . (9) resolution close to the hot spots. Fig.3 depicts the FS of the (c T + c0T 2) + c |q|2 + c ω + c0 ω2 model in Eq. (1) obtained from G(k, β/2) ∼ A(k, ω = 0) via t t q ω ω DQMC (panels (a) and (b)) and EQMC (panels (c) and (d)). −1 In particular, at low ω, χ (0, hc, 0, ω) exhibits a crossover The left panels are for h < hc, i.e inside AFM metallic phase, behavior from ω2 to ω and the susceptibility scales with q whereas the right panels are for h ∼ hc, i.e. at the AFM- −1 2 as χ (0, hc, q, 0) ∝ |q| , i.e., no anomalous dimension is QCP. The DQMC data are obtained from L = 28, β = 14 observed. The system acquires dynamic critical exponent z = simulations, it is clear that the momentum resolution is still too 2, consistent with the Hertz-Millis mean-field expectation of low to provided detailed FS structures near the hot spots. With the AFM-QCP at its upper critical dimension d + z = 4. EQMC, the system sizes are L = 60 and β = 14 in Fig.3 (c) and For the square lattice model in this paper, we expect the (d), and the momentum resolution is dramatically improved. dynamic spin susceptibility has the following asymptotic form For example, in Fig.3 (c), inside the AFM metallic phase, the in the quantum critical region(h = h ) gap at hot spots are clearly visualized. And in Fig.3 (d), at c the AFM-QCP the FS recovers the shape of the non-interacting χ(T, hc, q, ωn) one, and non-Fermi-liquid behavior emerges at the hot spots as 1 shown in the next section. To capture this important physics, = . (10) a 2 1−η 0 2 EQMC and its higher momentum resolution play a vital role. ctT t + (cq |q| + cωω) + cωω 6

0 0 4

-0.05 -0.005 -0.01 2 -0.1 -0.015 -0.15 Fermi-pockets -0.02 Fermi-pockets 0 Hot-spots Hot-spots -0.2 -0.025 0 1 2 3 4 5 0 1 2 3 4 5 -2 0 0 Fermi-pockets -0.05 -0.005 Hot-spots -4 -0.01 -2 -1.5 -1 -0.5 0 0.5 1 1.5 -0.1 -0.015 -0.15 Fermi-pockets -0.02 4 Hot-spots -0.2 -0.025 0 1 2 3 4 5 0 1 2 3 4 5 3 FIG. 4. (a) Self-energy obtained from EQMC inside the AFM-metal phase (h < hc). On the Fermi pockets, the system is in a Fermi liquid 2 state as shown by the linearly vanishing of Im(Σ(k, ω)) at small ω. At the hot spot, due to the Fermi surface reconstruction, an energy gap opens up, resulting in a divergent Im(Σ(k, ω)). (b) Self-energy 1 obtained from EQMC at the AFM-QCP (h ∼ hc). On the Fermi pockets, the system remains a Fermi liquid as shown by the linearly 0 vanishing Im(Σ(k, ω)) at small ω. At the hot spot, Im(Σ(k, ω)) has 0 1 2 3 4 5 6 a small but finite value as ω goes to 0, which is the signature of a non-Fermi-liquid. (c) and (d) show the same quantities produced in DQMC simulations with smaller system sizes, where qualitative behaviors remain the same. FIG. 5. (a) |q| dependence of the bosonic susceptibilities χ(T = 0, h = hc, q, ω = 0) at the AFM-QCP. The system sizes are L = 40, 50 and 60. The fitting line according to the form in Eq. (B1) −1 2(1−η) This functional form is similar to the Hertz-Millis theory, but reveals that there is anomalous dimension in χ (q) ∼ |q| we allow an anomalous dimension (η) as a free fitting param- with η = 0.125. (b) ω dependence of the bosonic susceptibilies χ(T = 0, h = h , q = 0, ω) at the AFM-QCP. The system size is eter. We used this functional form to guide our data analysis. c −1 L = 50 and the temperature is as low as β = 25 (Lτ = 500). The We first look at the q dependence of χ , as shown in Fig.5 fitting line according to the form in Eq. (B1) reveals that there is (a), the momentum |q| is measured with respect to the hot anomalous dimension in χ−1(ω) ∼ ω(1−η) at small ω and crossover spot K. Here we plot the susceptibility data by substract- to χ−1(ω) ∼ ω2 at high ω. −1 ing the finite temperature background as, χ (T, hc, |q|, 0) − −1 a χ (T, hc, 0, 0) = cq |q| q , where aq = 2(1 − η), and fit the curve to obtain the coefficient c and the anomalous dimen- q ior from the bare (2 + 1)D Ising universality with η = 0.036 sion η, as shown by the solid line in Fig.5 (a). Utilizing to a new one of AFM-QCP with η = 0.125. However, at high- EQMC, with the system size as large as L = 60, the powerlaw frequency, where the coupling between fermions and bosons behavior χ−1(|q|) ∝ |q|aq clearly manifests, with c = 1.04(1) q becomes irrelevant, the bare boson university comes back and and a = 2(1 − η) = 1.75(2) with η = 0.125. In DQMC sim- q the ω2-term dominates over the susceptibility, consistent with ulation, we observed the same exponents η = 0.11(2), with our observation of the bare boson susceptiblity. We also note a little bit lower accuracy due to smaller system sizes, the that because the frequency dependence here is polluted by the DQMC results are shown in the Sec. II of Appendix. IR irrelevant ω2 contributions, whose contribution is about As for the frequency dependence in χ, as shown in Fig.5 10% at ω ∼ 0.25, this frequency exponent has a lower accu- (b), we analyze the χ−1(T, h , 0, ω) − χ−1(T, h , 0, 0), to sub- c c racy, in comparison with the momentum dependence shown tract the finite temperature background and test the predicted in Fig.5 (a). Although the data are consistent with dynamical anomalous dimension η = 0.125 in Eq. (B1) and the data exponent z = 2, small corrections in the form of anomalous points fit very well the expected functional form dimension in the dynamical exponent as predicted in Ref. [26] −1 1−η 0 2 cannot be excluded. χ (ω) = (cωω) + c ω , (11) ω −1 1.96 In the absence of fermions, χ (0, hc, |q|, 0) ∝ |q| [(2 + where we obtained the values of the coefficients cω = 0.068(1) 1)D Ising]. According to the Hertz-Millis theory, this exponent 0 and cω = 0.071(3). It is worthwhile to note that the crossover should increase from 1.96 to 2 in the presence of fermions −1 −1 2 behavior in χ (ω) is very interesting, since at low-frequency, χ (0, hc, q, 0) ∝ |q| . Such an increase is indeed observed the anomalous dimension in ω0.875 dominates bosonic sus- in triangular lattice model (2Q , Γ)[40]. However, it is ceptibility and this means that the coupling of the fermions remarkable for the square lattice model (2Q = Γ), exactly with the critical bosons have changed the universality behav- the opposite was observed. Instead of increasing, this power 7

(b) results will hold and thus here we compare our numerical results with existing theoretical predictions for Heisenberg AFM QCPs, but it must also be emphasized that agreement at the quantitative level is not expected here because of this difference in symmetry. In the perturbative renormalization group calculation [23, 26], the anomalous dimension depends on the angle between (c) the hot spot Fermi velocity and the order wavevector Q. As shown below, for our model, this angle is close to 45◦. At this angle, the RG-prediction for the anomalous dimension is η = 1/Nh.s. where Nh.s. is the number of hot spots [23, (a) 26]. In our model, Nh.s. = 16, and thus the RG predicted value is η = 1/16. However, as shown above, although we observed the same qualitative behavior, the value of η that we FIG. 6. (a) Feynman diagram representing a four-boson interaction observed is close to 2/Nh.s. instead. Whether this quantitative vertex. Dashed lines, φ(k), represent spin fluctuations at momentum disagreement is due to the symmetry difference (Heisenberg k and we set q << Q. Because low-energy physics is dominated by vs Ising) or some other contributions is an interesting open fermionic excitations near the FS, two of the four boson legs must question. have momenta near Q, while the other two are near −Q to keep In addition, the RG analysis also predicts that near the QCP the fermions near the FS as shown in the figure. For 2Q = Γ, +Q the Fermi surface at hot spots shall rotate towards nesting [23]. − and Q becomes identical, and thus there exist two ways to contract This rotation of Fermi surface will further increases the anoma- the external legs as shown in (b) and (c). For 2Q , Γ, however, only the contraction shown in (b) is allowed, while the momentum lous dimension and can even renormalize the value of the dy- conservation law is violated in (c). namic critical exponent z [23, 26]. As will be shown below, our study indeed observed this Fermi surface rotation near the QCP. However, because this RG flow is very slow, our hot-spot −1 1.75 Fermi surface only rotated by about half degree in our simu- actually decreases from 1.96 to 1.75, χ (0, hc, q, 0) ∝ |q| . Such a significant contrast is beyond numerical error, and it lation before stoped by cut-off. For such a small rotation, the indicates that QCPs with 2Q , Γ and 2Q = Γ belong to resuling increase of anomalous dimension and the change in totally different universality classes, which is one of the key dynamic critical exponent is too weak to be observed. observation in our study. This difference can be understand in the following way. 0 TABLE I. The Fermi velocity vF at hot spot in K3 hot spot mesh Between 2Q = Γ and 2Q , Γ, the constraints that the mo- ( ) obtained from ω0ReG k,ω0 data showed in Fig.7 and the Fermi mentum conservation law enforces are different. As shown in ImG(k,ω0) Ref. [23], the QCPs with 2Q = Γ deviates from the Hertz- velocity in free fermion case. Millis theory already at the the level of four-boson vertex cor- Hot spots location kx ky rection, as shown in Fig.6 (a). For 2Q = Γ, this four-boson vertex gives two topologically different bosonic self-energy di- 2.5800 0.5615 agrams, Fig.6 (b) and (c), and in particular the diagram shown vF at hot spots vk v⊥ in Fig.6(c) results in logarithmic corrections and is responsible for the breakdown of the Hertz-Millis scaling. However, for Near QCP 1.523(8) 1.435(8) 2Q , Γ (e.g. in the triangular lattice model, we have 3Q = Γ Free fermion 1.506 1.468 instead), this crucial diagram is prohibited by the momentum conservation law, and thus, at least within the same level of We calculate the Fermi velocity vF as approximation, deviations from the Hertz-Millis picture is not expected. Further investigations, both analytical and numer- ∂ ω0ReG(k, ω0) ical, are needed to better understand the role of this subtle vF = , (12) ∂k ImG(k, ω0) k=k difference, as well as the RG flows in other cases like 3Q = Γ, F etc. where ω0 = π/β. To accurately compute the derivative, we first use simple functions to fit the discrete data points of ω0ReG(k,ω0) vs k as showed in Fig.7. And then, we compute the ImG(k,ω0) C. Comparison with RG analysis derivative for the fitting function to obtain the Fermi velocity, which is recorded in TableI. vk and v⊥ are the two components On the theory side, perturbative renormalization group cal- of the Fermi velocity at a hot-spot parallel and perpendicular culation has been performed for Heisenberg AFM QCPs with to Q respectively. In the table, we showed both the non- SU(2) symmetry [23, 26], while the same study for Ising spins interacting Fermi velocity (bare values) as well as the Fermi has not yet been carefully analyzed to our best knowledge. velocity measured at the QCP (renormalized values). However, because some of the key features in the RG analysis According to the RG analysis [23], vk and v⊥ shall flow to are insensitive of the spin symmetry [23, 26], many qualitative infinite and zero respectively, but in the same time their product 8

universality class, in contrast to the Hertz-Millis prediction, 2 which doesn’t rely on the value of Q. This observation supports L = 50, β = 25 (a) the 1/Nh.s. expansion and R.G. analysis discussed in Ref. [23]. f(kk) = a1cos(kk) + b1 In the study of criticality and anomalous critical scalings, ) 0 ) 1 0 the comparison between theory and numerical results plays a k,ω ( k,ω

( vital role. For QCPs in itinerant fermion systems, although G G

Re L/Lf = 5 non-mean-field scaling beyond the Hertz-Millis theory has 0 Im 0 ω a = −2.12(1) been predicted in theory and observed in QMC simulations, 1 it has been a long standing challenge to reconcile numerical -1 and theoretical results. Our study offers a solid example where 0 0.5 1 1.5 an agreement between theory and numerical simulations starts kk to emerge, which is one first step towards a full understanding about itinerant QCPs [32]. In particular, to pin point its exactly 2 value of the frequency exponent and to probe the predicted L = 50, β = 25 (b) anomalous dynamical critical exponents [26, 28, 29], lower g(k⊥) = a2cos(1.45 × k⊥) + b2 ) 0 ) 1 temperature and frequency range needs to be explored. As 0 k,ω

( pointed out in Refs. [31], future works along this line are k,ω ( G

G highly desirable, and are actively being pursued by us [32]. Re 0 Im 0 L/L = 5 ω f At the techinical level, a combination of DQMC and EQMC methodologies in this work shows a very promising direction a2 = −1.080(7) -1 in the numerical investigations of itinerant QCPs. Besides the 0 0.5 1 1.5 consistency check in Ref. [41] for triangular lattice AFM-QCP, the square lattice AFM-QCP investigated here provide the sec- k⊥ ond example of the consistency in DQMC and EQMC in terms of revealing critical properties. Such consistency suggests a ω0ReG(k,ω0) FIG. 7. variation along (a) Q = (π, π) direction k k and new pathway for future studies about quantum criticality in ImG(k,ω0) ( ) 0 fermionic systems, in that, one can use DQMC on small sys- (b) perpendicular to Q = π, π direction k⊥ at K3 hot spot mesh showed at Fig.1. We use the simple trigonometric function f (k k) tems to provide benchmark results, and the utilize EQMC to and g(k⊥) to ﬁt the data in (a) and (b). reveal IR physics at the thermodynamic limit.

vk × v⊥ shall remain a constant. In a numerical simulation, ACKNOWLEDGEMENT this RG flow will be stoped by numerical cutoffs, e.g. finite size effects. Because this RG flow is marginal at tree level, We acknowledge valuable discussions with Avraham Klein, the flow is expected to be very slow (logarithmic) and thus our Sung-Sik Lee, Yoni Schattner, Andrey Chubukov, Subir observed renormalized value shall not differ dramatically from Sachdev and Steven Kivelson on various subjects of itinerant the bare ones. As can be seen from TableI, this is indeed what quantum critcality. ZHL, GPP and ZYM acknowledge fund- we observed. The renormalized value of vk (v⊥) is slightly ings from the Ministry of Science and Technology of China larger (smaller) than its bare value, and the product of vk and through the National Key Research and Development Program v⊥ remains largely a constant (2.210 for the bare values and (2016YFA0300502), the Strategic Priority Research Program 2.186 for the renormalized ones), as the RG theory predicts. It of the Chinese Academy of Sciences (XDB28000000) and is worthwhile to highlight here that although changes in vk and from the National Science Foundation of China under Grant v⊥ are small, it is beyond numerical error as shown in TableI, Nos. 11421092, 11574359 and 11674370. XYX is thankful and theoretically, this small change in consistent with the slow for the support of HKRGC through grant C6026-16W. K.S. RG flow predicted by theory. acknowledges support from the National Science Foundation under Grant No. EFRI-1741618 and the Alfred P. Sloan Foun- dation. We thank the Center for Quantum Simulation Sciences DISCUSSIONS in the Institute of Physics, Chinese Academy of Sciences and the Tianhe-1A and Tianhe-II platforms at the National Super- If we compare the AFM-QCPs with 2Q , Γ and 2Q = computer Centers in Tianjin and Guangzhou for their technical Γ, QMC studies indicate that they belong to two different support and generous allocation of CPU time.

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Appendix A: THERMAL PHASE TRANSITION are determined. Fig.8 (a) and (b) are for the EQMC results at β = 5 and Fig.8 (c) and (d) are for the DQMC results at β = 4. The thermal phase boundary in Fig. 1(c) of the main text The left panels (a) and (c) plot the original data of the order is controlled by the 2D Ising critical exponents β = 1/8 and parameter |m(h, L)| vs h. The right panels (b) and (d) shows ν = 1. In our model, the AFM Ising field order parameter is the data collapse according to Eq. (A2). Note that hc is a free given by fitting parameter, and the best data collapse gives hc = 3.30(1) at β = 5.0 for EQMC and hc = 3.21(5) at β = 4.0 for DQMC. All the other critical field strength h at finite temperature c ¹ Õ phase transitions, given in pure boson model (the 2D transverse |m| dτ sz(τ)eiQ·ri , = i (A1) field Ising model), DQMC and EQMC simulations, presented i in the Fig.1 (c) of the main text, are obtained in the same procedure as shown here. where Q = (π, π) represented the finite 2Q = Γ SDW. Close to the thermal phase boundary, the order parameter |m| are expected to obey the finite size scaling forms Appendix B: Comparison of EQMC and DQMC at quantum critical region −β/ν 1/ν m(h, L) = L f (L (h − hc)), (A2) In the main text, we have discussed the dynamic Ising spin where f is the scaling function. To locate the finite temperature susceptibility, χ(T, h, q, ωn), and performed the quantum crit- classical critical point, we collect the m(h, L) with different size ical scaling analysis by the data obtained from EQMC. And L and transverse field h, and rescale the order parameter as for the sake of saving space, we didn’t show the correspond- ing DQMC results, which we present in this section of SI. The dynamic Ising spin susceptibility obtained from DQMC β/ν 1/ν L m(h, L) = f (L (h − hc)). (A3) at quantum critical region h = hc = 3.32 are shown in Fig.9, the fitting form of the χ is still as that in the main text,

1 0.35 0.6 χ(T, h , q, ω ) = , c n a 2 1−η 0 2 ctT t + (cq |q| + cωω) + cωω 0.3 0.5 0.25 with the momentum |q| is measured with respect to the hot (a) 0.4 (b) 0.2 spot K. 0.3 0.15 2.5 5 (a) (b) 0.2 2 0.1 EQMC EQMC 1.5 4 0.1 0.05 1 3 0.5 0 0 3.25 3.3 3.35 3.4 -1 -0.5 0 0.5 1 0 2 -0.5 0.7 1 -1 1 0.6 -1.5 0.8 -2 0 0.5 -1 -0.5 0 0.5 1 1.5 0123456

0.4 (c) 0.6 (d)

0.3 FIG. 9. Ising susceptibility data obtained form DQMC at the QCP 0.4 DQMC DQMC hc = 3.32. (a) |q| dependence of the bosonic susceptibilities χ(T = 0.2 0, h = hc, q, ω = 0) at the AFM-QCP. The fitting line according 0.2 to the form in Eq. (B1) reveals that there is anomalous dimension 0.1 in χ−1(q) ∼ |q|2(1−η) with η = 0.11(2). (b) ω dependence of the 0 0 bosonic susceptibilies χ(T = 0, h = hc, q = 0, ω) at the AFM-QCP. 2.833.2 3.4 3.6 -2 -1 012 The fitting line according to the form in Eq. (B1) reveals that there is anomalous dimension in χ−1(ω) ∼ ω(1−η) at small ω and crossover −1 2 FIG. 8. Finite-temperature AFM-to-PM phase transition and the date to χ (ω) ∼ ω at high ω. collapses according to the 2D Ising critical exponents for EQMC simulations (upper panels (a) and (b)) and DQMC simulations (lower In Fig.9, We detect the power-laws in momentum planes (c) and (d)). dependence and frequency dependence of the bosonic susceptibilities χ(T = 0, h = hc, q, ω = 0) by fit- −1 −1 2(1−η) In the main text, we show the finite temperature phase ting χ (T, hc, |q|, 0) − χ (T, hc, 0, 0) = cq |q| and −1 −1 1−η 0 2 boundary obtained by EQMC and DQMC. Fig.8 shows the χ (T, hc, 0, ω) − χ (T, hc, 0, 0) = (cωω) + cωω at QCP examples of how these finite temperature phase boundaries located by DQMC. We obtained the values of the coefficients 11

0.45 0.45 0.45 (a) (b) (c)

0.4 0.4 0.4

0.35 0.35 0.35 3.2 3.25 3.3 3.35 3.4 3.2 3.25 3.3 3.35 3.4 3.2 3.25 3.3 3.35 3.4

D E FIG. 10. Static pairing-correlation function C = 1 Í ∆†∆ for order parameters deﬁned in Eq. (C1) as a function of transverse ﬁeld h. L2 ij i j No enhancement of pairing correlation functions is observed.

0 tss aq = 2(1 − η) = 1.77(3), cω = 0.065(8) and cω = 0.077(3). where ∆i is orbital-triplet, spin-singlet channel s-wave order Most importantly, the anomalous dimension η = 0.11(2) is st0s parameter, ∆i is orbital-singlet, spin-triplet channel s-wave obtained from DQMC data. These results are consistent with st1s order parameter and ∆i is orbital-singlet, spin-triplet chan- that obtained in our EQMC data showed in the main text. nel s-wave order parameter, on lattice site i and 1 and 2 are the layer indices. In Fig. 10, we show of these three types of static pairing-correlation function C obtained from DQMC data at Appendix C: SUPERCONDUCTIVITY spin fermion coupling strength ξ = 1.0 and β = 25 across the AFM-QCP hc = 3.32 determined in DQMC simulation. We now discuss the superconducting properties close to The pairing-correlation function C in Fig. 10 shows that the the 2Q = Γ AFM QCP. As described in previous FM-QCP pairing correlations do not grow with the system size, indicat- study Ref. [13], in such type of two layered band structure of ing that the superconducting instabilities are still weak with spin fermion model, the obvious superconductivity instabili- our current parameter set. Of course, there is high possibility, ties usually happen at very strong coupling strength (ξ/t = 6 as we have seen in the FM-QCP case in Ref. [13], that en- for example) and relatively low temperature, and the spin ﬂuc- hancing the strength of spin-fermion coupling ξ/t much larger tuations could most likely induce an attractive interaction in than 1 would enhance the superconducting instabilities, but the on-site s-wave channel and after that other channels with the purpose of this paper is to have a pristine QCP region such even weaker strength. that the quantum critical scaling behaviors can be determined To probe for superconducting tendencies near the QCP, within accessible parameter sets, and we found that our current we calculate three types of pairing correlations C = choice of parameter served the purpose. D E 1 Í ∆†∆ L2 ij i j for on-site s-wave channels, and the superconducting order parameters ∆ is given by

tss 1 ∆i = √ cˆ1↑cˆ1↓ + cˆ2↑cˆ2↓ , 2 st0s 1 ∆i = √ cˆ1↑cˆ2↓ + cˆ1↓cˆ2↑ , 2 st1s 1 ∆i = √ cˆ1↑cˆ2↑ + cˆ1↓cˆ2↓ , (C1) 2