Scaling theory for Mott-Hubbard transitions

A comprehensive understanding of the physics of the Mott insulator has proved elusive due to the absence of any small parameter in the problem. We present a zero-temperature renormalisation group analysis of the one-band Hubbard model in two dimensions at, and away from, half-filling. We find that the transition in the half-filled system involves, for any Hubbard repulsion, passage from a non-Fermi liquid metallic state to a topologically-ordered gapped Mott liquid through a pseudogapped phase. The pseudogap is bookended by Fermi surface topology-changing Lifshitz transitions: one involving a disconnection at the antinodes, the other a final gapping at the nodes. Upon doping, we demonstrate the collapse of the Mott state at a quantum critical point possessing a nodal non-Fermi liquid with superconducting fluctuations, and spin-gapping away from the nodes. d-wave Superconducting order is shown to arise from this critical state of matter. Our findings are in striking agreement with results obtained in the cuprates, settling a long-standing debate on the origin of superconductivity in strongly correlated quantum matter.

exists only in one spatial dimension [1], while the status of the problem remains open in general.
While the Mott insulator is often associated with a (T = 0) first order transition leading to a Neél antiferromagnetic ground state [2], the search continues for an insulating state reached via a continuous transition and which breaks no lattice or spin-space symmetries. Indeed, there is some evidence for insulating spin-liquid ground states in layered organic conductors [3] and in Herbertsmithite [4]. Theoretical studies have not, however, identified unambiguously the order parameter for such interaction-driven metal-insulator transitions. These difficulties appear to be associated with an interplay of two complications: the fermion-sign problem limits numerical investigations at low-temperatures [5,6], while a lack of an identifiable small parameter makes most analytic approaches beyond various mean-field schemes intractable when studying the problem at strong coupling. Similar issues exist for the case of doped Mott insulators, with the case of the cuprates being most prominent (see [7] for a recent review). While it is by now widely accepted that the physics of the cuprates is associated with that of (almost) decoupled Cu-O layers in which the doped holes (or electrons) pair into a d-wave superconducting state, an overarching understanding of the mechanisms responsible for the observed complex phenomenology remains elusive. Challenges include understanding a non-Fermi liquid (NFL) metallic phase with a striking resistivity that varies linearly with temperature [8], a phase in which parts of the Fermi surface (FS) are gapped (the pseudogap (PG), whose origin and role remain unknown [9]) and the (much debated) existence of a quantum critical point (QCP) within the superconducting dome [10,11].
That these phases are reached by exiting the Mott insulating state suggest that their origins lie therein [12], highlighting the need for understanding the Mott transition in generality.
Scaling theory for Mott-Hubbard MIT at 1/2-filling. Our analysis adopts the view advocated by Kohn [13]: the FS and its immediate neighbourhood at T = 0 are witness to the localisation of electronic states during the passage through the transition. Thus, we implement a scaling approach where high-energy excitations are integrated out in reaching an effective Hamiltonian for the vicinity of the FS, and their influence appears through renormalised parameters of the theory. Analytic implementations of this approach to the Kondo [14] and Anderson impurity models [15], as well as the superconducting instability of the isotropic FS for the Landau Fermi liquid [16] have revealed how universal states of matter are emergent at low-energies and longwavelengths from complex microscopic Hamiltonians. Earlier attempts [17,18] at scaling for the Mott problem have been typically limited to weak-to intermediate-coupling, and have not offered conclusive insight into the physics at stronger couplings.
We analyse the Hubbard model on the two dimensional square lattice at 1/2-filling (ν = N e /2N = 1/2, N e : number of electrons, N : number of lattice sites) where the first term denotes electron conduction between nearest neighbours ij with hopping strength t, and the second an on-site (Hubbard) repulsion U > 0. At ν = 1/2 (i.e., one particle per site), it is expected that a critical value (U/t) c separates a metallic state (U/t < (U/t) c ) from an insulating state (U/t > (U/t) c ). A notable contrary view is that the insulating state is reached for any U > 0 [19,20]. A resolution of this debate must also offer insight into the nature of the transition, as well as the metallic and insulating states on either side. In developing a scaling approach, we describe the relevant electronic scattering processes away from the FS in terms of Anderson's pseudospin construction (see supplementary materials [21], [22]).
As shown in Fig.1(b), these pseudospin electron pairs are placed on high-energy surfaces that are placed parallel to the FS, such that they enjoy the same geometry. Their scattering in the directions along (forward), as well as directly opposite (backward) to, the local normal (ŝ) leads to renormalisation of the associated couplings at lower energies and longer lengthscales. We have benchmarked the results obtained from the eventual fixed-point theory at low-energies against those found from state of the art numerical methods [5] (see Supplementary materials (SM) [21]). Postponing technical details of the RG approach to the Methods section, we present below the major outcomes of the analysis.
Mott MIT at half-filling. The phase diagram obtained, Fig. 1(a), shows the absence of a critical (U/t) c [19,20] for the metal-Mott insulator transition for the 1/2-filled Hubbard model on the 2d square lattice. The y-axis of the plot represents an energy scale for quantum fluctuations (0 < 4 − ω < 8t) and the x-axis the bare value of the on-site Hubbard coupling (0 < U 0 ≤ 16 = 2W ) (with the hopping strength t = 1). A striking observation is that the transition involves a pseudogap phase (PG) arising from a differentiation of electrons based on the monotonic variation of their kinetic energy from node to antinode [24]. This involves a continuous gapping of the FS (Fig.1(e) -1(h), Video S1) via a steady conversion of poles of the one-particle Greens function into zeros, while respecting the f-sum rule. This is seen within our formalism from the fact that the zeros of the single-particle Greens function are concomitant with the appearance of poles in the two-particle (pseudo)spin Greens function [27,21]. The PG is initiated in the form of a FS topology changing Lifshitz transition that disconnects the connected FS at the antinodes for ω < ω P G ≡ 0 ( Fig.1(f)), and proceeds until the nodes are gapped via a second Lifshitz transition at ω = ω ins ( Fig. 1(h)). The transition can, thus, be seen via a topological invariant related to the sign of the (pseudo)spin Greens function: Nŝ(ω) = e i(π+γ ⇑ ) , γ ⇑ = −i z dz ∂ z ln(Gŝ ,(⇑) (z − ω)) (see Methods). For the metal (connected FS), Nŝ(ω < ω P G ) = 1 ∀ŝ, while for the insulator (gapped FS), Nŝ(ω > ω ins ) = −1 ∀ŝ. The PG is a coexistence of gapped and gapless parts of the FS, and is characterised by a different topological invariant: | and the parity operation Rŝ : s x ↔ŝ x , :ŝ y ↔ −ŝ y (or vice versa). It is easily seen that N P Ĝ s = 1∀ŝ in the PG phase, and vanishes in the metallic and insulating phases. Such non-local order parameters ensure that the two T = 0 Lifshitz transitions [21] leading from the metal to the insulator do not belong to the Ginzburg-Landau-Wilson paradigm [24]. Our results suggest that the nodal-antinodal dichotomy and Fermi arc phenomenology of the doped cuprates [9,25,26] originates from the physics at half-filling.
At energies above the entry into the PG (ω < ω P G ), the metal is found to have NFL character in the form of a linear dependence of the resistivity on the quantum energyscale (ρ(ω) ∼ 4 − ω, Fig. 1(c)), vanishing quasiparticle residue [21] as well as a highly broadened electronic spectral function (see, e.g., Fig. 1(d)) possessing an electron lifetime τ qp ∼ 1/|4 − ω|. This ensues from a relevant scaling of the forward scattering coupling, such that collective doublon-holon excitations (encoded in the dynamics of the charge pseudospins) destabilise the Landau quasiparticles and lead to the observed phenomenology of the marginal Fermi liquid [28] (see [21] for further evidence). The Mott insulating ground state reached for ω < ω ins is gapped for both charge as well as spin excitations, and possesses the full SU (2) charge × SU (2) spin symmetry of the parent Hamiltonian. That this liquid-like state possesses topological order can be seen as follows. The spectral gap is created via Umklapp processes associated with a flip of charge pseudospins diametrically across the FS, leading to a net momentum transfer of the center of mass: P cm → P cm + 2πQ π,π . This leads to a change in boundary conditions of the center of mass component of the many-body wavefunction [13], gathering a non-zero expectation value for the charge twist operator [29],Ô c x,y = exp [i(2π/N ) r r · (x,ŷ)n r ]: states are degenerate in the thermodynamic limit, and the passage between them involves topo-logical excitations with fractional charge eν ≡ e/2 close to E F (see [21]). This can be read from the total phase acquired by the many-body wavefunction from the non-commutativity of the twist operator O c x,y and the translation operators along the x and y-directions (T x,y ) : x/y T x/y , ν = 1/2 and an exchange statistical angle θ = πν = π/2 [30]. It is well understood that such ground states possess short-ranged resonating valence bond (RVB) order [12,31,11]. Further, our RG analysis shows that this topologically ordered Mott liquid ground state is replaced by a (π, π) charge density wave (CDW) broken symmetry ground state in the presence of a staggered chemical potential. Similar arguments establish the existence of topological order in the spin sector of the Mott liquid, and symmetry breaking to a (π, π) spin density wave (SDW) Neél ground state via a staggered magnetic field [21].
Mottness Collapse and quantum criticality with doping. We present the RG phase diagram for the 2D Hubbard model with doping away from 1/2-filling in Fig.2(a). This phase diagram results from the inclusion of effects of doping and tangential scattering processes across the Fermi surface. The primary effect of doping is in providing separate quantum energyscales for the pseudogapping effects of spin and charge fluctuations: while spin fluctuations start gapping the antinodes at ω > ω P G ≡ ω s P G for all doping, initiation of the pseudogapping of charge fluctuations (ω c P G ) falls linearly with increasing doping: ω c P G = −µ ef f , where the effective chemical potential with doping is tuned via a Hartree-shift arising from the bare Hubbard coupling, This tuning of the chemical potential immediately lowers the SU (2) charge symmetry of the 1/2-filled system to U (1) charge by explicitly breaking the particle-hole symmetry in the doublon-holon sector [32]. The collapse of Mottness [33,34] ends at a QCP at ω c, * P G = W/2 = −µ * ef f with a point-like Fermi surface at the nodes and spin-gapping which increases monotically from the nodes to the antinodes ( Fig.2(d), Videos S2, S3). Importantly, the QCP possesses emergent chiral (SU (2) charge × SU (2) spin ) R/L symmetries. Recall that similar symmetries exist for the particle-hole symmetric 1/2-filled Hubbard model. Indeed, these findings for the QCP are in striking agreement with the works of Phillips and co-workers [35].
For ω > ω c, * P G , at low energies and to the left of the QCP lies the Mott liquid ( Fig.2( For ω > ω t = µ ef f − 2µ * ef f , to the right of the QCP lies a correlated Fermi liquid (CFL) arising from RG relevant tangential scattering ( Fig.2(f)). The CFL is associated with well-defined electronic quasiparticles coexisting with NFL metal on different stretches of a connected FS [21]. These findings are consistent with results obtained from the dynamical cluster quantum Monte Carlo method applied to the 2D Hubbard model with doping away from 1/2-filling [36], and unveil the mechanism responsible for the experimentally observed topological reconstruction of the Fermi surface near critical doping [25,26,37].
At the QCP, while tangential scattering is irrelevant everywhere on the FS, Umklapp and spin backscattering are RG relevant everywhere but at the nodes [21]. This is a topological protection of the nodal degrees of freedom, arising from the phase of the charge pseudospin Greens function being γ ⇑ (node, ω c, * P G ) = 2π (in the doublon basis), leading to a sign N node (ω c, * P G ) = −1. The relevant charge and spin forward scattering couplings lead, instead, to a spin-pseudogapped NFL metallic state along the nodal directions ( Fig.1(b),(c)), extending to finite energyscales in the wedge-like quantum critical region directly above the QCP (ω < min(ω c, * P G , ω t ), Fig.2(a),(e)), merging finally into the NFL with a connected FS at very high energies [28]. Such a nodal liquid state appears to have been observed in ARPES measurements carried out within the PG phase of the slightly underdoped cuprate Bi2212 above the superconducting dome [38]. In addition, this spin-gapped metallic state possesses large pairing fluctuations with d-wave symmetry [39]. Our finding of pairing fluctuations prior to the onset of superconducting off-diagonal long-range order is in agreement with the findings from Nernst experiments on various members of the cuprates [40]. Below, these fluctuations will be seen to interplay with the spin-gap in leading to d-wave superconductivity [41].
We now establish the origin of the d-wave superconducting order as the QCP at µ * ef f = −4.
We have shown above the the nodal points support a gapless NFL metal, as the irrelevance of all gapping mechanisms at the nodes arise from the topological signature of the doublon propagator. Thus, the onset of superconductivity at critical doping [21] takes place in the backdrop of this protection for the nodal gapless states. We have also seen that the spin-gap at critical doping has d-wave structure, but without a sign change across the nodes. A U (1) symmetric breaking RG calculation [21] further reveals that the nodal points act as domain walls for the growth of the superconducting order upon scaling down to low energies: the RG-integrated Conclusions. We conclude with a few striking consequences of our analysis. First, even as the d-wave superconducting phase shields its origin from a QCP lying at critical doping, it possesses properties of that criticality (e.g., gapless nodes, gap with d-wave symmetry). Second, the QCP involves a drastic change in the nature of the ground state and low-lying excitations: from fractionally charged excitations (gapped Mott liquid at underdoping) to electronic quasiparticles (CFL at overdoping) through critical fermionic collective excitations emergent at the QCP (nodal NFL at critical doping). The associated change in the exchange statistics of the excitations has been called statistical quantum criticality [34]. Third, the qualitative agreement of RG phase diagram, Fig.3, with the experimentally obtained temperature versus doping phase diagram for the cuprates [7] is remarkable. This settles conclusively a long-standing debate on whether the physics of the one-band Hubbard model at and away from 1/2-filling is pertinent to the physics of high-temperature superconductivity [43]. Our results are likely pertinent to the ubiquitous presence of superconductivity in several other forms of strongly correlated quantum matter, e.g., the heavy-fermion systems [44]. Finally, our formalism predicts that the kinetic energy of nodal fermions is related to the optimal quantum fluctuation energy scale for the onset of d-wave superconducting order: depressing the former (e.g., by tuning the curvature of the Fermi surface via next-nearest neighbour hopping [45]) should enhance the latter.
terms of Anderson pseudospin operators constructed from electron-electron and electron-hole spinors respectively [22] placed at an off-set Λ for every direction normal to the Fermi surface (FS)ŝ (see Supplementary Fig.S1) This yields a two-particle spin-charge hybridised pseudospin Greens function where the dispersions for the Anderson pseudospins for charge and spin excitations are c,s Λŝ = Λŝ ± −ΛTŝ (Tŝ :ŝ x ↔ŝ y ) and the phases γ ⇓,⇑ (ŝ, ω) signify the signature of Gŝ ,(⇓,⇑) [23] The renormalisation group procedure is formulated via a recursive Gauss-Jordan folding of pseudospin states along everyŝ starting from energies far away from, and approaching, the FS.
As is common with all RG schemes, this yields a renormalisation of all couplings associated with various scattering processes. From these couplings, we construct a low-energy stable fixed point theory. (See Supplementary Materials [21] for a detailed discussion) The RG equations obtained, for instance, for the forward (V c,s Q ηη =(0),ŝ ) and backward scattering (V c,s Q ηη =(π),ŝ ) couplings for charge pseudospin (c) and spin (s) excitations for every direction normal to the FS (ŝ) are The RG equations have been cast as difference equations to take account of the non-linear dispersion and discrete nature of the electronic Hilbert space. The couplings ∆V c,s Q ηη =(0,π),ŝ represent, as always, changes in the bare values with changes in the logarithmic dimensionless energyscale ∆ log(Λ/Λ 0 ) as the discrete scaling transformations are carried out. Here η and η denote positions of a (pseudo)spin electron pair (⇑, ⇓) with respect to the FS before and after a scattering event, such that Q ηη gives a scattering wavevector for forward scattering (η = η , Q = 0) and backward scattering (η = −η, Q = π). The logarithmic dimensionless scale (log(Λ/Λ 0 )) denotes the distance from the FS in energy-momentum space. The RG equations are solved numerically in an iterative manner on a two-dimensional grid in momentum-space, obtaining a phase diagram as well as several physical observables from fixed point values of various couplings, spectral weights and gaps [21].

Schematics of the Renormalization Group
We analyze the 2d-Hubbard model with doping described by, where c † kσ /c kσ is the electron creation/annihilation operator with wave-vector k and spin σ, n rσ = c † rσ c rσ is the number operator at lattice site r = j 1x + j 2ŷ , and the bare dispersion is given by 0k = −2t(cos k x + cos k y ), with t 1 being the nearest neighbour hopping strength, U 0 is the strength of on-site interaction, µ ef f = µ − U 0 2 . In the presence of two-particle interactions states in the neighbourhood the Fermi surface get populated virtually even at T = 0K due to quantum mechanical two particle scattering processes. In order to understand the effects of such low energy fluctuations in the neighbourhood of the Fermi surface generated via such scattering processes we develop a family of shells around the Fermi surface imbibing its geometry. On such shells off-setted from the Fermi energy we develop a scaling approach to understand the effect of fluctuations for the low energy region.

Construction of curves off-setted parallel to the Fermi surface(FS)
We define a window of width 2|Λ 0 | around the Fermi surface FS by drawing lower and upper parallel(offset) curves around it. FS is constituted by family of wave-vectors F : The offset curves C F,Λ 's are defined as normal translations of the Fermi surface wave-vectors k Λ (ŝ) = k F (ŝ) + Λŝ , The fermionic creation/annhilation operators along . Pseudospin construction on offset curves: In order to take account of four fermionic longitudinal (forward and backward scattering) and tangential scattering terms we define a pseudospin basis for pair of electronic states about the nodal wave-vector around the neighbourhood of FS [22], where , and , In the above equations the basis states defined via |f c,s; † Λnŝσ f c,s Λnŝ refer to pseudo-spins created to take account of charge and spin fluctuations due to longitudinal forward and backward scattering. Similarly the basis states |N Λn , ± 1 2 c,s formed out by summing the pseudospins over all normal directionsŝ in the FS neighbourhood allows exploration of charge/spin fluctuations generated due to tangential scattering. The coefficient q l = 1 − q t refer to hybridisation proportions for a pair of electronic states to be involved in longitudinal(l),tangential(t) scattering respectively. While p c = 1 − p s refer to charge-spin hybridisation for the fluctuations in the longitudinal channel and r c = 1 − r s refer to charge-spin hybridisation for the fluctuations in the tangential channel.

Gauss-Jordan block diagonalization in the pseudospin Hilbert space
We define projection operators in the basis eq(9) given by Q <Λn , Q Λn , Q >Λn , with 0 < Λ n < Λ 0 where Λ 0 is the bare shell width around FS. Here Q <Λn , Q >Λn projects pseudospin states in the low (l) and high (h) energy-momentum windows |Λ| < Λ n and |Λ| > Λ n respectively while Q Λn projects pseudospin states on the high energy-momentum pivot (hp) |Λ| = Λ n . The sum of the projection operators are constrained to yield the completeness relation in the pseudospin basis, Q <Λn +Q Λn +Q >Λn = 1. In the pseudo-spin basis the Hubbard Hamiltonian eq(6) can be written as a composition of off-diagonal block matrices in the l,hp subspaces given byĤ l,hp n = Q <ΛnĤ Q Λn ,Ĥ hp,l n = Q ΛnĤ Q <Λn which corresponds to spin flip scattering between those sectors, see orange colored blocks in fig(5(b)). While the red blocks in fig(5(b)) corresponds to a pivot Hamiltonian H hp n = Q ΛnĤ Q Λn . By performing a Gauss-Jordan pivot folding step while removing connectivities between the l and hp sectors we can relate the Hamiltonians at the end of the nth and n+1th step by a recursive RG equation [14], where G hp,n (E) = (E − H hp n ) −1 .
Pivot Greens function in the pseudospin basis: The pivot Greens function,Ĝ hp,n , has poles in the E = ±ω branches for the pseudospin ⇑ / ⇓ states respectively. The quantum energy scale − W 2 < ω < W 2 probes electronic states of the non-interacting tight-binding metal. In the pseudospin basis,Ĝ hp,n can, therefore, be represented aŝ where ,Ĝ Renormalization group difference equations for the couplings associated with the forward (V c,s Q ηη =(0),ŝ ), backward (V c,s Q ηη =(π),ŝ ) and tangential (V c,s tŝ ) scattering can be deduced from the above operator relation eq(10) where γ ⇑,⇓ is the Topological phase given by γ ⇑,⇓ := e iπ(N ⇑,⇓ +1) and the topological invariant N Λŝ,P = dzG −1 Λŝ ∂ z G Λŝ . The hybridization proportions (p ω , q ω , r ω ) are determined from a maximization of the Greens function |G pqr,ŝ,⇑ (ω, Λ)| at every ω. This protocol leads to a dynamical determination of the most singular pole among G −1 pqr,ŝ,⇑ (ω, Λ) s at each and every ω. Specialising to the case of half-filling (µ ef f = 0), we find that the tangential scattering is marginal as q l = (1 − q t ) = 1 is associated with the fastest growing RG flow. In this way, the electronic states very naturally choose the longitudinal scattering channel for everyŝ along the FS. We use the Python language to solve the RG equations numerically in an iterative manner on a two-dimensional grid in momentum-space, obtaining a phase diagram from the fixed point values of various couplings, spectral weights and gaps. Some properties of various phases (e.g., resistivity) are also obtained from these final values.

Renormalization Group equations with hole doping (µ
Upon doping away from 1/2-filling, the RG equations are given by where feedback from tangential scattering has been taken account of by updating the kinetic energy of the charge and spin fluctuations via c Λ,avg . For µ ef f > µ * ef f = − W 2 , we find that tangential scattering is irrelevant, analytically continuing to the µ ef f → 0 case discussed above.
As discussed in the main text, this region displays the collapse of Mottness.

Quantum critical point
At critical doping µ ef f = µ * ef f , the RG equations for the nodalŝ have the form where the nodal excitation velocity scale is v We will see below that, due to forward scattering via doublon-holon and spinon collective excitations, the electronic quasi-particles attain a finite (i.e., non-diverging) lifetime as the FS is approached τ qp ∝ ω −1 . The presence of gapless collective excitations at the FS ensures that the QCP corresponds to a nodal non-Fermi liquid (also see later discussion on other properties of this metallic state).
2.4 (π, π) Charge density wave, (π, π) Spin density wave and d-wave Superconducting instabilities We include (π, π) charge density wave and (π, π) spin density wave symmetry breaking fieldŝ and perform a RG calculation for the instabilities associated with them. Thus, the RG equations for the CDW and SDW instabilities have the form where G −1 r,⇑,Λ = ω− (E r kin,Λ ) 2 + (U int (r)) 2 for r = 0, 1 respectively and the net kinetic energy of the electronic states in the high energy sector is given by, The fixed point value of the spin-charge hybridised interaction strength for the symmetry preserved spin(r=0)/charge(r=1) fluctuation dominated Mott insulator is where the gap scales in the spin and charge excitation sectors are and (U c f p , U s f p ) are computed from the first level of the RG equations eq (14).
Finally, we add the superconducting fluctuations through a bare uniform U(1) symmetry breaking field The renormalization group equations then leads to the following RG equations for the superconducting instability at a givenŝ as well as at the node (ŝ N ) , γ N = i ln sgn(G −1 hh,ωΛnŝ ) = π (24) leading to the fixed point values Λ f p (ŝ) = Λ * (ŝ) Λ 0 , Λ f p (ŝ N ) = 0 , and V * sc = Λŝ + Λ−Tŝ . From these, we can obtain the expectation value for the d-wave superconducting order parameter The superconducting fluctuations characterised by V sc,0 / Λŝ + Λ−Tŝ possess their largest value at the antinodes (AN) and smallest value infinitesmally close to the nodes (N): the gap is, therefore, the largest at AN and vanishes precisely at N. Indeed, the vanishing gap at the nodes arises from the irrelevance of spin backscattering renormalization, which is turn is associated with a hole-occupancy in the high energy sector at the node: |f † sc,N (Λ,ŝ)σ z f sc,N (Λ,ŝ) = − 1 2 . In this way, the effective d-wave structure of the superconducting gap and gapless nodal Dirac fermions associated with Lifshitz criticality is inherited from the state achieved at Mottness collapse (ω * c,P G = 4). Indeed, critical doping corresponds to the highest ("optimal") superconducting transition temperature (T c ). It is remarkable that U(1) phase-rotation symmetry-breaking leads to the spread of d-wave superconductivity to an entire region in the ω − µ ef f phase diagram, i.e., a "dome" that is centered around, but extends well beyond, the neighbourhood of critical doping.

Low-energy fixed-point theory
At the stable fixed point reached upon approaching low energies, the Hamiltonian has the form where the effective Hamiltonian for the degrees of freedom within the emergent low-energy withĤ (2) <Λ * (ŝ),ŝ = 1 4 The effective Hamiltonian for the degrees of freedom residing outside this low-energy window, Hŝ >Λ * (ŝ) , can be obtained from the fixed point of the RG procedure in an analogous manner.
Henceforth, we will focus our attention on the states lying within the window Λ * (ŝ).
The many body Hilbert space at the stable fixed point with SU (2) spin rotational invariance , U (1) global phase rotational invariance and translational invariance is given by where, and f Λŝ = [c Λŝσ c † −ΛTŝ−σ ] is the electron spinor shown earlier. For ν = N e /(2N ) = 1/2 (halffilling, N e is the total number of electrons in the system and N is the total number of sites), the emergent particle-hole symmetry at this filling arises in the basis states from A z ≤Λ * (ŝ) + A z ≤Λ * (−ŝ) = 0. This many-body basis corresponds to eigenstates of total number operator and total momentum.

Benchmarking against existing numerical results
In order to benchmark the results obtained from the effective low-energy Hamiltonian and wavefunctions given above against those found from existing numerical methods applied to the 2D Hubbard model on the square lattice [5], we present results for the ground state energy per particle E g and the fraction of bound pairs (Bp) in the gapped Mott liquid ground state. The analytic form for E g and Bp are computed from the spin and charge backscattering parts H <Λ * (ŝ),ŝ of the effective Hamiltonian given above, and are found to be where n E  Fig.(6)), is considerably close to the range −0.51t < E gs < −0.53t obtained in the thermodynamic limit from nine state-of-the-art numerical methods applied to the half-filled 2D Hubbard model at U = 8t [5]. Similarly, the average saturation value for the fraction of unbound pairs in the Mott liquid obtained from a finite-size scaling analysis (inset of Fig.(7)), U bp ∼ 0.051, is also found to be considerably close to the range 0.0535 < U bp < 0.0545 obtained in Ref. [5].

Lifetimes and the f-sum rule
Quasiparticles, doublon-holon and two-spinon collective excitations are found to reside within the low-energy window in the metallic regions. In the insulating region, the low-energy window is constituted of doublon-holon and/or two spinon bound pairs. The doublon-holon collective excitations are highly entangled objects formed out of pairs of electrons around the N points.
These are exact eigenstates of the two particle forward-scattering interactions. On the other hand, quasi-particle(hole) states are eigenstates of the Hartree and the kinetic energy parts of the final renormalized Hamiltonian H f p . Similarly, in the insulating regions, the doublon-holon bound pairs are also highly entangled objects, being exact eigenstates of the Umklapp scattering interaction. The lifetime for the quasiparticles (τ qp ), bound-pair/gapless collective excitations (τ coll ) and fractionally-charged topological excitations (τ top−exc ) are given by the following relations [13] τ qp (ω,ŝ) = Im wheren f p Λŝσ (t) = e iH f p tn Λŝσ e −iH f p t is the time-evolved fermion-number operator, and the state |Ψ l exc,Λ = Λ <Λ * (ŝ) c † Λŝσ c Λŝσ |Ψ(l), N e . Total spectral weight is conserved via the f-sum rule by integrating the lifetimes of the quasiparticles and collective excitations over the full frequency range. In this way, we are able to take account of the conversion of quasi-particle poles at and around the Fermi surface (within the low-energy window Λ * for everyŝ) into the poles of the two-particle pseudospin electron pairs [27]

Formation of Bound state
The RG relevant forward scattering coupling leads to the formation of a pole of two-particle (doublon-holon) gapless collective excitations at (and in the neighbourhood of) the Fermi surface. In Fig.(S8(a) (left panel)), this is observed through a crossing of the total phase Φ F S (= 0 Λ 0 G −1 s,⇓ (ω, Λ)) of the two-particle forward scattering Greens function in the doublon-holon basis (red line) crossing the energy scale for quantum fluctuations associated with the inverse bare interaction strength ∆U −1 0 . As will be discussed below, this leads to a renormalisation of the one-particle Greens function at a given point on the Fermi surface, turning them marginal in nature. Similarly, in Fig.(S8(a) (right panel)), we observe the formation of a two-particle doublon-holon bound state at a given point on the Fermi surface (blue line for total phase This is concomitant with a zero of the single-particle Greens function. Indeed, this is analogous to Cooper's demonstration of the formation of bound pairs of electrons leading to an instability of the Fermi surface as being responsible for the onset of superconductivity [47].

Quasiparticle Lifetime, residue and resistivity of the non-Fermi liquid phase
Our RG delivers microscopic evidence for several key aspects of the phenomenology of the marginal Fermi liquid (see [28]). This can be seen as follows. The single particle green function in the presence of forward scattering gets renormalized due to doublon-holon or two-spinon excitations being present at the vicinity of E F has the following form, where V Λ * ŝ(ω) = p c c,Λ * (ŝ) + p s s,Λ * (ŝ) − ω with Λ * (ŝ)/Λ 0 being a fraction of the spectral weight along a given normal directionŝ of the doublon-holon/two-spinon collective excitations (Fig.(S8)). The complementary fraction (1 − Λ * (ŝ) Λ ) constitute electronic quasiparticles whose spectral function get broadened due to these collective excitations, rendering them with a non-diverging liftime at low-energies (using eq.(32)) This is shown in Fig.(S9)(c), and displays an enhanced dissipation of quasiparticle excitations in the marginal Fermi liquid (see [28]) due to collisions with doublon-holon collective excitations discussed above.
Another striking feature of this marginal Fermi liquid is the vanishing of the quasiparticle residue Z Λŝσ at low energies. This can be computed from the Re(Σ) which can be in turn computed from the Im(Σ) via the Kramer Kronig relations, Indeed, a divergent enhancement in the effective mass Z −1 ∼ (m * /m) is shown in Fig.(S10). 4(c,d) of the main text). Further insights and results will be provided in a later section from the viewpoint of the renormalised single particle propagator and self-energy.

Pseudogap progression, Fermi arc and topological excitations of the Mott liquid
In Fig. (S9)(a) and (b), we observe the pseudogapping of the spectral function at (and near) the antinodes. The gradual growth in the extent of the pseudogap for charge excitations (shown as a width of the k-space window around the Fermi surfaceΛ) is shown in Fig.(S7), finally saturating at a finite value when the entire Fermi surface is gapped (the Mott liquid). The inset of Fig.(S7) shows the finite-size scaling of the saturated value of the window width,Λ s (proportional to the many-body gap of the Mott liquid), with log 2 √ N (where N is the system size): the plot appears to saturate at a finite value.
As the pseudogap for charge and spin excitations is bookended by Fermi surface topologychanging Lifshitz transitions at the antinode (initial) and node (final) respectively, we show below in Fig.(S12) the finite-size scaling of the energy scales for the entry (ω c1 (cyan), ω s1 (red)) and exit (ω c2 (violet) and ω s2 (blue)) of the charge (c) and spin (s) pseudogaps respectively with log 2 √ N (where N is the system size). We can clearly see that the extent of the charge and spin pseudogaps, i.e., the differences ω c2 − ω c1 and ω s2 − ω s1 saturate with increasing system size. Further, the size of the spin pseudogap is dominant over that of the charge pseudogap in the thermodynamic limit. This impacts the growth of the d-wave superconducting order upon doping. Further, the influence of the pseudogap on the resistivity has already been shown in the main manuscript, as well as in Fig.(S11).
In Fig.(S13), we show the gradual decrease of the length of the gapless Fermi surface (i.e., the Fermi arc) with gradual progression of the pseudogap phase towards the Mott liquid state (zero Fermi arc-length) [24]. Finally, in

Statistical quantum criticality and fermion signs
In the main text, we observed that the collapse of Mottness, i.e., the vanishing of the energyscale Here, we clarify how the well-known problem associated with fermion signs in Euclidean pathintegral formulations of fermionic quantum many-body systems allows for such a QCP. First, note that the Fermi liquid itself corresponds to a sign-full path integral. However, as shown in Refs. [48,49], this path integral can be evaluated in totality without the worrisome negative probabilities arising from the exchange of fermions. Further, the gapped Mott liquid state clearly does not suffer from any fermion sign issues. Finally, it was shown in Refs. [50,51] that the fermion sign problem is mitigated in systems involving non-relativistic interacting fermionic excitations on isolated patches of the Fermi surface. This shows that the nodal non-Fermi liquid metal emergent at the QCP (and which is adiabatically connected to the non-Fermi liquid living at a connected Fermi surface at high energies) is free from fermion sign-related negative probabilities. This offers hope that quantum Monte Carlo simulations of the low-energy effective Hamiltonian reached from our T = 0 RG procedure can be used to compute thermodynamic quantities at finite-T without encountering the sign problem.

Renormalized single particle green function
We attain the following longitudinal,tangential scattering interactions at the nth step of the RG flow (Λ n = Λ 0 b n ) and ω, The kinetic energy cost for the spin and charge pseudospins is given by, The recursive renormalization group equations for the longitudinal and tangential scattering events are given by, The green function written down in the pseudospin basis for the longitudinal and tangential where N F (ŝ) = ŝ 1 is the number of normal directions on the Fermi surface, p c = 1 − p s = p.
The hybridization proportions p ω is determined from the maximization of the green function |G p,ŝ,⇑ (ω, Λ)| overallŝ at every ω which is the quantum energy scale varying from 0 < W 2 −ω < W , with W=8t being the bandwidth . This protocol leads to the most singular pole among G −1 p,ŝ,⇑ (ω, Λ) s being dynamically determined at each and every quantum energy scale.

Representing four Fermi interaction in terms of pseudospins coupled to fermion
The four fermi interaction in the pseudospin basis can be rewritten in terms of the three fermion composite operator and a fermion, where M c,s Λ Λŝσ is given by,

Coupling Fermi surface pseudospins to Fermionic states in offset curves
We will now reduce down to a effective Hamiltonian involving the coupling between pseudospins at the Fermi energy with the offset curves at Λ for every normal directionŝ, this is motivated on the perspective of arriving at low energy single electron green function due to scattering fermions from the bare Fermi surface, where M c,s; † F Λ,ŝη,ŝ η ,σ = f c,s; †

Computing the imaginary piece of self energy matrix from the pseudospin fermion Hamiltonian
From the renormalized two particle vertex at the final fixed point we determine the single particle self energy by considering the integrating out of virtual states described as a pseudospin coupled to a fermion. This leads us contracting two two particle vertices and gathering a six fermion term 2×(2pseudospin+1fermion) which can be written as a product of three single particle green function and corresponds to the mathematical form of Σ (II) [53]. This integrating out leads to a two fermion scattering processes 1) forward scattering processes that causes lifetime broadening for the single fermion quasiparticles and residue Z decay , 2) backscattering processes that causes pole to zero conversion for single particle states. Below the figure fig(15) represents the scattering proccess. The connectivity term between single electronic states in the neighbourhood of one direction along the Fermi surfaceŝ and scattering to a opposite direction s → −ŝ can be arrived at by pivot folding the fermionic states coupled to the Fermi surface pseudospins eq(43) leading to a imaginary self energy matrix, Σ Im Λ,ŝ,n (ω) = f † Λ,ŝ,σK Λ,ŝ,σ f Λ,ŝ,σ ,K Λ,ŝ,σ = Σ Im++ Λ,ŝ,n (ω) Σ Im+− Λ,ŝ,n (ω) Σ Im−+ Λ,ŝ,n (ω) Σ Im−− Λ,ŝ,n (ω) where η = ±1 represents the directionsŝ, −ŝ respectively. The components of the self energy matrix is computed, where the green functionĜ c,s F,Λ,Λ ,ŝη,ŝ η for the doublon-holon/two spinon pseudospin coupled to fermion is written in the basis given by , The integrals I Λŝ in eq (45) can be computed in the following way, where the fixed point values of the coupling coefficients are given by, Here N Q=π,ŝ (ω)+N Q=0,ŝ (ω) = 1, when N Q=π,ŝ (ω) = 1 then the non-trivial topological number corresponds to gapping the electronic states due to backscattering, whereas when N Q=π,ŝ (ω) = 0 then that corresponds to gapless electronic states undergoing forward scattering.

Computing the complete single particle green function fromΣ I
The off-diagonal pieces of the self energy matrix generated due to Umklapp /spin backscattering process try to generate a gap for single electronic states by backscattering electronic states whereas the diagonal pieces generated from forward scattering determine the nature of poles for the gapless electronic states. This backscattering processes lead to a renormalized diagonal pieces. This renormalized pieces can be reached by decoupling the states alongŝ, −ŝ while taking account of the motion flip backscattering or the backflow of the electrons, Upon folding the off diagonal elements, we obtain Σ Im++ Λ,ŝ,n (ω) + Σ Im+− Λ,ŝ,n (ω) Therefore the renormalized self energy matrix is given by, The Re(Σ) which can be in turn computed from the Im(Σ) via the Kramers-Kronig relations, Now we can write the complete single particle green function G n * σ (Λ,ŝ, ω) = c n * ,Λŝσ (ω)c † n * ,Λŝσ (ω) , is the quasiparticle residue and the inverse lifetime of the single fermion quasiparticle is contained in the imaginary part of the self energy.

Pole to zero conversion in the single fermion propagator
At a given probe energy scale, there is continuous region with normal orientationsŝ s ranging from the AN toŝ ω upto which N Q=0,ŝ (ω) = 1 − N Q=π,ŝ (ω) = 0. This topological signature leads to a pole of the imaginary part of self energy, and in turn a vanishing lifetime (and a zero of the Greens function G) [24] N Q=π,ŝ (ω) = 1 → G n * σ (Λ,ŝ, ω) = 1 using this the f sum rule for the gapped regions can be represented as, Λ,ŝ,n * (ω) , Λ,ŝ,n * (ω) + G Λ,ŝ,n * (ω) (b) Recursive Gauss-Jordan diagonalisation of the many-body state space (in the two-particle pseudospin basis) by folding the pivot (red box) interpolating between the high-energy (brown box) and low-energy sector (green box). Orange boxes indicate connectivities between the pivot and low-energy sector. Fig. S2. Variation of ground state energy per particle (E g ) with the probe energyscale ω at half-filling and U = 8t during the passage from metal (ω < 0) to the Mott liquid (ω > 2.8) through the pseudogap (0 ≤ ω ≤ 2.8). A saturation value of E g = −0.495t for a k-space grid of 2 11 × 2 11 (and t = 1). Inset: Finite-size scaling of the saturation value for E g ≡ E gs with 1/ √ Volume with increasing k-space grid size from 2 9 × 2 9 to 2 15 × 2 15 . The saturation E gs for the largest grid is observed to be −0.507t. The error bar for all data points is ∼ O(10 −5 t) .     S7. Crossover in the resistivity (ρ) from a non-Fermi liquid metal at high-energies (linear against 4 − ω upto ω ≤ 0) to a correlated Fermi liquid (CFL, ω ≥ 2.5t) at various doping strengths µ ef f greater than critical doping µ ef f = −4t. The effects of a spin pseudogap at intervening energies is observed to be strong close to critical doping (blue and green curves) and weaken for larger dopings (red and cyan curves).