Yu-Min Hu of the Max Planck Institute for the Physics of Complex Systems, alongside colleagues at Harvard University and Princeton University, have identified a novel quantum state of matter exhibiting a large number of ground states.
Their work studies a one-dimensional spin model possessing an exponential U(1) symmetry where charge decays as 2 to the power of negative j, with site position j. As a result of their findings, they demonstrate that in a lattice of length L, this condensate possesses approximately 2 to the power of L spontaneous symmetry-breaking ground states originating from the approximately 2 to the power of L number of exponential U(1) charge sectors, leading to a finite entropy density of ln 2, and behaving as a disorder-free quantum glass beyond existing classifications of phases of matter. The condensate has a spontaneous symmetry-breaking order parameter being the local in-plane spin, which points in angles related by the chaotic Bernoulli map, and thus is effectively random.
The authors and their work find that the number of charge sectors is 2S(2 to the power of L minus 1) plus 1 for open boundary conditions, and 2 to the power of L minus 1 for periodic boundary conditions.
Exponential U(1) Symmetry-Breaking and the Quantum Glass Phase
A one-dimensional spin model exhibits an unusual property: its charge decays proportionally to 2 to the power of negative j with position j along the lattice, a relationship fundamental to a newly discovered quantum state of matter. Their work reveals that this condensate, possessing an exponential U(1) symmetry, behaves as a disorder-free quantum glass, possessing characteristics unlike any previously observed.
In a lattice of length L, this condensate has approximately 2 to the power of L spontaneous symmetry-breaking ground states originating from the approximately 2 to the power of L number of exponential U(1) charge sectors, leading to a finite entropy density of ln 2, and is observed to undergo a first-order spontaneous symmetry breaking phase transition, a finding verified through numerical analysis and an exactly solvable model on the Rokhsar-Kivelson line. The researchers employed both exact diagonalization and density matrix renormalization group techniques to map the phase diagram of this spin quantum breakdown model.
Crucially, the condensate’s order parameter, the local in-plane spin, points in angles related by the chaotic Bernoulli map, making it effectively random. This randomness is further supported by the observation of nondecaying local autocorrelations, indicating persistent, yet unpredictable, relationships between neighboring spins. Despite this apparent disorder, the condensate lacks off-diagonal long-range order, a property commonly associated with conventional quantum phases. The team demonstrated an edge mode existing only on the left edge when employing open boundary conditions, altering the expected energy spectrum.
This condensate exhibits a bulk gap, violating the Goldstone theorem which typically predicts gapless excitations in systems with broken continuous symmetries. The researchers and their work found that the number of charge sectors is 2S(2 to the power of L minus 1) plus 1 for open boundary conditions, and 2 to the power of L minus 1 for periodic boundary conditions.
Spin Quantum Breakdown Model Hamiltonian and Interactions
Recent investigations into one-dimensional spin systems have revealed a novel phase of matter exhibiting characteristics of both order and disorder, termed a quantum breakdown condensate. This symmetry is not merely a mathematical curiosity; it fundamentally alters how the system organizes itself at the quantum level. The condensate has a spontaneous symmetry-breaking order parameter being the local in-plane spin, which points in angles related by the chaotic Bernoulli map and thus is effectively random.
This unique combination of characteristics, finite entropy density, random order parameter, nondecaying autocorrelations, and the absence of long-range order, leads the researchers to classify the quantum breakdown condensate as a state of matter previously unobserved. The model, defined by the Hamiltonian in Eq. 1, possesses an exponential symmetry, U(φ), transforming the spin on site j in a specific manner.
This symmetry is particularly pronounced when considering the charge Q, defined as the sum of f_j times n_j, where f_j depends on site j and n_j is the particle number. This detailed characterization of the quantum breakdown condensate provides a new framework for understanding complex quantum systems and potentially opens avenues for exploring novel quantum technologies.
O(2^L) Ground States Define Finite Entropy Density
This proliferation of ground states originates from the condensate’s exponential U(1) charge sectors. This exponential increase in possibilities directly leads to the finite entropy density, a measure of the system’s inherent disorder despite its ordered condensate state. This combination, random order parameter and persistent correlations, is unusual and contributes to the “glass-like” behavior. The team’s calculations, particularly those performed near the Rokhsar-Kivelson line, provide strong evidence for the first-order nature of the phase transition into this condensate.
This analytical understanding, built upon a Holstein-Primakoff transformation, reinforces the numerical observations and demonstrates the robustness of the findings.
Rokhsar-Kivelson Line Enables Exact Solvability
The ability to precisely solve complex quantum systems has taken a step forward with the identification of the Rokhsar-Kivelson line within a one-dimensional spin model, enabling researchers to map its behavior with unprecedented accuracy. Despite this apparent randomness, the condensate exhibits nondecaying local autocorrelations, indicating persistent, underlying order. The discovery of this unique condensate, and the analytical tools used to understand it, represent an advance in the field of condensed matter physics.
Chaotic Bernoulli Map Generates Random In-Plane Spins
An unexpected degree of randomness emerges from a highly ordered symmetry in a newly studied quantum system. This isn’t simply disorder; it’s a specific type of order as disorder, leading the team to classify the resulting state as a phase of matter not previously accounted for in existing classifications. Unlike traditional glasses which arise from structural disorder, this glass-like behavior emerges solely from the symmetry and interactions within the model itself.
Exponential U(1) Symmetry Reduction with Boundary Conditions
The model’s defining characteristic is its exponential U(1) symmetry, where the associated “charge” diminishes as 2 to the power of negative j with position j along the lattice. These features distinguish the quantum breakdown condensate from more conventional ordered phases. Instead, the authors state that the spin directions at each site point in angles related by the chaotic Bernoulli map, effectively rendering the order parameter random.
👉 More information
🗞 Exponential U(1) Symmetry-Breaking Phase as a Disorder-Free Quantum Glass
✍️ Yu-Min Hu, Zhaoyu Han and Biao Lian
🧠 DOI: http://link.aps.org/doi/10.1103/ryb5-8ntp
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