Density matrix renormalization group on a cylinder in mixed real and momentum space

We develop a variant of the density matrix renormalization group (DMRG) algorithm for two-dimensional cylinders that uses a real space representation in the direction along the axis of the cylinder and a momentum space representation in the direction around the circumference. The mixed representation allows us to use the momentum around the cylinder as a conserved quantity in the DMRG algorithm. Compared with the traditional purely real-space approach, we find a significant speedup in computation time and a considerable reduction in memory usage. Applying the method to the interacting fermionic Hofstadter model, we demonstrate a reduction in computation time by over 20-fold, in addition to a sixfold memory reduction.

Figure 1

Numbering scheme in order to transform the 2D lattice into a 1D chain. The horizontal direction is the direction along the cylinder, the vertical one around the cylinder. As an example, we show a cylinder of width $L_y=12$ (a) real space, (b) mixed real and momentum space approach.

Figure 2

Single-particle spectrum of the Hofstadter model for a flux density of $\varphi /{\varphi }_0=1/3$. The magnetic unit cell is chosen to be $l_x\times l_y=3\times 1$.

Figure 3

Benchmark plots for the Hofstadter model on an infinite cylinder with $L_y=10,\varphi =2\pi /3$, and $V/t=0.1$ at one third filling. This corresponds to a fully occupied lowest band and weak interactions. Here, $\chi$ is the DMRG bond dimension; increasing $\chi$ leads to better accuracy, at the expense of greater computational cost. The subplots are (a) energy, (b) entanglement entropy, (c) computing time, and (d) memory usage, as a function of $\chi$ for the range $\chi =100\text{–}12800$.

Figure 4

Average charge value of the left Schmidt states as a function of flux going through the cylinder. After three flux quanta $\left(6\pi \right)$ have been adiabatically threaded through the cylinder, one charge has been pumped across the artificial cut of the system indicating a Hall conductivity of ${\sigma }_{xy}=1/3\times e^2/h$.

Figure 5

Momentum resolved entanglement spectrum for $L_y=12,\varphi =2\pi /3,V/t=7.0$, and $\chi =6400$ at one ninth filling (one third filling of lowest band). The different colors indicate the charge values of the corresponding entanglement eigenstates. The system is in a fractional Chern insulator phase and the chiral structure of the entanglement matching the edge theory state counting predicted from conformal field theory is clearly visible. The numbers indicate the counting for the zero charge sector depicted in green.

Figure 6

Part of the FSM at sites $i$ and $i+1$ generating the operator $\widehat{O}$ and MPO matrix for the matrix $M^{\left[n\right]}$ (8). The letters $R,A,B$, and $F$ label the states of the FSM as well as the rows/columns of the MPO matrix. Two paths of the FSM producing the term ${\widehat{A}}_i{\widehat{B}}_{i+1}+{\widehat{B}}_i{\widehat{A}}_{i+1}$ are highlighted in red. The gray rectangles indicate the six transitions in the FSM that exactly correspond to the six nonzero entries of $M$.

Figure 7

Finite-state machine creating the hoppings in the real space Hofstadter Hamiltonian for $L=4$ and $\varphi =\pi$. For creating the Hermitian conjugate and interaction part of the Hamiltonian, we need two more copies of the depicted graph. In the Landau gauge, the Hamiltonian is invariant under translation by eight sites (two rings), which means that there are eight different MPO matrices. The remaining four matrices $M^{\left[5\right]-\left[8\right]}$ can be obtained from $M^{\left[1\right]-\left[4\right]}$ by simply replacing $-{\sigma }^{+}$ by ${\sigma }^{+}$ in the paths describing the hopping around the cylinder.

Figure 8

Different cases of momentum transfer within one ring occurring in the MPO due to the interaction term $H_{\mathrm{int}}$. Four cases not shown in the figure are related to $A\text{–}D$ via Hermitian conjugation.

Figure 9

Coarse-grained version of the FSM generating all intra-ring interaction terms of type $A$ to $D$ from Fig. 8. The transitions between adjacent columns inserts a creation/annihilation operator at various momenta. The self-pointing “loops” places strings of $\mathbb{1}/{\sigma }^z$ operators in between the four creation/annihilation operators. In the actual FSM, there are multiple copies of the nodes $A_{\mu }$ and $C_{\mu }$ labeled by momenta in order to ensure momentum conservation (see Fig. 10).

Figure 10

(a) Application of the FSM from Fig. 9 for $L=8$ and the specific term proportional to $c_0^{†}{c}_2{c}_4{c}_6^{†}$, which following the Jordan-Wigner transformation (11) becomes ${\sigma }_0^{-}{\sigma }_1^z{\sigma }_2^{+}{\mathbb{1}}_3{\sigma }_4^{+}{\sigma }_5^z{\sigma }_6^{-}{\mathbb{1}}_7$. The red and blue portions of the path correspond, respectively, to case $A$ and case $B$ from Fig. 8. Following the path from $R$ to $F$ generates the desired coefficient of $\frac{V}{L}2(cos2\alpha -cos4\alpha )$ with $\alpha =2\pi /L$. The intermediate nodes are labeled $A_{\mu }\left(k\right)$, where $k$ denotes the cumulative momentum, modulo $L$. [Thus $6\equiv -2(mod8)$.](b) Modified path after rotating the basis which replaces the complex exponentials by real-valued sine and cosine terms.

Figure 11

(a) Scaling of the average MPO bond dimension $\langle D\rangle$ with the circumference $L$ of the cylinder. Despite the additional complexity of the $k$-space MPO, its bond dimension is smaller than that of the real space MPO. (b) Average ratio of the number of nonzero vs total number of entries in the MPO matrices for the real space and the exponential and sine/cosine formulation in mixed space.