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Quantum Monte Carlo Study of the Hubbard Ladder

We present quantum Monte Carlo results for the Hubbard model on a ladder using the re-structuring method which employ eigenstates of two-site system on a rung to construct a complete set. From technical reasons we concentrate on the case in which the hopping along the leg is much less than the hopping on the rung. We observe the ground state of the half-filled system is made of singlet-like states described in this paper and its first excited state contains one triplet rung. When the system is lightly doped states composed of an electron and a hole appear on some rungs. We find very few hole-pairing on the rung upon doping up to the quarter-filled case.


Section 1 Introduction
Lately strongly correlated systems of electrons on ladders with two or more legs have become quite intriguing from both theoretical and experimental points of view. Various models are numerically investigated by means of exact diagonalization [1,2,3] and renormalization group approaches [4,5,6]. Active studies on the quantum spin ladder systems, which correspond to the half-filled fermionic ladders with infinite Coulomb repulsion, are also stimulative [7]. Among their results, especially interesting are spin gaps, ground states as well as excited states of the systems and formation of the hole-pair bound state. Presence of spin gaps is reported for several spin systems and for fermion systems such as the t-J-t ′ -J ′ model on a single ladder (two coupled chains) studied in Ref. [1], the t-J model on a single ladder [2] and on multi-ladders with even number of coupled chains [3], and the two-chain Hubbard Model [4]. In Refs. [1,2] emergence of the hole-pairing upon doping, which would indicate the high T c superconductivity, is also claimed.
Quantum Monte Carlo techniques based on the Suzuki-Trotter formula, which prove useful for many spin systems [8,9], are difficult to apply to the fermion systems because of the negative sign problem except for those on a single chain [10]. Usually studies on the two-dimensional fermion systems have been done after integrating the fermionic degrees of freedom [11,12,13,14]. Yet it would be desirable to simulate such electron systems keeping fermionic degrees of freedom and vividly observe how electrons and holes behave on ladders. Recently dynamical properties of the halffilled Hubbard ladder are studied by Scalapino et al. [6] using grand canonical quantum Monte Carlo calculations with maximum entropy analytic continuation.
In this paper we formulate the Hubbard Model on a single ladder, with or without doping, applying the re-structuring approach [9]. Here we employ a complete set composed of sixteen eigenstates of electrons sitting two ends of a rung, including the double occupancy (See the Table). Although our simulations are limited to the case in which the hopping along each chain is much weaker compared to the hopping between sites on a rung, we obtain there Monte Carlo results with satisfactory statistical accuracy on the system's energy as well as what we call state distribution in this paper, which is defined in section 2, sorted according to values of the third component of the total spin. At half-filling we observe that in the ground state the singlet-like state, the ninth eigenstate in the Table, is sitting on all rungs of the system and the first excited state has a clear gap because of the triplet eigenstate which appears on one rung of the system. Upon doping the lowest lying state we have observed contains not a hole-pair eigenstate but some of one-electron eigenstates in addition to the singlet-like eigenstate. Here we see no spin gap. All our results are compatible with the corresponding ones in Ref. [4]. For the half-filled ladder our observations are also in agreement with the t-J results in Ref. [2].
In the next section we sketch the model and the method used in this paper. In section 3 the Monte Carlo results are presented. The final section is devoted to the summary.

Section 2 the Model
The model we study is the Hubbard ladder with two legs and N rungs. Its

Hamiltonian is
where c † jσ (i) is the creation operator for an electron of spin σ on the i-th rung of the j-th leg of the ladder and n jσ (i) = c † jσ (i)c jσ (i). From technical reasons we employ an open boundary condition with a hole at each edge of the legs (See Fig. 1). The partition function Z is given by with the inverse temperature β, the chemical potential µ and Let us describe our re-structuring approach briefly. In our representation the states of the system are given by where a m (b m ), which is one of the sixteen eigenstates of the two-site system shown in the Table, denotes the state on the m-th rung along the chain A(B) in Fig. 1.
We expect this choice is proper at least for strong hopping on rungs. It should be noted that a m and b m may differ by sign for the same occupants on the two sites of the m-th rung because the left site on the rung along the chain A is the right site along the chain B and vice versa.
Then we divide the above Hamiltonian (1) with even N into four parts, where Note that the Hamiltonian H A1 + H A2 (H B1 + H B2 ) describes the Hubbard chain A(B) with the hopping parameters t c and t r /2 and the Coulomb repulsion U/2.
Applying the Suzuki-Trotter formula we obtain where n is the Trotter number and µ ′ ≡ µ/4.

For later use let us introduce a hermitian operator O defined by
where C α denotes a real number. Since With the partition function Z n in (5)  We do not fix the number of electrons on a ladder so that the system would come to the thermal equilibrium swiftly.
Since one is not free from the negative sign problem with the formulations stated above, our study is limited to the parameter regions where the cancellation is not serious. To check it we measure the r ratio, where Z + (Z − ) is the number of configurations with positive (negative) weight. We refer to the part that survived the cancellation as the whole configurations. A physical quantity A is calculated by with contributions from positively (negatively) signed configurations A + (A − ).
The physical quantities we measure are the hole doping δ and the system's energy E . We obtain δ by N e being the number of electrons on the ladder. As for the system's energy, we measure the quantity G ≡ E − µN e using the standard technique in the quantum Monte Carlo method and obtain E by In order to shed some light on the system's wave function we also measure the frequency distribution for every eigenstate in the Table, which is the ratio of the number of rungs on a ladder occupied by that eigenstate to the total number of rungs N and will be called state distribution. It should be noted this quantity is physical too, because the operator to represent the number of each eigenstate on a ladder is expressed by (6). For the hole-pair eigenstate, for example, C α in (6) is zero if none of a i 's in the state | α are equal to | 00 , is one if | α contains one and only one | 00 state among its a i 's, and so on.
The third component of the total spin of the system, S z , is another measurable quantity in this formalism. We will see S z is very helpful to study system's energy spectra.

Section 3 Results
Now let us show our Monte Carlo results. Typically first ten thousand configurations are discarded for the thermalization and next a few ten thousand configurations are used for the measurement. Throughout the rest of the paper we take t r =1 and, unless stated otherwise, U=6.
We concentrate on the case t c ≪ 1, where the negative sign problem is not so serious. In this case the ground state at half-filling is expected to be made of singlet rungs. We simulate the system with t c = 0.1, namely t r /t c = 10. In Fig. 2 we plot the state distribution on the N = 16 ladder as a function of β, with values of µ adjusted to ensure δ small enough to realize the half-filled system. Values of the r ratio are between 0.58 and 1 for these β and µ. We see the mixed state (λ 1 ), the ninth state in the Table, Table Caption), which is about 0.92 for U = 6 and goes to 1 as U goes to infinity. We refer to this state as the singlet-like state.
In order to study the excitation spectrum at half-filling, we examine the value of S z at β = 13 and δ ∼ 0, where we find that 98.85% of the whole configurations has S z = 0 and 0.59 (0.56)% has S z = 1 (−1). The state distribution with non-zero S z shows that 6.235 (6.244)% of the S z = 1 (−1) part is occupied by the sixth  show us that the ratio of the second, the fourth and the ninth eigenstates in the Table (namely the even  respectively. It should be noted that these values of E are smaller than −10.48, the value with t c = 0 directly calculated from the eigenvalues of the rungs. We also simulate the N = 32 ladder with N 0 = 2 at β = 9 and obtain statistically meaningful results on S z = 0, −1 configurations which suggest that the system has 8 two (one) even one-electron rungs with up spin, two (three) even one-electron rungs with down spin and 28 singlet-like rungs for S z = 0 (−1). The energy on this ladder is E = −21.30 ± 0.05 (−21.29 ± 0.07) for S z = 0 (−1), which indicates that the finite-size effect is small.

Section 4 Summary
In this paper we carry out Monte Carlo simulations of the Hubbard model on a ladder by the re-structuring method we proposed in Ref. [9]. Having constructed the complete set in the Suzuki-Trotter formula from the eigenstates for two electrons on a rung, our method works satisfactorily when the hopping along the chain is much weaker compared to the hopping within rungs. At half-filling we obtain numerical results which indicate that the ground state is of singlet-like eigenstate only, the first excited state contains one triplet rung besides singlet-like ones and the spin gap persists in the thermodynamic limit. When the system is doped we observe that on some rungs the eigenstates which contain one electron and one hole appear instead of the singlet-like ones seen in the half-filled case.
We did not observe any hole-pair formation in the parameter region we could study. Whether the doped system realizes hole-pairing or not on ladders with larger hopping parameters along the chain is an open question so far. Although it seems difficult to obtain data for such ladders straightforwardly, detailed study on the system's wave function which is possible in this formulation would be useful to understand properties of the system.

Table Caption
Eigenstates, eigenvalues for the partial Hamiltonian of two-site system on one rung, with t r and U in the Hamiltonian (1) and the chemical potential µ in (2), and their names used in this study.
| S l , S r in the Table represents the state on a rung whose left (right) site is occupied by S l (S r ), where S l (S r ) is either 0 (hole), ↑ (an up-spin electron), ↓ (a down-spin electron) or (an electron pair).
Parameters u 1 and u 2 and eigenvalues λ 1 and λ 2 in the ninth and the tenth eigenstates are defined as