Effect of interactions, disorder and magnetic field in the Hubbard

PRAMANA

— journal of

c Indian Academy of Sciences °

physics

Vol. 64, No. 6 June 2005 pp. 1051–1061

Effect of interactions, disorder and magnetic field in the Hubbard model in two dimensions N TRIVEDI1,2 , P J H DENTENEER3 , D HEIDARIAN1 and R T SCALETTAR4 1

Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400 005, India 2 Department of Physics, The Ohio State University, Columbus, OH, USA 3 Lorentz Institute, Leiden University, Leiden, The Netherlands 4 Department of Physics, University of California at Davis, Davis, CA, USA E-mail: [email protected] Abstract. The effects of both interactions and Zeeman magnetic field in disordered electronic systems are explored in the Hubbard model on a square lattice. We investigate the thermodynamic (density, magnetization, density of states) and transport (conductivity) properties using determinantal quantum Monte Carlo and inhomogeneous Hartree Fock techniques. We find that at half filling there is a novel metallic phase at intermediate disorder that is sandwiched between a Mott insulator and an Anderson insulator. The metallic phase is highly inhomogeneous and coexists with antiferromagnetic long-range order. At quarter filling also the combined effects of disorder and interactions produce a conducting state which can be destroyed by applying a Zeeman field, resulting in a magnetic field-driven transition. We discuss the implication of our results for experiments. Keywords. Metal–insulator transition; correlations; disorder. PACS Nos 71.30.+h; 71.10.Fd; 72.15.Rn

1. Introduction and motivation The simplest disorder-driven quantum phase transition is the Anderson localization problem that describes the properties of non-interacting electrons in a random potential. It was shown by the scaling theory [1,2] that in dimensions d > 2 the disordered system shows a transition from a conducting to an insulating phase. In a conductor, the wave functions are extended across the system and respond to perturbations at the boundaries. On the other hand, in an insulating phase the wave functions are localized around deep potentials and are therefore insensitive to boundary conditions. The scaling theory further showed that for non-interacting electrons, the destructive quantum interference from time reversed paths localizes all the states in 2d and therefore no metallic behavior is possible in a 2d system. However, in a condensed matter system, interactions cannot be ignored. In fact, the effect of localizing the particles by disorder only enhances the effective 1051

N Trivedi et al interactions. Many studies have addressed whether correlations might change the possibility of a metallic phase in two dimensions [3–9]. In this paper we discuss the open question of the combined effect of repulsive interactions and disorder in a fermionic system. In particular, we would like to understand whether the scaling result of the absence of metallic behavior in 2d remains robust when interactions are included or whether we get something new. We have previously studied the interplay of attractive interactions and disorder in superconductors [10,11] as well as the interplay of repulsive interactions and disorder in superfluids [12]. Including the effects of interactions in a disordered system is completely nontrivial. Given the many body nature of basis states, the Hilbert space grows exponentially with system size, and exact diagonalization is limited to about 16–20 sites. Quantum Monte Carlo techniques are powerful as they can incorporate disorder and interaction effects exactly for about 100 fermions. However, in many cases the simulations are restricted to temperatures greater than T ' 0.2t because of the ‘sign problem’ [13]. This has motivated us to explore another technique in considerable detail in this paper – an inhomogeneous Hartree Fock approach [14]. Though it treats interactions within mean field, this method includes disorder effects exactly and has the virtue of being able to deal with ∼ 5000 fermions at T = 0 as well as at finite T . We present here results of our quantum Monte Carlo simulations which gave the first hint of some unusual features in the behavior of the conductivity with increasing disorder in the Hubbard model. We then discuss in detail our results of the inhomogeneous mean field theory which has been able to provide a very intriguing picture of the phases that emerge with increasing disorder. Experimentally, our motivation is derived from the behavior of the resistivity in 2d MOSFETs [15–17] which give a clear indication of a transition from an insulating behavior (with dρ/dT < 0) to metallic behavior (with dρ/dT > 0) as the density of carriers n is increased. The data can further be scaled by plotting the resistivity vs. a scaled variable |n − nc |/T 1/νz where nc is the critical density that marks the metal–insulator transition and ν and z are critical exponents. Previously we have shown that metal–insulator transitions driven by interactions and disorder also occurs in a simple model for 2d electrons [18]. Here we discuss the effect of a Zeeman magnetic field on the metallic state and find that it is also similar to that found in the experiments on various semiconductor heterostructures [19]. 2. The model For an accurate modeling of the disordered 2d electron gas, it may be desirable to include the long-range nature of the Coulomb interactions and also the changes in the screening properties of the electrons with varying density. In this paper we choose, however, to first analyse a model with short-range interactions for which the effects of disorder is also very challenging, and still open, but which is simpler to treat numerically. To this end, we consider interacting fermions in a 2d disordered system using a repulsive Hubbard Hamiltonian with potential disorder: X X X H=− tij (c†iσ cjσ + h.c.) + U ni↑ ni↓ + (Vi − µ)niσ , (1) hiji,σ

1052

i

i,σ

Pramana – J. Phys., Vol. 64, No. 6, June 2005

Hubbard model in two dimensions where the first term is the kinetic energy for near-neighbor hopping, c†iσ (ciσ ) the creation (destruction) operator for an electron with spin σ on a site ri of a square lattice of N sites, U is the on-site repulsion between electrons, niσ = c†iσ ciσ , and µ is the chemical potential which controls the density. Disorder can be introduced onsite through a random potential Vi chosen independently at each ri and uniformly distributed in the interval [−V, V ]; V thus controls the strength of the disorder. Disorder can also be introduced in the hopping amplitudes by choosing tij uniformly within an interval t(1 − ∆t /2) < tij < t(1 + ∆t /2). All energies are measured in units of t. This is a minimal model containing the interplay of electronic correlations and localization: for zero disorder V = 0 and ∆t = 0, it describes an antiferromagnetic Mott insulator at half filling, and for U = 0 it reduces to the (non-interacting) Anderson localization problem. The two types of disorder, site (chemical potential) and bond (hopping amplitudes) disorder, at half-filling have a very different effect on conductivity and the Mott gap. Studies of alternative types of disorder (next near neighbor hopping and random Zeeman fields) has demonstrated that these effects go hand-in-hand and can be attributed to whether the disorder breaks particle–hole symmetry [20]. Away from half-filling, where particle–hole symmetry is absent, the effects of bond and site disorder are therefore similar. 3. A puzzle As is well-known, at half filling the Hubbard model has a gap in the density of states which is proportional to U , the scale for repulsive interactions, for large U . Further, it can also be shown that the system has antiferromagnetic long-range order (AFLRO) generated by a superexchange mechanism of scale J ∼ t2 /U , where t is the hopping parameter. We ask a sharp question: With increasing disorder, which is killed first – the gap in the energy spectrum or the AFLRO, or do they get destroyed together? If your answer is that the lower energy scale, i.e. the AFLRO, will get destroyed first, then read on. You are in for a surprise! If your answer is that the gap will get destroyed first, you are right, but still read on to make sure you have the correct reasoning!! 4. Determinantal quantum Monte Carlo method We first use the determinantal quantum Monte Carlo technique for the numerical simulation of the disordered Hubbard model. The method is based on the Trotter decomposition in imaginary time and a decoupling of the four fermion interaction term by auxiliary Ising spins. It can be applied, in principle, to arbitrary random systems. This method is ‘exact’ (within statistical error) in that it includes all the fluctuations. However, as one approaches low temperatures, the algorithm can suffer from the minus-sign problem [13] which can limit the temperatures to those values greater than about 0.03W where W = 8t is the non-interacting bandwidth of the clean system. Pramana – J. Phys., Vol. 64, No. 6, June 2005

1053

N Trivedi et al

Figure 1. Conductivity σdc as a function of temperature T comparing U = 4 and U = 0 and different lattice sizes at hni = 0.5 and (hopping) disorder strength ∆t = 2.0. Data points are averages over many realizations for this disorder strength (see text). Error bars are determined by the disorder averaging and not by the Monte Carlo simulation.

One central result is shown in figure 1 which shows a remarkable feature: the conductivity for a disordered system is enhanced when the electrons are interacting strongly. Given the temperature limitations and finite-size effects we cannot conclude definitively if the interacting system is a metal or an insulator in the T → 0 limit. However, the data for U = 4 on lattice sizes L = 4 and L = 8 are encouraging in that they show the system becoming more metallic as one goes toward larger lattices. On the other hand, the non-interacting U = 0 values are turning more sharply towards insulating behavior as the lattice size increases, further suggesting that the conductivity enhancement may not be a finite-size effect. The conductivity is obtained from the current–current correlation function [10]. Using the fluctuation–dissipation relation we have Z ∞ dω exp(−ωτ ) ImΛxx (q, ω) (2) Λxx (q, τ ) = −∞ π [1 − exp(−βω)] for 0 ≤ τ ≤ β. From QMC we compute Λ(q, τ ). The dc conductivity σdc = limω→0 [ImΛxx (q = 0, ω)/ω] which requires a knowledge of ImΛ(ω), a quantity that requires a numerical inversion of the Laplace transform of the QMC data. In order to avoid this ill-defined procedure, we have developed an approximate procedure that is valid for T ¿ Ω, where Ω is the scale on which ImΛ deviates from the lowfrequency behavior ImΛ ≈ ωσdc . With the low-frequency input, eq. (2) simplifies to σc =

β2 Λxx (q = 0, τ = β/2) π

(3)

which yields the dc conductivity in terms of the current–current correlation calculated from QMC. 1054


Hubbard model in two dimensions We next explore an alternative mean field method which is studied at T = 0 and that eliminates the need for any extrapolation to zero temperature. It is further possible to study much larger systems (about 5000 electrons) and thereby perform a more extensive finite-size scaling. 5. Inhomogeneous Hartree Fock method We begin by treating the spatial fluctuations of the local densities niσ = hc†iσ ciσ i and † † − local magnetic fields h+ i = −U hci↑ ci↓ i and hi = −U hci↓ ci↑ i using a site-dependent Hartree Fock (HF) approximation. Starting with an initial guess for {h± i } and {niσ } we numerically solve for the eigenvalues ²n and eigenvectors ψnσ (ri ) of a 2N × 2N matrix, where N is the number of lattice sites. The local fields are then determined in terms of the eigenfunctions and the eigenvalue problem with these new local fields as input is then iterated until self consistency is achieved at each site. For a given realization of the disorder potential, the chemical potential µ is also adjusted to satisfy the half filling condition exactly. We report results primarily for U = 4t and N up to ∼ 50 × 50. The inhomogeneous Hartree Fock approach is crucial to reveal the spatial structure of the electronic density and magnetization induced by disorder. Since we solve for all the eigenstates, we also have dynamical information to extract the density of states and frequency-dependent conductivity [14]. 6. Results: Mott insulator + site disorder These results were first reported in ref. [14]. 6.1 Destruction of antiferromagnetism (AF) Figure 2a shows that the local magnetization m†i ≡ (−1)ix +iy hSz (i)i continues to have AF order for V = 2t with only a few ‘defects’. With increasing disorder, the defective regions with reduced antiferromagnetic order grow in size. This can also be seen in figures 2b and 2c where the distribution of m†i shows a growth of PM sites with m†i ∼ 0 as disorder increases and correspondingly increased fluctuations in the local density away from ni = 1. 6.2 Destruction of spectral gap We next look at the behavior of the two defining characteristics of a Mott insulator: the spectral gap and the AF order are shown in figure 3. The spectral gap is obtained from the lowest eigenvalue of the HF Hamiltonian, and the AF order parameter mOP is obtained from the large-distance behavior of the spin–spin correlation function. The first surprise is that even though the energy scale for charge Pramana – J. Phys., Vol. 64, No. 6, June 2005

1055

N Trivedi et al

0.48 0.20

V=2t

V=1 V=2 V=3 V=5

6 P(m†)

(a)

-0.09 0.48

4 2

(b)

0 -0.3 0 0.3 0.6 m†(i)

0.15 6

-0.19 0.57 0.10 V=5t

P(〈n〉

V=3t

(c)

4 2 0 0

-0.37

0.5

1 1.5 〈ni〉

2

Figure 2. (a) Local staggered magnetization m† (i) ≡ hSz (i)i(−1)ix +iy for the disordered Hubbard model at half filling with U = 4t on a 28 × 28 lattice for disorder strengths V = 2t; V = 3t; V = 5t for one realization of disorder. The regions in blue have AF order; the defective sites with reduced AF order are shown in yellow and red. (b) Probability distribution P (m† ) for different values of V . With increasing V , P (m† ) gets broader and develops weight near 0, indicating the growth of paramagnetic (PM) regions. (c) Probability distribution P (n) of the site-occupancy hni i showing a peak near hni i ≈ 1 for V = 1 which gets broader with increasing V and develops weight for doubly occupied and unoccupied sites.

fluctuations is larger than the scale for antiferromagnetic exchange, i.e. U À J ∼ t2 /U , the spectral gap vanishes at Vc1 ≈ 2t which is lower than the critical disorder Vc2 ≈ 3.4t where AF order vanishes. We have compared mOP obtained by the inhomogeneous HF method with QMC simulations [21] on similar size lattices and find good agreement, suggesting that this result is not simply a consequence of the mean field treatment. 6.3 Prediction of a novel metallic phase The occurrence of two transitions in figure 3 defines three distinct regions: In region I, defined by 0 ≤ V ≤ Vc1 , the system is a Mott insulator with a finite gap and AF order. The intermediate region II, Vc1 < V < Vc2 is extremely unusual with AF order but no gap and is discussed in greater detail below. In region III, V ≥ Vc2 , the system is an Anderson or localized insulator with gapless excitations and no magnetic order. We argue below that in region II the system is an inhomogeneous AF metal. The spatial extent of the HF eigenstates near the Fermi energy in figure 4 distinctly 1056


Hubbard model in two dimensions

Figure 3. (a) Density of states averaged over 10 realizations at half filling and U = 4t. For V = t there is a gap in the spectrum at the chemical potential E = 0 which closes for V ≥ 2t. (b) Single particle energy gap as a function of V showing the collapse of the gap at Vc1 ∼ 2t (scale on left). The AF staggered order parameter m† (scale on right) vanishes beyond Vc2 ∼ 4t > Vc1 . Note that at V ∼ 2t the system is still strongly AF even though the spectral gap has vanished.

shows that they are localized at low disorder V = 2t and high disorder V = 5t, but quite surprisingly at intermediate disorder V = 3t the states get more extended. It is evident that the metal is formed when the wave function localized in the region around the defects generate a ‘percolating’ network with reduced AF. Unlike the classical case where the percolating regions have to physically touch, in the quantum case they are connected by tunneling (see figure 4). The localization length ξloc (α) for state ψα (ri ) is related to the Pthe (normalized)−2 (α) [2]. One would expect inverse participation ratio IPR ≡ ri |ψα (ri )|4 ∝ ξloc that with increasing disorder, ξloc would decrease or correspondingly the IPR would increase. Instead we see in figure 5 a very definite decrease in the IPR for V ≈ U/2 signaling that even though disorder is increasing, the states are getting more extended. Such anomalous behavior continues until about V ≈ 3U/4 and then once again reverts to the usual behavior where the IPR increases with V . One simple effect of repulsive interactions between particles is to screen the random potential, thereby generating an effectively weaker random potential. Previous QMC simulations on disordered bosonic [12,22] and fermionic [18] systems have shown such screening effects. The appearance of an antiferromagnetic metal has also been discussed for the Hubbard model in D = ∞ [23], in a Hartree Fock treatment in D = 3 [24,25] and beyond [26]. Our crucial finding here of a non-monotonic behavior of the localization length seen in figure 5 cannot be understood on the basis of screening alone. Certainly, the effective localization length in the screened potential would be longer than in Pramana – J. Phys., Vol. 64, No. 6, June 2005

1057

N Trivedi et al

V=2t

0

V=3t

0.46 0

V=5t

0.05 0

0.19

Figure 4. Plot of |ψn (r)|2 corresponding to an energy eigenstate ²n at the Fermi energy. The eigenstate is localized for low disorder V = 2t and high disorder V = 5t but surprisingly is more extended at intermediate disorder V = 3t.

−2 Figure 5. Inverse participation ratio IPR ∝ ξloc , where ξloc is the localization length for the wave function at the Fermi energy, as a function of disorder V . In both regions I and III the IPR increases with disorder or ξloc decreases with disorder, as expected. In the intermediate region II, the IPR shows a very distinctive non-monotonic behavior around V ≈ U/2 which indicates that the wave function in fact gets less localized with increasing disorder. The states in the horizontally hatched region 2.8 < V /t < 4 are found to be extended.

the non-interacting case. However, this effective length would monotonically decrease with increasing V or correspondingly the IPR would monotonically increase with V , in contrast to the behavior seen in figure 5. 1058


Hubbard model in two dimensions

Figure 6. Conductivity σdc (in units of e2 /h) as a function of temperature T for various strengths of the Zeeman magnetic field Bk . As Bk increases, a transition from metallic to insulating behavior is seen in σdc . Calculations are performed on 8 × 8 lattices for U = 4 at density hni = 0.5 with hopping disorder strength ∆t = 2.0; error bars result from averaging over typically 16 quenched disorder realizations.

Figure 7. Degree of spin polarization P = (n↓ − n↑ )/(n↓ + n↑ ) as a function of Bk for fixed low T = t/8. The polarization is about 0.3 at the estimated critical field strength Bk = 0.4 for the metal–insulator transition.

The origin of the metal at intermediate disorder is crucially related to the spatially inhomogeneous and correlated nature of the effective screened disorder potential. 7. Effect of Zeeman magnetic field The effect of a Zeeman field on the electrons is modeled by adding to eq. (1) a term of the form X HZ = Bk (ni↑ − ni↓ ). (4) i

We calculate the conductivity as a function of T using QMC, for the case of disorder in the hopping amplitudes and a density well away from half filling at hni = 0.5, where the sign problem is less severe and we are away from the Mott and AF phases. The results shown in figure 6 indicate that the disordered system becomes relatively less conducting as the Zeeman field is increased. The effect of the Zeeman field is to polarize the system and thereby reduce the effective interaction. The critical field Bk = 0.4t destroys the metallic phase. Interestingly, the polarization near this magnetic field driven metal–insulator transition is close to 30% as seen in figure 7. Thus it appears that while the polarizing effects of the Zeeman field reduce the effect Pramana – J. Phys., Vol. 64, No. 6, June 2005

1059

N Trivedi et al of the Hubbard-U interaction, the transition is nevertheless driven by many body effects at a polarization value well below the non-interacting limit. These effects of a Zeeman magnetic field are in agreement with phenomena in various semiconductor heterostructures: see ref. [19] for more extensive discussion and comparison to experiments. In general, one cannot argue that the orbital effects are negligible. However, many of the experiments are done with the applied field in the plane of the 2d film. In this situation, only the Zeeman term is needed. Our results are most relevant to those experiments. Including orbital terms arising from the magnetic field requires the introduction of complex hopping amplitudes. The resulting ‘probabilities’ for the quantum Monte Carlo calculations also become complex, and the simulations become difficult. However, it is possible to include orbital effects in a self-consistent mean field theory. 8. Implication for experiments We propose that experiments on cuprates in their parent Mott insulating phase, could introduce disorder by isoelectronic substitution, or by bombardment, without changing the carrier doping. Recently, Vajk et al [27] have measured the spin correlations using neutron scattering in La2 CuO4 with Zn substitution on the Cu sites. It will be necessary to complement the magnetic information from neutron scattering with transport and spectroscopy to look for signatures of an inhomogeneous metallic phase. 9. Conclusions The disordered Hubbard model has been studied in two dimensions using two complementary techniques: quantum Monte Carlo simulations and inhomogeneous mean field theory. At half filling there is substantial evidence for a two-dimensional metal at intermediate disorder sandwiched between a Mott insulator at low disorder and an Anderson-type insulator at high disorder. The existence of such a metal violates the scaling theory of localization that was developed for non-interacting electrons and which suggested the absence of metallic behavior in 2d. The metal– insulator transition can be driven in a variety of ways, by turning interactions on from the non-interacting limit, by increasing the randomness at fixed interaction, or with a magnetic field. In the latter case, the transition occurs well away from full-spin polarization.

Acknowledgements DH was supported in part by a grant from the Kanwal and Ann Rekhi Foundation administered by the Alumni Association of TIFR. RTS acknowledges the support of NSF DMR 0312261 and NSF ITR 0313390. 1060


Hubbard model in two dimensions References [1] E Abrahams et al, Phys. Rev. Lett. 42, 673 (1979) [2] P A Lee and T V Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985) [3] F Wegner, Z. Phys. B36, 209 (1980) K B Efetov, A I Larkin and D E Khmelnitskii, Sov. Phys. JETP 52, 568 (1980) [4] A M Finkel’stein, Zh. Eksp. Teor. Fiz. 84, 168 (1983) [Sov. Phys. JETP 57, 97 (1983)] C Castellani, C Di Castro, P A Lee and M Ma, Phys. Rev. B30, 527 (1984) [5] D Belitz and T R Kirkpatrick, Rev. Mod. Phys. 66, 261 (1994) [6] V Dobrosavljevi´c, E Abrahams, E Miranda and S Chakravarty, Phys. Rev. Lett. 79, 455 (1997) [7] C Castellani, C Di Castro and P A Lee, Phys. Rev. B57, R9381 (1998) S Chakravarty, L Yin and E Abrahams, Phys. Rev. B58, R559 (1998) S Chakravarty, S Kivelson, C Nayak and K V¨ olker, preprint cond-mat/9805383 [8] P Phillips, Y Wan, I Martin, S Knysh and D Dalidovich, Nature (London) 395, 253 (1998) D Belitz and T R Kirkpatrick, Phys. Rev. B58, 8214 (1998) [9] B L Altshuler and D Maslov, Phys. Rev. Lett. 82, 145 (1999) T M Klapwijk and S Das Sarma, preprint cond-mat/9810349 [10] N Trivedi, R T Scalettar and M Randeria, Phys. Rev. B54, R3756 (1996) [11] A Ghosal, M Randeria and N Trivedi, Phys. Rev. Lett. 81, 3940 (1998); Phys. Rev. B63, 020505 (2000) [12] W Krauth, N Trivedi and D Ceperley, Phys. Rev. Lett. 67, 2307 (1991) [13] E Y Loh, J E Gubernatis, R T Scalettar, S R White, D J Scalapino and R L Sugar, Phys. Rev. B41, 9301 (1990) [14] D Heidarian and N Trivedi, Phys. Rev. Lett. 93, 126401 (2004) [15] S V Kravchenko et al, Phys. Rev. B50, 8039 (1994); ibid, 51, 7038 (1995); Phys. Rev. Lett. 77, 4938 (1996) E Abrahams et al, Rev. Mod. Phys. 73, 251 (2001) [16] D Popovi´c, A B Fowler and S Washburn, Phys. Rev. Lett. 79, 1543 (1997) [17] D Simonian, S V Kravchenko and M P Sarachik, Phys. Rev. B55, R13421 (1997) Y Hanien et al, Phys. Rev. Lett. 80, 1288 (1998) M Y Simmons et al, Phys. Rev. Lett. 80, 1292 (1998) [18] P J H Denteneer, R T Scalettar and N Trivedi, Phys. Rev. Lett. 83, 4610 (1999) [19] P J H Denteneer and R T Scalettar, Phys. Rev. Lett. 90, 246401 (2003) [20] P J H Denteneer, R T Scalettar and N Trivedi, Phys. Rev. Lett. 87, 146401 (2001) [21] M Ulmke et al, Adv. Sol. St. Phys. 38, 369 (1999) (Vieweg, Wiesbaden); Phys. Rev. B55, 4149 (1997); Phys. Rev. B51, 10411 (1995) [22] N Trivedi and S Ullah, J. Low Temp. Phys. 89, 67 (1992) [23] M Ulmke, V Janiˇs and D Vollhardt, Phys. Rev. B51, 10411 (1995) [24] M A Tusch and D E Logan, Phys. Rev. B48, 14843 (1993) [25] S Das Sarma and E H Hwang, Phys. Rev. Lett. 83, 164 (1999) V T Dolgopolov and A Gold, JETP Lett. 71, 27 (2000) I Herbut, Phys. Rev. B63, 113102 (2001) [26] D Tanaskovic et al, cond-mat/0303145 M C O Aguiar et al, cond-mat/0305511, cond-mat/0302389 [27] O P Vajk et al, Science 295, 1691 (2002)


1061