Finite Correlation Length Scaling with Infinite Projected Entangled-Pair States

We show how to accurately study two-dimensional quantum critical phenomena using infinite projected entangled-pair states (iPEPS). We identify the presence of a finite correlation length in the optimal iPEPS approximation to Lorentz-invariant critical states which we use to perform a finite correlation length scaling analysis to determine critical exponents. This is analogous to the one-dimensional finite entanglement scaling with infinite matrix product states. We provide arguments why this approach is also valid in 2D by identifying a class of states that, despite obeying the area law of entanglement, seems hard to describe with iPEPS. We apply these ideas to interacting spinless fermions on a honeycomb lattice and obtain critical exponents which are in agreement with quantum Monte Carlo results. Furthermore, we introduce a new scheme to locate the critical point without the need of computing higher-order moments of the order parameter. Finally, we also show how to obtain an improved estimate of the order parameter in gapless systems, with the 2D Heisenberg model as an example.

Popular Summary

Exotic phases of matter, such as high-temperature superconductivity, arise from strong interactions among the electrons in a material. Studying transitions from one phase to another, however, is challenging: Small changes in the strength of the interactions, the temperature, or the number of electrons often have dramatic consequences, leading researchers to rely on complex numerical analyses. Here, we demonstrate the ability of so-called tensor network methods—a numerical technique that has proven to be useful in studying phases of matter—to also locate and characterize quantum phase transitions.

Tensor networks provide an approximate description of the states of a many-body system. The accuracy of these models can be increased by enlarging the “bond dimension,” which controls the number of parameters in a tensor. Using two-dimensional tensor network simulations, we show that a finite bond dimension has an effect similar to a finite system size; namely, it cuts off the diverging correlation length at the quantum phase transition. By analyzing how quantities scale as a function of the effective correlation length obtained for different bond dimensions, we can accurately locate phase transitions and characterize their properties.

Our benchmark results demonstrate the applicability and potential of this approach to study phase transitions in two dimensions.

Figure 1

In the scaling regime, close to a Lorentz-invariant critical point, the low-energy physics of a 2D quantum lattice system is described by a continuous 3D field theory. In the same regime, the iPEPS wave function describes a more complicated landscape, where the 3D space is pierced by infinitely long valleys separated by distances of the order of the lattice spacing. The correlation length along those towers provides an infrared cutoff, as if the system had a finite imaginary correlation time, leading to a finite spatial correlation length, which we observe in our numerical simulations. These towers are generated by the finite bond dimension in iPEPS (shown in the lower left-hand corner of the illustration), as explained in detail in the main text.

Figure 2

(a) The ground space projector can be represented as an infinite 3D tensor network, spanning the three directions $x$, $y$, $t$. As usual, tensors are represented here by geometric shapes and lines attached to them are their indices. A line joining two tensors represents the tensor contraction. Here we represent a projection of the 3D tensor network in 2D, where the vertical direction is the Euclidean time $t$, while the horizontal direction is the spatial direction $x$. (b) By using a boundary state we can approximate the full 3D network as the computation of the norm of a 2D iPEPS state. We contract one layer of the tensor network, the transfer matrix $T$, with the boundary iPEPS state $|{\psi }_0⟩$ (first line). The bond dimension of the iPEPS state would increase exponentially number of time steps, and thus we approximate the iPEPS state by truncating its bond dimension back to $D$ (second to fourth line) using appropriate projectors (yellow tensors). We iterate these last two steps many times until the boundary state has actually converged. The converged iPEPS represents our best approximation of the ground state $|\mathrm{\Omega }⟩$ with finite bond dimension $D$.

Figure 3

(a) The fixed point iPEPS tensors are the result of many contraction and projection steps. Here we expand explicitly the iPEPS into its elementary constituents along the imaginary time direction (vertical in the figure), made by the original constituents of the 3D network (cyan tensors) and projectors (yellow tensors) used at each step to project the evolved iPEPS to a state with bond dimension $D$. The result is that the infinite 3D tensor network becomes very anisotropic, and we observe the appearance of quasi-one-dimensional time channels. (b) The equal space, nonequal time correlation function $⟨\mathrm{\Omega }|\mathcal{O}(t_0)\mathcal{O}(t_1)|\mathrm{\Omega }⟩$. The insertion of the local operators $\mathcal{O}$ at two different times $t_0$ and $t_1$ is indicated by purple tensors. The contraction of the network outside the selected time channel produces the two environment tensors represented as rectangles on the left and the right of the time channel. The correlation function thus assumes a one-dimensional structure. The projectors provide a timelike MPS state. Since the bond dimension of the MPS state is finite and equal to $D$, we expect that these correlations either do not decay or decay exponentially with the time separation $t_1-t_0$.

Figure 4

(a) CDW order parameter as a function of interaction strength $V/t$ for different values of $D$. The location of the critical point is indicated by the dashed line. (b) Order parameter $m$ and (c) correlation length $\xi$ as a function of inverse boundary dimension $\chi$ for different values of $D$ at the critical point $V/t=1.356$. (d) A power-law fit at the critical point yields an exponent $\beta /\nu =0.64(2)$, in agreement with the QMC result.

Figure 5

Results for $y=m{\xi }^{\beta /\nu }$ as a function of $D$ for fixed $\beta /\nu =0.64$. At the critical point $V_c/t=1.356$ the value of $y$ becomes independent of the bond dimension $D$, cf. Eq. (6). The standard deviation of the points is shown on the right-hand side, which clearly exhibits a minimum at the critical point.

Figure 6

Data collapse plots using Eqs. (7) and (8).

Figure 7

(a) Data collapse using Eq. (12) yielding $V_c/t=1.356(2)$. (b) Data collapse based on Eq. (13) giving $\beta =0.53(1)$.

Figure 8

iPEPS results for the 2D Heisenberg model. (a) Correlation length $\xi$ and (b) staggered magnetization $m$ as a function of inverse $\chi$ for different values of $D$. (c) Squared of the staggered magnetization as a function of inverse ${\xi }_D$. A linear extrapolation yields $m=0.307\pm 0.002$, which is in agreement with the QMC result $m=0.30743(1)$. (d) Order parameter as a function of inverse $D$.