Doublon-holon binding as origin of Mott transition and fractionalized spin liquid -- Asymptotic solution of the Hubbard model in the limit of large coordination
aa r X i v : . [ c ond - m a t . s t r- e l ] D ec Asymptotic solution of the Hubbard model in the limit of large coordination:Doublon-holon binding, Mott transition, and fractionalized spin liquid
Sen Zhou , Long Liang , and Ziqiang Wang CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China COMP Centre of Excellence, Department of Applied Physics,Aalto University School of Science, FI-00076 Aalto, Finland and Department of Physics, Boston College, Chestnut Hill, MA 02467, USA (Dated: March 5, 2018)An analytical solution of the Mott transition is obtained for the Hubbard model on the Bethelattice in the large coordination number ( z ) limit. The excitonic binding of doublons (doublyoccupied sites) and holons (empty sites) is shown to be the origin of a continuous Mott transitionbetween a metal and an emergent quantum spin liquid insulator. The doublon-holon binding theoryenables a different large- z limit and a different phase structure than the dynamical meanfield theoryby allowing intersite spinon correlations to lift the 2 N -fold degeneracy of the local moments. Weshow that the spinons are coupled to doublons/holons by a dissipative compact U(1) gauge field thatis in the deconfined phase, stabilizing the spin-charge separated gapless spin liquid Mott insulator. PACS numbers: 71.10.-w, 71.10.Fd, 71.27.+a, 74.70.-b
A Mott insulator is a fundamental quantum electronicstate protected by a nonzero energy gap for charge ex-citations that is driven by Coulomb repulsion but notassociated with symmetry breaking [1]. It differs fromthe other class of insulators and magnets, better termedas Landau insulators, that require symmetry breakingorder parameters produced by the residual quasiparticle(QP) interactions in a parent Fermi liquid. The moststriking feature of Mott insulators is the separation ofcharge and spin degrees of freedom of the electron thatcompletely destroys coherent QP excitations. A ubiqui-tous example of a Mott insulator is the quantum spinliquid where the spins are correlated but do not exhibitsymmetry-breaking long-range order [2–5]. The spin liq-uid states have been observed in the κ -organics near theMott metal-insulator transition [6–9]. The Mott insula-tor and the Mott transition are at the heart of the strongcorrelation physics. It is conceivable that the Mott in-sulator is the ultimate parent state of strong correlationfrom which many novel quantum states emerge [10–14].In this work, we provide a theory for the Mott in-sulator and the Mott transition in the Hubbard modelat half-filling. The Hilbert space here is local and con-sists of doubly occupied (doublon), empty (holon), andsingly occupied (spinon) states. The electron spectralfunction in different scenarios of the Mott transition issketched in Fig. 1. Focusing exclusively on the coher-ent QP, Gutzwiller variational wave function approaches[15] obtained a strongly correlated Fermi liquid [16] thatundergoes a Brinkman-Rice (BR) transition [17] to a“pathological” localized state with vanishing QP band-width and vanishing doublon (D) and holon (H) den-sity (Fig. 1a). The dynamical meanfield theory (DMFT)maps the Hubbard model to a quantum impurity em-bedded in a self-consistent bath [18, 20]. The mapping is exact only in a well-defined large- z limit. The obtained T = 0 Mott transition shown in Fig. 1b shows that theopening of the Mott gap at U c and the disappearance ofthe QP coherence at U c do not coincide such that theQP states in the metallic state for U c < U < U c areseparated from the incoherent spectrum by a preformedgap. This peculiar property [21–23] was shown to be cor-rect [24] for the large- z limit taken in DMFT where thespin-exchange interaction J ∼ t /U scales with 1 /z andforces the insulating state to be a local moment phasewith 2 N -fold degeneracy. The present theory builds on adifferent asymptotic solution on the Bethe lattice usingthe slave-boson formulation to capture the most essentialMott physics, i.e. the excitonic binding between oppo-sitely charged doublons and holons [25–30]. We constructa different large- z limit than DMFT and find a contin-uous Mott transition shown in Fig. 1c from a correlatedmetal to an insulating quantum spin liquid, where theopening of the Mott gap and the vanishing of the QPcoherence coincide at the same U c . The BR transitionis preempted by quantum fluctuations and replaced bythe Mott transition. A key feature of our asymptoticsolution is that on the insulating side of the Mott tran-sition, while all doublons and holons are bound in realspace into excitonic pairs with the Mott charge gap setby the D-H binding energy [29], the spinon intersite cor-relations remain and survive the large- z limit, forming agapless spin liquid by lifting the ground state degeneracy.We derive the compact gauge field action in the large- z limit and show that the emergent dissipative dynamicsdrives the gauge field to the deconfinement phase wherethe fractionalized U(1) spin liquid is stable.Consider the Hubbard model on the Bethe lattice H = − t X h ij i c † iσ c jσ + h . c . + U X i n i ↑ n i ↓ , (1) FIG. 1: Schematic diagram of the Mott transition in (a)Gutzwiller with U BR ≃ . D (a); (b) DMFT with U c ≃ . D and U c ≃ . D [18–20]; and (c) present D-H bindingtheory with U c ≃ . U BR = 2 . D . D is half bandwidth. where the t -term describes electron hopping on z nearestneighbor bonds and the U -term is the on-site Coulombrepulsion. To construct a strong-coupling theory that isnonperturbative in U , Kotliar and Ruckenstein [31] in-troduced a spin-1/2 fermion f σ and four slave bosons e (holon), d (doublon), and p σ to represent the lo-cal Hilbert space for the empty, doubly-occupied, andsingly occupied sites respectively: | i = e † | vac i , |↑↓i = d † f †↓ f †↑ | vac i , and | σ i = p † σ f † σ | vac i . The physical Hilbertspace obtains under the holomorphic constraints for com-pleteness e † i e i + P σ p † iσ p iσ + d † i d i = 1 and the consistencyof particle density f † iσ f iσ = p † iσ p iσ + d † i d i . The Hubbardmodel is thus faithfully represented by H = − t X h ij i Z † iσ Z jσ f † iσ f jσ + h . c . + U X i d † i d i , (2)where Z iσ = L − / iσ ( p † i ¯ σ d i + e † i p iσ ) R − / i ¯ σ . The operators L iσ = 1 − d † i d i − p † iσ p iσ and R i ¯ σ = 1 − e † i e i − p † i ¯ σ p i ¯ σ shouldbe understood as projection operators and the choice ofthe − / h c † iσ c jσ i ∼ / √ z . Hence the hopping t mustbe rescaled according to t → t/ √ z in order to maintaina finite kinetic energy in the large- z limit [32], such as inDMFT [33]. While natural for the metallic phase, on theMott insulating side the exchange interaction J ∼ t /U is forced to scale according to J → J/z which suppressesthe intersite correlations that may otherwise lift the 2 N -fold degeneracy in the ground state [18]. This ultimately leads to an immediate emergence of the local momentson the insulating side and eliminates the possibility ofa quantum spin liquid where charges are localized butspins form a correlated liquid state.The slave-boson formulation of the Hubbard model inEq. (2) offers a different large- z limit without invokingthe rescaling of t . This can be seen intuitively since thehopping of spinons in Eq. (2) between neighboring sites isalways accompanied by the co-hopping of the Z -bosonsand vice versa. Thus, the effective hopping amplitudeof the spinon/ Z -boson carries a dynamically generated1 / √ z from the Z -boson/spinon intersite correlator, re-sulting in a finite total kinetic energy in the large- z limit.A spin liquid can thus emerge on the Mott insulating side.We note that on the metallic side not discussed in detailhere, the D/H condensate contributions need to be scaledby 1 / √ z in the intersite correlator in order to be treatedon equal footing as the uncondensed part to keep the ki-netic energy finite, in close analogy to the recent formula-tion of the bosonic DMFT [34]. We focus on the Mott in-sulator at U > U c , where all holons ( e ) and doublons ( d )are bound into exciton pairs. Although the doublon andholon densities are nonzero ( n d = n e = 0), their single-particle condensates are absent. To leading order in 1 /z ,the single-occupation p σ bosons must therefore condense,i.e. p σ = p † σ = p . The operators L iσ and R iσ that en-ter Z iσ cannot introduce additional intersite correlationsand must depend only on the local densities. This leadsto L iσ = R iσ = 1 / Z iσ = 2 p ( d i + e † i ).Thus h Z † iσ Z jσ i ∼ / √ z . Together with h f † iσ f jσ i ∼ / √ z ,the electron correlator h c † iσ c jσ i ∼ /z and the kineticenergy is finite in the large- z limit.The Hamiltonian in Eq. (2) becomes, H = − p t X h ij i (cid:2) ( d † i d j + e † j e i + e i d j + d † i e † j ) f † iσ f jσ + h . c . (cid:3) + U X i d † i d i . (3)The condensation of the p σ bosons collapses two of theconstraints to n d + p = n fσ . The remaining one canbe written as e † i e i − d † i d i + P σ f † iσ f iσ = 1, which corre-sponds to the unbroken U (1) gauge symmetry and spec-ifies the gauge charges of the particles. Thus, increasingthe spinon number by one must be accompanied by ei-ther destroying a holon or creating a doublon at the samesite in the Mott insulator. The partition function can bewritten down as an imaginary-time path integral Z = Z D [ f † , f ] D [ d † , d ] D [ e † e ] D [ a , a ] D λe − R β L dτ . (4)The Lagrangian is given by L = X i [ f † iσ ( ∂ τ + ia ) f iσ + d † i ( ∂ τ − ia ) d i + e † i ( ∂ τ + ia ) e i ] + i X i λ i ( d † i d i + e † i e i + 2 p − − H, (5)where the Hamiltonian H = H f + H b , H f = − t f √ z X h i,j i ( e ia ij f † iσ f jσ + h . c . ) (6) H b = − t b √ z X h i,j i (cid:2) e − ia ij ( e † j e i + d † i d j (7)+ e i d j + d † i e † j ) + h . c . (cid:3) + U X i ( d † i d i + e † i e i ) , with t f = 8 tp √ z ( χ d +∆ d ), t b = 8 tp √ zχ f . In a station-ary state, χ d = h d † i d j i = h e † j e i i is the quantum averageof the D/H nearest neighbor hopping, χ f = h f † iσ f iσ i thefermion hopping per spin, and ∆ d = h d † i e † j i = h e i d j i is the D-H binding order parameter. In Eqs. (5-7), thespinons and the D/H are coupled by the emergent U(1)gauge fields a and a ij associated with the constraint.Physically, the instantons of this compact gauge field cor-respond to the tunneling events where the spinons andD/H tunnel in and out of the lattice sites [35].We will first obtain the stationary state solution with a = a ij = 0, and then study the properties of the gaugefield fluctuations. Eq. (6) shows that the spinon hop-ping amplitude is t f / √ z where t f scales with the D/Hdensity, resulting in a bandwidth on the order of ex-change coupling J ∼ t /U . The spinon kinetic energyper site is K f = ( t f /t ) K where K = 2 R D ρ ( ω ) ωdω is that for noninteracting electrons with hopping t/ √ z and ρ is the corresponding semicircle density of states ρ ( ω ) = πD p − ( ω/D ) on the infinite- z Bethe latticewith a half-bandwidth D = 2 t . Thus, K = 4 D/ π and K f = 8 t f / π . Alternatively, expressing K f = 4 t f √ zχ f ,we obtain χ f = √ z π . The effective boson hopping inEq. (7) is thus t b = 8 p t √ zχ f = 16 p t/ π which is ofthe order t . The spectrum of the charge excitations re-siding in the D/H sector has a bandwidth on the orderof the bare electron bandwidth, representing the largeincoherent spectral weight in the Mott insulator.From Eqs (5) and (7), the stationary state bosonicHamiltonian in the D/H sector is H D / H = Z D − D dωρ ( ω ) (cid:0) d † ω , e ω (cid:1) (cid:18) ε ω − ∆ ω − ∆ ω ε ω (cid:19) (cid:18) d ω e † ω (cid:19) , where ε ω = U + λ − t b t ω , ∆ ω = t b t ω are the D/H kineticand pairing energies; λ = h iλ i . Diagonalizing H D / H usingthe Bogoliubov transformation produces two degeneratebranches for the D/H excitations: Ω ω = p ε ω − ∆ ω . TheMott insulator is thus an excitonic insulator and the Mottgap is given by the charge gap in Ω ω , G Mott ( U ) = 2Ω D = 2 s(cid:18) U λ (cid:19) (cid:18) U λ − t b (cid:19) . (8)The physical condition for a real Ω requires U ≥ t b − λ and the equal sign determines the critical U c for the Motttransition where G Mott ( U c ) = 0. √1−ω/ D B o s on e n e r gy Ω / D U = 4 DU = 3.5 DU = 3 DU = 2.71 D ISF N ( Ω ) (a) (b) FIG. 2: The doublon/holon energy spectrum (a) and the cor-responding spectral density of state (b) for different U . Minimizing the energy leads to the self-consistentequations, p = − n d , λ = 4 K √ z ( χ d + ∆ d ), and n d = 12 Z D − D (cid:18) ε ω Ω ω − (cid:19) ρ ( ω ) dω, (9) χ d = 12 D √ z Z D − D ε ω Ω ω ωρ ( ω ) dω, (10)∆ d = 12 D √ z Z D − D ∆ ω Ω ω ωρ ( ω ) dω. (11)Eq. (9) shows that the nonzero D/H density is entirelydue to quantum fluctuations above the Mott gap in Ω ω for U > U c . Lowering U toward U c , G Mott must reduceto host the increased D/H density until G Mott = 0 at U = U c where the D/H condensation emerges and thecontinuous 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 U is reduced to-ward U c . Note that the spectral density in Fig. 2(b) van-ishes quadratically upon gap closing, which ensures thatthe Mott transition is continuous at zero temperature.Remarkably, the critical properties of the transition canbe determined analytically. First, using the expressionfor λ , the critical U c is obtained from Eq. (8), U c = U BR [1 − n cd − √ z ( χ cd + ∆ cd )] , (12)where U BR = 8 K = 32 D/ π is the critical value forthe BR transition on the Bethe lattice. Eq. (12) showsthat the Mott transition emerges as the quantum cor-rection to the BR transition due to D-H binding. Since U c < U BR , the BR transition is pre-emptied by the Motttransition. At U = U c , the D/H excitation spectrum be-comes Ω ω = 8 K p p − ω/D as in Fig. 2(a), which isindependent of χ d and ∆ d . The integrals in Eqs. (9-10)can all be evaluated analytically to obtain the criticaldoublon density n cd = (12 √ − π ) / π ≃ . √ zχ cd = 2 √ / π ≃ . √ z ∆ cd = 22 √ / π ≃ . U / D n d QMCED U / D G M o tt / D IPTDMRGD-H -2 0 2 ω / D D N σ ( ω ) ω / D Σ ( ω ) / D a)c) b)d) Im Σ Re Σ Re Σ N σ (ω) /10 f Hubbard III
FIG. 3: Comparison of the current theory (red lines) with theDMFT results (data from Ref.[18, 19]) obtained using quan-tum monte carlo (QMC - solid black circles), exact diagonal-ization (ED - blue lines), iterative perturbation theory (IPT-open squares), and dynamical density matrix renormalizationgroup (DMRG - open circles) as impurity solvers. (a) Thedoublon density as a function of U . (b) The Mott gap in thecharge sector as a function of U . (c) The spectral density ofstates at U = 4 D . Thin solid line: spinon density of states.(d) The real and imaginary parts of the electron self-energyat U = 4 D . Inset: Real part of self energy on log-log plot,showing the 1 /ω dependence. Mott transition is thus U c = 0 . · U BR = 2 . D , at whichthe charge gap closes and the QP coherence emerges withthe D/H condensate simultaneously.In Figs 3(a) and 3(b), the calculated doublon densityand Mott gap are plotted in red solid lines as a function of U/D . Various single-site DMFT results [18–20] are alsoplotted in Fig. 3 for comparison solely for the purposeto benchmark the results in the charge sector, despite ofthe different large- z limit and the continuous Mott transi-tion to a spin liquid at a single U c . The critical behaviorof the Mott gap near U c can be obtained analyticallyfrom Eq. (8), G Mott ( U ) = α √ U − U c , α = 2 √ t b , wherethe square-root singularity is clearly seen in Fig. 2(b).Figs 3(c) and 3(d) show the spectroscopic properties onthe Mott insulating side benchmarked with correspond-ing DMFT results. The local electron Green’s function isobtained by convoluting those of the spinon and D/H ( Z -boson) G σ ( iω n ) = P iν n G fσ ( iω n − iν n ) G Z ( iν n ). The lat-ter can be obtained readily from the spinon and the D/HGreen’s functions [29]: G fσ ( iω n ) = R dǫρ ( ǫ ) G fσ ( ǫ, iω n )and G Z ( iν n ) = R dǫρ ( ǫ ) G Z ( ǫ, iν n ). The electron spec-tral density is given by N σ ( ω ) = − π Im G σ ( iω n → ω + i + ) . Fig. 3(c) shows N σ ( ω ) obtained at U = 4 D , exhibit-ing the upper and the lower Hubbard bands separated bythe Mott gap, in quantitative agreement with the DMFTresults [18, 19]. The spectral density of the spinons N fσ ( ω ) remains gapless as shown in Fig. 3(c) and con- tributes to thermodynamic properties of the spin liquidat low temperatures. We note in passing that these prop-erties of the Mott transition/insulator are inaccessibleto Gaussian fluctuations around the Kotliar-Ruckensteinsaddle point for the putative BR transition at large U [36, 37]. The central quantity in the large- z limit is thelocal self-energy Σ( ω ), which can be extracted by castingthe local electron Green’s function in the form G σ ( ω ) = Z D − D dǫρ ( ǫ ) 1 ω − ǫ − Σ( ω ) . (13)Fig. 3(d) shows that the obtained Σ( ω ) in the D-H bind-ing theory is remarkably close to the real and imagi-nary part of the self-energy in DMFT at the same valueof U = 4 D [18, 19], including the scaling behaviorReΣ( ω ) ∝ /ω inside the Mott gap shown in the inset.The emergence of the spin-liquid Mott insulator withgapless spinon excitations requires the separation of spinand charge and is stable only if the gauge field that cou-ples them is deconfining. To derive the gauge field action,we integrate out the matter fields by the hopping expan-sion [38]. To leading order in 1 /z , the low energy effectivegauge field action is obtained, S eff = − ηzπ X h i,j i Z β d τ Z β d τ cos ( a ij ( τ ) − a ij ( τ ))( τ − τ ) + 1 zC X h i,j i Z β d τ ( ∂ τ a ij ) , (14)where the second term comes from integrating out thegapped D/H and corresponds to charging with the“charging energy” on a link C ∝ U /t b in the large- U limit. The first term with η = 1, which is nonlo-cal in imaginary time and corresponds to dissipation,comes from the contribution from the gapless fermionspionons. It is periodic in the gauge field consistent withits compact nature. Thus the gauge field action is dis-sipative. It has been argued under various settings thata large enough dissipation η can drive the compact U(1)gauge field to the deconfinement phase at zero temper-ature [39–41]. In the large- z limit, Eq. (14) shows thatspatial correlations of the link gauge field are suppressedand the dissipative gauge field theory becomes local, i.e. a ij ( τ ) = a ( τ ). As a result, the action becomes identi-cal to the dissipative tunneling action derived by Ambe-gaokar, Eckern, and Sch¨on [42] for a quantum dot coupledto metallic leads, or a Josephson junction with QP tun-neling [43]. The 2 π -periodicity of the compact gauge fieldrequires a ( τ ) = ˜ a ( τ )+2 πnτ /β where ˜ a ( τ ) is single-valuedand satisfies ˜ a (0) = ˜ a ( β ), and n is an integer windingnumber associated with charge quantization, i.e. the in-stantons in the electric field when charges tunnel in andout of the link. For a 2D array of dissipative tunnel junc-tions, it has been shown that there exists a confinement-deconfinement (C-DC) transition of the winding numberat a critical η Dc ≃ .
29 [44]. Using the Villain transfor-mation [45], one can show that the instanton action isdescribed by a dissipative sine-Gordon model, exhibitinga C-DC transition at a critical dissipation η c = 1 / η > η c , and the temporal proliferation ofthe instantons is suppressed by dissipation. Thus, thegauge 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 solutionof the Hubbard model in a novel large- z limit and ob-tained a continuous Mott transition from a PM metalto a spin liquid Mott insulator where the opening of theMott gap and the vanishing of the QP coherence coincideat the same critical U c . We elucidated the essential roleplayed by the D-H binding in such remarkable phenom-ena of strong correlation. The present theory provides aconcrete example for a gapless spin liquid Mott insulatorwhere the spin-charge separation is realized in the de-confinement phase of the dissipative compact gauge field.The simplicity of the D-H binding theory for the Mottphenomena holds promise to become a calculational toolfor studying Mott-Hubbard systems and materials withstrong correlation.We thank Y.P. Wang for useful discussions. Thiswork 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 ofExcellence Programme (2015-2017) under project num-ber 284621 (L.L.), and the Key Research Programof Frontier Sicences, CAS No. QYZDB-SSW-SYS012(S.Z.). Z.W. thanks the hospitality of Aspen Center forPhysics where this work was conceived, and the supportof ACP NSF grant PHY-1066293. [1] N. F. Mott, Metal-Insulator Transitions , 2nd Ed. Taylor& Francis, London 1990.[2] P. W. Anderson, Mater. Res. Bull. , 153 (1973).[3] X.-G. Wen, Phys. Rev. B , 2664 (1991).[4] P. A. Lee, Science , 1306 (2008).[5] L. Balents, Nature , 199 (2010).[6] Y. Shimizu, K. Miyagawa, K. Kanoda, M. Maesato, andG. Saito, Phys. Rev. Lett. , 107001 (2003).[7] Y. Kurosaki, Y. Shimizu, K. Miyagawa, K. Kanoda, andG. Saito, Phys. Rev. Lett. , 177001 (2005).[8] S. Yamashita, Y. Nakazawa, M. Oguni, Y. Oshima, H.Nojiri, Y. Shimizu, K. Miyagawa, and K. Kanoda, Nat.Phys. , 459 (2008).[9] M. Yamashita, N. Nakata, Y. Kasahara, T. Sasaki, N.Yoneyama, N. Kobayashi, S. Fujimoto, T. Shibauchi, andY. Matsuda, Nat. Phys. , 44 (2008).[10] P. W. Anderson, Science , 1196 (1987).[11] S. A. Kivelson, D. S. Rokhsar, J. P. Sethna, Phys. Rev. B , 8865 (1987).[12] P. Phillips, Ann. of Phys. , 1634 (2006).[13] P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys.
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