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$W$ state is not the unique ground state of any local Hamiltonian

Lei Gioia, Ryan Thorngren

Abstract

The characterization of ground states among all quantum states is an important problem in quantum many-body physics. For example, the celebrated entanglement area law for gapped Hamiltonians has allowed for efficient simulation of 1d and some 2d quantum systems using matrix product states. Among ground states, some types, such as cat states (like the GHZ state) or topologically ordered states, can only appear alongside their degenerate partners, as is understood from the theory of spontaneous symmetry breaking. In this work, we introduce a new class of simple states, including the $W$ state, that can only occur as a ground state alongside an exactly degenerate partner, even in gapless or disordered models. We show that these states are never an element of a stable gapped ground state manifold, which may provide a new method to discard a wide range of 'unstable' entanglement area law states in the numerical search of gapped phases. On the other hand when these degenerate states are the ground states of gapless systems they possess an excitation spectrum with $O(1/L^2)$ finite-size splitting. One familiar situation where this special kind of gaplessness occurs is at a Lifshitz transition due to a zero mode; a potential quantum state signature of such a critical point. We explore pathological parent Hamiltonians, and discuss generalizations to higher dimensions, other related states, and implications for understanding thermodynamic limits of many-body quantum systems.

$W$ state is not the unique ground state of any local Hamiltonian

Abstract

The characterization of ground states among all quantum states is an important problem in quantum many-body physics. For example, the celebrated entanglement area law for gapped Hamiltonians has allowed for efficient simulation of 1d and some 2d quantum systems using matrix product states. Among ground states, some types, such as cat states (like the GHZ state) or topologically ordered states, can only appear alongside their degenerate partners, as is understood from the theory of spontaneous symmetry breaking. In this work, we introduce a new class of simple states, including the state, that can only occur as a ground state alongside an exactly degenerate partner, even in gapless or disordered models. We show that these states are never an element of a stable gapped ground state manifold, which may provide a new method to discard a wide range of 'unstable' entanglement area law states in the numerical search of gapped phases. On the other hand when these degenerate states are the ground states of gapless systems they possess an excitation spectrum with finite-size splitting. One familiar situation where this special kind of gaplessness occurs is at a Lifshitz transition due to a zero mode; a potential quantum state signature of such a critical point. We explore pathological parent Hamiltonians, and discuss generalizations to higher dimensions, other related states, and implications for understanding thermodynamic limits of many-body quantum systems.
Paper Structure (12 sections, 3 theorems, 36 equations, 1 figure)

This paper contains 12 sections, 3 theorems, 36 equations, 1 figure.

Table of Contents

  1. Proof of the No-Go Theorem
  2. Consequences
  3. Gapless Hamiltonians & Lifshitz transitions
  4. Gapped Hamiltonians
  5. Pathological Parent Hamiltonians
  6. Explicit length dependence of parameters
  7. Locality-breaking
  8. Discussion
  9. Proof of Proposition 1
  10. Sketch of a generalization
  11. Proof of Corollary \ref{['cor:momentumguys']}
  12. Fun facts

Key Result

Theorem 1

The $W$ state (and more generally $|W_n\rangle$) is not the unique ground state of any finite-range 1d Hamiltonian system. In particular, if it is a ground state, $|0\rangle$ is also an exactly degenerate ground state.

Figures (1)

  • Figure 1: A simple Lifshitz transition is depicted. In (a) we have a fully empty band since the Fermi energy $\epsilon_F$ is below the band - this corresponds to the insulating phase. As we increase the chemical potential $\mu$ we arrive at a critical Lifshitz transition as depicted in (b). Here the ground state manifold contains both the $|0\rangle$ and $|W\rangle$ states, a hallmark of a Lifshitz transition. In (c) we increase the chemical potential further such that the Fermi energy lies within $O(1/L^2)$ above the $|W\rangle$ state we have the $|W\rangle$ state as the ground state. However to maintain this state as the sole ground state one has to continuously tune the chemical potential with increasing system size, otherwise one creates a finite density Fermi surface corresponding to a metal.

Theorems & Definitions (10)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • proof
  • proof
  • Conjecture 1
  • proof
  • proof
  • proof
  • proof