Papers

We study how thermal equilibrium states of infinite fermionic lattices respond to perturbations of the Hamiltonian. Our main result concerns extensive perturbations of a chosen KMS state. We show that exponential decay of correlations and exponential local indistinguishability under the extensive perturbation yield stretched-exponential decay of correlations for the perturbed state. Neither the initial nor the perturbed state is required to be unique, allowing stability to be studied in the presence of phase coexistence. Our approach uses Quantum Belief Propagation (QBP), which we establish directly on the underlying $C^*$-algebra. Combining QBP with Lieb–Robinson bounds, we show implications between decay of correlations, the principle that local perturbations perturb locally, and local indistinguishability, closing the cycle for perturbations of the tracial state under the stated decay, support and uniformity assumptions. For extensive perturbations under uniform clustering conditions, we also obtain a differentiable path of local equilibrium expectations and bound their change linearly in the interaction norm of the perturbation.

We derive a universal upper bound on the speed of multipartite entanglement generation. For arbitrary differentiable pure multipartite states undergoing parameter-dependent evolution, we prove that the rate of change of generalized concurrence is bounded by the square root of the quantum Fisher information (QFI). The proof follows directly from the symmetric logarithmic derivative formalism, revealing the bound as a consequence of quantum estimation theory, while a complementary Schmidt-spectrum analysis identifies the conditions for saturation and yields a tighter bound for rank-two bipartitions. We illustrate these results in interacting many-body spin models and investigate their behavior across the quantum phase transition of the transverse-field Ising model. At the critical point, we find that the QFI can be strongly enhanced while the generalized concurrence is nearly stationary and its sensitivity to the coupling is suppressed. This demonstrates that the enhanced parameter sensitivity associated with quantum criticality need not arise from rapid changes in the magnitude of multipartite entanglement.

Entanglement and nonlocal non-stabilizerness are distinct quantum resources encoded in the Schmidt spectrum of a bipartite pure state. For qubit systems, we establish a universal quadratic constraint on their simultaneous allocation, governed by the minimum number of logical qubits required on each side to encode the occupied Schmidt support. Additional spectral constraints reveal a strong asymmetry in their allocation with the Entanglement that can exhaust the logical capacity, whereas the maximal nonlocal non-stabilizerness grows only logarithmically with it. For a tensor-network cut crossed by a single separating virtual edge, the logical capacity is determined by the occupied bond dimension. We further derive a one-sided certificate for the resources of the untruncated state from a retained Schmidt spectrum and its discarded weight. For small discarded weight, the resulting corrections are linear in that weight. These results connect a universal resource budget to resource-aware control of tensor-network truncations.

Characterising physical devices typically requires comparison with more precise reference standards. For clocks, however, this approach is problematic because no physically perfect time reference exists; real clocks have finite, model-dependent uncertainty. We address whether multiple imperfect clocks can collectively determine their precision without an absolute reference. This paper provides analytic bounds and protocols for quantifying clock precision using only multiple copies of the same clock, which need not be synchronised and may have manufacturing variations or environmental coupling. Beyond clock characterisation, these methods enable precise measurements f sub-period intervals and relativistic effects such as time dilation and Doppler shifts, providing tools for quantum metrology and fundamental physics.

Determining a clock's precision usually requires comparison with a better time reference, yet no ideal reference is physically attainable. We show how a single clock can characterise its own precision using only a signal splitter and delayed trigger, whose uncertainty is explicitly accounted for. The resulting estimator depends only on tick-count statistics. A purpose-built electronic circuit validates the method without an external timekeeper or a second clock.

A confluence of recent works has shown that many quantum circuits and dynamics are efficiently simulable by classical algorithms that track local information, even when conventional complexity measures such as the entanglement and magic are high. Here, we introduce a novel measure of complexity, the reactivity, to capture this new method of classical attack. Unlike conventional complexity measures, the reactivity does not capture a property of a quantum state or operator in isolation, but rather a quantum experiment as a whole. We provide numerical and rigorous evidence that quantum experiments with low reactivity are simple by a host of measures: they are efficient to classically simulate, learn, and fast-forward. This motivates the search for quantum experiments with high reactivity, which may evade these simplistic features. To this end, we introduce easily implementable experimental protocols—dubbed Pauli path spectroscopy—that allow one to efficiently measure the reactivity of any quantum experiment of interest. Our protocols are applicable even when the experiment itself is beyond the reach of classical simulation.

Microwave power-to-frequency transduction is demonstrated using the emergent oscillation frequency of a magnetically initiated Rydberg dissipative time crystal (DTC). In a room-temperature 87Rb vapor, a ~14 G magnetic field establishes a self-sustained DTC near 21 kHz, while a near-resonant 8.325 GHz field coupling the 63D3/2 to 64P3/2 manifold continuously shifts the autonomous oscillation frequency. The resulting field-to-frequency transfer function is strongly nonlinear, with a total fundamental-frequency excursion of ~6.7 kHz and peak responsivity approaching 1 kHz/(mV/cm). The nonlinear transition is substantially steeper than a simple quadratic saturation response and is captured by a self-consistent mean-field picture in which microwave-driven Rydberg-population redistribution modifies the collective resonance condition. Higher DTC harmonics preserve the same normalized transfer function while exhibiting approximately linear growth of absolute responsivity with harmonic order. These results establish an intrinsic many-body atomic route to RF amplitude-to-frequency conversion, enabling continuous frequency-domain sensing through an emergent atomic frequency without an externally engineered RF feedback oscillator, repeated optical spectral scans, or resolved Autler Townes splitting.

We derive explicit, absolutely convergent series formulas for the quantum, unassisted classical, and entanglement-assisted classical capacities of the qubit amplitude damping channel, eliminating residual optimizations and implicit roots. We also obtain analytical upper and lower bounds on two-way assisted quantum and private communication capacities by evaluating a balanced-squashing bound and optimizing the reverse coherent information. Our results characterize the optimal input populations, establish convergence and truncation properties, and provide analytical benchmarks for quantum and classical communication over dissipative channels.

Coherent uncomputation can be as costly as the computation itself, creating substantial spacetime overheads that can be prohibitive on early fault-tolerant quantum computers. We develop an algebraic framework for measurement-based uncomputation (MBU) through a quantum error-correction picture: measurements of the garbage register induce relative phases analogous to errors causing non-trivial syndromes. Accepting only zero-syndrome outcomes achieves post-selected MBU, removing the need for an uncomputation oracle at the cost of a probabilistic success rate determined by the affine rank of the populated garbage support. This rank can be much smaller than the number of garbage qubits, as illustrated through an example of the HHL algorithm. We then derive a structural condition under which the original uncomputation oracle can be reused to correct the phase for all measurement outcomes. No correction is required for the zero syndrome, while corrections for non-zero syndromes can be further adapted to individual measurement outcomes for simpler circuits, as illustrated by existing modular-arithmetic, QROM, and QRAM MBU constructions. Phase-correction circuits can be further simplified by co-optimising their implementation with the representation of the measurement-induced phases. Selecting which syndrome classes to correct interpolates between post-selected, partially corrected, and deterministic MBU, enabling trade-offs among correction cost, workspace, and repetitions.

The quantum Cramér-Rao bound constrains the precision of parameter estimation through the symmetric logarithmic derivative quantum Fisher information (SLD QFI). It is asymptotically achievable for regular single-parameter estimation models, but generally not in the multiparameter setting, where optimal measurements for different parameters may be incompatible. For multiparameter estimation, allowing collective measurements, the corresponding asymptotically achievable precision limit is the Holevo bound, whose standard formulation is an optimization over auxiliary Hermitian operators. We bridge this apparent difference in formulation by giving an exact characterization of the Holevo bound in terms of a family of QFIs interpolating between the SLD and right logarithmic derivative (RLD) QFIs. Specifically, for any locally identifiable finite-dimensional estimation problem, we prove that the operator-feasible region in the standard variational definition of the Holevo bound coincides with the intersection of the regions defined by the corresponding Cramér-Rao-type matrix constraints for the entire QFI family. This characterization also applies to rank-deficient states and does not require any prior choice of weight matrix. Consequently, the conventional weight-dependent Holevo bound is recovered by minimizing the corresponding weighted cost over the resulting common feasible region.

Scalable photonic circuits are essential for realizing practical quantum networks beyond proof-of-principle demonstrations. Although diamond-based integrated photonics has been extensively investigated over the past two decades for quantum networking applications, its integration into scalable photonic platforms remains challenging. Quantum memories based on rare-earth-ion atomic frequency combs (AFCs) offer an attractive alternative; however, most demonstrations to date have relied on bulk free-space crystals, limiting their scalability for practical implementations. In this work, we systematically investigate the potential for hybrid integration of AFC-based quantum memories with a scalable integrated photonic platform. We propose a device architecture combining thin-film lithium tantalate (TFLT) nanophotonic circuits with ytterbium-doped yttrium orthosilicate (Yb:YSO) crystals for long-distance quantum communication. Our analysis shows that the hybrid TFLT-YSO platform offers a promising and scalable pathway toward integrated quantum memories and, ultimately, large-scale quantum networks.

Learning and sensing are among the most promising applications of quantum technology, often with provable quantum advantages. In any realistic experiment, however, noise threatens to erase these advantages as the problem size grows. We introduce oracle distillation, a procedure that distills many queries to a noisy oracle into a single high-fidelity oracle. Here the oracle, an unknown unitary from a known family, models the interface through which a quantum device learns about an unknown system. We construct an explicit protocol that distills any Boolean oracle, both under adversarial noise that corrupts a constant fraction of the input qubits and under i.i.d. depolarizing noise at a constant rate on every qubit. At the heart of the protocol is the weak query, in which the device deliberately queries the oracle only weakly, trading response strength for the ability to correct errors. Our protocol is near-optimal among all protocols that distill the oracle while keeping its errors correctable, revealing a fundamental tension between these two requirements. An important implication of the protocol is a threshold theorem: every Boolean oracle problem whose quantum advantage is polynomial in the domain size, including Grover search, $k$-forrelation, and Simon's problem, retains a quantum advantage whenever the per-qubit depolarizing rate is below a constant threshold. In particular, the quadratic quantum advantage of Grover search, long believed fragile under noise, survives nearly intact at low error rates even when almost every query involves errors on one or more qubits. We further extend the protocol to fractional and continuous-time Boolean oracles. Quantum error correction can thus protect not only prescribed operations but also unknown dynamics. Our results make a large family of quantum advantages in learning robust against noise and open a path toward robust computational sensing.

We investigate the two-dimensional Dunkl–Pauli equation for a spin–$1/2$ particle in the presence of an external magnetic field within an algebraic framework. While previous studies have provided exact solutions of the Dunkl–Pauli system, its underlying dynamical symmetry and algebraic structure remain largely unexplored. By employing Schrödinger factorization, we construct the generators of the $\mathfrak{su}(1,1)$ algebra and establish the associated symmetry of the radial sector. The energy spectrum is derived using irreducible unitary representations, and the corresponding Sturmian radial basis is obtained analytically. Furthermore, $\mathrm{SU}(1,1)$ coherent states are constructed and their time evolution is analyzed, revealing a characteristic radial breathing behavior. The results show that the Dunkl deformation introduces parity-dependent modifications in the spatial structure of the system while preserving its underlying algebraic dynamics.

Sipser-Spielman codes and Dijkgraaf-Witten twisted gauge theories are among the most profound ideas in the last few decades in the areas of computer science and mathematical physics respectively. This work unifies them in the same framework using the language of sheaf cohomology, and produces a new framework of twisted sheaf gauge theories describing a class of quantum low-density parity-check (qLDPC) codes with non-Abelian $D_4$ topological order. The corresponding twisted Clifford-stabilizer qLDPC code can be obtained from gauging a new type of 0-form sub-complex symmetry from a qLDPC code defined on a general sheaf complex, and can be interpreted as a topological defect network of non-Abelian $D_4$ patches glued together with proper gapped interfaces. As an application, one can use this to realize a magic state fountain via the gauging measurement of the addressable logical CZ gates as 0-form subcomplex symmetries in a 2D hypergraph-product (HGP) sheaf code. This includes a scheme of subdividing an arbitrary constant-rate 2D HGP code with parameters $[[n,\Theta(n), \Omega(n^{1/2})]]$ into a quantum sheaf code which allows preparation of $\Theta(n^{1/2})$ logical CZ magic states in parallel, equivalent to the recent geometric construction using the code-to-manifold mapping in (arXiv:2601.06736). Moreover, using the recent sheaf complex and algebraic code constructions by Golowich-Tamo-Zhu (arXiv:2609.27801) with parameters $[[n,\Theta(n^{1-\epsilon}), \Omega(n^{(1-\epsilon)/2})]]$ for arbitrary small $\epsilon$, one can prepare $\Theta(n^{1-\epsilon})$ CZ magic states in parallel and hence achieve an almost-constant magic rate.

We describe the Quantum-HPC hybrid system comprising the supercomputer Fugaku, the QPUs Quantinuum Reimei and ibm_kobe, and the Tierkreis hybrid workflow software. We present two demonstrations of the system at work. In the first, we compute the excited states of a biomolecular system using Fugaku and Reimei (simulator) to handle classical and quantum parts of the workflow respectively. In the second, Fugaku coordinates Reimei and ibm_kobe working together in a combined VQE + QPE workflow to compute the ground state energy of a simple system.

For many important optimization problems we are restricted to approximate solutions in practice due to computational complexity. Distinct from the exact optimization setting, approximate optimization admits performance measures beyond whether the optimum is found, with different tradeoffs and complexity. For MaxCut, a near-unity (ordinary) approximation ratio can coexist with near-zero improvement (gain) over a random cut. For the standard encoding, the unconditional classical MaxCut-Gain hardness gap implies that any uniformly efficient quantum or hybrid procedure recovering a fixed positive fraction of the optimal classical gain on every input, with at least inverse-polynomial success probability, would place NP in BQP. Such a procedure is therefore believed impossible under standard assumptions. We broadly address where our worst-case barriers do or do not apply across the quantum algorithm landscape. We prove that the barrier survives quantum random access optimization (QRAO) compression and applies between the classical and relaxed optimal values. For every input, a product state attains the classical optimum. Thus the barrier to reaching the classical threshold does not arise from a need for entanglement. For $d\in\{2,3\}$ variables per qubit, the known decoder transfers encoded energy gain to decoded mean gain by the exact factor $1/d^2$. Combining this identity with MaxCut-Gain hardness gives an operational preparation barrier for QRAO. We also construct hard $n$-qubit families with relative quantum relaxation excess $\Theta(1/n)$, while the maximally mixed state has energy approximation ratio $1-\Theta(1/n)$, zero encoded energy gain, and hence zero decoded mean gain. Our results separate the effects of relaxation tightness and energy approximation from operational accessibility, motivating more comprehensive accounting in benchmarking and performance assessment.

We demonstrate a simple and efficient method to parallelize and distribute Trotterized Hamiltonian simulation of fermionic systems across multiple QPUs. Using combinatorial covering designs to define a minimal set of fermion-qubit encodings, we demonstrate communication cost scaling as $\mathcal{O}(M q^4 r)$ for a system of $M$ fermionic modes and Trotter number $r$, improving on the static encoding bound for $q$ QPUs, $\mathcal{O}(M^4 q r)$. We compare this approach to dynamic encoding using a randomised method, Pauli-weight based optimisation and hypergraph partitioning. Applying these to the Hamiltonians of a range of molecular systems split across two QPUs, we find the combinatorial covering approach results in the lowest communication cost in all but the sparsest Hamiltonians.

We study the geometric spread of discrete-time quantum walks using Connes' distance formula. We introduce its discrete analogue by adapting the continuous spectral metric to the lattice, with the Dirac operator replaced by the unitary step operator. Focusing first on a (2+1)-dimensional quantum walk, we evaluate the dynamical spread of localized wavepackets to compare propagation in free space against that under a uniform magnetic field. We find that while the free walk shows linear ballistic growth, gauge-induced phases in the magnetic walk constrain the admissible test operators, suppressing the long-time geometric expansion. Furthermore, we apply this discrete framework to a (1+1)-dimensional plastic quantum walk simulating an inhomogeneous spatial metric. The results demonstrate that the discrete Connes distance naturally captures the underlying spatially varying metric, providing a robust measure of the wavepacket's spread that adapts directly to the local simulated geometry.

In 2025, Jordan et al. introduced Decoded Quantum Interferometry (DQI), a quantum algorithm for combinatorial optimization based on decoding. They considered Reed-Solomon decoding and its associated optimization problem, called Optimal Polynomial Intersection (OPI), which may be viewed as a polynomial regression problem over a finite field. DQI exhibits provable speedups on certain problem instances and establishes a connection between decoding problems and optimization tasks. Leveraging the well-understood dual structure and decoding theory of algebraic geometry codes from other curve families, we introduce COFI: Curve-based Optimal Function Intersection. By considering two-point Hermitian codes, Suzuki codes, and extended norm-trace codes, we broaden the range of algebraic geometry codes used in DQI and identify families that offer further improvements over one-point Hermitian codes in the Hermitian Optimal Polynomial Intersection framework considered by Jordan and Gu. Depending on the family and parameter regime, these curves can reduce quantum resource requirements or increase the number of constraints that can be considered.

Non-Hermitian skin effect, the accumulation of bulk eigenstates at edges under open boundary conditions, has been observed across classical, quantum, and synthetic platforms. Whether it can be generalized to the genuine time dimension remains open. We propose a spatiotemporal NHSE in the Floquet phase space via Floquet engineering of a ring-trapped ultracold atomic gas. Placing spatial and temporal degrees of freedom on an equal footing, we show that a single parent skin effect projects onto either domain. The spatial projection yields an NHSE with arbitrarily tunable skin locations, eliminating the need for physical open boundaries. The temporal projection gives time-domain nonreciprocity and periodic temporal funneling, consistent with causality. This establishes a unified spatiotemporal paradigm for NHSE and opens a route to non-Hermitian topological physics in time.

We develop a new three-party quantum secret-sharing (QSS) protocol based on a three-qubit W state. This protocol encodes the secret-sequence bits using X gates and employs randomly selected Hadamard operations and measurement bases to generate information and security-test rounds. We evaluate the efficiency of the proposed protocol and analyze its security against an internal adversary performing intercept-and-resend and entangle-and-measure attacks, deriving the corresponding detection probabilities. The analysis of the entangle-and-measure attack shows that extracting information through an ancillary system leads to a nonzero probability of forbidden outcomes in the test rounds, thereby enabling the detection of the attack. The proposed protocol is implemented in Python using the Qsikit framework and executed on quantum processors. Its performance is evaluated in terms of the quantum bit error rate (QBER) for information rounds under different secret-bit encoding configurations. The experimental results demonstrate that the QBER depends noticeably on the quantum processor used.

Teleportation-based Gottesman-Kitaev-Preskill (GKP) error correction typically assumes a decoherence-free GKP ancilla Bell pair, an idealization that is difficult to reconcile with the ubiquitous environmental decoherence in bosonic systems. Here, we analyze the performance of the GKP protocol subject to pure-dephasing processes, ubiquitous in realistic situations. We show that, under this class of interactions, teleportation, instead of projecting the input onto the GKP subspace, transforms GKP error correction into a correlated Pauli-Gaussian noise process that can accumulate over successive correction rounds. Thus, unlike in previously studied noisy GKP scenarios, the code no longer self-corrects. Moreover, repeated noisy GKP teleportation drives the encoded modes out of the logical subspace, progressively degrading the encoded codewords. We show that the resulting leakage rapidly saturates with the number of correction rounds. Finally, we investigate the implications of our results for motional-oscillator-based GKP error-correction platforms.

Nonstabilizerness is a necessary resource for quantum systems to lie beyond the classically simulable regime. With the advent of stabilizer $\alpha$-Rényi entropies, nonstabilizerness has also become a many-body diagnostic, complementary to entanglement. While deciding whether an arbitrary quantum state has nonstabilizerness is provably hard, such states already require a description exponentially large in the number of qubits and are thus out of reach for many-body physics. Here we instead consider 2D tensor network (TN) states, which compactly capture states obeying an entanglement area law, and ask whether they admit a simpler algorithm for the same task. In contrast to the 1D case, we prove that, even at small, constant bond dimension, computing the stabilizer entropy of 2D TN states is $\#\mathrm{P}$-hard for any integer $\alpha\ge 2$, and deciding stabilizer membership is $\mathrm{C_=P}$-complete. The corresponding constant-accuracy problems remain $\mathrm{C_= P}$-hard. We further prove that deciding whether a PEPS can be transformed into a stabilizer state by local unitaries over a specified bipartition is $\mathrm{C_= P}$-hard. Under standard complexity assumptions, these results rule out classical or quantum algorithms with polynomial resources for all three tasks in the worst case.

Two-atom superradiance is studied for atoms having a J=0 ground state and a J=1 excited state. The atoms lie on the x-axis and are excited by a \sigma_{+}-polarized laser pulse propagating in the z-direction. We use two, complementary approaches to obtain the final angular momentum in the field, a source-field approach and a Schrodinger picture approach. Both approaches give identical answers, but it is necessary to include terms in both calculations that, at first glance, might not be expected to contribute. We show that, owing to an exchange between a state in which atom 1 is in its m=1 excited state and atom 2 is in its ground state to one in which atom 2 is in its m=-1 excited state and atom 1 in its ground state, the final angular momentum in the field is not necessarily equal to the initial angular momentum in the two-atom system. The missing angular momentum appears as orbital angular momentum of the two atoms.

The Quantum Approximate Optimization Algorithm (QAOA), whose design is underpinned by the adiabatic theorem, is one of the leading variational quantum algorithms for preparing the ground states of gauge theories. A recently proposed variant, DC-QAOA, incorporates counterdiabatic driving to accelerate the adiabatic process, reducing both circuit depth and runtime. In this work, we present a systematic analysis, from both theoretical and computational perspectives, of DC-QAOA and related counterdiabatic variants for preparing the ground state of the Schwinger model, the paradigmatic (1+1)-dimensional lattice gauge theory. Benchmarking these methods against standard QAOA, we show that counterdiabatic driving improves ground-state preparation while lowering circuit depth requirements. Our results establish counterdiabatic protocols as a practical route towards efficient quantum simulation of gauge theories.

We study how the spatial profile of a deterministic coherent gate perturbation affects finite-size quasienergy statistics and operator dynamics in U(1)-symmetric XXZ Floquet circuits. Using a shared homogeneous reference, we compare a single modified bond with an alternating perturbation applied across the whole chain. After separating symmetry sectors and checking several known integrable forms, we find that the alternating circuits move toward circular orthogonal ensemble level statistics and change operator-space entanglement and out-of-time-order correlations at smaller local amplitudes than the single defects. For intermediate numbers of modified bonds, the response is nonmonotonic and also depends on where the bonds are placed, so bond density alone is not enough. A perturbative analysis identifies the additive weight of many modified bonds, while coherent spatial interference reshapes the coupling between nearby eigenstates.

We present quantum algorithms to approximate the exponential of banded Toeplitz matrices. These matrices have central importance in many PDE related problems, but their quantum implementation is hindered by their possibly large norms. Constructing directly the exponential we can circumvent this limitation. By relating the lower/upper shift operators to circulant and skewcirculant generators, which are diagonalised by the QFT, we construct (i) an LCU-based block encoding of banded Toeplitz matrices, (ii) an efficient, controllably truncated Pauli-string decomposition of the circulant eigenphases with a closed-form error bound, and (iii) a specialized QFT-Trotter product formula. As an application we build a block encoding of the propagator of the discretised heat equation with periodic, Dirichlet and Neumann boundary conditions, using a frequency cutoff.

Dissipative algorithms approach the problem of ground state preparation by mimicking the natural thermalization of a quantum system in contact with a large, low-temperature thermal environment. The environment can be efficiently simulated by a single ancilla qubit with a variable energy gap that is repeatedly coupled to the system qubits to generate a dissipative channel, and then reset after each interaction. Here we present an adaptive implementation of the dissipative algorithm, RL-Adapt, that uses single-shot reinforcement learning to optimize the selection of ancilla frequency and system-bath interaction operator to maximize energy dissipation without relying on a priori knowledge of the system spectrum. The adaptive implementation results in significantly reduced convergence times and can successfully find the ground state even for non-ideal operator pools that fail to converge using non-adaptive, uniform random operator and ancilla frequency selection.

We study the three-parameter family of quantum conditional Rényi entropies $H_{\alpha,z}^\lambda$ introduced in Commun. Math. Phys. 407, 180 (2026). We prove that the previously established data-processing region is sharp. We then characterize the club-sandwiched family through a classical coarse-graining principle: within the full data-processing region, universal monotonicity under deterministic coarse-graining of the classical conditioned register holds if and only if $z = \alpha$. Finally, we establish rigidity of entropic duality by showing that any universal duality relation for finite-dimensional tripartite pure states must satisfy the known parameter relations. Thus, the data-processing region, the club-sandwiched subfamily, and the duality transformation are each uniquely characterized by their respective universal properties.

In a recent paper Nahum argued on the basis of a renormalization-group analysis that, contrary to standard lore, ground states of 1D spin chains with short-range interactions can spontaneously break U(1) “easy-plane” spin rotation symmetry. True long-range order of $(S^x,S^y)$ arises at the phase transition between two quasi-long-range-ordered phases. Here we present detailed numerical results for an anisotropic spin-1 chain model. We find a magnetization exponent $\beta\approx0.29$, consistent with the $\epsilon$-expansion prediction $\beta\approx0.28$, a divergence of the Luttinger parameter on approaching the transition from both sides with an exponent that we relate to Nahum's transition, and finite-size spectra consistent with a dynamical critical exponent $z\approx2$. Our results support the critical behavior predicted by Nahum's field theory.