Deconfined critical point in a doped random quantum Heisenberg magnet

We describe the phase diagram of electrons on a fully connected lattice with random hopping, subject to a random Heisenberg spin exchange interactions between any pair of sites and a constraint of no double occupancy. A perturbative renormalization group analysis yields a critical point with fractionalized excitations at a non-zero critical value $p_c$ of the hole doping $p$ away from the half-filled insulator. We compute the renormalization group to two loops, but some exponents are obtained to all loop order. We argue that the critical point $p_c$ is flanked by confining phases: a disordered Fermi liquid with carrier density $1+p$ for $p>p_c$, and a metallic spin glass with carrier density $p$ for $p<p_c$. Additional evidence for the critical behavior is obtained from a large $M$ analysis of a model which extends the SU(2) spin symmetry to SU($M$). We propose that key aspects of cuprate phenomenology are realized by the vicinity of this deconfined quantum critical point.

However, it appears that the restoration of the broken symmetry cannot be the driving mechanism for a quantum phase transition at p = p c : the broken symmetries are weak and differ among the cuprates, and the transport [5], thermodynamic [6][7][8], electronic [2][3][4]10], and spin dynamics [11][12][13] signatures are strong. This paper will study a model with all-to-all randomness (see (1.1) below) which exhibits a deconfined quantum critical point [14] with many similarities to the mysterious cuprate phenomenology. Our model has a quantum critical point at p = p c with fractionalized excitations, separating metallic states with carriers densities of p and 1 + p (see Fig. 1). The overdoped state is a conventional disordered Fermi liquid, the underdoped 'pseudogap' phase with carrier density p has spin glass order, but the quantum critical point is described by fractionalized excitations. Our critical theory is distinct from a Landau-Hertz-Millis-type theory [15,16] of the quantum fluctuations of the spin glass order in a metal; the latter theory has no fractionalization at criticality and does not exhibit a change in carrier density across the transition. Moreover, our p = p c critical theory is expected to maximally chaotic [17], similar to the Sachdev-Ye-Kitaev (SYK) [18,19] models, and unlike the Landau theories [20].
Our results provide a simple rationale for the existence of a quantum phase transition in correlated metals with 'Mottness'. Broken symmetries are not essential, and only play a secondary role. At low doping, we have fermionic 'holons' of density p, moving in a background of condensed bosonic 'spinons' (see Fig. 1). At higher doping, we have condensed bosonic holons, so that the fermionic spinons behave like a Fermi liquid of hole-like carrier density 1 + p. This statistical transmutation, and corresponding transformation in the many-body state, is accomplished by a strongly coupled deconfined critical point which exhibits a boson-fermion duality. Note that, because of the presence of the Higgs-like condensates on both sides of the critical point, there is no true fractionalization for either p > p c or p < p c .
There have been discussions [21] of deconfined critical points between a magnetic metal with 'small' Fermi surfaces, and a non-magnetic heavy Fermi liquid with a 'large' Fermi surface (a review of related ideas is in Ref. 22). However, to date, no tractable realization of this scenario has been found in non-random systems. Our results show that a similar scenario is naturally realized in models with random couplings. We also note the study of Ref 23, which found an evolution between small and large Fermi surface across optimal doping in a plaquette dynamical mean field theory.
Our model is in the class of SYK models [18,19] with random all-to-all couplings, which have been extensively exploited recently for descriptions of strange metals and the quantum information theory of black holes. Specifically, we generalize the insulating random Heisenberg magnet originally studied in Ref. 18 to metallic states of a t − J ij model at non-zero doping, along the lines of Ref. 24. We consider electrons, annihilated by c iα , spin α =↑, ↓ on sites i = 1 . . . N with double occupancy prohibited α c † iα c iα ≤ 1. The Hamiltonian is the familiar t-J model with where P is the projection on non-doubly occupied sites, µ is the chemical potential and S i = (1/2)c † iα σ αβ c iβ is the spin operator on site i, with σ the Pauli matrices. The complex hoppings t ij = t * ji and the real exchange interactions J ij are independent random numbers with zero mean and mean-square values |t ij | 2 = t 2 and J 2 ij = J 2 . We will work at a variable hole density p, defined by 1 N i c † iα c iα = 1 − p . (1. 2) The insulating p = 0 case of H tJ was studied in Ref. 18, and in Ref. 24 for non-zero p, after generalizing the SU(2) spin symmetry to SU(M ) and taking the large M limit (see Appendix C). A gapless critical ground state was found [18] at large M for p = 0. However, subsequent numerical studies [25,26] of the insulating SU(M = 2) case found a spin glass ground state, and such insulating spin glass states had also been examined in the large M limit [27,28]. At non-zero p,

Disordered
Fermi liquid. Condense holon b, f ↵ carrier density 1 + p < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 a L W m L R L Y L d I h s s a R L f s 5 y 6 H B + k = " > A A A C Y 3 i c d V B d a x Q x F M 2 M r d a p H 2 v 1 T Y T g j i B Y h p n 9 q P a t t C I + V n D b w m Z Z M s m d 3 d B M E p N M Y R n 2 Z / r g Q x / 9 F b 4 0 u 9 2 C i h 4 I H M 4 5 N z c 5 p Z H C + T z / E c X 3 t r b v P 9 h 5 m O w + e v z k a e f Z 3 p n T j W U w Y l p q e 1 F S B 1 I o G H n h J V w Y C 7 Q u J Z y X l y c r / / w K r B N a f f U L f carrier density p < l a t e x i t s h a 1 _ b a s e 6 4 = " C z T 5 Z 3 9 M p v 1 M j B F q j U k D 8 N 1 M D 4 e f 0 g 9 m E 5 H 2 X G G s y T t a 4 h 2 d b Y c / C C F Y n X l 7 a x b b p 6 l 2 i 0 a a h x n A t q I 1 B Y 0 Z R u 6 g r m H k l Z g j 4 o r r m 0 P F 0 0 f W o s P v V j g U h n / p M M 9 e 9 v c 0 M r a b Z X 7 z u 4 g + 7 f W k f / S 5 r U r 3 y 8 a L n X t Q L L r Q W U t s F O 4 + w F c c A P M i a 0 H l B n u 1 8 Z s T Q 1 l P g 4 b + T x u j s b / B + e j J B s n 0 y + j 4 c n H X T J 7 6 B V 6 g 9 6 i D L 1 D J + g T O k M z x N D P Y D 9 4 E b w M f o W v w 2 F 4 e N 0 a B j v P c / R H h c l v 4 g y + z g = = < / l a t e x i t > b † " |vi < l a t e x i t s h a 1 _ b a s e 6 4 = " v E C y c C T t e y k y w P j 6 S C P C L A B 1 c / k h n r 3 P g O 9 L t 7 g Z 7 A Q / a / g I t t s T x s P k r S g p R 5 a h J K L C 2 H / g l D W o w J I X C W S O q L J Y g x p B h 3 1 E N O d q P y U S W d k E H 9 S L o j L 9 3 Z s L T w r i n i S / U f 5 d r y K 2 d 5 r G b n M e y j 7 2 5 + J T X r y j 9 N K i l L i t C L e 4 P p Z X i V P B 5 a z y R B g W p q S M g j H T f 5 m I E B g S 5 b h u u j 7 + h + f / J y W 4 7 6 L S 7 X 8 P W 4 e d l M + t s h 7 1 j H 1 j A 9 t k h O 2 L H r M c E u 2 Q / 2 E 9 2 7 V 1 5 N 9 6 t d 3 c / u u I t d 7 b Z A 3 i / / w B t R a g 6 < / l a t e x i t > b † # |vi < l a t e x i t s h a 1 _ b a s e 6 4 = " z 0 N n q 6 d M e H I M T M Z k u 1 J t g F u 2 L X 7 M a 7 9 G 6 9 O + / + c X T F W + 5 s s D / g / X 4 A L B C p I Q = = < / l a t e x i t > f † |vi < l a t e x i t s h a 1 _ b a s e 6 4 = " l X g T R A 1 N H c D D R U e Z o n / r k M t K M n 4 = " > A A A C H 3 i c d V D L a t t A F B 2 l z a P O y 2 m X 3 Q w x h U C C k f x I 2 l 1 o N 1 2 m E C c B y z F X o y t 5 8 G g k Z q 4 M R v V H 9 B P 6 F d 2 2 q + 5 C t l n 0 X z p 2 H G h C e m D g c M 6 9 3 D k n K p S 0 5 P t 3 3 s q L l 6 t r 6 x u v a p t b 2 z u 7 9 b 3 X F z Y v j c C e y F V u r i K w q K T G H k l S e F U Y h C x S e B m N P 8 3 9 y w k a K 3 m p s k y S F F 4 V B y C K F 5 9 H l 4 d g / H 6 C x M t e n N C y w m 0 G q Z S I F k J N 6 1 W 9 J L 4 z z K w 3 G 5 F c / w h j S F A 0 P F S Z 0 z Q c 8 N D L t U 2 h A p w p 7 1 Z p f 9 1 v N R r P J H T R 2 G t 9 9 B 6 3 W d r A T 8 K D u T 6 r G p n X c q / 5 z d 4 s y Q 0 q v 2 k l X 6 a b 0 a z 8 0 W F k p Z 8 / 6 e 3 s r r 2 7 P n 6 x m Z l 6 8 X L V 9 v V 1 z u n N i  numerical studies of multi-orbital Hubbard models [29,30], and at the metal-insulator transition of a disordered Hubbard model at half-filling (p = 0) [31].
are all physical observables which realize the superalgebra SU(1|2) [32] (see Appendix A). We are interested in the 3-dimensional representation of physical states obeying Hence, the physical states are invariant under the U(1) gauge transformation f α → f α e iφ , b → be iφ , while individual spinon and holon excitations carry U(1) gauge charges.
Alternatively we can use a representation with bosonic spinons b α and fermionic holons f. Now the gauge-invariant operators are This realizes the same superalgebra SU(2|1) ≡ SU(1|2) as (1.3), and the same 3-dimensional representation is obtained by the constraint Note that while we find it convenient to refer to the superalgebra, there will be no supersymmetry in our results: H tJ does not commute with all SU(1|2) generators.
We can now describe the structure of our main results illustrated in Fig. 1. We find a deconfined critical point p = p c at which the 3 spinon and holon states are nearly degenerate. Assuming all three states are equally probable at criticality, we obtain a critical density p c = 1/3. Indeed, as in the 2+1 dimensional theories, we will find that a Wess-Zumino-Witten [32,[35][36][37][38] term (S B in (2.4) and (2.6) below) plays a central role in the criticality.
Away from the critical point, there is a runaway RG flow to states in which either the spinon or holon states are lower in energy. As illustrated in Fig. 1 We will describe the nature of the infinite volume (N → ∞) limit of H tJ in Section II, and map possible critical states of the large N limit to quantum impurity models in Section II A. The II. LARGE VOLUME LIMIT The limit of large volume (N → ∞) of H tJ is obtained by the methods described in Refs. [18,27,28] for the insulating model at p = 0. We introduce field replicas in the path integral, and average over t ij and J ij . At the N = ∞ saddle point, the problem reduces to a single site problem, with the fields carrying replica indices. The replica structure is important in the spin glass phase [27,28], which we will explore in subsequent work. In the interests of simplicity, we drop the replica indices here as they play no significant role in the critical theory and the RG equations.
Within the imaginary time path integral formalism (with τ ∈ [0, 1/T ], with T the temperature), the solution of the model involves a local single-site effective action which reads: In this expression, µ is the chemical potential determined to satisfy (1.2) and S ∞ is the action associated with the constraint of no double occupancy (U = ∞). Decoupling the path integral introduces fields analogous to R and Q which are initially off-diagonal in the spin SU(2) indices.
We have assumed above that the large-volume limit is dominated by the saddle point in which spin rotation symmetry is preserved on the average, and so R and Q were taken to diagonal in spin indices. The path integral Z is a functional of the fields R(τ ) and Q(τ ), and we define its In the thermodynamic (N → ∞) limit, the solution of the model is obtained by imposing the two self-consistency conditions: These equations and the mapping to a local effective action are part of the extended dynamical mean-field theory framework (EDMFT), which becomes exact for random models on fully connected lattices [16]. They can also be viewed as an EDMFT approximation to the non-random t-J model [23,[39][40][41]. To make contact with notations often used in the (E)DMFT literature, we note This path-integral representation can be formulated in the SU(1|2) representation (1.3) as: where S(τ ) is to be represented by (1.3). Here S B is the kinematic Berry phase (i.e. the Wess-Zumino-Witten term [35]) of the SU(1|2) superspin at each site [32], S tJ is the action containing the terms arising from H tJ , λ is the Lagrange multiplier imposing Eq. (1.4) and the chemical potential s 0 is determined to satisfy (1.2), which now becomes Note that Z is a U(1) gauge theory, and under the U(1) gauge transformation λ → λ − ∂ τ φ.
Let us also present the exactly equivalent formulation of the large N saddle point in terms of the SU(2|1) superspin. Now the Berry phase S B in (2.4) is replaced by while S tJ has the same form, apart from representing c α (τ ) and S(τ ) by (1.5), and replacing the The density constraint determining s 0 in (2.5) is replaced by Appendix C analyzes the path integral (2.4) using a large M expansion in an approach which generalizes the SU(2) spin symmetry to SU(M ); a similar large M method has been used previously for a Hubbard model [42][43][44] and other phases of a disordered t-J model [45].
The body of the paper will focus on an RG analysis of Z. This is performed by expressing it in terms of an auxiliary quantum impurity problem, which we will now set up.
A. Mapping to a quantum impurity problem In our RG analysis, we find it useful to consider the path integral as a functional of the fields R(τ ) and Q(τ ) with an arbitrary time dependence, and to defer imposition of the self-consistency conditions in (2.3). As we are looking for critical states, we assume that these fields have a power-law decay in time with where, for now, d and r are arbitrary numbers determining exponents, which will only be determined after imposing (2.3). Our analysis exploits the freedom to choose d and r: we will show that a systematic RG analysis of the path integral Z is possible to all orders in andr, where (2.9) The analysis assumes andr are of the same order, and expands order-by-order in homogeneous polynomials in andr. Such RG analyses were carried out in Refs. [46][47][48] for an insulating spin model in which t = 0, and by Fritz and Vojta [49][50][51] for a pseudogap Anderson impurity model in which J = 0 (see also Refs. [52,53]); we note that ther expansion of Refs. [49][50][51] was in good agreement with numerical studies [54]. We will combine these analyses here, and our notations here for andr follow these earlier works.
We proceed by decoupling the last two terms in S by introducing fermionic (ψ α ) and bosonic (φ a , a = x, y, z) fields respectively, and then the path integral Z reduces to a quantum impurity problem. The 'impurity' is a SU(1|2) superspin realizing the 3 states on each site of the t-J model, and this is coupled to a 'superbath' of the ψ α and φ a excitations. The quantum impurity problem is specified by the Hamiltonian For completeness let us also explicitly present the Hamiltonian using a SU(2|1) impurity of bosonic spinons and fermionic holons We note several features of H imp , which apply equally to (2.10) and (2.11): • The bosonic bath is realized by a free massless scalar field in d spatial dimensions, as in Refs. [46][47][48]. The field π a is canonically conjugate to the field φ a . The impurity spin S couples to the value of φ a at the spatial origin, φ a (0) ≡ φ a (x = 0, τ ). It is easy to verify that upon integrating out φ a from H imp , we obtain the J term in S tJ , with Q(τ ) obeying (2.8).
• The fermionic bath is realized by free fermions ψ kα with energy k and a 'pseudogap' density of states ∼ |k| r . The impurity electron operator c α is coupled to ψ α (0) ≡ |k| r dk ψ kα .
Integrating out ψ kα from H imp yields the t term in S tJ , with R(τ ) obeying (2.8).
• We have replaced the path integral over the Lagrange multiplier iλ in S B by a constant real λ. The constraint (1.4) can be conveniently and exactly imposed by the Abrikosov method of sending λ → ∞ [48][49][50][51], as we will see in Section III. So the consequences of S B will be accounted for exactly, and that is also the case for the alternative analysis in Appendix B, where S B is accounted for by the exact implementation of the superalgebras.
• The two formulations of H imp in (2.10) and (2.11) are equivalent, and lead to identical RG equations. This is because, ultimately, the quantum dynamics depends only upon the superspin algebra and the representation of the superspin on each site, and these are the same for SU(2|1) formulation by (1.3,1.4) and SU(1|2) formulation by (1.5,1.6). An explicit derivation of the one-loop RG equations using only the superspin algebra and representation appears in Appendix B.
• The model is now characterized by 3 couplings constants, s 0 , γ 0 , and g 0 , we will derive the RG equations for these couplings in Section III. The coupling of the superspin to the fermionic bath is g 0 , and to the bosonic bath is γ 0 : we will see that the RG flow of these couplings is marginal for small andr, and they are attracted to a deconfined critical point.
• The coupling s 0 acts like a 'Zeeman field' on the superspin, which breaks the degeneracy between the spinon and holon states. The flow of s 0 is strongly relevant at the deconfined critical point, and this drives the system into one of the two phases in Fig. 1.
We note that impurity models with both fermionic and bosonic baths have been considered earlier by Sengupta [55], and by Si and collaborators [56][57][58], but not for the 'superspin' case with significant particle-hole asymmetric charge fluctuations on the impurity site. Specifically, we fully project out doubly occupied states, while keeping holon states low energy, and these features are crucial to the structure of our critical theory, as in Refs. [49][50][51]. Also, the effect of a Zeeman field in an impurity spin model with a bosonic environment was studied in Refs. [59][60][61] in the context of the superfluid-insulator transition.

III. RG ANALYSIS
This section will present the RG analysis of the impurity model defined by (2.10). The RG analysis will initially not account for the self-consistency conditions (2.3). We will apply them later in Section III E.
We will employ the SU(1|2) superspin formulation, although essentially the same analysis can be applied to the SU(2|1) superspin, with exactly the same results. A key feature of the computation is that we impose the local constraint (1.4) exactly. This implemented by the Abrikosov method of taking the λ → ∞ limit, as described in earlier analyses [48][49][50][51].
An alternative approach to obtain the RG equations generalizes the method of Refs. [46,47] for SU(2) spins to superspins in either SU(2|1) or SU(1|2). This method utilizes only gaugeinvariant information contained in the superspin algebra and its representation, and is presented in Appendix B. The RG equations are identical to those obtained in this section.
At the tree-level, we can identify the scaling dimensions from (2.10): This establishes r = 1 and d = 3 as upper critical dimensions.
We define the following renormalized fields and couplings, The renormalization factors are to be obtained from self-energy and vertex corrections, as we will show below. We will work with s 0 = 0 and subsequently derive the flow away from it. Also, we work at zero temperature, i.e., β → ∞. coupling. Here we have two relevant diagrams and we quote the self-energy below: with γ E being the Euler's constant.
For the bosonic self energy there is only one diagram (Fig. 2(c)) at the one-loop level. The self energy is evaluated as follows: A factor of 2 is due to the spin index of internal f and ψ-line.
The expressions for Σ f 2(a) and Σ f 2(c) agree with those in Refs. [49][50][51], while that for Σ f 2(b) agrees with that in Ref. 48. Similarly, the self-energy diagrams at two-loop level are evaluated in a straightforward manner, as shown in Appendix D.

B. Vertex correction
There is no one-loop vertex correction to the fermionic bath coupling g 0 . However, it does acquire corrections at two-loop level and the corresponding diagrams are shown in Fig. 12. The bosonic bath coupling γ 0 has vertex corrections both at the one-loop and two-loop level. The one-loop diagram is shown in Fig. 2(d), while the two-loop diagrams are shown in Fig. 13. Here we explicitly evaluated the one-loop vertex correction to γ 0 , This expression agrees with that in Ref. 48. We can similarly evaluate the two-loop level corrections and these are quoted in the Appendix D.

C. Beta functions and fixed points
We now demand the cancellation of poles in the expression for the renormalized vertex and the f /b Green's functions at the external frequency, iν − λ = µ. This leads to the following expressions of the renormalization factors. Note that Z φ = 1 exactly, owing to the absence of bulk interaction terms such as φ 4 . For the rest we have, .
Using Eqns. (D3) and (D4), we obtain the beta functions as follows: We can find the fixed points to two-loop order by setting the beta functions to zero. This gives us four fixed points (g * 2 , γ * 2 ): The stability of the fixed points can be analyzed by looking at the eigenvalues of the stability matrix. We find that the Gaussian fixed point is always unstable. Importantly, we find that the non-trivial fixed point, F P 4 , with g * = 0 and γ * = 0 is stable for a range of values in the parameter space of andr. In Fig. 3 we plot the RG flow in the g − γ plane at one-loop level and show the different fixed points.
These fixed points corresponds to the underlying t-J model at a non-zero doping density p.
However, the precise value of p depends upon high energy details, and cannot be deduced from the fixed point couplings, as we discuss in Appendix D 2. With the beta function at hand, it is straight forward to calculate the anomalous dimension of the fermion and boson propagators, defined as follows: Note that these anomalous dimensions are gauge-dependent, and not physically observable. We have defined them in the gauge λ = constant. In terms of the coupling constants, At the fixed points, we obtain the following expressions for the anomalous dimension, (3.26)

E. Anomalous dimensions of the electron and spin operators
We now calculate the anomalous dimensions η c and η S of the physical and gauge-invariant composite operators, the electron c α and the spin S specified in (1.3), defined such that at large τ . We will show that it is possible to determine these anomalous dimensions to all orders in the andr expansions, as was also the case in previous analyses [46][47][48][49][50][51].
To compute these scaling dimensions, we add source terms to the action Within the field-theoretic RG scheme, we have are renormalized as follows: It turns out that the diagrams contributing to the vertex corrections to Λ S and Λ c are exactly those we encountered while evaluating Z γ and Z g respectively. Thus we have, It is these identities which enable use to compute the anomalous dimensions exactly. We evaluate the required anomalous dimensions as, We can now make an exact statement for η S for fixed points with γ = 0. From Eqns. (3.32) and (3.31) we obtain, Substituting the above equation in Eqn. (D4), we obtain, which leads to The first term on the r.h.s. of (3.35) arises from the tree-level scaling dimension, while the second term contains potential corrections higher order in . However, at the fixed point where γ = γ * = 0, we have β(γ * ) = 0 and therefore, η S = , to all orders in andr. (3.36) The same value of η S is also obtained in the large M expansion in (C44) and (C47).
Similarly, using Eqns. (3.32) and (3.31) in combination with Eqn. (D3) we obtain the following relation: Thus at the fixed point, β(g * ) = 0, such that g * = 0, we obtain η c = 2r , to all orders in andr. (3.38) The same value of η c is also obtained in the large M expansion in (C34) and (C36).
We can now state the main result of this subsection: at the non-trivial fixed point F P 4 (g * = 0, γ * = 0) we have η S = and η c = 2r to all orders in andr.
Finally, we can impose the self-consistency condition, Eq.  c α ∼ f α , and the hopping term t ij in H tJ in (1.1) reduces to a renormalized hopping term for the f α spinons. Indeed, the resulting theory for the f α fermions is similar to that studied by Parcollet and Georges [24], and more recently in SYK-like extensions [62][63][64][65][66].
Note that this disordered Fermi liquid has a hole carrier density of 1+ p. This follows from c α ∼ f α , and the density of f α fermions obtained from (1.4) and (2.5).
As we approach the critical point, with p p c , we expect E c and b to both vanish algebraically.
However, we do not expect the large M theory of Ref. 24 to properly describe the approach to the critical point: in this large M theory, we obtain an insulating state as b vanishes, whereas our p = p c critical point is metallic. Indeed, as b is gauge-charged field, the value of b is not a gauge-invariant quantity which can be directly compared between different approaches. However, the crossover scale E c is better defined, and we can deduce the behavior of E c near p = p c by the RG analysis of Section III. We expect where λ s is the relevant RG eigenvalue with which s flows away from the F P 4 fixed point, specified in (D13).

B. Pseudogap region
For p < p c , we use the SU(2|1) theory, and condense the b α spinons to obtain spin glass order, as described in Refs. [27,28]. The presence of the mobile f fermions will make this a metallic spin glass with carrier density p, as determined by (2.7).
We need to extend the insulating spin glass theory of Refs. [27,28] to the metallic spin glass, and this will be studied in greater detail in future work. Here we note that a systematic description Right at p = p c , the critical theory is expected [28] to have a non-vanishing extensive entropy S 0 as T → 0. This follows from the similarity of the random insulating magnet [18], and many other models in the SYK class.
Away from p = p c , we expect that the entropy follows the behavior of the critical point at temperatures above the coherence scale E c , before vanishing lineary with T at temperatures below E c , as shown in Fig. 4. We can therefore estimate that the linear-in-T coefficient of the specific So we expect γ to diverge as |p − p c | −1/λs in the infinite range model H tJ . It is notable that this behavior resembles experimental observations [8].  ; a linear-in-T resistivity was also found in numerical studies [40,41] of lattice models without disorder described by equations closely related to those of the large M limit of Appendix C. And we note that there is a recent report of spin glass correlations in the pseudogap phase [13], extending earlier impurity-induced observations [11,12].

l a t e x i t s h a 1 _ b a s e 6 4 = " S 2 E m c x 5 f H z H c + C E L O u G A t 2 X Z T b A = " > A A A C G H i c d Z C x T h t B E I b 3 C A Q w S T B J S b P C i k S R n P b w w b m K U G h S E i k G J N u y 9 v b G 9 o q 9 v d X u H G A d f o w 0 F P A q d C g t H W + S M m v j S A G R X x r p 1 / w z m t G X G i U d M v Y Q L L x a X H q 9 v L J a W 3 v z 9 t 1 6 f e P 9 k S t K K 6 A t C l X Y k 5 Q 7 U F J D G y U q O D E W e J 4 q O E 5 P D 6 b 5 8 R l Y J w v 9 A 8 c G e j k f a j m Q g q N v d S 7 N Z 9 M X l / Q L Z f 1 6 g 4 V J 0 t z Z a 1 I W R i x J W s w b F s f x b o t G I Z u p Q e Y 6 7 N d / d 7 N C l D l o F I o 7 1 4 m Y w V 7 F L U q h Y F L r l g 4 M F 6 d 8 C B 1 v N c / B f c r O p H E z 2 6 t m 3 0 / o R x 9 m d F B Y X x r p r P v v c s V z 5 8 Z 5 6 i d z j i P 3 P J s 2 X 8 o 6 J Q 5 a v U p q U y J o 8 X h o U C q K B Z 2 i o J m 0 I F C N v e H C S v 8 2 F S N u u U A P r N Z 1 4 G n q I Y 6 q L s I F n s v M 3 6 l i q S c e 1 V 8 e 9 P / m a C e M m u H u 9 7 i x / 3 U O b Y V s k i 2 y T S K S k H 3 y j R y S N h G k I D / J N b k J r o L b 4 C 7 4 9 T i 6 E M x 3 P p A n C u 7 / A L F n o W
It is useful to compare the structure of H tJ in (1.1) with that of SYK lattice models [62][63][64][65].
The SYK models have a random 4-fermion interaction term, and a random 2-fermion hopping term of strength t, but no on-site fermion constraint. At the lowest energies, the 2-fermion term is always relevant and drives the system away from SYK criticality to a Fermi liquid state. In Consequently, we find p c = 1/3 at zeroth order (see Fig. 1). Finally, we comment on the extent to which a model with all-to-all randomness can be mapped to the cuprates. Randomness is present in the experimental systems, and also serves important simplifying purposes in our theoretical analysis. Moreover, certain approximations to models without randomness lead to closely related saddle point equations [23,[39][40][41]. Several of the broken symmetries do not exist in the random model, and subtle questions [74] about the structure of the Fermi surface in momentum space can be avoided. However, the central issues of carrier density, fractionalization, emergent gauge charges, and associated quantum phase transitions remain well defined even in the presence of randomness. Given the recent spin glass observations [13], and as we noted in Section IV B, it will be useful to study the metallic pseudogap state, and the interplay between the spin glass order and charge transport. A possible approach is the extend the large M theory of Refs. [27,28] to include fermionic holons, as well as numerical studies for M = 2.

Appendix C presents
The constraint (1.4) commutes with all operators of the superalgebra. Imposing this constraint yields the fundamental representation of SU(1|2).
Alternatively, we can use the operators in (1.5). These realize the SU(2|1) algebra, and it can be verified that these operators also obeys the superalgebra in Eq. (A1). The constraint projecting to the fundamental representation is now (1.6).

Larger symmetries
We consider the model for general M and M , with the electron operator operator where the matrices T a obey The operators c α , S a , the operator A general constraint fixing the representation is with P a positive integer, and our interest in the case P = M/2 which realizes the representation in which the SU(M ) subalgebra is self-conjugate. Note that the fundamental representation of the superalgebra is P = 1, but this does not lead to a convenient large M limit.
We can also consider a bosonic spinon and fermionic holon decomposition for general M , M The analogous steps will lead to a realization of the SU(M |M ) superalgebra, which is identical to the SU(M |M ) superalgebra. However, the constraint now leads to a different representation of the superalgebra from (A6) for P = 1 [75].

Operator expectation values
We will compute the only RG equations for the SU(M |M ) theory in Appendix B following the method in Appendix C of Ref. 47. After using the identity (A4), the computations in Appendix B can be reduced to the following operator traces.
First, let us compute the dimension, D(M, M , P ), of the superspin Hilbert space. To compute this, it is useful to compute the grand-canonical partition function, while ignoring the constraint (A6).
where z is the common fugacity. Then we can impose the constraint (A6), and the dimension of The values for M = 2, P = 1, and M = 1 case of interest to us are simple: This simplicity is the reason Section III was able to compute the RG equations using Feynman diagrams and the Abrikosov method. This appendix generalizes the method of Refs. [46,47] for SU (2) spins to superspins in SU(M |M ). This method utilizes only gauge-invariant information contained in the superspin algebra and its representation; thus the Berry phase S B (see (2.4) and (2.6)) of the supergroup [32] is exactly accounted for by the commutation and anti-commutation relations. The RG equations obtained here reduce to those of Section III at M = 2, P = 1, M = 1.

Some other random values
We consider here the Hamiltonian where c α and S a are defined in (A2)  The setup of the renormalization factors in the present perturbation theory is somewhat different from (3.2). We now write, using the operators defined in (A2) and (A3), The renormalization constants Z S and Z c are the same as those defined in (3.30), but we will now compute them in a different manner. The notation of our renormalization constants also differs from that in Ref. 48, and we provide a translation in Table I  interactions of the bosonic bath field φ, and hence we have Z φ = 1, as we noted in Section III C.
For the same reasons, it was argued in Refs. [46][47][48] that (in our notation) Z γ = 1 in the absence of bulk interactions. The reasoning extends also to the fermionic bath, and so we have Z γ = 1 and Z g = 1 exactly. These identities can also be understood from the statement below (3.30) that the vertex corrections Λ S,c arise from the same diagrams as Z γ,g . We will now compute Z S and Z c by renormalizing the two-point correlators of S a and c α , and this will sufficient to obtain the needed beta functions.     Also, Note we evaluate the above integrals at T = 0, by extending the integrals appropriately as explained in Ref. 47. Here, Similarly, From (B3) and (B4) we obtain, It is then straightforward to write, where , Note that for M = 2 , M = 1, we obtain L γ = L g = 2 which agrees with the result that can be obtained from (3.31) and the results in Section III.

Electron correlator
Next we evaluate the electron correlation, O 2 ≡ c(τ )c † (0) = N 2 /D. The diagrams contributing to the numerator are shown in Fig. 7, while those contributing to the denominator have been already evaluated in (B3). Thus we obtain, where, Thus we have, Similarly, it is the straightforward to write, where Note that for M = 2 , M = 1, we obtain P g = 3 and P γ = 3/4 which agrees with the result that can be obtained from (3.31) and the results in Section III.

RG flow
We are now in a position to write the beta functions for the coupling constants. Using (B2) we find two equations, We have used the exact relations Z g = Z γ = 1 in obtaining these equations. Solving these two equations and using the expressions for the renormalization factors found above we obtain the following one-loop beta functions, Recall that at M = 2, M = 1, we have P g = 3, P γ = 3/4, L g = 2, and L γ = 2. With this the above expressions match the one-loop beta functions derived earlier in Sec. III.
We can also calculate the anomalous dimension for the spin and electron operators, defined in (3.32). From (B47) and (B48) we obtain exactly the same equations derived before, i.e., (3.35) and (3.37). Thus at the non-trivial fixed point where γ * = 0, g * = 0 we obtain η S = and η c = 2r to all orders in andr.

Appendix C: Large M limit
In this appendix we consider the large M limit examined originally in the insulating spin model in Ref. 18. To extend the large M limit to the t-J model, we also need to endow the electron with an additional orbital index = 1 . . . M as in (A2), and take the large M limit at fixed using SU(M |M ) superspin formulation of Appendix A 1 while imposing the constraint (A6) at P = M/2. Similar large M limits were taken in particle-hole symmetric models in Refs. [42][43][44][45] and for a non-random t-J model in Refs. [40,41].
A sketch of our proposed large M phase diagram is shown in Fig. 8. This applies to the theory < l a t e x i t s h a 1 _ b a s e 6 4 = " v W 6 V 0 A e t X n V 6 s n p s g v n E K e U V M i g = " > A A A C J n i c d V D L S g M x F M 3 4 r O O r 6 t J N s A g u Z J j p w 8 e u 6 M a N U M G q 0 C k l k 7 l T g 5 n M k G Q K Z e h / + A l + h V t d u R N x I f g p p m M F F T 0 Q O J x z b p J 7 g p Q z p V 3 3 1 Z q a n p m d m y 8 t 2 I t L y y u r 5 b X 1 C 5 V k k k K b J j y R V w F R w J m A t m a a w 1 U q g c Q B h 8 v g 5 n j s X w 5 A K p a I c z 1 M o R u T v m A R o 0 Q b q V e u + g H 0 m c g p C A 1 y Z J + C J p w z 6 v t Y p U z g P i d K O b Y P I v z K 9 M o V 1 3 E b 9 V q 9 j g 2 p 7 d U O X U M a j a q 3 5 2 H P c Q t U 0 A S t X v n N D x O a x W a c j q / r e G 6 q u z m R m l E O I 9 v P F K S E 3 p A + d A w V J A a 1 G w 5 Y q g r a z Y s 1 R 3 j b m C G O E m m O 0 L h Q v w / n J F Z q G A c m G R N 9 r X 5 7 Y / E v r 5 P p 6 K C b M 5 F m G g T 9 f C j K O N Y J H n e G Q y a B a j 4 0 h F D J z L c x v S a S U F O H s k 0 f X 0 v j / 8 l F 1 f H q T u O s W m k e T Z o p o U 2 0 h X a Q h / Z R E 5 2 g F m o j i m 7 R P X p A j 9 a d 9 W Q 9 W y + f 0 S l r M r O B f s B 6 / w C T y q a p < / l a t e x i t > hb ↵ i 6 = 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " V / N L 9 Z E E 2 0 c o L 3 8 n N X 5 T J 7 X 7 K C U = " > A A A C K X i c d V D B T t t A F F x T K D S F E t o j l x V R p R 5 Q Z B M 7 w C 2 i l x 5 T q Q G k O I q e N 8 / J K u u 1 2 X 1 G i q x 8 C Z / Q r + g V T t z a S p z 6 I 9 2 E V I K q H W m l 0 c x 7 e j u T F E p a 8 v 0 f 3 t q L 9 Y 2 X m 1 u v a q + 3 d 9 7 s 1 v f e n t u 8 N A J 7 I l e 5 u U z A o p I a e y R J 4 W V h E L J E 4 U U y / b j w L 6 7 R W J n r L z Q r c J D B W M t U C i A n D e t R r D C l W I E e K + R x B j R J D U y r Z D 6 M Q R U T 4 L G R 4 w n F Z j W h 8 Y r 7 w 3 r D b / p R 2 A p D 7 k i r 3 T r 1 H Y m i o 6 A d 8 K D p L 9 F g K 3 S H 9 Y d 4 l I s y Q 0 1 C g b X 9 w

r M = " > A A A B + n i c d V D L T g I x F O 3 g C / E F u n T T C E b c T D q 8 l 0 Q 3 b k g w O k A C h H R K g Y b O I 2 1 H Q w Y + x Y 0 L j X H r l 7 j z b y w P E z V 6 k p u c n H N v 7 r 3 H C T i T C q E P I 7 a 2 v r G 5 F d 9 O 7 O z u 7 R 8 k U 4 c N 6 Y e C U J v 4 3 B c t B 0 v K m U d t x R S n r U B Q 7 D q c N p 3 x 5 d x v 3 l E h m e / d q k l
A u y 4 e e m z A C F Z a 6 i V T N 3 Y 2 U z u b 1 j L n U I 2 o L y a 9 Z B q Z V q W U L 5 Q g M o s W Q u W 8 J r k c q l h l a J l o g T R Y o d 5 L v n f 6 P g l d 6 i n C s Z R t C w W q G 2 G h G O F 0 l u i E k g a Y j P G Q t j X 1 s E t l N 1 q c P o O n W u n D g S 9 0 e Q o u 1 O 8 T E X a l n L i O 7 n S x G s n f 3 l z 8 y 2 u H a l D p R s w L Q k U 9 s l w 0 C D l U P p z n A P t M U K L 4 R B N M B N O 3 Q j L C A h O l 0 0 r o E L 4 + h f + T R s 6 0 C m b x O p e u X q z i i I N j c A K y w A J l U A V X o A 5 s Q M A 9 e A B P 4 N m Y G o / G i / G 6 b I 0 Z q 5 k j 8 A P G 2 y d l t J L U < / l a t e x i t > SU(M |M 0 ) theory < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 g 6 n r o G H R Q 5 G g V u 3 W G 0 p + 7 b H d J A = " > A A A B + n i c d V D L T g I x F O 3 g C / E F u n T T C E b c T D q 8 l 0 Q 3 b k g w O k A C h H R K g Y b O I 2 1 H Q w Y + x Y 0 L j X H r l 7 j z b y w P E z V 6 k p u c n H N v 7 r 3 H C T i T C q E P I 7 a 2 v r G 5 F d 9 O 7 O z u 7 R 8 k U 4 c N 6 Y e C U J v 4 3 B c t B 0 v K m U d t x R S n r U B Q 7 D q c N p 3 x 5 d x v 3 l E h m e / d q k l A u y 4 e e m z A C F Z a 6 i V T N 3 Y 2 U 5 v W z j L n U I 2 o L y a 9 Z B q Z V q W U L 5 Q g M o s W Q u W 8 J r k c q l h l a J l o g T R Y o d 5 L v n f 6 P g l d 6 i n C s Z R t C w W q G 2 G h G O F 0 l u i E k g a Y j P G Q t j X 1 s E t l N 1 q c P o O n W u n D g S 9 0 e Q o u 1 O 8 T E X a l n L i O 7 n S x G s n f 3 l z 8 y 2 u H a l D p R s w L Q k U 9 s l w 0 C D l U P p z n A P t   for details of a similar computation with particle-hole symmetry. In our case, both the boson and fermion Green's functions will have to be particle-hole asymmetric, as in Refs. [18,27,28,76].
We will perform an analytic low energy analysis of the large M equations, and find a critical solution which is in close correspondence with the RG fixed point F P 4 in Section III. However, the large M solution appears to be present for a range of dopings, and not at a critical doping as in the RG analysis. We expect that either constraints from the higher energy structure of the large M theory, or corrections higher order in 1/M , will convert the critical phase to a critical point.
It appears that the numerical studies of Haule et al. [40,41] examined the finite temperature behavior about the critical phase described here as their model of the pseudogap. This contrasts with our model of the pseudogap in Fig. 1 as a metallic spin glass flanking a critical point. We also note that the present large M limit, with fermionic spinons, cannot obtain a metallic spin glass; instead we have to use the bosonic spinon approach outlined in Section IV B.

Green's functions
We follow the condensed matter notation for Green's functions in which We drop indices α, , and all Green's functions are diagonal in these indices. It is useful to make ansatzes for the retarded Green's functions in the complex frequency plane, because then the constraints from the positivity of the spectral weight are clear. At the Matsubara frequencies, the Green's function is defined by So the bare Green's functions are The Green's functions are continued to all complex frequencies z via the spectral representation For fermions, the spectral density obeys for all real Ω and T , and for bosons the constraint is The retarded Green's function is G R (ω) = G(ω + iη) with η a positive infinitesimal, while the advanced Green's function is G A (ω) = G(ω − iη). It is also useful to tabulate the inverse Fourier transforms at T = 0 where C +R > 0 and C −R > 0, but they need not be equal. This corresponds to the ansatz in (2.8) which has C +R = C −R . We can allow these amplitudes to be distinct in the large M limit. We also examined generalization of the RG analysis in Section III to the case C +R = C −R ; we found that the perturbative RG then gave inconsistent renormalizations of the coupling g, and so C +R = C −R in the context of the andr expansion.
For the boson Green's function, we write at a complex frequency |z| J expressed in terms of the three real parameters, C b , ∆ b and θ b . The constraint (C9) becomes Using (C10) we obtain in τ space for |τ | 1/J Finally, we can write expressions similar to (C18) for Q(τ ) for |τ | 1/J (C22) where C Q > 0. This corresponds to the ansatz for Q(τ ) in (2.8) We can relate the parameters in the ansatzes for the bosonic bath Q(τ ) and the fermionic bath R(τ ) to the parameters in the ansatzes for the Green's functions G f and G b , by using the selfconsistency conditions (C13). This yields expressions forr, , C ±R , and C Q in terms of ∆ f,b , θ f,b , and C f,b :r However, in keeping with RG computation, we will defer application of the relations in (C23).
Now we see that the large M result (C34) is precisely the result (3.38) for the electron anomalous dimension obtained to all orders in the andr expansions.
Comparing the amplitudes of (C31), (C32) and (C33) we obtain The comparison of this with (C29), (C30) leads to two possible solutions, appearing as the two intermediate critical phases in Fig. 8.
The second J 2 terms in (C29) and (C30) are much smaller than the t 2 terms when ∆ f − /2 > ∆ b −r; using (C34), we obtain the condition ∆ f > /4. So the J 2 terms can be neglected. Indeed, the low energy solution is then entirely independent of the strength of the exchange interaction, which is rather different from the structure of the F P 4 fixed point in Section III with both g * and γ * non-zero. Instead, it is the F P 3 fixed point, with γ * = 0, which matches the structure of the present large M solution, and this fixed point was found to be unstable in the RG analysis for M = 2, M = 1. We will therefore only write down the saddle point equations here, and not consider this case further.
We can also apply the self-consistency relations in (C23) to the exponents, and obtainr = 1/2 Now the t 2 and J 2 terms in (C29) and (C30) are equally important, and we will see that the structure of this large M solution is very similar to that of the critical point found in the RG analysis in Section III.
In the large M limit, the spin correlator is given by and so the anomalous dimension of the spin operator is We now see that the spin anomalous dimension implied by the large M equations (C44) and (C47) is consistent with the result (3.36) obtained to all orders in the andr expansion.
Note that the values of ∆ f and ∆ b above, combined with (C36) and (C47) yield the self-consistent values in (3.39). For the last two equations in (C50), notice the bounds |θ f | < π/4 and π/4 < θ b < π/2 below (C24); so all the co-efficients on the left hand sides of (C50) are positive, and the last two equations determine the values of C f and C b . The values of θ f and θ b are then determined by the particle density p from (C24). So this low energy solution can exist at a variable particle density, and the present low energy M = ∞ theory describes a potential critical phase, rather than a critical point.
Finally, let us note the form of the electron Green's function from (C35) , for τ > 0 and T = 0 The exponent and signs of (C51) agree with the self-consistent electron Green's function obtained in (3.40) (recall (C16) and (C20)), but it appears that the magnitudes of the amplitudes in (C51) can be different between τ > 0 and τ < 0. This is a subtle feature of the large M theory which is not reproduced by the andr expansion in the body of the paper. This is related to the discussion below (C18).
Also note that the 1/τ decay of (C51) is similar to that of a Fermi liquid. Nevertheless, this Similarly, we have

Flow away from criticality
For the flow equation of s at one-loop, we will follow the momentum-shell RG procedure, where the cut-off D is kept explicitly. In this case, we introduce masses for bosons and fermions, but keeping in mind that only their difference is physically relevant. To this end we consider the Fourier-transformed action, where the self energies are evaluated as follows: with Σ F = Σ a + Σ b and Σ B = Σ c . The scaling factor is l = 1 + δD/D such that under the scaling k = lk and iω = liω. Thus we have, Thus we have the following expressions for the renormalized masses: Note that along with this the fermionic and bosonic operators, bosonic field and the coupling constants are also renormalized. For instance, f = l −1+g 2 /2+3γ 2 /8 f and b = l −1+g 2 b. In addition to the self-energy corrections there is also a vertex correction to γ at this order. However, this does not influence the mass renormalization and thus we can already proceed to calculate the flow equation for the mass. In our notation introduced earlier, We can compute the relevant eigenvalue associated with the flow of s at the fixed points of the beta functions, and find at the non-trivial fixed point F P 4 . At the self-consistent values, i.e., = 1 andr = 1/2 we have λ s = 1/6, although we cannot trust the result at such large values of andr. Similarly, at F P 1 , F P 2 , and F P 3 we find λ s to be 1, 1, and 1 − 2r respectively.
Within the momentum-shell RG, we get the same beta functions for g and γ, after considering the vertex correction as well. We can also calculate the particle densities (n f /b ). This can be done at any s. We will first make the following identification: s =s/β, such thats is small. This will facilitate us to do a q(λ) = q f (λ) + q b (λ), such that,

Particle density
So, we have, Note that we still satisfy the particle density constraint n f + n b = 1 exactly. It is also interesting to note that n b = 1/3 at zeroth order, which corresponds to p c = 1/3.

Two-loop self energy
We first evaluate the fermionic self energies to two-loop order. The relevant Feynman diagrams are shown in Fig. 10.
Σ f 10(c) (iν) = −3 16 We will now evaluate the bosonic self energies to two-loop order. The relevant Feynman diagrams are shown in Fig. 11.

Two-loop vertex corrections
Let us first evaluate the vertex correction to the fermionic bath coupling g 0 at two-loop level.
The relevant Feynman diagrams are shown in Fig. 12,.
We will now evaluate the two-loop vertex correction to the bosonic bath coupling γ 0 . The corresponding diagrams are shown in Fig. 13.
We get C sf = 0. Using the results in this appendix we obtain the renormalization factors and beta