RAPID COMMUNICATIONS
PHYSICAL REVIEW B 95, 161106(R) (2017)
Mott physics beyond the Brinkman-Rice scenario Marcin M. Wysoki´nski1,2,* and Michele Fabrizio1,† 1
2
International School for Advanced Studies (SISSA), via Bonomea 265, IT-34136 Trieste, Italy Marian Smoluchowski Institute of Physics, Jagiellonian University, ulica prof. S. Łojasiewicza 11, PL-30-348 Kraków, Poland (Received 22 December 2016; revised manuscript received 10 March 2017; published 11 April 2017) The main flaw of the well-known Brinkman-Rice description, obtained through the Gutzwiller approximation, of the paramagnetic Mott transition in the Hubbard model is in neglecting high-energy virtual processes that generate, for instance, the antiferromagnetic exchange J ∼ t 2 /U . Here, we propose a way to capture those processes by combining the Brinkman-Rice approach with a variational Schrieffer-Wolff transformation, and apply this method to study the single-band metal-to-insulator transition in a Bethe lattice with infinite coordination number, where the Gutzwiller approximation becomes exact. We indeed find for the Mott transition a description very close to the real one provided by the dynamical mean-field theory, an encouraging result in view of possible applications to more involved models. DOI: 10.1103/PhysRevB.95.161106
A metal-to-insulator transition driven by electron-electron repulsion was envisioned by Mott more than 50 years ago [1]. Since then, the underlying physics of this phenomenon has been studied by a large variety of quantum many-body tools in models for strongly correlated systems [2–5]. One of the earliest microscopic descriptions of Mott localization was made by Brinkman and Rice [2], and obtained through the Gutzwiller approximation applied to the halffilled Hubbard model. In their scenario the transition to the insulating state occurs when the hopping is fully hampered by repulsion, i.e., its expectation value in the variational wave function strictly vanishes. This result is elegant in many ways. It is fully analytical and provides a very intuitive and physically transparent, almost classical, interpretation of the Mott phenomenon. Nonetheless, this description, frequently called the Brinkman-Rice transition, has a severe drawback: The expectation value of the hopping cannot be zero, and it is so in the Gutzwiller approximation only because there is a complete, static and dynamic, locking of the charge degrees of freedom. In reality, dynamical charge fluctuations do play a role even deep in the Mott phase, and, in particular, they mediate the antiferromagnetic spin exchange, as is clear by the large U mapping onto the Heisenberg model that can be formally derived through the Schrieffer-Wolff transformation [6]. Since the result of Brinkman and Rice, a variety of quantum many-body tools have been constructed that are generically able to sensibly capture those dynamical processes [3,5,7,8]. In particular, when applied to the Hubbard model, they provide satisfying descriptions of the Mott transition, though relying on a heavy numerical computation. Nowadays, the scenario provided by the dynamical mean-field theory (DMFT) [3], which becomes exact in infinite dimensions [9], has become an invaluable benchmark to be compared. In the present Rapid Communication we revisit the problem of the Brinkman-Rice transition, and complement it with the inclusion of the dynamical processes in a semianalytic manner. In order to achieve this goal, we construct a method
that combines the Gutzwiller’s variational approach with a variational Schrieffer-Wolff transformation. As a case study, we apply our technique to the half-filled Hubbard model in the paramagnetic phase on the infinitely coordinated Bethe lattice. The energy functional to be minimized can be obtained fully analytically. Its minimization leads to a significantly improved description of the Mott transition as compared to the standard Brinkman-Rice scenario, and much closer to the exact DMFT one [3]. The improvement is, in particular, highlighted by (i) a sizable lowering of the critical interaction strength for a transition, (ii) a lower value of the insulator energy that includes a nonzero expectation value of the hopping ∼ −t 2 /U , and (iii) a proper balance of kinetic and potential energies at the transition. The starting point of our analysis is the half-filled singleband Hubbard model on the infinitely coordinated Bethe lattice, t U H = −√ Tij + (ni↑ + ni↓ − 1)2 , (1) z ij 2 i where z → ∞ is the coordination number of the Bethe lattice. † † Tij = Tji ≡ σ (ciσ cjσ + cjσ ciσ ) is the Hermitian hopping †
operator between neighboring sites i and j , and niσ = ciσ ciσ is the local density of spin σ = ↑,↓ electrons. We rewrite the interaction, the last term in Eq. (1), as U Hint = (n − 1)2 Pi (n), (2) 2 i n where Pi (n) is the projector at site i onto the subspace with n electrons. In order to construct a partial Schrieffer-Wolff transformation [6] that accounts for an incomplete projection of double occupancies, we separately define the components of the hopping operator Tij projected on the right or on the left onto the configurations where both sites i and j are singly occupied, T˜ij ≡ [Pi (2)Pj (0) + Pi (0)Pj (2)]Tij [Pi (1)Pj (1)], † T˜ij ≡ [Pi (1)Pj (1)]Tij [Pi (2)Pj (0) + Pi (0)Pj (2)].
* †
[email protected] [email protected]
2469-9950/2017/95(16)/161106(4)
(3)
Their sum is gathered under the form of the new operator † Tij ≡ T˜ij + T˜ij , while the remaining part of the hopping 161106-1
©2017 American Physical Society
RAPID COMMUNICATIONS
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PHYSICAL REVIEW B 95, 161106(R) (2017)
operator is gathered under Tij ≡ Tij − Tij . We construct the partial Schrieffer-Wolff transformation U() = exp √ S (4) z
of H is approximated by a variational Gutzwiller wave function |ψG constructed from the uncorrelated Fermi sea |ψ0 through |ψG ≡ Pi |ψ0 . (9)
through the anti-Hermitian operator † Sij = (T˜ij − T˜ ). S=
Pi is a linear operator that, in the presence of the particle-hole symmetry, can be parametrized as √ (10) Pi = 2[sin θ Pi (0) + cos θ Pi (1) + sin θ Pi (2)],
ij
ij
(5)
ij
The transformed Hamiltonian reads H = U()† H U() H − √ [S,H ] z 2 3 [S,[S,H ]] − 3/2 [S,[S,[S,H ]]], (6) 2z 6z where is variationally determined so as to minimize the energy. In the following, we shall assume that for any value of U the optimal is small enough to safely neglect higherorder terms O( 4 ) in Eq. (6). A posteriori, we shall check the validity of such an assumption. We rewrite the transformed Hamiltonian in a more useful form, ⎛ ⎞ t U H H + ⎝ [Sij ,Tkl ] + √ Tij ⎠ z ijkl z ij ⎞ ⎛ +
−
⎟ 2 ⎜ U ⎟ ⎜ t [Sij ,[Skl ,Tmn ]] + [Sij ,Tkl ]⎟ ⎜ 3/2 ⎠ 2 ⎝z z ijkl
+
3 ⎜ ⎜t [Sij ,[Skl ,[Smn ,Tpq ]]] ⎜ 6 ⎝ z2 ijkl mnpq
+
⎞
⎟ U ⎟ [S ,[S ,T ]] ⎟, ij kl mn 3/2 ⎠ z
(7)
ijk lmn
where we made use of the following equality, ⎡ ⎤ ⎣ Sij ,Hint ⎦ = −U Tij . ij
(8)
ij
The transformed low-energy Hamiltonian (7) is then analyzed by a variational approach. Specifically, the ground state
F(,θ )
where θ is a variational parameter bounded by θ ∈ {0,π/4}, where θ = π/4 corresponds to the uncorrelated (metallic) state, whereas θ = 0 projects out of |ψ0 all configurations with doubly occupied and empty sites. In other words, the actual variational wave function for the ground state of the original Hamiltonian H is | = U()|ψG ,
(11)
and depends both on θ and . The variational energy functional per lattice site (where N is the total number of sites) F(,θ ) can be now obtained as the expectation value in the Gutzwiller wave function (9) of the Hamiltonian (7), which can be analytically computed in the infinitely coordinated Bethe lattice, 8T0 N F(,θ ) = |H | = ψG |H|ψG ,
(12)
in reduced units of 8T0 . Here, −T0 is the hopping energy per site of |ψ0 , which in a Bethe lattice reads T0 = 8t/3π . Already at this point, qualitative differences with respect to the standard Brinkman-Rice transition clearly emerge. In our approach, θ = 0, providing vanishing double occupancies in |ψG do not yield the same for the actual wave function |. Explicitly, the density of doubly occupied sites d can be calculated as 1 † d ≡ ψG |U() Pi (2) U()|ψG , (13) N i
ijk lmn
⎛
i
which generically does not provide d = 0 even if θ = 0. In order to evaluate (12) as well as (13) we apply the Wick’s theorem. Each expectation value resulting from this procedure can be conveniently visualized by a diagram with nodes denoting sites and edges being the averages of the intersite single-particle density matrix. We shall shortly denote the sum of diagrams with the same number x of nodes as an x vertex. We checked that a satisfying accuracy is obtained by keeping all x vertices up to x = 4. The resulting energy functional reads
u 1 u − 1− sin2 2θ + 2u(1 − cos 2θ ) − 8τ 1 − cos 2θ 8 2τ 2τ 3 u 32 3 τ u 2 2 9 − − cos 2θ 1− sin 2θ cos 2θ + sin2 2θ + 8τ 4τ 4 2τ 3 2 u 7 4 155 3 2 2 2 − 1− − sin 2θ + 9 sin 2θ cos 2θ + sin 2θ cos 2θ , 128τ 2 6τ 2 192τ
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MOTT PHYSICS BEYOND THE BRINKMAN-RICE SCENARIO
PHYSICAL REVIEW B 95, 161106(R) (2017)
where the interaction strength U and the hopping amplitude t are rescaled as U t 3π u= , τ= = . (15) 8T0 8T0 64
0.06
The first line in Eq. (14) includes the 2-vertex contribution, the second line is the 3-vertex one, and finally the third is the 4-vertex correction. Additionally, the expectation value of the double occupancy d reads 1 2 − (2 − 4 2 ) cos 2θ + (1 − 2 ) sin2 2θ d(,θ ) = 8 2τ
0 E
-0.04 Energy GA+SW total kinetic potential
-0.06
0.3
(17)
under the condition that the Hessian is positive definite. We start observing that for u = 0 the minimum of the functional (14) is correctly determined by = 0 and θ = π/4, and corresponds to the fully uncorrelated metal. For an interaction strength roughly up to u 0.4, the optimized energy is almost coincident with that obtained either by the Gutzwiller approximation or by DMFT. For stronger correlations, u 0.4, the Gutzwiller approximation starts to deviate appreciably with respect to DMFT, while our variational energy remains quite close. In Fig. 1(a) we plot the total energy, as well as kinetic and potential energies separately, of the minimum of functional Eq. (14), as compared with DMFT [10], and with the sole Gutzwiller approximation, for which we just show the total energy. Following Brinkman and Rice [2], we associate the Mott insulating state with θ = 0, which is always a saddle point of the functional (14). However, this saddle point becomes a minimum only when the metal becomes unstable; the metal-to-insulator transition is thus continuous and occurs at a critical interaction uc 0.822, which is sizably lower than the Brinkman-Rice value uBR = 1, and quite close to DMFT, uc2;DMFT 0.854. In Fig. 1(b) we show the values of the variational parameters and θ on the both sides of the transition. In the same figure, we also plot the average double occupancy d [from Eq. (16)], which is nonzero in the insulating phase and decreases almost linearly in the metallic state. In the insulating phase u uc , when the optimal θ = 0, we can analytically calculate several quantities. For instance, the saddle-point value of can be obtained in a power series of τ/u, 3 5 τ τ τ (18) +O 5 , ins = − 4 u u u
DMFT
EBR
-0.1
From (14) and (16) we can easily recover the results of the standard Gutzwiller approximation applied to the Hubbard model by setting = 0. In this case the Brinkman-Rice transition takes place for uBR = 1 and the insulating state is characterized by d = 0. We search for minima of the functional F with increasing u by standard methods. Namely, for each u we look for the pair of variables {,θ } satisfying ∂F/∂ = ∂F/∂θ = 0,
-0.02
-0.08
(16)
(a)
0.02
θ, ε, d
3 2 3 2 + sin 2θ cos 2θ + 32τ 2 768τ 3 7 4 2 2 × − sin 2θ + 9 sin 2θ cos 2θ . 2
MIT 0.04
(b)
θ ε d
0.2
uc1;DMFT
0.1
uc2;DMFT
uBR
0.9
1
0 0.4
0.5
0.6
0.7
0.8
u
FIG. 1. (a) The equilibrium energy balance across the metal-toinsulator transition (MIT). For a comparison we have provided data points of the real energies from DMFT calculations [10]. Additionally, for a reference, we have also included the energy corresponding to the Brinkman-Rice result (EBR ) [2]. The transition takes place for a quite similar critical interaction as for DMFT (uc2;DMFT ). Also, as for DMFT [3], potential and kinetic energies are characterized with pronounced kinks while the total energy remains smooth. (b) The equilibrium values of θ, , and d vs u across the metal-to-insulator transition. We marked the critical values of interaction for the Brinkman-Rice transition (uBR ) as well as those obtained by DMFT, in which case at uc1;DMFT both metallic and insulating solutions begin to coexist. As for DMFT predictions [3], we obtain a nonvanishing double occupancy also in the insulating phase.
whereas the energy per site is 2 6 τ τ t2 4τ 4 4t 4 − 3 +O 5 − + Eins = −8T0 . 2u 3u u 2U 3U 3 (19) Additionally, the average double occupancy in powers of τ/u reads 4 6 1 τ 2 τ τ (20) −4 +O 6 , dins = 2 u u u which is indeed nonzero. We emphasize that, since our calculation is performed in a Bethe lattice with infinite coordination number, the quasiparticle residue, which can be identified by Z = sin2 2θ and vanishes at the transition, remains, just as in DMFT, inversely proportional to the mass enhancement m∗ /m ∼ 1/Z, which thus diverges. Another physical quantity easy to calculate that further highlights the difference between our method and the simple Gutzwiller approximation is the uniform magnetic susceptibility χ (T ) at zero temperature T = 0 at and beyond the Mott transition. We recall that χ (0) → ∞ within the Gutzwiller
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PHYSICAL REVIEW B 95, 161106(R) (2017)
approximation, whose description of a Mott insulator as a collection of independent spins is evidently very poor. By contrast, χ (0) is finite by DMFT, which reveals the antiferromagnetic correlations within the true Mott insulators that hamper the uniform magnetic field. We calculate at leading 2-vertex order the energy functional F (,m) of the Mott insulator θ = 0 in the presence of a uniform Zeeman field h, which now depends variationally on the magnetization m = n↑ − n↓ besides the Schrieffer-Wolff parameter , 2u (1 − m2 ) − hm. F (,m) = − τ − (21) 2 The saddle solution of (21) is characterized by = τ 2 /u and m u (22) = 2 ≡ χ (0). h τ Indeed, our approach yields a finite magnetic susceptibility of the same order as in DMFT. Let us further compare our results with the exact DMFT ones [3]. First, the critical interaction strength at the continuous metal-insulator transition is quite similar to DMFT. However, we do not find any insulator metal coexistence, unlike DMFT where coexistence does occur and spreads over a significant region uc1 u uc2 [see Fig. 1(b)]. Despite that, we do find an energy balance across the transition that is qualitatively similar to DMFT, and quite different from the Gutzwiller approximation. Indeed, hopping and potential energies have kinks at the transition, though the total energy is smooth, and the energy of the Mott insulator scales at the leading order as ∼ −t 2 /2U [cf. Fig. 1(a)]. The critical component A of the total energy E can be extracted through the double occupancy ∂E u − uc − d(u → uc ) = ∼ −A + B, (23) ∂u u→u−c uc
[1] N. F. Mott, Philos. Mag. 6, 287 (1961). [2] W. F. Brinkman and T. M. Rice, Phys. Rev. B 2, 4302 (1970). [3] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996). [4] M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 70, 1039 (1998). [5] M. Capello, F. Becca, M. Fabrizio, S. Sorella, and E. Tosatti, Phys. Rev. Lett. 94, 026406 (2005).
where B is the noncritical term. We find A 0.317 and B 0.013, as compared to the exact DMFT results [3], A 0.235 and B 0.015, and the analytical Gutzwiller approximation values, A = 0.250 and B = 0. We note that our estimate of A is actually worse than the Gutzwiller approximation one, though we find a quite accurate noncritical coefficient B, which is the leading component of the energy and is completely missed by the Gutzwiller approximation. In summary, we have analyzed a very simple variational wave function for the ground state of a correlated electron system that consists of the Gutzwiller wave function combined with a variational Schrieffer-Wolff transformation. We have benchmarked this wave function against exact DMFT results [3] for the paramagnetic Mott transition in the half-filled single-band Hubbard model on a Bethe lattice with infinite coordination number. Although there are obviously differences with the exact results, nevertheless our variational wave function provides a description of the Mott transition that is much closer to the reality than the Brinkman-Rice scenario. More importantly, our wave function is able to portray a Mott insulator where charge fluctuations are not completely suppressed as in the Brinkman-Rice transition, and which therefore has a nonzero expectation value of the hopping energy. This variational technique might open different possibilities to access the Mott physics or related phenomena in more realistic models with minimal computational effort. We are grateful to Adriano Amaricci for providing us the DMFT data. M.M.W. acknowledges support from the Polish Ministry of Science and Higher Education under the “Mobilno´sc´ Plus” program, Agreement No. 1265/MOB/IV/2015/0. M.F. acknowledges support from European Union under the H2020 Framework Programme, ERC Advanced Grant No. 692670 “FIRSTORM.”
[6] K. A. Chao, J. Spałek, and A. M. Ole´s, J. Phys. C 10, L271 (1977). [7] U. Schollwöck, Rev. Mod. Phys. 77, 259 (2005). [8] A. Toschi, A. A. Katanin, and K. Held, Phys. Rev. B 75, 045118 (2007). [9] W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989). [10] C. Weber, A. Amaricci, M. Capone, and P. B. Littlewood, Phys. Rev. B 86, 115136 (2012).
161106-4
PHYSICAL REVIEW B 95, 161106(R) (2017)
Mott physics beyond the Brinkman-Rice scenario Marcin M. Wysoki´nski1,2,* and Michele Fabrizio1,† 1
2
International School for Advanced Studies (SISSA), via Bonomea 265, IT-34136 Trieste, Italy Marian Smoluchowski Institute of Physics, Jagiellonian University, ulica prof. S. Łojasiewicza 11, PL-30-348 Kraków, Poland (Received 22 December 2016; revised manuscript received 10 March 2017; published 11 April 2017) The main flaw of the well-known Brinkman-Rice description, obtained through the Gutzwiller approximation, of the paramagnetic Mott transition in the Hubbard model is in neglecting high-energy virtual processes that generate, for instance, the antiferromagnetic exchange J ∼ t 2 /U . Here, we propose a way to capture those processes by combining the Brinkman-Rice approach with a variational Schrieffer-Wolff transformation, and apply this method to study the single-band metal-to-insulator transition in a Bethe lattice with infinite coordination number, where the Gutzwiller approximation becomes exact. We indeed find for the Mott transition a description very close to the real one provided by the dynamical mean-field theory, an encouraging result in view of possible applications to more involved models. DOI: 10.1103/PhysRevB.95.161106
A metal-to-insulator transition driven by electron-electron repulsion was envisioned by Mott more than 50 years ago [1]. Since then, the underlying physics of this phenomenon has been studied by a large variety of quantum many-body tools in models for strongly correlated systems [2–5]. One of the earliest microscopic descriptions of Mott localization was made by Brinkman and Rice [2], and obtained through the Gutzwiller approximation applied to the halffilled Hubbard model. In their scenario the transition to the insulating state occurs when the hopping is fully hampered by repulsion, i.e., its expectation value in the variational wave function strictly vanishes. This result is elegant in many ways. It is fully analytical and provides a very intuitive and physically transparent, almost classical, interpretation of the Mott phenomenon. Nonetheless, this description, frequently called the Brinkman-Rice transition, has a severe drawback: The expectation value of the hopping cannot be zero, and it is so in the Gutzwiller approximation only because there is a complete, static and dynamic, locking of the charge degrees of freedom. In reality, dynamical charge fluctuations do play a role even deep in the Mott phase, and, in particular, they mediate the antiferromagnetic spin exchange, as is clear by the large U mapping onto the Heisenberg model that can be formally derived through the Schrieffer-Wolff transformation [6]. Since the result of Brinkman and Rice, a variety of quantum many-body tools have been constructed that are generically able to sensibly capture those dynamical processes [3,5,7,8]. In particular, when applied to the Hubbard model, they provide satisfying descriptions of the Mott transition, though relying on a heavy numerical computation. Nowadays, the scenario provided by the dynamical mean-field theory (DMFT) [3], which becomes exact in infinite dimensions [9], has become an invaluable benchmark to be compared. In the present Rapid Communication we revisit the problem of the Brinkman-Rice transition, and complement it with the inclusion of the dynamical processes in a semianalytic manner. In order to achieve this goal, we construct a method
that combines the Gutzwiller’s variational approach with a variational Schrieffer-Wolff transformation. As a case study, we apply our technique to the half-filled Hubbard model in the paramagnetic phase on the infinitely coordinated Bethe lattice. The energy functional to be minimized can be obtained fully analytically. Its minimization leads to a significantly improved description of the Mott transition as compared to the standard Brinkman-Rice scenario, and much closer to the exact DMFT one [3]. The improvement is, in particular, highlighted by (i) a sizable lowering of the critical interaction strength for a transition, (ii) a lower value of the insulator energy that includes a nonzero expectation value of the hopping ∼ −t 2 /U , and (iii) a proper balance of kinetic and potential energies at the transition. The starting point of our analysis is the half-filled singleband Hubbard model on the infinitely coordinated Bethe lattice, t U H = −√ Tij + (ni↑ + ni↓ − 1)2 , (1) z ij 2 i where z → ∞ is the coordination number of the Bethe lattice. † † Tij = Tji ≡ σ (ciσ cjσ + cjσ ciσ ) is the Hermitian hopping †
operator between neighboring sites i and j , and niσ = ciσ ciσ is the local density of spin σ = ↑,↓ electrons. We rewrite the interaction, the last term in Eq. (1), as U Hint = (n − 1)2 Pi (n), (2) 2 i n where Pi (n) is the projector at site i onto the subspace with n electrons. In order to construct a partial Schrieffer-Wolff transformation [6] that accounts for an incomplete projection of double occupancies, we separately define the components of the hopping operator Tij projected on the right or on the left onto the configurations where both sites i and j are singly occupied, T˜ij ≡ [Pi (2)Pj (0) + Pi (0)Pj (2)]Tij [Pi (1)Pj (1)], † T˜ij ≡ [Pi (1)Pj (1)]Tij [Pi (2)Pj (0) + Pi (0)Pj (2)].
* †
[email protected] [email protected]
2469-9950/2017/95(16)/161106(4)
(3)
Their sum is gathered under the form of the new operator † Tij ≡ T˜ij + T˜ij , while the remaining part of the hopping 161106-1
©2017 American Physical Society
RAPID COMMUNICATIONS
´ MARCIN M. WYSOKINSKI AND MICHELE FABRIZIO
PHYSICAL REVIEW B 95, 161106(R) (2017)
operator is gathered under Tij ≡ Tij − Tij . We construct the partial Schrieffer-Wolff transformation U() = exp √ S (4) z
of H is approximated by a variational Gutzwiller wave function |ψG constructed from the uncorrelated Fermi sea |ψ0 through |ψG ≡ Pi |ψ0 . (9)
through the anti-Hermitian operator † Sij = (T˜ij − T˜ ). S=
Pi is a linear operator that, in the presence of the particle-hole symmetry, can be parametrized as √ (10) Pi = 2[sin θ Pi (0) + cos θ Pi (1) + sin θ Pi (2)],
ij
ij
(5)
ij
The transformed Hamiltonian reads H = U()† H U() H − √ [S,H ] z 2 3 [S,[S,H ]] − 3/2 [S,[S,[S,H ]]], (6) 2z 6z where is variationally determined so as to minimize the energy. In the following, we shall assume that for any value of U the optimal is small enough to safely neglect higherorder terms O( 4 ) in Eq. (6). A posteriori, we shall check the validity of such an assumption. We rewrite the transformed Hamiltonian in a more useful form, ⎛ ⎞ t U H H + ⎝ [Sij ,Tkl ] + √ Tij ⎠ z ijkl z ij ⎞ ⎛ +
−
⎟ 2 ⎜ U ⎟ ⎜ t [Sij ,[Skl ,Tmn ]] + [Sij ,Tkl ]⎟ ⎜ 3/2 ⎠ 2 ⎝z z ijkl
+
3 ⎜ ⎜t [Sij ,[Skl ,[Smn ,Tpq ]]] ⎜ 6 ⎝ z2 ijkl mnpq
+
⎞
⎟ U ⎟ [S ,[S ,T ]] ⎟, ij kl mn 3/2 ⎠ z
(7)
ijk lmn
where we made use of the following equality, ⎡ ⎤ ⎣ Sij ,Hint ⎦ = −U Tij . ij
(8)
ij
The transformed low-energy Hamiltonian (7) is then analyzed by a variational approach. Specifically, the ground state
F(,θ )
where θ is a variational parameter bounded by θ ∈ {0,π/4}, where θ = π/4 corresponds to the uncorrelated (metallic) state, whereas θ = 0 projects out of |ψ0 all configurations with doubly occupied and empty sites. In other words, the actual variational wave function for the ground state of the original Hamiltonian H is | = U()|ψG ,
(11)
and depends both on θ and . The variational energy functional per lattice site (where N is the total number of sites) F(,θ ) can be now obtained as the expectation value in the Gutzwiller wave function (9) of the Hamiltonian (7), which can be analytically computed in the infinitely coordinated Bethe lattice, 8T0 N F(,θ ) = |H | = ψG |H|ψG ,
(12)
in reduced units of 8T0 . Here, −T0 is the hopping energy per site of |ψ0 , which in a Bethe lattice reads T0 = 8t/3π . Already at this point, qualitative differences with respect to the standard Brinkman-Rice transition clearly emerge. In our approach, θ = 0, providing vanishing double occupancies in |ψG do not yield the same for the actual wave function |. Explicitly, the density of doubly occupied sites d can be calculated as 1 † d ≡ ψG |U() Pi (2) U()|ψG , (13) N i
ijk lmn
⎛
i
which generically does not provide d = 0 even if θ = 0. In order to evaluate (12) as well as (13) we apply the Wick’s theorem. Each expectation value resulting from this procedure can be conveniently visualized by a diagram with nodes denoting sites and edges being the averages of the intersite single-particle density matrix. We shall shortly denote the sum of diagrams with the same number x of nodes as an x vertex. We checked that a satisfying accuracy is obtained by keeping all x vertices up to x = 4. The resulting energy functional reads
u 1 u − 1− sin2 2θ + 2u(1 − cos 2θ ) − 8τ 1 − cos 2θ 8 2τ 2τ 3 u 32 3 τ u 2 2 9 − − cos 2θ 1− sin 2θ cos 2θ + sin2 2θ + 8τ 4τ 4 2τ 3 2 u 7 4 155 3 2 2 2 − 1− − sin 2θ + 9 sin 2θ cos 2θ + sin 2θ cos 2θ , 128τ 2 6τ 2 192τ
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where the interaction strength U and the hopping amplitude t are rescaled as U t 3π u= , τ= = . (15) 8T0 8T0 64
0.06
The first line in Eq. (14) includes the 2-vertex contribution, the second line is the 3-vertex one, and finally the third is the 4-vertex correction. Additionally, the expectation value of the double occupancy d reads 1 2 − (2 − 4 2 ) cos 2θ + (1 − 2 ) sin2 2θ d(,θ ) = 8 2τ
0 E
-0.04 Energy GA+SW total kinetic potential
-0.06
0.3
(17)
under the condition that the Hessian is positive definite. We start observing that for u = 0 the minimum of the functional (14) is correctly determined by = 0 and θ = π/4, and corresponds to the fully uncorrelated metal. For an interaction strength roughly up to u 0.4, the optimized energy is almost coincident with that obtained either by the Gutzwiller approximation or by DMFT. For stronger correlations, u 0.4, the Gutzwiller approximation starts to deviate appreciably with respect to DMFT, while our variational energy remains quite close. In Fig. 1(a) we plot the total energy, as well as kinetic and potential energies separately, of the minimum of functional Eq. (14), as compared with DMFT [10], and with the sole Gutzwiller approximation, for which we just show the total energy. Following Brinkman and Rice [2], we associate the Mott insulating state with θ = 0, which is always a saddle point of the functional (14). However, this saddle point becomes a minimum only when the metal becomes unstable; the metal-to-insulator transition is thus continuous and occurs at a critical interaction uc 0.822, which is sizably lower than the Brinkman-Rice value uBR = 1, and quite close to DMFT, uc2;DMFT 0.854. In Fig. 1(b) we show the values of the variational parameters and θ on the both sides of the transition. In the same figure, we also plot the average double occupancy d [from Eq. (16)], which is nonzero in the insulating phase and decreases almost linearly in the metallic state. In the insulating phase u uc , when the optimal θ = 0, we can analytically calculate several quantities. For instance, the saddle-point value of can be obtained in a power series of τ/u, 3 5 τ τ τ (18) +O 5 , ins = − 4 u u u
DMFT
EBR
-0.1
From (14) and (16) we can easily recover the results of the standard Gutzwiller approximation applied to the Hubbard model by setting = 0. In this case the Brinkman-Rice transition takes place for uBR = 1 and the insulating state is characterized by d = 0. We search for minima of the functional F with increasing u by standard methods. Namely, for each u we look for the pair of variables {,θ } satisfying ∂F/∂ = ∂F/∂θ = 0,
-0.02
-0.08
(16)
(a)
0.02
θ, ε, d
3 2 3 2 + sin 2θ cos 2θ + 32τ 2 768τ 3 7 4 2 2 × − sin 2θ + 9 sin 2θ cos 2θ . 2
MIT 0.04
(b)
θ ε d
0.2
uc1;DMFT
0.1
uc2;DMFT
uBR
0.9
1
0 0.4
0.5
0.6
0.7
0.8
u
FIG. 1. (a) The equilibrium energy balance across the metal-toinsulator transition (MIT). For a comparison we have provided data points of the real energies from DMFT calculations [10]. Additionally, for a reference, we have also included the energy corresponding to the Brinkman-Rice result (EBR ) [2]. The transition takes place for a quite similar critical interaction as for DMFT (uc2;DMFT ). Also, as for DMFT [3], potential and kinetic energies are characterized with pronounced kinks while the total energy remains smooth. (b) The equilibrium values of θ, , and d vs u across the metal-to-insulator transition. We marked the critical values of interaction for the Brinkman-Rice transition (uBR ) as well as those obtained by DMFT, in which case at uc1;DMFT both metallic and insulating solutions begin to coexist. As for DMFT predictions [3], we obtain a nonvanishing double occupancy also in the insulating phase.
whereas the energy per site is 2 6 τ τ t2 4τ 4 4t 4 − 3 +O 5 − + Eins = −8T0 . 2u 3u u 2U 3U 3 (19) Additionally, the average double occupancy in powers of τ/u reads 4 6 1 τ 2 τ τ (20) −4 +O 6 , dins = 2 u u u which is indeed nonzero. We emphasize that, since our calculation is performed in a Bethe lattice with infinite coordination number, the quasiparticle residue, which can be identified by Z = sin2 2θ and vanishes at the transition, remains, just as in DMFT, inversely proportional to the mass enhancement m∗ /m ∼ 1/Z, which thus diverges. Another physical quantity easy to calculate that further highlights the difference between our method and the simple Gutzwiller approximation is the uniform magnetic susceptibility χ (T ) at zero temperature T = 0 at and beyond the Mott transition. We recall that χ (0) → ∞ within the Gutzwiller
161106-3
RAPID COMMUNICATIONS
´ MARCIN M. WYSOKINSKI AND MICHELE FABRIZIO
PHYSICAL REVIEW B 95, 161106(R) (2017)
approximation, whose description of a Mott insulator as a collection of independent spins is evidently very poor. By contrast, χ (0) is finite by DMFT, which reveals the antiferromagnetic correlations within the true Mott insulators that hamper the uniform magnetic field. We calculate at leading 2-vertex order the energy functional F (,m) of the Mott insulator θ = 0 in the presence of a uniform Zeeman field h, which now depends variationally on the magnetization m = n↑ − n↓ besides the Schrieffer-Wolff parameter , 2u (1 − m2 ) − hm. F (,m) = − τ − (21) 2 The saddle solution of (21) is characterized by = τ 2 /u and m u (22) = 2 ≡ χ (0). h τ Indeed, our approach yields a finite magnetic susceptibility of the same order as in DMFT. Let us further compare our results with the exact DMFT ones [3]. First, the critical interaction strength at the continuous metal-insulator transition is quite similar to DMFT. However, we do not find any insulator metal coexistence, unlike DMFT where coexistence does occur and spreads over a significant region uc1 u uc2 [see Fig. 1(b)]. Despite that, we do find an energy balance across the transition that is qualitatively similar to DMFT, and quite different from the Gutzwiller approximation. Indeed, hopping and potential energies have kinks at the transition, though the total energy is smooth, and the energy of the Mott insulator scales at the leading order as ∼ −t 2 /2U [cf. Fig. 1(a)]. The critical component A of the total energy E can be extracted through the double occupancy ∂E u − uc − d(u → uc ) = ∼ −A + B, (23) ∂u u→u−c uc
[1] N. F. Mott, Philos. Mag. 6, 287 (1961). [2] W. F. Brinkman and T. M. Rice, Phys. Rev. B 2, 4302 (1970). [3] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996). [4] M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 70, 1039 (1998). [5] M. Capello, F. Becca, M. Fabrizio, S. Sorella, and E. Tosatti, Phys. Rev. Lett. 94, 026406 (2005).
where B is the noncritical term. We find A 0.317 and B 0.013, as compared to the exact DMFT results [3], A 0.235 and B 0.015, and the analytical Gutzwiller approximation values, A = 0.250 and B = 0. We note that our estimate of A is actually worse than the Gutzwiller approximation one, though we find a quite accurate noncritical coefficient B, which is the leading component of the energy and is completely missed by the Gutzwiller approximation. In summary, we have analyzed a very simple variational wave function for the ground state of a correlated electron system that consists of the Gutzwiller wave function combined with a variational Schrieffer-Wolff transformation. We have benchmarked this wave function against exact DMFT results [3] for the paramagnetic Mott transition in the half-filled single-band Hubbard model on a Bethe lattice with infinite coordination number. Although there are obviously differences with the exact results, nevertheless our variational wave function provides a description of the Mott transition that is much closer to the reality than the Brinkman-Rice scenario. More importantly, our wave function is able to portray a Mott insulator where charge fluctuations are not completely suppressed as in the Brinkman-Rice transition, and which therefore has a nonzero expectation value of the hopping energy. This variational technique might open different possibilities to access the Mott physics or related phenomena in more realistic models with minimal computational effort. We are grateful to Adriano Amaricci for providing us the DMFT data. M.M.W. acknowledges support from the Polish Ministry of Science and Higher Education under the “Mobilno´sc´ Plus” program, Agreement No. 1265/MOB/IV/2015/0. M.F. acknowledges support from European Union under the H2020 Framework Programme, ERC Advanced Grant No. 692670 “FIRSTORM.”
[6] K. A. Chao, J. Spałek, and A. M. Ole´s, J. Phys. C 10, L271 (1977). [7] U. Schollwöck, Rev. Mod. Phys. 77, 259 (2005). [8] A. Toschi, A. A. Katanin, and K. Held, Phys. Rev. B 75, 045118 (2007). [9] W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989). [10] C. Weber, A. Amaricci, M. Capone, and P. B. Littlewood, Phys. Rev. B 86, 115136 (2012).
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