Twisted Boundary Conditions Research Articles

Fractional spins, unfolding, and holography. Part I. Parent field equations for dual higher-spin gravity reductions

In this work and in the companion paper arXiv:2403.02301, we initiate an approach to holography based on the AKSZ formalism. As the first step, we refine Vasiliev’s holography proposal in arXiv:1203.5554 by obtaining 4D higher-spin gravity (HSG) and 3D coloured conformal higher-spin gravity (CCHSG) — i.e., coloured conformal matter fields coupled to conformal higher-spin gauge fields and colour gauge fields — as two distinct and classically consistent reductions of a single parent theory. The latter consists, on-shell, of a flat superconnection valued in a fractional-spin extension of Vasiliev’s higher-spin algebra. The HSG and CCHSG reductions are characterized by dual structure groups and two-form cohomology elements, and their embedding in a common parent model provides a rationale for deriving holographic relations from multi-dimensional AKSZ partition functions on cylinders with dual boundary conditions, to appear separately. In this work we i) construct the underlying non-commutative geometry as a metaplectic operator algebra represented in a Hermitian module of a pair of conformal particles; ii) identify a discrete modular group, arising from twisted boundary conditions of the first-quantized system, and connecting different boundary conditions of the second-quantized system; and iii) identify the holonomies, structure groups and two-form cohomology elements that characterize the HSG and CCHSG reductions, and equate the dual second Chern classes.

Decorated defect construction of gapless-SPT states

Symmetry protected topological (SPT) phases are one of the simplest, yet nontrivial, gapped systems that go beyond the Landau paradigm. In this work, we study an extension of the notion of SPT for gapless systems, namely, gapless symmetry protected topological states. We construct several simple gapless-SPT models using the decorated defect construction, which allow analytical understanding of non-trivial topological features including the symmetry charge under twisted boundary conditions, and boundary (quasi)-degeneracy under open boundary conditions. We also comment on the stability of the gapless-SPT models under symmetric perturbations, and apply small-scale exact diagonalization when direct analytic understanding is not available.

Exact zeros of fidelity in finite-size systems as a signature for probing quantum phase transitions.

The fidelity is widely used to detect quantum phase transitions, which is characterized by either a sharp change of fidelity or the divergence of fidelity susceptibility in the thermodynamical limit when the phase-driving parameter is across the transition point. In this work, we unveil that the occurrence of exact zeros of fidelity in finite-size systems can be applied to detect quantum phase transitions. In general, the fidelity F(γ,γ[over ̃]) always approaches zero in the thermodynamical limit, due to the Anderson orthogonality catastrophe, no matter whether the parameters of two ground states (γ and γ[over ̃]) are in the same phase or different phases, and this makes it difficult to distinguish whether an exact zero of fidelity exists by finite-size analysis. To overcome the influence of orthogonality catastrophe, we study finite-size systems with twist boundary conditions, which can be introduced by applying a magnetic flux, and demonstrate that exact zeros of fidelity can be always accessed by tuning the magnetic flux when γ and γ[over ̃] belong to different phases. On the other hand, no exact zero of fidelity can be observed if γ and γ[over ̃] are in the same phase. We demonstrate the applicability of our theoretical scheme by studying concrete examples, including the Su-Schrieffer-Heeger model, Creutz model, and Haldane model. Our work provides a practicable way to detect quantum phase transitions via the calculation of fidelity of finite-size systems.

A study of the SYK2 model with twisted boundary conditions

We study a version of the 2-body Sachdev-Ye-Kitaev (SYK2) model whose complex fermions exhibit twisted boundary conditions on the thermal circle. As we show, this is physically equivalent to coupling the fermions to a 1-dimensional external gauge field A(t). In the latter formulation, the gauge field itself can be thought of as arising from a radial symmetry reduction of a (2 + 1)-dimensional Chern-Simons gauge field Aμ(t,x). Using the diagnostic tools of the out-of-time-order correlator (OTOC) and spectral form factor (SFF), which probe the sensitivity to initial conditions and the spectral statistics respectively, we give a detailed and pedagogical study of the integrable/chaotic properties of the model. We find that the twisting has no effect on the OTOCs and, by extension, the early-time chaos properties of the model. It does, however, have two notable effects on the spectral form factor; an enhancement of the early-time slope and the emergence of an explicit twisting-dependent timescale needed for the manifestation of zero modes. These zero modes are responsible for the late-time exponential ramp in the quadratic SYK model.

Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries

We study ’t Hooft anomalies of global symmetries in 1+1d lattice Hamiltonian systems. We consider anomalies in internal and lattice translation symmetries. We derive a microscopic formula for the “anomaly cocycle” using topological defects implementing twisted boundary conditions. The anomaly takes value in the cohomology group H^3(G,U(1)) × H^2(G,U(1))H3(G,U(1))×H2(G,U(1)). The first factor captures the anomaly in the internal symmetry group G, and the second factor corresponds to a generalized Lieb-Schultz-Mattis anomaly involving G and lattice translation. We present a systematic procedure to gauge internal symmetries (that may not act on-site) on the lattice. We show that the anomaly cocycle is the obstruction to gauging the internal symmetry while preserving the lattice translation symmetry. As an application, we construct anomaly-free chiral lattice gauge theories. We demonstrate a one-to-one correspondence between (locality-preserving) symmetry operators and topological defects, which is essential for the results we prove. We also discuss the generalization to fermionic theories. Finally, we construct non-invertible lattice translation symmetries by gauging internal symmetries with a Lieb-Schultz-Mattis anomaly.

Nonrelativistic Dirac fermions on the torus

Two dimensional conformal field theories have been extensively studied in the past. When considered on the torus, they are strongly constrained by modular invariance. However, introducing relevant deformations or chemical potentials pushes these theories away from criticality, where many of their aspects are still poorly understood. In this note we make a step towards filling this gap, by analyzing the theory of a Dirac fermion on the torus, deformed by a mass term and a chemical potential for the particle number symmetry. The theory breaks conformal and Lorentz invariance, and we study its spectrum and partition function. We also focus on two limits that are interesting on their own right: a massless relativistic fermion with nonzero chemical potential (a simple model for CFTs at finite density), and nonrelativistic Schrodinger fermions (of relevance in condensed matter systems). Taking inspiration from recent developments in massive modular forms, we obtain a representation of the torus free energy based on Fourier-transforming over a twisted boundary condition. This dual representation fullfills many properties analogous to modular invariance in CFTs. In particular, we use this result to derive Cardy-like formulas for the high energy density of states of these theories.

Bethe-state counting and Witten index

We count the physical Bethe states of quantum integrable models with twisted boundary conditions using the Witten index of 2d supersymmetric gauge theories. For multi-component models solvable by the nested Bethe ansatz, the result is a novel restricted occupancy problem. For the SU(3) spin chain and the t-J model, we propose formulae for the solution count on singular loci in the space of twist parameters.

Multi-fractional instantons in SU(N) Yang-Mills theory on the twisted T4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbbm{T}}^4 $$\\end{document}

We construct analytical self-dual Yang-Mills fractional instanton solutions on a four-torus T4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbbm{T}}^4 $$\\end{document} with ’t Hooft twisted boundary conditions. These instantons possess topological charge Q=rN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ Q=\\frac{r}{N} $$\\end{document}, where 1 ≤ r < N. To implement the twist, we employ SU(N) transition functions that satisfy periodicity conditions up to center elements and are embedded into SU(k) × SU(ℓ) × U(1) ⊂ SU(N), where ℓ + k = N. The self-duality requirement imposes a condition, kL1L2 = rℓL3L4, on the lengths of the periods of T4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbbm{T}}^4 $$\\end{document} and yields solutions with abelian field strengths. However, by introducing a detuning parameter ∆ ≡ (rℓL3L4 – kL1L2)/L1L2L3L4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{L_1{L}_2{L}_3{L}_4} $$\\end{document}, we generate self-dual nonabelian solutions on a general T4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbbm{T}}^4 $$\\end{document} as an expansion in powers of ∆. We explore the moduli spaces associated with these solutions and find that they exhibit intricate structures. Solutions with topological charges greater than 1N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{1}{N} $$\\end{document} and k ≠ r possess non-compact moduli spaces, along which the O∆\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}\\left(\\Delta \\right) $$\\end{document} gauge-invariant densities exhibit runaway behavior. On the other hand, solutions with Q=rN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ Q=\\frac{r}{N} $$\\end{document} and k = r have compact moduli spaces, whose coordinates correspond to the allowed holonomies in the SU(r) color space. These solutions can be represented as a sum over r lumps centered around the r distinct holonomies, thus resembling a liquid of instantons. In addition, we show that each lump supports 2 adjoint fermion zero modes.

Ground state energy of twisted AdS3 × S3 × T4 superstring and the TBA

We use the lightcone AdS3 × S3 × T4 superstring sigma model with fermions and bosons subject to twisted boundary conditions to find the ground state energy in the semi-classical approximation where effective string tension h and the light-cone momentum L are sent to infinity in such a way that J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{J} $$\\end{document} ≡ L/h is kept fixed. We then analyse the ground state energy of the model by means of the mirror TBA equations for the AdS3 × S3 × T4 superstring in the pure RR background. The calculation is performed for small twist μ with L and h fixed, for large L with μ and h fixed, and for small h with μ and L fixed. In these limits the contribution of the gapless worldsheet modes coming from the T4 bosons and fermions can be computed exactly, and is shown to be proportional to hL/(4L2 – 1). Comparison with the semi-classical result shows that the TBA equations involve only one Y0-function for massless excitations but not two as was conjectured before. Some of the results obtained are generalised to the mixed-flux AdS3× S3× T4 superstring.

Lieb-Schultz-Mattis, Luttinger, and 't Hooft - anomaly matching in lattice systems

We analyze lattice Hamiltonian systems whose global symmetries have ’t Hooft anomalies. As is common in the study of anomalies, they are probed by coupling the system to classical background gauge fields. For flat fields (vanishing field strength), the nonzero spatial components of the gauge fields can be thought of as twisted boundary conditions, or equivalently, as topological defects. The symmetries of the twisted Hilbert space and their representations capture the anomalies. We demonstrate this approach with a number of examples. In some of them, the anomalous symmetries are internal symmetries of the lattice system, but they do not act on-site. (We clarify the notion of “on-site action.”) In other cases, the anomalous symmetries involve lattice translations. Using this approach we frame many known and new results in a unified fashion. In this work, we limit ourselves to 1+1d systems with a spatial lattice. In particular, we present a lattice system that flows to the c=1c=1 compact boson system with any radius (no BKT transition) with the full internal symmetry of the continuum theory, with its anomalies and its T-duality. As another application, we analyze various spin chain models and phrase their Lieb-Shultz-Mattis theorem as an ’t Hooft anomaly matching condition. We also show in what sense filling constraints like Luttinger theorem can and cannot be viewed as reflecting an anomaly. As a by-product, our understanding allows us to use information from the continuum theory to derive some exact results in lattice model of interest, such as the lattice momenta of the low-energy states.

Integrability treatment of AdS/CFT orbifolds

We elaborate on the treatment of orbifolds of type IIB string theory on and their dual gauge theories with integrability techniques. The implementation of orbifolds via twisted spin-chains, thermodynamic Bethe ansatz equations with chemical potentials and Y- and T-systems with modified asymptotics is confronted with twisted boundary conditions of the string sigma-model. This allows us to consistently twist the quantum spectral curve, which is believed to bridge the two sides of the AdS/CFT duality. We discuss Abelian orbifolds of and treat the special cases of supersymmetric -orbifolds and type 0B string theory on as primary examples. This opens a pathway to probe the validity of the duality and to study the long-standing question of tachyon stabilisation in non-supersymmetric AdS/CFT. We comment on the current understanding of this issue and point out the next steps in this challenge.

Nonintegrable Floquet Ising model with duality twisted boundary conditions

Results are presented for a Floquet Ising chain with duality twisted boundary conditions, taking into account the role of weak integrability breaking in the form of four-fermion interactions. In the integrable case, a single isolated Majorana zero mode exists which is a symmetry in the sense that it commutes both with the Floquet unitary and the $Z_2$ symmetry of the Floquet unitary. When integrability is weakly broken, both in a manner so as to preserve or break the $Z_2$ symmetry, the Majorana zero mode is still found to be conserved for small system sizes. This is reflected in the dynamics of an infinite temperature autocorrelation function which, after an initial transient that is controlled by the strength of the integrability breaking term, approaches a plateau that does not decay with time. The height of the plateau agrees with a numerically constructed conserved quantity, and is found to decrease with increasing system sizes. It is argued that the existence of the plateau and its vanishing for larger system sizes is closely related to a localization-delocalization transition in Fock space triggered by the integrability-breaking interactions.

The Casimir effect in a dilute Bose gas at zero-temperature taking into account the contribution of self-energy within CJT theory

The Casimir effect in a weakly interacting Bose-Einstein condensate (BEC) gas is investigated at zero temperature within the Cornwall-Jackiw-Tomboulis (CJT) effective action approach under elastic and twisted boundary conditions (BCs) of the wave vector. After discussing the rationales for applying the CJT theory to a confinement BEC system, we establish the analytical formulae and then evaluate the numerical values of the Casimir force, which are essential for the observation experiment of the theoretical results.

Dynamical singularity of the rate function for quench dynamics in finite-size quantum systems

The dynamical quantum phase transition is characterized by the emergence of nonanalytic behaviors in the rate function, corresponding to the occurrence of exact zero points of the Loschmidt echo in the thermodynamical limit. In general, exact zeros of the Loschmidt echo are not accessible in a finite-size quantum system except for some fine-tuned quench parameters. In this work, we study the realization of the dynamical singularity of the rate function for finite-size systems under the twist boundary condition, which can be introduced by applying a magnetic flux. By tuning the magnetic flux, we illustrate that exact zeros of the Loschmidt echo can be always achieved when the postquench parameter is across the underlying equilibrium phase transition point, and thus the rate function of a finite-size system is divergent at a series of critical times. We demonstrate our theoretical scheme by calculating the Su-Schrieffer-Heeger model and the Creutz model in detail and exhibit its applicability to more general cases. Our result unveils that the emergence of dynamical singularity in the rate function can be viewed as a signature for detecting dynamical quantum phase transition in finite-size systems. We also unveil that the critical times in our theoretical scheme are independent on the systems size, and thus it provides a convenient way to determine the critical times by tuning the magnetic flux to achieve the dynamical singularity of the rate function.

Topological invariants for interacting systems: From twisted boundary conditions to center-of-mass momentum

Beyond the well-known topological band theory for single-particle systems, it is a great challenge to characterize the topological nature of interacting multiparticle quantum systems. Here, we uncover the relation between topological invariants defined through the twisted boundary condition (TBC) and the center-of-mass (c.m.) momentum state in multiparticle systems. We find that the Berry phase defined through the TBC can be equivalently obtained from the multiparticle Wilson loop formulated by c.m. momentum states. As the Chern number can be written as the winding of the Berry phase, we consequently prove the equivalence of Chern numbers obtained via TBC and c.m. momentum state approaches. As a proof-of-principle example, we study topological properties of the Aubry-Andr\'e-Harper model. Our numerical results show that the TBC approach and c.m. approach are well consistent with each other for both the many-body case and the few-body case. Our work lays a concrete foundation and provides insights for exploring multiparticle topological states.

Time-resolved single-particle spectrum of the one-dimensional extended Hubbard model after interaction quenches

We investigate the non-equilibrium dynamics of the one-dimensional extended Hubbard model after interaction quenches. In strong-coupling regime with large on-site interaction, the ground states of this model with small and large nearest-neighbor interactions are in spin-density-wave and charge-density-wave phases, respectively. Combining twisted boundary conditions with the time-dependent Lanczos method, we obtain snapshots of the time-dependent single-particle spectrum after quenches. We find that for quench within the same phase, the single-particle spectrum becomes close to that of the quenched Hamiltonian immediately after the quench. While for quench across the critical point, the afterward evolution process depends mainly on the distribution of the initial state among the eigenstates of the quenched Hamiltonian. Our finding may serve as a way to detect the phase transition in ultracold atom systems with interactions.

Systematically improvable mean-field variational ansatz for strongly correlated systems: Application to the Hubbard model

A systematically improvable wave function is proposed for the numerical solution of strongly correlated systems. With a stochastic optimization method, based on the auxiliary field quantum Monte Carlo technique, an effective temperature ${T}_{\mathrm{eff}}$ is defined, probing the distance of the ground-state properties of the model in the thermodynamic limit from the ones of the proposed correlated mean-field ansatz. In this way, their uncertainties from the unbiased zero temperature limit may be estimated by simple and stable extrapolations well before the so-called sign problem gets prohibitive. At finite ${T}_{\mathrm{eff}}$, the convergence of the energy to the thermodynamic limit is indeed shown to already be possible in the Hubbard model for relatively small square lattices with linear dimension $L\ensuremath{\simeq}10$, thanks to appropriate averages over several twisted boundary conditions. Within the estimated energy accuracy of the proposed variational ansatz, two clear phases are identified, as the energy is lowered by spontaneously breaking some symmetries satisfied by the Hubbard Hamiltonian: (a) a stripe phase where both spin and translation symmetries are broken and (b) a strong coupling $d$-wave superconducting phase when the particle number is not conserved and global $U(1)$ symmetry is broken. On the other hand, the symmetric phase is stable in a wide region at large doping and small coupling.

Phase structure of linear quiver gauge theories from anomaly matching

We consider the phase structure of the linear quiver gauge theory, using the 't Hooft anomaly matching condition. This theory is characterized by the length $K$ of the quiver diagram. When $K$ is even, the symmetry and its anomaly are the same as those of massless QCD. Therefore, one can expect that the spontaneous symmetry breaking similar to the chiral symmetry breaking occurs. On the other hand, when $K$ is odd, the anomaly matching condition is satisfied by the massless composite fermions. We also consider the thermal partition function under the twisted boundary conditions. When $K$ is even, from the anomaly at finite temperature, we estimate the relation between the critical temperatures associated with the confinement/deconfinement and the breaking of the global symmetry. Finally we discuss the anomaly matching at finite temperature when $K$ is odd.

Quantum spin liquids of Rydberg excitations in a honeycomb lattice induced by density-dependent Peierls phases

We show that the nonlinear transport of bosonic excitations in a two-dimensional honeycomb lattice of spin-orbit-coupled Rydberg atoms gives rise to disordered quantum phases which are topological and may be candidates for quantum spin liquids. As recently demonstrated by Lienhard et al. [Phys. Rev. X 10, 021031 (2020)] the spin-orbit coupling breaks time-reversal and chiral symmetries and leads to a tunable density-dependent complex hopping of spin excitations which behave as hard-core bosons. Using exact diagonalization (ED), we numerically investigate the phase diagram resulting from the competition between density-dependent and direct transport terms as well as density-density interactions. In mean-field approximation there is a phase transition from a condensate to a ${120}^{\ensuremath{\circ}}$ phase when the amplitude of the complex hopping exceeds that of the direct one. In the full model a new phase emerges close to the mean-field critical point as a result of quantum correlations induced by the density dependence of the complex hopping. We show that without density-density interactions this phase is a genuine disordered one, has large spin chirality, and is characterized by a nontrivial many-body Chern number. The Chern number is found to be robust to disorder. ED simulations of small lattices with up to 30 lattice sites give indications for a nondegenerate ground state with finite spin and collective gaps and thus for a bosonic integer quantum Hall phase, protected by $\mathrm{U}(1)$ symmetry. On the other hand, while staying finite, the many-body gap varies substantially when different twisted boundary conditions are applied, which points to a gapless phase. For very strong negative nonlinear hopping amplitudes we find another disordered regime with vanishing spin gap. This phase also has a large spin chirality and could be a gapless spin liquid but lies outside the parameter regime experimentally accessible in the Rydberg system.

Phase transitions in the Haldane-Hubbard model with ionic potentials

By employing the exact-diagonalization method, we revisit the ground-state phase diagram of the Haldane-Hubbard model on the honeycomb lattice with staggered sublattice potentials. The phase diagram includes the band insulator, Mott insulator, and two Chern insulator phases with Chern numbers C=2 and C=1, respectively. The character of transitions between different phases is studied by analyzing the lower-lying energy levels, excitation gaps, structure factors, and fidelity metric. We find that the C=1 phase can be continuously deformed into the C=2 phase without a gap closure in the periodic boundary condition, while a further analysis on the Berry curvatures indicates that the excitation gap closes at the phase boundary in a twisted boundary condition, accompanied by the discontinuities of structure factors. All the other phase transitions are found to be first-order ones as expected.