Gapped and gapless phases of frustration-free spin-1/2 chains (1503.04035v3)

Abstract: We consider a family of translation-invariant quantum spin chains with nearest-neighbor interactions and derive necessary and sufficient conditions for these systems to be gapped in the thermodynamic limit. More precisely, let $\psi$ be an arbitrary two-qubit state. We consider a chain of $n$ qubits with open boundary conditions and Hamiltonian $H_n(\psi)$ which is defined as the sum of rank-1 projectors onto $\psi$ applied to consecutive pairs of qubits. We show that the spectral gap of $H_n(\psi)$ is upper bounded by $1/(n-1)$ if the eigenvalues of a certain two-by-two matrix simply related to $\psi$ have equal non-zero absolute value. Otherwise, the spectral gap is lower bounded by a positive constant independent of $n$ (depending only on $\psi$). A key ingredient in the proof is a new operator inequality for the ground space projector which expresses a monotonicity under the partial trace. This monotonicity property appears to be very general and might be interesting in its own right. As an extension of our main result, we obtain a complete classification of gapped and gapless phases of frustration-free translation-invariant spin-1/2 chains with nearest-neighbor interactions.

• The paper establishes precise conditions under which frustration-free spin-1/2 chains exhibit gapped or gapless phases based on eigenvalue analysis.
• The research introduces a novel operator inequality that demonstrates monotonicity under partial trace to classify phase behavior.
• The study reveals finite volume criticality, showing that gapless regions may exist over parameter ranges with positive measure.

A Comprehensive Analysis of Gapped and Gapless Phases in Frustration-Free Spin-$\frac{1}{2}$ Chains

This paper addresses a fundamental problem in quantum many-body physics: determining when quantum spin chains exhibit a spectral gap in the thermodynamic limit. Authored by Sergey Bravyi and David Gosset, the paper presents a detailed examination of translation-invariant quantum spin-$\frac{1}{2}$ chains with nearest-neighbor interactions, focusing on conditions under which these chains are gapped or gapless.

Key Contributions

The authors focus on Hamiltonians composed of rank-$1$ projectors onto arbitrary two-qubit states, designated as the "forbidden state," effectively penalizing adjacent qubits for being in that state. The spectral gap—a critical indicator of quantum phase transitions—is analyzed under varying conditions related to these projectors.

1. Main Theorems:
   • Gapless Phases: The spectral gap vanishes when the eigenvalues of a $2x2$ matrix, derived simply from the two-qubit state $\psi$, possess equal non-zero absolute values. Under these circumstances, the gap is upper bounded by $1/(n-1)$, indicating a critical scaling with system size.
   • Gapped Phases: Conversely, when the eigenvalues have differing magnitudes or both are zero, the chain remains gapped, with a positive spectral gap that is independent of the system size.
2. Monotonicity Under Partial Trace: A novel operator inequality is introduced, establishing that the projector onto the ground state of $n+1$ qubits is greater than or equal to the partial trace over a single qubit of the $n$-qubit ground state projector. This monotonicity under partial trace plays a significant role in demonstrating the classifications of phases.
3. Finite Volume Criticality: Unlike typical cases where quantum systems are gapless along lines separating distinct gapped phases, the models explored here can be gapless over regions in parameter space with positive measure.
4. Classification of Special Cases:
   • The special case where $T_\psi$ has degenerate eigenvalues results in distinct classes of gapped and gapless behavior. The criticality, akin to quantum phase transitions, is tied to more complex internal parameter dependencies.

Methodology and Results

The derivation of conditions for gapped and gapless phases involves mathematically rigorous treatment and verifying with a variety of cases. The authors employ analytical techniques and indirect proofs where explicit formulae of ground state projectors are not readily available. The result is a complete classification of phases of 1D spin-$\frac{1}{2}$ chains, establishing a cornerstone for understanding critical behavior and entanglement in quantum spin systems.

Broader Implications

This research underscores the intricate relation between the structure of Hamiltonians and resulting spectral properties. The ability to map out phase diagrams with precision aids in the design and prediction of quantum materials and quantum information systems, particularly where control over entanglement and decoherence is essential.

Moreover, the findings present an avenue for investigating higher-dimensional and more complex quantum systems, where direct application of Knabe's or Nachtergaele's methodologies may not be feasible.

Future Directions

Potential future research could explore extending these classifications to higher-dimensional qudit systems and identifying scalable algorithms to efficiently determine ground states in complicated Hamiltonian landscapes. The established results provide a platform for further exploration of gapless phases and their potential calculations related to emergent phenomena such as topological order.

In summary, Bravyi and Gosset have contributed substantially to our understanding of phase transitions in one-dimensional quantum systems. Their methodological innovations and theoretical insights lay the groundwork for future advancements in quantum simulation and computing technology.