Quantum Criticality with the Multi-scale Entanglement Renormalization Ansatz

TL;DR

This work introduces the scale-invariant MERA as a tensor-network ansatz tailored for 1D quantum critical systems, enabling direct access to scale-invariant operators and conformal data from lattice Hamiltonians. It contrasts MERA with MPS, offering benchmarks on Ising, Potts, XX, and Heisenberg zig-zag models, and demonstrates accurate extraction of scaling dimensions and OPE coefficients of local and non-local operators. The authors detail symmetry handling, transitional layers, and methods for obtaining local densities and conformal data, illustrating that MERA excels at long-range correlations and identifying the underlying CFT. The results suggest MERA as a powerful tool for characterizing quantum criticality and conformal data, with extensions to boundaries, interfaces, and higher dimensions discussed as future directions.

Abstract

The goal of this manuscript is to provide an introduction to the multi-scale entanglement renormalization ansatz (MERA) and its application to the study of quantum critical systems. Only systems in one spatial dimension are considered. The MERA, in its scale-invariant form, is seen to offer direct numerical access to the scale-invariant operators of a critical theory. As a result, given a critical Hamiltonian on the lattice, the scale-invariant MERA can be used to characterize the underlying conformal field theory. The performance of the MERA is benchmarked for several critical quantum spin chains, namely Ising, Potts, XX and (modified) Heisenberg models, and an insightful comparison with results obtained using a matrix product state is made. The extraction of accurate conformal data, such as scaling dimensions and operator product expansion coefficients of both local and non-local primary fields, is also illustrated.

Figures (20)

• Figure 1: (i) The coarse-graining transformation $U$, a specific implementation of entanglement renormalization, is comprised of isometries $w$ and disentanglers $u$ and maps blocks of three sites from the initial lattice $\mathcal{L}$ into a site of the coarser lattice $\mathcal{L} '$. (ii) The tensors $w$ and $u$ are constrained to be isometric, see also Eq. \ref{['eq:sRe3']}.
• Figure 2: (i) Under coarse-graining with entanglement renormalization, an operator $o(r_1, r_2)$ supported on two sites of lattice $\mathcal{L}$ is transformed into a new operator $o'(r_1 ', r_2 ')$ supported on two sites of the coarser lattice $\mathcal{L '}$, see Eq. \ref{['eq:sRe4']}. The coarse-graining of local operators can be implemented directly via the (iia) left, (iib) center and (iic) right ascending superoperators, denoted $\mathcal{A}_L, \mathcal{A}_C$ and $\mathcal{A}_R$ respectively. Notice that the coarse-graining of $o(r_1, r_2)$ in (i) corresponds to application of the left ascending superoperator $\mathcal{A}_L$.
• Figure 3: The ternary MERA for a lattice $\mathcal{L}^{[0]}$ of $N=54$ sites. Each layer $U^{[\tau]}$ of the MERA can be interpreted as a coarse-graining transformation between an initial lattice $\mathcal{L} ^{[\tau]}$ and a coarser lattice $\mathcal{L} ^{[\tau+1]}$. The past causal cone of two sites $(r_1,r_2)$ in lattice $\mathcal{L}^{[0]}$ is shaded.
• Figure 4: (i) The causal cone (shaded) of four sites $(r_1, r_2, r_3, r_4)$ in lattice $\mathcal{L}^{[\tau-1]}$ involves two sites $(r_1 ', r_2 ')$ in lattice $\mathcal{L}^{[\tau]}$. Starting from the reduced density matrix $\rho^{[\tau]} (r_1 ', r_2 ')$ on lattice $\mathcal{L}^{[\tau]}$, the reduced density matrix on any pair of contiguous from $(r_1, r_2, r_3, r_4)$ can be obtained using the (iia) left, (iib) center and (iic) right descending superoperators, denoted $\mathcal{D}_L, \mathcal{D}_C, \mathcal{D}_R$ respectively.
• Figure 5: Three different MERA schemes for a $D=1$ dimensional lattice. An example of a causal cone is shaded for each scheme. (i) The binary MERA scheme, based upon a 2-to-1 coarse-graining step, has a causal width of three sites and a cost of contraction that scales with the bond dimension $\chi$ as $O(\chi^9)$. (ii) The ternary MERA scheme, based upon a 3-to-1 coarse-graining step, has a causal width of two sites and a cost of contraction that scales as $O(\chi^8)$. (iii) The modified binary MERA scheme, equivalent to the binary MERA scheme with every second disentangler removed, has a causal width of two sites and a cost of contraction that scales as $O(\chi^7)$.