Spin and Charge Gaps

To deepen our understanding of the quantum phase transition we perform an analysis of the size-dependence of the spin and charge excitation gaps. We define the gaps as follows:

∆_s = E₀(N, S_z = 1/2)−E₀(N, S_z = 0), (10.4)

∆₁ = E₁(N, S_z = 0)−E₀(N, S_z= 0), (10.5)

∆2 = E2(N, Sz = 0)−E0(N, Sz= 0). (10.6) In these definitionsE₀ is the ground state in the subspace under consideration and E₁ andE₂ are the first and second highest excited states, respectively. Note that in a finite

-5 -4 -3 -2 -1 0

ln(|δ − 1.28|)

-0.8 -0.6 -0.4 -0.2 0

ln(m CDW(δ − 1.28))

CDW

|δ − 1.28|^0.126

Figure 10.2: Fit of the CDW order parametermCDW with the power law|δ−δc|^β plotted on a logarithmic scale. The best fit is obtained with δc= 1.28 andβ = 0.126≈1/8 which is very close to the critical exponent of the classical two-dimensional Ising model.

system states that are degenerate in the thermodynamic limit acquire a non-zero energy difference. The boundaries act as a perturbation that split the levels which will become exactly degenerate in the limit L→ ∞. This is true also for periodic systems.

To determine the excitation gaps ∆1 and ∆2, the three lowest-lying eigenstates are included as targets in the reduced density-matrix of the subsystem. The spin gap ∆_s is determined similarly by targeting the ground states of the S_z = 0 andS_z= 1/2 sectors of the Hilbert space.

In contrast to the previous section, we do not employ open boundaries to calculate the excitation gaps. It turns out that localized bound states occur at the system boundaries when we use open boundary conditions. These oscillations of the local spin and charge densities are localized at one chain end. This indicates that charge and spin is localized at the system boundary, or in other words, that a charged bound state forms at the chain ends. Since we are not interested in the energy of such surface effects we have to use periodic boundaries. This makes the simulations considerably more difficult as DMRG is most efficient on open chains. In this study it was possible to determine the excitation gaps in periodic chains with lengths up toL= 128. Computationally this is very costly since it requires keeping as much as m = 3072 density-matrix eigenstates. We verified our DMRG results for spin and charge gaps by exact diagonalizations of small systems (L≤14).

Figure 10.3 shows a plot of the gaps as a function of the dimerization. For clarity, results for small systems are omitted. The gaps ∆₁ and ∆₂ are strongly size dependent.

Below the critical value of the dimerization the gap ∆₁ extrapolates to values very

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 δ

0 1 2 3 4

∆1/t, ∆2/t, ∆s/t

spin gap (L = 14, ED) spin gap (Lmax = 26) spin gap (extrapolated)

∆₁ (L = 14, ED)

∆₁ (Lmax = 32)

∆₁ (L_max = 64)

∆₁ (Lmax = 128)

∆₁ (extrapolated)

∆₂ (L = 14, ED)

∆₂ (Lmax = 32)

∆₂ (Lmax = 64)

∆2 (Lmax = 128)

∆₂ (extrapolated) 4.6 - 3.74 δ -1.9 + 1.46 δ

Figure 10.3: Dimerization dependent gaps ∆1, ∆2, and ∆s for different system sizes (L = 14,64,128). Other system sizes have been omitted for clarity. In the regionδ < δc the ground state is degenerate and ∆1= 0. The gap ∆2is reduced linearly (red line) as we approachδc and extrapolates to zero at the critical point. Above the transition the gap to the first excited state,

∆1, opens linearly (dashed red line). The spin gap ∆sis non-zero for allδand is equal to ∆2for very small dimerization, as expected. Note that the one-particle gap (not shown) is finite with values between ∆c/t≈5−6. TheL= 14 data were obtained using exact diagonalizations and agree perfectly with the DMRG data.

close to zero in the thermodynamic limit. This means that the ground-state is twofold degenerate in this dimerization regime. This corresponds to our expectation that the CDW state in the classical picture is invariant under the translation l→ l+ 2. Above the critical dimerization the gap ∆₁ opens linearly and the ground state is no longer degenerate and has no long-range CDW order.

Atδ = 0 the extrapolated gap to the second excited state ∆2is very close to the value of the spin gap ∆_s since both are expected to be equal in a CDW insulator. They stay close also for small dimerizations which indicates that the CDW phase of the extended Hubbard model is not strongly perturbed by a small dimerization. This is true only on a lattice, since it is known that in the low-energy continuum limit [105, 106] any non-zero dimerization is a relevant perturbation. Tuningδ to larger values the spin gap ∆_sis not much affected in contrast to ∆2 which is now linearly reduced with growing dimerization (cf. figure 10.3). Above the critical dimerization ∆₂ increases with the dimerization. It is difficult to reliably extrapolate ∆₂ to L→ ∞ numerically because of the strong size dependence of the gap. However, figure 10.3 suggests that ∆2 is at most slightly larger than ∆₁ in the thermodynamic limit or possibly degenerate.

In the previous section we argued that the onset of the CDW order parametermCDW

is compatible with an Ising-type phase transition. Now, we can go further to show that the excitation of the system that becomes critical atδ_calso suggests this interpretation.

Asδ approaches δc we expect that the gap to the lowest excitation vanishes like [108]

∆^±∼A^±|δ−δ_c|^zν , (10.7)

below (−) and above (+) the critical point. The non-universal constant A^± is a typical energy scale of the system and zν is a universal critical exponent. This gap is the characteristic energy scale of the quantum phase transition. In figure 10.3 we show a linear fit of the extrapolated gap to the lowest excited state above (∆⁺= ∆1) and below (∆⁻= ∆₂) the transition point. We obtain

∆₂(δ)/t = 4.6−3.74δ , (10.8)

∆₁(δ)/t = −1.9 + 1.46δ . (10.9)

Both fits are a good description of our data since we can derive a critical dimerization

δ_c≈1.25, (10.10)

which is consistent with our estimate (10.3) for the open chains. This suggest that the gap to the lowest excitation indeed vanishes as in equation (10.7). We can now infer that

zν ≈1. (10.11)

In order to fixzwe note that the characteristic length scaleξ(δ) of the critical fluctuations diverges at the critical point such that

ξ ∼ |δ−δ_c|^−ν (10.12)

holds [108]. The length scale ξ(δ) can be estimated by considering the critical dimer-ization δ_c(L) as a function of the system length L. By inverting this relation we obtain a critical system size L_c(δ) which is an estimate of of the length scale ξ(δ) of the crit-ical fluctuations. We find that ν = 0.98 ≈1 in equation (10.12). Comparing with the vanishing characteristic energy scale in (10.7) we conclude that the dynamical critical exponent

z= 1. (10.13)

Of course, this is what we expected for a critical quantum system that belongs the universality class of a (1 + 1)-dimensional classical system.

Since we could only access system sizes ofL= 128 lattice sites with periodic boundary conditions the extrapolations toL→ ∞were difficult. We therefore extrapolate a more robust quantity in order to extract the critical value of δ in a periodic system. To this end we calculate the dimerization dependence of ∆₂ for many different system sizes.

The results are shown in figure 10.4. We determine the minimum ∆₂(δ_min) by fitting second order polynomials to the curves ∆2(δ) to obtain δmin(L) as a function of L.

Extrapolating this quantity toL→ ∞should yield the critical value of the dimerization in the thermodynamic limit. This is shown in figure 10.5 where observe that ∆₂(δ_min) = 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 δ

0 1 2 3

∆2/t

L=10L=12 L=14L=16 L=18L=20 L=22L=24 L=26L=28 L=32L=64 L=128

Figure 10.4: Dimerization dependence of the gap ∆2(δ) for periodic systems with ring lengths L= 10, . . . ,128. The position and value of the gap minimum is strongly size-dependent.

within the numerical precision of our extrapolation. An extrapolation yields a critical dimerization

δ^PBC_c = 1.3, (10.14)

of the periodic system. This value agrees well with the result in the open chainsδ^OBC_c = 1.28 obtained in (10.3) with open boundary conditions and is consistent with the result (10.10) for the periodic system.

0 0.02 0.04 0.06 0.08 0.1

1/L 0

0.5 1 1.5 2 2.5 3

∆2(1/L)/t , δmin(1/L)

δ_min(1/L)

∆₂(1/L)

Figure 10.5: Extrapolation of the positionδminand value of the minimum of the gap ∆2to the limit 1/L→ 0. Within numerical precision ∆2(L → ∞) = 0 at δ^PBCc (L → ∞) = 1.3 in good agreement with open boundary conditions.