A Effective Hamiltonian

d-wave superconductivity in boson+fermion dimer models


We present a slave-particle mean-field study of the mixed boson+fermion quantum dimer model introduced by Punk, Allais, and Sachdev [PNAS 112, 9552 (2015)] to describe the physics of the pseudogap phase in cuprate superconductors. Our analysis naturally leads to four charge fermion pockets whose total area is equal to the hole doping , for a range of parameters consistent with the model for high temperature superconductivity. Here we find that the dimers are unstable to d-wave superconductivity at low temperatures. The region of the phase diagram with d-wave rather than s-wave superconductivity matches well with the appearance of the four fermion pockets. In the superconducting regime, the dispersion contains eight Dirac cones along the diagonals of the Brillouin zone.

Introduction. The Rokhsar-Kivelson quantum dimer model (QDM) was originally introduced to describe a possible magnetically-disordered phase – the resonating valence bond (RVB) phase – in high-temperature superconducting materials (1). The arena where the QDM has been deployed has greatly expanded since its inception, and the model has taken on a key role in the study of a variety of magnetic quantum systems. Quantum dimers show up prominently in the study of hard-core bosons hopping on frustrated lattices (2), of arrays of Josephson junctions with capacitative and Josephson couplings (3), of frustrated Ising models with an external field or with perturbative XY couplings (4), of various types of gauge theories (5), and of models with large spin-orbit couplings (6) and various cold atom setups (7). The study of QDMs led to an abundance of new phenomena including deconfined quantum criticality and new routes to deconfinement (8). It also provided one of the earliest known examples of topologically ordered states in a lattice model (9).

Recently QDMs have been revisited as models of high-temperature superconductivity (10); (11); (12). This was motivated by the need to reconcile transport experiments (13); (14); (15); (16) and photoemission data (17); (18); (19) in the underdoped region of cuprate superconductors: while photoemission data show Fermi arcs enclosing an area (with being the doping), transport measurements indicate plain Fermi-liquid properties consistent with an area . In order to resolve this issue and produce a Fermi liquid which encloses an area , the authors of Refs. (10); (11); (12) introduced a model for the pseudogap region of the cuprate superconductors which consists of two types of dimers: one spinless bosonic dimer – representing a valence bond between two neighboring spins – and one spin fermionic dimer representing a hole delocalized between two sites. Fig. 1 shows an example of a boson+fermion dimer covering of the square lattice and depicts the dimer moves dictated by the quantum Hamiltonian in Eq. (2). The boson+fermion QDM (bfQDM) was introduced and studied numerically in Ref. (10) using exact diagonalization, supporting the existence of a fractionalized Fermi liquid enclosing an area .

In this work we present a slave boson and fermion formulation of the bfQDM. We find that four symmetric fermion pockets, located in the vicinity of in the Brillouin zone, naturally appear at mean-field level. The total area of the four pockets is given by the hole (fermionic) doping. We find that the system is unstable to d-wave superconductivity at low temperatures. The region of the phase diagram with d-wave superconductivity matches well the region with four fermion pockets. In the superconducting phase, the fermionic dimers (holes) acquire a Dirac dispersion at eight points along the diagonals of the Brillouin zone.

Figure 1: The boson+fermion quantum dimer model of Ref. (10). (A) A particular dimer configuration. The lattice is shown in black. The bosonic dimers representing the valence bonds are shown in blue while the spinful fermionic dimers representing a single electron delocalized over two sites are shown in green. (B) Diagrams representative of the various terms in the dimer Hamiltonian Eq. (2).

Mapping onto slave boson/fermion model. The quantum dimer model can be mapped exactly onto a slave boson+fermion model by considering a secondary Hilbert space where we assign to each link () of the lattice () a bosonic mode and a spinful fermionic mode (). We associate the number of dimers on a link with the occupation numbers of the bosons or fermions on that link. As such we have embedded the dimer Hilbert space in a larger boson/fermion Hilbert space. The constraint that each site of the lattice has one and exactly one dimer attached to it may be rephrased in the boson/fermion language as:


Here, for convenience of notation, labels the four links that are attached to vertex . Any Hamiltonian for the dimers has a boson/fermion representation; in particular the terms illustrated in Fig. 1B can be written as:


where we included a chemical potential for the holes (fermionic dimers), which is important for the connection with doped high-temperature superconductors (20); (21). The terms not written explicitly in Eq. (2) are simply obtained from those shown by translational symmetry, four fold rotational symmetry, and reflection symmetry about the two diagonals. This Hamiltonian also has a local gauge symmetry


with a phase associate to each vertex . Any Hamiltonian that preserves the constraint given in Eq. (1) is invariant under this gauge transformation (22); (23).

A slave boson/fermion formulation of the bfQDM is obtained by introducing a Lagrange multiplier: a real field that enforces the dimer constraint Eq. (1) at all times , and shifting the action by .

Slave boson/fermion mean-field decoupling. A systematic mean-field approach can be obtained by taking the saddle point with respect to the Lagrange multiplier field , with a time-independent value that enforces the average constraint . This procedure is accompanied by Hubbard Stratonovich (HS) transformations of every term in the Hamiltonian in Eq. (2) separately. We begin with the purely bosonic potential term:


where and are auxiliary fields to be integrated over and is arbitrary. At mean-field level we can drop the integrals over the auxiliary fields and replace them with their saddle point values , and . The hopping term may be decoupled in a similar manner:


where, again, at mean-field level we use the saddle point values , , , and is arbitrary. Other HS decouplings, and linear combinations thereof, are also possible.

We can make substantial progress in understanding the fermionic component of the theory without detailed analysis of the bosonic component. Indeed, any translationally invariant (liquid-like) bosonic ansatz, naturally expected in gauge theories coupled to fermions with a Fermi surface (24); (25); (26), yields similar fermionic effective theories. The fermionic mean-field Hamiltonian reads


which is effectively a tight-biding model with renormalized hoppings , and .

The resulting model is defined on the bipartite checkerboard lattice that is medial to the original square lattice. The horizontal () and vertical () links make up the two sublattices where the fermions reside. We define (in momentum space) the spinor that encodes these two flavors as and



The eigenvalues are given by , where and . For hole doping (the number of fermions in our model) the lower band will be partially occupied. The total area enclosed by the Fermi surface in the lower band is equal to the hole doping (multiplied by ).

The Hamiltonian Eq. (7) has four-fold rotational symmetry, and , and reflection symmetry about the diagonals and as well as and . Depending on the relative values of , and , the band minima will be located at different points in the Brillouin zone, and the Fermi surface topology will vary accordingly. In Fig. 2A we show the position of the minima along the directions (or line), as a function of the ratios and . We identify two regions in parameter space, where the dispersion minima are (i) at the point (blue-colored region), and (ii) in between the and points, varying continuously with (faded region). An example of dispersion where the minima are at is shown in the bottom inset of Fig. 2A. Case (ii) is clearly conducive to the appearance of four Fermi pockets in an appropriate range of the chemical potential.

Figure 2: (A) Location of the band minima as a function of and . The color scale corresponds to the distance along the line in the Brillouin zone: blue corresponds to the point, , and red corresponds to the point, . The insets show contours of the dispersion of the lower band of the Hamiltonian Eq. (6) for specific choices of parameters in the corresponding regions. (B) Dominant superconductivity instability as a function of and for doping and : d-wave (white) vs. s-wave (black). Note the good correlation between d-wave superconductivity and the appearance of four band minima.

d-wave Superconductivity. To study superconducting instabilities we need to include four-fermion terms in the Hamiltonian, i.e., go beyond the model introduced in Refs. (10); (11); (12) and summarized in Fig. 1B and Eq. (2). Consider the Hamiltonian on the square lattice (27),


subject to the constraint that . Here and are the electron creation and annihilation operators () of the model, (with summed over), and .

We can identify the dimer Hilbert space with a subspace of the Hilbert space for the model, where the zero dimers state corresponds to the state with zero electrons, and the rest of the Hilbert space can be introduced via the operators and . The phases represent a gauge choice and we shall follow the one by Rokhsar and Kivelson (1) and define and , where is the -component of the 2D square lattice site index .

Given the conventional inner product for the electron Hilbert space, the dimer basis is not orthonormal. This issue can be addressed in general by Gram-Schmidt orthogonalization (28); however, it is customary to use the leading order approximation and to assume that the dimer states are orthogonal (and normalized) (23). The relevant Hamiltonian can then be determined by projecting Eq. (8) onto this basis. The pairing term (four-fermion interaction) comes from the spin-spin term in the model, namely . Let us focus on a single plaquette term and consider eight relevant states for this plaquette, and , . The Hamiltonian is non-zero only in the singlet channel and therefore we must restrict the spins to be in a singlet state, thereby the effective Hamiltonian for the dimers is given by  (23). As such we add to our Hamiltonian in Eq. (2) the term:


For convenience we define and , whereby d-wave pairing corresponds to (which in turn can be chosen real with an appropriate choice of phase). Using a HS transformation on Eq. (9), , where


and . Here and . The eigenvalues of this Hamiltonian are given by



When , and are such that there are four Fermi pockets (in the absence of superconductivity), there are eight Dirac points in the dispersion, i.e., there are eight nodes where the gap . These points are located along the diagonals of the Brillouin zone. When , vanishes, and the gap closing condition is equivalent to , where and . Notice that the Fermi surface in the absence of superconductivity is given by . Therefore, whenever there are four Fermi pockets, for a range of there will be two nodes for each pocket, slightly shifted along the diagonal from the original Fermi surface (23).

Using self consistent equations for the superconducting order parameter, we can then compare s-wave and d-wave instabilities. Up to an unimportant constant energy shift, the Gibbs free energy is given by


Minimizing the free energy with respect to , we obtain:


and similarly for . From the symmetries of this equation we see that there are two solutions, , corresponding to d-wave and extended s-wave superconductivity.

We numerically compare the two solutions at zero temperature and find that d-wave superconductivity wins for a large range of ratios and , as illustrated in Fig. 2B. The correlation between the region with fermion pockets depicted in Fig 2A and the region with d-wave superconductivity in Fig. 2B is evident. This can be qualitatively understood as the largest change in the Gibbs free energy upon entering the superconducting state comes from the contribution of the integral around the FS. Since the shape of the four Fermi pockets follows largely the nodal lines of the s-wave order parameter, and it anti-correlates with the d-wave nodal lines, one expects the appearance of the pockets to favor d-wave superconductivity.

Whereas the horizontal boundaries match very well in the two panels in Fig. 2, the vertical boundaries less so. Indeed, along the horizontal boundary the dispersion transitions smoothly from having a single minimum at the point to having four minima along the direction in the Brillouin zone, i.e., the minima move continuously away from the point (which thus becomes a maximum). On the other hand, along the vertical boundary, the minima jump discontinuously from the point to the new four minima, as four local minima at finite momenta dip down to become the global minima. Depending on the value of the chemical potential, there is a region in the vs. plane near the vertical boundary where the Fermi surface has five sheets, four pockets coexisting with a surface surrounding the point. The latter favors s-wave superconductivity as it has no nodes at the point, and it is therefore expected to shift the position of the boundary between d-wave and s-wave superconductivity, as observed.

Conclusions. We presented a slave particle formulation of a mixed boson+fermion quantum dimer model recently proposed in the context of high-T superconductors (10); (11); (12). A key finding of this work is that substantial progress can be made using a mean-field analysis that simply assumes a translational and rotational invariant (liquid) state for the bosonic component. We analyze the effective theory for the remaining fermionic degrees of freedom, and distinguish between two regimes of Fermi surface topology, depending on the effective couplings obtained from both microscopic parameters and correlations of the bosonic liquid state. The two regimes correspond to one Fermi surface around the point, or four Fermi pockets centered along the lines. By including additional interactions that arise from the model, we find that the system is unstable to superconductivity. The symmetry of the superconducting order parameter, s-wave vs. d-wave, is shown to correlate strongly with the Fermi surface topology, with d-wave being favored when four Fermi pockets are present.

Acknowledgements: This work was supported in part by Engineering and Physical Sciences Research Council (EPSRC) Grants No. EP/G049394/1 (C.Ca.) and No. EP/M007065/1 (C.Ca. and G.G.), by DOE Grant DEF-06ER46316 (C.Ch.), and by the EPSRC Network Plus on “Emergence and Physics far from Equilibrium”. Statement of compliance with the EPSRC policy framework on research data: this publication reports theoretical work that does not require supporting research data. C.Ca. and G.G. thank the BU visitor program for its hospitality.

Supplementary Online Information

Appendix A Effective Hamiltonian

Here we provide more details about the derivation of the effective two body (four fermion) interaction introduced in Eq. (9) in the main text. The procedure to obtain this term for the dimer model is to identify the dimer Hilbert space with a subspace of the model Hilbert space and project the Hamiltonian Eq. (8) accordingly.

We identify the state with zero dimers of any kind with the state with zero electrons for the model. The rest of the Hilbert space for the dimers can be introduced via the operators and . The phases represent a gauge choice and we shall follow the one by Rokhsar and Kivelson (1) and define and , here is the y-component of the dimer co-ordinate. Given the conventional inner product for the electron Hilbert space, the dimer basis is not orthonormal and therefore does not serve as a convenient basis to calculate matrix elements. This can be resolved by Gram-Schmidt orthogonalization. In general, if we denote the basis elements of the dimer Hilbert space by and the overlap matrix between states , then an orthonormal basis for the Hilbert space is given by (2):


It is not too hard to check that the matrix is a real symmetric matrix and therefore . From this it follows that , i.e., the new states are orthonormal.

The Hamiltonian projected onto this basis is given by (2):


To leading order, as the dimers are nearly orthogonal. To show this consider two states and . We can form the loop graph of and by deleting all the dimers that and have in common. The rest of the dimers will form loops (with dimers from state and state alternating along a loop). If there is a loop of length 2, that is two dimers of different type on the same link then , so we have for those states. Assuming there are no such links we have that all loops are at least length four. Now the overlap of and is the product of overlaps over all loops. Furthermore it is known that the overlap between two loops is exponential in the length of the loop (1); (2). Since all loops are of at least length four (rather long) to leading order we may set the overlap matrix to zero if there is at least one loop or the states and are different. Now the states are normalized to unity so we have .

The pairing term (four-fermion interaction) comes from the spin-spin term in the model, namely . Let us focus on a single plaquette term and consider eight relevant states for the dimers on this plaquette, and . We notice that the Hamiltonian is zero in the triplet channel. This means that the effective Hamiltonian for the dimers is also zero in the triplet channel. Indeed, the spins of the dimers are the same as the spins of the electrons for the model, so the projected Hamiltonian has the same spin structure. As such we might as well restrict the spins of the dimers to lie in a singlet (there are two such states per plaquette with the two dimers lying either along the x-axis or along the y-axis). Moreover, the projected Hamiltonian is diagonal in this basis. Indeed the bare Hamiltonian contains no hopping terms for the electrons, only spin flip terms. As such the only terms that could contribute to off diagonal matrix elements come from states of the form and (and linear combinations thereof) which belong to both dimer configurations (along the x-axis and along the y-axis). However the Hamiltonian annihilates such states and the projected Hamiltonian has no corresponding hopping terms. By symmetry the Hamiltonian when restricted to the singlet subspace is a multiple of the identity matrix. Its value is given by:

and, within the spin-singlet channel, the Hamiltonian is . Correspondingly, we can add to our Hamiltonian in Eq. (2) the term:


This is a spin spin Hamiltonian for the fermionic dimers.

Appendix B Dirac Cones

Figure 3: Particular realization of the Dirac Cones for , , , and (top), (middle), and (bottom panel). The superconducting paring was chosen to optimize the zero temperature Gibbs free energy , respectively. The whole Brillouin zone from to is shown on the left and a cross section along the major diagonal on the right, which highlights of the Dirac cones.

To verify the existence and robustness of the Dirac cones predicted in the main text, we plot numerically the function near the values of the hopping matrix elements , , , for various values of near the that minimizes the free energy. We focus on the case of large (see Fig. 3), and we find eight cones in all cases, even though the exchange coupling is taken to be times larger then the relevant value for cuprate superconductors.

To get an estimate of the ratio of we note that for the cuprates  eV, the nearest neighbor hopping  eV (5); (3); (4),  (6), and in general , as any boson bilinear must be less then the largest occupation number . Recalling that , we get for that , likely of the order of .

Appendix C Large N arbitrary S

The considerations given in the main text can be extended to multiple species () and different occupation number () of dimers. The meanfield studied in the main text becomes arbitrarily accurate in this limit. We consider the case of the square lattice but generalizations to different lattice geometries is straightforward. We consider different species of bosonic dimers living on the links of the lattice and species of fermionic dimers also living on the same links. We represent the dimers using species of bosons living on the links of the lattice and species of fermions . The dimer Hilbert space can be mapped onto a subspace of the boson/fermion Hilbert space via the correspondence where bosonic dimers of species living on the link are identified with the state for the slave particles where the occupation number of the boson on link is given by . Similarly for the fermions. Rather than enforcing the constraint of at most one dimer per link (which is redundant in the physical case where ), we introduce the constraint


Here, for convenience of notation, labels the four links that are attached to vertex . In the dimer language this corresponds to the constraint that the total number of dimers of any species on all the links touching the vertex is given by . can be an arbitrary number and in the limit where becomes large the bosonic part of the dimer model becomes semiclassical. In the path integral formulation, the constraint can be written as


where is the length of a time slice.

There is no prescription to write a Hamiltonian for a large theory. However, in order to proceed further we need to write down Hamiltonians for our slave bosons/fermions that reduce to the Hamiltonian given in the main text in the case and are amenable to large expansion (7). All the Hamiltonians written in this section have a direct interpretation in terms of Hamiltonians for the dimers (they all correspond to various dimer hopping terms and terms that count the number of flippable dimer plaquettes). Note that it does not matter whether the Hamiltonians we produce respect the constraint in Eq. (17) as in the path integral formulation we insert projectors onto the physical space at every time slice. The hint towards how to do this extension comes from the expressions derived around Eqs. (4) and (5) in the main text. Indeed in order to apply a HS transformation to our expressions we need to write our Hamiltonian (when restricted to a single plaquette) schematically in the form where and are single particle operators for one species of dimer, either bosonic or fermionic (see the main text e.g. Eqs. (4), (5) and (6)). The main idea is to replace . Here , are single particle operators either bosonic or fermionic which are identical to and except they now carry an index . As such each HS transformation given in the main text corresponds to a different large Hamiltonian. In the large limit, with this extension, we present models where the HS mean-field becomes arbitrarily accurate. Qualitatively we expect mean-field theory to become more and more accurate as each particle interacts with particles with an interaction strength that is attenuated by . We now proceed to give several examples of this procedure. We note that none of the Hamiltonians have any dependence on . In particular we can replace


where is arbitrary and each value of produces a different Hamiltonian. Similarly we have


where again