2 Tensor Networks and Geometry
Reference: Tensor networks and geometry (Evenbly and Vidal 2011). This works presents a perspective on the entanglement and correlator properties of tensor-network states by examining the geometry of the tensor network ansatz.
Critical v. non-critical systems
Definition 2.1 (two-point correlator) Given local operators \(P_{x_1}, Q_{x_2}\) supported on sites \(x_1, x_2\in \mathcal L\), the two-point correlator is defined as \[ C(x_1, x_2) = \langle P_{x_1}Q_{x_2}\rangle- \langle P_{x_1}\rangle\langle Q_{x_2}\rangle \] Where \(\langle O\rangle\) denotes the expectation w.r.t. the ground state of a given Hamiltonian.
Known results for the decay of correlators, when the Hamiltonian features local interactions:
- Gapped Hamiltonian (i.e. finite separation between ground and excited spectrums): \[ C(x_1, x_2) \approx e^{|x_1 - x_2|/\xi} \] here \(\xi\geq 0\) is the correlation length. Such systems are also known as non-critical systems
- Gapless Hamiltonian i.e. critical systems: \[ C(x_1, x_2) \approx |x_1 - x_2|^{-q} \]
Definition 2.2 (entanglement entropy) Given a block \(A\subset \mathcal L\) of the lattice, the entanglement entropy is \[ S(A) = -\mathrm{tr}(\rho_A \log_2 \rho_A) \]
The following results are also known for the entanglement entropy in \(D\) dimensions:
- For gapped systems, \(S(A) \approx |\partial A| \approx L^{D-1}\).
- Gapless systems exhibit mixed behavior:
- For Fermi surface dimension \(\Gamma < D-1\), the boundary law holds.
- Systems with \(\Gamma = D - 1\) exhibit a logarithmic correction to the boundary law: \[ S(A) \approx L^{D-1}\log L \]
Connection to tensor networks
Assuming a generic homogeneous tensor network i.e. all tensors of the same type are copies of a single tensor, with coefficients chosen randomly, connectivity determines important properties about the correlators and entanglement entropy.
Proposition 2.1 (geodesic distance and correlators) Given two sites \(x_1, x_2\in \mathcal L\) with geodesic distance \(D(x_1, x_2)\), \[ C(x_1, x_2) \approx e^{-\alpha D(x_1, x_2)} \]
Proof: The transfer matrix which determines contraction has positive eigenvalues, maximally \(1\), and the correlator is obtained by evaluating an expression involving the \(D(x_1, x_2)\)-th power of the transfer matrix.
This implies the following correlator bounds for MPS and MERA:
- \(C_{\mathrm{MPS}}(x_1, x_2) \approx e^{-\alpha |x_1 - x_2|}\).
- \(C_{\mathrm{MERA}}(x_1, x_2) \approx e^{-\alpha \log_2 |x_1 - x_2|} \approx |x_1 - x_2|^{-q}\).
Proposition 2.2 (entanglement entropy and boundary size) Let \(\bar A = \mathcal L - A\) denote the complementary system with \(|A|\leq |\bar A|\), given a Schmidt decomposition of the ground state w.r.t. the partition with Schmidt rank \(\chi\): \[ |\psi\rangle= \sum_{j=1}^{\chi} \lambda_j |\psi^A_j\rangle|\psi^{\bar A}_j\rangle, \quad \sum \lambda_j^2 = 1, \quad \chi\leq |A| \] The entanglement entropy is maximized when \(\lambda_{\forall j} = 1/\sqrt \chi\) \[ S(A) = -\sum_{j=1}^{\chi} \lambda_j^2 \log \dfrac 1 {\lambda_j^2} \leq \log_2 \chi \]
Given \(A\subset \mathcal L\), for every partition \(\Omega_A\sqcup \Omega_B\) of the nodes in the tensor network with such that \(\Omega_A\) contains all the free legs of \(A\), we can upper-bound the entanglement entropy according to \[ S(A) \leq |\partial \Omega_A| \log_2\chi \] where \(|\partial \Omega_A|\) is the number of legs across the cut \(\Omega_A\sqcup \Omega_B\), and \(\chi\) is the bond dimension of the homogeneous tensor.
- In 1D-MPS, \(|\Omega_A|=2\), yielding the saturation of entanglement entropy.
- In 1D-MERA, \(|\Omega_A| = \sum_{z=1}^{\log L} 2 \approx \log L\), yielding the logarithmic violation.
- In \(D>1\) MERA, \(|\Omega_A| = \sum_{z=1}^{\log L} L^{D-1}/2^z \approx L^{D-1}\). Only the shortest length-scale (i.e. least coarse-graining) contributes meaningfully to the entropy.
Branched MERA
The concluding section of (Evenbly and Vidal 2011) mentioned the idea of branching, subsequently developed in (Evenbly and Vidal 2014), Scaling of entanglement entropy in the (branching) MERA. The main contributions are:
- Generalization of MERA by introducing a branching parameter \(b\), with \(b=1\) corresponding to the known MERA.
- The \(b=2^{D-1}\)-branched MERA supports logarithmic area law in \(D>1\) dimensions: \(S(A) \approx L^{D-1}\log L\).
- Numerical evidence that the upper-bound for entanglement entropy is saturated up to leading order.
The paper identifies two regimes in the coarse-graining procedure:
- Shrinking regime: the size of the coarse-grained block shrinks as a result of the coarse-graining isometry. At this scale, the coarse-grained system is entangled with the complementary system.
- Stationary regime: the size of the coarse-grained block stays constant (or vanishes). At this scale and above, the coarse-grained system no longer contributes entanglement.
The total entanglement entropy can be computed as
\[
S(A) \propto |\partial \Omega_A|
= \sum_{z=0}^{\bar z} |\partial A_z|
\]
where \(\partial A_z\) is the size of the boundary of the coarse-grain
of \(A\) at scale \(z\). For the traditional MERA on lattice dimension \(D\),
each coarse-graining maps \(L \mapsto L/2\), so
\[
|\partial A_{z+1}| = |\partial A_z| 2^{-(D-1)}
\]
with \(A\) being a hypercube of size \(L\), we have \(\bar z = \log_2 L\)
and \(|\partial A_z|=L^{D-1}/2^z\).
In order to yield the desired \(L^{D-1}\log L\) entropy, we need to multiplicatively correct \(|\partial A_z| \mapsto |\partial A_z| 2^{(D-1)z}\) to obtain logarithmic violation \[ S(A) \propto \sum_{z=0}^{\bar z} \dfrac{|\partial A_z| 2^{(D-1)z}}{2^{(D-1)z}} = L^{D-1} \log_2 L \] This corresponds to introducing \(2^{(D-1)}\) copies of the coarse-grained site at each coarse-graining step.