From the pseudogap metal to the Fermi liquid using ancilla qubits

We propose a new parton theory of the hole-doped cuprates, describing the evolution from the pseudogap metal with small Fermi surfaces to the conventional Fermi liquid with a large Fermi surface. We introduce 2 ancilla qubits per square lattice site, and employ them to obtain a variational wavefunction of a fractionalized Fermi liquid for the pseudogap metal state. We propose a multi-layer Hamiltonion for the cuprates, with the electrons residing in the 'physical' layer, and the ancilla qubits in two 'hidden' layers: the hidden layers can be decoupled from the physical layer by a canonical transformation which leaves the hidden layers in a trivial gapped state. This Hamiltonian yields an emergent gauge theory which describes not only the fractionalized Fermi liquid, but also the conventional Fermi liquid, and possible exotic intermediate phases and critical points. The fractionalized Fermi liquid has hole pockets with quasiparticle weight which is large only on"Fermi arcs", and fermionic spinon excitations which carry charges of the emergent gauge fields.

The structure of the pseudogap metal state in the hole-doped cuprate superconductors has long been the focus of much theoretical and experimental attention [1]. In more recent experiments, it has become clear that the pseudogap state is present only for a hole doping p smaller than a critical value p c [2][3][4][5][6][7][8][9][10][11][12][13][14]. For p > p c , many observables indicate the presence of a conventional Fermi liquid (FL) state, with a large Fermi surface enclosing a volume associated with a hole density 1 + p. While there have been many theoretical proposals for the pseudogap metal, there is as yet no framework which can capture the essential physics of both the pseudogap metal and the conventional Fermi liquid as different mean-field solutions of the same theory. Such a framework is surely needed as a starting point for a theory of the mysterious strange metal found near p = p c .
We present such a framework here. We show that the introduction of 2 ancilla qubits per square lattice site leads to a valuable flexibity in describing possible correlated states of mobile electrons on the square lattice. It should be noted that the ancilla qubits are not physical degrees of freedom which can be directly observed; rather, they are theoretical tools which enable us to capture new varieties of entangled states of the electrons.
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Our approach with ancilla qubits is illustrated in Fig. 1. We are interested in the electrons, c iσ , in the physical layer, where i labels a square lattice site, and σ =↑, ↓ is the electron spin.
We use the ancilla qubits in the hidden layers to generate wavefunctions and field theories for observables in the physical layer. The spin operators S i;1 , S i;2 act on the qubits in the two layers.
It is convenient (but not required) to represent the spins by fermions f i;1σ , f i;2σ using where σ are the Pauli matrices, and there must be exactly one fermion on each hidden layer site We can now introduce our trial wavefunctions for the electrons in the physical layer. We propose a trial Slater determinant state |Slater[c, f 1 , f 2 ] for c, f 1 and f 2 fermions. Then we project out components of this wavefunction in which the ancilla qubits are locked in a spin singlet on each The wavefunction on the physical layer is states using an auxiliary band with total fermion density 2 − p have also been discussed earlier [20,32]). Concomitantly the f 2 layer realizes the needed fractionalized spinons of the FL* state.
In the FL state, we choose |Slater[c, f 1 , f 2 ] with decoupled the physical and hidden layers. Then the c layer, with fermion density 1 − p will have a large hole-like Fermi surface of volume 1 + p, and the hidden layers will form a trivial gapped insulator.
For analytic progress, we need the Hamiltonian of Fig. 1, from which we can derive gauge theories. We consider the Hamiltonian where H U is a Hubbard model of electrons c iσ on the sites i of a square lattice and H a describes 2 'hidden' layers of ancilla spin S = 1/2 qubits S i;1 , S i;2 (see Fig. 1) where H 1 represents exchange interactions between the first hidden layer qubits S i;1 (which we do not specify), and similarly for H 2 . When J 2 is large, the two ancilla layers will form a spin gap state; so we can safely 'integrate out' the ancilla qubits, and induce near-neighbor exchange interactions between the c iσ electrons on the physical layer. Alternatively stated, a canonical transformations decouples the physical and hidden layers, at the cost of additional exchange interactions in the physical layer. Rather than accounting for these exchange interactions explicitly, more progress is possible by keeping the hidden layers 'alive', and considering possible states and gauge theories in the expanded Hilbert space. In the end we can take the J 2 → ∞ limit (corresponding to projection onto the singlet hidden layer state |s in (1.3)), which reduces the above model to the standard Hubbard model for the cuprates. A similar approach using auxiliary degrees of freedom has been taken by Refs. 33 and 34 for a description of the bosonic composite Fermi liquid in the lowest Landau level.
The outline of the rest of the paper is follows. We present a gauge theory description of phases of H in Section II. This leads to a mean-field description of the FL* and FL phases.
The physical properties of the FL* phase, including its photoemission spectrum, are described in Section III. The critical region between the FL* and FL phases is discussed in Section IV, including a possible intermediate phase, or a direct transition. Some directions for future research are noted in Section V, and we summarize in Section VI.

II. GAUGE STRUCTURE AND MEAN FIELD THEORY
The spins in the hidden layers can be represented by the standard fermionic partons [1] in (1.1).
Naively we can just form mean field theories using c, f 1 , f 2 . However, this kind of analysis does not incorporate the large J 2 . For example, let us consider mean field ansatz for which c decouples from f 1 , f 2 ; then f 1 , f 2 can form gapless spin liquids. However, in the large J 2 region, the spin carried by f 1 and f 2 must be gapped. This is similar to a "Mott" gap in spin channel. In the familiar Hubbard model, we use slave boson theory to describe the Mott transition and incorporate the charge gap at large U . Here we can use a similar slave spin approach to incorporate the spin gap at large J 2 .
Therefore , we perform a further fractionalization of f 1 and f 1 in (1.1): where a = 1, 2 and the slave spin R is a SU (2) matrix, similar to that introduced in Ref. 27.
Basically this means the "spin" indexσ carried byf can be freely rotated by a SU (2) gauge transformation: where U i;S ∈ SU (2). Accordingly the slave spin transforms as R i → R i U † i;S . We label this gauge transformation as SU (2) S . See also Appendix A for further discussion on the origin of this expanded gauge structure.
It is well known [1] that the parton representation for each layer in (1.1) has another SU (2) gauge transformation in the particle-hole channel: where a = 1, 2 and U i;a ∈ SU (2). The U i;1 and U i;2 are two independent gauge transformations for f 1 andf 2 respectively, which commute with U i,S [35]. The slave spin R i remains unchanged under both U i;1 and U i;2 . We label these two gauge transformations as SU (2) 1 and SU (2) 2 respectively.
Here we need to mod out the transformation:f i;aσ → −f i;aσ . The combined transformations in The fermionsf 1 ,f 2 are neutral under both physical charge and spin probes because they couple to the (SU (2) 1 × SU (2) 2 × SU (2) S )/Z 2 gauge fields. If the boson R is gapped, and there is no further Higgs term, in the strong gauge field coupling limitf 1 ,f 2 will be confined to form on-site spin singlets. A more interesting possibility can be obtained by a mean field ansatz in which we Higgs the gauge field by couplingf a;σ to the physical electron c: we will explore such an ansatz in the remainder of the paper.

B. Mean field theory
Let We define ρ a and µ a as Pauli matrices acting on the spin and particle-hole channels respectively.
Then the SU (2) S gauge transformation U i;S is generated by ρ a , and acts on both Ψ 1 and Ψ 2 .
The SU (2) 1 gauge transformation U i;1 is generated by µ a , and acts only on Ψ 1 . Similarly, the SU (2) 2 gauge transformation U i;2 is generated by µ a and acts only on Ψ 2 . In this basis, the gauge transformations U i;1 , U i;2 and U i;S are 4 × 4 matrices.
After condensation of approriate Higgs fields, we obtain our proposed mean field theory: Note that the U i;a commute with the U i;S [35], and so their ordering is unimportant.
We also have a global U (2) C symmetry corresponding to charge conservation and spin rotation.
In the basis of C, the U (1) part is generated by µ z while the SU (2) spin rotation is generated remain unchanged. Hence B i;1 carries both physical charge-spin and gauge charges.
The mean-field characterizations of the phases are where Φ and Φ are real numbers. Here we chose a specific gauge. Equivalent ansatzes can be obtained through gauge transformations. In the FL phase, B 2 = 0 does not need to be put by hand; once B 1 = 0 and R i = 0 , gauge fluctuation can confine Ψ 1 and Ψ 2 automatically, which is equivalent to the effect of non-zero B 2 .
The condensate B 1 = Φρ 0 ⊗ µ z higgses the (SU (2) 1 × SU (2) 2 × SU (2) S )/Z 2 gauge fields (the hopping terms in H 1 and H 2 also Higgs parts of the gauge symmetry even with B 1 = 0). In a more precise language, it locks the SU (2) C ⊂ U (2) C background field to the internal gauge fields corresponding to SU (2) S . Thus the spin indexσ carried byf 1σ andf 2σ can now be identified as a physical spin index. The internal U (1) gauge field generated by µ z in SU (2) 1 is locked to the physical electromagnetic field; after the condensation of B 1 ,f 1 can be viewed as electron andf 2 can be identified as a spinon. The fermionf 2 is still charge neutral because SU (2) 2 is not locked to the physical background field.

III. PROPERTIES OF THE PSEUDOGAP METAL
In this section we provide details of the FL* mean field ansatz for the pseudogap metal in the underdoped region and discuss its properties.
When B 1 = Φρ 0 ⊗ µ z , B 2 = 0, we have the following mean field theory The Hamiltonian H c,f 1 describes the electron Fermi surface while Hf 2 describes the phase of the spinonf 2 . Small Fermi surfaces with Luttinger volume A F S = p/2 can be obtained by where µ 1 is added to fix nf 1 = σ f † i;1σf i;1σ = 1, and µ c fixes n c = 1 − p. For the hopping parameters t c;ij , we use t = 1, t = −0.22, t = 0.19 which can reproduce the shape of the Fermi surface in the overdoped regime. The hopping parameters forf 1 should be determined by minimizing the energy for the wavefunction in (1.4). Alternatively, we can also view them as phenomenological parameters and fit them from experimental data. Here we choose t 1 = 1, t 1 = −0.1, t 1 = 0.1 for the purpose of illustration. In practice, the hopping parameters t 1;ij can also have dependences on doping level p. Later we will provide an intuitive explanation why the hoppings off 1 have the opposite sign to that of c.
For the spinonf 2 , we use the familiar d wave pairing ansatz [1]: where, ∆x = −∆ŷ = ∆. This ansatz is equivalent to the staggered flux ansatz, and the spinonf 2 is in a U (1) Dirac spin liquid phase.
The gap at the anti-node is opened by Φ = 0. We choose to use Φ(p) = 0.25 √ 0.23 − p.
Then we can calculate spectral densities A c (ω, k) = (1/π)Im c † (ω, k)c(ω, k) and Af 1 (ω, k) = (1/π)Im f † 1 (ω, k)f 1 (ω, k) . We show plots of the calculated spectral densities in Fig. 2. Between p l ≈ 0.19 and p c = 0.23, the anti-node is not gapped even if Φ = 0. Instead, there is one hole Fermi surface dominated by c and one electron pocket dominated byf 1 . The total Hall number is close to p and this region still belongs to the pseudogap phase. When we decrease p below p l , there is a Lifshitz transition and the Fermi surfaces are reconstructed to four small hole pockets centering at node K N = (± π 2 , ± π 2 ). For each pocket, one side is dominated byf 1 and thus has almost vanishing spectral weight in terms of c. As a result, only Fermi arcs are visible in an ARPES measurement.
Note also the similarity of Fig. 2 to the STM observations in Refs. 4 and 5.
In Fig. 3 we show A c (ω, k = (π, k y )). We define anti-node to be at K AN = (π, ±δ) which separates the n c (k) = 1 and n c (k) = 0 regions along the cut of k x = π. Then at k = K AN , A c (ω, K AN ) has a peak at ω = −∆. Here the sharp quasi particle peak at ω = −∆ is an artifact of the mean field calculation which ignores gauge fluctuation. Even inside the pseudogap phase, the mass of the higgsed gauge fields is at order of Φ ∼ ∆. For high energy region |ω| > ∆, gauge fluctuations can not be ignored and may completely destroy the quasi-particle peak at anti-node. In contrast, around the node, the Fermi arc is at zero energy and the gauge fluctuations do not have strong influences on spectral function here.
We define ∆ as the gap of the anti-node. The dependence of ∆ and Φ on the doping level is shown in Fig. 4. The doping p c is the true quantum critical point QCP at the end of the pseudogap phase. For our parameters, the anti-node gap ∆ onsets at a smaller doping p l through a Lifshitz transition. In slightly overdoped region p c < p < p d , there may be ghost Fermi surfaces decoupled from the Fermi liquid, which get confined and disappear at a larger doping p d .

FIG. 3. Spectral function
A c (ω, k = (π, k y )) calculated from mean field theory at various doping levels.
The doping is (a) p = 0.15; (b) p = 0.19 and (c) p = 0.21. In ARPES only the ω < 0 region can be measured at low temperature. Because of the gauge fluctuations at high energy, the true spectral function may be quite different and no sharp quasi-particle peak exists.

A. Physical meaning of auxiliary fermions
At the J 2 → ∞ limit, the wavefunction of (1.4) can be written as Slater[f 2 ] is the ground state of Hf 2 in (3.3). The above wavefunction is a state purely in the Hilbert space of the physical layer. Thusf 1 and f 2 should correspond to physical degrees of freedom. A natural question is: what is the physical meaning of these auxiliary fermions inside the physical Hilbert space? To gain intuition, let us look at the zero doping case first. In this case, Φ can gap out both c andf 1 and we have a Mott insulator. Therefore we should interpret Φ as Mott gap. The Φ condensate binds electron c † to holef 1 , and so we should viewf 1 as creation operator for a "correlation hole". This correlation hole may be quite nonlocal and should not be confused with on-site hole operator c i . In a certain sense, the physics is similar to that of fractional quantum Hall effect (FQHE). In the FQHE system, because of Coulomb interaction, electron also binds with its correlation hole (also called a vortex) and only the the bound state (a composite fermion) can move coherently. In our case, the Hubbard U also favors the binding between electron and correlation hole, which causes Mott localization.
Unlike the FQHE, here in (3.2) we use a BCS description of the exciton binding instead of viewing the exciton as the fundamental particle. The exciton binding means that the exciton c †f 1 moves coherently, and thus the hopping off 1 should be similar to that of c † . Thus it is natural that the hopping off 1 has the opposite sign to that of c. The critical theory is The Lagrangian L C is the action for the fermi liquid theory of physical electron. L Ψ 2 ,a 2 ,α is the action for the spinon coupled to (U (1) 2 × SU (2) S )/Z 2 gauge field. For our Dirac spin liquid ansatz, it has N = 2 Dirac fermions in the fundamental representation coupled to the U (2) gauge field.
The theory for the critical boson B is which is a non-linear sigma model coupled to U (2) gauge field. We used Einstein summation for α, β, β =↑, ↓.
The fermion Ψ 1 forms a Fermi surface and couples to both a 1 and α: The quantum critical point is tuned by the mass term m for the critical boson. At m < 0, B = Φρ 0 and this is a FL* phase. When m > 0,Ψ 1 andΨ 2 decouple from the physical electron.
In this caseΨ 1 forms ghost a Fermi surface which couples to neither charge nor spin probes.Ψ 1 couples to a U (1) and a SU (2) gauge field. It is known that U (1) gauge field suppresses pairing, while SU (2) gauge field induces pairing [36]. A recent experimental measurement of specific heat indeed found that γ = C/T close to critical region is almost four times larger that that of the overdoped FL [9]. An independent measurement of the effective mass is needed to subtract the contribution from the Fermi liquid part and test the existence of ghost Fermi surface.

V. FUTURE DIRECTIONS
There are several directions to generalize our theory. In this paper we restricted our analysis to the symmetric FL* phase, but it is easy to incorporate symmetry breaking orders. For example, we can let the ansatz off 1 have nematic order or loop current order. In this case, the nematic order or loop current order will onset when Φ = 0, and coincides with the onset of the pseudogap phase. However, the symmetry breaking order is just a byproduct of the pseudogap phase and does not play any essential role. Alternatively,f 2 can be put in an ansatz with antiferromagnetic order.
Thus the theory describes evolution from an antiferromagnetic metal with small Fermi surfaces towards the Fermi liquid phase with large Fermi surface. In contrast to the standard Hertz-Millis theory of antiferromagnetic critical point, this theory allows a jump of carrier density across the critical point and may be relevant for the quantum critical point in heavy fermion systems.
Our framework can also be easily generalized to SU (N ) Hubbard model with any N . At integer filling n c , there is a Mott insulator with n c electrons per site. We also introduce an auxiliary fermioñ f 1 at density n f 1 = N − n c and an auxiliary fermionf 2 with density nf 2 = n c . The fermionsf 1 and f 2 can again form trivial SU (N ) singlet per site, and there is a U (1) 1 × U (1) 2 × SU (N ) S gauge structure. Then upon doping at filling n = n c − x, we can have a coupling like −Φc † αf 1α , which can lead to small Hall number η H = −x. The fermionf 2 can be viewed as a spinon, and be put in spin liquid phase or ordered phase. This implies that pseudogap metal is a quite universal phenomena upon doping a generic Mott insulator. Recently an approximate SU (4) Hubbard model is shown to be realized in graphene moiré superlattice [37,38] and thus this generalized theory may be relevant there.

VI. SUMMARY
We have proposed a new framework for describing the pseudogap phase obtained from doping a Mott insulator, which can also be extended towards understanding its evolution towards the conventional Fermi liquid at larger doping. We showed that the use of ancilla qubits allows us to address the complete doping evolution in a single mean-field framework, which has not been possible in previous work. We applied our theory to the hole doped cuprates. At small doping, we provide a simple parton mean field theory of the fractionalized Fermi liquid (FL*) with small Fermi surfaces and reproduce the "Fermi arc" in ARPES experiments. We also provide a critical theory at the end of the pseudogap phase across which the carrier density jumps from p to 1 + p.
Our theory finds a "ghost" Fermi surface close to the optimal doping which should significantly enhance the density of states.
Finally, we note the relationship of the present gauge theory of optimal doping criticality to a recent SU (2) gauge theory [26,39] of the same regime. The common features are a SU (2) S gauge field and a large Fermi surface of gauge-neutral electrons c σ . The differences are that the other theory [26,39] has (i ) multiple Higgs fields that transform in the adjoint of SU (2) S , (ii ) the Higgs fields transform non-trivially under the space group of the square lattice, (iii ) bosonic spinon excitations which remain gapped across the transition. In contrast, our present theory has (i ) a single Higgs field that transforms as a SU (2) S fundamental , and also under a separate emergent U (1) gauge field, (ii ) the Higgs fields is also a fundamental of the global spin rotation SU (2) and the electromagnetic U (1), (iii ) there are gapless "ghost" fermionic excitations which carry neither spin nor charge, but which carry the charges of the SU (2) S gauge field, the emergent U (1) gauge field, and another SU (2) 2 gauge field. For our present theory, it is possible to take a linear combination of the emergent and electromagnetic U (1)'s and transfer the electromagnetic charge from the Higgs field to a ghost fermion [40].
We also have the Hermitian conjugates of the constraints in the second line, and so (A1) contains a total of 6 real constraints on each lattice site i. In the limit J 2 → ∞, the spins in the hidden layers are projected onto singlets on each site, and so we have 3 additional constraints after replacing the f σ in (1.1) byfσ. We note here that these 9 constraints per site correspond precisely to the 9 generators of the (SU (2) 1 × SU (2) 2 × SU (2) S )/Z 2 gauge symmetry.
As discussed in Ref. 1 (see also Ref. 41), the constraints in (A1) can be expressed as the vanishing of the Nambu pseudospin operator for each i and a = 1, 2. This pseudospin operator transforms as an adjoint under SU (2) 1 × SU (2) 2 , and so its vanishing is maintained under these transformations.
The expressions in (A1) are explicitly spin rotation invariant, and so are also invariant under the SU (2) S transformation in (2.2).
Another feature of the SU(2) gauge transformation in Ref. 1 is that it leaves the spin operator invariant. So (A2) is invariant under SU (2) 1 × SU (2) 2 . Finally, we note that the SU (2) S spin rotation in (2.2) performs an adjoint rotation of (A2) in spin space, and so the vanishing of the total spin per site is also obeyed after the SU (2) S transformation.