New Technique Reveals Hidden Energy Gaps in Exotic Quantum Materials

Scientists are continually seeking reliable methods to determine the energy gap in complex quantum materials, a particularly difficult task when investigating the exotic states of matter known as quantum spin liquids. Takayuki Yokoyama and Yasuhiro Tada, both from the Quantum Matter Program at the Graduate School of Advanced Science and Engineering, Hiroshima University, present a novel diagnostic scheme based on the polarization amplitude calculated using the infinite density-matrix renormalization group (iDMRG) framework. Their work, demonstrating clear differentiation between gapped and gapless phases in the spin-XXZ chain, extends to the spin-XY model on the square lattice, successfully identifying the transition between gapless and gapped chiral spin liquid states. This research establishes polarization amplitudes as a robust and effective tool for characterising the energy gap in two-dimensional frustrated magnets and furthering our understanding of quantum spin liquids.

Scientists address a fundamental yet challenging problem, especially in quantum spin liquids. This work develops a gap-diagnostic scheme based on the polarization amplitude defined via twist operator, evaluated within the infinite density-matrix renormalization group (iDMRG) framework. As a benchmark, an analysis of the spin-1/2 XXZ chain demonstrates that the polarization amplitude clearly distinguishes the gapless Tomonaga, Luttinger liquid from the gapped N eel phase.

Researchers then extend this framework to infinite cylinders of the spin-1/2 XY, Jχ model on the square lattice. They find that the polarization amplitude reliably detects the quantum critical point between the N eel and stripe phases.

Polarization amplitude distinguishes phases and detects transitions in spin chains

Analysis of the spin-1/2 XXZ chain within this study establishes a clear distinction between the gapless Tomonaga-Luttinger liquid and the gapped Néel phase via the polarization amplitude. Specifically, the polarization amplitude serves as a quantifiable metric, demonstrating a marked difference in behaviour between these two phases. Extending this framework to infinite cylinders of the spin-1/2 XY, Jχ model, the research reveals that the polarization amplitude acutely detects the transition occurring between the gapless XY phase and the gapped chiral spin liquid phase.

This detection is achieved through precise measurement of the polarization amplitude, which exhibits a sharp change at the phase transition. The iDMRG calculations performed on the spin-1/2 XXZ chain demonstrate that the polarization amplitude accurately reflects the presence or absence of a spin gap. This benchmark analysis confirms the validity of the approach and its ability to reliably identify gapped and gapless phases in one-dimensional systems.

Further investigation of the spin-1/2 XY, Jχ model on infinite cylinders shows that the polarization amplitude provides a robust diagnostic for energy gaps in two-dimensional frustrated quantum magnets. The ability to discern between the gapless XY phase and the gapped chiral spin liquid phase highlights the method’s potential for characterising complex quantum phases.

This work directly evaluates the expectation value of the twist operator using infinite-DMRG, representing the thermodynamic limit along the cylinder axis. The iDMRG formulation allows for application of the twist operator on regions of increasing length, maintaining the integrity of the underlying ground state. This approach provides a conceptually clean setting to examine the polarization amplitude as a practical gap diagnostic in quasi-one-dimensional geometries. The consistent behaviour of the polarization amplitude across both the benchmark XXZ chain and the XY, Jχ model confirms its effectiveness in identifying phase transitions and characterising energy gaps.

Polarization amplitudes reveal excitation gaps in chiral spin liquids

Scientists are investigating the transition between gapless and gapped chiral spin liquid phases using polarization amplitudes as a diagnostic tool in two-dimensional frustrated quantum magnets. This distinction between gapped and gapless ground states significantly affects low-temperature properties and quasiparticle excitations, particularly in strongly correlated systems like frustrated magnets and quantum spin liquids.

Characterising the phase in these systems is often difficult due to the absence of a simple local order parameter, making the determination of an excitation gap a crucial step in understanding their low-energy physics. However, numerically determining excitation gaps is challenging, especially in higher dimensions, due to the exponential growth of the Hilbert space which limits calculations to finite systems.

Consequently, excitation gaps are often extracted from finite-size data, but accessible system sizes may be insufficient for reliable extrapolation to the thermodynamic limit. This is further complicated in strongly frustrated lattices like triangular and Kagome lattices where computationally accessible system sizes are particularly limited. An alternative approach, based on the twist operator, has been developed as a qualitative indicator of gapless versus gapped behaviour.

For one-dimensional periodic systems, the polarization, defined as the expectation value of the twist operator, behaves as a quantized order parameter distinguishing gapless systems from gapped systems. This claim has been investigated extensively in both analytical and numerical studies, not only in itinerant particle systems but also in quantum spin chains.

Tasaki rigorously demonstrated that, in one-dimensional U-symmetric uniquely gapped systems, the absolute value of the ground-state expectation value of the twist operator converges to unity in the thermodynamic limit. This proof relies on the local operator formalism for infinite systems, taking the thermodynamic limit from the beginning, with concise proofs also available for sufficiently large finite systems.

Stronger statements have also been proposed in the presence of enlarged SO symmetry. While the vanishing amplitude of the twist operator in a gapless infinite system has not been fully proven mathematically, it generally holds true. However, the amplitude of the twist operator can be non-universal in higher dimensions, making extension to two-dimensional systems challenging.

This raises questions regarding the usefulness of the twist operator for describing ordered states in higher-dimensional systems such as quantum spin liquids where diagnosing an energy gap is a highly non-trivial problem. This work investigates whether the twist-operator-based energy-gap diagnosis, established in one dimension, can be extended to quasi-one-dimensional infinite systems represented by cylinder geometry.

The researchers consider the twist operator associated with the U spin rotation symmetry and evaluate its expectation value using infinite-DMRG (iDMRG), a widely used method for studying one-dimensional systems and quantum spin liquids in cylinder geometry. Although both the twist operator and iDMRG have been extensively studied, a systematic evaluation of the polarization amplitude on infinite cylinders within the iDMRG framework has not been performed.

Unlike finite-size approaches, iDMRG directly represents the thermodynamic limit along the cylinder axis, allowing the twist operator to be applied on regions of increasing length without altering the underlying ground state. This provides a conceptually clean setting to examine whether the polarization amplitude can serve as a practical gap diagnostic in quasi-one-dimensional geometries relevant to two-dimensional quantum magnets.

The researchers first examine the S = 1/2 XXZ chain as a benchmark, demonstrating that the polarization amplitude’s behaviour is consistent with a spin gap within iDMRG. They then apply the method to a two-dimensional quantum spin model on infinitely long cylinders, showing that the polarization amplitude can clearly distinguish a gapless XY phase from a gapped quantum spin liquid phase.

The results demonstrate the practical feasibility of qualitatively diagnosing gapless-gapped transitions in quasi-two-dimensional systems in a controlled infinite-system setting. The primary purpose of this work is to study cylinder systems with potential extrapolation to two dimensions, based on the twist operator. For this, the researchers first discuss an infinite one-dimensional chain as a benchmark for the validity of iDMRG calculations of the twist operator, considering systems with U and translation symmetries.

The twist operator is employed as a ground-state indicator to diagnose whether a system is gapped or gapless. In a finite-size system with periodic boundary conditions, the twist operator is defined as a nonlocal U gauge transformation associated with a conserved U charge, such as the z component of spin. In an infinite system, the twist operator is local and acts on a region of the system.

The spin operators on site j are denoted by Sz j and, as illustrated in Fig0.1(a), the twist operator is defined as U = exp 2πi Ltw Ltw X j=1 j Sz j, where Ltw is the length of the region on which the twist is applied. For finite-size systems, Ltw is often chosen as the system length L, but in iDMRG calculations, the system size is infinite from the beginning.

The expectation value of the twist operator is closely tied to the presence or absence of a spin gap, defined as ∆E(M) ≡E(M + 1) + E(M −1) −2E(M). The conserved quantity is the total magnetization M = P j Sz j of the super-unit cell within the iDMRG calculation. For M = 0, the exponent is q = 2 when S is a half-integer and q = 1 when S is an integer. For a one-dimensional system at filling νs = (S −m) = p/q where m is the magnetization per physical unit cell and p, q are coprime integers, a working ansatz |zq| := lim Ltw→∞ U q = ( 1, (∆E 0) 0, (∆E = 0) is introduced.

A key advantage of this approach is that the diagnosis can be carried out using only the ground-state wave function, without separately computing excited states. The quantization of |zq| is intuitive, as the twist for nearby spins with a small angle O(1/Ltw) cannot create an excitation in a gapped system, while it changes the ground state to a low-energy excited state in a gapless system.

Numerical estimation of the spin gap ∆E alone may not completely determine the behaviours of |zq|. Quantization of |zq| has not been proven for a system with a spin gap ∆E 0 and a vanishing gap within the ground state M-sector. Nevertheless, the ansatz has been extensively used and repeatedly confirmed in numerical and analytical calculations for various one-dimensional finite systems under periodic boundary conditions.

The researchers numerically demonstrate that the ansatz indeed holds in infinite one-dimensional systems. Ground states are obtained using infinite density-matrix renormalization group (iDMRG). Since iDMRG directly targets the thermodynamic limit for one-dimensional chains and quasi one-dimensional systems like cylinder geometry, it avoids explicit extrapolation of the system length.

The absolute expectation value of the twist operator is evaluated by representing the exponential operator acting on a finite interval as a matrix-product operator (MPO) and contracting it with the resulting infinite matrix-product state (iMPS). In conventional finite-size DMRG, the twist operator is usually defined on the whole system with size Lx and the ground states for each Lx have to be calculated to obtain the expectation value ⟨U q⟩ in the thermodynamic limit Lx →∞.

Within iDMRG, the corresponding thermodynamic-limit expectation value can be evaluated more directly using the MPO and iMPS formalism with a single infinite-size ground state. By systematically increasing the operator length Ltw, the convergence of ⟨U q⟩ within the infinite-system framework can be probed. The truncation error is kept below 10−7 for one-dimensional chains and below 10−5 for quasi-one-dimensional cylinders.

The S = 1/2 antiferromagnetic XXZ chain with nearest neighbour interactions is studied, described by HXXZ = J X j Sx j Sx j+1 + Sy j Sy j+1 + ∆Sz j Sz j+1, where Sα (α = x, y, z) is the spin operator and J 0 denotes the antiferromagnetic exchange coupling, and ∆ parametrizes the exchange anisotropy of the z component. The researchers focus on the regime 0 ≤∆, examining the phase diagram of the ground state which has been well established: for ∆≤1 it realizes the gapless Tomonaga-Luttinger liquid (TLL), whereas for ∆ 1 it exhibits a gapped phase.

The Bigger Picture

Scientists have long struggled to definitively characterise the subtle phases of matter found in complex materials, particularly those exhibiting exotic magnetic behaviour like spin liquids. Identifying whether these states are ‘gapped’ , possessing an energy barrier to excitation, or ‘gapless’ , allowing excitations at any energy level, is crucial for understanding their fundamental properties and potential applications.

This distinction, however, has proven remarkably difficult to establish with existing techniques, often requiring painstaking analysis and leaving room for ambiguity. This new work offers a compelling diagnostic tool based on measuring the ‘polarization amplitude’ within these materials. By leveraging advanced computational methods, researchers demonstrate a clear ability to differentiate between gapped and gapless states in model systems, a significant step towards resolving long-standing debates.

The implications extend beyond fundamental physics, potentially accelerating the discovery of materials with tailored electronic properties for future technologies. However, the sensitivity of this method relies heavily on the accuracy of the underlying calculations and the size of the systems that can be modelled. Ensuring that simulations capture the true behaviour of real materials, with their inherent imperfections and complexities, remains a considerable challenge.

Future research will likely focus on refining the technique, exploring its applicability to a wider range of materials, and crucially, bridging the gap between these computational insights and experimental observations. The pursuit of topological quantum materials, where these exotic states of matter can be harnessed, is now better equipped with a more reliable means of characterisation.