The novel metallic states of the cuprates: Fermi liquids with topological order, and strange metals
PPTEP (2016) 12C102; arXiv:1605.03579
Progress of Theoretical and Experimental Physics
The novel metallic states of the cuprates:Fermi liquids with topological order, and strange metals
Subir Sachdev
1, 2 and Debanjan Chowdhury Department of Physics, Harvard University, Cambridge MA 02138, USA Perimeter Institute for Theoretical Physics,Waterloo, Ontario, Canada N2L 2Y5 (Dated: May 11, 2016)
Abstract
This article is based on a talk by S.S. at the Nambu Memorial Symposium at the University ofChicago. We review ideas on the nature of the metallic states of the hole-doped cuprate high temperaturesuperconductors, with an emphasis on the connections between the Luttinger theorem for the size of theFermi surface, topological quantum field theories (TQFTs), and critical theories involving changes in thesize of the Fermi surface. We begin with the derivation of the Luttinger theorem for a Fermi liquid, usingmomentum balance during a process of flux-insertion in a lattice electronic model with toroidal boundaryconditions. We then review the TQFT of the Z spin liquid, and demonstrate its compatibility with thetoroidal momentum balance argument. This discussion leads naturally to a simple construction of Fermiliquid-like states with topological order: the fractionalized Fermi liquid (FL*) and the algebraic chargeliquid (ACL). We present arguments for a description of the pseudogap metal of the cuprates using Z -FL* or Z -ACL states with Ising-nematic order. These pseudogap metal states are also describedas Higgs phases of a SU(2) gauge theory. The Higgs field represents local antiferromagnetism, but theHiggs-condensed phase does not have long-range antiferromagnetic order: the magnitude of the Higgsfield determines the pseudogap, the reconstruction of the Fermi surface, and the Ising-nematic order.Finally, we discuss the route to the large Fermi surface Fermi liquid via the critical point where the Higgscondensate and Ising nematic order vanish, and the application of Higgs criticality to the strange metal. a r X i v : . [ c ond - m a t . s t r- e l ] N ov . INTRODUCTION Nambu’s early papers [1–3] laid down the close connection between fundamental questions insuperconductivity and high energy physics. These connections have continued to flourish to thepresent day, to the mutual benefit of both fields. In Ref. [1], Nambu clarified the manner in whichgauge-invariance was maintained in the BCS theory of the Meissner effect of superconductivity,and this paved the way for the proposal of the Higgs-Anderson mechanism. The subsequent papers[2, 3] treated the BCS theory in a slightly different manner: it was viewed as a theory with a global
U(1) symmetry, rather than with the U(1) gauge invariance of Maxwell electromagnetism. Thebreaking of the global U(1) symmetry led to the appearance of Nambu-Goldstone bosons, and thisinspired ideas on chiral symmetry breaking in nuclear physics. These global and gauge perspectiveson electromagnetism turn out to be closely related because the electromagnetic theory is weaklycoupled, but it is important to keep the distinction in mind.In the present article, in the hopes of continuing the tradition pioneered by Nambu, we willreview recently discussed connections between the high temperature superconductors and gaugetheories. The gauge theories will all involve strongly-coupled emergent gauge fields, while theU(1) gauge invariance of electromagnetism will be treated as a global symmetry. In this context,the emergent gauge fields do not reflect any underlying symmetry of the Hamiltonian, but area manifestation of the long-range quantum entanglement of the states under consideration. Wewill illustrate how emergent gauge fields are powerful tools for deducing the physical properties ofentangled many-body quantum states, and for connecting theories to experimental observations.In Fig. 1a, we show the quasi-two dimensional layers of CuO . For the purposes of this article,we can regard the O p orbitals as filled with pairs of electrons and inert. Only one of the Cu orbitalsis active, and in a parent insulating compound, this orbital has a density of exactly one electronper site. The rest of this article will consider the physical properties of this Cu orbital residingon the vertices of a square lattice. Fig. 1b shows a schematic phase diagram of the hole-dopedcopper oxide superconductors. The AF state in Fig. 1b is the antiferromagnet shown in Fig. 2a,in which there is one electron on each Cu orbital, and their spins are polarized in a checkerboardpattern. This state is referred to as a Mott insulator, because it is primarily the Coulomb repulsionwhich prevents the electrons from becoming mobile. This AF insulator should be contrasted fromthe band insulator with 2 electrons per Cu site, which is shown in Fig. 2b; the latter state is aninsulator even for non-interacting electrons because all electron motion is impeded by the Pauliexclusion principle.The rich phases of the cuprates appear when we remove a density of p electrons from the AFstate, as illustrated in Fig. 3a. It is important to note that relative to the band insulator in Fig. 2b,the state in Fig. 3a has a density of 1 + p holes. So if we described the ground state at this density2 M FL
CuO plane CuO (a) (b)
FIG. 1. (a) The square lattice of Cu and O atoms found in every copper-based high temperature su-perconductor. (b) A schematic phase diagram of the YBCO superconductors as a function of the holedensity p and the temperature T adapted from Ref. [4]. The phases are discussed in the text: AF–insulating antiferromagnet, PG–pseudogap, DW–density wave, dSC– d -wave superconductor, SM–strangemetal, FL–Fermi liquid. The critical temperature for superconductivity is T c , and T ∗ is the boundary ofthe pseudogap regime. by adiabatic continuity from a free electron ground state, the Luttinger theorem states that weshould obtain a metal with a Fermi surface of size equivalent to 1 + p holes. This turns out to beprecisely the case in the larger p region labeled FL (for Fermi liquid) in Fig. 1b. The corresponding‘large’ Fermi surface observed in photoemission experiments is shown in Fig. 3b.The focus of this article will be on the metallic phases in Fig. 1b, labeled by PG, SM, and FL. Ofthese, only the FL appears to be well understood as a conventional Fermi liquid. The traditionalproof of the Luttinger theorem is given in terms of conventional diagrammatic and Ward identityarguments. However, it was argued more recently by Oshikawa [6] that the Luttinger theorem hasa topological character, and a proof can be given using a momentum balance argument that followsthe many-electron wavefunction on a torus geometry in the presence of a flux penetrating one ofthe cycles of the torus. We will review this argument in Section II. The subsequent Section IIIwill turn to spin liquid states of the insulator at p = 0: these states are described at low energiesby a topological quantum field theory (TQFT). We will describe key characteristics of the TQFTwhich enable the spin liquid to also satisfy the momentum balance constraints of Section II.We will describe a model for the pseudogap (PG) metal as a Z -FL* state (and the related3 F (a) (b) FIG. 2. (a) The insulating AF state at hole density p = 0. (b) The band insulator with 2 electrons persite. (a) (b) FIG. 3. (a) State obtained after removing electrons with density p from the AF state in Fig. 2a. Relativeto the fully-filled state with 2 electrons per site in Fig. 2b, this state has a density of holes equal to 1 + p .(b) Photoemission results from Ref. [5] showing a Fermi surface of size 1 + p in the FL region of Fig. 1b.This is the Fermi surface size expected by the Luttinger theorem for a Fermi liquid without AF order orother broken symmetry. Z -ACL state) in Section IV, along with its connections to recent experimental observations. Thestrange metal (SM) appears to be a metal without quasiparticle excitations, and we will discusscandidate critical field theories for such a state in Section V.A small part of the discussion in Sections III and IV overlaps with a separate, less technical,recent article by one of us [7]. 4 xy FIG. 4. Torus geometry with a flux quantum inserted.
II. MOMENTUM BALANCE ON THE TORUS AND THE LUTTINGER THEOREM
Consider an arbitrary quantum system, of bosons or fermions, defined on (say) a square latticeof unit lattice spacing, and placed on a torus. The size of the lattice is L x × L y , and we imposeperiodic boundary conditions. Assume the system has a global U(1) symmetry, and all the localoperators carry integer U(1) charges. Pick an eigenstate of the Hamiltonian (usually the groundstate) | G (cid:105) . Because of the translational symmetry, this state will obeyˆ T x | G (cid:105) = e iP x | G (cid:105) , (2.1)where ˆ T x is the translational operator by one lattice spacing along the x direction, and P x is themomentum of the state | G (cid:105) . Note that P x is only defined modulo 2 π . The state | G (cid:105) will also havedefinite total U(1) charge, which we denote by the integer N .Now we gauge the global U(1) symmetry, and insert one flux quantum (with flux 2 π ) through oneof the cycles of the torus (see Fig. 4). After the flux insertion, the Hamiltonian is gauge equivalentto the Hamiltonian without the flux. So we gauge transform to the original Hamiltonian; the newstate of the system, | G (cid:48) (cid:105) will not, in general, be the same as the original state | G (cid:105) . Indeed, itsmomentum P (cid:48) x will differ from P x by ∆ P x with∆ P x = 2 πL x N (mod 2 π ) . (2.2)A general proof of (2.2) can be found in Refs. [6, 8, 9]. But we can easily deduce the result by firstconsidering the case of non-interacting particles. Then, an elementary argument shows that eachparticle picks up momentum 2 π/L x from the flux insertion, and so (2.2) is clearly valid. Now turnon the interactions: these cannot change the total momentum, which is conserved (modulo 2 π )both by the interactions and the flux insertion; so (2.2) applies also in the presence of interactions.So far, we have been quite general, and not specified anything about the many-body system,apart from its translational invariance and global U(1) symmetry. In the subsequent discussion,we will make further assumptions about the nature of the ground state and low-lying excitations,5nd compute ∆ P by other methods. Equating such a result to (2.2) will then lead to importantconstraints on the allowed structure of the many-body ground state.In the present section, following Oshikawa [6], we assume the ground state is a Fermi liquid.So its only low-lying excitations are fermionic quasiparticles around the Fermi surface. For oursubsequent discussion, it is important to also include the electron spin index, α = ↑ , ↓ , and so we willhave a Fermi liquid with 2 global U(1) symmetries, associated respectively with the conservationof electron number and the z -component of the total spin, S z . Consequently, there will be twoLuttinger theorems, one for each global U(1) symmetry. The action for the fermionic quasiparticles, c k α , with dispersion ε ( k ) is S F L = (cid:90) dτ (cid:90) d k π (cid:88) α = ± c † k α (cid:18) ∂∂τ − i αA sτ − iA eτ + ε ( k − α A s / − A e ) (cid:19) c k α , (2.3)where τ is imaginary time. The Fermi surface is defined by ε ( k ) = 0, and S F L only applies for k near the Fermi surface, although we have (for notational convenience) written it in terms of anintegral over all k . We have also coupled the quasiparticles to 2 probe gauge fields A eµ = ( A eτ , A e )and A sµ = ( A sτ , A s ) which couple to the 2 conserved U(1) currents associated, respectively, withthe conservation of electron number and S z .We place the Fermi liquid on a torus, and insert a 2 π flux of a gauge field that couples onlyto the up-spin electrons. So we choose A sµ = 2 A eµ ≡ A µ . Then the general momentum balance in(2.2) requires that ∆ P x = 2 πL x N ↑ (mod 2 π ) = 2 πL x N π ) , (2.4)where we assume equal numbers of up and down spin electrons N ↑ = N ↓ = N/
2. Now let usdetermine ∆ P x by using the description of the quasiparticles described by S F L . As illustratedin Fig. 5, each quasiparticle near the Fermi surface will behave like a free fermion, and have itsmomentum shifted by δp x = 2 π/L x . We add up the contributions of all the quasiparticles byintegrating in the vicinity of the Fermi surface. After using the divergence theorem, or pictoriallyby the sketch in Fig. 5, we can convert the integral to a volume integral inside the Fermi surface[6, 9], and so show ∆ P x = 2 πL x (cid:18) L x L y V F S π (cid:19) (mod 2 π ) , (2.5)where V F S is the momentum space area enclosed by the Fermi surface; the factor within thebrackets on the right-hand-side equals the number of momentum space points inside the Fermisurface. Note that the entire contribution to the right-hand-side of (2.5) comes from the vicinity ofthe Fermi surface where the quasiparticles are well-defined; we have merely used a mathematicalidentity to convert the result to the volume, and we are not assuming the existence of quasiparticlesfar from the Fermi surface. 6 p FIG. 5. Response of a Fermi liquid to flux insertion. Each quasiparticle near the Fermi surface acquiresa momentum shift δ p = ( δp x , Now we use (2.4) and (2.5), along with the corresponding expressions for flux inserted in theother cycle of the torus, to deduce the Luttinger theorem. The complete argument requires carefulattention to the (mod 2 π ) factors using situations where L x and L y are mutually prime integers[6, 9]. But ultimately, naively equating (2.4) and (2.5) gives the correct result V F S π = NL x L y (mod 2) = (1 + p ) (mod 2) . (2.6)In the final step, we have applied the Luttinger theorem to the holes in the cuprates, with adensity of holes of (1 + p ) relative to the filled band insulator in Fig. 2b. The expression (2.6) isexperimentally verified in the FL region in Fig. 3b. III. TOPOLOGICAL QUANTUM FIELD THEORY OF THE Z SPIN LIQUID
We now return to the insulator at p = 0. In Fig. 1b, the insulator breaks translational and spinrotation symmetries in the AF state shown in Fig. 2a. However, as AF order disappears at rathersmall values of p , it is useful to begin the analysis of doped states by examining insulating states at p = 0 which preserve both translation and spin rotation symmetries. An example of such a state isthe ‘resonating valence bond’ (RVB) insulator [10–13], illustrated in Fig. 6a. A trial wavefunctionfor the RVB state takes the form | Ψ (cid:105) = (cid:88) i c i | D i (cid:105) (3.1)7 (b) | D i i = ( |" ) / p | D i i X (a) (b)(c) (d) | D i i FIG. 6. (a) Illustration of a component, | D i (cid:105) , of the RVB wavefunction in (3.1). (b) A pair of S = 1 / Z spin liquid. In terms of (3.1), the co-efficients c i are modified so that each singlet bond crossing the ‘branch-cut’ (dashed line) picks up a factor of − Q ij and P ij non-zero onlybetween nearest-neighbor sites: the wavy lines indicate the Q ij and P ij with a change in their sign in thepresence of a vison. where i extends over all possible pairings of electrons on nearby sites, and a state | D i (cid:105) associatedwith one such pairing is shown in Fig. 6a. Note that the electrons in a valence bond need not benearest-neighbors. Each | D i (cid:105) is a spin singlet, and so spin rotation invariance is preserved. Wealso assume that the c i respect the translational and other symmetries of the square lattice.A theory for a stable RVB state with time-reversal symmetry and a gap to all excitations firstappeared in Refs. [14–16], which described a state now called a Z spin liquid. It is helpful todescribe the structure of the Z spin liquid in terms of a mean-field ansatz. We write the spinoperators on each site, S i(cid:96) ( (cid:96) = x, y, z ), in terms of Schwinger bosons b iα ( α = ↑ , ↓ ) S i(cid:96) = 12 b † iα σ (cid:96)αβ b iβ , (3.2)8here σ (cid:96) are the Pauli matrices, and the bosons obey the local constraint (cid:88) α b † iα b iα = 2 S (3.3)on every site i . Here we are primarily interested in the case of spin S = 1 /
2, but it is useful to alsoconsider the case of general S . Schwinger fermions can also be used instead, but the description ofthe S > / Z spin liquid is described by an effectiveSchwinger boson Hamiltonian [14, 17] H b = − (cid:88) i 2; more specifi-cally, the spinon is the Bogoliubov quasiparticle obtained by diagonalizing H b in terms of canonicalbosons.( ii ) The second quasiparticle, the ‘vison’, shown in Fig. 6c,d, is spinless and it has a more subtletopological character of a vortex in an Ising-like system (hence its name [20]). The vison state isthe ground state of a Hamiltonian, H vb , obtained from H by mapping Q ij → Q vij , P ij → P vij ; thenthe vison state | Ψ v (cid:105) has a wavefunction as in Eq. (3.5), but with f ij → f vij . Far from the centerof the vison, we have | Q vij | = | Q ij | , | P vij | = | P ij | , while closer to the center there are differences inthe magnitudes. However, the key difference is in the signs of the link variables, as illustrated inFig. 6c,d: there is a ‘branch-cut’ emerging from the vison core along which sgn( Q vij ) = − sgn( Q ij )9 X X X (a) (b)(c) (d) FIG. 7. Adiabatic motion of a vison (denoted by the X) around a single site of the triangular lattice(denoted by the filled circle). The initial state is in (a), and the final state is in (d), and these differ by agauge transformation under which b iα → − b iα only on the filled circle site. and sgn( P vij ) = − sgn( P ij ). This branch-cut ensures that the Z magnetic flux equals -1 on all loopswhich encircle the vison core, while other loops do not have non-trivial Z flux.The spinons and visons have two crucial topological properties.( i ) A spinon and a vison are mutual semions [21]. In other words, adiabatically moving a spinonaround a vison (or vice versa) yields a Berry phase of π . This is evident from the structure of thebranch-cut in Q vij and P vij : these Q vij and P vij are the hopping amplitudes for the spinon, and theyyield an additional phase of π (beyond those provided by P ij and Q ij ) every time a spinon crossesthe branch cut.( ii ) A less well-known and distinct property involves the motion of a single vison without anyspinons present: adiabatic motion of a vison around a single lattice site yields a Berry phase of2 πS [18–20]. This property is illustrated in Fig. 7, and see Ref. [22] for a complete computation.The initial and final states of the adiabatic motion differ by a Z gauge transformation, b iα → − b iα ,only on the site which has been encircled. From the projection operator P S in (3.5) we find thatthe wavefunction | Ψ (cid:105) has picked up a factor of ( − S , and this is the only contribution to agauge-invariant Berry phase.The background Berry phase of 2 πS per site for vison motion implies that there are two distincttypes of Z spin liquids [18–20, 23–25]. As was first pointed out in Refs. [18, 19], these are ‘odd Z spin liquids’, which are realized in the present model by half-integer S antiferromagnets, and ‘even10 spin liquids’, realized here by integer S antiferromagnets. In the Z gauge theory framework(or the related ‘toric code’ [26]), there is a unit Z electric charge on each lattice site of an odd- Z gauge theory. A. Topological quantum field theory All of the above properties of the Z spin liquid can be described elegantly using a topolog-ical quantum field theory (TQFT). The TQFT presentation also highlights the robustness andgenerality of the structure we have described above.The TQFT is obtained by implementing the mutual semion statistics between the spinon andthe vison using U(1) Chern-Simons gauge fields. We introduce two ‘emergent’ gauge fields, a µ and b µ . We couple the visons to a µ with unit charge. This implies that the branch-cut emanating fromthe vison in Fig. 6c, d is the Wilson line operatorexp (cid:18) i (cid:90) B dx i a i (cid:19) , (3.6)taken along the branch cut B . We couple the spinons to b µ , also with unit charge. We also notethat the external gauge field A sµ coupling to the z -component of the spin S z (see (2.3)) will alsocouple to the spinons which carry S z = ± / 2. Then standard methods [27] yield the followingaction for the TQFT (in imaginary time, τ ) [ ? ] S CS = (cid:90) d xdτ (cid:20) iπ (cid:15) µνλ a µ ∂ ν b λ + i π (cid:15) µνλ A sµ ∂ ν a λ (cid:21) . (3.7)This theory can be exactly quantized [28], and this yields interesting information on the structureof Z spin liquids on topologically non-trivial manifolds. On a torus, the only non-trivial gauge-invariant observables are the Wilson loops around the two cycles of the torus, which we denoteby ˆ W x = exp (cid:18) i (cid:73) dx a x (cid:19) , ˆ W y = exp (cid:18) i (cid:73) dy a y (cid:19) ˆ V x = exp (cid:18) i (cid:73) dx b x (cid:19) , ˆ V y = exp (cid:18) i (cid:73) dy b y (cid:19) . (3.8)The quantization of (3.7) at A sµ = 0 is characterized by the operator algebraˆ W x ˆ V y = − ˆ V y ˆ W x , ˆ W y ˆ V x = − ˆ V x ˆ W y , (3.9)and all other combinations of operators commute. This operator algebra is easily realized by 2independent sets of Pauli matrices. This implies that the ground state of the Z spin liquid has a4-fold degeneracy on the torus. This degeneracy can also be obtained from the trial wavefunctions11 x L y ˆ W y ( i ) ˆ W y ( i + ˆ x ) FIG. 8. Square lattice on a torus. The Wilson loop ˆ W y is translated by one lattice spacing in the ˆ x direction. for the spin liquid [29, 30] in Section III: the degenerate ground states are obtained by applyingthe branch-cut around the cycles of the torus, a connection evident from (3.6) and (3.8).The TQFT can also implement the second crucial property of the vison described above, andillustrated in Fig. 7. As in Section II, we place the Z spin liquid on a square lattice of size L x × L y with toroidal boundary conditions. Now consider the impact of translation by one lattice spacing,ˆ T x , on the Wilson loop operator ˆ W y , as shown in Fig. 8. The motion of the Wilson loop encloses L y lattice sites, and so this operation is equivalent to a vison having encircled L y sites. FromSection III, we conclude that such a process yields a Berry phase of 2 πSL y . The net result is thefollowing non-trivial operator relation ˆ T x ˆ W y = e πiSL y ˆ W y ˆ T x , (3.10)and a second relation with x ↔ y . Note that for L y odd, ˆ T x and ˆ W y anti-commute (commute) forodd (even) Z spin liquids. These relationships are not part of the TQFT structure per se, butinstead show how global symmetries of the underlying quantum system are realized in a non-trivialmanner by the TQFT operators. In other words, they describe the ‘symmetry enriched topological’structure, or the ‘symmetry fractionalization’ by gapped excitations, [31–33] of the Z spin liquid. B. Momentum balance The general results in (2.2) and (2.4), describing flux insertion through the cycle of torus, applyto any lattice quantum system with a global U(1) symmetry, and so should also apply to the Z spin liquid. We will now show, using (3.10), that (2.2) and (2.4) are indeed satisfied.As in Section II, we insert a flux, Φ, which couples only to the up spin electrons, which requireschoosing A sµ = 2 A eµ ≡ A µ . We work in real time, and thread a flux along the x -cycle of the torus.12 xy ˆ W y FIG. 9. As in Fig. 4. For a Z spin liquid, the flux insertion is equivalent to an operator acting on thered line: this is the branch-cut operator acting on the RVB state, or equivalently, the operator ˆ W y of theTQFT. So we have A x = Φ( t ) L x (3.11)where Φ( t ) is a function which increases slowly from 0 to 2 π . In (3.7), the A x gauge field couplesonly to a y , and we parameterize a y = θ y L y . (3.12)Then, from (3.7), the time evolution operator of the flux-threading operation can be written asˆ U = exp (cid:18) i π (cid:90) dt ˆ θ y d Φ dt (cid:19) = e i ˆ θ y ≡ ˆ W y (3.13)So the time evolution operator is simply the Wilson loop operator ˆ W y . If the state of the systembefore the flux-threading was | G (cid:105) , then the state after the flux threading will be ˆ W y | G (cid:105) . This isillustrated in Fig. 9.Now we can easily determine the difference in momenta of the states | G (cid:105) and ˆ W y | G (cid:105) . From(3.10) we obtain ∆ P x = 2 πSL y (mod 2 π ) = 2 πL x ( SL x L y ) (mod 2 π ) . (3.14)In the second form above, we see that (3.14) is consistent with (2.4) for N ↑ = SL x L y . This isindeed the correct total number of up spin electrons in a spin S antiferromagnet. IV. FERMI LIQUID-LIKE METALS WITH TOPOLOGICAL ORDER FOR THE PSEU-DOGAP STATE A simple picture of the fractionalized Fermi liquid (FL* metal) [8, 9, 34–37] is that it is acombination of the systems described in Sections II and III. The low energy excitations of such a13tate on a torus are given by the action S F L ∗ = S F L + S CS (4.1)which is the direct sum of the action for fermionic quasiparticles in (2.3), and of the action forthe TQFT in (3.7). Consequently the momentum balance also involves the direct sum of thequasiparticle contribution in (2.5) and the TQFT contribution in (3.14) for S = 1 / 2; and theseshould add up to the total number of up spin electrons N ↑ = N/ Z spin liquid leads to a modified constraint on the volume of the Fermi surface enclosed bythe quasiparticles. For the cuprate case, with a total density of 1 + p holes, we have in the FL*metal a modification from (2.6) to V F S π = p (mod 2) . (4.2)The simplest realizations of FL* are in 2-band Kondo-Heisenberg lattice models [8, 34, 35, 37].Then the origin of the direct sum picture described above can be understood in a simplifiedpicture: the local moments with Heisenberg exchange interactions can form the spin liquid, whilethe conduction electrons form the ‘small’ Fermi surface. This simple picture assumes the Kondoexchange between the local and itinerant electrons can be neglected, but it can be important fordetermining whether FL* is realized in a specific model [37].However, for the cuprates we need a realization of FL* in a 1-band model, as only a singleband of electronic excitations is observed. Such a realization has appeared in a series of works[38–49]. Here we briefly describe the simplified model of Ref. [49], which extends the RVB pictureto include mobile fermionic carriers which have the same quantum numbers as the electron. Asshown in Fig. 10a, we construct a trial wavefunction as a superposition of valence bond coveringsof the square lattice with two distinct categories of pairs of sites: ( i ) the blue bonds in Fig. 10a,which represent a pair of electrons in a singlet bond, and ( ii ) the green bonds in Fig. 10a, whichrepresent a single electron in a bonding orbital between the sites. The density of green bonds is p , and relative to the RVB background of blue bonds, each green bond is a fermion which carriescharge + e and spin S = 1 / i.e. the same quantum numbers as a hole in a band insulator.These mobile green fermions can then form a ‘small’ Fermi surface of volume given by (4.2). Thebackground of resonating blue and green bonds still preserves the topological order of the spinliquid, and forms a sector described by the TQFT of a Z spin liquid [50].We also show in Fig. 10b a related state called the ‘holon metal’ [51, 52], or more generallyan ‘algebraic charge liquid’ (ACL) [43]. In this case, in addition to the blue singlet bonds, wehave a density, p , of spinless, fermionic vacancies (the ‘holons’, or more generally the ‘chargons’)each carrying charge + e . Now the chargons can form a Fermi liquid-like state with a small Fermisurface of size p , but the quasiparticles at the Fermi surface will not be electron-like, as they carryonly charge but no spin. Note that although the number of quasiparticle states inside the Fermi14 ( |" i + | "i ) / p = ( |" ) / p (a) FL* (b) ACL FIG. 10. (a) A component of a resonating bond wavefunction for FL* in a single-band model on thesquare lattice [49]. The density of the green bonds is p , and these are fermions which form Fermi surfaceof volume (4.2) with electron-like quasiparticles. (b) A component of a wavefunction for an ACL. Thevacancies are the ‘holons’, or more generally, the ‘chargons’; they are assumed to be fermions which forma Fermi liquid-like state with a Fermi surface of spinless quasiparticles of charge e . surface is the ‘small’ value p , determination of the Fermi wavevector requires accounting for thespin or other quantum numbers carried by the chargons: for the model in Section V, the chargonsalso carry a pseudospin index ( s = ± ) which has the same degeneracy as electronic spin.The momentum balance argument for an ACL works just like for FL*. The chargons carrycharge but no spin, and so they couple to the electromagnetic gauge field A eµ . As we saw inSection III B, flux insertion coupling only to spin-up particles is carried out using A sµ = 2 A eµ ≡ A µ ,and the net result is that the chargon contribute just as naively expected: as spinless fermions ofcharge + e , making up a metal with a total charge density of pe mobile carriers. In general, bothchargon and electron-like Fermi surfaces can be present, and their sizes should sum to p [43].Turning to the phase diagram of the cuprates in Fig. 1b, we now summarize the evidence thata FL* (or an ACL) model describes the PG regime. • Model computations [45, 49] of the Fermi surface configuration for FL* yield hole pock-ets centered near, but not exactly at, ( π/ , π/ A T -independent positive Hall coefficient R H corresponding to carrier density p in the highertemperature pseudogap [54]. This is the expected Hall co-efficient of the hole pockets in theFL* or ACL phase. • The frequency and temperature dependence of the optical conductivity has a Fermi liquidform ∼ / ( − iω + 1 /τ ) with 1 /τ ∼ ω + T [55]. This Fermi liquid form is present althoughthe overall prefactor corresponds to a carrier density p . • Magnetoresistance measurements obey Kohler’s rule [56] with ρ xx ∼ τ − (1 + a ( Hτ ) ), againas expected by Fermi pockets of long-lived charge-carrying quasiparticles. • Density wave modulations have long been observed in STM experiments [57] in the regionmarked DW in Fig. 1b. Following theoretical proposals [58, 59], a number of experiments[4, 60–63] have identified the pattern of modulations as a d -form factor density wave. Com-putations of density wave instabilities of the FL* metal lead naturally to a d -form factordensity wave, with a wavevector similar to that observed in experiments [64, 65] (very sim-ilar results would be obtained in a computation starting from an ACL metal because thedensity wave instabilities are not sensitive to the spin of the quasiparticles). In contrast,computation of density wave instabilities of the large Fermi surface FL metal lead to densitywave order along a ‘diagonal’ wavevector not observed in experiments [59, 66, 67]. • Finally, very interesting recent measurements by Badoux et al. [68] of the Hall co-efficientat high fields and low T for p ≈ . 16 in YBCO clearly show the absence of DW order, unlikethose at lower p . Furthermore unlike the DW region, the Hall co-efficient remains positiveand corresponds to a density of p carriers. Only at higher p ≈ . 19 does the FL Hall co-efficient of 1 + p appear: in Fig. 1b, this corresponds to the T ∗ boundary extending past theDW region at low T . A possible explanation is that the FL* or ACL phase is present at p = 0 . V. FLUCTUATING ANTIFERROMAGNETISM AND THE STRANGE METAL The strange metal (SM) region of Fig. 1b exhibits strong deviations in the temperature andfrequency dependence of its transport properties from those of a Fermi liquid. Its location in thetemperature-density phase diagram suggests that the SM is linked to the quantum criticality of azero temperature critical point (or phase) near p = 0 . 19. We interpret the experiments as placinga number of constraints on a possible theory: • The quantum transition is primarily “topological”. The main change is in the size of the16ermi surface from small (obeying (4.2)) to large (obeying (2.6)) with increasing p . This isespecially clear from the recent Hall effect observations of Badoux et al. [68]. • Symmetry-breaking and Landau order parameters appear to play a secondary role. A conven-tional order which changes the size of the Fermi surface must break translational symmetry,and the only such observed order is the charge density wave (DW) order. However, thecorrelation length of this order is rather short in zero magnetic field, and in any case it seemsto disappear at a doping which is smaller than p = 0 . 19; see Fig. 1b. • The main symmetry breaking which could be co-incident with the transition at p = 0 . • The small doping side of the critical point exhibits significant spin fluctuations at wavevectorsclose to but not equal to ( π, π ), and these become anisotropic when the Ising-nematic orderis present. • The Hall effect observations of Badoux et al. [68] show a smooth evolution of the Hallresistance between values corresponding to a density of p carriers at p = 0 . 16, to that corre-sponding to a density of 1 + p carriers at p = 0 . 19. Such a smooth evolution is very similarto that obtained in a model of the reconstruction of the Fermi surface by long-range antifer-romagnetism [69]. It is possible that magnetic-field-induced long-range antiferromagnetismis actually present in the high field measurements of Badoux et al. [68], but fluctuating anti-ferromagnetism with a large correlation length is a more likely possibility. We will describebelow a model of a ACL/FL* metal based upon a theory of fermionic chargons in the pres-ence of local antiferromagnetism without long-range order: the evolution of the Hall effectin this model has little difference from that in the case with long-range antiferromagnetism.We note that theories of a change in Fermi surface size involving bosonic chargons [8, 46, 70]lead to a jump in the Hall co-efficient at the critical point [35] (when the half-filled bandof fermionic spinons discontinuously acquires an electromagnetic charge upon the transitionfrom FL* to FL [8]), and this appears to be incompatible with the data.It appears we need a gauge theory for a topological transition from a deconfined Z -FL* state(or the related Z -ACL state) to a confining FL with a large Fermi surface involving fermionic chargons. Significant non-( π, π ) spin correlations should be present in the deconfined Z state.Moreover, we would like Ising-nematic order to be present as a spectator of the deconfined Z state, and disappear at the confining transition to the FL state. This situation is the converse of17hat found in models of ‘deconfined criticality’ [14, 18, 71], where the spectator order parameterappears in the confining phase, and not in the deconfined phase wanted here. Also, the criticalpoint should not be simply given by a theory of the Ising-nematic order, as this cannot accountfor the charge in the Fermi surface size.We now describe a model compatible with these constraints. We will begin with a lattice modelof electrons coupled to spin fluctuations. When the spin fluctuations can be neglected, we have aconventional FL state with a large Fermi surface. Conversely, when spin fluctuations are condensed,we have antiferromagnetic long-range order with small pocket Fermi surfaces. However, our focuswill be on possible ‘deconfined’ intermediate phases where there is no long-range antiferromagneticorder, but the local magnitude of the antiferromagnetic order is nevertheless finite: the local orderdetermines the magnitude of the pseudogap and leads to small pocket Fermi surfaces even withoutlong-range order. We will argue that the concept of ‘local antiferromagnetic order’ can be madeprecise by identifying it with the Higgs phase of an emergent gauge theory. The Higgs phase willrealize the small Fermi surface Z FL* or ACL phases discussed above as models of the PG metal.The model of electrons coupled to spin fluctuations has the Lagrangian L = L c + L c Φ + L Φ . (5.1)The first term describes the fermions, c iα , hopping on the sites of a square lattice. L c = (cid:88) i c † iα (cid:20)(cid:18) ∂∂τ − µ (cid:19) δ ij − t ij (cid:21) c jα . (5.2)We describe the interactions between the fermions via their coupling to spin fluctuations at thewavevectors K x and K y which are close to but not equal to ( π, π ), and are related by 90 ◦ rotation.Along K x this is characterized by a complex vector in spin space Φ x(cid:96) , and similarly for K y so that (cid:68) c † iα σ (cid:96)αβ c iβ (cid:69) ∼ Φ ix(cid:96) e i K x · r i + c.c. + Φ iy(cid:96) e i K y · r i + c.c. . (5.3)Then the Lagrangian coupling the electrons c iα to the spin fluctuations is given by L c Φ = − λ (cid:88) i (cid:2) Φ ix(cid:96) e i K x · r i + c.c. + Φ iy(cid:96) e i K y · r i + c.c. (cid:3) c † iα σ (cid:96)αβ c iβ . (5.4)The coupling λ is expected to be large, and our discussion below will implicitly assume so. Finally,we have the Lagrangian describing the spin fluctuations L Φ = (cid:2) | ∂ τ Φ x(cid:96) | + v |∇ Φ x(cid:96) | + | ∂ τ Φ y(cid:96) | + v |∇ Φ y(cid:96) | + r (cid:0) | Φ x(cid:96) | + | Φ y(cid:96) | (cid:1) + . . . (cid:3) , (5.5)where v is a spin-wave velocity.The theory L is often referred to as a ‘spin-fermion’ model [72], and it provides the theory forthe direct onset of antiferromagnetism in a Fermi liquid. There has been a great deal of work on18his topic, starting with the work of Hertz [58, 73–75]. Recent sign-problem-free quantum MonteCarlo simulations of spin-fermion models [76–80] have yielded phase diagrams with remarkablesimilarities to those of the pnictides and the electron-doped cuprates. The spin-fermion problemcan also be applied at half-filling ( p = 0) with ( π, π ) antiferromagnetic correlations, and then thebackground half-filled density of c fermions yields [81] the correct Berry phases of the ‘hedgehog’defects in the N´eel order parameter [82–84]. The latter Berry phases are characteristic of theinsulating Heisenberg antiferromagnet at p = 0, and so a judicious treatment of the spin-fermionmodel at large λ can also describe Mott-Hubbard physics.Here, we want to extend the conventional theoretical treatments of the spin-fermion model toreach more exotic states with Mott-Hubbard physics at non-zero p . The formalism we present belowcan yield insulators at p = 0 both with and without AF order, with the latter being topologicalphases with emergent gauge fields. Moreover, the topological order will also extend to metallicphases at non-zero p , with gauge-charged Higgs fields describing local antiferromagnetism in thepresence of a pseudogap and small pocket Fermi surfaces, but without long-range antiferromagneticorder.The key step in this process [85–87] is to transform the electrons to a rotating reference framealong the local magnetic order, using a SU(2) rotation R i and (spinless-)fermions ψ i,s with s = ± , (cid:32) c i ↑ c i ↓ (cid:33) = R i (cid:32) ψ i, + ψ i, − (cid:33) , (5.6)where R † i R i = R i R † i = 1. Note that this representation immediately introduces a SU(2) gaugeinvariance (distinct from the global SU(2) spin rotation) (cid:32) ψ i, + ψ i, − (cid:33) → U i ( τ ) (cid:32) ψ i, + ψ i, − (cid:33) , R i → R i U † i ( τ ) , (5.7)under which the original electronic operators remain invariant, c iα → c iα ; here U i ( τ ) is a SU(2)gauge-transformation acting on the s = ± index. So the ψ s fermions are SU(2) gauge fundamentals,carrying the physical electromagnetic global U(1) charge, but not the SU(2) spin of the electron:they are the fermionic “chargons” of this theory, and the density of the ψ s is the same as thatof the electrons. The bosonic R fields also carry the global SU(2) spin (corresponding to leftmultiplication of R ) but are electrically neutral: they are the bosonic “spinons”, and are related[83, 85, 88] to the Schwinger bosons in (3.4). Later, we will also find it convenient to use theparameterization R = (cid:32) z ↑ − z ∗↓ z ↓ z ∗↑ (cid:33) (5.8)with | z ↑ | + | z ↓ | = 1. 19 ield Symbol Statistics SU(2) gauge SU(2) spin U(1) e . m . charge Electron c fermion -1AF order Φ boson ψ fermion -1Spinon R or z boson ¯ H boson L and L g . The transformations under the SU(2)’sare labelled by the dimension of the SU(2) representation, while those under the electromagnetic U(1)are labeled by the U(1) charge. The antiferromagnetic spin correlations are characterized by Φ in (5.3).The Higgs field determines local spin correlations via (5.12). A summary of the charges carried by the fields in the resulting SU(2) gauge theory, L g , isin Table I. This rotating reference frame perspective was used in the early work by Shraimanand Siggia on lightly-doped antiferromagnets [89, 90], although their attention was restricted tophases with antiferromagnetic order. The importance of the gauge structure in phases withoutantiferromagnetic order was clarified in Ref. [85].Given the SU(2) gauge invariance associated with (5.6), when we express L in terms of ψ wenaturally obtain a SU(2) gauge theory with an emergent gauge field A aµ = ( A aτ , A a ), with a = 1 , , L g = L ψ + L Y + L R + L H . (5.9)The first term for the ψ fermions descends directly from the L c for the electrons L ψ = (cid:88) i ψ † i,s (cid:20)(cid:18) ∂∂τ − µ (cid:19) δ ss (cid:48) + iA aτ σ ass (cid:48) (cid:21) ψ i,s (cid:48) + (cid:88) i,j t ij ψ † i,s (cid:20) e iσ a A a · ( r i − r j ) (cid:21) ss (cid:48) ψ j,s (cid:48) , (5.10)and uses the same hopping terms for ψ as those for c , along with a minimal coupling to the SU(2)gauge field. Inserting (5.6) into L cn , we find that the resulting expression involves 2 complex Higgsfields, H ax and H ay , which are SU(2) adjoints; these are defined by H ax ≡ 12 Φ x(cid:96) Tr[ σ (cid:96) Rσ a R † ] , (5.11)and similarly for H ay . Let us also note the inverse of (5.11)Φ x(cid:96) = 12 H ax Tr[ σ (cid:96) Rσ a R † ] , (5.12)20nd similarly for H ay , expressing the antiferromagnetic spin order in terms of the Higgs fields and R . Then L c Φ maps to the form of a ‘Yukawa’ coupling equal to λ , L Y = − λ (cid:88) i (cid:0) H aix e i K x · r i + H a ∗ ix e − i K x · r i + H aiy e i K y · r i + H a ∗ iy e − i K y · r i (cid:1) ψ † i,s σ ass (cid:48) ψ i,s (cid:48) . (5.13)We note again that our discussion below will implicitly assume large λ . The remaining terms inthe Lagrangian involving the bosonic Higgs field, H , the bosonic spinons R , and the gauge field A aµ follow from gauge invariance and global symmetries, and are similar to those found in theoriesof particle physics. In particular the spinon Lagrangian is L R = 12 g Tr (cid:20) ( ∂ τ R − iA aτ Rσ a )( ∂ τ R † + iA aτ σ a R † ) + v ( ∇ R − i A a Rσ a )( ∇ R † + i A a σ a R † ) (cid:21) . (5.14)For the Higgs field, we have L H = (cid:12)(cid:12) ∂ τ H ax + 2 i(cid:15) abc A bτ H cx (cid:12)(cid:12) + ˜ v (cid:12)(cid:12) ∇ H ax + 2 i(cid:15) abc A b H cx (cid:12)(cid:12) + (cid:12)(cid:12) ∂ τ H ay + 2 i(cid:15) abc A bτ H cy (cid:12)(cid:12) + ˜ v (cid:12)(cid:12) ∇ H ay + 2 i(cid:15) abc A b H cy (cid:12)(cid:12) + V ( H ) , (5.15)with the Higgs potential V ( H ) = h (cid:0) | H ax | + | H ay | (cid:1) + u (cid:0) [ H a ∗ x H ax ] + [ H a ∗ y H ay ] (cid:1) + u (cid:0) [ H ax ] [ H b ∗ x ] + [ H ay ] [ H b ∗ y ] (cid:1) + u [ H a ∗ x H ax ][ H b ∗ y H by ] + u [ H a ∗ x H bx ][ H a ∗ y H by ] + u [ H a ∗ x H bx ][ H b ∗ y H ay ] . (5.16)Despite the apparent complexity of the gauge theory Lagrangian, L g , described above, it shouldbe noted that its structure follows largely from the quantum number assignments in Table I, andthe transformations of the fields under lattice translation. Under translation by a lattice vector r , c , ψ , and R transform trivially, while H ax → H ax e i K x · r , H ay → H ay e i K y · r (5.17)(and similarly for Φ x(cid:96) and Φ y(cid:96) ). The physical interpretations are obtained from the mappings in(5.3), (5.6), and (5.12) between the physical observables and the gauge-charged fields in Table I.Also note that, while we can take the continuum limit for the bosonic fields, the fermionic fieldshave to be described on a lattice to account for the Fermi surface structure.The main innovation of the above description [85–87] is the introduction of the Higgs fields H ax and H ay as a measure of the local antiferromagnetic order along wavevectors K x and K y . As theseHiggs fields only carry SU(2) gauge charges (see Table I), their condensation does not break theglobal SU(2) spin rotation symmetry. However, their magnitude is a gauge invariant observable,and this does measure the magnitude of the local ‘pseudogap’ created by the Higgs condensate,and changes the dispersion of the fermionic charge carriers into small pocket Fermi surfaces. Sothe Higgs phase, with no other fields condensed, will realize the PG metal in a theory of localantiferromagnetic correlations. 21 /g (A) Antiferromagnetic metal (B) Fermi liquid with large Fermi surface(C) Z FL* or ACL with small Fermi surfaces and long-range Ising-nematic order (D) SU(2) ACL unstable to pairing and confinement h R i = 0 , h H a i = 0 h R i 6 = 0 , h H a i = 0 h R i 6 = 0 , h H a i 6 = 0 h R i = 0 , h H a i 6 = 0 Hertz criticalityof antiferromagnetismHiggs criticality h FIG. 11. Mean-field phase diagram of the SU(2) gauge theory L g , as a function of the coupling g in (5.14)and the ‘mass’ h in (5.16). Phase A has antiferromagnetic order at wavevectors close to, but not equalto, ( π, π ). Phase C is our candidate for the PG metal, and the ‘Higgs criticality’, between phases C andD, is our candidate for the description of the strange metal. The boundary between phases B and D doesnot remain a phase transition [91] after confinement of the SU(2) gauge theory has been accounted for:the boson R carries fundamental SU(2) gauge charge, and its Higgs (B) and confinement (D) phases aresmoothly connected [92]. This phase diagram shows how the conventional physics of Hertz criticality,applicable to the pnictides, evolves naturally to the ‘topological’ physics of Higgs criticality, applicableto the hole-doped cuprates. The Fermi surface reconstruction across the B-A boundary is due to theantiferromagnetic order, and it involves changes in the band structure of electron-like quasiparticles, c .A nearly identical Fermi surface reconstruction, with similar transport properties, takes place across theD-C boundary, except that it involves spinless chargons, ψ . A. Phase diagram We now return to the full model with SU(2) spin rotation symmetry, and discuss the possiblephases of the SU(2) gauge theory L g .It is useful to proceed by sketching the mean field phase diagram in terms of possible condensatesof the bosons R and H , and follow it by an analysis of the role of gauge fluctuations. Such a phasediagram is sketched in Fig. 11 as a function of the coupling g in (5.14) and the ‘mass’ h in (5.16).Phases A and B: The phases in which the spinon, R , is condensed are the familiar Fermi liquid22hases. This is evident from (5.6), which implies that with R condensed c ∼ ψ ; also from (5.11)the Higgs fields H is related by a global rotation to the antiferromagnetic order parameter Φ.Consequently, the phase B in Fig. 11 is the conventional Fermi liquid with a large Fermi surfaceof size 1 + p . The condensation of R ∼ Φ leads to the onset of antiferromagnetic order in phase Avia a Hertz type critical point [58, 73–75]; this condensation will reconstruct the Fermi surface toyield a Fermi liquid with ‘small’ Fermi surfaces.We therefore turn our attention to the possibly exotic phases C and D in Fig. 11.Phase D: There is no Higgs condensate in phase D, and so all the SU(2) gauge fields are active.The gauge-charged matter sector includes a large Fermi surface of ψ ± fermions which transform asa SU(2) doublet. The attractive SU(2) gauge force is expected to pair these fermions, leading toa superconducting state [93, 94]. The resulting gapping of the fermionic excitations will unscreenthe SU(2) gauge force, which will confine all gauge-charged excitations. Ultimately, we thereforeexpect phase D to be a superconductor without topological order or fractionalized excitations, anda conventional Fermi liquid could appear in a magnetic field or at higher temperatures. Also, asindicated in Fig. 11, we expect phase D to be smoothly connected to the Fermi liquid phase B, asthe latter is also unstable to pairing induced by the spin fluctuations [91].Phase C: Finally, we turn our attention to phase C. Here we have a H condensate, and this willbreak the SU(2) gauge invariance down to a smaller gauge group. But, because R is not condensed,by (5.3), global spin rotation invariance is nevertheless preserved. The case of particular interestto us here is a residual gauge invariance of SU(2) / SO(3) ∼ = Z . This will be the situation as long asthe Higgs potential V ( H ) in (5.16) is such that all four of the real 3-vectors (cid:104) Re( H ax ) (cid:105) , (cid:104) Im( H ax ) (cid:105) , (cid:104) Re( H ay ) (cid:105) , (cid:104) Im( H ay ) (cid:105) are not parallel to each other, so that the Higgs condensate transforms underSO(3) global rotations. (For the case with all four vectors parallel, there is a residual U(1) gaugeinvariance associated with rotations about the common direction [85, 86].) A simple case withresidual Z gauge invariance is H ax ∼ (1 , i, , H ay = (0 , , , (5.18)or its global SO(3) rotations. We then obtain an effective Z gauge theory, with the same structureas the TQFT of Section III. In particular, the π (SO(3)) = Z vortices in the Higgs field H correspond to vison excitations [14, 88, 95], which are gapped in phase C.For the Higgs condensate in (5.18), writing the spinons R as in (5.8), (5.12) becomesΦ x(cid:96) = − ε αγ z γ σ (cid:96)αβ z β , (5.19)where ε is the unit anti-symmetric tensor (the z α spinons are connected [83, 85, 88] to the Schwingerbosons in (3.4)). In terms of the real and imaginary components of Φ x(cid:96) = n (cid:96) + in (cid:96) , (5.19) yields apair of orthonormal vectors n (cid:96) and n (cid:96) , describing the SO(3) antiferromagnetic order parameter.23o by (5.3), (5.18) represents spiral spin correlations along the wavevector K x , with no corre-sponding correlations along K y . Such a state has long-range Ising-nematic order, as correlationsof spin-rotation invariant observables will be different along the x and y lattice directions.Our description of phase C so far leads to a Z -ACL state, described earlier in simple termsin Section IV. From the Yukawa coupling in (5.13), the Higgs condensate reconstructs the Fermisurface of the ψ fermions into a filled band along with small pockets: this reconstruction hasan identical structure to that of the c fermions across the B-A phase boundary, and so chargetransport across the D-C transition should be similar to that across the B-A transition [86]. Thefilled band in phase C corresponds to a density of a unit Z gauge charge on every site, andso this phase is described by an odd Z gauge theory. The quasiparticles around the Fermisurfaces of the small pockets are the ψ chargons (see Table I), and we obtain the ACL staterepresented earlier in Fig. 10b. To obtain a Z -FL* state (see Fig. 10a), we need the reconstructed ψ quasiparticles to form bound states with the R spinons, and for the resulting bound stateof electron-like quasiparticles to form a Fermi surface: the hopping t ij in (5.2) is an attractiveinteraction between the chargons and spinons which can lead to such bound states. In general,both ψ and ψ - R Fermi surfaces will be present [43], and their combined size is restricted bythe Luttinger constraint [96, 97]. Computations of models of this bound-state formation havebeen presented elsewhere [42, 43, 45, 47]. Transport measurements on the PG metal do notdistinguish between ψ and ψ - R quasiparticles, as they are only sensitive to the charge carriedby the fermionic quasiparticles. However, photoemission only sees ψ - R quasiparticles, and moredetailed photoemission observations could determine the situation in the cuprates.Finally, we comment on the Higgs criticality between phases C and D. Here the theory consistsof a critical Higgs field tuned to the edge of the Higgs phase, by taking the ‘mass’ h in V ( H ) to itscritical value. Because the Higgs condensate is absent, the ψ fermions form a large Fermi surface,and there is no Ising-nematic order. There could also be a spectator small Fermi surface of ψ - R quasiparticles, but this is not expected to be important for the critical theory. The R spinonsare gapped, and can also be neglected in the critical theory. So the final proposed theory for theSM is a large Fermi surface of ψ chargons and a critical Higgs field coupled to SU(2) gauge field.Such a theory includes the quantum fluctuations of visons, and their Berry phases, as it allowsfor amplitude fluctuations of the Higgs fields, and the lines of zeros in the Higgs field correspondto the π (SO(3)) = Z vortices. Transport properties of such a theory, and their connection toexperiments in the cuprates, have been discussed recently elsewhere [86]. Note that in this scenario,the SU(2) gauge excitations are not deconfined in either phase C or phase D, and only apparentin the non-Fermi liquid behavior in the finite-temperature quantum critical region [94]; so this isan example of ‘deconfined criticality’ [71]. 24 . Simplified Z and U(1) lattice gauge theories A notable feature of the phase diagram in Fig. 11 is that none of the ground state phases havedeconfined SU(2) electric gauge charges, which appear only in a deconfined quantum critical regionat non-zero temperature. However, deconfined Z electric gauge charges are present in phase C.This raises the question of whether it is possible to formulate the theory purely as a Z gaugetheory. As was shown in recent work [98], it is indeed possible to do so. The new formulation isdefined on the square lattice, and it does not yield a direct route to a continuum theory for possiblequantum critical points towards confinement. Continuum formulations of confinement transitionsin Z gauge theories require duality transforms to vison fields via mutual Chern-Simons terms [99],but we will not discuss this duality here; it is possible that such an analysis of the criticality willlead back to the deconfined SU(2) gauge theory discussed above.For simplicity, we consider the case with spiral spin correlations only along the wavevector K x ;it is not difficult to extend the action below to also include the K y direction. We assume the Higgsfield is quenched as in (5.18), and so write the antiferromagnetic order parameter Φ x(cid:96) using (5.19).Then the action for the Z lattice gauge theory is [98] L Z = L c + L cz + L z + L µ , (5.20)where the electron Lagrangian L c was specified in (5.2), and the coupling between the electronsand the spinons z α is obtained by combining (5.4) and (5.19) L cz = − λ (cid:88) i (cid:2) − ε αγ z iγ σ (cid:96)αβ z iβ e i K x · r i + c.c. (cid:3) c † iα σ (cid:96)αβ c iβ . (5.21)The spinons have the Lagrangian L z = 1 g | ∂ τ z α | − v g (cid:88) (cid:104) ij (cid:105) µ zij ( z ∗ iα z jα + c.c.) , (5.22)where we have introduced an Ising spin, µ zij = ± 1, on the links of the square lattice as a Z gaugefield. This gauge field is necessary because the z α spinon carries a Z gauge charge. Finally, we givean independent dynamics to the Z gauge fields, via a standard [100] Z gauge theory Hamiltonian H µ , associated with the Lagrangian L µ in (5.20) H µ = − K (cid:88) (cid:3) (cid:34)(cid:89) (cid:3) µ zij (cid:35) − h (cid:88) (cid:104) ij (cid:105) µ xij , (5.23)where µ xij is a Pauli matrix which anti-commutes with µ zij . The theory L Z in (5.20) can be viewedas a reformulation of the spin-fermion model in (5.1), using additional Z gauge degrees of freedomthat allow for the possibility of fractionalized phases. For small K in (5.23), we can trace over25he Z gauge degrees of freedom in powers of K , and obtain terms with same structure as in thespin-fermion model in (5.1). The unusual feature of the degrees of freedom in L Z , not presentin earlier treatments [20, 70, 81, 85, 86, 101], is partial fractionalization: gauge charges are onlyexplicitly present in the spinon sector, while the charged degrees of freedom are gauge-invariantelectrons.The main point of Ref. 98 is that, despite the partial fractionalization in the presentation of L Z , the large K and λ phases of L Z have the same topological order and fractionalization as thosereviewed earlier in the present paper. At large K , π (SO(3)) = Z vortices in the antiferromagneticorder parameter are suppressed, and this leads to phases with Z fractionalization [95]. At p = 0,insulating Z spin liquids like those discussed in Section III can appear. In the degrees of freedomin (5.20), the fermionic chargon, ψ , is a bound state of c and z (via ψ = R − c from (5.6) and(5.8)), and its formation can be established in a large λ perturbation theory [98]. Although the Z gauge sector in (5.23) appears to be even, we noted above that the Z fractionalized phase C (inFig. 11) has a background filled band of ψ fermions carrying Z electric charges (doping this bandleads to small Fermi surfaces), and this converts it to Z -odd [9], as was required in Section IVfor a small Fermi surface. So at p = 0, L Z describes a Mott insulator with odd Z topologicalorder, similar to those described in Section III. At non-zero p , at large λ and large K , L Z exhibitsthe fractionalized phase C with all the same characteristics as the SU(2) theory; the conventionalphases A and B in Fig. 11 appear at small K . Phase D of the SU(2) gauge theory in Fig. 11 issmoothly connected to phase B, and it does not appear initially as a separate phase in the Z gaugetheory. Finally, the transition from phase C to phase B/D will appear as a confinement transitionin the Z gauge theory upon decreasing K , at the same time as the gauge theory changes from Z -odd to Z -even [98].We close this subsection by noting in passing the generalization of L Z to the case of the U(1)gauge theory of collinear antiferromagnetism considered in Refs. [85, 86]. Now the potential V ( H )in (5.16) is such that the Higgs condensates are all collinear, and we choose (cid:104) H ax (cid:105) ∼ (0 , , 1) and (cid:104) H ay (cid:105) = (0 , , x(cid:96) = z ∗ α σ (cid:96)αβ z β , from (5.12) and (5.8),and this is invariant under the U(1) gauge transformation z α → e iφ z α . The Lagrangian for theU(1) gauge theory, replacing (5.20), is [98] L U (1) = L c + L cz + L z + L A , (5.24)where L c remains as in (5.2), L cz in (5.21) is replaced by L cz = − λ (cid:88) i (cid:2) z ∗ iα σ (cid:96)αβ z iβ e i K x · r i + c.c. (cid:3) c † iα σ (cid:96)αβ c iβ . (5.25)Strictly speaking, such a parameterization applies only at commensurate K x , including the casewith N´eel order at K x = ( π, π ); the case with incommensurate collinear antiferromagnetism has26n additional ‘sliding charge mode’ [98], which we do not treat here. The spinon Lagrangian L z in (5.22) is replaced by L z = 1 g | ∂ τ z α | − v g (cid:88) (cid:104) ij (cid:105) (cid:0) e iA ij z ∗ iα z jα + c.c. (cid:1) , (5.26)where A ij is the connection of a compact U(1) gauge field. The action of the U(1) gauge field isthe standard generalization of the Maxwell action L A = K (cid:88) (cid:3) cos (cid:32)(cid:88) (cid:3) A ij (cid:33) + 12 h (cid:88) (cid:104) ij (cid:105) ( ∂ τ A ij ) . (5.27)For p = 0 and large λ , we obtain the insulating N´eel and valence bond solid states [81, 83, 84]. For p (cid:54) = 0 and large λ , the deconfined U(1)-ACL phase can appear at large K , while the conventionalphases A and B in Fig. 11 appear at small K . We note, however, that the U(1)-ACL is expectedto be unstable to pairing and confinement, as was the case for the SU(2)-ACL [94]. VI. CONCLUSIONS We have reviewed candidate theories for describing the unconventional metallic phases observedover a wide region in the phase diagram of the cuprate high temperature superconductors.The key idea in the discussion has been to encapsulate the strongly correlated nature of theproblem in terms of emergent gauge theories and topological order. In metals with well-definedquasiparticle excitations, we have shown how Luttinger’s theorem allows us to sharply distinguishbetween phases with and without topological order. Specifically, we described a connection betweenthe size of the Fermi surface and the odd/even nature of the ‘symmetry enriched’ [32] TQFTdescribing the Z topological order.A central mystery in the study of cuprate superconductors concerns the nature of the strangemetal without quasiparticle excitations and its relation to an underlying quantum critical point. Wehave argued that the critical point is best described in terms of a transition between a metal withtopological order and small Fermi surfaces, to a confining Fermi liquid with a large Fermi surface.Such a transition necessarily falls outside of the conventional Landau-Ginzburg-Wilson paradigmof symmetry-breaking phase transitions. Starting from a lattice model of electrons coupled tostrongly fluctuating antiferromagnetic spin fluctuations, we proposed a deconfined critical theoryfor the strange metal, with a SU(2) gauge field coupled to a large Fermi surface of chargons and acritical Higgs field. On the low doping side of the critical point, the Higgs field condenses to leave aresidual odd Z gauge theory describing the pseudogap metal with small Fermi surfaces and long-range Ising-nematic order, and the magnitude of the pseudogap determined by the magnitude of27he Higgs field. On the high doping side, the Higgs correlations are short-ranged, and the confiningphase of the SU(2) gauge field leads to a large Fermi surface with no Ising-nematic order. Althoughthe long-range Ising-nematic order vanishes at the critical point, the critical theory is not simplythat of the onset of this order in a Fermi liquid.An overall perspective is provided by the phase diagram in Fig. 11. This shows how the conven-tional physics of Hertz criticality, applicable to the pnictides, evolves smoothly to the ‘topological’physics of Higgs criticality, applicable to the hole-doped cuprates: the Higgs field theory can beunderstood as a ‘SU(2) gauged’ version of the Hertz theory. (Another perspective on the phases inFig. 11 appears in a separate paper [98]). The Fermi surface reconstruction of electrons across theHertz transition (from phase B to A) is identical, at the saddle point level, to the Fermi surfacereconstruction of chargons across the Higgs transition (from phase D to phase C): this followsfrom the similarity between (5.4) and (5.13). Consequently, the evolution of the charge transportacross the conventional transition between A and B will be similar to that across the topologicaltransition between C and D. Specifically, the evolution of the Hall effect as a function of p in amodel of the reconstruction of the Fermi surface by long-range antiferromagnetism [69] also appliesto the evolution of the Hall effect from C to D. The remarkable agreement of such a model withrecent observations [68] makes the topological Higgs theory an attractive candidate for the opti-mally hole-doped cuprates. Note that the use of fermionic chargons is important in this theory,and we argued that other approaches involving bosonic chargons [70, 101] lead to rather differentresults for the Hall effect.Future experiments will no doubt explore more completely the nature of the low T , high field,pseudogap metal discovered in Ref. [68]. Quantum oscillations could yield more precise informationon the nature of the Fermi surface, including whether the quasiparticles are spinful (as in FL*) orspinless (as in ACL). Nuclear or muon spin resonance experiments can determine if there is anyfield-induced magnetic order [102] in this regime. Also, studies of transport in the vicinity of thecritical point between the pseudogap metal and the Fermi liquid, with techniques borrowed fromhydrodynamics and holography, are promising avenues to explore. 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