We describe a construction of topological orders from coupled lower dimensional symmetry-protected topological orders, which is closely related to gauging a subsystem symmetry. Our construction yields both conventional topological orders and exotic fracton topological orders of type-I and type-II. In particular, we find a coupled spin chain construction of Haah's cubic code, and a coupled layer construction of Yoshida's fractal spin liquids.
We provide a new perspective on fracton topological phases, a class of three-dimensional topologically ordered phases with unconventional fractionalized excitations that are either completely immobile or only mobile along particular lines or planes. We demonstrate that a wide range of these fracton phases can be constructed by strongly coupling mutually intersecting spin chains and explain via a concrete example how such a coupled-spin-chain construction illuminates the generic properties of a fracton phase. In particular, we describe a systematic translation from each coupled-spin-chain const…
In three dimensions, gapped phases can support "fractonic" quasiparticle excitations, which are either completely immobile or can only move within a low-dimensional submanifold, a peculiar topological phenomenon going beyond the conventional framework of topological quantum field theory. In this work we explore fractonic topological phases using three-dimensional coupled wire constructions, which have proven to be a successful tool to realize and characterize topological phases in two dimensions. We find that both gapped and gapless phases with fractonic excitations can emerge from the models.…
In this work, we develop a coupled layer construction of fracton topological orders in $d=3$ spatial dimensions. These topological phases have sub-extensive topological ground-state degeneracy and possess excitations whose movement is restricted in interesting ways. Our coupled layer approach is used to construct several different fracton topological phases, both from stacked layers of simple $d=2$ topological phases and from stacks of $d=3$ fracton topological phases. This perspective allows us to shed light on the physics of the X-cube model recently introduced by Vijay, Haah, and Fu, which …
The Chamon model is an exactly solvable spin Hamiltonian exhibiting nontrivial fracton order. In this work, we dissect two distinct aspects of the model. First, we show that it exhibits an emergent fractonic gauge theory coupled to a fermionic subsystem symmetry-protected topological state under four stacks of $\mathbb{Z}_2$ planar symmetries. Second, we show that the Chamon model hosts 4-foliated fracton order by describing an entanglement renormalization group transformation that exfoliates four separate stacks of 2D toric codes from the bulk system.
We propose a coupled-layer construction of a class of fracton topological orders in three spatial dimensions, which is characterized by spatially anisotropic mobility of quasiparticle excitations constrained in subdimensional manifolds. The simplest model is obtained by stacking and coupling layers of the two-dimensional toric codes on the square lattice and can be exactly solved in the strong-coupling limit. The resulting fracton excitations are understood as a consequence of anyon pair condensation induced by the coupling between layers. We also present generalizations of the construction fo…
Anomaly inflow for CSS and fractonic lattice models and dualities via cluster state measurement
Graduate School of Arts and Sciences, University of Tokyo（东京大学大学院人文社会系研究科）；C. N. Yang Institute for Theoretical Physics & Department of Physics and Astronomy, State University of New York at Stony Brook（纽约州立大学石溪分校陈省身理论物理研究所及物理与天文学系）
Institute for Theoretical Physics, University of Amsterdam（阿姆斯特丹大学理论物理研究所）；Dutch Institute for Emergent Phenomena（荷兰涌现现象研究所）；Institute for Advanced Study, University of Amsterdam（阿姆斯特丹大学高等研究院）
We study p-string condensation mechanisms for fracton phases from the viewpoint of higher-form symmetry, focusing on the examples of the X-cube model and the rank-two symmetric-tensor U(1) scalar charge theory. This work is motivated by questions of the relationship between fracton phases and continuum quantum field theories, and also provides general principles to describe p-string condensation independent of specific lattice model constructions. We give a perspective on higher-form symmetry in lattice models in terms of cellular homology. Applying this perspective to the coupled-layer constr…
We consider a vector gauge theory in 2 + 1 dimensions of the type recently proposed by Radzihovsky and Hermele [1] to describe fracton phases of matter. The theory has U(1)XU(1) vector gauge fields coupled to an additional vector field with a non conventional gauge symmetry. We added to the theory scalar matter in order to break the gauge symmetry. We analyze non trivial configurations by reducing the field equations to first order self dual (BPS) equations which we solved numerically. We have found vortex solutions for the gauge fields which in turn generate for the extra vector field non-tri…
Fracton topological order describes a remarkable phase of matter which can be characterized by fracton excitations with constrained dynamics and a ground state degeneracy that increases exponentially with the length of the system on a three-dimensional torus. However, previous models exhibiting this order require many-spin interactions which may be very difficult to realize in a real material or cold atom system. In this work, we present a more physically realistic model which has the so-called X-cube fracton topological order but only requires nearest-neighbor two-spin interactions. The model…
We present a coupled-wire construction of a model with chiral fracton topological order. The model combines the known construction of $ν=1/m$ Laughlin fractional quantum Hall states with a planar p-string condensation mechanism. The bulk of the model supports gapped immobile fracton excitations that generate a hierarchy of mobile composite excitations. Open boundaries of the model are chiral and gapless, and can be used to demonstrate a fractional quantized Hall conductance where fracton composites act as charge carriers in the bulk. The planar p-string mechanism used to construct and analyze …
Fracton Spin Liquid and Exotic Frustrated Phases in Ising-like Octochlore Magnets
University of California, Irvine（加州大学尔湾分校）；University of Waterloo（滑铁卢大学）；Waterloo Institute for Nanotechnology, University of Waterloo（滑铁卢大学滑铁卢纳米技术研究所）；Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Rice University（莱斯大学）
In this work, we present a collection of three-dimensional higher-order symmetry protected topological phases (HOSPTs) with gapless hinge modes that exist only in strongly interacting systems subject to subsystem symmetry constraints. We use a coupled wire construction to generate three families of microscopic lattice models: insulators with helical hinge modes, superconductors with chiral Majorana hinge modes, and fractionalized insulators with helical hinge modes that carry fractional charge. In particular, these HOSPTs do not require spatial symmetry protection, but are instead protected by…
We propose a quantum field theory description of the X-cube model of fracton topological order. The field theory is not (and cannot be) a topological quantum field theory (TQFT), since unlike the X-cube model, TQFTs are invariant (i.e. symmetric) under continuous spacetime transformations. However, the theory is instead invariant under a certain subgroup of the conformal group. We describe how braiding statistics and ground state degeneracy are reproduced by the field theory, and how the the X-cube Hamiltonian and field theory can be minimally coupled to matter fields. We also show that even o…
There and Back Again: A Gauging Nexus between Topological and Fracton Phases
University of Chicago（芝加哥大学）；Princeton University（普林斯顿大学）；Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）；Bard College（巴德学院）；Institute for Advanced Study（高等研究院）；Stony Brook University（石溪大学）；California Institute of Technology（加州理工学院）；The University of Sydney（悉尼大学）
We propose an unconventional topological quantum phase transition connecting a higher-order topological insulator (HOTI) and a featureless Mott insulator sharing the same symmetry patterns. We construct an effective theory description of the quantum critical point (QCP) by combining a bosonization approach and the coupled-stripe construction of 1D critical spin ladders. The phase transition theory is characterized by a critical dipole liquid theory with subsystem $U(1)$ symmetry whose low energy modes contain a Bose surface along the $k_x,k_y$ axis. Such a quantum critical point manifests frac…
We propose that the fracton models with subsystem symmetry can be a class of toy models for the holographic principle. The discovery of the anti-de Sitter/conformal field theory correspondence as a concrete construction of holography and the subsequent developments including the subregion duality and Ryu-Takayanagi formula of entanglement entropy have revolutionized our understanding of quantum gravity and provided powerful tool sets for solving various strongly-coupled quantum field theory problems. To resolve many mysteries of holography, toy models can be very helpful. One example is the ho…
We study the role of "$θ$ terms" in the action for three-dimensional $U(1)$ symmetric tensor gauge theories, describing quantum phases of matter hosting gapless higher-spin gauge modes and gapped subdimensional particle excitations, such as fractons. In Maxwell theory, the $θ$ term is a total derivative which has no effect on the gapless photon, but has two important, closely related consequences: attaching electric charge to magnetic monopoles (the Witten effect) and leading to a Chern-Simons theory on the boundary. We will find that a similar story holds in the higher-spin $U(1)$ gauge theor…
We study complex scalar theories with dipole symmetry and uncover a no-go theorem that governs the structure of such theories and which, in particular, reveals that a Gaussian theory with linearly realised dipole symmetry must be Carrollian. The gauging of the dipole symmetry via the Noether procedure gives rise to a scalar gauge field and a spatial symmetric tensor gauge field. We construct a worldline theory of mobile objects that couple gauge invariantly to these gauge fields. We systematically develop the canonical theory of a dynamical symmetric tensor gauge field and arrive at scalar cha…
By gauging a higher-moment polynomial degree global symmetry and a discrete charge conjugation (i.e., particle-hole) symmetry coupled to matter fields (two symmetries mutually non-commutative), we derive a new class of higher-rank tensor non-abelian gauge field theory with dynamically gauged fractonic matter fields: Non-abelian gauged fractons interact with a hybrid class of higher-rank (symmetric or generic non-symmetric) tensor gauge theory and anti-symmetric tensor topological field theory, generalizing [arXiv:1909.13879, 1911.01804]. We also apply a quantum phase transition similar to that…
By using the variational matrix product state method, we numerically study the interacting Kitaev chain with spatially varying periodic and quasi-periodic potentials and the latter follows the Fibonacci sequence. The edge correlation functions of Majorana fermions and low-lying ground states are computed to explore the robustness of topological superconducting phase. It is found that the original topological nontrivial phase is separated into to two branches by an emergent topological trivial phase, as a result of the competition among spatially varying potential, electronic Coulomb interactio…
We study the crossover from the one-dimensional to the two-dimensional Hubbard model in the photoemission spectra of weakly coupled chains. The chains with on-site repulsion are treated using the spin-charge factorized wave function, that is known to provide an essentially exact description of the chain in the strong coupling limit. The hoppings between the chains are considered as a perturbation. We calculate the dynamical spectral function at all energies in the random-phase approximation, by resuming an infinite set of diagrams. Even though the hoppings drive the system from a fractionalize…
Three-coupled chains of quarter-filled Hubbard model with dimerization have been studied for periodic boundary conditions. By use of a renormalization group method, it is found that the interchain hopping is renormalized to zero leading to confinement when charge gap becomes larger than the effective interchain hopping energy. Such a result of confinement is compared with that of two-coupled chains.
The influence of bond randomness on long range magnetic ordering in the weakly coupled S = 1/2 antiferromagnetic spin chain materials Cu(py)2(Cl1-xBrx)2 is studied by muon spin rotation and bulk measurements. Disorder is found to have a strong effect on the ordering temperature TN, and an even stronger one on the saturation magnetization m0, but considerably more so in the effectively lower-dimensional Br-rich materials. The observed behavior is attributed to Random Singlet ground states of individual spin chains, but remains in contradiction with chain mean field theory predictions. In this c…
The conductivity properties between Luttinger liquids are analyzed by exact Renormalization Group methods. We prove that in a two chain system or in a model of bilayer graphene, described by two coupled fermionic honeycomb lattices interacting with a gauge field, the transverse optical conductivity at finite temperature is anomalous and decreasing together with the frequency as a power law with Luttinger liquid exponent.
Using a recently proposed perturbative numerical renormalization-group algorithm, we explore the connection between quantum criticality and the emergence of Luttinger liquid physics in $t-J$ chains coupled by frustrated interactions. This study is built on an earlier finding that at the maximally frustrated point, the ground state of weakly-coupled Heisenberg chains is disordered, the transverse exchanges being irrelevant. This result is extended here to transverse couplings up to $J_{\perp}=0.6$, and we argue that it may also be valid at the isotropic point. A finite size analysis of coupled …
We study interplay of electronic correlation, randomness and dimensionality effects in half-filled random Hubbard chains weakly coupled via an interchain one-particle hopping. Based on the two-loop renormalization-group approach, phase diagrams are given in terms of temperature vs. strengths of the intrachain electron-electron umklapp scattering, the random scattering and the interchain one-particle hopping.
We calculate the exact Green function of a special model of N coupled Luttinger chains with arbitrary interchain hopping t_{perp}. The model is exactly solvable via bosonization if the interchain interaction does not fall off in the direction perpendicular to the chains. For any finite N we find Luttinger liquid behavior and explicitly calculate the anomalous dimension gamma^(N). However, the Luttinger liquid state does not preclude coherent interchain hopping. We also show that gamma^(N) -> 0 for N -> infinity, so that in the limit of infinitely many chains we obtain a Fermi liquid.
We present a many chain generalization of a recent work of ours, wherein an arbitrary number of fermionic chains are coupled via a Gauge interaction. Central to this construction is the role of an antisymmetric tensor which enters the couplings and helps define a pseudo-unitary transformation, that maps the models on to ``free fermi'' models with twisted boundary conditions.
A role of umklapp scattering has been examined for two-coupled chains with both forward and backward scatterings by applying renormalization group method to bosonized Hamiltonian. It has been found that a state with relevant interchain hopping changes into a state with irrelevant (confined) one when the magnitude of umklapp scattering becomes larger than that of interchain hopping. Critical value of umklapp scattering for such a confinement-deconfinement transition is calculated as the function of interchain hopping and intrachain interactions. A crossover from one-dimensional regime into that…
We consider the Hall effect in a system of weakly coupled Luttinger chains. We obtain the full conductivity tensor in the absence of dissipation along the chains. We show that while the dependence of the Hall and transverse conductivities on temperature and frequency are affected by the Luttinger interaction very strongly, the Hall resistivity is given by a simple formula corresponding to the noninteracting fermions. We compute the frequency, temperature and field dependence of the transverse conductivity. Consequences for the quasi-one-dimensional organic conductors are discussed.
The role of various magnetic inter-chain couplings is investigated by numerical methods in doped frustrated quantum spin chains. A non-magnetic dopant introduced in a gapped spin chain releases a free spin-1/2 soliton. The formation of a local magnetic moment is analyzed in term of soliton confinement. A four-spin coupling which might originate from cyclic exchange is shown to produce such a confinement in contrast to transverse magnetic exchange. Dopants on different chains experience an effective space-extended non-frustrating pairwise spin interaction.
Magnetic excitations in the quasi-one-dimensional quantum antiferromagnets PbNi2V2O8 and SrNi2V2O8 are measured all the way up to the zone boundary energy using inelastic neutron scattering from powder samples. An estimate for next-nearest-neighbor in-chain interactions is obtained. The role played by these interactions in spin-vacancy induced magnetic ordering in PbNi2V2O8 is discussed.
We show that numerical quasi-one-dimensional renormalization group allows accurate study of weakly coupled chains with modest computational effort. We perform a systematic comparison with exact diagonalization results in two and three-leg spin ladders with a transverse Hamiltonian that can involve frustration. Due to the variational nature of the algorithm, the accuracy can be arbitrarily improved enlarging the basis of eigenstates of the density matrix defined in the transverse direction. We observe that the precision of the algorithm is directly correlated to the binding of the chains. We al…
We adapt the block-Lanczos density-matrix renormalization-group technique to study the spin transport in a spin chain coupled to two non-interacting fermionic leads. As an example, we consider leads described by two-dimensional tight-binding models on a square lattice. Although the simulations are carried out using a chain representation of the leads, observables in the original two-dimensional lattice can be calculated by reversing the block-Lanczos transformation. This is demonstrated for leads with Rashba spin-orbit coupling.
By applying renormalization group method to the bosonized Hamiltonian of two-coupled chains with repulsive intrachain interaction, we have examined a role of backward scattering with a spin-anisotropy which competes with interchain hopping. From calculation of a dominant state in the limit of low energy, it is found that superconducting state moves into spin density wave state when the anisotropy becomes larger than a critical value. Further phase diagram is shown on the plane of g-ology.
Quasi-one-dimensional quantum antiferromagnets formed by a d-dimensional hypercubic lattice of weakly coupled spin-1/2 antiferromagnetic Heisenberg chains are studied by combining exact results in one-dimension and renormalization group analyses of the interchain correlations. It is shown that d-dimensional magnetic long-range order develops at zero-temperature for infinitesimal antiferromagnetic or ferromagnetic interchain couplings. In the presence of weak bond alternations, the order-disorder transition occurs at a finite interchain coupling. Relevances to the lightly doped quantum antiferr…
We study spin-half and spin-one Heisenberg models in the limit where one dimensional (1-D) linear chains, with exchange constant J1, are weakly coupled in an anisotropic triangular lattice geometry. Results are obtained by means of linked-cluster series expansions at zero temperature around different magnetically ordered phases. We study the non-colinear spiral phases that arise classically in the model and the colinear antiferromagnet that has been recently proposed for the spin-half model by Starykh and Balents using a Renormalization Group approach. We find that such phases can be stabilize…
We analyze a tight binding model of two coupled chains with strongly interacting fermions. Depending on the parameter $w$, the many body lowest energy band consists of either single particles or bound pairs. A topological quantum pump can be created by periodically varying the coupling strengths under adiabatic conditions. We numerically show that the single particles and bound pairs are pumped in opposite directions along topologically non-trivial paths. The exact quantization of the pumped charge can be expressed by a (many body) Chern number.
Effects of the intrachain dimerization and the interchain one-particle hopping, $t_{b}$, in a quasi-one-dimensional dimerized chain system at quarter filling have been studied, based on the perturbative renormalization group (PRG) approach. Based on the results, we discuss difference in the low-energy properties between TMTTF and TMTSF compounds.
By means of density functional theory, we investigate the magnetic ground state of edge-sharing CuO_2 spin-chains, as found in the (La,Ca,Sr)_14Cu_24O_41system, for instance. Our data rely on spin-polarized electronic structure calculations including onsite interaction (LDA+U) and an effective model for the interchain coupling. Strong doping dependence of the magnetic order is characteristic for edge-sharing CuO_2 spin-chains. We determine the ground state magnetic structure as function of the spin-chain filling and quantify the competing exchange interactions.
We study the low-temperature transport of a doped two-chain ladder system of interacting fermions in the presence of a barrier or of a low concentration of impurities. Above a certain value of the interaction, the conductance is suppressed, like for a single chain, despite the presence of dominant superconducting correlations. There is, however, a region of repulsive interaction where perfect transmission across the barrier occurs unlike the single-chain case. We provide a possible explanation for the temperature maximum of the resistivity in the normal state of \srca.
The effect of umklapp scattering on interchain hopping has been investigated for two-coupled chains of interacting electrons with half-filled band. By analyzing in terms of renormalization group method, we have found that interchain hopping is renormalized to zero and is confined when a gap induced by umklapp scattering becomes larger than a critical value. From a phase diagram calculated on a plane of the interchain hopping and the gap, we discuss a role of the correlation gap which has been studied in metallic state at temperatures above spin density wave state in organic conductors.
Using a perturbative renormalization group approach, we show that the extended ($J_1$-$J_2$-$J_d$) Heisenberg model on the kagome lattice with a staggered chiral interaction ($J_χ$) can exhibit a gapless chiral quantum spin liquid phase. Within a coupled-chains construction, this phase can be understood as a chiral sliding Luttinger liquid with algebraic decay of spin correlations along the chain directions. We calculate the low-energy properties of this gapless chiral spin liquid using the effective field theory and show that they are compatible with the predictions from parton mean-field the…
Long range magnetic ordering in the quasi-one-dimensional random-bond antiferromagnet BaCu$_2$(Si$_{1-x}$Ge$_{x}$)$_2$O$_7$ is studied in $μ$SR experiments as a function of disorder strength. Compared to the disorder-free parent materials, the saturation ordered moment is found to be considerably reduced. Moreover, even in weakly disordered species, the magnetically ordered state is shown to be highly inhomogeneous. The results are interpreted in terms of weakly coupled random spin chains, governed by the ``infinite randomness`` fixed point.
We develop an analytical approach based on a unitary transformation to investigate S=1/2 antiferromagnetic Heisenberg chains coupled to phonons, and find a new quantum phase transition at zero temperature. Although the usual phase transition occurs depending on the strengths of the spin-spin and spin-phonon interactions, the phase transition in the present work is induced by a change in the geometrical structure of the spin-phonon interaction. The magnetic properties of the phase transition can be understood in relation to a phase transition that has already been found in the frustrated J1-J2 …
One-particle interchain hopping in a system of coupled Luttinger liquids is investigated by use of exact diagonalizations techniques. Firstly, the two chain problem of spinless fermions is studied in order to see the behaviour of the band splitting as a function of the exponent αwhich characterizes the 1D Luttinger liquid. We give numerical evidence that inter-chain coherent hopping (defined by a non-vanishing splitting) can be totally suppressed for α\sim 0.4 or even smaller αvalues. The transverse conductivity is shown to exhibit a strong incoherent part even when coherent inter-chain hoppin…
We consider hardcore bosons in two coupled chain of one dimensional lattices at half filling with repulsive intra-chain interaction and inter-chain attraction. This can be mapped on to a coupled chain of spin-1/2 XXZ model with inter chain ferromagnetic coupling. We investigate various phases of hardcore bosons (and related spin model) at zero temperature by density matrix renormalization group method. Apart from the usual superfluid and density wave phases, pairing of inter chain bosons leads to the formation of novel phases like pair-superfluid and density wave of strongly bound pairs. We di…
A low-field spin flop transition in the quasi one-dimensional antiferromagnet \ba is exploited to study the polarization dependence of low-energy magnetic excitations. The measured longitudinal spectrum is best described as single broad continuum, with no sharp ``longitudinal mode'', in apparent contradiction with the commonly used chain-MF/RPA theories. The observed behavior is also quite different than that previously seen in the related \kcuf material, presumably due to a large difference in the relative strength of inter-chain interactions. The results highlight the limitations of the chai…
