Fermion Dynamical Symmetry and Strongly-Correlated Electrons: A Comprehensive Model of High-Temperature Superconductivity

The lack of agreement concerning the mechanism for high-temperature superconductivity, and the limited understanding of how these mechanisms are connected to the various other occurrences of superconductivity noted above, have two ready explanations:

1. 1.

   In each case the issues are complex but most models emphasize only limited aspects of the overall problem.

2. 2.

   Often the superconducting mechanism is obscured by the complex behavior because the dearth of solvable models with broad physics content makes it difficult to separate essential features from secondary ones.

Thus, we believe that the primary issue is not that there are key measurements remaining to be made that will magically unravel the HTSC problem, but rather that most theoretical tools are inadequate to assess properly the implications of the quite sophisticated data sets already in existence. Stated concisely, theoretical models that deploy a sufficiently broad arsenal of physics tend to not be solvable, while those models that are solvable tend to be so at the cost of emphasizing certain aspects of the problem relative to others. This latter feature of solvable models then tends to produce strong either/or dichotomies around which various camps rally.

We propose here that a solvable model with an adequate range of physics for the HTSC problem is possible by applying a set of mathematical tools originating in the theory of Lie algebras and generalized coherent states. The methods that we shall use to accomplish this might be viewed as unconventional in that they have not found broad use in condensed matter physics prior to the work reviewed here. However, their validity is well established in strongly-correlated fermion and boson systems for fields such as nuclear FDSM ; IBM , elementary particle Bi94 , molecular Ia95 , and polymer physics Ia99 , and—truth be told—they draw substantially on ideas that originated in condensed matter and related fields. The dynamical symmetry methods discussed here may be viewed as sophisticated generalizations of pseudospin models, which were introduced in application to the BCS model by Anderson ande58 , and later used extensively in both condensed matter and nuclear physics. Likewise, the method of generalized coherent states is a sophisticated extension of Glauber coherent states gla63 , which have been employed often in condensed matter contexts.

By making use of these powerful methods, we shall arrive at a physical picture that is surprisingly conventional relative to many proposed explanations of high-temperature superconductivity. We shall show that a theory unifying self-consistently superconductivity built on a BCS-like wavefunction (possibly with unconventional order) and the Néel model of antiferromagnetism in the presence of strong Coulomb repulsion leads, with few further assumptions, to physics very similar to that observed in actual high-temperature superconductors.

It is useful to begin the discussion with some things that are are not so contentious. Despite the absence of general agreement on the mechanism for high-temperature superconductivity in the cuprates and iron-based compounds, we believe that there is fairly broad consensus on four issues.

(1) High-temperature superconductivity is related intimately to an interplay among pairing, antiferromagnetism, and charge degrees of freedom for strongly-correlated electrons, as illustrated schematically in Fig. 1.

(2) The superconducting state is a condensate of spin-singlet Cooper pairs behaving in many respects as an ordinary BCS superconductor, but with important differences, particularly in states with low doping. The orbital pairing formfactor for the cuprate superconductors is of $d_{x^{2}-y^{2}}$ form and dominated by contributions from a single band. Figure 2 illustrates.

For the iron-based superconductors there is less certainty but it is generally believed that multiple bands near the Fermi surface contribute and that there may be more than one pairing gap. The orbital symmetry of the gap often appears to be a modified $s$-wave form, but it is possible that the symmetry of the gap may vary with the compound being examined hirs2011 .

(3) The precursor to the cuprate superconducting state is an antiferromagnetic (AF) Mott insulator (an insulating state in which the insulator properties derive specifically from strong onsite Coulomb repulsion), unlike for ordinary superconductors where the precursor is a normal Fermi liquid (an interacting system of fermions having excitations that can be put into one-to-one correspondence with those of a non-interacting fermion system). Figure 3 illustrates.

The precursor to the iron-based superconductors is typically a poor AF metal that might (or might not) be near a Mott transition.

(4) The pseudogap, corresponding to an observed partial gapping of states in the underdoped region above the superconducting transition temperature pseudogap ; pseudogap2 , is a well-established enigma for the cuprate superconductors (Fig. 4).

Whether a pseudogap exists for the iron-based compounds is not settled (positive evidence for one is given in Refs. shee2010 ; mert2009 ; ahil2008 ; sato2008 ; ning2008 ). The pseudogap state is not a “normal” normal state, and it is not well described by traditional Fermi liquid concepts. There is a general feeling that understanding the pseudogap state is necessary to understanding HTSC, at least for the cuprates.

In contrast to the areas enumerated in the previous section where there is relatively uniform agreement, there are a number of important issues for which there is little agreement. These share two common features, in our opinion: (1) Their resolution may be central to understanding the high-temperature superconducting mechanism. (2) A resolution of these issues requires a multiphysics approach capable of integrating concepts on a similar footing that could be treated as approximately independent in many simpler problems.

In this paper we use dynamical symmetries implemented in terms of Lie algebras and generalized coherent state to address all of these issues simultaneously. Specifically, we demonstrate in a solvable model motivated by cuprate phenomenology that the ground state of a minimal implementation of competing singlet pairing, antiferromagnetism, charge, and spin is an antiferromagnetic Mott insulator that is fundamentally unstable with respect to condensing electron hole pairs (and thus becoming a superconductor) at any finite hole doping. Furthermore, the same solution implies pseudogap states having many properties that are in quantitative agreement with data. We shall show that this ground state is unique near half filling and near optimal doping and beyond, but can become highly degenerate in the underdoped region. This degeneracy implies extreme sensitivity to perturbations and thus to a variety of induced inhomogeneity and other emergent behavior, but only in a narrow range of doping for underdoped compounds.

Then, we shall argue that the dynamical symmetry motivated by cuprate phenomenology is—despite superficial differences—also appropriate for a description of the Fe-based superconductors, thus providing a unified description of cuprate and iron superconductors. Indeed we shall argue that superficial differences between the cuprate and iron superconductors (for example, different orbital pairing symmetry) are not essential to a unified description at the broad level of understanding emergent superconductivity in these systems.

We shall then discuss the relationship of the theory derived in the present paper to various other proposals to explain high-temperature superconductors. We shall show, for example, that SU(4) coherent states contain much of the physics of resonating valence bond (RVB) models, but without some specifically RVB assumptions and with broader physics than RVB, that the present dynamical symmetry methodology provides a microscopic derivation of the Zhang SO(5) model as one of its approximate solutions, and that Hubbard or $t$-$J$ models and the present methodology are related by the adoption of fundamentally different (but equally valid) approaches to truncation of the Hilbert space for the full problem.

Finally, based on the experience with normal and unconventional superconductors discussed in this review and in the review of nuclear structure physics discussed in Ref. FDSM , we shall suggest an even more sweeping conjecture. Although superconductivity occurs in many forms in a variety of disciplines, we believe that all of these forms may represent a single basic mechanism involving a Cooper pairing instability that may occur in the presence of other strongly collective modes, and that has a common algebraic description across diverse systems in terms of fermion dynamical symmetries.

Microscopic-properties truncation identifies key microscopic physical features of the idealized weakly-interacting system that are expected to be valid for the actual correlated many-body system, and uses that as a guide for truncating the full space. Typically Hubbard or $t$–$J$ model approaches are of this form. An assumption is made about the microscopic form of the important physical interactions and this is used to construct a simple Hamiltonian. In principle this implies no truncation of the configuration space, but in practice calculations are possible only if a small-enough subspace is chosen. The choice of this space is guided by microscopic physical insight and symmetries of the interactions (for example, a basis of spin states), often assuming that only states below some energy cutoff in the non-interacting basis contribute.

However, it is often not feasible to implement microscopic-properties truncation without drastic assumption. In high-$T_{\mbox{\scriptsize c}}$ superconductors, the correlations may be so strong that dynamics can no longer be given a meaningful description in terms of individual fundamental particles of the weakly-interacting system (see, for example, the discussion in Refs. An00 ; pines00 ; laug00 ). The essential physics then tends to be governed by a few collective modes that are emergent. (An emergent mode is a new collective state that emerges because of interactions and not because it exists already in the microscopic constituents of the non-interacting system). Then a quite different kind of simplification is possible, based on what we shall term emergent-symmetry truncation.

Truncation based on emergent symmetries is tailored for collective modes and long-range order. It identifies essential forms of the collective modes of interest and uses that as a guide to remove from the full space all states inconsistent with these forms, leaving a small collective subspace that is highly sympathetic to the relevant collective modes. In picturesque terms, we might also call this Michelangelo truncation, since the famous sculptor is said to have replied to a query about how he made such beautiful statues that he looked at the block of stone, envisioned the statue trapped within it, and then chipped away everything that wasn’t statue.

The most powerful systematic method of determining the essential form of collective modes is to identify a symmetry associated with them. We shall call this a symmetry of emergent modes, as opposed to symmetries of the weakly-interacting system. Note that if a mode is emergent there is no reason to expect its dynamical symmetries to have any direct relationship with the symmetries of the Hamiltonian for the weakly-interacting system in the full space before truncation.

This latter point is sufficiently important to merit further attention. A symmetry that is important at both the microscopic and collective levels is that associated with angular momentum (isotropy of space), which is conserved microscopically and also in the collective mode if it is treated exactly. But if we demand that angular momentum be conserved both microscopically and in the collective mode, the use of symmetry to truncate the space is limited to using the Clebsch–Gordan series to decompose the space into a finite number of subspaces that can be solved independently because each is labeled by a conserved angular momentum quantum number—in matrix language, the full matrix is transformed into one with block-diagonal form by a similarity transformation, permitting each block-diagonal submatrix to be solved independently.

A far more spectacular simplification results if we give up the requirement of angular momentum conservation for the collective state and treat it as geometrically deformed, thereby breaking rotational invariance. Then the symmetries of the emergent collective state do not include that of rotational invariance and the solution fails to conserve angular momentum. We then say that the emergent state breaks the symmetry of the true Hamiltonian for the system (which certainly does conserve angular momentum) spontaneously, because the Hamiltonian is symmetric but the wavefunction is not. Strictly, such a state is unphysical, because it breaks a symmetry observed by the exact solution. However, the broken-symmetry approximate solution can often be a very useful simplification because (1) the symmetry violation may not be very important for many physical properties of the system, and (2) there are established techniques to restore the broken symmetry of the collective state by the use of projection integrals when it is important to do so.

The Bohr–Mottelson approach to nuclear structure physics utilizes as approximate solutions to the nuclear many-body problem deformed states that may break rotational invariance bohr-mottelson . In the language employed here, these imply emergent symmetries. A second example of the emergent symmetries described above is the ordinary BCS superconducting state, which is separated by a phase transition from the normal state and cannot be reached by perturbation of the normal state. If we view the SC state as being described by a symmetry [pseudospin SU(2) in the case of a simple BCS superconductor], this symmetry is a symmetry of the collective many-body state, not of the underlying microscopic system. It is emergent. In this example, the BCS state may be viewed as a state that breaks rotational invariance in gauge space; more prosaically, it fails to conserve particle number ¹¹1 The broken particle number symmetry can be restored by particle-number projection, but in practice this procedure may not be necessary as we are dealing with a system having a very large number of fermions. .

The SU(4) symmetry described in this review may be understood as a very sophisticated pseudospin model embodying emergent symmetries associated with the collective modes of the system. In the symmetry limits it is an exact many-body solution, but in the coherent state approximation that we shall often employ it becomes a spontaneously-broken symmetry.

Assuming the validity of the preceding conjecture, we may implement the following algorithm.

The only approximation to the full quantum-mechanical problem in our approach is the space truncation. If all degrees of freedom are incorporated the resulting theory is microscopic and exact. Of course, practically only a few carefully-selected degrees of freedom can be included and the effect of the excluded space must be incorporated through renormalized (effective) interactions operating in the truncated space.

Thus, the utility of this approach depends on making a wise choice for the relevant collective degrees of freedom and on the availability of sufficient phenomenological or theoretical information to specify the effective interactions that operate within the truncated space. The validity of the resulting formalism then stands on whether predicted matrix elements agree with corresponding physical observables, once a small set of effective interaction parameters has been fixed by comparison with the global data set.

Motivated by phenomenology of the cuprates, our basic physical assumption is that the configuration space for a minimal theory of high-temperature superconductivity is built from coherent pairs representing superpositions of particles or holes centered on different lattice sites (which we shall term bondwise pairs; see Fig. 7b). In the interest of constructing a minimal theory, we shall neglect pairing between next nearest neighbors. We shall also consider an extended theory including in addition pairs of particles or holes defined on the same lattice sites (which we shall term onsite pairs; see Fig. 7a). However, as will be discussed further in §VII.2 below, such onsite pairing configurations are important in conventional superconductivity but are likely of less importance for the low-lying states in high-temperature superconductors. The next step is to use phenomenology as a guide to identify a set of operators that can be associated with the relevant physical degrees of freedom exhibited by this collective subspace.

We propose to solve for the doping and temperature dependence of observables in a theory that incorporates on an equal footing antiferromagnetism and (possibly unconventional) superconductivity. To construct a Hamiltonian embodying these degrees of freedom, and conservation laws for charge and spin, we employ the concept of a complete set of quantum operators, which may be defined in either physical or mathematical terms. (1) Physically, a complete set of quantum operators represents all degrees of freedom produced if an initial set of operators is allowed to undergo all possible (those not forbidden by fundamental principles) interactions among themselves. (2) Mathematically, a complete set of quantum operators corresponds to a set of creation and annihilation operators closed under the operation of commutation, implying that the operators form a Lie algebra.

For the high temperature superconductor problem we require at a minimum three staggered magnetization operators $\vec{Q}$ to describe antiferromagnetism, creation and annihilation operators $p^{\dagger}$ and $p$ for bondwise singlet pairs plus a number operator $\hat{n}$ to describe superconductivity, and three spin operators $\vec{S}$ to describe electron spin. However, this set of nine operators is physically incomplete since scattering of singlet pairs (antiparallel spins on adjacent sites) from the AF particle–hole degrees of freedom can produce triplet pairs (parallel spins on adjacent sites), which are not part of the operator set. This is illustrated in Fig. 8.

The mathematical statement of this incompleteness is that the set $\{\vec{Q},p^{\dagger},p,\hat{n},\vec{S}\}$ does not close a Lie algebra under commutation of set members, because commuting singlet-pair operators with antiferromagnetic operators produces triplet pair operators and commuting triplet-pair operators with antiferromagnetic operators produces singlet pair operators (inset to Fig. 8). The physical picture of Fig. 8 suggests that a self-consistent Hilbert space containing singlet pairs must contain triplet pairs also if antiferromagnetic interactions are present. As we shall now demonstrate, a minimal physically-complete operator set results if we add to the original nine operators three creation and three annihilation operators for bondwise spin-triplet pairs defined on adjacent lattice sites.

The operators that we shall use are particle–hole symmetric (though our predictions for observables generally will not be; see the discussion in note particle-hole ). Unless specified explicitly in the following, we shall use “electrons" to reference either electrons or electron holes. We begin by introducing the following 16 operators

in which we define

where $\alpha_{\bm{k}b}$ is the amplitude to find an electron in a band labeled by $b$ with momentum $\bm{k}$, $c_{\bm{k},b,i}^{\dagger}$ creates a fermion of momentum $\bm{k}$ and spin projection $i,j=1{\rm\ or\ }2=\ \uparrow$ or $\downarrow$ in band $b$, $\bm{Q}$ is an AF ordering vector, $\Omega$ is the effective lattice degeneracy, which is the maximum allowed number of doped electrons that can form coherent SU(4) pairs (explained further below), and $g(\bm{k})$ is a pairing formfactor. We may attach a simple physical interpretation to the operators in Eqs. (1):

• •

  The vector $\vec{S}$ is the electron spin operator.

• •

  The vector $\vec{Q}$ is the staggered magnetization characterizing the antiferromagnetism.

• •

  $\vec{\pi}^{\dagger}$ ($\vec{\pi}$) is a vector of creation (annihilation) operators for bondwise spin-triplet pairs.

• •

  $p^{\dagger}$ ($p$) is a creation (annihilation) operator for bondwise singlet pairs.

• •

  $\hat{n}$ is the electron number operator.

• •

  $Q_{+}$ is a commensurate charge density wave operator.

It will sometimes prove useful to replace the number operator $\hat{n}$ in Eq. (1f) with

where we may interpret $M$ physically as the charge operator.

The preceding operators may receive contributions from more than one band. If we introduce effective one-band creation and annihilation operators through

then Eqs. (2) may we written as

which is the form of the operators for the original single-band SU(4) model introduced in Ref. guid01 . Although Eqs. (5) are an adequate starting point for discussion of cuprate superconductivity, the multiband expressions in Eqs. (2) are more appropriate for applications such as the iron-based superconductors where multiband pairing is important.

Inserting the AF ordering vector $\bm{Q}=(Q_{x},Q_{y})=(0,\pi)$ appropriate for the observed FeAs magnetic structure, or $\bm{Q}=(Q_{x},Q_{y})=(\pi,\pi)$ appropriate for the cuprates, and calculating all commutators for the 16 operators in Eqs. (1), the set is found to be closed under commutation if three conditions are satisfied by the pairing formfactor:

(We shall elaborate on the physical meaning of these constraints in §VII.1.) If these conditions are met, the operators defined in Eq. (1) close a ${\rm U(4)}\supset{\rm U(1)}\times{\rm SU(4)}$ Lie algebra, where the U(1) factor is generated by the commensurate charge density wave operator $Q_{+}$, which commutes with all other generators guid01 ; guid04 . Because of the direct-product structure, one can without loss of generality view the theory as an SU(4) theory describing superconductivity and antiferromagnetism, and global charge and spin conservation, with the U(1) charge-density wave sector treated independently. This SU(4) group has three independent subgroup chains

ending in the subgroup ${\rm SU(2)}_{\mbox{\scriptsize s}}\times{\rm U(1)}$ representing spin and charge conservation, with the U(1) subgroup being generated by the charge operator $M$ and the ${\rm SU(2)}_{\mbox{\scriptsize s}}$ subgroup being generated by the three spin operators $S_{1}$, $S_{2}$, and $S_{3}$. Some important properties of this group and subgroup structure are summarized in Tables 1 and 2 of Appendix A.

As we shall discuss further below, the subgroup chains defined in Eqs. (7) imply three fermion dynamical symmetries FDSM that will permit exact many-body solutions to be obtained for particular ratios of antiferromagnetic and pairing coupling strengths. The methodology is that of symmetry-dictated truncation, as outlined in §III. The subgroup chains of Eqs. (7) and the corresponding many-body solutions imply that charge and spin are conserved. In §VI we shall find it useful to introduce approximate solutions through coherent state methods that are generalizations of the BCS solution and lead to spontaneous symmetry breaking and to intrinsic states violating particle number conservation, but the starting point (7) conserves both charge and spin.

The group SU(4) is rank-3 and the irreducible representations (irreps) may be labeled by three weight-space quantum numbers, $(\sigma_{1},\sigma_{2},\sigma_{3})$ su4so6 . We assume a collective subspace illustrated in Fig. 9

that is spanned by the vectors

If there are no unpaired particles, the collective subspace is associated with “fully-stretched”, and therefore maximally collective, irreducible representations (irreps) of the form

(More general representations having broken pairs of particles are discussed in Ref. sun05 .) A subspace associated with such maximally-collective configurations is an obvious candidate for describing the lowest-energy states of the system. This wavefunction represents a coherent superposition of pairs and implies a rather rich structure, as illustrated for the real space in Fig. 10.

The commensurate charge density wave operator $Q_{+}$ defined in Eq. (1g) is the generator of the ${\rm U(1)}$ factor in ${\rm U(4)}\supset{\rm U(1)}\times{\rm SU(4)}$. It commutes with all generators so it annihilates the state $|S\rangle$ and

which means that charge-density wave excitations are excluded in the symmetry limits for the low-lying collective subspace of the effective theory.

Thus we shall ignore the U(1) charge-density wave for much of the subsequent discussion and concentrate on the antiferromagnetic–superconductivity competition embodied in the SU(4) symmetry. However, we shall address the issue of possible charge degrees of freedom induced by perturbations of the SU(4) subspace later in §XI.2.

The group SU(4) has a quadratic Casimir operator (see Table 1 of Appendix A),

and the corresponding expectation value of the SU(4) quadratic Casimir operator evaluated in the irreducible representations (9) is a constant,

The most general 2-body Hamiltonian within the collective pair subspace consists of a linear combination of (lowest-order) Casimir operators $C_{\mbox{\scriptsize g}}$ for all subgroups g (Table 1 of Appendix A) threebodynote . These are

We may utilize that the SU(4) Casimir expectation value (12) is constant to eliminate terms in $\vec{\pi}^{\dagger}\cdot\vec{\pi}$, and the most general SU(4) 2-body Hamiltonian can be expressed as guid01 ; guid11

where $H_{0}$, $\tilde{G}_{0}$ and $g^{\prime}$ are effective interaction parameters, $p^{\dagger}$ creates singlet pairs, $\vec{Q}$ is the staggered magnetization, $\vec{S}$ is spin, $\tilde{G}_{0}=\chi(x)+G_{0}(x),$ and the parameter $\sigma$, given by

with $G_{0}(x)$ and $\chi(x)$ the effective SC and AF coupling strengths, respectively, governs the relative strength of antiferromagnetic and pairing interactions in the subspace sun06 ; sun07 . In these expressions, doping is characterized by a parameter

for an $n$-electron system, with $\Omega$ the maximum number of doped holes (or doped electrons for electron-doped compounds) that can form coherent pairs, assuming the normal state at half filling ($n=\Omega$, implying $M=0$) to be the vacuum. Since $\Omega-n$ is the hole number when $n<\Omega$, positive $x$ represents the case of hole-doping, with $x=0$ corresponding to half filling (no doping) and $x=1$ to maximal hole-doping. Negative $x$ ($n>\Omega$) is then the relative doping fraction for electron-doping.

The parameter $x$ is the effective doping concentration, corresponding to the ratio of the number of doped pairs $\tfrac{1}{2}(\Omega-n)$ to the pair degeneracy $\tfrac{1}{2}\Omega$ of the collective subspace. The physical doping fraction, defined as $P=(\Omega-n)/\Omega_{e}$, where $\Omega_{e}$ is the number of physical lattice sites, is related to $x$ through

with $P>0$ for hole doping and $P<0$ for electron doping. $P_{\mbox{\scriptsize f}}$ can be regarded as the maximum value of the physical doping fraction $P$ that is consistent with SU(4) symmetry, as we shall explain further below.

The SU(4) subgroup chain

which we term the SO(4) dynamical symmetry for brevity, is the symmetry limit of Eq. (14) when $\sigma=1$. As we now argue, it corresponds to a collective state having long-range antiferromagnetic order and Mott insulator character. The SO(4) subgroup is locally isomorphic to the product group ${\rm SU(2)}_{F}\times{\rm SU(2)}_{G}$ that is generated by the linear combinations

of the original SO(4) generators $\vec{Q}$ and $\vec{S}$. The new generators $\vec{F}$ and $\vec{G}$ may be interpreted physically by noting that if $Q_{ij}$ and $S_{ij}$ defined in momentum space in Eq. (2) are transformed to the physical coordinate lattice we obtain,

implying that $\vec{F}$ is the generator of total spin on even sites and $\vec{G}$ is the generator of total spin on odd sites, as illustrated schematically in Fig. 13.

Thus, SO(4) is generated by two independent spin operators: one that is the total spin on all sites and one that is the difference in spins on even and odd sites of the spatial lattice. This clearly is an algebraic version of the physical picture associated with antiferromagnetic long-range order.

The SO(4) Casimir operator may be expressed as

The SO(4) representations can be labeled by the spin-like quantum numbers $(F=\tfrac{1}{2}w,G=\tfrac{1}{2}w)$, where $w=N-\mu$ with $\mu=0,2,\ldots,N$. Eigenstates are labeled by $w$ and the spin $S$,

and are of dimension $(w+1)^{2}$.

The ground state corresponds to $\omega=N$ and $S=0$, and has $\tfrac{1}{2}n$ spin-up electrons on the even sites ($F=N/2$) and $\tfrac{1}{2}n$ spin-down electrons on odd sites ($G=N/2$), or vice versa. Thus it has maximal staggered magnetization,

and a large energy gap associated with the antiferromagnetic correlation $\vec{Q}\cdot\vec{Q}$,

In addition, the pairing gap

is small near half filling ($x=0$), and we shall demonstrate below that the SU(4) symmetry requires that the lattice have no double occupancy at half filling and thus be a Mott insulator. We conclude that near half filling these SO(4) states are identified naturally with a collective, antiferromagnetic, Mott insulating state. This identification will be strengthened below by examination of the ground-state energy surface evaluated in this limit (illustrated in Fig. 23 and discussed in §VIII.1).

The SU(4) subgroup chain

which we shall term the SU(2) dynamical symmetry for brevity, corresponds to the $\sigma=0$ symmetry limit of Eq. (14). The eigenstates are labeled by $v$ and spin $S$,

and are of dimension $\tfrac{1}{2}(v+1)(v+2)$. The seniority-like quantum number $v$ is the number of particles that do not form singlet pairs (see Table 2 of Appendix A). The ground state has $v=0$, implying that all electrons are singlet-paired. In addition, there exists a large pairing gap

(Table 2 of Appendix A), the pairing correlation is the largest among the three symmetry limits, and the staggered magnetization vanishes in the ground state:

Thus we interpret this dynamical symmetry as a $d$-wave pair condensate associated with a collective spin-singlet superconducting state. This identification will be strengthened below by examination of the ground-state energy surface evaluated in this limit (illustrated in Fig. 23 and discussed in §VIII.2).

The SU(4) subgroup chain

which we shall term the SO(5) dynamical symmetry, has the nature of a transitional or critical dynamical symmetry. This symmetry limit results when $\sigma=\tfrac{1}{2}$ in Eq. (14). The SO(5) irreps are labeled by a quantum number $\tau$ and the eigenstates may be labeled by $\tau$ and the spin $S$,

with $N=\Omega/2-\tau+\lambda$, where $\lambda$ is the number of $\pi$ pairs. For given $N$ the irreducible representations are of dimensionality $\tfrac{1}{2}(\lambda+1)(\lambda+2)$ and the ground state has $\lambda=0$ and $S=0$.

The SO(5) dynamical symmetry has very unusual character. Although the expectation values of $\Delta$ and $Q$ for the ground state in this symmetry limit are the same as those of Eq. (25) for the SU(2) case, there exist many states with different values of $\lambda$ (the number of $\pi$ pairs) that can mix easily with the ground state when $x$ is small because the excitation energy in this symmetry limit is

(see Table 2 of Appendix A). In particular, at half filling ($x=0$) the ground state is highly degenerate with respect to $\lambda$ and mixing different numbers of $\pi$ pairs in the ground state costs no energy. The $\pi$ pairs must be responsible for the antiferromagnetism in this phase, since within the model space only $\pi$ pairs carry spin if there are no unbroken pairs. Thus the ground state in this symmetry limit has large-amplitude fluctuation in the AF (and SC) order. As such, it will be seen to play a role as a doorway between antiferromagnetic and superconducting order, as illustrated in Fig. 14.

This identification will be strengthened below by examination of the ground-state energy surface evaluated in this limit (illustrated in Fig. 23 and discussed in §VIII.3).

The work of Gilmore gil72 ; gil74 and Perelomov per72 (see also Klauder kla63 ) showed that Glauber coherent states gla63 of the electromagnetic field could be generalized to coherent states specified by an arbitrary Lie algebra. Specifically, it was found that the original Glauber theory for coherent photon states could be expressed in terms of an SU(2) Lie algebra by examining the commutation properties of the second-quantized operators of the theory, and then this formalism could be generalized to encompass a set of such operators closed under any Lie algebra. The resulting theory can apply to either fermion or boson fields, provided that the relevant creation and annihilation operators close a Lie algebra. This extension of the Glauber theory to arbitrary Lie algebras is termed the generalized coherent state method. A comprehensive review exists zhan90 , so we proceed directly to application of the generalized coherent state method to fermions described by the SU(4) algebra of §IV.3.

The SU(4) coherent state $|\psi\rangle$ can be expressed as

where the operator ${\cal T}$ is given by

In Eq. (27), $|0^{*}\rangle$ is the physical vacuum (ground state) of the system, the real parameters $\eta_{00}$ and $\eta_{10}$ are symmetry-constrained variational parameters, and h. c. stands for hermitian conjugation. The variational parameters weight the elementary excitation operators $p^{\dagger}_{12}$ and $q^{\dagger}_{12}$ in Eq. (28), so they represent collective state parameters for a pair subspace truncated under the SU(4) symmetry coherent-parameters . We then express a symmetry-constrained variational Hamiltonian as

where $H$ is the Hamiltonian given in Eq. (14) and $\lambda$ is the chemical potential, determined by requiring that the average particle number be conserved. The parameters $\eta_{00}$ and $\eta_{10}$ in Eq. (28) are determined by the variational condition $\delta\langle H^{\prime}\rangle=0$, where

is the expectation value of $H^{\prime}$ with respect to the ground state $|0^{*}\rangle$.

As shown in Appendix A of Ref. sun05 , it is convenient to evaluate the variation $\delta\langle H^{\prime}\rangle=0$ using a 4-dimensional matrix representation that was employed in Refs. lawu03 ; wmzha88 . In this representation the unitary operator ${\cal T}$ implements a transformation from the original particle basis to a quasiparticle basis and the variational parameters $\eta_{00}$ and $\eta_{10}$ are replaced, respectively, by $u_{\pm}$ and $v_{\pm}$, subject to a unitarity condition

Under this transformation the physical vacuum state $|0^{*}\rangle$ is transformed to a quasiparticle vacuum state $|\psi\rangle$ and the basic fermion operators $c_{\bm{r}i}$ are converted to new quasiparticle operators $a_{\bm{r}i}$ through the transformation

(see Appendix A of Ref. sun05 ), with

Thus this is a transformation of the Bogoliubov form: each quasiparticle state is a mixture of a particle and a hole, and the coherent state $|\psi\rangle$ is a quasiparticle vacuum constrained to respect SU(4) symmetry. It follows that the generalized coherent state method described here is equivalent to the most general Hartree–Fock–Bogoliubov (HFB) variational method, but subject to an SU(4) symmetry constraint guid99 ; guid01 ; lawu03 . By using the 4-dimensional matrix representation, the expectation value for any operator $\hat{O}$ in the coherent state representation can be calculated through the transformation

as detailed in Appendix A of Ref. sun05 .

At finite temperature $|\psi\rangle$ will generally no longer be a quasiparticle vacuum state and the quasiparticle annihilation operators acting on $|\psi\rangle$ do not necessarily give zero. In Appendix B of Ref. sun05 a formalism is derived to deal with the finite-temperature case, by replacing the state $|\psi\rangle$ with a state $|\psi(T)\rangle$ in which the quasiparticles at temperature $T$ may be thermally excited.

For a temperature $T$ we assume that the single-particle levels $\varepsilon_{r\pm}$ [which are defined in Eq. (42) below] are degenerate and contain $\tilde{n}_{r+}+\tilde{n}_{r-}$ quasiparticles. The quasiparticle number densities are assumed given by the Fermi–Dirac distribution

where $e_{\pm}$ is the quasiparticle energy defined in Eq. (41) below and $R$ is an empirical scaling factor of order one that corrects on average for approximations that we shall make for the single-particle spectrum.

We may then evaluate the expectation value of one-body operators at finite temperature, with the result (see Appendix B of Ref. sun05 )

where in these expressions we have employed the definitions

If a large-$\Omega$ approximation is invoked by ignoring terms of order $1/\Omega$ relative to the leading terms, the scalar products of these one-body operators reduce to products of the corresponding one-body ones,

If $T\rightarrow 0$, then $P_{\pm}(T)\rightarrow 1$ in Eq. (36) and Eqs. (35) and (37) reduce to the simpler forms

which are valid only for $T=0$ (and large $\Omega$ for the 2-body terms).

The preceding results may be used to find the expectation value of the variational Hamiltonian $\langle H^{\prime}\rangle$ in the coherent state representation. In terms of the energy gaps defined by

one obtains from Eqs. (29), (30), and (14),

where $\epsilon$ is the single-particle energy. Variation of $\langle H^{\prime}\rangle$ with respect to $u_{\pm}$ or $v_{\pm}$ yields

which is satisfied by

where

and

Inserting Eq. (40) into Eqs. (35)–(37) and employing the gap definitions (39), one obtains the temperature-dependent gap equations

where we define

Solution of the above algebraic equations yields all the gaps and the chemical potential $\lambda^{\prime}$ (we shall discuss methods of solution in §VI.7 and §VI.8 below). Then the total energy is

To simplify the discussion, we shall ignore the single-particle energy in the above equation by setting $\varepsilon=0$, since this term has been approximated as a state-independent constant and thus plays no role in the phase competition. The energy density $E/\Omega$ is then

The three gaps $\Delta_{d}$, $\Delta_{\pi}$ and $\Delta_{q}$ in the above equations are defined in (39) and represent the characteristic energy scales of spin-singlet pairing correlations, spin-triplet pairing correlations, and antiferromagnetic correlations, respectively. Hence, these correlations determine the ground state energy. Once the gaps and the chemical potential $\lambda^{\prime}$ are known from solution of the gap equations (43), the quasiparticle energies $e_{\pm}$ and the amplitudes $u_{\pm}$ and $v_{\pm}$ can all be determined through Eqs. (40)–(42), permitting other ground state properties to be calculated.

In the following sections we shall give analytical solutions for the gap equations (43), first for zero temperature and then for finite temperatures. As we shall see, a rich phase structure emerges naturally in these solutions as a consequence of competition between the various energy scales.

The preceding results are formally analogous to those of the BCS theory with $v_{\pm}^{2}$ the probability of single particle levels $\varepsilon_{\pm}$ being occupied, $\Delta_{\pm}$ the energy gaps, and $e_{\pm}$ the quasiparticle energies. The essential difference from normal BCS theory is that conventional pairing theories deal with one energy gap and one kind of quasiparticle; here we have two kinds of quasiparticles and several energy gaps, implying a large variety of new physics. As should be clear from the derivations leading to Eq. (45), this new physics arises as a natural consequence of a BCS-like theory in which the pairing may be unconventional, Coulomb repulsion plays a significant role, and antiferromagnetic correlations are allowed to enter on a footing equal with that of the pairing correlations.

In the formalism describing this more sophisticated pairing the quantities $e_{\pm}$ are energies for two kinds of quasiparticle excitation, corresponding to two sets of non-degenerate single particle energy spectra $\{\varepsilon_{\pm}\}$ separated by an energy $2\Delta_{q}$, as illustrated in Fig. 16.

Each level can be occupied by only one electron of either up or down spin. The corresponding pairing gaps are $\Delta_{\pm}$, which are linear combinations of the two gaps $\Delta_{d}$ and $\Delta_{\pi}$. The probabilities for single-particle levels to be occupied or unoccupied are $v^{2}_{\pm}$ and $u^{2}_{\pm}$, respectively.

The resulting theory has the limits that we would expect for a symmetry that unites superconductivity and antiferromagnetism at a fundamental level. The gap equations reduce exactly to the gap equations of the ordinary BCS theory in the limit that the antiferromagnetic interactions can be neglected. Conversely, in the limit that the pairing interactions can be neglected we obtain the equations expected for a Néel antiferromagnet.

The three parameters, $\chi$, $G_{0}$, and $G_{1}$ in the algebraic equations (43) correspond to the three basic effective interactions in the SU(4) model:

1. 1.

   The antiferromagnetic correlation, with strength $\chi$.

2. 2.

   The spin-singlet pairing correlation, with strength $G_{0}$.

3. 3.

   The spin-triplet pairing correlation, with strength $G_{1}$.

Experimental evidence suggests that these three interactions in cuprates are all attractive, and that their strengths are ordered