Holographic dynamics from multiscale entanglement renormalization ansatz

The multiscale entanglement renormalization ansatz (MERA) is a tensor network based variational ansatz that is capable of capturing many of the key physical properties of strongly correlated ground states such as criticality and topological order. MERA also shares many deep relationships with the AdS/CFT (gauge-gravity) correspondence by realizing a UV complete holographic duality within the tensor networks framework. Motivated by this, we have repurposed the MERA tensor network as an analysis tool to study the real-time evolution of the 1D transverse Ising model in its low-energy excited state sector. We performed this analysis by allowing the ancilla qubits of the MERA tensor network to acquire quantum fluctuations, which yields a unitary transform between the physical (boundary) and ancilla qubit (bulk) Hilbert spaces. This then defines a reversible quantum circuit, which is used as a “holographic transform” to study excited states and their real-time dynamics from the point of the bulk ancillae. In the gapped paramagnetic phase of the transverse field Ising model, we demonstrate the holographic duality between excited states induced by single spin-flips (Ising “magnons”) acting on the ground state and single ancilla qubit spin flips. The single ancillae qubit excitation is shown to be stable in the bulk under real-time evolution and hence defines a stable holographic quasiparticle, which we have named the “hologron.” Their bulk 2D Hamiltonian, energy spectrum, and dynamics within the MERA network are studied numerically. The “dictionary” between the bulk and boundary is determined and realizes many features of the holographic correspondence in a non-CFT limit of the boundary theory. As an added spin-off, this dictionary together with the extension to multihologron sectors gives us a systematic way to construct quantitatively accurate low-energy effective Hamiltonians.

Figure 1

(a) A MERA quantum circuit defined on a graph $\mathrm{\Gamma }$ with periodic boundary conditions and 16 inputs $\{s_{(0,\tau =0)},...,s_{(15,\tau =0)}\}$ at the bottom. The flow of “time” $\left[\tau \right]$ of the circuit is from bottom to top. At the nodes of the graph are disentanglers (blue rectangles $\square$) and isometries (red triangles $▴$). Located at the isometries are hanging edges which we identify as the bulk qubits ${\left\{t_{(i,\tau )}\right\}}_{(i,\tau )}$. They are the outputs of the MERA circuit computation. The dashed lines and tensors denote connections that go around the periodic graph. On the lower right corner are some labeled tensors and edges (qubits) where we have suppressed the time index for clarity. Note that at the highest level or latest $\left[\tau \right]$, there is no need for a final disentangler gate. (b) The basic unit of a MERA network from which the entire network is constructed, where the edges are decorated with arrows to emphasize the direction of information flow. Our convention for the site ordering between adjacent $\tau$ layers is also described in this basic unit. (c) The decomposition of the “isometry” $W_{\mu }$ into low $W_{l\mu }$ and high $W_{h\mu }$ isometries.

Figure 2

(a) The decomposition of the disentangled two-site reduced density matrix ${\rho }_{(i,\tau )}$ by the $W$ unitary tensor. (b) The $W$ tensor is further decomposed into the low and high isometries $W_{l,h}$. (c) The tensor $D$ is a diagonal matrix with entries of the eigenvalues of ${\rho }_u^A$ ordered by the $|s,t\rangle =|0,0\rangle ,|0,1\rangle ,|1,0\rangle ,|1,1\rangle$ states. The $t=0$ states correspond to the low $W_l$ isometry with the two largest Schmidt eigenvalues ${\lambda }^{(1,2)}$ and the converse for $t=1$.

Figure 3

The time development of a locally excited ground state in the paramagnetic phase with $h^z=3$ in a periodic chain of length $L=16$. The initial ground state uniform magnetization is $\langle {\sigma }_i^z\rangle \approx -0.972$. The system is excited by flipping a spin at position $i=8$ at time $\mathrm{t}=0$. (a) Time development of $\langle {\sigma }_i^z\left(\mathrm{t}\right)\rangle$ at selected sites $i=0,8,12$. (b) Variation of $\langle {\sigma }_i^z\left(\mathrm{t}\right)\rangle$ in position at shifted times. (c) Overall time evolution of $\langle {\sigma }_i^z\left(\mathrm{t}\right)\rangle$ as a 2D surface by interpolating between discrete sites $i$ of the chain.

Figure 4

Combined plot of the physical $\langle {\sigma }_i^z\left(\text{t}\right)\rangle$ spins and bulk $\langle t_{\mu }\left(\text{t}\right)\rangle$ qubits at several times during the time evolution. Interior triangles denote the bulk $t_{\mu }$ qubits where the isometries $W_{\mu }$ reside. The exterior circles are the physical boundary spins ${\sigma }_i$. The numbers are used to label boundary $\left\{i\right\}$ and bulk sites $\left\{\mu \right\}$. For clarity, the disentanglers are not shown and the set of bulk sites are labeled in a cyclic order. (a) The initial ground state $|{\mathrm{\Omega }}_0\rangle$ under the MERA circuit with uniform $\langle {\sigma }_i^z\rangle \approx -0.972$. The bulk qubits $t_{\mu }$ are strongly polarized toward zero with values in the $<{10}^{-3}$ regime. (b) Directly after acting with ${\sigma }_i^x$ at position $i=8$. With the exception of the excited spin, the rest of the physical spins ${\sigma }_i^z$ or rather their expectations values are little affected by the sudden excitation. But in the bulk degrees of freedom, there is a string of excited $\langle t_{\mu }\rangle >0$ qubits (located at $\mu =2,9,13,14,15$) emanating from the excitation at $i=8$. (c) After some time has elapsed $t=0.5$. The bulk excitations with $\langle t_{\mu }\rangle >0$ have now de-localized into the rest of the interior. (d) At the final time $t=1.0$. (e) Several time traces of $\langle t_{\mu }\left(\mathrm{t}\right)\rangle$ taken from selected bulk sites.

Figure 5

Spectral distribution of the total hologron number operator $N_t={\sum }_{\mu }{t}_{\mu }$, for the transformed ground state $\mathbf{M}|{\mathrm{\Omega }}_0\rangle$ (a), and the transformed spin-flipped excited state $\mathbf{M}{\sigma }_8^x|{\mathrm{\Omega }}_0\rangle$ (b). For $\mathbf{M}|{\mathrm{\Omega }}_0\rangle$ the spectral weight is strongly localized in the $N_t=0$ vacuum sector where the trivial product state $|0\rangle$ lies. For the excited state $\mathbf{M}{\sigma }_8^x|{\mathrm{\Omega }}_0\rangle$, the spectral weight is now localized in the $N_t=1$ sector which is an 16 dimensional subspace of ${\mathcal{H}}_t$.

Figure 6

Plots of the 1-hologron bulk states $|J_i^x\rangle$ sourced by the spin-flip operators ${\sigma }_i^x$ acting on ground state $|{\mathrm{\Omega }}_0\rangle$. Due to translational symmetry, only a pair of odd (a) and even (b) boundary sites are shown that are marked by the red disks. Note that the MERA network itself breaks the one-site translational symmetry of the physical chain down to a two-site translation. The histograms are probability density distributions of $|J_i^x\rangle$ within the bulk, with the bulk sites arranged along the binning axis. In either case, $|J_i^x\rangle$ is a stringlike excitation in the bulk emanating from the physical boundary excited site, just as seen in Fig. 4.

Figure 7

Spectral distribution of the total hologron number operator $N_t={\sum }_{\mu }{t}_{\mu }$, for the transformed spin-flipped excited state $\mathbf{M}{\sigma }_8^y|{\mathrm{\Omega }}_0\rangle$ (a) and the spin-flipped excited state $\mathbf{M}{\sigma }_8^z|{\mathrm{\Omega }}_0\rangle$ (b). For $\mathbf{M}{\sigma }_8^y|{\mathrm{\Omega }}_0\rangle$, the spectral weight remains strongly localized in the $N_t=1$, while for the state $\mathbf{M}{\sigma }_8^z|{\mathrm{\Omega }}_0\rangle$, the spectral weight is now distributed over the $N_t=0,2$ sectors.

Figure 8

The boundary states of local bulk hologrons ${\mathbf{M}}^{†}{\chi }_{\mu }^x|0\rangle \in {\mathcal{H}}_s$. The dark triangles mark the location $\mu$ of the hologron in the bulk. In the paramagnetic phase $h^z=3$, the physical observable $\langle {\sigma }_i^z\rangle$ is strongly polarized towards $-1$. So shown in the bar chart are the observables $(\langle {\sigma }_i^z\rangle +1)/2$ as a function of site $i$, and are strongly polarized towards 0 in the ground state. Due to translational symmetry only four chosen hologron states (a)–(d) are shown, with increasing circuit depth $\tau$. However, there is a slight left-right asymmetry from the MERA tensors network and is not a feature of the physical model. As was mentioned in Sec. 4b, the hologrons closer (further) to the boundary are more localized (nonlocal) in physical space. Again, this can be understood purely from the structure of their causal cones. For the two deepest hologrons located at $\mu =14\text{and}15$, their physical influence extends to the entire boundary. Otherwise, when the physical extent of a local bulk hologron excitation is bounded on the boundary, its physical state resembles a stretched magnon waveform or wavelet.

Figure 9

A schematic of the bulk-boundary correspondence, which is just a unitary isomorphism between physical ${\mathcal{H}}_s$ and logical or ancillae ${\mathcal{H}}_t$ Hilbert spaces. The vertical axis denotes real-time $\mathrm{t}$ direction and the unitary maps $Texp\left(-i\int h_i^x{\sigma }_i^x\right)$ and $Texp\left(-i\int j_{\mu }^x{\chi }_{\mu }^x\right)$ act on $|{\mathrm{\Omega }}_0\rangle$ and $|0\rangle$, respectively. The respective partition functions or quantum amplitudes are given by the overlaps ${\mathcal{Z}}_{\text{bnd}}=\langle {\mathrm{\Psi }}_{\text{bnd}}|{\mathrm{\Omega }}_0\rangle$ and ${\mathcal{Z}}_{\text{blk}}=\langle {\mathrm{\Psi }}_{\text{blk}}|0\rangle$. The holographic equivalence is the statement that ${\mathcal{Z}}_{\text{bnd}}={\mathcal{Z}}_{\text{blk}}$ which derives from $|{\mathrm{\Omega }}_0\rangle ={\mathbf{M}}^{†}|0\rangle$ and the relation (21) between the source fields.

Figure 10

The spectrum of $H_{\mathbf{M}}^{\left(1\right)}$ which is $H_{\mathbf{M}}$ truncated to the 1-hologron subspace ${\mathcal{H}}_t^{\left(1\right)}$. The model remains in integrable paramagnetic phase with $h^z=3$, and the overall agreement between the exact and effective is excellent. (a) The comparison between the exact low-energy spectrum of $H$ and $H_M^{\left(1\right)}$, as well the variational ansatz energy $\langle 0\left|\mathbf{M}H{\mathbf{M}}^{†}\right|0\rangle$. (b) and (c) Wave functions of the first $|{\psi }_1\rangle$ and ${16}^{\text{th}} |{\psi }_{16}\rangle$, excited eigenstates of $H_{\mathbf{M}}$ computed in the 1-hologron sector. The histograms display the probability densities at each bulk site. $|{\psi }_1\rangle$ is strongly held at the bulk center while $|{\psi }_{16}\rangle$ is evenly distributed near boundary.

Figure 11

The matrix elements of $H_{\mathbf{M}}^{\left(1\right)}$ interpreted as a hopping Hamiltonian in the bulk. Recall also that disentanglers are present but not drawn here. All matrix elements are all real-valued because the $H$ and the MERA tensors are real-valued. For clarity, only the traceless part of $H_{\mathbf{M}}^{\left(1\right)}$ is plotted. The solid (green) circle marks the bulk site from which the hologron hops from. The triangles are colored according to the value of the relevant matrix element, and the triangle within the (green) circle corresponds to the on-site energy matrix element. The bar charts display the same data in quantitative values. For example, in (d), the large negative value at site 14 indicates a deep potential well for the hologron. The shaded region bounded by the dashed (red) lines denotes the bulk sites accessible by hopping from the circle. The range of hopping is limited by both the causal cone structure (kinematics) and the specific values of the Hamiltonian (dynamics). For example, the “forbidden pockets” around sites 12, 13, 14, and 15 in (c) and (d) are dynamical in origin.

Figure 12

The square of the dynamical correlator $\langle {\mathrm{\Omega }}_0\left|{\sigma }_i^x\left(\mathrm{t}\right){\sigma }_0^x\left(0\right)\right|{\mathrm{\Omega }}_0\rangle$ computed exactly using the time-evolved many body wave function $\left|\mathrm{\Psi }\right(\mathrm{t})\rangle$ (red lines), and using the effective (blue line) 1-hologron Hamiltonian $H_{\mathbf{M}}^{\left(1\right)}$ and the source coefficient $\left\{J_{i\mu }^x\right\}$ of Fig. 6. (a)–(c) Several time-dependent correlators from different physical sites of varying distance from $i=0$. (d) The MERA network with sites labeled for quick reference.

Figure 13

Comparisons between the exact physical energies and the effective holographic bulk Hamiltonian energies for a nonintegrable deformation of the transverse Ising model. The model is generalized by adding a perturbation $V{\sum }_i{\sigma }_i^z{\sigma }_{i+1}^z$ to $H$. The specific parameters are $h^z=3$ and $V=0.1$ (a) The case where the bulk Hamiltonian contains at most a single hologron. (b) The case where up to three hologrons ($N_t\le 3$) are included in the constructing the effective bulk Hamiltonian. The effective Hilbert space remains much smaller than $\text{dim}\left({\mathcal{H}}_s\right)$, yet correctly captures the low-energy spectra. (c) Comparison between the first 137 low-energy exact eigenstates and the spectra computed with a hologron bulk Hamiltonian with up to four hologrons ($N_t\le 4$). The agreement or accuracy of the bulk effective hologron Hamiltonian greatly improves with increasing numbers of hologrons included.

Figure 14

A basic four-site block (solid green) acted by a left/right disentanglers $u_{L,R}$ and an isometry $W$. The hashed green regions denote the rest of the sites that act as the environment. Of the 4 sites, the central two sites denote the $A$ region, and the left/right sites, the $B$ region. A four-site block reduced density matrix ${\rho }_0^{AB}$ is obtained by tracing over the environment (hashed) degrees of freedom. The left/right disentanglers act to minimize the short-ranged entanglement between the central $A$ sites and its boundary $B$ sites. This entanglement is quantified by the von-Neumann entropy $S_{\text{ent}}\left[{\rho }_u^A\right]=-\text{Tr}({\rho }_u^{A}log{\rho }_u^A)$ of the disentangled $A$ reduced density matrix ${\rho }_u^A={\text{Tr}}_B\left(u{\rho }_0^{AB}u\right)$ with $u=u_L\otimes u_R$. The $W$ isometry tensor then acts to separate the high and low entanglement Schmidt states producing new bulk $t_{\mu }$ and $s_{\mu }$ output qubits. Here $u_L\equiv U_{(i,\tau )}, u_R\equiv U_{(i-1,\tau )}$ and $W\equiv W_{(i,\tau +1)}$.

Figure 15

(a) and (b) The minimal entanglement entropy $S_{\text{ent}}$ of a 2RDM ${\rho }_u^A$ after optimal disentanglers have been applied at the first layer $\tau =1$. Data are shown in an (a) absolute and (b) $log$ scale. The two-site block resides in the $\tau =1$ layer of the MERA and peaks near the critical point $h^z=1$, which is shifted due to finite size effects. Linear extrapolation of the maximum entanglement entropy from for log-log plot (b) yields $S_{\text{ent}}\approx 0.195$. This should be contrasted with the maximum possible entanglement entropy of $log0.25\approx 1.39$ for two qubit density matrices.

Figure 16

Spectral distribution of the total hologron number operator $N_t={\sum }_{\mu }{t}_{\mu }$, for the transformed ground state $\mathbf{M}|{\mathrm{\Omega }}_0\rangle$ at the CFT point $h^z=1$ but at finite system size of $L=16$ with PBC. (a) The distribution using tensors of $\mathbf{M}$ that are numerically optimized to maximize fidelity. (b) The distribution using the analytic expressions for the tensors from Evenbly and White [37].

Figure 17

Spectral distribution of the total hologron number operator $N_t={\sum }_{\mu }{t}_{\mu }$, for transformed eigenstates at the nonintegrable point $(h^z,V)=(3,0.1)$. (a) The ground state's $\mathbf{M}|{\mathrm{\Omega }}_0\rangle$ spectral distribution remains highly localized in the vacuum sector. (b) The third excited state spectral distribution is spread between $N_t=1$ and $N_t=2$ subspaces indicating a significant amount of hologron interactions which are not $N_t$ conserving.