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Non-zero momentum requires long-range entanglement

Lei Gioia, Chong Wang

Abstract

We show that a quantum state in a lattice spin (boson) system must be long-range entangled if it has non-zero lattice momentum, i.e. if it is an eigenstate of the translation symmetry with eigenvalue $e^{iP}\neq1$. Equivalently, any state that can be connected with a non-zero momentum state through a finite-depth local unitary transformation must also be long-range entangled. The statement can also be generalized to fermion systems. Some non-trivial consequences follow immediately from our theorem: (1) several different types of Lieb-Schultz-Mattis-Oshikawa-Hastings (LSMOH) theorems, including a previously unknown version involving only a discrete $\mathbb{Z}_n$ symmetry, can be derived in a simple manner from our result; (2) a gapped topological order (in space dimension $d>1$) must weakly break translation symmetry if one of its ground states on torus has nontrivial momentum - this generalizes the familiar physics of Tao-Thouless; (3) our result provides further evidence of the "smoothness" assumption widely used in the classification of crystalline symmetry-protected topological (cSPT) phases.

Non-zero momentum requires long-range entanglement

Abstract

We show that a quantum state in a lattice spin (boson) system must be long-range entangled if it has non-zero lattice momentum, i.e. if it is an eigenstate of the translation symmetry with eigenvalue . Equivalently, any state that can be connected with a non-zero momentum state through a finite-depth local unitary transformation must also be long-range entangled. The statement can also be generalized to fermion systems. Some non-trivial consequences follow immediately from our theorem: (1) several different types of Lieb-Schultz-Mattis-Oshikawa-Hastings (LSMOH) theorems, including a previously unknown version involving only a discrete symmetry, can be derived in a simple manner from our result; (2) a gapped topological order (in space dimension ) must weakly break translation symmetry if one of its ground states on torus has nontrivial momentum - this generalizes the familiar physics of Tao-Thouless; (3) our result provides further evidence of the "smoothness" assumption widely used in the classification of crystalline symmetry-protected topological (cSPT) phases.
Paper Structure (13 sections, 7 theorems, 21 equations, 5 figures)

This paper contains 13 sections, 7 theorems, 21 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Proof
  3. Proof in $1$d
  4. Higher-dimensional extension
  5. Fermion systems
  6. Consequences
  7. LSMOH constraints
  8. Topological orders: weak CDW
  9. Crystalline symmetry-protected topological phases
  10. Discussions
  11. Higher dimensional FD quantum circuit
  12. Proof for fermion systems
  13. Weak CDW example - Toric code

Key Result

Theorem 1

If a quantum state $|\Psi\rangle$ in a lattice spin (boson) system is an eigenstate of the lattice translation operator $T:|\Psi\rangle\to e^{iP}|\Psi\rangle$ with a non-trivial momentum $e^{iP}\neq1$, then $|\Psi\rangle$ must be long-range entangled, namely $|\Psi\rangle$ cannot be transformed to a

Figures (5)

  • Figure 1: (Color online) Depiction of finite-depth quantum circuits applied on $|\Psi_P\rangle$. Here qudits are depicted as solid circles while unitaries are depicted as rectangles. (a) A SRE state $|\Psi_P\rangle$ is always connected to the $|\mathbf{0}\rangle$ trivial state via a FD quantum circuit $U$. From $U$ a lightcone-like 'adiabatic cut' $\tilde{U}$ can be created (framed in blue). (b) $\tilde{U}$ connects $|\Psi_P\rangle$ to a state that is completely decoupled across the cut.
  • Figure 2: (Color online) Illustration of the adiabatic cutting procedure on a periodic length $L=mn$ chain. Here we take $m=4$ example to demonstrate how four identical cuts, applied by $\tilde{U}$ (blue rectangle) at every $n$th link, on a length $L=4n$ state $|\Psi_{P(L)}\rangle$ (purple circle) produces four decoupled length $n$ SRE states.
  • Figure 3: (Color online) Illustration of splitting $TV_L^\dag T^\dag V_L=\tilde{V}_{L,1}\tilde{V}_{L,2}$ with $\tilde{V}_{L,1}\tilde{V}_{L,2}|\mathbf{0}\rangle=e^{-iP(L)}|\mathbf{0}\rangle$. Here we have taken a snapshot of the circuit to focus on $\tilde{V}_{L,1}$ (framed in blue), however the support of $\tilde{V}_{L,1}$ (in the depicted example 16 qudits) is actually much smaller than the system length. Recall that the circuit is periodic such that the orange arrows, corresponding to components of $\tilde{V}_{L,2}$ (framed in orange), eventually connect on the far side of the ring.
  • Figure 4: (Color online) A sample 2d FD quantum circuit $U$ decomposed along $\hat{x}$ into 'extended lightcone' unitaries $\{V_{i}\}$ (shaded red) and 'extended reverse lightcone' unitaries $\{W_{i}\}$ (shaded blue). The exact position to begin the lightcone cut is variable, although here we have done so symmetrically.
  • Figure 5: (Color online) The Toric code system on a periodic lattice with $L_y=1$. There are two spin degrees of freedom (d.o.f) per unit cell in $\hat{x}$.

Theorems & Definitions (7)

  • Theorem 1
  • Corollary 1.1
  • Theorem 2
  • Corollary 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Corollary 2.4