Topological Order in the Pseudogap Metal

Home , Fermi liquid theory, Topological order

Downloaded by guest on September 27, 2021 epiaiycnie,tetplgclodri soitd(14, associated is order topological case U(1) the the consider, for systems; primarily gapless we in order of topological examples of inition describe will and “Hig- U(1) We is with group. and group U(1) smaller to gauge down a the Higgsing to when metal down surfaces pseudogap the Fermi gsed” a in small yields outline theory only will be gauge with we formally SU(2) as The can model, section. theory Hubbard next gauge lattice of a a background from Such the derived order. in AF freedom proper of fluctuating keep degrees to the fermionic required is the formulation of theory track gauge a where then case and the in interested are We oe urts(3,btti ean ai tlwtemperatures low at valid electron- remains the this (T describe but to (13), used cuprates been doped has of parameter fluctuations order of AF treatment the semiclassical ordered alternative, topologically An the the nonzero phase. of indicate local, theory entanglement the a quantum of preserving long-range fields the while gauge in emergent order, The fluctuations AF magnitude. describes the theory of gauge orientation a pseu- the Such describe metal. to 12) dogap (11, metals far in (AF) so antiferromagnetism has pseudogap the in order lacking. topological been for of evidence independent presence quan- But the in order. possible topological sym- only with that broken are states tum volume (6–10) with Luttinger arguments states the nonperturbative discuss is from deviations are not state There will pseudogap here.] we the metry and which 5), over (1, range wide doping present a over and it of temperature sign no allow of is can there symmetry but surfaces, translational the- Fermi liquid lattice “small” Fermi square of [Broken surface theorem pho- (4). Fermi Luttinger a ory that the “large” from is However, the expected cuprates show is 3). that the not (2, of do experiments study theory toemission the conven- liquid in a Fermi mystery of long-standing obeying dependence metal frequency transport electrical and tional of temperature evidence the clear with exhibits It (1). conductors T superconductors high-temperature pseudogap phase. Higgs holds that the theorem mod- in Luttinger a zeros the derive of of also We version lines model. nonperturbative these Hubbard ified, the how of describe theory our and in theory appear gauge the topological underlying of zero-frequency the order the of signs of are zeros function (approximate) Green’s electronic of self-energy. lines electronic that and show function We spectral doping agree- and the hopping, the Good of frequency, momentum, dependencies of model. the in phase found Hubbard pseudogap is ment hole-doped the on interacting calculations, strongly Carlo quan- compared and Monte theory, are field tum mean results dynamical of The anti- extensions fluctuating lattice. cluster of with square theory the gauge on topologically SU(2) Tremblay)ferromagnetism a the Andre-Marie of and of phase Kotliar function Gabriel Higgs Green’s by ordered reviewed electronic 2017; the 27, November compute review We for (sent 2018 9, March Sachdev, Subir by Contributed and 2Y5; N2L Canada Ontario, Waterloo, Physics, metal Scheurer S. pseudogap Mathias the in order Topological www.pnas.org/cgi/doi/10.1073/pnas.1720580115 10010; NY York, New a 12 aasa,France; Palaiseau, 91128 eateto hsc,HradUiest,CmrdeM 02138; MA Cambridge University, Harvard Physics, of Department nti ae,w s U2 ag hoyo fluctuating of theory gauge SU(2) a use we paper, this In nyi h Fcreainlength correlation AF the if only ) epedgpmtli oe tt feetoi matter electronic super- of high-temperature state cuprate novel hole-doped, the a in is found metal pseudogap he | oooia order topological Z 2 oooia re.See order. topological e eateto unu atrPyis Universit Physics, Matter Quantum of Department c a,1 ntttd hsqe Coll Physique, de Institut hbauChatterjee Shubhayu , | Z ubr model Hubbard 2 n hs ilyedmtli states metallic yield will these and , ξ AF IAppendix, SI ξ ean nt at finite remains AF g | g eFac,705Prs France; Paris, 75005 France, de ege eateto hsc,Safr nvriy tnod A94305 CA Stanford, University, Stanford Physics, of Department ` iegsas diverges a e Wu Wei , A o def- a for b T T eted hsqeTh Physique de Centre b,c → 0 = ihlFerrero Michel , eGen de e ´ 0. , 1073/pnas.1720580115/-/DCSupplemental h otx fti etraiepofo h utne theorem. Luttinger the of in proof even perturbative permitted, this are of that surfaces context argued Fermi the therefore small electron was with the It states when zeros: metallic of surface lines Fermi has the function by Green’s perturbative enclosed contribution additional conventional volume an the yields The to theorem surface.” Luttinger the “Luttinger of func- proof a Green’s electron lines on the of in dimensions) tion presence spatial the two to (in linked zeros of been have theorem Luttinger the function gauge companion a a SU(2) as in (20). appear computations, the paper hopping, DQMC second-neighbor between and and Addi- doping comparison DCA of computations. the the theory and gauge on temperature theory the find results accessible with and numerically tional well calculations the agree determi- (DQMC) in range, performed Carlo that, also correla- Monte self-energies have strong we quantum of reason, momentum-space nant regime limited this has the (16–19). For it temperature, study resolution. (DMFT) low to to cluster theory down us tions a field allows (DCA), mean DCA While dynamical approximation obtained of cluster those gauge extension dynamical resemble the of the closely function order from Green’s topological electron with and the theory real the of both reasonable parts theory, a gauge for imaginary SU(2) that the be in Hub- will parameters for results the of main range allow on our results of computations U(1) One Such numerical model. the bard theory. with in gauge comparison direct SU(2) zone a the Brillouin of entire phase the Higgs across function Green’s order. spacetime AF the fluctuating in the defects of “hedgehog” configuration of suppression the with 15) ulse nieArl2 2018. 2, April online Published at online information supporting contains article This 1 the under Published interest. of conflict no declare Sherbrooke. authors of The University A.-M.T., and and research, University; Rutgers performed G.K., S.S. Reviewers: and paper. the A.G., wrote M.F., S.S. W.W., and M.S.S. S.C., M.S.S., contributions: Author owo orsodnemyb drse.Eal [email protected] or [email protected] Email: addressed. be may correspondence [email protected]. whom To v,C-21Gnv,Switzerland; Geneva, CH-1211 eve, hrceitc ftemn-oyqatmstate. quantum many-body topological the the these of of characteristics detection in direct entanglement the to quantum and under-superconductors long-range deeper the a the to of of route much a standing up model opens can study Our fields topolog- data. with gauge numerical metal emergent a and of order theory a ical that show We has electron lacking. understanding the been theoretical of fundamental nature a the but studies low on corrections, of numerical information regime much and yielded a experimental have in Extensive temperature tempera- density. critical at hole the phase above metal just “pseudogap” tures superconductors mysterious high-temperature a display oxide-based copper The Significance ` nsvrldsusosi h ieaue(12) iltosof violations (21–29), literature the in discussions several In electronic the of computation mean-field a present will We d etrfrCmuainlQatmPyis ltrnInstitute, Flatiron Physics, Quantum Computational for Center eorique, ´ b,c non Georges Antoine , NSlicense. PNAS cl oyehiu,CR,Universit CNRS, Polytechnique, Ecole ´ PNAS . f eiee nttt o Theoretical for Institute Perimeter b,c,d,e | o.115 vol. n ui Sachdev Subir and , www.pnas.org/lookup/suppl/doi:10. | o 16 no. Paris-Saclay, e ´ | E3665–E3672 a,f,g,1

PHYSICS But this argument appears problematic because the real part of (30–32). To this end, we decompose the electronic fields in terms the Green’s function (and hence the positions of the zeros) can of the spin and charge degrees of freedom: be changed by modifying the spectral density at high frequen- † † † cies, and so it would appear that high-energy excitations have ci (τ) = Ri (τ)ψi (τ), ci (τ) = ψi (τ)Ri (τ). [5] an undue influence on the low-energy theory. Indeed, specific computations of higher order corrections (21) do find that lines Here, the unitary 2 × 2 matrices, Ri (τ), are the bosonic spinon of zeros in metallic Green’s functions do not contribute to the and ψi (τ), the two-component fermionic chargon operators. volume enclosed by the Fermi surface. This parameterization introduces an additional redundancy We maintain that violations of the Luttinger theorem in met- leading to an emergent local SU(2) gauge invariance (11):

als cannot appear in states that are perturbatively accessible from † the free electron state but are only possible in nonperturbative Ri (τ) → Ri (τ)Vi (τ), ψi (τ) → Vi (τ)ψi (τ). [6] metallic states with topological order (6–9). While the SU(2) 6 gauge theory is shown to violate the conventional Luttinger the- By design, the transformation in Eq. leaves the electron field c (τ) orem, we derive a modified sum rule on the Fermi surfaces of the operators i invariant and hence is distinct from spin rotation. ψ metallic states with topological order. Note that the chargon field, , does not carry spin. In contrast, R We also find that the nonzero-temperature SU(2) gauge the- carries spin 1/2, and global spin rotations act via left multipli- R 6 ory has lines of approximate zeros of the electron Green’s cation on (in contrast to the right multiplication in Eq. ). Inserting the transformation (Eq. 5) into the electron–boson function in a suitable regime. These lines are remnants of lines of x y z coupling Sint and introducing the Higgs field H = (H , H , H ) zeros in the mean-field Green’s functions of fractionalized parti- † cles (“chargons”) in the theory. So we claim that the approximate via σ · Hi (τ) = Ri (τ)σRi (τ) · Φi (τ), we obtain (11) zeros can be interpreted as heralds of the underlying topological Z β order. Moreover, the good agreement between the SU(2) gauge X † Sint = dτ ψiασαβ ψiβ · Hi . [7] theory and the numerical computations in DCA and DQMC, 0 i noted above, appears in the regime where the approximate lines of zeros are present. Taken together, we reach one of our main Note that the Higgs field, Hi , transforms under the adjoint of conclusions: There is evidence for topological order in the DCA the gauge SU(2), while the spinons and chargons transform as and DQMC studies of the pseudogap state of the Hubbard gauge SU(2) fundamentals. Furthermore, a crucial feature is that model. the Higgs field does not carry any spin since it is invariant under global SU(2) spin rotation. SU(2) Gauge Theory of the Hubbard Model The needed metallic (or insulating) phases with topological We are interested here in the Hubbard model with Hamiltonian order are obtained simply by entering a Higgs phase where hHi i 6= 0, while maintaining hRi i = 0. Any such phase will pre- ˆ X † X † X HU = − tij cîαcˆj α − µ cîαcîα + U nî↑nî↓ [1] serve spin-rotation invariance, and we will focus on such phases i,j i i throughout this paper. The vanishing of hRi i arises from large fluctuations in the local rotating reference frame, and so we are on a square lattice of sites, i, describing electrons cîα with hop- considering states with local magnetic order whose orientation ping parameters tij = tji ∈ R, chemical potential µ, and on-site undergoes large quantum fluctuations. However, the magnitude repulsion U (summation over Greek indices appearing twice is of the local magnetic order remains large, and this is captured by † hH i 6= 0 implied, and nîα ≡ cîαcîα). Let us begin by writing the exact the Higgs field with i . Depending upon the spatial configuration of hHi i, different path integral of HÛ in the “spin-fermion” form with action “flavors” of topological order with residual gauge group U (1) or S = Sc + Sint + SΦ. Using τ to denote imaginary time, we have (β = T −1 is inverse temperature) Z2 and potentially also broken discrete symmetries are realized (11, 12, 33, 34). In this paper, we will focus on the simplest case, Z β " # with no broken symmetries and U (1) topological order, where X † X † the Higgs field resembles AF order Sc = dτ ciα(∂τ − µ)ciα − tij ciαcj α . [2] 0 i i,j T ix +iy hHi i = (0, 0, ηi H0) , ηi = (−1) , [8] The electrons are coupled locally to a bosonic field Φ = (Φx , y z as this scenario has the minimal number of independent param- Φ , Φ ), which describes spin fluctuations, according to (σ = T eters (only H0) that can be adjusted to fit the numerical data (σx , σy , σz ) are Pauli matrices) presented below. More complicated Higgs field configurations, with one or more additional parameters, leading to 2 topolog- Z β Z X † ical order (see SI Appendix, A for an overview), can be treated Sint = dτ ciασαβ ciβ · Φi . [3] 0 i similarly, but we will not present explicit results because the current momentum space resolution of our DCA computations 3 R β P 2 and DQMC results do not allow us to distinguish between the While taking the action for Φ to be SΦ = 2U 0 dτ i Φi leads to an exact representation of the Hubbard model (Eq. 1), we will different phases. use the more general spin-fermion model form Lines of (Approximate) Zeros Z β Our central results for the electron spectral functions and 1 hX 2 X X 2 i SΦ = dτ (∂τ Φi ) + Jij Φi · Φj + V (Φi ) , the near zeros of the Green’s function can be understood 4g0 0 i i,j i qualitatively by considering the effective Hamiltonian Hˆψ for [4] deconfined chargons ψ in the Higgs phase; in the metallic case, this phase corresponds to an “algebraic charge liq- which we imagine arises as an effective low-energy theory of fluc- uid” (ACL) (35). To obtain the effective Hamiltonian, we tuating AF from the Hubbard model (Eq. 1). To describe phases insert the transformation (Eq. 5) into the quadratic elec- with topological order, we rewrite the path integral as a SU(2) tronic action Sc in Eq. 2. We decouple the resulting quartic t c† c = t ψ† R† (R ) ψ gauge theory by transforming to a “rotating reference frame” hopping ij iα j α ij iβ i βγ j γδ j δ into two quadratic

E3666 | www.pnas.org/cgi/doi/10.1073/pnas.1720580115 Scheurer et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 ae.W ilntaayeteipc fbudsaeformation state bound of impact work. the this in analyze quantitatively not will bound We leads (FL electron-like faces. This (7–9) liquid (35–39): binding into Fermi order fractionalized the a topological spinons to account the retaining and into while take chargons states, theory Fur- the ACL. the an between of (35) labeled refinements previously ther phase the to corresponds Hamiltonian tive Eq. in derivative ucinatracnouinwt hto h utaigspinons fluctuating 1B), the Fig. of in that diagram with (see convolution Eq. a to after Eq. back function from go weight to spectral have electron We the obtain directly cannot components, off-diagonal momentum has Eq. is neither Fig. in illustrated is function 1A Green’s chargon the of dependence is function Green’s configu- chargon Higgs the For Eq. order. in struc- long-range ration band no is the there So though electrons. even the on the of does ture condensate order Higgs magnetic the a that like is observation key The potential chemical The electrons. bare the from amplitudes ited hopping the of renormalization space, SU(2) placing While frequencies. subara (q respec- chargons, and tively; spinons the of function Green’s Matsubara {k|ξ notation the Within Σ zeros. has Eq. also of but surfaces Fermi structed Eq. with Combined chargons. and obtain electrons for identical 2 parameters mean-field mutual the by (U related consistently hc epcsalsmere ftesur atc (see lattice square the of symmetries all Appendix, respects which Z where terms, cerre al. et Scheurer i sntrnraie u oteidentity the to due renormalized not is −j fcus,tecagn r o h hsclelectrons, physical the not are chargons the course, Of (see shown be can It H n cu when occur and (i = ij ˆ o aaeesrlvn otehl-oe cuprates. hole-doped the to relevant parameters for k+Q ψ ) G t αβ ij = c η Ω ω h re’ ucinntol a oe tterecon- the at poles has only not function Green’s The . h atraeascae ihplso h self-energy the of poles with associated are latter the 10, (k t = ω 0 = − → n ij n , = ) + [ψ h .Here C). X opiemmnaadfrinc(ooi)Mat- (bosonic) fermionic and momenta comprise q) r ooi asbr rqece;and frequencies; Matsubara bosonic are ψ 0 i } ,j i R i † + α η emosi eosrce nosalFrisurfaces, Fermi small into reconstructed is fermions h oetmdaoa lmn fteretarded the of element momentum-diagonal the 8, tzr nryti steLtigrsrae The surface. Luttinger the is energy—this zero at G G T i † , (Z 2 (U U → 10 Σ R Q ψ R αβ αα ij X i ,r j ψ Ω r ij (π = −j ω h ulcagnGensfnto,wihalso which function, Green’s chargon full the (q H = (ω (ω n ) 2 ˆ αβ ω n αβ t G ψ Z tholds it , = ) ij Z , , = nasmlrwy eoti reeffec- free a obtain we way, similar a in i R ψ BZ , = k) = k) i + htgvrstecagndnmc.This dynamics. chargon the governs that and −j π ξ j and β g k+Q and ), δ (2π IAppendix, SI 1, ij (χ + d Ω (χ ω ω 2 µ) npriua,teei ieo zeros of line a is there particular, In . 5 G q G ) Z n 2 + + 2 ij ψ n opt h hsclspectral physical the compute and ψ ij ˆ ψ + ij ξ ) X i i i † α αβ k ∈ α eoetemomentum-diagonal the denote ) H η η Ω αβ ,β D olw rmEq. from follows stedseso neie from inherited dispersion the is ψ ˆ n ec nyla oa to lead only hence and R, − − 0 n 2 2 = q+ 2 j G α + ξ ξ R δ hψ R αα k+Q k + αβ i † E − that B) Ω R i † (q X q+Q α 2 Σ ∗ j n , 2 i ψ )G ihsalFrisur- Fermi small with ) R + hψ ψ r j αβ ψ β ˆ (ω i † ψ t D ββ raigtetime the Treating i. k i † R hc r self- are which ], ij α ψ , q− 2 U i σ (k → k+Q † k) = ij αβ , − , Z 1 hH ,ads we so and i), r rva in trivial are ψ 10 i ˆ k q −j n sthus is and i i ), β (i = i t pnre- upon µ h · ij csjust acts H c nEq. in inher- ω i 7 (and n [12] [11] [10] i. [9] , 10. we SI k) k D ob eofrcnrtns.(B concreteness. for zero be to hopping (red neighbor chargons nearest the the of of of taken units surface have dashed) in Luttinger (white energies the absence all as Measuring and well line). line) as (blue condensate presence Higgs the the (Eq. in function chargons Green’s the energy of chargon zero retarded at the shown is of elements diagonal the ee ehv introduced have we Here, branches, dispersion two the introduced have we where scales. energy other all than differ- smaller much two real for the positive where is and red) function of Green’s (light values electronic space ent retarded momentum the in of region part the and (spinon) chargons chargon the to referring function. lines Green’s (dashed) solid with function, Green’s t 1. Fig. where h rgno h oie utne hoe ntedecon- the the compute identity in We operator described. the theorem easily by be density Luttinger electron now modified can phase the Higgs fined of origin The and DCA Theorem in Luttinger Modified found also are function; below. Green’s zeros discussed electron as approximate DQMC the these of of height) signs finite of (peaks asso- the zeros of Eq. poles in the self-energy) equivalently ciated (or function Green’s chargon “smeared” a as gap seen be the can of function version Green’s electron the that is good a is it plots, the we E in calculations, use used our to parameters in the approximation used for that, is out function point Green’s spinon full the G and chargons. the to coupling the from resulting contribution from herited AB 0 q qs = c 2 h anqaiaieosraini h vlaino Eq. of evaluation the in observation qualitative main The in found be can calculation this of Details ,r stesio dispersion. spinon the as J .Truhu hswr,alfrhrniho opnsaetaken are hoppings further-neighbor all work, this Throughout −0.3. = ∆ (ω 2 n h eutn eaddeetoi re’ function Green’s electronic retarded resulting the and C, = 1 2 + stesals nrysae npriua,tezrso the of zeros the particular, In scale. energy smallest the is , h pcrlweight spectral The (A) .T ecietezr-eprtr ii,w aetaken have we limit, zero-temperature the describe To 0.3.

k) E s q E q H Φ ilb oprdwt C n QCblw While below. DQMC and DCA with compared be will 0 q 2 H (E = 2 + 0 S = 0.5, ψ sidctdadfie oedniyof density hole fixed and indicated as q C 2 Φ E −2J − hw h eosrce em ufc bu ie fthe of line) (blue surface Fermi reconstructed the shows re’ ucini h ii hr h spinon the where limit the in function Green’s q+Q E 2 nEq. in µ q E = = q+Q 2 (g + 2 ω ,adanx-onaetniho opn of hopping neighbor next-to-nearest a and −0.8, (cos q G = ) R clrpo)tgte ihteFrisurface Fermi the with together plot) (color 0 αα 2 4, χ 10 E 2(E + q n-opdarmyedn h electronic the yielding diagram One-loop ) Ω (q q ∆ x Φ A eeal eoeol approximate only become generally ) cos + k Ch 2 ) 2 stesio a,and gap, spinon the is ≈ (ω + q 2 C ) PNAS g + = /(Ω E q − y E q c − q+Q π 1 2 n 2 | 2 ImG 2) + + o.115 vol. )(g E ψ αα ∆ eoe h atin- part the denotes s ,r q 2 χ 2 (ω hc identifies which ), = , Ω , ) ftespinon the of ±, p soitdwith associated k) IAppendix, SI | 2 = (g + o 16 no. 0.1, E χ t q 0 Ω c | = T ) sthe is 4 −0.3, E3667 obe to [13] [14] t ! we , 10) 11 . B

PHYSICS † ˆ† ˆ cîαcîα = ψiαψiα, [15] Results for the Pseudogap Metal To allow for a direct and systematic comparison of the predic- and we apply the standard Luttinger analysis (40, 41) to the right- tions of the SU(2) gauge theory and the Hubbard model (Eq. hand side of Eq. 15. The structure of the complete perturbation 1), we have performed DCA and DQMC calculations (see SI theory in the deconfined Higgs phase for the ψ correlator is Appendix, D for more details on the numerical methods). In the formally the same as that in a conventional long-range ordered SU(2) gauge theory, the main fitting parameter is the magni- AF phase for the c correlator. In the Higgs case, we do have tude H0 of the Higgs field, which we choose so as to have a additional interactions from the fluctuations of the gauge field, similar size of the antinodal pseudogap in DCA and the gauge but the stability of the Higgs phase in the metal implies that theory. The spinon gap ∆ is constrained to be of the order of or these can be treated perturbatively. So we can simply transfer smaller than the temperature to allow for zero-frequency spec- all of the Luttinger theorem arguments in ref. 21 for the case tral weight in the nodal region, as seen in the experiment and our of AF order to the deconfined Higgs phase. These arguments numerical calculations. Note that ∆ plays the role of a Lagrange then imply that the ψ Fermi surfaces are small—that is, in a multiplier in the mean-field theory and is hence uniquely deter- Higgs state with fluctuating AF order, the ψ Fermi surfaces obey mined by all other system parameters; in particular, it depends the same Luttinger sum rule as those of the electron Fermi on the spin stiffness g0 in Eq. 4. However, we take the formally surfaces in a state with long-range AF order: At zero tempera- equivalent view of specifying ∆ instead of g0 (which is adjusted ture, the hole density p is given by p = Shole − Sdouble, where Shole accordingly) in the following, as ∆ is physically more insightful (Sdouble) denotes the fraction of the Brillouin zone where both in the present context. Except for J and η, which only have a bands of the chargon Hamiltonian (Eq. 9) are above (below) the minor impact on the qualitative shape of the spectral function Fermi level. (SI Appendix, E), all other parameters of the SU(2) gauge theory We note that the exact operator identity in Eq. 15 is only were determined by solving the mean-field equations. For con- † ˆ† ˆ creteness, we focus on nearest (t) and next-to-nearest neighbor fulfilled “on average,” hcîαcîαi = hψiαψiαi, in our mean-field 0 treatment of the gauge theory since the unitarity constraint hopping (t ). Since we are eventually interested in understand- † † ing the pseudogap phase in the hole-doped cuprates, we consider R R = 1 of the spinons is only treated on average, hR R i = 1. i i i i small hole dopings p > 0 in the regime of large onsite repulsion However, this is sufficient for the modified Luttinger theorem U (taking U = 7t for concreteness). All energies are measured in discussed above to hold exactly, as it is only a statement about units of t. the expectation value of the particle number. The presence of a finite gap ∆ =6 0 in the spinon spectrum Antinodal Point and Lifshitz Transition. The gauge theory result leads to a gap at ω = 0 in the electronic spectral function Ak(ω) at zero temperature. The vanishing zero-frequency spectral weight for the spectral function at the antinodal point, k = (π, 0), is of the electrons does not allow us to define an interacting analog shown in Fig. 2A and displays the strong suppression of the of an electronic Fermi surface at zero temperature. Nonetheless, low-energy spectral weight characterizing the pseudogap phase. the chargons are described by the effective Hamiltonian (Eq. To understand this behavior, we first note that the q-integrand 9) and hence exhibit well-defined Fermi surfaces. This not only in the expression (Eq. 11) for the electronic Green’s function k,q s0 allows us to conveniently fix the particle number in the system as exhibits poles at the energies ωss0 = sEq + ρk−q after perform- explained above but also shows that the Higgs phase is charac- ing the Matsubara summation and the analytic continuation and αα 2 2 terized by Fermi liquid-like charge and thermal transport at low using the simplified expression GR (q) ≈ g/(Ωn + Eq ) for the s temperatures. spinon Green’s function discussed above. Here ρk are the two Due to the vanishing imaginary part of the zero-frequency chargon bands (s = ±) of the Hamiltonian Hˆψ in Eq. 9 and Eq electronic Green’s function Gc,r (ω = 0, k) at T → 0, the afore- the spinon dispersion as given in Eq. 14. In the relevant param- mentioned lines of approximate zeros of Gc,r (ω = 0, k) in the eter regime ∆ < T H0, the energies for which the spectral Brillouin become exact. Notwithstanding the presence of elec- weight is suppressed are determined by the q = 0 components tronic Luttinger surfaces, the perturbative Luttinger theorem (π,0),q=0 ω 0 = ω 0 , as can be seen in Fig. 2A and as discussed in (4), relating the particle density n = 1 − p to the area in k space ss ss more detail in SI Appendix, C. Using the explicit form of Eq and where Re Gc,r (ω = 0, k) > 0 s k,q ρk entering ωss0 , we estimate a gap of size 2H0 centered around 0 ω = ξ = 4Z 0 t − µ Z 0 Z Z 2 0 (π,0) t , where t denotes i−j for next-to- d k nearest neighbors i and j . The same “Mott-insulating” behavior n = 2 2 Θ (Re Gc,r (ω = 0, k)), [16] BZ (2π) at the antinodal point is found in our DCA result shown in Fig. 2B. By comparison, we extract a value of about H0 = 0.3. Note is violated in the Higgs phase. As can be seen in Fig. 1C, the that while the precise position of the minimum of the spectral function differs in the two approaches, the asymmetry of size of the area with Re Gc,r (ω = 0, k) > 0 changes with H0 at the peaks in Fig. 2 A and B with respect to ω = 0 is qualita- fixed electron density (keeping the area enclosed by the chargon 0 0 Fermi surface fixed). The violation of Eq. 16 is a manifestation tively the same. Increasing t (t > −0.15), the minimum of the of the nonperturbative nature of the Higgs phase. antinodal spectral function moves toward positive values of ω in This reveals the crucial conceptual differences to the phe- DCA as well. nomenological Yang–Rice–Zhang (YRZ) theory (25): While For a more detailed comparison of DCA and the gauge YRZ introduced an ansatz for the electronic Green’s func- theory, we also extract the retarded electronic self-energy r r −1 tion of the pseudogap state that respects the identity (Eq. 16), Σk (ω) from the Green’s function, Σk (ω) = −(Gc,r (ω, k)) + 0 we provide a gauge-theory description of fluctuating AF that ω − k. Here, k = −2t(cos kx + cos ky ) − 4t cos kx cos ky − µ0 is shows that the perturbative result (Eq. 16) is not required to the bare electronic dispersion, with µ0 denoting the bare elec- hold in strongly coupled systems. Due to Eq. 16, the YRZ tronic chemical potential—that is, in the absence of interactions, theory requires a fine-tuned position of the Luttinger surface. U = 0. As can be seen in Fig. 2 C and D, we also find very good In our gauge theory, the lines of (approximate) zeros change agreement between the gauge theory and DCA for the real and r continuously with system parameters, without obeying the con- imaginary part of the antinodal self-energy Σ(π,0)(ω): At small straint (Eq. 16), and are a consequence of the underlying topo- negative t 0 (solid lines), the imaginary part of the self-energy logical order. is peaked (and the real part changes sign) at positive energies

E3668 | www.pnas.org/cgi/doi/10.1073/pnas.1720580115 Scheurer et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 n h pnnGensfnto ob osatfralvle of values all for constant be taken to have function we Green’s Therefore, spinon the unknown. doping and priori on a param- depend is all system dependence principle, in the that, of Note DCA. eters (B) in and theory gauge t value doping hole of tion even of tendencies potentially function, above. discussed Green’s already as electronic formation bound-state inducing the cor- to direct yield rections also fluctuations gauge Furthermore, band-renormalization treatment. the lack of the value of accurate factors consequence the a of predominantly knowledge be of order to of expected is factor This a by fers lse xesoso MT(6 8 9 42–45). using 29, 28, works (26, several DMFT of in self-energy extensions antinodal emphasized cluster the previously of was part (quasi-pole) imaginary with the associated is in pseudogap peak the a that fact The lines). dashed (see ω function. spectral the “smoothen” and integration uation cerre al. et Scheurer assumed and ato h lcrnczr-rqec re’ ucin(e oi ie,teLtigrsrae(re ahdln) n em ufc bu ahdln)o the of line) dashed (blue surface Fermi and line), dashed (green surface Luttinger the points line), distinct solid four (red the function for Green’s chargons zero-frequency electronic the of part ftecagn oce h o ftelwrcagnbn teLfht rniino h hro utne ufc cus.Tedpn eedneof dependence doping The occurs). surface Luttinger chargon for the transition of same transition the Lifshitz of (the location band the chargon show lower lines the dashed of black top position The peak the (B). the touches where DCA is, chargons and changes—that the self-energy (A) the of theory of asymmetry gauge particle-hole the SU(2) where the indicate ω lines from red calculated the Furthermore, lines electrons. blue noninteracting solid as shown is t i.3. Fig. model. Hubbard the on to DCA refer in lines obtained (C (dashed) as point solid and antinodal The theory the gauge at SU(2) self-energy the electronic retarded the 2. Fig. ˜ B A BCEAB 0 (π 0 peak nrae (from increases ) = = oeipratta xc ueia aus efidtesame the find we values, numerical exact than important More oeta h xc au of value exact the that Note ,0) T 5(t −0.15 ω and fteatndlsl-nrycagssg.In sign. changes self-energy antinodal the of = peak p i ω The oprsno h lcrncseta weight spectral electronic the of Comparison 0frteDAcluain.I diin o h ag hoy we theory, gauge the for addition, In calculations. DCA the for 1/30 peak Z H n ω i 0 → > peak −j = p-t 0 rdsldln)a which at line) solid (red while 0, ω = 0.3, 0 u ocretosto corrections to due ntegueter r hw in shown are theory gauge the in + ω )frtegueter oe.W used We model. theory gauge the for −0.5) eedneo h itrcigLfht rniin”dfie ytesg hneo h eomlzdqaiatceenergy quasiparticle renormalized the of change sign the by defined transition,” Lifshitz “interacting the of dependence peak i η J 2 ftegueter octofplsi h numerical the in poles off cut to theory gauge the of = o nya ucinof function a as only not p 0.1, p ω peak peak ssmaie nFg 3 Fig. in summarized as ∆ 0 = = hne infrsfcetylarge sufficiently for sign changes 1 1 and 0.01, nDAadi h ag theory. gauge the in and DCA in at t 0 = a–d t t 0 0 5(t −0.15 D C 0 = where niae in indicated Z η ω i with ) −j = peak 4i h nltclcontin- analytical the in 0.04 p 0 eodormean-field our beyond ω = and C hne in(tfixed (at sign changes peak h ahdbu rd iecrepnst h aaee ofiuain hr h hmclpotential chemical the where configurations parameter the to corresponds line (red) blue dashed the A, 5 o C n to and DCA for −0.25) t and −t 0 nti gr,w used we figure, this In A. u loa func- a as also but h doping The A–D: A t hne indif- sign changes 0 D 0 (π oee,this However, . the (A) in both o w ifrn n-iesoa usof cuts one-dimensional different two for ,0) U (A = and 7, and p Σ = and B) for D) ( r 0.05, −t π H ,0) 0 0 U nFg 3A Fig. in when solution vanishes frequency this to antinode, the self-energy Eq. At in electronic chargons the the equating of that naively by proceeds sible obt h eomlzto factor renormalization the both 3 to dashed Fig. (black in electrons the line of of transition transition Lifshitz by Lifshitz [given noninteracting surface the Luttinger between also chargons’ difference (cf. the the chargons that the Note of 3E). surface Luttinger the of change sign companion the to the refer with we well (20). discussion, agrees paper further already For and result. parameters DCA tuning free of ber t hmclpotentials chemical isiztasto 2)btenafittoshl-ie( ( hole-like electronic-like fictitious case, a and noninteracting between the (20) transition to of Lifshitz analogy change By sign point. the antinodal the at surface. Fermi chargon the h neatn isiztransition, Lifshitz interacting the utne ufc ftecagn.Telte sdfie by defined is latter The chargons. the of func- surface Green’s chargon the Luttinger of part real the tion of small sign for the and gap hence, and the gap, in a stays have of chargons chargons values the the of the potential theory, the chemical in gauge the limit, the However, of noninteracting electron-like. phase the becomes Higgs In directly potential. surface chemical Fermi the of behav- ues this of shares increasing interpretation at which qualitative Starting theory, a gauge ior: admits The feature, doping. low this at dashed) (U noninteracting (black its Lifshitz to interacting blue) the (solid of line deviation transition strong the is details) more for DCA. and theory 3 gauge the Fig. between in line) blue 0 = ipewyt aetedpn eedneof dependence doping the make to way simple A ehv losuidternraie uspril energy, quasiparticle renormalized the studied also have We atclrysrkn seto h C eut(e e.20 ref. (see result DCA the of aspect striking particularly A and 7, D G T ˜ k = ψ p = ,r 1/30, p sti eurstefws num- fewest the requires this as 3A, Fig. at arrive to (ω p n lsl follows closely and = k p nti eie hr sn hro em surface, Fermi chargon no is there regime, this In . + 0, = p a fixed (at H (t Re A 0 ω 0 = p ) and peak k) Σ Finally, A. 0.2, 0 = fti qaini hw sarddse line dashed red a as shown is equation this of k r ˜ ssll eemndb h oiino the of position the by determined solely is µ (ω A ˜ sascae ihteLfht rniinof transition Lifshitz the with associated is ( ,wihcrepnsto corresponds which B), 0 , π J ( t 2 π ,0) 6 = and 0 t )= 0) = = ,0) 0 0 = µ 0 = 0.1, a eue odfiea interacting an define to used be can E > hc xiisapa at peak a exhibits which 10, eutn rmtercntuto of reconstruction the from resulting ed oicesnl eaieval- negative increasingly to leads ) hw h oiino h eo ftereal the of zeros the of position the shows gi,w n odagreement good find we Again, B. em ufc.Telcto of location The surface. Fermi 0) point , ∆ p −Re peak = PNAS 1 and 0.01, p (t FS Z (G 0 nti nepeain the interpretation, this In ). t a (t 0 c 6 1 = | 0 nFg 3 Fig. in ,r sas niae (solid indicated also is ), o.115 vol. (ω µ(p η n h ifrnein difference the and = 0, = 0.04. 4 = ) µ(p µ b k)) | ˜ 4Z = ) A 0 (π 0 = Z and o 16 no. 4 = −1 ,0) t and 0 ω t analogue ) at , ˜ 0 t ω peak ( n the and ] d 0 t π ω sdue is , 0 = ,0) and E, peak | t nFig. in 0 plau- ξ The . E3669 < [17] k+Q > 0) 0, .

PHYSICS ABLuttinger surface of the chargons), as can be more clearly seen in Fig. 4B. We expect that lines of zeros of the real part of the Green’s function will still be present after q integration for sufficiently small spinon gaps, albeit with deformed shape. Indeed, this is what we see in Fig. 4 C and D, where the resulting electronic C Green’s function is shown; Note, however, that the zeros associated with the Luttinger surface and with the Fermi surface of the chargons have merged to a single, “hybrid,” line of zeros of Re Gc,r (0, k). Another consequence of the q integration is that it also “washes out” the peaks in the spectral function as can be seen by comparing Fig. 4 A and D. However, for the small value of ∆ assumed in the plot, we recover the metallic, Fermi arc, behavior D E in the vicinity of k = (π/2, π/2) in coexistence with the suppres- a sion of low-energy spectral weight at the antinode. Depending b on whether we consider momenta inside (point a in Fig. 4D) or outside (point b) the Fermi arc, the spectral weight is peaked at positive or negative energies (see Fig. 4E). This is consistently reflected in the large value of the low-frequency spectral F weight at the patch centered around k = (π/2, π/2) of the DCA calculations (blue dotted line). Note that the much broader dis- tribution of the spectral weight in DCA (with maximum at ω = 0) is attributed to the fact that the DCA result is an average over the entire momentum patch around the nodal point (π/2, π/2). At higher precision, we expect that it is important to account for chargon–spinon interactions in the nodal region, and these could lead to the formation of a FL∗ state (38). Fig. 4. In A, the momentum dependence of the imaginary part (color We emphasize that the spectral weight is nonzero in the entire r scale) and the zeros of the real part (red lines) of the q integrand gc are Brillouin zone (in the physically relevant regime where temper- shown. One-dimensional cuts of its real part, normalized to the value at ature is of the order of or larger than the spinon gap), and k = 0, can be found in B. C and D show the analogous plots for the full the aforementioned lines of zeros of the real part, which stem electronic Green’s function of the gauge theory—that is, after integration in part from the Luttinger surface of the chargons, only corre- over q. In E, the spectral weight at the two momenta a (red solid line) and b (red dashed line) indicated in D are shown together with the spec- spond to approximate zeros of the Green’s function. Nonethe- tral function of DCA (blue dotted line) averaged over the patch centered less, it provides a natural explanation for the suppression of the around the nodal point k = (π/2, π/2). Part F shows the real part of the spectral weight at the “backside of the Fermi arc” as result- DCA Green’s function at k = (0, 0) and k = (π, π) as a function of t0. Here ing from its proximity to an approximate zero of the Green’s we used the same parameters U, p, T, H0, J, ∆, and η, as in Fig. 2 except function. for t0 = −0.25. As follows from our discussion, a prediction of the gauge theory, which is robust in the sense that it does not change upon small changes of system parameters, is the sign change of the real ξk+Q = 0 and thus becomes more hole-like upon reducing the chemical potential (see point b). Viewing the electronic Green’s part of the low-frequency Green’s function from positive at k = 0 to negative at k = (π, π). We have verified that this sign change function as a smeared version of G explains why Re Gc,r (ω = ψ,r is present in our DCA calculations and stable under variation of 0, k = (π, 0)) > 0, and thus, ˜(π,0) < 0 (hole-like) in this regime. Upon further increasing p, the chemical potential touches the top of the lower chargon band. For t 0 ≥ 0, the resulting chargon Fermi surfaces are located in the vicinity of (π, 0), changing the AB sign of the real part of the Green’s function at the antinode (see point c). As expected from this qualitative picture, the line in p– t 0 space where the chemical potential enters the lower chargon band (dashed blue line) roughly follows the interacting Lifshitz transition (solid blue line).

Behavior in the Full Brillouin Zone. Having established good agreement with DCA results at the antinodal point, we can now use the gauge theory to calculate the electronic Green’s function in the entire Brillouin zone. To gain qualitative understanding of the result, let us ﬁrst go one step back and investigate the q-loop integrand g r of the Green’s function, deﬁned via c Fig. 5. The imaginary part of the self-energy at the lowest Matsubara fre- R d2q r Gc,r (ω, k)= BZ (2π)2 gc (k, q, ω), which is obtained from Eq. 11 quency ω0 = πT determined from DQMC on the Hubbard model (U = 7, 0 after performing the Matsubara sum and the analytic continua- t = −0.1, T = 0.25, p = 0.042) and from the SU(2) gauge theory is shown in A and B, respectively. The remaining free parameters of the gauge tion. For small ∆, its q = 0 component is expected to yield the 2 G ω = 0 theory have been chosen to be H0 = 0.3, J = 0.1, η = 0.04, and ∆ = 0.01 main contribution to c,r and is plotted at as a function of as in Fig. 2. To avoid too much broadening, we have applied a slightly k in Fig. 4A. We see that its momentum space structure closely smaller temperature of T = 0.15 for the gauge theory. The Inset in B shows resembles that of the chargons (cf., Fig. 1). In particular, its the gauge theory prediction at zero frequency and low temperature (as real part shows sign changes both at poles (inherited from the before T = 1/30). The black dashed line corresponds to the position of the Fermi surface of the chargons) and at zeros (stemming from the Luttinger surface of the chargons.

E3670 | www.pnas.org/cgi/doi/10.1073/pnas.1720580115 Scheurer et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 4 edN ahe 19)Si-eel,vlnebn oi,adN and solid, valence-bond Spin-Peierls, (1990) S Sachdev N, Read 14. 5 eml ,e l(04 tblt fU1 pnlqisi w dimensions. two in liquids spin U(1) of Stability (2004) al et M, Hermele 15. Dar V, Hankevych B, Kyung bro- and 13. order topological Intertwining (2017) metals. M in arXiv: Scheurer waves S, density reconstruction. Sachdev spin S, Fluctuating Chatterjee surface (2009) C 12. Xu Fermi Y, Qi and MA, Metlitski order S, Sachdev Topological 11. (2018) S Sachdev 10. cerre al. et Scheurer o cesbetmeaue.Orcluain r efre at limit performed lower are a calculations provides T of Our which temperatures. limitation problem, accessible main sign for the The is self-energy. approach electronic this the the about of information calculations complementary Green’s structure DQMC and performed the unbiased also interpolat- of obtain have of to we instead part data, However, similar imaginary DCA 4. very our Fig. the ing in yield and shown that real 46–50) as the 29, function both 28, of 26, dence refs. e.g., (see, the of evidence indirect remnant hence, a of order. and, as topological chargons line interpreted for the is additional of part approximate this surface real an Luttinger theory, the is, in gauge changes of part—that the sign part imaginary In real its the surface. in of Luttinger changes peak pres- sign the a of necessitates without arc line Fermi a The of the point. ence in nodal surface the Fermi hole of a vicinity by description band conventional the t nDM.Ti iceac ih gi edet h akof lack the to due be again might as discrepancy chosen 5 This have Fig. DQMC. in we in theory that gauge Note the for axis. Im on temperature imaginary both reduced the evaluated slightly as on a reliably data, and be DQMC real can not the the theory chosen of gauge a have continuation mean-field We as analytical the surface. interpreted the (Eq. Luttinger be perform self-energy chargon can to chargon the it the along where of 10) divergence 5B, gauge the SU(2) Fig. of the in consequence in shown feature as in this line theory, recover straight We approximately space. 5 an Fig. along momentum (evalu- in peak DQMC shown characteristic is from a frequency) has obtained Matsubara lowest part), the real at ated the of these discussion at the visible already char- DQMC are self-energy our phase the and in pseudogap temperatures. theory signatures the the gauge of that SU(2) acteristic believe the the between we to of below, results, find Due phase will (20). Higgs we data the which agreement, DCA qualitative our good from estimated temperature .SnhlT ot ,ScdvS(04 ekmgeimadnnFrilqisnear to theorem liquids Luttinger’s non-Fermi Extending (2004) and A magnetism Vishwanath A, Weak Paramekanti (2004) liquids. 9. S fermi Sachdev M, Fractionalized Vojta (2003) T, M fermi Senthil Vojta the 8. S, and Sachdev theorem T, Luttinger’s Senthil to approach 7. a Topological of point (2000) critical pseudogap M the at Oshikawa density carrier 6. of Change (2016) al. et S, Badoux 5. .Ca K ta 21)I-ln antrssac by olrsrl nthe in rule II. system. Kohler’s many-fermion a obeys of energy Ground-state magnetoresistance (1960) JC In-plane Ward JM, Luttinger (2014) 4. al et MK, Chan tem- 3. and energy quantum liquid-like Fermi From for evidence (2015) Spectroscopic (2013) J al. et Zaanen SI, Mirzaei S, 2. Uchida MR, Norman SA, Kivelson B, Keimer 1. 0 seFg 4F Fig. (see eas oethat note also We h o-nrysatrn ae Im rate, scattering low-energy The 0 = utain ntenra tt fteeeto-oe cuprates. antiferromagnets. quantum electron-doped low-dimensional the of state normal 93:147004. the in fluctuations 70:214437. waves. density spin fluctuating of theory a 227002. in symmetry ken B Rev Phys 1801.01125. matter. of phases ized points. critical heavy-fermion 90:216403. lattice. Kondo a of surface superconductor. cuprate Rev superconductors. cuprate of phase pseudogap cuprates. the of phase USA pseudogap Sci the Acad in Nat rate Proc relaxation the of dependence perature oxides. copper in superconductivity 186. high-temperature to matter Σ k 118:1417–1427. .25t eoe o mae u hntkn h au of value the taking when out smeared too becomes hc so h re ftepedgptransition pseudogap the of order the of is which , 80:155129. .Nt httesg hnecno eepandin explained be cannot change sign the that Note ). hsRvB Rev Phys 110:5774–5778. Nature k hsRvLett Rev Phys hsRvB Rev Phys pc neplto cee fDMFT of schemes interpolation space M rmlyAS(04 suoa n spin and Pseudogap (2004) AMS Tremblay AM, e ´ 531:210–214. 70:245118. 84:3370–3373. 69:035111. hsRvB Rev Phys hsRvLett Rev Phys Σ k (see 42:4568–4589. IAppendix, SI 113:177005. e rudsae of states ground eel ´ hsRvLett Rev Phys G Nature c ,r hsRvLett Rev Phys hsRvLett Rev Phys Z (ω k 2 hsRvB Rev Phys fractional- T depen- 0, = 518:179– A F used and Phys 119: for k) k 8 ola ,Svao Y P SY, Savrasov G, Kotliar 18. 0 hamnB,Sgi D(98 oievcnisi unu esnegantiferro- Heisenberg quantum a in vacancies Mobile (1988) ED Siggia BI, mott Shraiman doped of 30. Fermi structure electronic the of Evolution of (2009) M Breakup Imada Y, (2006) Motome A S, Sakai Georges 29. S, Biermann T, in Giamarchi zeros to C, due theorem Berthod Luttinger’s of in 28. Absence function (2013) Green CL the Kane PW, of Phillips zeros KB, hidden Dave and arcs 27. Fermi (2006) G Kotliar TD, state. Stanescu pseudogap the 26. of theory Phenomenological (2006) low FC Zhang and TM, rule Rice sum KY, Luttinger Yang Liquid: spin 25. Doped (2006) AM Tsvelik Luttinger TM, Rice The RM, theorem: Konik Luttinger the 24. of consequences Some (2003) insulators. I Mott Dzyaloshinskii one-dimensional coupled 23. Weakly (2002) AM Luttinger Tsvelik (1998) FH, DK Essler Morr AM, 22. Finkel’stein two-dimensional A, Dashevskii the AV, in Chubukov topology BL, surface Altshuler Fermi and 21. Pseudogap (2017) al. et theories. W, cluster Wu Quantum (2005) 20. MH Hettler T, Pruschke M, Jarrell T, Maier 19. S B, Kyung AMS, Tremblay theory 17. mean-field Dynamical (1996) MJ Rozenberg W, Krauth G, Kotliar A, Georges 16. nutyCnd n ytePoic fOtrotruhteMnsr of Ministry the through through Ontario Canada of of Province Government the Innovation. at and the by Research Research and by Foundation. work- Canada Simons supported Industry the 2017 is by May Canada. Institute supported Jouvence, the Perimeter is in of Quantique Institute Institut atmosphere Flatiron the The stimulating by organized the cuprates and on for A.G. shop University. grateful Stanford 2016-12. at and are program Institute LPDS S.S. Professor Perimeter Visiting Grant at Hanna Energy through the Cenovus from German from Leopoldina the support Sciences acknowledges from S.S. support of acknowledges Academy from M.S.S. Mul- National Office. W911NF-14-1-0003 and Research Grant s575), Army Initiative Project the (CSCS, Research Center University Col- Supercomputing tidisciplinary ERC-319286-“QMAC”), National Many-Electron (project Swiss Council Foundation the Research Simons European the the laboration, DMR-1664842, Grant dation topological ACKNOWLEDGMENTS. of flavors required be other theory. will the studies but refine precise to theory, more and gauge Higgs- possible, remain U(1) SU(2) order simplest a the of with studied. numerics ranges ing the doping the compared and on have temperature model We the Hubbard over hole-doped lattice the square agree- in evidence good order provides this the topological theory, for with gauge the combination and DCA In between ment DQMC. with agrees preserving also while fermions, the charged in symmetry; the of translational peak of location a expected surface the yields Luttinger of vicinity the theory the this in self-energy results, electron the DQMC with magnitude. orientational local agreement with well-established a metal In with a order of AF theory of gauge fluctuations a presented have We Summary of value increasing to the closer increasing move U to upon found DQMC is rate in scattering the in peak calcu- 5 when Fig. seen clearly more Im be lating can surface Luttinger chargon of of enhancement values accurate the of knowledge h oncinbtentepa ntesatrn aeadthe and rate scattering the in peak the between connection The praht togycreae systems. correlated strongly to approach (review theory quantitative a Towards coupling. strong article). to weak from conductivity nuaos eosrcino oe n eo fGensfunction. magnet. Green’s of zeros and poles 102:056404. of Reconstruction insulators: systems. low-dimensional in transition Mott 136401. the near surface function. Green single-particle the state. pseudogap the B Rev Phys order. temperature insulators. Mott and liquids non-Fermi in surfaces B Rev state. spin-density-wave a for theorem arXiv:1707.06602. model. Hubbard Phys dimensions. infinite of limit the and Phys systems fermion correlated strongly of ∼ 6t B, 77:1027–1080. 68:13–125. 65:115117. n h aetedi on ntegueter upon theory gauge the in found is trend same the and , .A icse in discussed As Inset). o epPhys Temp Low J hsRvLett Rev Phys Σ 73:174501. H k r 0 (ω . 0) = T hsRvLett Rev Phys hsRvB Rev Phys 61:467–470. eaiet h hro nryscales. energy chargon the to relative hswr a upre yNtoa cec Foun- Science National by supported was work This ntegueter tlwtmeaue(see temperature low at theory gauge the in 32:424–451. lsnG ioiG(01 ellrdnmclma field mean dynamical Cellular (2001) G Biroli G, alsson en ´ ´ ca 20)Pedgpadhg-eprtr super- high-temperature and Pseudogap (2006) D echal ´ 74:125110. 96:086407. hsRvLett Rev Phys U IAppendix, SI uohsLett Europhys eedneo h eklocation peak the of dependence hsRvLett Rev Phys PNAS Z i 110:090403. −j hsRvB Rev Phys | edn oa effective an to leading , o.115 vol. 41:401–406. 87:186401. U h oiino the of position the E, rudmoderate around 68:085113. | hsRvLett Rev Phys o 16 no. hsRvLett Rev Phys k (π = | e Mod Rev e Mod Rev E3671 , Phys π 97: )

PHYSICS 31. Schulz HJ (1990) Effective action for strongly correlated fermions from functional 42. Maier TA, Pruschke T, Jarrell M (2002) Angle-resolved photoemission spectra of the integrals. Phys Rev Lett 65:2462–2465. Hubbard model. Phys Rev B 66:075102. 32. Schrieffer JR (2004) Pairing, magnetic spin fluctuations, and superconductivity near a 43. Gull E, Parcollet O, Werner P, Millis AJ (2009) Momentum-sector-selective metal- quantum critical point. J Supercond 17:539–543. insulator transition in the eight-site dynamical mean-field approximation to the 33. Chowdhury D, Sachdev S (2015) Higgs criticality in a two-dimensional metal. Phys Rev Hubbard model in two dimensions. Phys Rev B 80:245102. B 91:115123. 44. Gull E, Ferrero M, Parcollet O, Georges A, Millis AJ (2010) Momentum-space 34. Chatterjee S, Sachdev S (2017) Insulators and metals with topological order and anisotropy and pseudogaps: A comparative cluster dynamical mean-field analysis of discrete symmetry breaking. Phys Rev B 95:205133. the doping-driven metal-insulator transition in the two-dimensional Hubbard model. 35. Kaul RK, Kim YB, Sachdev S, Senthil T (2008) Algebraic charge liquids. Nat Phys Phys Rev B 82:155101. 4:28–31. 45. Lin N, Gull E, Millis AJ (2010) Physics of the pseudogap in eight-site cluster dynam- 36. Mei JW, Kawasaki S, Zheng GQ, Weng ZY, Wen XG (2012) Luttinger-volume violat- ical mean-field theory: Photoemission, Raman scattering, and in-plane and c-axis ing Fermi liquid in the pseudogap phase of the cuprate superconductors. Phys Rev B conductivity. Phys Rev B 82:045104. 85:134519. 46. Civelli M, Capone M, Kancharla SS, Parcollet O, Kotliar G (2005) Dynamical 37. Punk M, Sachdev S (2012) Fermi surface reconstruction in hole-doped t-J models breakup of the fermi surface in a doped Mott insulator. Phys Rev Lett 95: without long-range antiferromagnetic order. Phys Rev B 85:195123. 106402. 38. Punk M, Allais A, Sachdev S (2015) Quantum dimer model for the pseudogap metal. 47. Kyung B, et al (2006) Pseudogap induced by short-range spin correlations in a doped Proc Nat Acad Sci USA 112:9552–9557. Mott insulator. Phys Rev B 73:165114. 39. Sachdev S, Berg E, Chatterjee S, Schattner Y (2016) Spin density wave order, 48. Haule K, Kotliar G (2007) Strongly correlated superconductivity: A plaquette topological order, and Fermi surface reconstruction. Phys Rev B 94:115147. dynamical mean-field theory study. Phys Rev B 76:104509. 40. Potthoff M (2006) Non-perturbative construction of the Luttinger-Ward functional. 49. Kancharla SS, et al (2008) Anomalous superconductivity and its competition with Condens Mat Phys 9:557. antiferromagnetism in doped Mott insulators. Phys Rev B 77:184516. 41. Kozik E, Ferrero M, Georges A (2015) Nonexistence of the Luttinger-Ward func- 50. Liebsch A, Tong NH (2009) Finite-temperature exact diagonalization cluster dynamical tional and misleading convergence of skeleton diagrammatic series for Hubbard-like mean-field study of the two-dimensional Hubbard model: Pseudogap, non-fermi- models. Phys Rev Lett 114:156402. liquid behavior, and particle-hole asymmetry. Phys Rev B 80:165126.

E3672 | www.pnas.org/cgi/doi/10.1073/pnas.1720580115 Scheurer et al. Downloaded by guest on September 27, 2021