Algorithms for Entanglement Renormalization: Boundaries, Impurities and Interfaces

Abstract

We propose algorithms, based on the multi-scale entanglement renormalization ansatz, to obtain the ground state of quantum critical systems in the presence of boundaries, impurities, or interfaces. By exploiting the theory of minimal updates (Evenbly and Vidal, arXiv:1307.0831, 2013), the ground state is completely characterized in terms of a number of variational parameters that is independent of the system size, even though the presence of a boundary, an impurity, or an interface explicitly breaks the translation invariance of the host system. Similarly, computational costs do not scale with the system size, allowing the thermodynamic limit to be studied directly and thus avoiding finite size effects e.g. when extracting the universal properties of the critical system.

Appendices

Appendix 1: Introduction to MERA

This appendix contains a brief introduction to entanglement renormalization and the MERA, focusing mostly on a system that is both translation invariant and scale invariant.

1.1 Appendix 1.1: Coarse Graining Transformation

We start by reviewing the basic properties of entanglement renormalization and the MERA in a finite, one-dimensional lattice $\mathcal {L}$ made of $N$ sites, where each site is described by a Hilbert space $\mathbb {V}$ of finite dimension $\chi$.

Let us consider a coarse-graining transformation $U$ that maps blocks of three sites in $\mathcal {L}$ to single sites in a coarser lattice $\mathcal {L}'$, made of $N'=N/3$ sites, where each site in $\mathcal {L}'$ is described by a vector space $\mathbb {V}'$ of dimension $\chi '$, with $\chi '\le \chi ^3$, see Fig. 25a. Specifically, we consider a transformation $U$ that decomposes into the product of local transformations, known as disentanglers $u$ and isometries $w$. Disentangles $u$ are unitary transformations that act across the boundaries between blocks in $\mathcal {L}$,

$$\begin{aligned} u^{\dagger }:\mathbb {V}^{ \otimes 2} \mapsto \mathbb {V}^{ \otimes 2},\;\;\; u^\dag u = \mathbb {I}^{ \otimes 2}, \end{aligned}$$

(66)

where $\mathbb {I}$ is identity on $\mathbb {V}$, while isometries $w$ implement an isometric mapping of a block of three sites in $\mathcal {L}$ to a single site in $\mathcal {L}'$,

$$\begin{aligned} w^{\dagger }:\mathbb V^{ \otimes 3} \mapsto \mathbb V',\;\;\; w^\dag w = \mathbb I', \end{aligned}$$

(67)

where $\mathbb I'$ is the identity operator on $\mathbb {V}'$. The isometric constraints on disentanglers $u$ and isometries $w$ are expressed pictorially in Fig. 25b.

Fig. 25

a The coarse-graining transformation $U$, based on entanglement renormalization, maps a lattice $\mathcal {L}$ made of $N$ sites into a coarse-grained lattice $\mathcal {L}'$ made of $N'=N/3$ sites. b The isometries $w$ and disentanglers $u$ that constitute the coarse-graining transformation $U$ are constrained to be isometric, see also Eqs. 66 and 67. c An operator $o_{\mathcal {R}}$, supported on a local region $\mathcal {R}\in \mathcal {L}$ made of two contiguous sites, is coarse-grained to a new local operator $o'_{\mathcal {R}'}$, supported on a local region $\mathcal {R}' \in \mathcal {L}'$ made also of two contiguous sites. d A nearest neighbor Hamiltonian $H=\sum \nolimits _r h(r,r+1)$ is coarse-grained to a nearest neighbor Hamiltonian $H'=\sum \nolimits _r h'(r,r+1)$. e The left, center and right ascending superoperators ${\mathcal {A}}_L$, ${\mathcal {A}}_C$ and ${\mathcal {A}}_R$ can be used to compute the new coupling $h'$ from the initial coupling $h$, see also Eq. 70.

An important property of the coarse-graining transformation $U$ is that, by construction, it preserves locality. Let $o_{\mathcal {R}}$ be a local operator defined on a region $\mathcal {R}$ of two contiguous sites of lattice $\mathcal {L}$. This operator transforms under coarse-graining as,

$$\begin{aligned} o_{\mathcal {R}} \mathop {\longrightarrow }\limits ^{U} o'_{\mathcal {R}'}, \end{aligned}$$

(68)

where the new operator $o'_{\mathcal {R}'}$ is supported on a region $\mathcal {R}'$ of two contiguous sites in lattice $\mathcal {L}'$, see Fig. 25c. The coarse-grained operator $o'_{\mathcal {R}'}$ remains local due to the specific way in which transformation $U$ decomposes into local isometric tensors $u$ and $w$. Indeed, in $U^{\dagger } o_{\mathcal {R}} U$, most tensors in $U$ annihilate to identity with their conjugates in $U^{\dagger }$. The causal cone ${\mathcal {C}}(\mathcal {R})$ of a region $\mathcal {R}$ is defined as to include precisely those tensors that do not annihilate to identity when coarse-graining an operator supported on $\mathcal {R}$, and it thus tracks how region $\mathcal {R}$ itself evolves under coarse-graining.

In particular, a local Hamiltonian $H$ on $\mathcal {L}$ will be coarse-grained into a local Hamiltonian $H'$ on $\mathcal {L}'$,

$$\begin{aligned} H=\sum \limits _r h(r,r+1) \mathop {\longrightarrow }\limits ^{U} H'=\sum \limits _r h'(r,r+1), \end{aligned}$$

(69)

see Fig. 25d. The local coupling $h'$ of the coarse-grained Hamiltonian $H'$ can be computed by applying the (left, center, right) ascending superoperators $\mathcal A_L$, $\mathcal A_C$ and $\mathcal A_R$ to the coupling $h$ of the initial Hamiltonian,

$$\begin{aligned} h' = \mathcal A_L \left( h\right) + \mathcal A_C \left( h\right) + \mathcal A_R \left( h\right) , \end{aligned}$$

(70)

see Fig. 25e.

The coarse-graining transformation $U$ can be repeated $T\approx \log _3(N)$ times to obtain a sequence of local Hamiltonians,

$$\begin{aligned} H_0 \mathop {\longmapsto }\limits ^{U_1} H_{1} \mathop {\longmapsto }\limits ^{U_2} \cdots \mathop {\longmapsto }\limits ^{U_{T}} H_{T}, \end{aligned}$$

(71)

where each of the local Hamiltonian $H_s$ is defined on a coarse-grained lattice $\mathcal {L}_s$ of $N_s=N/(3^s)$ sites. Notice the use of subscripts to denote the level of coarse-graining, with the initial lattice $\mathcal L_0 \equiv \mathcal L$ and Hamiltonian $H_0 \equiv H$. The final coarse-grained Hamiltonian $H_T$ in this sequence, which is defined on a lattice $\mathcal {L}_T$ of $N_T\approx 1$ sites, can be exactly diagonalized so as to determine its ground state $| \psi _T \rangle$. As a linear (isometric) map, each transformation $U_s$ can also be used to fine-grain a quantum state $|\psi _s\rangle$ defined on $\mathcal {L}_{s}$ into a new quantum state $|\psi _{s-1}\rangle$ defined on $\mathcal {L}_{s-1}$,

$$\begin{aligned} \left| {\psi _{s-1} } \right\rangle = U_s \left| {\psi _s } \right\rangle . \end{aligned}$$

(72)

Thus a quantum state $|\psi _0\rangle$ defined on the initial lattice $\mathcal L_0$ can be obtained by fine graining state $| \psi _T \rangle$ with the transformations $U_s$ as,

$$\begin{aligned} \left| {\psi _0 } \right\rangle = U_1 U_2 \cdots U_T \left| {\psi _T } \right\rangle . \end{aligned}$$

(73)

If each of the transformations $U_s$ has been chosen as to properly preserve the low energy subspace of the Hamiltonian $H_{s-1}$, such that $H_s$ is a low-energy effective Hamiltonian for $H_{s-1}$, then $\left| {\psi _0 } \right\rangle$ is a representation of the ground state of the initial Hamiltonian $H_0$. More generally, the MERA is the class of states that can be represented as Eq. 73 for some choice of $\{ U_1, U_2, \ldots , U_T \}$ and $| \psi _T \rangle$.

For a generic choice of local Hilbert space dimensions $\chi _0, \chi _1, \cdots , \chi _{T-1}$ (where $\chi _0 \equiv \chi$), only a subset of all states of lattice $\mathcal {L}$ can be represented in Eq. 73, whereas the choice $\chi _s = \chi ^{3^s}$ allows for a (computationally inefficient) representation of any state of the lattice.

1.2 Appendix 1.2: Scale Invariant MERA

We now move to discussing the MERA for a quantum critical system that is both scale invariant and translation invariant. We describe how universal information of the quantum critical point can be evaluated, by characterizing the scaling operators and their scaling dimensions. We also review the power-law scaling of two-point correlators. In this appendix, fixed-point objects (e.g. $U$, $H$, $\{u,w\}$, etc) are denoted with a star superscript (as $U^*$, $H^*$, $\{u^*,w^*\}$, etc), whereas in the main text of this manuscript we did not use a star superscript to ease the notation.

Let $\mathcal {L}_0$ be an infinite lattice and let $H_0$ denote a translation invariant, quantum critical Hamiltonian. We assume that this Hamiltonian tends to a fixed point of the RG flow of Eq. 71, such that all coarse-grained Hamiltonians $H_s$ are proportionate to a fixed-point Hamiltonian $H^*$ for some sufficiently large $s$. Specifically, the coarse-grained Hamiltonians in the scale invariant regime are related as $H_s=H_{s-1}/\Lambda$, where $\Lambda =3^z$ with $z$ is the dynamic critical exponent of the Hamiltonian (i.e. $z=1$ for a Lorentz invariant quantum critical point). Equivalently, the local couplings that define that Hamiltonians are related as $h_s=h_{s-1}/\Lambda$. For concreteness, let us assume that the initial Hamiltonian $H_0$ reaches the scale invariant (Lorentz invariant) fixed point after $s=2$ coarse-grainings, such that its RG flow can be written,

$$\begin{aligned} H_0 \mathop {\longmapsto }\limits ^{U_1} H_{1} \mathop {\longmapsto }\limits ^{U_2} H^* \mathop {\longmapsto }\limits ^{U^*} \frac{1}{3} H^* \mathop {\longmapsto }\limits ^{U^*} \frac{1}{9} H^* \mathop {\longmapsto }\limits ^{U^*}\cdots , \end{aligned}$$

(74)

where $U^*$ represents the scale invariant coarse-graining transformation for $H^*$. In this case, the ground state $| \psi _0 \rangle$ of the Hamiltonian $H_0$ can be represented by the infinite sequence of coarse-graining transformations,

$$\begin{aligned} \left| {\psi _0 } \right\rangle = U_1 U_2 U^* U^* U^* \cdots \end{aligned}$$

(75)

see Fig. 26. The class of states that can be represented as Eq. 75 are called scale invariant MERA. The scale-dependent transformations before scale invariance, here $U_1$ and $U_2$, correspond to transitional layers of the MERA. These are important to diminish the effect of any RG irrelevant terms potentially present in the initial Hamiltonian, which break scale invariance at short distances. In general, the number $M$ of transitional layers required will depend on the specific critical Hamiltonian under consideration. [Strictly speaking, scale invariance is generically only attained after infinitely many transitional layers, but in practice a finite number $M$ of them often offers already a very good approximation]. We call the fixed-point coarse-graining transformation $U^*$ scale invariant. Notice that the scale invariant MERA, which describes a quantum state on an infinite lattice, is defined in terms of a small number of unique tensors. Each transitional map $U_s$ is described by a pair of tensors $\{u_s, w_s \}$ and the scale invariant map $U^{*}$ is described by the pair $\{u^*, w^* \}$.

Fig. 26

a A scale invariant MERA consists of some number $M$ of transitional layers with coarse-graining maps $\{U_1, U_2,\ldots , U_M \}$, here $M=2$, followed by an infinite sequence of scaling layers, with a scale invariant map $U^*$. b Each $U_s$ of the scale invariant MERA is a coarse-graining transformation composed of local tensors $\{u_s, w_s \}$

We now discuss how scaling operators and their scaling dimensions can be evaluated from the scale-invariant MERA. This is covered in more detail in e.g. Refs. [8, 9] and [15]. For simplicity, let us consider a scale invariant MERA with no transitional layers, that is composed of an infinite sequence of a scale invariant map $U^*$, described by a single pair $\{u^*,w^*\}$. As shown in Fig. 27a, a one-site operator $o$, placed on certain lattice sites, is coarse-grained under the action of layer $U^*$ into new one-site operator $o'$. This coarse-graining is implemented with the one-site scaling superoperator $\mathcal S$,

$$\begin{aligned} o' = \mathcal {S}\left( o \right) , \end{aligned}$$

(76)

where $\mathcal S$ is defined in terms of the isometry $w^*$ and its conjugate, see also Fig. 27b. The (one-site) scaling operators $\phi _{i}$ are defined as those operators that transform covariantly under action of $\mathcal S$,

$$\begin{aligned} \mathcal {S}(\phi _{i}) = \lambda _{i} \phi _{i}, \quad \Delta _{i} \equiv -\log _3 \lambda _{i}, \end{aligned}$$

(77)

where $\Delta _{i}$ is the scaling dimension of scaling operator $\phi _{i}$. As is customary in RG analysis, the scaling operators $\phi _{i}$ and their scaling dimensions $\Delta _{i}$ can be obtained through diagonalization of the scaling superoperator ${\mathcal {S}}$.

One can obtain explicit expressions for two-point correlation functions of the scale invariant MERA based upon their scaling operators, as we now describe. Let us suppose that two scaling operators $\phi _{i}$ and $\phi _{j}$ are placed on special sites $r$ and $r+l$ that are at a distance of $l = 3^q$ sites apart for positive integer $q$, as shown in Fig. 27c. The correlator $\left\langle \phi _i \left( r \right) \phi _j \left( r+l\right) \right\rangle$ can be evaluated by coarse-graining the scaling operators until they occupy adjacent sites, where the expectation value

$$\begin{aligned} C_{ij} \equiv \left\langle \phi _{i}(r) \phi _{j}(r +1)\right\rangle = \mathrm Tr \big ( (\phi _{i}\otimes \phi _{j}) {\rho } \big ). \end{aligned}$$

(78)

can then be evaluated with the local two-site density matrix $\rho$ (which is the same at every level of the MERA due to scale invariance).

For each level of coarse-graining applied to the scaling operators $\phi _{i}$ and $\phi _{j}$, we pick up a factor of the eigenvalues of the scaling operators, as described Eq. 77, and the distance $l$ between the scaling operators shrinks by a factor of 3, see Fig. 27c, which leads to the relation

$$\begin{aligned} \left\langle \phi _i \left( r \right) \phi _j \left( r+l\right) \right\rangle = \lambda _i \lambda _j ~ \left\langle \phi _i \left( r \right) \phi _j \left( r+l/3\right) \right\rangle . \end{aligned}$$

(79)

Notice that the scaling operators are coarse-grained onto adjacent sites after $T = \log _3 |l|$ levels, thus through iteration of Eq. 79 we have

$$\begin{aligned} \left\langle {\phi _i (r )\phi _j (r+l )} \right\rangle&= \left( {\lambda _i \lambda _j } \right) ^{\log _3 |l|} \left\langle {\phi _i (r)\phi _j (r+1)} \right\rangle \nonumber \\&= \left( {3^{ - \Delta _i } 3^{ - \Delta _j } } \right) ^{\log _3 |l|} C_{i j } \nonumber \\&= \frac{{C_{i j } }}{{\left| {l } \right| ^{\Delta _i + \Delta _j } }} . \end{aligned}$$

(80)

where constant $C_{\alpha \beta }$ is the expectation value of the correlators evaluated on adjacent sites,

$$\begin{aligned} C_{ij} \equiv \left\langle \phi _{i}(r) \phi _{j}(r +1)\right\rangle = \hbox {tr}\big ( (\phi _{i}\otimes \phi _{j}) {\rho } \big ). \end{aligned}$$

(81)

Thus it is seen that the correlator of two scaling operators $\phi _{i}$ and $\phi _{j}$ scales polynomially in the distance between the operators, with an exponent that is the sum of their corresponding scaling dimensions $\Delta _i$ and $\Delta _j$, in agreement with predictions from CFT [34, 35].

Notice that Eq. 80 was derived from structural considerations of the MERA alone and, as such, holds regardless of how the tensors in the scale invariant MERA have been optimized. This argument is only valid for the chosen special locations $r$ and $r+l$. For a generic pair of locations, the polynomial decay of correlations may only be obtained after proper optimization (for instance, via energy minimization) of the MERA so as to approximate the ground state of a translation invariant, quantum critical Hamiltonian $H$.

Fig. 27

a Scale invariant MERA composed of an infinite sequence of scale invariant maps $U^*$, which are defined in terms of a single pair of tensors $\{ u^*, w^* \}$. A one-site operator $o$ is coarse-grained into new one-site operators $o'$ and $o''$. b The scaling superoperator $\mathcal S$ acts covariantly upon scaling operators $\phi _i$, see also Eq. 77. c Two scaling operators $\phi _i$ and $\phi _j$ that are separated by $l$ lattice sites are coarse-grained onto neighboring sites after $\log _3 (l)$ maps $U^{*}$

Appendix 2: Reflection Symmetry

In this appendix we describe how symmetry under spatial reflection can be exactly enforced into the MERA. This is done by directly incorporating reflection symmetry in each of the tensors of the MERA (note that an equivalent approach, dubbed inversion symmetric MERA, was recently proposed in Ref. [22]). Such a step was found to be key in applications of the modular MERA to quantum critical systems with a defect, as considered in Sect. 4. Indeed, we found that in order for the modular MERA to be an accurate representation of the ground state of a quantum critical system with a defect, the homogeneous system (that is, the system in the absence of the defect) had to be addressed with a reflection invariant MERA.

Let us describe how the individual tensors of the MERA, namely the isometries $w$ and disentanglers $u$, can be chosen to be reflection symmetric, i.e.

$$\begin{aligned} w = \mathrm{{Rft}}\left( w \right) ,\; \; u = \mathrm{{Rft}}\left( u \right) , \end{aligned}$$

(82)

see Fig. 28. Here $\mathrm{{Rft}}\left( \cdot \right)$ is a superoperator that denotes spatial reflection, which squares to the identity. The spatial reflection on a tensor involves permutation of its indices, as well as a ‘reflection’ within each index, as enacted by a unitary matrix $R$ such that $R^2 = I$. The latter is needed because each index of the tensor effectively represents several sites of the original system, which also need to be reflected (permuted). Matrix $R$ has eigenvalues $p=\pm 1$ corresponding to reflection symmetric and reflection antisymmetric states, respectively. It is convenient, though not always necessary, to work within a basis such that each $\chi$-dimensional index $i$ decomposes as $i=(p,\alpha _p)$, where $p$ labels the parity ($p=1$ for even parity and $p=-1$ for odd parity) and $\alpha _p$ labels the distinct values of $i$ with parity $p$. In such a basis, $R$ is diagonal, with the diagonal entries corresponding to the eigenvalues $p=\pm 1$.

Fig. 28

a The definition of reflection symmetry for a ternary isometry $w$, which involves spatial permutation of indices as well as enacting a unitary matrix $R$ on each index. b The definition of reflection symmetry for a disentangler $u$

Let us turn our attention to the question of how reflection symmetry, as described in Eq. 82, can be imposed on the MERA tensors. For concreteness, we consider an isometry $w$ (analogous considerations apply to a disentangler). Notice that we cannot just symmetrize $w$ under reflections directly,

$$\begin{aligned} w' = \frac{1}{2} \left( w+ \mathrm Rft (w) \right) , \end{aligned}$$

(83)

because the new, reflection symmetric tensor $w'$ will no longer be isometric. Instead, we can include an additional step in the optimization algorithm that symmetrizes the environment of the tensors before each tensor is updated. In the optimization of the MERA [10], in order to update an isometry $w$ one first computes its linearized environment $\Upsilon _w$. Now, to obtain an updated isometry that is reflection symmetric, we first symmetrize its environment,

$$\begin{aligned} \Upsilon _w \mapsto \Upsilon '_w = \Upsilon _w + \mathrm{{Rft}}\left( {\Upsilon _w } \right) . \end{aligned}$$

(84)

In this way we ensure that the updated isometry $w'$ (which is obtained through a SVD of $\Upsilon '_w$, see Ref. [10]), is reflection symmetric, yet also retains its isometric character. Likewise the environments $\Upsilon _u$ of disentanglers $u$ should also be symmetrized.

Appendix 3: Decomposition of Isometries

In the formulation of modular MERA described in Sect. 2 it was convenient to decompose some of the isometries $w$ of the MERA used to describe the homogeneous system into pairs of upper and lower isometries $w_{U}$ and $w_{L}$, as depicted in Fig. 29a. In this section we discuss how this can be accomplished.

Fig. 29

a A isometry $w$ from the ternary MERA, which coarse-grains three $\chi$-dimensional lattice sites into a single $\chi$ dimensional lattice site, is decomposed into upper and lower binary isometries, $w_{U}$ and $w_{L}$. The index connecting the upper and lower binary isometries is chosen at an independent dimension $\chi '$. b The upper and lower binary isometries $w_{U}$ and $w_{L}$ should be chosen to maximize their overlap with the ternary isometry $w$ against the one-site density matrix $\rho$, see Eq. 85

Let $\chi$ denote the bond dimension of the indices of the isometry $w$, and let $\chi '$ denote the index connecting the upper and lower isometries $w_{U}$ and $w_{L}$. Since $\chi '$ effectively represents two sites with bond dimension $\chi$, we have that the isometric character of $w_U$ requires $\chi '\le \chi ^2$. We should perform this decomposition such that it does not change the quantum state described by the MERA (perhaps to within some very small error). Therefore the best choice of upper $w_{U}$ and lower $w_{L}$ isometries follows from maximizing their overlap with the isometry $w$ against the one-site density matrix $\rho$. That is, we choose them such that they maximize

$$\begin{aligned} \mathrm {Tr} \left( \rho w_U w_L w ^\dag \right) , \end{aligned}$$

(85)

see Fig. 29b. Given the density matrix $\rho$ and isometry $w$, one can obtain $w_U$ and $w_L$ by iteratively maximizing the above trace over each of the two tensors, one at a time. Ideally, we would like the decomposition of $w$ into the product of $w_U$ and $w_L$ to be exact, that is, such that such that $\mathrm {tr} \left( \rho w_U w_L w ^\dag \right) =1$. This is typically only possible for $\chi ' = \chi ^2$. However, in practice we find that for choice of bond dimension $\chi '$ between one or two times the dimension $\chi$, i.e. $\chi < \chi '< 2\chi$, the above trace is already $1-\epsilon$ with $\epsilon$ negligibly small. The use of a $\chi '$ smaller than $\chi ^2$ results in a reduction of computational costs.