Downloaded by guest on September 26, 2021 tnrn unu rtclsseshv ensuid uhas wave such density charge studied, liq- (13), of been ferromagnetic that variety Fermi 34), have wide 32 (33, systems a a Ising-nematic ref. critical fluctuations, coupling in quantum critical review Via bosonic itinerant a various progress). and with recent uid 31 (see ref. the problem in challenging summarize commentary this concise about a understanding sharp- toward our pathway additional ening an paved has techniques Carlo non- its puzzles to and due questions chal- open. important remain physics, most many matter the and nature, among condensed perturbative still in 23– is subjects (1–9, criticality lenging decades quantum recent itinerant (20), in 30), efforts MnP extensive superconductiv- (19), despite and However, CrAs monopnictides, order critical and CrAs transition-metal quantum magnetic in Cu- pressure-driven between ity 15), well discovered (QCP) as (14, recently point anoma- (16–18) the materials of behav- superconductors as understanding heavy-fermion non–Fermi-liquid high-temperature the in and Fe-based in metal, (9–13) role strange iors vital transport, a lous plays It (1–8). I transition phase quantum understand- critical final gapless toward with interacting stone fermions theoreti- excitations. correlated stepping 2D recent criticality the the quantum of the as ing metallic serve bridge in can results opens developments and gap These numerical energy hotspots. and the cal the as at observed metallic are ordered up a pockets antiferromagnetically into fermion the evolve phase, In hotspots at liquid. fermions non-Fermi and dimension propagator, critical bosonic anomalous and quantum the univer- finite class the different universality a At Ising a prediction. point, RPA bare into Hertz–Mills–Moriya the point the both critical from bosonic different sality, bare return in the fermions the render and introduce fermions fluctuations among spin interactions antiferromagnetic effective the that found precision. high We unprecedented with revealed are behaviors scaling 60 of to relevant sizes be (L System might behaviors. which non–Fermi-liquid metals, critical their other and high- cuprates in Tc at fluctuations antiferromagnetic fluctuations low-energy and spin setup face on antiferromagnetic point with Q critical lattice wavevector square quantum state-of-the-art 2D itinerant sys- a a and the develop technique investigate are we simulation tematically work, Carlo approaches Monte this numerical quantum In large-scale and converg- review. analytical and under the still between systems, in results electronic themes central ing correlated 2019) the 31, of January among review for is understanding (received criticality 2019 1, quantum July approved Metallic and MA, Cambridge, University, Harvard Sachdev, Subir by Edited China 523808, Guangdong Dongguan, China; 100190, China; www.pnas.org/cgi/doi/10.1073/pnas.1901751116 China; 100190, Beijing Sciences, China; of Kong, Academy Chinese of University Sciences, a Liu Hong Zi pockets hotspots fermion and with point critical quantum Itinerant ejn ainlLbrtr o odne atrPyisadIsiueo hsc,CieeAaeyo cecs ejn 010 China; 100190, Beijing Sciences, of Academy Chinese Physics, of Institute and Physics Matter Condensed for Laboratory National Beijing rn lcrnssesi fgetiprac n interest and importance great of is itin- in systems criticality quantum electron materials, correlated erant of study the n × h eetdvlpeto inpolmfe unu Monte quantum sign-problem-free of development recent The L 1−x f × hns cdm fSine etro xelnei oooia unu optto,Uiest fCieeAaeyo cecs Beijing Sciences, of Academy Chinese of University Computation, Quantum Topological in Excellence of Center Sciences of Academy Chinese L P τ x r ofral cesd n h unu critical quantum the and accessed, comfortably are ) d eateto hsc,Uiest fMcia,AnAbr I48109; MI Arbor, Ann Michigan, of University Physics, of Department a,b 2) n te rM-deeto ytm (22). systems electron Cr/Mn-3d other and (21), g colo hsclSine,Uiest fCieeAaeyo cecs ejn 010 hn;and China; 100190, Beijing Sciences, of Academy Chinese of University Sciences, Physical of School = apiPan Gaopei , (π , π — rbe htrsmlsteFrisur- Fermi the resembles that problem )—a | o-em liquid non-Fermi a,b ioYnXu Yan Xiao , | η rtclexponent critical ∼ 2 sosre in observed is 0.125 c a Sun Kai , × 60 d n iYn Meng Yang Zi and , × 320 c eateto hsc,Hn ogUiest fSineadTcnlg,Hong Technology, and Science of University Kong Hong Physics, of Department ihfiieodrn wavevector QCPs itinerant ordering on finite focusing with by studies numerical and theoretical numerically be to remain verified. and still observed behaviors Hertz– the unpre- scaling beyond or Millis–Moriya properties observed proposed 40) are theoretically theories, (33, while Hertz– existing (13), found the from deviating is either exponents, behavior QCPs, dicted scaling itinerant mean-field studied simulations, Millis QMC theory recently in far, between all So gap among remains. still major studies numerical a beyond and points theory, critical Hertz–Millis–Moriya quantum novel the of search the for obtained, genuine quantum the itinerant for access limit (IR) criticality. sizes to infrared the us system in quantum behaviors allowing larger scaling determinantal consequently explore conventional Carlo, to with Monte handled possible those becomes (EQMC) than Carlo now Monte and it quantum (48–54) (41), (SLMC) ultrasize in fast Carlo momentum techniques, the Monte elective (QMC) With self-learning Carlo (42–47). the Monte particular fields quantum gauge of and development topolog- transitions interaction-driven phase and (36–41), ical wave density spin (35), 1073/pnas.1901751116/-/DCSupplemental. y at online information supporting contains article This 1 BY-NC-ND) (CC 4.0 NoDerivatives License distributed under is article access open This Submission. y Direct PNAS a is article This interest.y of conflict no declare authors The these of study the in question key One (CDW/SDW). waves n ...promdrsac;XYX,KS,adZYM nlzddt;adKS n Z.Y.M. and K.S. paper. and the data; K.S., analyzed wrote Z.Y.M. X.Y.X., G.P., and K.S., Z.H.L., X.Y.X., research; research; performed Z.Y.M. designed Z.Y.M. and and K.S., X.Y.X., contributions: Author owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To et ohi nltclter n nnmrclsimulation numerical in develop- and QCPs. future theory itinerant bridge of analytical in could in observed they both being and ments are manner results unbiased These evolve an liquid. hotspots anoma- non-Fermi at finite fermions a a and into QCP, observed, the is At dimension relations accuracy. lous high scaling with critical revealed quantum are are cuprate and temperature as accessed low such and comfortably size a systems, system to correlated Large relevance strongly superconductors. immediate of has range model wide antiferromag- The by fluctuations. generated authors QCPs netic The itinerant simu- examine Carlo (QCP). research Monte to point quantum lation developed in critical and model progress quantum a designed major itinerant the summarizes on work present The Significance nti ae,w i tipoigtecnegnebetween convergence the improving at aim we paper, this In been have insights and results intriguing many Although e eateto hsc,TeUiest fHn og ogKong, Hong Kong, Hong of University The Physics, of Department e,a,f,g,h,1 y . y h oghnLk aeil Laboratory, Materials Lake Songshan raieCmosAttribution-NonCommercial- Commons Creative Q 6 0 = .. hresi density charge/spin e.g., , www.pnas.org/lookup/suppl/doi:10. NSLts Articles Latest PNAS b colo Physical of School | f8 of 1

PHYSICS QCPs is about the universality class, i.e., whether all these types (28, 29). For 2Q =6 Γ, on the other hand, these exotic behav- of QCPs, e.g., commensurate and incommensurate CDW/SDW iors are not expected, at least up to the same order in the 1/N QCPs, belong to the same universality class or not. To the expansion, and thus presumably they follow the Hertz–Millis leading order, within the random phase approximation (RPA), mean-field scaling relation. as long as the ordering wavevector Q is smaller than twice On the numerical side, a QCP with 2Q 6= Γ was recently the Fermi wavevector 2kF or, more precisely, as we shift the studied (40, 41) and the results are in good agreement with Fermi surface (FS) by the ordering wavevector Q in the momen- the Hertz–Millis–Moriya theory. For the more exotic case with tum space, the shifted FS and the original one shall cross 2Q = Γ, the numerical result is less clear because in QMC at hotspots, instead of tangentially touching with or overrun- simulations, a superconducting dome usually arises and covers ning each other, and the same (linear ω) Landau damping the QCP (38, 39, 56). Outside the superconducting dome, at and critical dynamics are predicated regardless of microscopic some distance away from the QCP, mean-field exponents are details, implying the dynamic critical exponent z = 2. For a observed to be consistent with the Hertz–Millis–Moriya theory. 2D system, this makes the effective dimensions d + z = 4, coin- However, whether the predicted anomalous (non–mean-field) ciding with the upper critical dimension. As a result, within behaviors (23, 26, 28, 29) will arise in the close vicinity of the Hertz–Millis approximation, mean-field critical exponents the QCP remains an open question, which requires the sup- shall always be expected, up to possible logarithmic correc- pression of the superconducting order. In addition, due to the tions, and thus all these QCPs belong to the same universality divergent length scale at a QCP, to obtain reliable scaling expo- class (1–3, 55). nents, large system sizes are necessary to overcome the finite-size On the other hand, more recent theoretical developments effect. point out that this conclusion becomes questionable once higher- In this paper, we perform large-scale quantum Monte Carlo order effects are taken into account. In particular, 2 differ- simulations to study the antiferromagnetic metallic quantum ent universality classes need to be distinguished, depending on critical point (AFM-QCP) with 2Q = Γ. In this study, 2 main whether 2Q coincides with a lattice vector or not, which are efforts are made to accurately obtain the critical behavior in dubbed as 2Q = Γ and 2Q =6 Γ to demonstrate that 2Q, mod the close vicinity of the QCP. 1) We design a lattice model that a reciprocal lattice vector, coincides or not with the Γ point. realizes the desired AFM-QCP with the superconducting dome Among these 2 cases, 2Q = Γ (e.g., antiferromagnetic QCP with greatly suppressed to expose the quantum critical regions and Q = (π, π)) is highly exotic. As Abanov et al. (23) pointed out 2) we use the determinantal quantum Monte Carlo (DQMC) explicitly, in this case the Hertz–Millis mean-field scaling law as well as the EQMC, both with self-learning updates to access breaks down and a non-zero anomalous dimension emerges. In much larger system sizes beyond existing efforts. The more con- addition, the critical fluctuations will also change the fermion ventional DQMC technique allow us to access system sizes up dispersion near the hotspots, resulting in a critical-fluctuation– to 28 × 28 × 200 for L × L × Lτ for a 2D square lattice, while induced Fermi surface nesting: i.e., even if one starts from a EQMC can access much larger sizes (60 × 60 × 320) to further Fermi surface without nesting, the renormalization group (RG) reduce the finite-size effect and confirms scaling exponents with flow of the Fermi velocity will deform the Fermi surface at higher accuracy. These 2 efforts (1 and 2) allow us to access hotspots toward nesting (23). This Fermi surface deformation the metallic quantum critical region and to reveal its IR scaling will further increase the anomalous dimension and make the behaviors with great precision, where we found a large anoma- scaling exponent deviate even farther from Hertz–Millis predic- lous dimension significantly different from the Hertz–Millis the- tion (23, 24), and even modifies the dynamic critical exponent z, ory prediction, and we also observed that the Fermi surface near as pointed out explicitly by Metlitski and Sachdev (26) and others the hotspots rotates toward nesting at the QCP, as predicted in

A fermion site Ising spin coupling B ’ C 4 1 Pure Boson hc=3.044(3) Ising site fermion DQMC hc=3.32(2) EQMC hc=3.355(5) -t1 ’ 2 3 -t2 -t3 1 4

λ=2 T SDW metal Fermi liquid ky

’ 2 3 h 1 4 J ’ λ=1 3 2 kx h

Fig. 1. (A) Illustration of the model in Eq. 1. Fermions reside on 2 of the layers (λ = 1,2) with intralayer nearest-, second-, and third-neighbor hoppings t1, z 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 on-site Ising coupling is introduced between fermion and Ising spins (ξ). (B) Brillouin zone (BZ) of the model in Eq. 1. The blue 0 lines are the FS of Hf and Qi = (±π, ±π), i = 1, 2, 3, 4 are the AFM wavevectors, and the 4 pairs of {Ki, Ki }, i = 1, 2, 3, 4 are the position of the hotspots (red circles), where each pair is connected by a Qi vector. The folded FS (gray lines) comes from translating the bare FS by momentum Qi. The green patches show the k mesh built around hotspots, and the 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 circle) at hc = 3.044(3) (57, 58) with 3D Ising universality. After coupling with fermions, the QCP shifts to higher values. The green solid circle is the QCP obtained with DQMC (hc = 3.32(2)). The violet solid circle is the QCP obtained from EQMC (hc = 3.355(5)); although the position of the QCP shifts, as it is a 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 those in DQMC with 28 × 28 × 200. The procedure of how the phase boundary is determined is shown in SI Appendix, Thermal Phase Transition.

2 of 8 | www.pnas.org/cgi/doi/10.1073/pnas.1901751116 Liu et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 and akfrtennrva tnrn unu rtclt.I sknown (H is QCP It boson bench- criticality. pure a quantum the as that itinerant serves nontrivial which the fermions, for without mark limit in QCP Model. boson the pure discuss Boson first the we AFM-QCP, Bare itinerant the the about results for Scaling Ising is fermiology antiferromagnetic folding 1B. band an Fig. of in Such result summarized ordering. a as AFM layers) to fermion due 2 from comes 2 factor hs fIigsis h Sehbt i em okt and pockets Fermi big 4 exhibits number FS (hotspot AFM hotspots the the of in pairs spins, deep 4 that Ising such of 59, ref. phase to according chosen are eters strength 58). (57, class wavevector universality a has Ising order 3D antiferromagnetic an the Such in transition phase tum until temperature, transition finite (3. lower a a with at transition finite Ising a transition on thermal ing Ising 2D a J the by denoted temperature of 1C Fig. axes in the shown along spanning diagram aldhtpt,wihaecosn onsbtenteoriginal the as between labeled FSs points folded crossing the ordering and are the which by hotspots, (shifted called folding zone from wavevector FSs and FS coupling original fermion-spin the via along where coupling Ising as on-site given interlayer an via i tal. et Liu field spins coupling Ising ferromagnetic of composed subject is are layer hoppings Fermions third-neighbor between. and in t second-, layer nearest-, spin intralayer Ising to 1 1A and Fig. in layers shown schematically is designed we that Fermiology. Antiferromagnetic Method and Model 2D. understanding in final criticality toward quantum metallic stone the stepping of precious the These are developments performed. numerical comparison and and also theoretical quantitative recent is And the results bridge 29). results numerical 28, and 26, theory (23, between analysis RG the 2 and , 1 = h emosin fermions The H ntesmlto,w set we simulation, the In ,weetetasto un noa into turns transition the where 04438(2)), H b H H h , ecie Dtases-edIigmdladhsaphase a has and model Ising transverse-field 2D a describes σ fb z b f µ i z along . ,λ = = = t = ξ 3 swl steceia potential chemical the as well as , = −ξ J −t − 1 = −1. Q X hij t 1 2 1 3 s X n leave and .0, (c i hij 11856 i x hhhij i Q (i X emosadIigsisaeculdtogether coupled are spins Ising and Fermions . i s † h i,λσ ,λ↑ i z lgtbu ie,at line), blue (light i s h ytmeprecstesm F 2D AFM same the experiences system the , X ,2 ,4 3, 2, 1, = s i iii,λσ z with j z H c − eeto density (electron c i σ f ,λ↑ H i † i z ,λσ h ,1 xeineteAMflcutosin fluctuations AFM the experience i = c ,2 ,4 3, 2, 1, = X J + − i † i c ,λσ H h j σ (J c b )fr em okt n h so- the and pockets Fermi form )) ,λσ f s i i z † scnrlprmtr.Teparam- The parameters. control as eog othe to belongs ) ,2 ,λ↓ K i x c + h qaeltieAMmodel AFM lattice square The > j t
i ,λσ 1 H − H , and n rnvremagnetic transverse a and 0) c s 1 = fb b i i t z ,λ↓ 2 the 1B, Fig. in shown As . + N nFg 1B. Fig. in + h ihnaetniho anti- nearest-neighbor with hh .0, K h H 0 = h hn ) ij .s X i 0 .c fb ii,λσ . i with t eoepeetn our presenting Before ξ ,λ stefrinspin fermion the is 8 = , h ytmundergoes system the , . 2 h aitna is Hamiltonian The . − ∼ i = T µ × c 2+1) + (2 −0.32, i rdal turn- Gradually . h sn spin Ising The µ. i † ,tecoupling the 0.8), i ,2 ,4 3, 2, 1, = ,λσ X ,λσ 16 = 2 ih2fermion 2 with Q c T T n j sn uni- Ising D ,λσ i (π = ,λσ t 0 = and 3 hr the where 0 = . , h quan- π h .128, = As . as ), [3] [4] [2] [1] H t h 1 b c , ycluaigtednmcsi susceptibility spin numerically dynamic demonstrated the be can calculating This by 58). 57, (13, class versality oee,i h unu rtclrgo,a eset form we explicitly. asymptotic of as the write form region, to functional critical hard quantum scaling dynamic the and the in of complicated However, form is functional the susceptibility QCP, spin the near principle, In etblte r hw nFg .T xlr h momentum the sus- explore dynamic To The 2. Fig. temperature. in low shown and are sufficiently sizes ceptibilities access Ising to system 3D (61) updates large Wolff cluster anisotropic performed (62) this we Swendsen–Wang simulations, solve and Carlo To Monte with (60). model a model to model Ising Ising transverse-field classical 2D the map to scheme point. critical Ising the Eq. at form symmetry Lorentz scaling emergent This and coefficients. class, sal universal Ising 3D the where eedneof dependence model boson bare 2. Fig. hog h aapit is points data the through is size tem is points data the scaling, critical quantum achieve to A B ihu emos ecnuetesadr path-integral standard the use can we fermions, Without χ(T a , oetmdpnec fthe of dependence Momentum (A) q h χ(T , 2 = L ~ q = , 0wt increasing with 20 ω , χ(q − n h a c = ) η q H , = ln(|q 1 = b q, h ytmszsare sizes system The . 0, L ω 1 ω .964(2) |) 2 n at ) + a = ) X q ij a ln(ω h 2 q Z c = ln(c χ(T t 0 ) h T steuieslciia xoetof exponent critical universal the is β β + c q β o h aebsnmodel boson bare the for , = 2 d ,with ), a ∝ , 2 (c + τ q 0 0 0 0 n 0 h iegoing line The 60. and 50, 40, 30, 20, h e L c ln(c c i t sapid h iegigthrough going line The applied. is , ω , ~ q q n a ω c L |q , q τ ,with ), q χ(q, ω = −i = and , 6 1 NSLts Articles Latest PNAS n 2 8 0 2 epciey and respectively, 72, 60, 48, ~ 2 q ) + xlctyrset the respects explicitly · − ω ~ r a edsrbdby described be can ij c = a η hs ω q c = )at 0) ω = ω i z 6 (B 1.96. 2 (τ 2 r nonuniver- are ) − a )s q h η /2 j z = h = (0)i. H Frequency ) , h = 2+1) + (2 b 1.96. c h sys- The . o the for , h | c the , f8 of 3 [5] [6] D

PHYSICS −1 dependence of the susceptibility, we plot χ (T , hc , q, ω = 0) − In the square lattice model, as shown in Fig. 1B, the AFM −1 χ (T , hc , q = 0, ω = 0), where the momentum q is measured wavevectors Qi connect 4 pairs of hotspots (Nh.s. = 8 in 1 layer from Q = (π, π) and the substraction is to get rid of the finite- and Nh.s. = 16 in 2 layers). In the IR limit, only fluctuations temperature background, such that the following scaling relation connecting each pair of hotspots are important to the univer- is expected at low T : sal scaling behavior in the vicinity of the QCP (7, 9, 23–26, 66). Hence, to study this universal behavior, we draw 1 patch around −1 −1 aq /2 aq each Kl and keep fermion modes therein and neglect other χ (T , hc , q, ω = 0) − χ (T , hc , q = 0, ω = 0) = cq |q| . [7] parts of the BZ. In this way, instead of the original N = L × L momentum points, EQMC keeps only Nf = Lf × Lf momentum As shown in Fig. 2A such a scaling relation is indeed observed points for fermions inside each patch. Here, L and Lf denote with L = 48, 60, and 72 with β = 1/T ∝ L. The power-law diver- the linear size of the original lattice and the size of the patch, gence of the (2 + 1)D Ising quantum critical susceptibility respectively. with power aq = 2 − η = 1.96 is clearly revealed (the 3D Ising DQMC and EQMC are complementary to each other; the for- anomalous dimension η = 0.04). mer provides unbiased results with relatively small systems and A similar scaling relation is also observed in the frequency the latter, as an approximation, provides results closer to the dependence as shown in Fig. 2B, where we plot QCP with finite-size effects better suppressed. One other benefit of EQMC is that it provides much higher momentum resolution −1 −1 aq /2 aq close to the hotspots. Fig. 3 depicts the FS of the model in Eq. 1 χ (T , hc , 0, ω) − χ (T , hc , 0, ω = 0) = cω ω [8] obtained from G(k, β/2) ∼ A(k, ω = 0) via DQMC (Fig. 3 A and B) and EQMC (Fig. 3 C and D). Fig. 3 A and C is for h < hc , for L = 20 with increasing β. The expected power-law decay i.e., inside the AFM metallic phase, whereas Fig. 3 B and D is with the small anomalous dimension aq = 2 − η = 1.96 is clearly for h ∼ hc , i.e., at the AFM-QCP. The DQMC data are obtained obtained. Hence, the data in Fig. 2 A and B confirm the QCP for from L = 28, β = 14 simulations, and it is clear that the momen- the pure boson part H in Eq. 1 belongs to the (2 + 1)D Ising b tum resolution is still too low to provide detailed FS structures universality class. near the hotspots. With EQMC, the system sizes are L = 60 and β = 14 in Fig. 3 C and D, and the momentum resolution is dra- DQMC and EQMC. To solve the problem in Eq. 1 we use 2 C complementary fermionic quantum Monte Carlo schemes. matically improved. For example, in Fig. 3 , inside the AFM The first one is the standard DQMC (13, 63–65) with the metallic phase, the gap at hotspots is clearly visualized. And in D SLMC update scheme (48–54) to speed up the simulation. In Fig. 3 , at the AFM-QCP the FS recovers the shape of the non- SLMC, we first perform the standard DQMC simulation on the interacting one, and non–Fermi-liquid behavior emerges at the model in Eq. 1 and then train an effective boson Hamiltonian hotspots as shown in the next section. To capture these impor- that contains long-range 2-body interactions both in spatial and tant physics, EQMC and its higher-momentum resolution play a in temporal directions. The effective Hamiltonian serves as the vital role. proper low-energy description of the problem at hand with the Results fermion degree of freedom integrated out. We then use the ef- fective Hamiltonian to guide the Monte Carlo simulations; i.e., Non-Fermi Liquid. As we emphasized above, the dramatically we perform many sweeps of the effective bosonic model (as improved momentum resolution in EQMC enables us to study the computational cost of updating the boson model is O(βN ), dramatically lower than the update of the fermion determinant 3 which scales as O(βN )) and then evaluate the fermion determi- 3 3 nant of the original model in Eq. 1 such that the detailed balance A B 2 of the global update is satisfied. As shown in our previous works 2 (40, 49, 50, 53, 54), the SLMC can greatly reduce the autocorre- High 1 1 lation time in the conventional DQMC simulation and make the ky 0 k 0 larger systems and lower temperature accessible. y -1 The other method is the EQMC (41). EQMC is inspired by -1 the awareness that critical fluctuations mainly couple to fermions -2 -2 near the hotspots. Thus, instead of including all of the fermion -3 -3 degrees of freedom, we ignore fermions far away from the -3-2 -1 0 1 2 3 -3-2 -1 0 1 2 3 kx kx hotspots and focus only on momentum points near the hotspots C 3 D 3 in the simulation. This approximation will produce different 2 2 results for nonuniversal quantities compared with the original 1 1 model, such as hc or critical temperature. However, for universal

quantities, such as scaling exponents, which are independent of ky 0 ky 0 microscopic details and the high energy cutoff, EQMC has been -1 -1 shown to generate values consistent with those obtained from standard DQMC (41). Low -2 -2 In EQMC, because a local coupling (in real space) becomes -3 -3 nonlocal in the momentum basis, one can no longer use -3-2 -1 0 123 -3-2 -1 0 1 2 3 kx k the local update as in standard DQMC, as that would cost x 3 βN · O(βNf ) computational complexity. Fortunately, cumula- Fig. 3. (A–D) Fermi surface obtained from DQMC (A and B) and EQMC tive update schemes in the SLMC have been developed recently (C and D). Here we show the Fermi surface by plotting the fermion spec- (49, 50). Such a cumulative update is a global move of the Ising trum function at 0 energy A(k, ω = 0) using the standard approximation 3 β/ ∼ ω = < spins and gives rise to the complexity O(βNf ) for computing G(k, 2) A(k, 0). (A and C) FS in the AFM ordered phase (h hc), where Fermi pockets are formed from zone folding. DQMC and EQMC the fermion determinant. Since Nf can be much smaller than N , speedup of the order ( N )3 ∼ 103 of EQMC over DQMC, with results are consistent with each other, while EQMC (with system size L = 60) Nf gives much higher resolution in comparison with DQMC (L = 28). (B and D) N ∼ 10, can be easily achieved. Similar comparison at the QCP (h = hc). Nf

4 of 8 | www.pnas.org/cgi/doi/10.1073/pnas.1901751116 Liu et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 i tal. et Liu a ml u nt au as value finite liquid. but non-Fermi small a has Im(Σ vanishing linearly the by eairfrom behavior a with revealed 4 are liquid, expectations Fig. Fermi Such hotspots. a in at remains up system opening gap the band 1B, Fig. the to studied according at We and 4. Fig. precisely. phase in shown AFM-metal more are the results FS The in AFM-QCP. the self-energies on fermion the modes fermionic the remain behaviors qualitative where sizes, same. the system smaller with simulations a in resulting up, opens (h gap energy an Im(Σ divergent reconstruction, surface Im(Σ Fermi vanishing the linearly the by npriua,a low at particular, In has one case 4 Fig. in non– shown show as the hotspots Im(Σ(k, namely, however, the at AFM-QCP, behavior; Fermions the Fermi-liquid different. Near very is hotspot). situation the (at diverges Im(Σ(k, the (h 4. Fig. 30 Eqs. in (defined with (40) AFM-QCP Exponents. lattice triangle Critical on and Class QCPs Universality finite itinerant a near such emerge where to 41). seems studies, 40, (13, always QMC term similar constant with or However, the consistent understood. reduce theoretically is part yet we imaginary it not as the is in 0 self-energy) term the to is constant of decay self-energy (a to behavior fermion seem This in frequency. not part does imaginary which finite observed, a hotspot, the same at the with results consistent away behavior. give qualitative fermions simulations DQMC again, The EQMC Once liquid-like. observed. and Fermi remain is hotspots the diverging from or vanishing either C A ∼ < nteAMmtlpae lhuhtebnsaefolded are bands the although phase, AFM-metal the In ti otwiet on u ht tteQP o fermions for QCP, the at that, out point to worthwhile is It × -0.15 -0.05 -0.2 -0.1 -0.15 -0.05 h h -0.2 -0.1 600 c c 0 .O h em okt,tesse ean em iuda shown as liquid Fermi a remains system the pockets, Fermi the On ). .O h em okt,tesse si em-iudsaea shown as state Fermi-liquid a in is system the pockets, Fermi the On ). 0 0 0 efeeg bandfo QCisd h F-ea phase AFM-metal the inside EQMC from obtained Self-energy (A) χ(T ω (L A )) 1 1 = × , (k, ω and ost ml osata low at constant small a to goes 3Q h L )) (c c ω ω 2 C , 2 × t efeeg bandfo QCa h AFM-QCP the at EQMC from obtained Self-energy (B) )). q, Γ = 5 2 ihrge o0lnal o h okt)or pockets) the (on linearly 0 to goes either T and h asbr-rqec eedneof dependence Matsubara-frequency The C. L and ω to + τ 3 ,tebsncsusceptibilities bosonic the ), n Hot-spots Fermi-pockets 3 rmDM,fi otefr of form the to fit DQMC, from ) Hot-spots Fermi-pockets D ) c ω ω t ls oteQP eeldwith revealed QCP, the to close 6) 0 hwtesm uniispoue nDQMC in produced quantities same the show T , n h ucpiiiysae with scales susceptibility the and 4 4 (k, 2 χ (k, + ) ω −1 ω ω )a small at )) 5 ost ,wihi h intr fa of signature the is which 0, to goes 5 (0, )a small at )) c D B q 1 | h q| -0.025 -0.015 -0.005 -0.025 -0.015 -0.005 c -0.02 -0.01 -0.02 -0.01 2 0, , + 0 0 ω 0 0 ω c ttehtpt Im(Σ hotspot, the At . ω ω ) 2Q norpeiu work previous our In ω ttehtpt u to due hotspot, the At . xiisacrossover a exhibits 1 1 + Γ = 6 ω c ω 0 n osg of sign no and , 2 2 ω i ati that in fact (in 2 χ(T . 3 3 Hot-spots Fermi-pockets B Hot-spots Fermi-pockets , h and c , 4 4 (k, q, 30 q [9] ω ω D, as × )) 5 ) 5 yai pnssetblt a h olwn smttcfr in (h form region asymptotic critical following quantum the the has susceptibility spin dynamic AFM-QCP dimension the critical upper of its expectation at mean-field Hertz–Millis the exponent with critical dynamic a acquires system The χ thigh at Eq. in in form dimension the lous to according line fitting is size system susceptibilities bosonic the tigln codn otefr nEq. in in dimension form the lous to according line fitting h sys- the with EQMC, Using as 5A. |q large Fig. as in size line tem solid the by shown coefficient the obtain to i.5. Fig. in shown are results DQMC the sizes; system smaller to due exponents same the 1.75(2) as background χ finite-temperature the ing momentum hotspot the 5A, analysis. Fig. data our guide to form functional this (η used we dimension We but theory, anomalous Hertz–Millis an the to allow similar is form functional This B A c −1 −1 , o h qaeltiemdli hsppr eepc htthe that expect we paper, this in model lattice square the For efis oka the at look first We a q, q (0, (T ω lal aiet,with manifests, clearly = ω , h (A) . h )a h F-C.Tesse ie are sizes system The AFM-QCP. the at 0) c eew lttessetblt aab substract- by data susceptibility the plot we Here K. with χ(T c , ,0 = 0) 0, , 0) q, |q L = = , η eedneo h ooi susceptibilities bosonic the of dependence h 0adtetmeauei slwas low as is temperature the and 50 χ |q ∝ c c χ 0 = −1 t , −1 T q, c (ω (q) nDM iuain eobserved we simulation, DQMC In .125 q a ω 2 L t |q ) .. oaoaosdmnini observed. is dimension anomalous no i.e., ; n ∼ η (c + |q ∼ h oe-a behavior power-law the 60, = ) a 0 = ω χ(T q c (1−η where , q q q |q 2(1−η ihsihl oe accuracy lower slightly with .11(2), |q = n h nmlu dimension anomalous the and eedneof dependence ) 0, tsmall at 2 c d smaue ihrsett the to respect with measured is = ) h q + + with 1 h = 1 = a c c z q 10 10 safe tigparameter. fitting free a as ) h ω ): c 4 = 2(1 = .04(1) ω , η ω eel htteei nanoma- an is there that reveals eel htteei nanoma- an is there that reveals q ) = 1−η = n rsoe to crossover and . 2.(B) 0.125. NSLts Articles Latest PNAS 0, − ω + and tteAMQP The AFM-QCP. the at ) L η χ χ c β = n ttecurve the fit and ), −1 ω 0 −1 = 0 0 n 0 The 60. and 50, 40, ω z a (T 5(L 25 ω ssonin shown as ; 2 2 = q . eedneof dependence 2(1 = , h τ consistent , χ χ χ(T c = −1 −1 , 0) |q, 0) The 500). (ω = − (| | ) q|) 0, η f8 of 5 η [10] ∼ as , = ) h ω ∝ − = 2

PHYSICS This difference can be understood in the following way. Between 2Q = Γ and 2Q 6= Γ, the constraints that the momentum conservation law enforces are different. As shown in ref. 23, the QCPs with 2Q = Γ deviate from the Hertz–Millis theory already at the level of 4-boson vertex correction, as shown in Fig. 6A. For 2Q = Γ, this 4-boson vertex shows 2 topologically different bosonic self-energy diagrams, Fig. 6 B and C, and in particular the diagram shown in Fig. 6C results in logarithmic corrections and is responsible for the breakdown of the Hertz–Millis scal- ing. However, for 2Q 6= Γ (e.g., in the triangular lattice model, we have 3Q = Γ instead), this crucial diagram is prohibited by the momentum conservation law, and thus, at least within the same level of approximation, deviations from the Hertz–Millis picture are not expected. Further investigations, both analyti- cal and numerical, are needed to better understand the role of this subtle difference, as well as the RG flows in other cases like Fig. 6. (A) Feynman diagram representing a 4-boson interaction vertex. 3Q = Γ, etc. Dashed lines, φ(k), represent spin fluctuations at momentum k and we set q << Q. Because low-energy physics are dominated by fermionic excita- Comparison with RG Analysis. tions near the FS, 2 of the 4 boson legs must have momenta near Q, while On the theory side, perturbative the other 2 are near −Q to keep the fermions near the FS as shown. For renormalization group calculation has been performed for 2Q = Γ, +Q and −Q become identical, and thus there exist 2 ways to con- Heisenberg AFM-QCPs with SU(2) symmetry (23, 26), while the tract the external legs as shown in B and C. For 2Q =6 Γ, however, only the same study for Ising spins has not yet been carefully analyzed contraction shown in B is allowed, while the momentum conservation law is to our best knowledge. However, because some of the key fea- violated in C. tures in the RG analysis are insensitive to the spin symmetry (23, 26), many qualitative results will hold and thus here we com- pare our numerical results with existing theoretical predictions SI Appendix, Comparison of EQMC and DQMC at Quantum for Heisenberg AFM-QCPs, but it must also be emphasized that Critical Region. agreement at the quantitative level is not expected here because For the frequency dependence in χ, as shown in Fig. 5B, we of this difference in symmetry. −1 −1 analyze the χ (T , hc , 0, ω) − χ (T , hc , 0, 0) to subtract the In the perturbative renormalization group calculation (23, finite-temperature background and test the predicted anomalous 26), the anomalous dimension depends on the angle between dimension η = 0.125 in Eq. 10 and the data points fit very well the the hotspot Fermi velocity and the order wavevector Q~ . As expected functional form shown below, for our model, this angle is close to 45◦. At this

−1 1−η 0 2 χ (ω) = (cωω) + cωω , [11] A where we obtained the values of the coefficients cω = 0.068(1) 0 and cω = 0.071(3). It is worthwhile to note that the crossover behavior in χ−1(ω) is very interesting, since at low frequency, the anomalous dimension in ω0.875 dominates bosonic suscepti- bility and this means that the coupling of the fermions with the critical bosons has changed the universality behavior from the bare (2 + 1)D Ising universality with η = 0.036 to a new one of AFM-QCP with η = 0.125. However, at high frequency, where the coupling between fermions and bosons becomes irrelevant, the bare boson universality comes back and the ω2 term dom- inates over the susceptibility, consistent with our observation of the bare boson susceptibility. We also note that because the frequency dependence here is polluted by the IR irrelevant ω2 contributions, whose contribution is about 10% at ω ∼ 0.25, this B frequency exponent has a lower accuracy, in comparison with the momentum dependence shown in Fig. 5A. Although the data are consistent with dynamical exponent z = 2, small corrections in the form of an anomalous dimension in the dynamical exponent as predicted in ref. 26 cannot be excluded. −1 1.96 In the absence of fermions, χ (0, hc , |q|, 0) ∝ |q| [(2 + 1)D Ising]. According to the Hertz–Millis theory, this expo- nent should increase from 1.96 to 2 in the presence of fermions −1 2 χ (0, hc , q, 0) ∝ |q| . Such an increase is indeed observed in the triangular lattice model (2Q =6 Γ) (40). However, it is remarkable that for the square lattice model (2Q = Γ), exactly the opposite was observed. Instead of increasing, this power −1 1.75 actually decreases from 1.96 to 1.75, χ (0, hc , q, 0) ∝ |q| . ω0ReG(k,ω0) Such a significant contrast is beyond numerical error, and it Fig. 7. (A and B) variation (A) along the Q = (π, π) direction kk ImG(k,ω0) indicates that QCPs with 2Q 6= Γ and 2Q = Γ belong to totally 0 and (B) perpendicular to the Q = (π, π) direction k⊥ at the K3 hotspot mesh different universality classes, which is one of the key observations shown in Fig. 1. We use the simple trigonometric function f(kk) and g(k⊥) in our study. to fit the data in A and B.

6 of 8 | www.pnas.org/cgi/doi/10.1073/pnas.1901751116 Liu et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 refermion Free aeoe.A a ese rmTbe1 hsi nedwa we what indeed is this 1, Table from seen our be thus can and As the (logarithmic) level, ones. from slow dramatically tree bare differ very the not will be finite- at value to renormalized marginal e.g., observed expected is cutoffs, is flow numerical flow prod- RG by the this their stopped Because be time effects. will size same flow the RG at this but respectively, uct 0, and QCP the infinity veloc- at measured Fermi velocity noninteracting Fermi the the values). both and (renormalized to values) showed (bare perpendicular we ity and 1, parallel Table hotspot In a at in ity recorded is which 1. velocity, Fermi Table the obtain to function fitting of points data discrete vs. the fit to functions simple use QCP Near location Hotspots case free-fermion the in velocity Fermi where is dimension anomalous the for prediction 1/N RG the angle, v from obtained velocity mesh Fermi The 1. Table model, i tal. et Liu of observed. be increase critical to dynamic resulting weak the the too in are rotation, change exponent the small and a surface dimension anomalous such Fermi the For hotspot cutoff. our a slow, by 0.5 very However, about by is QCP. only rotated flow the near RG rotation this surface because dynamic Fermi the this of observed value the renormalize exponent critical even anomalous can the increase and further dimension will surface This Fermi (23). the nesting toward of rotate rotation will hotspots at surface Fermi the question. other open some interesting or an Ising) is vs. contributions (Heisenberg difference symmetry the to of value qual- 2/N same the the behavior, observed we itative although above, shown as However, .G .Seat o-em-iudbhvo in behavior Non-Fermi-liquid Stewart, R. G. 4. itinerant in points critical Moriya, quantum on T. temperature 3. nonzero a of Effect Millis, J. phenomena. A. critical Quantum 2. Hertz, A. J. 1. .H .L quantum v. influences H. invariance scale 7. generic How Vojta, T. Kirkpatrick, R. T. Belitz, D. 6. P C. Chubukov, V. A. 5. .A .Cuuo,D .Mso,Si osrainadFrilqi eraferromagnetic a near liquid Fermi and conservation Spin Maslov, L. D. Chubukov, V. A. 8. F codn oteR nlss(23), analysis RG the to According ecluaeteFrivelocity Fermi the calculate We QCP the near that predicts also analysis RG the addition, In thotspots at elnHiebr,Gray 1985). Germany, Berlin/Heidelberg, systems. fermion n lsia hs transitions. phase classical and ferromagnets. (2001). 797–855 73, unu rtclpoint. critical quantum transitions. phase quantum k v h h ssoni i.7 hn ecmuetedrvtv o the for derivative the compute we Then, 7. Fig. in shown as k .s .s × . . ω hesn .Rsh .Vja .W P. Vojta, M. Rosch, A. ohneysen, ¨ where , N nta.Wehrti uniaiedsgemn sdue is disagreement quantitative this Whether instead. 0 v v h = ⊥ .s k pnFutain nIieatEeto Magnetism Electron Itinerant in Fluctuations Spin . π/β and ilrmi osat nanmrclsimulation, numerical a In constant. a remain will n hsteR rdce au is value predicted RG the thus and 16, = hs e.Lett. Rev. Phys. hs e.B Rev. Phys. N oacrtl opt h eiaie efirst we derivative, the compute accurately To . pn .Rc,Isaiiyo h unu-rtclpito itinerant of point quantum-critical the of Instability Rech, J. epin, v v z ´ h F ⊥ .s 2,2) si hw eo,orsuyindeed study our below, shown is As 26). (23, hs e.Lett. Rev. Phys. . = r h opnnso h em veloc- Fermi the of components 2 the are ω stenme fhtpt 2,2) nour In 26). (23, hotspots of number the is ImG 0 e.Md Phys. Mod. Rev. ∂ ReG ∂ 1379 (1993). 7183–7196 48, k ◦ 403(2004). 147003 92, (k e.Md Phys. Mod. Rev. ω (k norsmlto eoebigstopped being before simulation our in ,ω Im 0 ,ω v 0 ReG F ) 0 G ) ttehtpti the in hotspot the at 141(2009). 216401 103, aasoni i.7adthe and 7 Fig. in shown data η (k 2.5800 1.506 1.523(8) le em-iudisaiiisa magnetic at instabilities Fermi-liquid olfle, ¨ hs e.B Rev. Phys. (k htw bevdi ls to close is observed we that d , v 0517 (2007). 1015–1075 79, and - v k ω F , k x ω 7–3 (2005). 579–632 77, 0 as ) 0 ) f eeto metals. -electron v

k= k 1518 (1976). 1165–1184 14, and F , respectively. Q, v ⊥ K (Springer-Verlag, ilflwto flow will ω 3 e.Md Phys. Mod. Rev. 0 ImG 0 hotspot ReG η 1 = 0.5615 1.468 1.435(8) ( k ( ,ω v k k [12] /16. ⊥ ,ω y η 0 ) 0 = ) 3 .Y u .Sn .Shtnr .Br,Z .Mn,NnFrilqi t(2 at liquid Non-Fermi Meng, Y. Z. Berg, E. Custers J. Schattner, Y. 14. Sun, K. Xu, liquids. Y. non-Fermi u(1) in X. pairing Cooper and 13. Senthil, T. Sachdev, nematic S. Mross, F. D. from Metlitski, A. scaling M. 12. dynamical Anomalous Metzner, W. Holder, T. 11. metals. liquid non-Fermi and surfaces Fermi Critical Senthil, T. 10. 5 .Steppke A. 15. bevd h eomlzdvleof value renormalized The observed. fw opr h F-Cswith AFM-QCPs the compare we If slow Discussion the with theory. consistent by and predicted are 1, flow Table changes RG in small shown as these error in theoretically, worth- numerical changes beyond is although are It they that predicts. small, here theory highlight RG of to the product while as for 2.186 ones), the and values renormalized bare and the the for value, (2.210 constant a bare largely remains its than (smaller) ineI ltom tteNtoa uecmue etr nTajnand time. CPU Tianjin and of in allocation Tianhe-1A generous Centers and the support Supercomputer and technical Sloan their National Sciences for P. the the Guangzhou of in at Alfred Academy Sciences platforms Chinese Simulation the Tianhe-II Quantum Physics, and for of Center EFRI-1741618 Institute the National Council thank the Grant Grants We from Research under Foundation. support Kong Foundation acknowledges Hong of K.S. Science C6026-16W. support Grant the Sci- through for National 11674370. and thankful the 11574359, Develop- is 11421092, and Grants X.Y.X. (XDB28000000); under and China Sciences of Research of Program Foundation ence and Academy Research Key Chinese Priority Science National the Strategic of of the the Ministry (2016YFA0300502); through Program the ment China from Z.H.L., criticality. funding of quantum acknowledge Technology itinerant Z.Y.M. and of Sachdev, and subjects Subir G.P., various Chubukov, on Andrey Kivelson Schattner, Steven Yoni Lee, Sung-Sik Klein, the at physics ACKNOWLEDGMENTS. IR reveal to provide to EQMC limit. systems use thermodynamic small and on DQMC results use benchmark can one fermionic that in of in criticality terms quantum systems, another about in suggests studies future EQMC consistency for and pathway Such DQMC properties. the in critical AFM-QCP, consistency revealing second lattice the the provides of triangular here example for investigated AFM-QCP 41 lattice ref. square con- in the in Besides check direction QCPs. promising sistency itinerant very of investigations a numerical shows the work this in methodologies actively are 31, and (32). ref. desirable us highly in by are pursued out being line pointed this As along explored. works be future to and temperature need anomalous lower range predicted 29), frequency 28, the (26, probe exponents of to critical value about dynamical and exact understanding the exponent full pinpoint an frequency to a the particular, where toward In step (32). example to QCPs first starts itinerant solid 1 simulations is a numerical which and emerge, offers theory has study the- between it and agreement Our numerical simulations, results. reconcile been QMC to oretical in has challenge non– long-standing observed theory a although and been Hertz–Millis theory systems, the in fermion predicted beyond itinerant vital scaling in a mean-field plays QCPs results For numerical role. and theory between comparison 23. ref. universality in does discussed analysis different which RG and prediction, 2 of expansion value Hertz–Millis the to the on rely to belong not contrast they in that classes, indicate studies .W eze,D oe .Adrasn otFrisrae n radw fFermi- of breakdown and surfaces Fermi Soft Andergassen, S. Rohe, D. Metzner, W. 9. ttetcncllvl obnto fDM n EQMC and DQMC of combination a level, technical the At the scalings, critical anomalous and criticality of study the In hs e.B Rev. Phys. metals. two-dimensional in fluctuations (2015). field gauge (2008). behavior. liquid YbNi4(P (2003). 524–527 424, point. critical quantum ferromagnetic 1-x h ra-po ev lcrn taqatmciia point. critical quantum a at electrons heavy of break-up The al., et As ermgei unu rtclpiti h ev-emo metal heavy-fermion the in point critical quantum Ferromagnetic al., et 111(2015). 115111 91, x )2. hs e.Lett. Rev. Phys. Science eakoldevlal icsin ihAvraham with discussions valuable acknowledge We 3–3 (2013). 933–936 339, hsosrainspot the supports observation This Q. 642(2003). 066402 91, hs e.X Rev. Phys. 2Q v 308(2017). 031058 7, k NSLts Articles Latest PNAS 6 Γ = (v ⊥ hs e.B Rev. Phys. and ssihl larger slightly is ) hs e.B Rev. Phys. 2Q v k and v Γ = k 041112 92, 035103 78, and 1/N QMC , v | ⊥ Nature f8 of 7 + h are v 1)D .s ⊥ .

PHYSICS 16. W. Zhang et al., Effect of nematic order on the low-energy spin fluctuations in 42. X. Y. Xu, K. S. D. Beach, K. Sun, F. F. Assaad, Z. Y. Meng, Topological phase transitions detwinned bafe1.935ni0.065as2. Phys. Rev. Lett. 117, 227003 (2016). with SO(4) symmetry in (2+1)d interacting Dirac fermions. Phys. Rev. B 95, 085110 17. Z. Liu et al., Nematic quantum critical fluctuations in bafe2−x nix as2. Phys. Rev. Lett. (2017). 117, 157002 (2016). 43. F. F. Assaad, T. Grover, Simple fermionic model of deconfined phases and phase 18. Y. Gu et al., Unified phase diagram for iron-based superconductors. Phys. Rev. Lett. transitions. Phys. Rev. X 6, 041049 (2016). 119, 157001 (2017). 44. S. Gazit, M. Randeria, A. Vishwanath, Emergent Dirac fermions and broken symme- 19. W. Wu et al., Superconductivity in the vicinity of antiferromagnetic order in CrAs. tries in confined and deconfined phases of z2 gauge theories. Nat. Phys. 13, 484–490 Nat. Commun. 5, 5508 (2014). (2017). 20. J. G. Cheng et al., Pressure induced superconductivity on the border of magnetic 45. Y. Y. He et al., Dynamical generation of topological masses in Dirac fermions. Phys. order in MnP. Phys. Rev. Lett. 114, 117001 (2015). Rev. B 97, 081110 (2018). 21. M. Matsuda et al., Evolution of magnetic double helix and quantum criticality near a 46. X. Y. Xu et al., Monte Carlo study of lattice compact quantum electrodynamics with dome of superconductivity in CrAs. Phys. Rev. X 8, 031017 (2018). fermionic matter: The parent state of quantum phases. Phys. Rev. X 9, 021022 (2019). 22. J. Cheng, J. Luo, Pressure-induced superconductivity in CrAs and MnP. J. Phys. 47. C. Chen, X. Y. Xu, Y. Qi, Z. Y. Meng, Metals’ awkward cousin is found. arXiv:1904.12872 Condens. Matter 29, 383003 (2017). (29 April 2019). 23. A. Abanov, A. V. Chubukov, J. Schmalian, Quantum-critical theory of the spin-fermion 48. J. Liu, Y. Qi, Z. Y. Meng, L. Fu, Self-learning Monte Carlo method. Phys. Rev. B 95, model and its application to cuprates: Normal state analysis. Adv. Phys. 52, 119–218 041101 (2017). (2003). 49. J. Liu, H. Shen, Y. Qi, Z. Y. Meng, L. Fu, Self-learning Monte Carlo method and 24. A. Abanov, A. Chubukov, Anomalous scaling at the quantum critical point in itinerant cumulative update in fermion systems. Phys. Rev. B 95, 241104 (2017). antiferromagnets. Phys. Rev. Lett. 93, 255702 (2004). 50. X. Y. Xu, Y. Qi, J. Liu, L. Fu, Z. Y. Meng, Self-learning quantum Monte Carlo method in 25. M. A. Metlitski, S. Sachdev, Quantum phase transitions of metals in two spatial interacting fermion systems. Phys. Rev. B 96, 041119 (2017). dimensions. I. Ising-nematic order. Phys. Rev. B 82, 075127 (2010). 51. Y. Nagai, H. Shen, Y. Qi, J. Liu, L. Fu, Self-learning Monte Carlo method: Continuous- 26. M. A. Metlitski, S. Sachdev, Quantum phase transitions of metals in two spatial time algorithm. Phys. Rev. B 96, 161102 (2017). dimensions. II. Spin density wave order. Phys. Rev. B 82, 075128 (2010). 52. H. Shen, J. Liu, L. Fu, Self-learning Monte Carlo with deep neural networks. Phys. Rev. 27. S. Sur, S. S. Lee, Anisotropic non-Fermi liquids. Phys. Rev. B 94, 195135 (2016). B 97, 205140 (2018). 28. A. Schlief, P. Lunts, S. S. Lee, Exact critical exponents for the antiferromagnetic 53. C. Chen et al., Symmetry-enforced self-learning Monte Carlo method applied to the quantum critical metal in two dimensions. Phys. Rev. X 7, 021010 (2017). Holstein model. Phys. Rev. B 98, 041102 (2018). 29. S. S. Lee, Recent developments in non-Fermi liquid theory. Annu. Rev. Condens. 54. C. Chen, X. Y. Xu, Z. Y. Meng, M. Hohenadler, Charge-density-wave transitions of Matter Phys. 9, 227–244 (2018). Dirac fermions coupled to phonons. Phys. Rev. Lett. 122, 077601 (2019). 30. A. Schlief, P. Lunts, S. S. Lee (2018) Noncommutativity between the low-energy 55. K. Sun, B. M. Fregoso, M. J. Lawler, E. Fradkin, Fluctuating stripes in strongly corre- limit and integer dimension limits in the -expansion: A case study of the lated electron systems and the nematic-smectic quantum phase transition. Phys. Rev. antiferromagnetic quantum critical metal. arXiv:1805.05252 (19 July 2018). B 78, 085124 (2008). 31. A. Chubukov, Solving metallic quantum criticality in a casino. J. Club Condens. Matter 56. E. Berg, S. Lederer, Y. Schattner, S. Trebst, Monte Carlo studies of quantum critical Phys., 201802 (2018). metals. Annu. Rev. Condens. Matter Phys. 10, 63–84 (2019). 32. X. Y. Xu et al., Revealing fermionic quantum criticality from new Monte Carlo 57. H. W. J. Blote,¨ Y. Deng, Cluster Monte Carlo simulation of the transverse Ising model. techniques. arXiv:1904.07355 (15 April 2019). Phys. Rev. E 66, 066110 (2002). 33. Y. Schattner, S. Lederer, S. A. Kivelson, E. Berg, Ising nematic quantum critical point 58. S. Hesselmann, S. Wessel, Thermal Ising transitions in the vicinity of two-dimensional in a metal: A Monte Carlo study. Phys. Rev. X 6, 031028 (2016). quantum critical points. Phys. Rev. B 93, 155157 (2016). 34. S. Lederer, Y. Schattner, E. Berg, S. A. Kivelson, Superconductivity and non-Fermi liq- 59. D. Chowdhury, S. Sachdev, Density-wave instabilities of fractionalized Fermi liquids. uid behavior near a nematic quantum critical point. Proc. Natl. Acad. Sci. U.S.A. 114, Phys. Rev. B 90, 245136 (2014). 4905–4910 (2017). 60. Y. C. Wang, Y. Qi, S. Chen, Z. Y. Meng, Caution on emergent continuous symmetry: 35. Z. X. Li, F. Wang, H. Yao, D. H. Lee, The nature of effective interaction in cuprate A Monte Carlo investigation of the transverse-field frustrated Ising model on the superconductors: A sign-problem-free quantum Monte-Carlo study. ArXiv:1512.04541 triangular and honeycomb lattices. Phys. Rev. B 96, 115160 (2017). (14 December 2015). 61. U. Wolff, Collective Monte Carlo updating for spin systems. Phys. Rev. Lett. 62, 361– 36. E. Berg, M. A. Metlitski, S. Sachdev, Sign-problem–free quantum Monte Carlo of the 364 (1989). onset of antiferromagnetism in metals. Science 338, 1606–1609 (2012). 62. R. H. Swendsen, J. S. Wang, Nonuniversal critical dynamics in Monte Carlo simula- 37. Z. X. Li, F. Wang, H. Yao, D. H. Lee, What makes the Tc of monolayer FeSe on SrTiO3 tions. Phys. Rev. Lett. 58, 86–88 (1987). so high: A sign-problem-free quantum Monte Carlo study. Sci. Bull. 61, 925–930 63. R. Blankenbecler, D. J. Scalapino, R. L. Sugar, Monte Carlo calculations of coupled (2016). boson-fermion systems. I. Phys. Rev. D 24, 2278–2286 (1981). 38. Y. Schattner, M. H. Gerlach, S. Trebst, E. Berg, Competing orders in a nearly 64. J. E. Hirsch, Discrete Hubbard-Stratonovich transformation for fermion lattice models. antiferromagnetic metal. Phys. Rev. Lett. 117, 097002 (2016). Phys. Rev. B 28, 4059–4061 (1983). 39. M. H. Gerlach, Y. Schattner, E. Berg, S. Trebst, Quantum critical properties of a metallic 65. F. Assaad, H. Evertz, “World-line and determinantal quantum Monte Carlo methods spin-density-wave transition. Phys. Rev. B 95, 035124 (2017). for spins, phonons and electrons” in Computational Many-Particle Physics, Lecture 40. Z. H. Liu, X. Y. Xu, Y. Qi, K. Sun, Z. Y. Meng, Itinerant quantum critical point with Notes in Physics, H. Fehske, R. Schneider, A. Weisse, Eds. (Springer, Berlin/Heidelberg, frustration and a non-Fermi liquid. Phys. Rev. B 98, 045116 (2018). Germany, 2008), vol. 739, pp. 277–356. 41. Z. H. Liu, X. Y. Xu, Y. Qi, K. Sun, Z. Y. Meng, Elective-momentum ultrasize quantum 66. M. A. Metlitski, S. Sachdev, Instabilities near the onset of spin density wave order in Monte Carlo method. Phys. Rev. B 99, 085114 (2019). metals. New J. Phys. 12, 105007 (2010).

8 of 8 | www.pnas.org/cgi/doi/10.1073/pnas.1901751116 Liu et al. Downloaded by guest on September 26, 2021