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arXiv:1903.09179

Stable Luttinger liquids and emergent $U(1)$ symmetry in constrained quantum chains

Ruben Verresen, Ashvin Vishwanath, Frank Pollmann

Published Mar 21, 2019• 12 citations
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Abstract

We explore the effect of local constraints on one-dimensional bosonic and fermionic ground state phases. Motivated by recent experiments on Rydberg chains, we constrain the occupation of neighboring sites in known phases of matter. Starting from Kitaev's topological superconductor wire, we find that a soft constraint induces a stable gapless Luttinger liquid phase. While Luttinger and Fermi liquids are usually unstable to superconducting proximity effects, the constraint suppresses pair creations, allowing for an emergent $U(1)$ symmetry and gaplessness. We substantiate this intuitive picture using field theoretical and Bethe ansatz methods. In particular, in the hard constraint limit, the model is explicitly $U(1)$-symmetric and integrable. For the corresponding spin-$1/2$ chains related by a Jordan-Wigner transformation, the Luttinger liquid is stabilized by the $\mathbb Z_2$ spin flip symmetry. Longer-range constraints stabilize gapless phases even without $\mathbb Z_2$ symmetry, connecting to the seminal work by Fendley, Sengupta and Sachdev [Physical Review B 69, 075106 (2004)], clarifying how the gapless floating phase observed therein can be vastly extended.

DOI
10.48550/arxiv.1903.09179

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Stable Luttinger liquids and emergent U(1)U(1)U(1) symmetry in constrained quantum chains
We explore the effect of local constraints on one-dimensional bosonic and fermionic ground state phases. Motivated by recent experiments on Rydberg chains, we constrain the occupation of neighboring sites in known phases of matter. Starting from Kitaev's topological superconductor wire, we find that a soft constraint induces a stable gapless Luttinger liquid phase. While Luttinger and Fermi liquids are usually unstable to superconducting proximity effects, the constraint suppresses pair creations, allowing for an emergent U(1)U(1)U(1) symmetry and gaplessness. We substantiate this intuitive picture using field theoretical and Bethe ansatz methods. In particular, in the hard constraint limit, the model is explicitly U(1)U(1)U(1)-symmetric and integrable. For the corresponding spin-1/21/21/2 chains related by a Jordan-Wigner transformation, the Luttinger liquid is stabilized by the Z2\mathbb Z_2Z2​ spin flip symmetry. Longer-range constraints stabilize gapless phases even without Z2\mathbb Z_2Z2​ symmetry, connecting to the seminal work by Fendley, Sengupta and Sachdev [Physical Review B 69, 075106 (2004)], clarifying how the gapless floating phase observed therein can be vastly extended.
The properties of stable Luttinger liquid phases in models with a nonconserved number of particles are investigated. We study the Luttinger liquid phases in one-dimensional models of hard-core boson and spinless fermion chains where particles can be created and annihilated three by three on adjacent sites. We provide an intuitive and systematic method based on the flow equation approach, which accounts for additional terms in the correlations generated by the Z3{\mathbb{Z}}_{3}Z3​-symmetric interactions. We find that despite the emergence of U(1) symmetry under renormalization, the observables are still affected by its breaking in the bare Hamiltonian. In particular, the standard bosonization mapping becomes insufficient to capture the full behavior of correlation functions.
We study quantum phases and phase transitions in a one-dimensional interacting fermion system with a Lieb-Schultz-Mattis (LSM) type anomaly. Specifically, the inversion symmetry enforces any symmetry-preserving gapped ground state of the system to be a Kitaev chain, following an LSM type theorem that we prove. Alternatively, via the Jordan-Wigner transformation, this system describes a spin system whose gapped ground states must break either the inversion or the Ising symmetry associated with fermion parity. We obtain a phase diagram using analytical methods and variational matrix product state simulations and study the critical behaviors of the quantum phase transitions therein using entanglement entropy, energy variance, and finite-size scaling of order parameters. In particular, we observe a continuous phase transition between two ordered phases that are beyond the Ginzburg-Landau-Wilson paradigm, in analogy to the deconfined quantum critical points in two spatial dimensions. We show this type of a 1D deconfined quantum critical point is described by the Tomonaga-Luttinger liquid theory and extract its Luttinger parameter and critical exponents. We also identify a gapless phase that emerges from a deconfined quantum critical point between two ordered phases, which cannot be described by a U(1) Luttinger liquid.
We study translationally invariant spin chains where each unit cell contains an nnn-state projective representation of a ${\mathbb{Z}}_{n}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{n}$ internal symmetry, generalizing the spin-1/2 XYZ chain. Such spin chains possess a generalized Lieb-Schultz-Mattis (LSM) constraint, and we demonstrate that certain $(n\ensuremath{-}1)$-component Luttinger liquids possess the correct anomalies to satisfy these LSM constraints. For n=3n=3n=3, using both numerical and analytical approaches, we find that such spin chains with nearest-neighbor interactions appear to be gapless for a wide range of microscopic parameters and described by a two-component conformally invariant Luttinger liquid. This implies the emergence of $n\ensuremath{-}1$ conserved U(1) charges from only discrete microscopic symmetries. Remarkably, the system remains gapless for an unusually large parameter regime despite the apparent existence of symmetry-allowed relevant operators in the field theory. This suggests that either these spin chains have hidden conserved quantities not previously identified, or the parameters of the field theory are simply unusual due to frustration effects of the lattice Hamiltonian. We argue that similar features are expected to occur in (1) ${\mathbb{Z}}_{n}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{n}$ symmetric chains for nnn odd and (2) ${\mathbb{S}}_{n}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{n}$ symmetric chains for all n>2n>2n>2. Finally, we suggest the possibility of a lower bound growing with nnn on the minimum central charge of field theories that possess such LSM anomalies.
Understanding the stability of strongly correlated phases of matter when coupled to environmental degrees of freedom is crucial for identifying the conditions under which these states may be observed. Here, we focus on the paradigmatic one-dimensional Bose-Hubbard model, and study the stability of the Luttinger-liquid and Mott insulating phases in the presence of local particle exchange with site-independent baths of noninteracting bosons. We perform a numerically exact analysis of this model by adapting the recently developed wormhole quantum Monte Carlo method for retarded interactions to a continuous-time formulation with worm updates; we show how the wormhole updates can be easily implemented in this scheme. For an Ohmic bath, our numerical findings confirm the scaling prediction that the Luttinger-liquid phase becomes unstable at infinitesimal bath coupling. We show that the ensuing phase is a long-range-ordered superfluid with spontaneously broken U(1) symmetry. While the Mott insulator remains a distinct phase for small bath coupling, it exhibits diverging compressibility and noninteger local boson occupation in the thermodynamic limit. Upon increasing the bath coupling, this phase undergoes a transition to a long-range-ordered superfluid. Finally, we discuss the effects of super-Ohmic dissipation on the Luttinger-liquid phase. Our results are compatible with a stable dissipative Luttinger-liquid phase that transitions to a long-range-ordered superfluid at a finite system-bath coupling.
Arrays of Rydberg atoms constitute a highly tunable, strongly interacting venue for the pursuit of exotic states of matter. We develop a strategy for accessing a family of fractionalized phases known as quantum spin liquids in two-dimensional Rydberg arrays. We specifically use effective field theory methods to study arrays assembled from Rydberg chains tuned to an Ising phase transition that famously hosts emergent fermions propagating within each chain. This highly entangled starting point allows us to naturally access spin liquids familiar from Kitaev's honeycomb model---albeit from an entirely different framework. In particular, we argue that finite-range repulsive Rydberg interactions, which frustrate nearby symmetry-breaking orders, can enable coherent propagation of emergent fermions between the chains in which they were born. Delocalization of emergent fermions across the full two-dimensional Rydberg array yields a gapless Z2{\mathbb{Z}}_{2}Z2​ spin liquid with a single massless Dirac cone. Here, the Rydberg occupation numbers exhibit universal power-law correlations that provide a straightforward experimental diagnostic of this phase. We further show that explicitly breaking symmetries perturbs the gapless spin liquid into gapped, topologically ordered descendants: Breaking lattice symmetries generates toric-code topological order, whereas introducing Floquet-mediated chirality generates non-Abelian Ising topological order. In the toric-code phase, we analytically construct microscopic incarnations of non-Abelian defects, which can be created and transported by dynamically controlling the atom positions in the array. Our work suggests that appropriately tuned Rydberg arrays provide a cold-atoms counterpart of solid-state ``Kitaev materials'' and, more generally, it spotlights a different angle for pursuing experimental platforms for Abelian and non-Abelian fractionalization.
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