Asymptotic solution of the Hubbard model in the limit of large coordination: Doublon-holon binding, Mott transition, and fractionalized spin liquid
An analytical solution of the Mott transition is obtained for the Hubbard model on the Bethe lattice in the large coordination number () limit. The excitonic binding of doublons (doubly occupied sites) and holons (empty sites) is shown to be the origin of a continuous Mott transition between a metal and an emergent quantum spin liquid insulator. The doublon-holon binding theory enables a different large- limit and a different phase structure than the dynamical meanfield theory by allowing intersite spinon correlations to lift the -fold degeneracy of the local moments. We show that the spinons are coupled to doublons/holons by a dissipative compact U(1) gauge field that is in the deconfined phase, stabilizing the spin-charge separated gapless spin liquid Mott insulator.
pacs:71.10.-w, 71.10.Fd, 71.27.+a, 74.70.-b
A Mott insulator is a fundamental quantum electronic state protected by a nonzero energy gap for charge excitations that is driven by Coulomb repulsion but not associated with symmetry breaking Mottbook (). It differs from the other class of insulators and magnets, better termed as Landau insulators, that require symmetry breaking order parameters produced by the residual quasiparticle (QP) interactions in a parent Fermi liquid. The most striking feature of Mott insulators is the separation of charge and spin degrees of freedom of the electron that completely destroys coherent QP excitations. A ubiquitous example of a Mott insulator is the quantum spin liquid where the spins are correlated but do not exhibit symmetry-breaking long-range order anderson73 (); wen91 (); palee08 (); balents10 (). The spin liquid states have been observed in the -organics near the Mott metal-insulator transition kanoda03 (); kanoda05 (); kanoda08 (); matsuda08 (). The Mott insulator and the Mott transition are at the heart of the strong correlation physics. It is conceivable that the Mott insulator is the ultimate parent state of strong correlation from which many novel quantum states emerge anderson87 (); kivelson (); phillips (); leermp (); weng ().
In this work, we provide a theory for the Mott insulator and the Mott transition in the Hubbard model at half-filling. The Hilbert space here is local and consists of doubly occupied (doublon), empty (holon), and singly occupied (spinon) states. The electron spectral function in different scenarios of the Mott transition is sketched in Fig. 1. Focusing exclusively on the coherent QP, Gutzwiller variational wave function approaches gutzwiller () obtained a strongly correlated Fermi liquid dv-rmp84 () that undergoes a Brinkman-Rice (BR) transition br () to a “pathological” localized state with vanishing QP bandwidth and vanishing doublon (D) and holon (H) density (Fig. 1a). The dynamical meanfield theory (DMFT) maps the Hubbard model to a quantum impurity embedded in a self-consistent bath dmftrmp96 (); dv2012 (). The mapping is exact only in a well-defined large- limit. The obtained Mott transition shown in Fig. 1b shows that the opening of the Mott gap at and the disappearance of the QP coherence at do not coincide such that the QP states in the metallic state for are separated from the incoherent spectrum by a preformed gap. This peculiar property nozieres (); kehrein (); gebhard () was shown to be correct kotliar () for the large- limit taken in DMFT where the spin-exchange interaction scales with and forces the insulating state to be a local moment phase with -fold degeneracy. The present theory builds on a different asymptotic solution on the Bethe lattice using the slave-boson formulation to capture the most essential Mott physics, i.e. the excitonic binding between oppositely charged doublons and holons kaplan (); yokoyama (); capello (); leigh (); zhouwangwang (); mckenzie (). We construct a different large- limit than DMFT and find a continuous Mott transition shown in Fig. 1c from a correlated metal to an insulating quantum spin liquid, where the opening of the Mott gap and the vanishing of the QP coherence coincide at the same . The BR transition is preempted by quantum fluctuations and replaced by the Mott transition. A key feature of our asymptotic solution is that on the insulating side of the Mott transition, while all doublons and holons are bound in real space into excitonic pairs with the Mott charge gap set by the D-H binding energy zhouwangwang (), the spinon intersite correlations remain and survive the large- limit, forming a gapless spin liquid by lifting the ground state degeneracy. We derive the compact gauge field action in the large- limit and show that the emergent dissipative dynamics drives the gauge field to the deconfinement phase where the fractionalized U(1) spin liquid is stable.
Consider the Hubbard model on the Bethe lattice
where the -term describes electron hopping on nearest neighbor bonds and the -term is the on-site Coulomb repulsion. To construct a strong-coupling theory that is nonperturbative in , Kotliar and Ruckenstein kr () introduced a spin-1/2 fermion and four slave bosons (holon), (doublon), and to represent the local Hilbert space for the empty, doubly-occupied, and singly occupied sites respectively: , , and . The physical Hilbert space obtains under the holomorphic constraints for completeness and the consistency of particle density . The Hubbard model is thus faithfully represented by
where The operators and should be understood as projection operators and the choice of the power reproduces the results of the Gutzwiller approximation at the meanfield level kr (); although it can be argued to arise from the hardcore nature of the slave bosons. To study the Mott transition and the Mott insulating state, we focus on the spin SU(2) symmetric phases of the Hubbard model.
When the quantum states in Eq. (1) are spatially extended, the nearest neighbor single-particle correlator scales as . Hence the hopping must be rescaled according to in order to maintain a finite kinetic energy in the large- limit metzner89 (), such as in DMFT gk92 (). While natural for the metallic phase, on the Mott insulating side the exchange interaction is forced to scale according to which suppresses the intersite correlations that may otherwise lift the -fold degeneracy in the ground state dmftrmp96 (). This ultimately leads to an immediate emergence of the local moments on the insulating side and eliminates the possibility of a quantum spin liquid where charges are localized but spins form a correlated liquid state.
The slave-boson formulation of the Hubbard model in Eq. (2) offers a different large- limit without invoking the rescaling of . This can be seen intuitively since the hopping of spinons in Eq. (2) between neighboring sites is always accompanied by the co-hopping of the -bosons and vice versa. Thus, the effective hopping amplitude of the spinon/-boson carries a dynamically generated from the -boson/spinon intersite correlator, resulting in a finite total kinetic energy in the large- limit. A spin liquid can thus emerge on the Mott insulating side. We note that on the metallic side not discussed in detail here, the D/H condensate contributions need to be scaled by in the intersite correlator in order to be treated on equal footing as the uncondensed part to keep the kinetic energy finite, in close analogy to the recent formulation of the bosonic DMFT bdmft (). We focus on the Mott insulator at , where all holons () and doublons () are bound into exciton pairs. Although the doublon and holon densities are nonzero (), their single-particle condensates are absent. To leading order in , the single-occupation bosons must therefore condense, i.e. . The operators and that enter cannot introduce additional intersite correlations and must depend only on the local densities. This leads to at half-filling and . Thus . Together with , the electron correlator and the kinetic energy is finite in the large- limit.
The Hamiltonian in Eq. (2) becomes,
The condensation of the bosons collapses two of the constraints to . The remaining one can be written as , which corresponds to the unbroken gauge symmetry and specifies the gauge charges of the particles. Thus, increasing the spinon number by one must be accompanied by either destroying a holon or creating a doublon at the same site in the Mott insulator. The partition function can be written down as an imaginary-time path integral
The Lagrangian is given by
where the Hamiltonian ,
with , . In a stationary state, is the quantum average of the D/H nearest neighbor hopping, the fermion hopping per spin, and is the D-H binding order parameter. In Eqs. (5-Asymptotic solution of the Hubbard model in the limit of large coordination: Doublon-holon binding, Mott transition, and fractionalized spin liquid), the spinons and the D/H are coupled by the emergent U(1) gauge fields and associated with the constraint. Physically, the instantons of this compact gauge field correspond to the tunneling events where the spinons and D/H tunnel in and out of the lattice sites ioffelarkin ().
We will first obtain the stationary state solution with , and then study the properties of the gauge field fluctuations. Eq. (6) shows that the spinon hopping amplitude is where scales with the D/H density, resulting in a bandwidth on the order of exchange coupling . The spinon kinetic energy per site is where is that for noninteracting electrons with hopping and is the corresponding semicircle density of states on the infinite- Bethe lattice with a half-bandwidth . Thus, and . Alternatively, expressing , we obtain . The effective boson hopping in Eq. (Asymptotic solution of the Hubbard model in the limit of large coordination: Doublon-holon binding, Mott transition, and fractionalized spin liquid) is thus which is of the order . The spectrum of the charge excitations residing in the D/H sector has a bandwidth on the order of the bare electron bandwidth, representing the large incoherent spectral weight in the Mott insulator.
From Eqs (5) and (Asymptotic solution of the Hubbard model in the limit of large coordination: Doublon-holon binding, Mott transition, and fractionalized spin liquid), the stationary state bosonic Hamiltonian in the D/H sector is
where , are the D/H kinetic and pairing energies; . Diagonalizing using the Bogoliubov transformation produces two degenerate branches for the D/H excitations: . The Mott insulator is thus an excitonic insulator and the Mott gap is given by the charge gap in ,
The physical condition for a real requires and the equal sign determines the critical for the Mott transition where .
Minimizing the energy leads to the self-consistent equations, , , and
Eq. (9) shows that the nonzero D/H density is entirely due to quantum fluctuations above the Mott gap in for . Lowering toward , must reduce to host the increased D/H density until at where the D/H condensation emerges and the continuous Mott transition takes place (Fig. 1c).
The D/H excitation spectrum is plotted in Fig. 2(a), showing the closing of the Mott gap as is reduced toward . Note that the spectral density in Fig. 2(b) vanishes quadratically upon gap closing, which ensures that the Mott transition is continuous at zero temperature. Remarkably, the critical properties of the transition can be determined analytically. First, using the expression for , the critical is obtained from Eq. (8),
where is the critical value for the BR transition on the Bethe lattice. Eq. (12) shows that the Mott transition emerges as the quantum correction to the BR transition due to D-H binding. Since , the BR transition is pre-emptied by the Mott transition. At , the D/H excitation spectrum becomes as in Fig. 2(a), which is independent of and . The integrals in Eqs. (9-10) can all be evaluated analytically to obtain the critical doublon density , D/H hopping , and D-H binding . The critical value for the Mott transition is thus , at which the charge gap closes and the QP coherence emerges with the D/H condensate simultaneously.
In Figs 3(a) and 3(b), the calculated doublon density and Mott gap are plotted in red solid lines as a function of . Various single-site DMFT results dmftrmp96 (); karski (); dv2012 () are also plotted in Fig. 3 for comparison solely for the purpose to benchmark the results in the charge sector, despite of the different large- limit and the continuous Mott transition to a spin liquid at a single . The critical behavior of the Mott gap near can be obtained analytically from Eq. (8), , , where the square-root singularity is clearly seen in Fig. 2(b). Figs 3(c) and 3(d) show the spectroscopic properties on the Mott insulating side benchmarked with corresponding DMFT results. The local electron Green’s function is obtained by convoluting those of the spinon and D/H (-boson) . The latter can be obtained readily from the spinon and the D/H Green’s functions zhouwangwang (): and . The electron spectral density is given by Fig. 3(c) shows obtained at , exhibiting the upper and the lower Hubbard bands separated by the Mott gap, in quantitative agreement with the DMFT results dmftrmp96 (); karski (). The spectral density of the spinons remains gapless as shown in Fig. 3(c) and contributes to thermodynamic properties of the spin liquid at low temperatures. We note in passing that these properties of the Mott transition/insulator are inaccessible to Gaussian fluctuations around the Kotliar-Ruckenstein saddle point for the putative BR transition at large raimondi (); castellani (). The central quantity in the large- limit is the local self-energy , which can be extracted by casting the local electron Green’s function in the form
Fig. 3(d) shows that the obtained in the D-H binding theory is remarkably close to the real and imaginary part of the self-energy in DMFT at the same value of dmftrmp96 (); karski (), including the scaling behavior inside the Mott gap shown in the inset.
The emergence of the spin-liquid Mott insulator with gapless spinon excitations requires the separation of spin and charge and is stable only if the gauge field that couples them is deconfining. To derive the gauge field action, we integrate out the matter fields by the hopping expansion im1995 (). To leading order in , the low energy effective gauge field action is obtained,
where the second term comes from integrating out the gapped D/H and corresponds to charging with the “charging energy” on a link in the large- limit. The first term with , which is nonlocal in imaginary time and corresponds to dissipation, comes from the contribution from the gapless fermion spionons. It is periodic in the gauge field consistent with its compact nature. Thus the gauge field action is dissipative. It has been argued under various settings that a large enough dissipation can drive the compact U(1) gauge field to the deconfinement phase at zero temperature nagaosa1993 (); wang2004 (); ksk2005 (). In the large- limit, Eq. (14) shows that spatial correlations of the link gauge field are suppressed and the dissipative gauge field theory becomes local, i.e. . As a result, the action becomes identical to the dissipative tunneling action derived by Ambegaokar, Eckern, and Schön aes1982 () for a quantum dot coupled to metallic leads, or a Josephson junction with QP tunneling kampf (). The -periodicity of the compact gauge field requires where is single-valued and satisfies , and is an integer winding number associated with charge quantization, i.e. the instantons in the electric field when charges tunnel in and out of the link. For a 2D array of dissipative tunnel junctions, it has been shown that there exists a confinement-deconfinement (C-DC) transition of the winding number at a critical mooji (). Using the Villain transformation villain (), one can show that the instanton action is described by a dissipative sine-Gordon model, exhibiting a C-DC transition at a critical dissipation longliang (). In our case, , and the temporal proliferation of the instantons is suppressed by dissipation. Thus, the gauge electric field is deconfining and the gapless U(1) spin-liquid is indeed the stable Mott insulating state.
In summary, we have provided an asymptotic solution of the Hubbard model in a novel large- limit and obtained a continuous Mott transition from a PM metal to a spin liquid Mott insulator where the opening of the Mott gap and the vanishing of the QP coherence coincide at the same critical . We elucidated the essential role played by the D-H binding in such remarkable phenomena of strong correlation. The present theory provides a concrete example for a gapless spin liquid Mott insulator where the spin-charge separation is realized in the deconfinement phase of the dissipative compact gauge field. The simplicity of the D-H binding theory for the Mott phenomena holds promise to become a calculational tool for studying Mott-Hubbard systems and materials with strong correlation.
We thank Y.P. Wang for useful discussions. This work is supported by the U.S. Department of Energy, Basic Energy Sciences Grant No. DE-FG02-99ER45747 (Z.W.), the Academy of Finland through its Centres of Excellence Programme (2015-2017) under project number 284621 (L.L.), and the Key Research Program of Frontier Sicences, CAS No. QYZDB-SSW-SYS012 (S.Z.). Z.W. thanks the hospitality of Aspen Center for Physics where this work was conceived, and the support of ACP NSF grant PHY-1066293.
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