Emergent gauge fields and their nonperturbative effects in correlated electrons

Emergent gauge fields and their nonperturbative effects in correlated electrons

Ki-Seok Kim    Akihiro Tanaka
Abstract

The history of modern condensed matter physics may be regarded as the competition and reconciliation between Stoner’s and Anderson’s physical pictures, where the former is based on momentum-space descriptions focusing on long wave-length fluctuations while the latter is based on real-space physics emphasizing emergent localized excitations. In particular, these two view points compete with each other in various nonperturbative phenomena, which range from the problem of high T superconductivity, quantum spin liquids in organic materials and frustrated spin systems, heavy-fermion quantum criticality, metal-insulator transitions in correlated electron systems such as doped silicons and two-dimensional electron systems, the fractional quantum Hall effect, to the recently discussed Fe-based superconductors. An approach to reconcile these competing frameworks is to introduce topologically nontrivial excitations into the Stoner’s description, which appear to be localized in either space or time and sometimes both, where scattering between itinerant electrons and topological excitations such as skyrmions, vortices, various forms of instantons, emergent magnetic monopoles, and etc. may catch nonperturbative local physics beyond the Stoner’s paradigm. In this review article we discuss nonperturbative effects of topological excitations on dynamics of correlated electrons. First, we focus on the problem of scattering between itinerant fermions and topological excitations in antiferromagnetic doped Mott insulators, expected to be relevant for the pseudogap phase of high T cuprates. We propose that nonperturbative effects of topological excitations can be incorporated within the perturbative framework, where an enhanced global symmetry with a topological term plays an essential role. In the second part, we go on to discuss the subject of symmetry protected topological states in a largely similar light. While we do not introduce itinerant fermions here, the nonperturbative dynamics of topological excitations is again seen to be crucial in classifying topologically nontrivial gapped systems. We point to some hidden links between several effective field theories with topological terms, starting with one dimensional physics, and subsequently finding natural generalizations to higher dimensions.

Revised Day Month Year

Keywords: SO(5) Wess-Zumino-Witten theory, Deconfined quantum criticality, QED and QCD, Pseudogap phase in high T cuprates, Nonlinear model, Topological term, Symmetry protected topological phase

1 Introduction

Strongly coupled field theories lie at the heart of unsolved fundamental problems not only in particle physics but also in condensed matter physics, which cover from confinement in quantum chromodynamics (QCD) to mechanism of high T superconductivity in doped Mott insulators. An important feature of strongly coupled field theories is that the function is negative in the renormalization group analysis, indicating that effective interactions between elementary excitations introduced in the UV (ultraviolet) limit are enhanced and such excitations become strongly coupled in the IR (infrared) limit. However, this does not necessarily mean that we cannot solve such strongly coupled field theories. Although it is negative the function of an effective interaction for superconducting instability in the Landau’s Fermi-liquid state , we all know that this problem can be solved in the framework of the BCS (Bardeen, Cooper, and Schrieffer) mean-field theory . On the other hand, even if essentially the same situation occurs in the Kondo problem , we do not have any mean-field types of effective theories which describe Fermi-surface instability due to a single magnetic impurity successfully except for exact methods based on Bethe ansatz and numerical renormalization group .

There exist other types of strongly coupled field theories, where corresponding functions vanish. In metals, most effective interactions between electron quasiparticles are irrelevant due to the presence of a Fermi surface while forward scattering channels remain marginal in the renormalization group sense, identified with Landau’s Fermi-liquid fixed point and described by Landau’s Fermi-liquid theory . This “strongly” coupled field theory is solved within the technique, which allows us to neglect vertex corrections, where is an enhanced spin degeneracy from to . On the other hand, when the spatial dimension is reduced to be one, vertex corrections should be introduced to play a central role in the treatment of IR divergences, which lead the electron-quasiparticle peak to split into double peaks of spinons and holons with their continuum, identified with Luttinger-liquid fixed point and described by Luttinger-liquid theory . Recently, effective field theories remain to be strongly coupled in the vicinity of quantum phase transitions from Landau’s Fermi-liquid state, where all planar diagrams are shown to be the same order in the technique , which implies that vertex corrections should be incorporated appropriately as the case of the Luttinger-liquid state.

These discussions give us an interesting question. When do vertex corrections become relevant in such strongly coupled field theories? In the above we have discussed two cases: (1) Fermi-surface instability toward the BCS superconducting state vs. Fermi-surface instability toward the local Fermi-liquid state (the Kondo effect) in the case of negative functions and (2) Landau’s Fermi-liquid theory vs. Luttinger-liquid theory and quantum criticality from the Landau’s Fermi-liquid state in the case of zero functions. Our speculation is that vertex corrections may encode the information of scattering between emergent localized excitations and itinerant electrons, where such localized excitations are identified with topologically nontrivial fluctuations, referred to as vortices in superconductivity, skyrmions in magnetism, and various forms of instantons localized even in time. Consistent introductions of vertex corrections in strongly coupled field theories mean that effects of topological excitations are incorporated into effective field theories appropriately. This scattering physics is expected to be responsible for Fermi-surface instabilities associated with orthogonality catastrophe . However, the absence of vertex corrections does not mean that the role of topological excitations is not introduced. If one considers the boson-vortex duality in the superfluid to Mott-insulator transition, the perturbative renormalization group analysis based on the charge description gives essentially the same critical physics as that based on the vortex picture aaaUnfortunately, it is not straightforward to prove explicitly that their critical physics are same, where the vortex description involves noncompact U(1) gauge fluctuations., implying that the information of topological excitations is introduced within the perturbative analysis.

The question is when the perturbative framework fails to incorporate physics of topological excitations. Here, the perturbative framework means that a given field theory can be solved within the self-consistent RPA (random phase approximation), equivalently the approximation or Eliashberg theory, where only self-energy corrections are introduced self-consistently. We recall that vertex corrections are introduced self-consistently through the Ward identity in one-dimensional interacting electrons, where the resulting Green’s function in the nonperturbative diagrammatic approach gives essentially the same expression as that in the bosonization framework which introduces spinons and holons explicitly, identified with topological excitations (solitons) . This implies that dimensionality which controls quantum fluctuations may play an important role for nonperturbative physics. We speculate that the perturbative framework may work near the upper critical dimension while it breaks down, which requires nonperturbative approaches, in low dimensions near the lower critical dimension or much below the upper critical dimension bbbKS enjoyed intensive discussions with Prof. V. Dobrosavljevic when he visited POSTECH in the summer season of 2014.. Here, the nonperturbative framework means to introduce topological excitations explicitly into the strongly coupled effective field theory and to deal with scattering physics between such localized excitations and itinerant electrons on equal footing .

In this review article we discuss nonperturbative effects of topological excitations on dynamics of correlated electrons. First, we focus on the problem of scattering between itinerant fermions and topological excitations in antiferromagnetic doped Mott insulators, where dynamics of localized magnetic moments and their localized excitations are described by emergent gauge fields and their topologically nontrivial configurations. We propose that nonperturbative effects of topological excitations can be incorporated within the perturbative framework, where an enhanced global symmetry allows us to introduce effects of topological excitations into an effective field theory explicitly in the presence of a topological term. Second, we discuss rich interplays between topological excitations and topological terms without itinerant fermions, where the nonperturbative dynamics of topological excitations is essential in classifying interacting topological insulators protected by symmetries. We clarify some hidden links between several effective field theories with topological terms, generalizing one dimensional physics into higher dimensions.

2 How to simulate nonperturbative physics from topological excitations within the perturbative framework?

2.1 Organization of this section

In section 2.2 we review an origin of non-Fermi liquid physics in antiferromagnetic doped Mott insulators, describing effective interactions between doped holes and hedgehog-type instanton excitations, expected to be involved with the pseudogap phase of high T cuprates. In section 2.2.2 we construct an effective gauge-field theory from the t-J Hamiltonian, regarded to be one of the standard models for strongly correlated electrons, where dynamics of localized magnetic moments is described by SO(5) Wess-Zumino-Witten (WZW) theory and that of doped holes is given by QED (quantum electrodynamics in one time and two spatial dimensions) with a finite chemical potential, referred to as QED and coupled to the SO(5) WZW theory . We discuss various limits of this emergent gauge theory. In section 2.2.1 we discuss the case of half filling, where hole concentration vanishes, thus reduced to the SO(5) WZW theory, which describes competing fluctuations between antiferromagnetic (three components) and valence bond (two components) order parameters. An essential aspect in this effective field theory is that space-time hedgehog fluctuations (magnetic monopoles as instantons) of the antiferromagnetic order parameter carry the quantum number of valence bond ordering near its core , which originates from the WZW term. Such topological excitations can be incorporated within the perturbative framework, where valence bond fluctuations are introduced explicitly and naturally through the SO(5) enhanced symmetry with the WZW term . We review physics of deconfined quantum criticality based on the SO(5) WZW theory , which argues how deconfinement of fractionalized spin excitations referred to as spinons, regarded to be quark-like objects, can be realized near quantum criticality, where magnetic monopole excitations as instantons become suppressed to preserve the total number of skyrmions . In section 2.2.3 we apply the QED coupled to the SO(5) WZW theory into one dimension, where the corresponding effective field theory is given by QED coupled to SO(4) WZW theory. We discuss that this effective field theory recovers the Luther-Emery phase , where spin excitations are gapped while superconducting correlations between doped holes are enhanced. In section 2.2.4 we discuss dynamics of doped holes near the deconfined quantum criticality of the SO(5) WZW theory, where the interplay between doped holes and space-time hedgehog excitations is encoded into the perturbative framework, i.e., scattering between itinerant fermions and valence bond fluctuations . We propose the role of valence bond fluctuations in dynamics of doped holes for their non-Fermi liquid physics in the pseudogap phase of high T cuprates.

Not only the situation of deconfinement but also that of confinement is discussed in section 2.3, based on a recently developed effective field theory for QCD at low energies in Hadron physics, referred to as Polyakov-loop extended Nambu-Jona-Lasinio (NJL) model , where such quark-like objects correspond to holons and spinons, representing doped holes and fractionalized spin excitations roughly speaking. Applying the Polyakov-loop extended NJL (PNJL) model to the problem of paramagnetic doped Mott insulators, we describe non-Fermi liquid transport phenomena near optimal doping of high T cuprates outside the pseudogap state, based on the confinement of spinons and holons .

Deep inside the Mott insulating phase, spin fluctuations are only relevant degrees of freedom at half filling. However, charge fluctuations are expected to play a central role in metal-insulator transitions, which may suppress magnetic ordering to allow spin liquid states, described by emergent SU(2) gauge theories. In section 2.4 we discuss metal-insulator transitions, generalizing the t-J Hamiltonian to the Hubbard model, where charge fluctuations are introduced. Constructing an effective SU(2) gauge theory to describe interactions between spinons and holons through SU(2) gauge fluctuations , we discuss possible spin liquid states near the metal-insulator transition on honeycomb and triangular lattices. In particular, we speculate how physics of spin liquids, metal-insulator transitions, and unconventional superconductivity will emerge from such nonabelian gauge theories beyond the saddle-point analysis, where gluon condensation consistent with the lattice symmetry is suggested to play an essential role.

In section 2.5 we conclude the first part of this review article, speculating that gauge field theories can appear rather commonly than expected in strongly coupled field theories . We discuss an antiferromagnetic quantum phase transition from the Landau’s Fermi-liquid state, where a critical field theory describes scattering between itinerant electrons and antiferromagnetic spin fluctuations . Recently, the technique turns out to fail to describe non-Fermi liquid physics near antiferromagnetic quantum criticality , where the critical field theory lies in the strongly coupled regime, meaning that vertex corrections should be incorporated consistently. We suggest that some types of instanton excitations may keep such nonperturbative physics, constructing an effective field theory with the introduction of instantons. Integrating out contributions of topological excitations, we speculate that an effective gauge-field theory emerges, regarded to generalize the scenario of the SO(5) WZW theory.

2.2 Emergent gauge fields and their nonperturbative effects in antiferromagnetic doped Mott insulators

2.2.1 SO(5) Wess-Zumino-Witten theory from Heisenberg model

Let’s start from an extended Heisenberg model on square lattice,

 H=HJ+HQ,     HJ=J∑ijSi⋅Sj, HQ=−Q∑{ijkl}∈□(Si⋅Sj−1/4)(Sk⋅Sl−1/4), (1)

where is an antiferromagnetic () Heisenberg model to describe dynamics of localized magnetic moments, and is an extended part to favor the formation of valence bond ordering () . It is not difficult to speculate that an antiferromagnetic phase appears in the case of , breaking SO(3) symmetry involved with spin rotation, while a valence bond ordered state emerges in the case of , breaking Z associated with lattice translation. In this respect one may expect that a critical field theory would enjoy SO(3) SO(2) symmetry in terms of both antiferromagnetic and valence bond order parameters, where the Z symmetry can be enhanced to SO(2) in the continuum limit. However, it has been proposed that the SO(3) SO(2) symmetry may be enlarged to SO(5), where both order parameters form a superspin vector at this antiferromagnetic to valence bond quantum critical point . This scenario is in parallel with the well-known physics of an antiferromagnetic quantum spin chain, where an effective field theory is given by SO(4) WZW theory although its microscopic lattice model enjoys SO(3) Z . This effective field theory turns out to be critical due to the existence of the WZW term, allowing fractionalized spin excitations referred to as spinons . Emergence of an enhanced symmetry is suggested to play a central role in deconfined quantum criticality above one time and one space dimensions . See Fig. 1, which shows a schematic phase diagram of the SO(5) WZW theory and a possible connection to the pseudogap phase of high T cuprates.

In order to take into account the role of Berry phase in the path-integral representation with the spin coherent basis , one may consider a projective representation for the spin operator as follows

 Si=12f†iασαβfiβ, (2)

backup by the single occupancy constraint . Here, we use the Einstein convention. Inserting this expression into the Heisenberg model and decomposing the four-fermion effective-interaction term into particle-hole and particle-particle channels within the singlet domain, we find an effective UV theory in this parton construction as follows

 ZUV=∫DψiαDχijDηijDakiτe−∫β0dτLeff, LUV=L0+Ls,     L0=Jr∑ijtr[U†ijUij], Ls=12∑iψ†iα(∂τ−iakiττk)ψiα+Jr∑ij(ψ†iαUijψjα+H.c.) (3)

with . is a two-component Nambu-spinor, where is an antisymmetric tensor. is an order-parameter matrix, where represents an effective hopping parameter and does a pairing order parameter. is a Lagrange multiplier field to impose the single occupancy constraint with , which may be identified with a time component of an SU(2) gauge field, where two constraint equations from are satisfied trivially by that from .

Performing the saddle-point analysis for the order-parameter matrix, the ground state turns out to be a flux phase , where flux penetrates each plaquette, given by

 UπFij=−χτ3exp[i(−1)ix+iyπ4τ3] (4)

with equal amplitudes between hopping and pairing order parameters as . See Fig. 2. Although amplitude fluctuations of the order-parameter matrix-field are frozen, there exist low-lying transverse excitations, which can be identified with SU(2) gauge fields. Introducing such low energy fluctuations into an effective lattice field theory within the flux phase, we obtain

 Z=∫DψiαDakijDakiτexp[−∫β0dτ{12∑iψ†iα(∂τ−iakiττk)ψiα +Jr∑ij(ψ†iαUπFijeiakijτkψjα+H.c.)+Jr∑ijtr[UπF†ijUπFij]}], (5)

where is a spatial component of an SU(2) gauge field.

It is straightforward to find a continuum field theory for this lattice gauge theory. Turning off the lattice gauge field in the saddle-point approximation, performing the Fourier transformation to the momentum space, and taking the long-wavelength limit near the chemical potential, one finds an effective SU(2) gauge-field theory

 Zeff=∫DψDakμe−∫β0dτ∫d2rLeff, Leff=¯ψγμ(∂μ−iakμτk)ψ−14e2fkμνfkμν, (6)

where SU(2) gauge fluctuations have been recovered. is an eight-component Dirac spinor, composed of four SU(2) doublets living at each site of a plaquette (Fig. 2), where Dirac matrices are given by , , and . is an SU(2) field-strength tensor, where this Yang-Mills dynamics is expected to appear from UV fluctuations of the lattice scale. We emphasize that both the Dirac structure and SU(2) gauge field emerge from the flux fixed-point ansatz.

In order to discuss spontaneous chiral symmetry breaking and find an effective field theory for low energy spin fluctuations, it is necessary to consider the physical symmetry of the matter sector. It is interesting to notice that lattice symmetries such as translations, rotations, and etc. are translated into internal symmetries given by Dirac matrices, for example, where and are associated with lattice translations along the and directions, respectively cccMore precisely, the translational symmetry should be backup by an appropriate gauge transformation in the projective representation . If one sees Fig. 2, he realizes immediately that the translational symmetry is broken explicitly for the configuration of Eq. 4. However, this should be regarded to be an artifact of the mean-field ansatz, where the order parameter field is not gauge invariant, allowing us to perform an appropriate gauge transformation and to recover the translational symmetry. W T, where T is the lattice-translation operator and W is the corresponding gauge transformation, is an element of the projective symmetry group, suggested to classify mean-field ground states of symmetric spin liquids described by emergent gauge theories.. In this way we have SU(2) chiral symmetry with three generators of , , and in addition to the SU(2) spin rotational one . This leads one to propose the SU(2) SU(2) symmetry, where the former is associated with spin rotations and the latter is involved with chiral symmetry. However, an actual global symmetry turns out to be more enlarged as follows . It is clear that this symmetry is closely connected with both spin and Dirac spaces. Since the spin SU(2) symmetry is hidden in the present eight-component representation, one may consider the redundant representation of sixteen-components with a Dirac spinor , regarded to be a time-reversal partner of . Noting that the group space is composed of , one sees ten generators associated with SO(5) symmetry given by , , , . Since SO(8) is the largest global symmetry, considering the Majorana fermion representation for the four-component SU(2) doublet Dirac spinor, the SO(5) symmetry can be regarded to be the largest subgroup, where the emergent Lorentz symmetry forms SO(3) and the SO(8) symmetry group can be decomposed as follows SO(8) SO(5) SO(3).

The above discussion implies that symmetry equivalent operators via the SO(5) rotation have the same strength for instability in this critical spin-liquid state, i.e., the same critical exponent for each correlation function, which suggests an SO(5) superspin vector through the following fermion-mass term with , where the former three components form Neel vectors and the latter two represent and valance bond fluctuations. As a result, one reaches the following Lagrangian for spontaneous chiral symmetry breaking

 Zeff=∫DΨDviDakμexp[−∫β0dτ{¯Ψγμ(∂μ−iakμτk)Ψ−m¯Ψ(v⋅Γ)Ψ−14e2fkμνfkμν}],

where the mechanism of this symmetry breaking is not clarified dddOne may demonstrate that SU(2) gauge fluctuations are responsible for this chiral symmetry breaking.. Integrating over massive fermion excitations and performing the gradient expansion for the superspin vector field, one finds an SO(5) WZW theory for the competing physics between antiferromagnetism and valence bond ordering as follows

 Zeff=∫Dvie−Seff,    Seff=SNLsM+SWZW, SNLsM=∫d3x12g(∂μvi)2,     SWZW=i2πArea(S4)∫10dt∫d3xϵabcdeva∂tvb∂τvc∂xvd∂yve,

where . Although the WZW term can be nicely derived in the absence of gauge fluctuations, an additional imaginary term may arise, a coupling term between gauge fields and Goldstone-Wilczek currents , which correspond to skyrmion currents in the present case if we restrict ourselves only in antiferromagnetic fluctuations instead of the superspin vector. When we represent the eight-component Dirac spinor as , where is a two-component SU(2) doublet with an isospin index , we can see that each sector in the Dirac space gives rise to such a term. However, their signs are opposite, thus such terms are canceled. This is well-known to be cancelation of parity anomaly in the lattice model .

Although it is not straightforward to solve this effective field theory, it would be helpful to revisit the one-dimensional version of this field theory, referred to as SO(4) WZW theory

 SWZW=∫d2x{12g4∑k=1(∂μvk)2+i2πArea(S3)∫10dtϵabcdva∂tvb∂τvc∂xvd}. (9)

Performing the renormalization group analysis in the one-loop level, one finds a conformal invariant stable fixed point, which originates from the existence of the WZW term , given by

 dlngdlnΛ=0, (10)

where is a UV cutoff. Actually, the SO(4) WZW theory is exactly solvable, characterized by the central charge , where such critical boson excitations are identified with fractionalized spin fluctuations called spinons . The SO(3) nonlinear model with a term (Berry phase) at UV flows into the SO(4) WZW theory at IR, where valence bond fluctuations carry exactly the same conformal dimension as antiferromagnetic spin fluctuations.

An important point that we would like to emphasize is as follows. One may try to solve the UV field theory directly, resorting to the CP representation for the SO(3) vector field, where the SO(3) nonlinear model is mapped into an emergent U(1) gauge theory with two flavors of bosonic spinons and the Berry-phase term is identified with an effective electric potential . Taking the easy-plane limit to map this problem into a two-flavor abelian Higgs model with an effective electric potential and performing the duality transformation to map the abelian Higgs model into an effective Sine-Gordon theory for skyrmion excitations as instantons, one may argue that such skyrmions in one-time and one-space dimensions carry the quantum number of valence bond ordering and their dynamics becomes critical, both of which originate from the Berry-phase term . Although this argument is far from being rigorous, where the enhancement of symmetry at IR is difficult to prove, we can find how such nonperturbative physics involved with instanton excitations at UV is revealed in the perturbative analysis (renormalization group) at IR, where valence bond fluctuations are introduced explicitly into an effective field theory through the symmetry enhancement eeeOne may write down the SO(3) nonlinear model, introducing skyrmion (instanton) and anti-skyrmion (anti-instanton) fluctuations explicitly, as follows () represents the number of skyrmions (anti-skyrmions), set to be equal in the respect of energy cost and thus, their total number is . is fugacity of single skyrmion excitations, where is an instanton action. means the moduli space of skyrmions such as their sizes, center-of-mass coordinates, and so on, referred to as collective coordinates and utilized for the first quantization . may be regarded to be the mass of spin fluctuations with spin quantum number , introduced to describe the unimodular constraint . describes dynamics of skyrmions, where the Berry phase term denoted by should be incorporated. describes scattering physics between smooth spin fluctuations and instanton fluctuations, regarded to be an essential part in this field theory. Unfortunately, the procedure until this field theory has not been clarified yet. This expression may be regarded to be formal. Nonperturbative physics would be encoded in this effective field theory, taking into account both topological excitations and smooth fluctuations on equal footing. An interesting point is that the Berry-phase term assigns the quantum number of valence bond ordering to the core of a skyrmion. As a result, scattering between spin fluctuations and skyrmion excitations may be translated into that between antiferromagnetic fluctuations and valence bond excitations. In this respect one may say that the SO(4) WZW theory encode the nonperturbative physics of the SO(3) nonlinear model with the Berry-phase term into the perturbative framework, where the renormalization group analysis in the one-loop level reveals essential physics qualitatively.. Figure 3 shows one mechanism how the nonperturbative physics becomes translated into the perturbative physics.

One may extend the above discussion into two dimensions. Performing the duality transformation for the two-flavor abelian Higgs model in the easy-plane approximation which reduces SO(3) to SO(2) Z, one can find another abelian Higgs model in terms of half-skyrmion (meron) excitations with two flavors, where magnetic monopole excitations give rise to meron and anti-meron pair excitations (hedgehog configurations) localized in time (instantons). However, the Berry-phase term has been proposed to make such instanton events suppressed, preserving the topological charge of meron currents and stabilizing meron excitations at the quantum critical point of this effective field theory . Such meron fluctuations may be identified with spinon excitations in the original representation. Since the magnetic-monopole excitation carries the valence bond order near its core, assigned from the Berry-phase term, their condensation transition identifies the nature of the quantum critical point between the antiferromagnetic state and the valence bond ordered phase. One may go beyond the easy-plane limit. In this case the duality transformation is not clarified, making it difficult to describe the deconfined quantum critical point explicitly. However, it is clear that meron excitations in SO(2) Z should turn into skyrmions (solitons) in SO(3). As a result, the skyrmion current is conserved at the quantum critical point, where fluctuations of magnetic monopoles (instantons) become suppressed, but the conservation law breaks down in the valence-bond solid state, where the proliferation of magnetic monopoles breaks the U(1) global symmetry associated with the conservation of the skyrmion current. In this case spinon excitations can be identified with an emergent spin degree of freedom in a Z vortex core, regarded to be a topological excitation in the valence-bond solid state, where the condensation of Z vortices have been argued to be responsible for the quantum phase transition from the valence bond solid state to the antiferromagnetic phase . An essential point is that this nonperturbative physics from topological excitations at UV can be incorporated by the perturbative physics of the SO(5) WZW theory at IR, where the conformal dimension of the valence bond order parameter is the same as that of the antiferromagnetic one. The renormalization group analysis in the one-loop level is expected to allow a conformal invariant fixed point as the SO(4) WZW theory, which gives rise to deconfined critical spinon excitations. Unfortunately, we do not know an explicit result on the perturbative renormalization group analysis of the SO(5) WZW theory in two dimensions. On the other hand, the emergence of the symmetry enhancement at IR seems to be confirmed by explicit numerical simulations for the extended Heisenberg model although it is difficult to avoid the nature of weakly first ordering in the simulation .

2.2.2 μ−Qed3 coupled with SO(5) WZW theory from t-J Hamiltonian

Effects of hole doping on the SO(5) WZW theory can be investigated, based on the t-J Hamiltonian

 HtJ=−t∑ij(c†iσcjσ+H.c.)+J∑ij(Si⋅Sj−14ninj), (11)

where double occupancy is prohibited. This constraint can be solved, resorting to the projective representation referred to as the SU(2) slave-boson representation for an electron operator ,

 ci↑=1√2h†iψi+=1√2(b†i1fi1+b†i2f†i2), ci↓=1√2h†iψi−=1√2(b†i1fi2−b†i2f†i1), (12)

where spinon and holon doublets are given by , , and , respectively. Resorting to this parton construction, one may rewrite the t-J Hamiltonian as follows

 ZUV=∫DψiαDhiDχijDηijDakiτe−∫β0dτLeff, LUV=L0+Ls+Lh,     L0=Jr∑ijtr[U†ijUij], Ls=12∑iψ†iα(∂τ−iakiττk)ψiα+Jr∑ij(ψ†iαUijψjα+H.c.), Lh=∑ih†i(∂τ−μ−iakiττk)hi+tr∑ij(h†iUijhj+H.c.), (13)

where the Hubbard-Stratonovich transformation has been performed for particle-hole and particle-particle channels in the singlet domain, giving rise to the following order-parameter matrix field with and , as discussed in the half-filled case. The time component of an SU(2) gauge field is to impose the single-occupancy constraint, and is a chemical potential to control hole concentration.

Following the strategy of the half-filled case, the variational analysis for the order-parameter matrix field gives rise to a staggered flux state , given by

 USFij=−√χ2+η2τ3exp[i(−1)ix+iyΦτ3], (14)

where a flux through a plaquette is and alternating. Although the staggered flux ansatz breaks translational invariance, this formal symmetry breaking is restored via SU(2) gauge transformation between nearly degenerate U(1) mean-field states . For example, one possible U(1) ground state, the d-wave pairing state can result from the staggered flux phase through the SU(2) rotation given by , where the corresponding SU(2) matrix is . Thus, this variational state should be regarded as one possible gauge choice, preserving both time reversal and translational symmetries. One can show that the staggered flux phase allows only one low-lying transverse fluctuations, identified with the third component of the SU(2) gauge field. As a result, an effective lattice field theory in the staggered flux state is given by

 +Jr∑ij(ψ†iαUSFijeia3ijτ3ψjα+H.c.)+∑ih†i(∂τ−μ−ia3iττ3)hi +tr∑ij(h†iUSFijeia3ijτ3hj+H.c.)+Jr∑ijtr[USF†ijUSFij]}]. (15)

An idea is to fermionize the holon sector attaching a fictitious flux to a holon field ,

 Lh=∑iη†i(∂τ−μ−ia3iττ3)ηi+tr∑ij(η†iUSFijeia3ijτ3eicijτ3ηj+H.c.) −i∑ici0(η†iτ3ηi−12Θ(∂xcy−∂ycx)i), (16)

where a bosonic field variable now becomes a fermionic one with . It is important to notice that our flux attachment is performed in an opposite way for each isospin sector, confirmed by the presence of in . As a result, there is no net flux in the mean-field approximation of this construction, considering that the density of bosons is the same as that of bosons in the staggered flux phase. This observation is interesting since it suggests a connection with an SU(2) slave-fermion representation . If is shifted to , the Chern-Simons flux is transferred to spinons, turning their statistics into bosons. Then, we have bosonic spinons with fermionic holons, nothing but the slave-fermion representation.

Following the strategy of the half-filled case, we find an effective continuum field theory

 Zeff=∫DψDηDa3μDcμe−∫β0dτ∫d2rLeff, Leff=¯ψγμ(∂μ−ia3μτ3)ψ+12e2(ϵμνγ∂νa3γ)2 +¯ηγμ(∂μ−ia3μτ3−icμτ3)η−μh¯ηγ0η+i4Θcμϵμνλ∂νcλ, (17)

where the Maxwell dynamics of is expected to appear from UV fluctuations of spinons. The Dirac structure results from the staggered flux ansatz, where both and are eight-component Dirac spinors and Dirac gamma matrices are , , and , the same as the half-filled case. It is important to understand that spinons are still at half filling even away from half filling in the SU(2) formulation . The single-occupancy constraint in the SU(2) slave-boson representation is . Thus, if the condition of with hole concentration is satisfied, we see , i.e., spinons are at half filling. As a result, a chemical potential term does not arise in the spinon sector. On the other hand, a chemical potential term appears in the holon sector, allowing four Fermi pockets around each Dirac node, consistent with the observed Fermi surface in the pseudogap phase of high T cuprates. See Fig. 4.

Spontaneous chiral symmetry breaking in this QED can be investigated, taking into account an emergent enhanced symmetry as the half-filled case of the flux state. It turns out that the group structure of enjoys SU(4) symmetry , allowing fifteen generators which correspond to , , , , , , and . There exist additional five generators in addition to the first ten generators of the SO(5) symmetry, satisfying SU(4) algebra. A novel spin-liquid fixed point has been proposed that such an SU(4) symmetry is broken down to SO(5) , where most relevant spin fluctuations are Neel vector and valence bond fluctuations, giving rise to the competition between them. Such spin fluctuations are symmetry equivalent operators via chiral rotation at this emergent novel fixed point. As a result, one is allowed to construct the following effective field theory

 Zeff=∫DΨDηDviDa3μDcμexp[−∫β0dτ{¯Ψγμ(∂μ−ia3μτ3)Ψ−m¯Ψ(→v⋅→Γ)Ψ +¯ηγμ(∂μ−ia3μτ3−icμτ3)η−μh¯ηγ0η−mη¯η(iγ3v4+iγ5v5)η +12e2(ϵμνγ∂νa3γ)2+i4Θcμϵμνλ∂νcλ}], (18)

where of sixteen-components with a Dirac spinor has been introduced for the SO(5) superspin vector field. We point out that dynamics of doped holes couples to valence bond fluctuations in the form of Yukawa coupling since they do not carry spin degrees of freedom. Valence bond fluctuations may be responsible for high T superconductivity in this formulation.

Integrating over massive fermion excitations and performing the gradient expansion for the superspin field, one finds an effective field theory, composed of QED coupled to SO(5) WZW theory,

 Zeff=∫DviDηDa3μDcμe−Seff, Seff=∫d3x{12g(∂μvk)2−mη¯η(iγ3v4+iγ5v5)η}+SWZW +∫d3x{¯ηγμ(∂μ−ia3μτ3−icμτ3−iAμ)η−μh¯ηγ0η+i4Θcμϵμνλ∂νcλ+12e2(ϵμνγ∂νa3γ)2}, SWZW=i2πArea(S4)∫10dt∫d3xϵabcdeva∂tvb∂τvc∂xvd∂yve. (19)

An important observation is that the Chern-Simons contribution becomes irrelevant if the holon dynamics is in a critical phase. Shifting the slave-boson gauge field as and performing the integration of Chern-Simons gauge fields, we obtain . This contribution is irrelevant since it has a high scaling dimension owing to the presence of an additional derivative. Considering that the density of holons is finite to allow Fermi surfaces (pockets around Dirac points), it is natural to assume that the fermion sector lies at quantum criticality. As a result, we find an effective field theory for antiferromagnetic doped Mott insulators

 Seff=∫d3x{12g(∂μvk)2−mη¯η(iγ3v4+iγ5v5)η}+SWZW +∫d3x{¯ηγμ(∂μ−ia3μτ3−iAμ)η−μh¯ηγ0η+12e2(ϵμνγ∂νa3γ)2}, (20)

which describes mutual effects on valence bond fluctuations and charge dynamics in the presence of the topological term.

We would like to emphasize that this field theoretic formulation makes effective interactions between doped holes and valence bond fluctuations explicit, allowing us to perform the perturbative analysis. If we do not take into account the valence bond order parameter explicitly, i.e., resorting to the SO(3) nonlinear model description instead of the SO(5) WZW theory, we may obtain the following effective field theory for dynamics of doped holes, which scatter with magnetic monopole excitations as follows

 Zeff=∑Nt=N+¯N∈evenN!¯N!Nt!yN+¯N∫DMDnDχDa3μ exp[−∫d3x{12g(∂μn)2+m2(|n|2−1)+¯χγμ(∂μ−ia3μτ3−iAμ)χ−μh¯χγ0χ +12e2(ϵμνγ∂νa3γ)2}−Sm[M;Θ]−Sχ−meff[χ,M]−Sn−meff[n,M]]. (21)

Since most mathematical symbols have been explained in the footnote for the SO(3) nonlinear model, we do not repeat them here, where instanton excitations are identified with magnetic monopole fluctuations. is a Dirac spinor to represent doped holes, where such a field variable becomes modified from due to scattering with a pair of monopole and anti-monopole. An important point in this effective field theory is the scattering term between instanton fluctuations and doped holes, where such topological excitations carry the quantum number of valence bond ordering, given by the Berry-phase term. However, it is not straightforward to derive such effective interactions at all. On the other hand, they are incorporated by

 Sint=−∫d3xmη¯η(iγ3v4+iγ5v5)η (22)

explicitly in the SO(5) WZW theoretical formulation. It would be quite appealing to find any signatures of the effective field theory Eq. (20) from Eq. (21).

2.2.3 Application to one dimension: Luther-Emery phase

It is interesting to apply the QED coupled to the SO(5) WZW theory to one dimension. In one dimension the spin sector is described by the SO(4) WZW theory, and the charge sector is represented by QED without the chemical potential term . Accordingly, the coupling term between valence bond fluctuations and holons is adjusted. The resulting effective field theory is given by

 S=∫d2x{12g4∑k=1(∂μvk)2+i2πArea(S3)∫10dtϵabcdva∂tvb∂τvc∂xvd} +∫d2x{¯ηγμ(∂μ−ia3μτ3−iAμ)η−mη¯η(iγ5v4)η+12e2(ϵ3μν∂μa3ν)2}, (23)

where is a four-component superspin vector field with three Neel components and one valence bond order parameter and is a four-component Dirac spinor with two-by-two Dirac matrices of and . The physical origin of the SO(4) WZW term has been attributed to non-abelian chiral anomaly within the path integral formulation, where classically conserved non-abelian chiral currents with Pauli spin matrices turn out to be not preserved in a background with a nontrivial topology .

Performing the abelian bosonization for the fermion sector , we obtain the following expression

 S=∫d2x{12g4∑k=1(∂μvk)2+i2πArea(S3)∫10dtϵabcdva∂tvb∂τvc∂xvd} +∫d2x{12(∂μϕ+)2+12(∂μϕ−)2+(Λπmη)v4sin(√4πϕ+)+(Λπmη)v4sin(√4πϕ−) −ia3μ(12πϵμν∂νϕ+−12πϵμν∂νϕ−)−iAμ(12πϵμν∂νϕ++12πϵμν∂νϕ−)+12e2(ϵμν∂μa3ν)2},

where the subscript in the bosonic field represents the SU(2) doublet involved with , and is a cutoff associated with band linearization. Performing integration for U(1) gauge fields, we find a mass-type term . This allows us to set in the low energy limit. Shifting with , we are led to

 S=∫d2x{12g4∑k=1(∂μvk)2+i2πArea(S3)∫10dtϵabc<